THE LIBRARY 
 
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 The University 
 
 of 
 North Carolina 
 
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 Music Wfc>. 
 
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 in 2013 
 
 http://archive.org/details/treatiseofnaturaOOhold 
 

TREATISE 
 
 O F T H E 
 
 Natural Grounds, and Principles 
 
 HARMONY. 
 
 By William Holder, D. D. Fellow of the Royal 
 Society \ and late Sub-Dean of their MAJESTY 5 * Chapel 
 Royal. 
 
 To which Is Added, by way oi APPENDIX; 
 
 RULES for Playing a Thorow-Bafs ; with Variety 
 of Proper Lejfons* Fuges, and Examples to Explain the faid 
 RULES. Alfo Dire&ions for Tuning an Harpjichord or 
 Spinnet. 
 
 By the late Mr. Godfrey Keller. 
 
 With feveral new Examples, which before were wan-. 
 ting, the better to explain fome Paffages in the for- 
 mer Impreffions. 
 
 The whole being Revised, and Corrected from many 
 grofs Miftakes committed in the firft Publication of 
 thefe Rules. 
 
 LONDON: 
 
 Printed by W. Pearson, over againft Wright's Coffee- 
 Houfe in Alder fgate-ftreet ; for J. Wilcox in Little Britain; 
 and T. Osbqrne in Grafs- Inn. 173 1. \ 
 
Jl .iTl 1l!v 
 
 PUBLISHER. 
 
 TO THE 
 
 READ E R. 
 
 THE Intention of Tullijhing Mr. Keller^ Rules, 
 (by way of Appendix to Dr. Holder's c BookJ 
 was chiefly to refcue them from many Mifiakes 
 and Errors, which were oceafioned by the Ignorance 
 of the firfi Tublipers of them on Plates, which wou*d 
 never have happened, if the judicious Author had liv y d 
 to have corrected the Plates himfelf : Nor cou J d he have 
 fufferd thofe Examples, which are now Added, to have 
 *leen wanting, the letter to explain fome of the faid 
 Rules, which before were only printed in Figures^ with** 
 out a prober Illufiration of the fame in Muftcal Notes ; 
 as is evident by ma?iy In/lances of the fame hind through- 
 out the faid Work. 
 
 And as this Book may fall into the Hands of fome, who 
 have (not only a Tafi for *I)r* Holder^ Treatife, but 
 alfoj a Genius for Compofing as well as for Playing a 
 Through Bafs j it is not improper to Obferve y that there 
 are many excellent Rules contained in it, which will be 
 found of great Advantage to young Compofers, as well 
 as to thofe who pra&ife a Through Ba/s ; efpeciaUy with 
 Regard to the various Ways of taking T)(fcords f which 
 is one of the mofi difficult Parts of Compofition. 
 
 And I humbly pref time, that for thefe Reafons $ it will 
 become at leafi as Ufeful and IntfruEiive, as any thing 
 that has hitherto been Vubliffd of this hind. 
 
 6 I • vj 
 
 ValeJ 
 [Vfc THE 
 
INDEX. 
 
 Chap, Pag. 
 
 North Carolina 
 THE 
 
 HE Introduclion. 
 
 Of Sound in General. I i. 
 
 Of Sound Harmonic k, H. $ • 
 appendix to Chap. II. concerning the Motions and 
 
 Meafures of a Tendulum. 1 5 
 
 Of Confonancy and Diffonancy. III. 3 1 
 
 Of Concords. IV. 3 8 
 
 Of (proportion. V. 67 
 
 Of Dif cords and Degrees. \h 04 
 
 Digreffion, concerning the? ^ 0Q 
 
 Ancient Greek Mufick.Ji 
 
 Of Dif cords. VII. 126 
 
 0/ Vffirences . VIII. 1 40 
 
 Conclujion. IX. 1 47 
 
 |The Index to Mr. Kellers 
 
 I Rules. 
 
 V^\F Concords and T)if cords. Page 159 
 
 V^Jf 0/ Common Chords differently taken. 1 60 
 
 Of Common Chords and Sixes differently taken. 1 60 
 
 Of Cadences. 1 6 1 
 
 A 2 Of 
 
Ik INDEX. 
 
 Of the federal Difcords and Manner of (playing them. 
 
 Page 163 
 How to move the Hands when the !Bafs afcends or de* 
 
 f vends. 1 64 
 
 Of Dividing upon Notes in Common Time. 1 6 51 
 
 Of 'Dividing in Triple Time. 1 67 
 
 Of Natural Sixes, and Proper Cadences in a flnrpl^ey. 
 
 169 
 Of Natural Sixes, 8cc. in a flat J{ey. 176 
 
 <$$ules how Sixes may he ufed in Compofing. 171 
 
 (Rules about Sevenths and Ninths. 173 
 
 Several Ways of Accompanying when the Bafs Afcends, 
 
 and Defcends by Degrees. 1 7 % 
 
 Of playing all Sorts of D if cords in aflat IQty. 174 
 Of playing Dij cords in a fharp ^ey. 180 
 
 Of making Chords eafy to the Memory. 1 8 5 
 
 Of playing fome Notes the fame way, which yet have a 
 
 different appearance in Writing. 185 
 
 Of Tranfpofition. 1 84 
 
 Of Difcords, how prepaid and refolVd. 186 
 
 Some Examples for playing a Thorow-^afs. 194 
 Short Leffons hy way of Fugeing. 200 
 
 <fiuks for Tuning an Harpfichrd, &c 
 
 Rules 
 
THE 
 
 Natural Grounds and Principles 
 
 OF 
 
 HARMONY. 
 
 The INTRODUCTION. 
 
 H 
 
 ARMONT conjiftsofCaafes, Na- 
 tural and eArtificial, as of Matter 
 
 and Form. The Material Fart of 
 it j is Sound or Voice. The Formal Part 
 is,, The 'Diffofetion of Sound or Voice into 
 Harmony ; which requires, as a prepara- 
 tive Caufe, skilful Compofitio'i ; and, as an 
 immediate Efficient, Artful 'Performance. 
 
 The former Part, viz. The Matter, lies 
 deep in Nature, and requires much Re- 
 fearch into Natural Phikfofhy to unfold it ; 
 to find how Sounds are made i mid how 
 they are fir ft -fitted by Nature for Harmony^ 
 lefore they he diffofed hy oArt - Both toge- 
 ther ma-he Harmony comfkat* 
 
 B ■ " ffart 
 
The INTRODUCTION, 
 
 Harmony j then^ refults from fraEtick Mu- 
 fick, and is made by the Natural and Arti* 
 ficial Agreement of different Sounds, (viz* 
 Grave and Acute) by which the Senfe of 
 Hearing is delighted. 
 
 This is proper in Symphony ^ u e. Con- 
 fent of more Voices in different Tones ; but 
 is found alfo in folitary Mufick of one Voice \ 
 hy the Ohfervation and Expectation of the 
 Ear, comparing the Habitudes of the fol- 
 lowing Notes to thofe which did precede* 
 
 Now the Theory in Natural Philofophy? 
 of the Grounds and Reafons of this oAgree-* 
 ment of Sounds^ and confequent ^Delight 
 and Pleafure of the Ear y (leaving the Ma- 
 nagement of thefe Sounds to the Mafiers of 
 Harmonick Compofure 3 and the skilful Ar- 
 tifis in Performance) is the Sub]eB of this 
 Difcourfe* The Defign whereof (for the 
 Sake and Service of all Lovers of Mufick^ 
 and particularly the Gentlemen of Their 
 Maje/ly^s Chapel Royal) is, to lay down thefe 
 Principles as port 7 and intelligible^ as tw 
 StiljeH Matter will bean 
 
 Where the fir ft thing Neceffary^ is a Con* 
 fideration of fomewhat of the Nature of 
 Sound in General; andthen^ more part icu« 
 larljz of Harmonick Sounds^ &c* 
 
 CHAP. 
 
Of Sound in General 
 
 CHAR I. 
 
 Of Sound in General. 
 
 IN General to pafs by what is not per* 
 tinent to this Defign) Sence and Ex- 
 perience confirm thefe following Pro- 
 perties of Sound. 
 
 1. All Sound is made by Motion, viz. 
 by Percuilxon with Collifion of the A in 
 
 2. That Sound may be propagated^ 
 and carried to Diftance ? it requires a Me- 
 dium by which to pafs. 
 
 3. This Medium (to our Purpofe) is 
 Air. 1 
 
 4. As far as Sound is propagated along 
 the Medium * fo far alfo the Motion paffeth. 
 For (if we may not fay that the Motion 
 snd Sound are one and the fame thing* 
 
 B % yet 
 
% Of Sound in General 
 
 yet at leaft) it is neceffariiy confeqtlent f 
 that if the Motion ceafe, the Sound muil 
 alfo ceafe. 
 
 5. Sound, where It meets with no 
 dbftacle, paffeth in a Sphere of the Me- 
 dium, greater or lefs, according to the 
 Force and Greatnefs of the Sound ; of 
 which Sphere the fonarous Body is as the 
 Centre. 
 
 6. Sound, (o far as it reacheth, paf- 
 feth the Medium j not in an Inftant, but in 
 a certain uniform Degree of Velocity, cal- 
 culated by Gaffendus, to be about the rate 
 of 276 Paces, in the fpace of a fecond Mi- 
 nute of an Hour. And where it meets 
 with any Obftacle, it is fubjeft to the Laws 
 of Reflexion, which is the Caufe of Ec- 
 cho's Meliorations, and Augmentations of 
 Sound* 
 
 7. Sound, /. e* the Motion of Sound, 
 or founding Motion, is carried through the 
 Medium or Sphere of Activity, with an 
 Impetus or Force, which fhakes the free 
 Medium, and ftrikes and fhakes every Ob- 
 ftacle it meets with, more or lefs, accord- 
 ing to the vehemency of the Sound, and 
 Nature of the Obftacle, and Nearnefs of 
 it to the Centre, or fonorous Body* Thus 
 
 the 
 
Of Sound in General % 
 
 the impetuous Motions of the Sound of 
 Thunder, or of a Canon, fhake all before 
 it, even to the breaking of Glafs Windows^ 
 
 i The Parts of the founding Body 
 are moved with a Motion of Trembling, 
 or Vibration, as is evident in a Bell or 
 Pipe, and mod manifeft in the firing of a 
 Mufical Xnftrument* 
 
 9. This Trembling, or Vibration, is 
 either equal and uniform, or elfe unequal 
 and irregular ; and again, fwifter or flower, 
 according to the Conftitution of the fono- 
 rous Body, and Quality and Manner of 
 of Percuflion ; and from hence arife Differ 
 rences of Sounds, 
 
 10. The Trembling, or Vibration of 
 the fonorous Body, by which the particu- 
 lar Sound is conftituted and difcriminated, 
 is impreffed upon, and carried along the 
 Medium in the fame Figure and Meafure, 
 otherwife it would not be the fame Sound, 
 when it arrives at a more diftant Ear, i. e* 
 the Tremblings and Vibrations (which 
 may be called Undulations) of the Air or 
 Medium, are all along of the fame Velo- 
 city and Figure, with thofe of the forio-* 
 yous Body, by which they are qaufed. 
 
 B z The. 
 
4 Of Sound Harmonic^ 
 
 The Differences of Sounds, as of one, 
 Voice from another, gffV. (befides the Dif- 
 ference of Tune, which is caufed by the 
 Difference of Vibrations) arife from the 
 Conftitution and Figure, and other Acck 
 dents of the fonorous Body. 
 
 ii f If the fonorous Body be requifitely 
 conftkuted, /, e. of Parts folid, or tenfe, 
 and regular, fit, being ftruck, to receive 
 and exprefs the tremulous Motions of 
 Sounds equally arid fwifily, then it will ren- 
 der a certain and even Harmonica! Tone 
 or Tune, received with Pleafure, and judg- 
 ed and meafured by the Ear ; Otherwife 
 it will produce an obtufe or uneven Sound 5 
 not giving any certain or difcernable Tune, 
 
 Now this Tune, or Tuneable Sound, 
 
 An agreeable Cadence of Voice, at one 
 Pitch or Tenfion, This Tuneable Sound 
 (I lay) as it is capable of other Tenfions 
 towards Acutenefs, or Gravity, /. e. the 
 Tenfions greater or lefs, the Tune graver 
 or more acute, /. e» lower or higher, is the 
 firft Matter or Element of MulicL And 
 t&is Harmonic k Sound comes next to be 
 confidereda 
 
 CHAR 
 
Of Sound Harmonkk. 
 
 CHAP. IL 
 Of Sound Harmonic^ 
 
 THE firft and great Principle upon 
 which the Nature of Harmonica! 
 Sounds is to be found out and difcovered, is 
 this : That the Tune of a Note (to fpeak in 
 our vulgar Phrafe) is conftituted. by the 
 Meafiire and Proportion of Vibrations of 
 the fonorous Body ; I mean, of the Velo^ 
 city of thole Vibrations in their Rccourfes* 
 
 FoRj the frequenter the Vibrations are, 
 the more Acute is the Tune •, the flower 
 and fewer they are in the fame Space of, 
 Time, by fo much the more Grave is the 
 Tune. So that any given Note of a Tune* " 
 is made by one certain Meafure of Ve- 
 locity of Vibrations ; viz* Such a certain 
 Number of Courfes and Recourfes. e*g. 
 of a Chord or String, in fuch a certain Space 
 of Time, doth constitute fuch a certain de- 
 terminate Tune. And all fuch Sounds as 
 are Unifons, or of the fame Tune with that 
 given Note, though upon whatfoever dif- 
 ferent Bodies, (as Strings "Bell, Pipe % La-*- 
 rj%x % &c.) are made with Vibrations cm? 
 
 B 4 Trem- 
 
6 Of Sound Harmonic^ 
 
 Tremblings of thole Bodies, all equal each 
 to the other: And whatfoever Tuneable 
 Sound is more Acute, is made with Vibra- 
 tions more fwlft ; and whatfoever is more 
 grave, is made with more flow Vibrations : 
 And this is univerfally agreed upon, as 
 moll evident to Experience, and will be 
 more manifeft through the whole Theory, 
 
 And, That the Continuance of the 
 Sound in the fame Tune, to the laft, (as 
 may be perceived in Wire-firings, which 
 being once {truck will hold their Sound 
 long) depends upon the Equality of Time 
 of the Vibrations, from the greateft Range 
 till they come to ceafe : And this perfectly 
 makes out the following Theory of Confo- 
 nancy, and DijQTonancy. 
 
 So m e of the ancient Greek Authors of 
 Mufick, took Notice of Vibrations : And 
 that the fwifter Vibrations caufed Acuter, 
 and the flower, graver Tones, And that 
 the Mixture, or not Mixture of Motions 
 creating feveral Intervals of Tune, was the 
 Reafpn of their being Concord or Dif* 
 cord* And likewife, they found out the 
 feveral Lengths of a Monochord. propor- 
 tioned to the feveral Intervals or Harmo- 
 nick Sounds : But they did not make out 
 the Equality of Meafure of Time, of the 
 
 Yfc 
 
Of Sound Harmonick. y 
 
 Vibrations laft fpoken of, neither could be 
 prepared to anfwer_ fuch Objections, as 
 might be made againft the Continuity of 
 the famenefs of Tune, during the Continu- 
 ance of the Sound of a Stringy or a Bell 
 after it is ftruck. Neither did any of them 
 offer any Reafons for the Proportions affign- 
 ed, only it is faid, that Pythagorus found 
 them out by Chance. 
 
 But now, Thefe (fince the Acute Ga-> 
 Jileo hath obferved, and difcovered the Na- 
 ture of Pendulums) are eafy to be explain- 
 ed, which I fhall do, premifing fome Con- 
 iideration of the Properties of the Motions 
 of a Pendulum. 
 
 H a i* g a Plumbet C on a String or 
 Wire, fixed at 0. Bear C to A : Then let 
 it range freely, and it will move toward 
 £, and from thence fwing back towards 
 4*. The Motion from A to B, I call the 
 
 Courfe, 
 
8 Of Sound Harmonic^ 
 
 Courfe, and back from B to J^ the Re- 
 courfe of the "Pendulum^ making almoft a 
 Semi-Circle, of which is the Centre, 
 Then fuffering the Pendulum to move of 
 itfelf, forwards and backwards, the Range 
 of it will at very Courfe and Recourfe a- 
 bate, and diminifh by degrees, till it come 
 to reft perpendicular at C 
 
 Now that which Galileo firft obferved, 
 was, that all the Courfes and Recourfes 
 of the Pendulum^ from the greateft Range 
 through all Degrees till it come to reft, 
 were made in equal Spaces of Time. That 
 is, e. g. The Range between A and S, is 
 made in the fame Space of Time, with the 
 Range between D and £; the Piumbet 
 moving fwifter between A and J3, the great- 
 er Space ; and flower between D and £, 
 the leffer; m fuch Proportions, that the 
 Motions between the Terms A B and D 
 £, are performed in equal Space of Time, 
 
 And here it is to be Noted, that where 
 ever in this Treatife, the fwiitnefs or flow-, 
 nefs of Vibrations is fpoke of, it muft be 
 always underftood of the frequency of their 
 Courfes and Recourfes, and not of the Mo- 
 tion by which it paffeth from one fide to 
 another. For it is true, that the fame Pen- 
 dulum under the fame Velocity of Returns,, 
 
 moves 
 
Of Sound Harmonic^ p 
 
 moves from one fide to the other, with 
 greater or lefs Velocity, according as the 
 Range is, greater or lefs. 
 
 And hence it is, that the Librations of 
 a Pendulum are become fo excellent, and 
 ufeful a Meafure of Time ; efpecially when 
 a fecond Obfervation is added, that, as you 
 fhorten the Tendulum^ by bringing C near- 
 er to its Centre 0, fo the Librations will 
 be made proportionally in a fhorter Mea- 
 fure of Time, and the contrary if you 
 lengthen it. And this is found to hold in 
 a Duplicate proportion of length to Velo- 
 city. That is, the length quadrupled, will 
 fubduple the Velocity of Vibrations ; And 
 the Length fubquadrupled, will duple the 
 Vibrations, for the Proportion holds reci- 
 procally. As you add to the length of the 
 Pendulum, fo/you diminifh the frequency 
 of Vibrations, and increafe them by fhort- 
 ning it. 
 
 Now therefore to make the Courfes of 
 a Pendulum doubly fwift, k e. to move 
 twice in the fameSpace of Time, in which 
 it did before move once; youmuft fubqua- 
 druple the Length of it, /. e. make the 
 Pendulum but a quarter fo long as it was 
 before- And to make the Librations dou- 
 bly (I0W5 t0 P a ^ s °JP.ce in the Time they 
 
 die! 
 
to Of Sound 'Harmonic f£ 
 
 did pafs twice ; you muft quadruple the 
 Length ; make the Pendulum four Times 
 as long as it was before, and fo on in what 
 Proportion you pleafe. 
 
 N o w to apply this to Mufick, make 
 twoT ^endulums, .4 Band CT>, fatten toge- 
 ther the Plumbets B and 7), and ftretch 
 them at length, (fixing the Centers A and 
 C) Then, being ftruck, and put into Mo- 
 tion ; the Vibrations, which before were 
 diftind, made by A c B y and C D 9 will now 
 be united (as of one entire String) both 
 backward and forward, between £ and R 
 Which Vibrations (retaining the aforefaid 
 Analogy to a Tendulum) will be made in 
 equal Spaces of Time, from the firft to the 
 laft ; /. e. from the greateft Range to the 
 leaft, until they ceafe. Now, this being a 
 double Pendulum^ to fubduple the fwiftnefs 
 of the Vibrations, you do but double the 
 length from A to C, which will be qua- 
 druple to A B. The lower Figure is the 
 fame with that above, only the Plymmets 
 taken off. 
 
 And 
 
Of Sound Harmonic^, i i 
 
 And here you have the Nature of the 
 String of a Mufical Inftrument, refembling 
 a double Pendulum moving upon two Cen- 
 ters, the Nut and the Bridge, and Vibra- 
 ting with the greateft Range in the mid- 
 dle • of its Length ; and the Vibrations e- 
 qual even to the laft, which muft make it 
 keep the fame Tune fo long as it Sounds. 
 And becaufe it doth manifeftly keep the 
 fame Tune to the laft ; it follows that the 
 Vibrations are equal, confirming one ano- 
 ther by two of our Senfes ; in that we fee 
 the Vibrations of a Pendulum move equally, 
 and we hear the Tune of a String, when 
 it is ftruck, continue the fame. 
 
 The Meafure of fwiftnefs of Vibra- 
 tions of the String or Chord, (as hath beea 
 faid,) conftitutes and determines the Tune ? 
 as to the Acutenefs and Gravity of the Note 
 which it founds j And the lengthning or 
 
 fliort- 
 
' 12 Of Sound Harmonkk* 
 
 fhortning of the String, under the fame 
 Tendon, determines the Meafure of the 
 Vibrations which it makes. And thus^ 
 Harmony comes under mathematical Cal- 
 culations of Proportions, of the length of 
 Chords, of the Meafure of Time in Vibra- 
 tions ; of the Intervals of Tuned Sounds* 
 As the length of one Chord to another, 
 Ceteris paribus , I mean, being of the fame 
 Matter, thicknefs and tendon, fo is the 
 Meafure of the Time of their Vibrations. 
 As the Time of Vibrations of one String 
 to another, fo is the Interval or Space of 
 Acutenefs or Gravity of the Tune of that 
 one, to the Tune of the other : And con- 
 fequently, as the length is (Ceteris ^arihus) 
 fo is the determinate Tune* 
 
 And upon tliefe Proportions m the Dif- 
 ferences of Lengths of Vibrations, and of 
 Acutenefs and Gravity ; I Oiall infift all a- 
 long in this Treatife, very largely and parti- 
 cularly, for the full Information of all fucll 
 ingenious Lovers of Mufick, as fhall have 
 the Curiofity to inquire into the Natural 
 Caufes of Harmony, and of the Th&no* 
 mena which occurr therein, though other- 
 wife, to t;he more learned in Mufick and 
 Mathematical Proportions, all might be 
 expreiled very much fliotter, and Hill' be 
 more JQbortned by the help of Symbols. 
 
 Am® 
 
Of Sound Harmonic^ J f 
 
 And here we may fix our Foot : Con- 
 cluding, that what is evident to Senfe, of 
 the Ph£nomena J in a Chord, is equally 
 (though not fo difcernably) true of the 
 Motions of all other Bodies which render 
 la tuneable Sound, as the Trembling of a 
 Bell or Trumpet, the forming of the Larynx 
 m our felves, and other Animals, the throat 
 of Pipes and of thofe of an Organ, &c. 
 All of them in feveral Proportions fenfibly 
 trembling and impreffing the like Undula- 
 tions of the Medium, as is done b^ the fe- 
 veral (more mauifeft) Vibrations of Strings 
 or Chords. 
 
 In thefe other Bodies, lafl: fpoken of, we 
 manifeftly fee the Reafon of the Difference 
 of the fwiftnefs of their Vibrations (though 
 we cannot fo well meafure them) from 
 their Shape, and other Accidents in their 
 Conftitution ; and chiefly from the Propor- 
 tions of their Magnitudes ; the Greater ge- 
 nerally Vibrating flower, and the Lefs 
 more fwiftly, which give the Tunes accord- 
 ingly* We fee it in the Greatnefs of a 
 String ; a greater and thicker Chord will 
 give a graver and lower Tone, than one 
 that is more flender, of the fame Tenfion 
 and Length ; but they may be made Uni- 
 fan by altering their Length and Tenfion, 
 
14 Of Sound Harmonic^. 
 
 Tension is proper to Chords or 
 Strings (except you will account a Drum 
 for a Miifical Inftrument, which hath a 
 Tenfion not in Length, but in the whole 
 furface) as when we wind up, or let down 
 the Strings, i.e. give them a greater or lefs 
 Tenfion,, in tuning a Viol, Lute^ or Harp- 
 fichordj and is of great Concern, and may 
 be meafured by hanging Weights on the 
 String to give it Tenfion but not eafily^ 
 nor fo certainly* 
 
 But the lengths of Chords (becaufe of 
 their Analogy to a Pendulum) is chiefly 
 confidered, in the difcovery of the Propor- 
 tions which belong to Harmony, it being 
 moft eafie to meaiure and defign the Parts 
 of a Monochord, in relation to the whole 
 String ; and therefore all Intervals in Har» 
 mony may firft be defcribed, and under- 
 flood, by the Proportions of the length 
 of Strings, and confequently of their Vi- 
 brations* And it is for that Reafon, that 
 in this Treatife of the Grounds of Harmo- 
 ny, Chords come fo much to be confi- 
 dered, rather than other founding Bodies, 
 and thofe, chiefly in their Proportions of 
 Length. It is true, that in Wind-Inftru- 
 ments, there is a Regard to the Length 
 of Pipes, but they are not fo well acco- 
 modated (as our Chords) to be examined^ 
 
Of Sound Harmonic 4 if 
 
 neither are their Vibrations, nor the mea* 
 fure of them fo manifefb 
 
 There are fome Mufical Sounds which 
 feem to be made, not by Vibrations but 
 by Pulfes as by whisking fwiftly over fome 
 Silk or Camblet-fluffs, or over the Teeth 
 of a Comb, which render a kind of Tunei 
 more Acute or Grave, according to the 
 fwiftnefs of the Motion, Here the Sound 
 is made, not by Vibrations of the fame 
 Body, but by Percuffion of feveral equal^ 
 and equidiftant Bodies ; as Threads of the 
 Stuff, Teeth of the Comb palling over 
 them with the fame Velocity as Vibrations 
 are made. It gives the fame Modification 
 to the Tune, and to the Undulations of the 
 Air, as is done by Vibrations pf the fame 
 Meafure j the Multiplicity of Pulfes or 
 Percuffi-ons, anfwering the Multiplicity of 
 Vibrations. I take this Notice of it, be- 
 caufe others have done fo ; but 1 think it 
 to be of no ufe in Mufick. 
 
 mJmmmam*mqmm*»" \ .njnninuw »'t*%m i 
 
 &t* 
 
•5 
 
 APPPENDIX. 
 
 Before I conclude this Chapter, it may feem 
 needful^ letter to confirm the Foundation 
 vce have laid 7 and give the Reader fome 
 more ample Satisfaction about the Moti- 
 ons and Meafures of a Pendulum, and the 
 Application of it to Harmonick Motion. 
 
 FIRST then, it is manifeft to Sence 
 and Experience, and out of all dif- 
 pute; that the Courfes and Recourfes re- 
 turn fooner or later, i. e« more or lefs fre- 
 quently, according as the 'Pendulum is fliort- 
 fied, or made longer. And that the Pro- 
 portion by which the Frequency increaf- 
 eth, is (at leaft) very near duplicate, viz* 
 of the length of the Pendulum^ to the Num- 
 ber of Vibrations, but is in reverfe, i. el as 
 the Length encreafeth, fo the Vibrations 
 decreafe ; and contrary, quadruple the 
 Length, and the Vibrations will be fub- 
 dupled. Subquadruple the Lengthy and 
 the Vibrations will be dupled. And laft- 
 ly, that the Librations, the Courfes and 
 Recourfes of the fame Pendulum , are all 
 made in equal Space of Time, or very 
 near to it, from the greateft Range to the 
 
 leaits 
 
Jeaft. Now though the duplicate Propor- 
 tion affigned, and the equality of Time, 
 are a little called in queftioa, as not per- 
 fectly exa£t 5 though very near it ; yet in a 
 Monochord we find them perfectly agree, 
 viz as to the length, Duple iniiead of Du- 
 plicate, becaufe a String faftned at both 
 ends is as a double Penduhm^ each of 
 which is quadrupled by dupling the whole 
 String. And on this duple Proportion, de- 
 pend all the Rations found in Harmony; 
 And again, the Vibrations of a String are 
 exa&ly equal, becaufe they continue to 
 give the fame Tune. 
 
 Supposing then fonne little difference 
 may fometime feem to be found in either of 
 thefe Motions of a Pendulum, yet the near- 
 nefs to Truth is enough to iupport our 
 Foundation, by fhewing what is intended 
 by Nature, though it fometimes meet with 
 fecret Obftacles in the Pendulum, which 
 it does not in a well made String. We 
 may juftly make fome Allowance for the 
 Accidents, and unfeen Caufes, which hap- 
 pen to make fome little Variations in Tri- 
 als of Motion upon grofs Matter, and con- 
 fequently the like for nicer Experiments 
 made upon the Pendulum* It is difficult 
 to find exactly the determinate Point of 
 the Plutnbet,- which regulates the Motions 
 B % of 
 
VB A <P<P END IX. 
 
 of the Tendulum, and confignsits juft length* 
 Then obferve the Varieties which happen 
 through various forts of Matter, upon which 
 Experiments are made. Merfennus tells us, 
 that heavier Weights of the fame length 
 move flower, fo that whilft a Lead Plum- 
 bet makes 39 Vibrations, Cork or Wood 
 will make at leaft 40. 
 
 Again, that a ftiff Pendulum vibrates 
 more frequently, than that which hangs 
 upon a Chord. So that a Bar of Iron, or 
 Staff of Wood ought to be half as long a- 
 gain as the other, to make the Vibrations 
 equal. Yet in each of thefe refpe&ively 
 to itfelf, you will find the duplicate Pro- 
 portion and Equality of Vibration, or as- 
 near as may be. And (as to Equality) 
 though in the Extreams of the Ranges of 
 Librations, viz. the greateft compared to 
 the leaft, there may (from unfeen Cau- 
 fes) appear fome Difference, yet there Is 
 no difcernable Difference of the Time of 
 Vibrations of a Pendulum in Ranges, that 
 are near to one another, whether greater 
 or lefs ; which is the Cafe of the Ranges 
 of the Vibration of a String being made 
 in. a very fmall Compafs : And therefore 
 the Librations of a Pendulum, limitted tq 
 a fmall Difference of Ranges, do well cor- 
 ■refpond with the Vibrations of a String. 
 
 As 
 
A s to Strings, the Whole of Harmony 
 depends upon this experimented and un* 
 questioned Truth, that Diapafon is duple 
 to its Unifon, and confequently Diapente 
 is Sefquialterum, DiatefTeron Sefquiteitmm, 
 &c. Yet if you happen to divide a faul- 
 ty String of an Inflrument, you will not 
 find the O&ave juft in the Middle, nor 
 the other Intervals in their due Proportion, 
 which is no default in Nature, but the 
 Matter we apply. A falfe String is that, 
 which is thicker in one Part of its Length, 
 than in another. The thicker Part natu- 
 rally vibrates flower, and founds graver • 
 the more {lender Part vibrates fwifter, and 
 founds more acute. Thus whilit two 
 Sounds fo near one another, are at once, 
 made upon the fame String, they make 
 a rough difcording Jarr, being a hoarfe 
 Tune mixed of both, more or iefs, as the 
 String is more or lefs unequal : And if the 
 thicker Part be next the Frets, then the. 
 Fret (for Example D. F. H. &c. in a Viol 
 or Lute) will render the Tune of the Note 
 too fharp ; and the contrary, if the {len- 
 der Part of the String be next the Frets ; 
 becaufe in the former, the thicker Part is 
 flopped, and the thinner founds more of 
 the acuter Part of this unhappy Mixture : 
 As in the later, the thicker Part is left to 
 found the graver Tune, and thus the Freft 
 C J wit 
 
20 J<P<P ENT> IX. 
 
 will give a wrong Tone though the Fault 
 be not in the Fret, but in the String ; which 
 yet, by an unwary Experimenter, may hap- 
 pen to caufe the Setao Canonis to be cal- 
 led in queftion, as well as the Meafuresof 
 a Tenatilum are difputecL 
 
 But all this does not difprove the Mea- 
 fures found out, and affigned to Harmo- 
 nick Intervals, which are verified upon a 
 true String or Wire as to their Lengths, 
 and as to the Equality of Recourfes in 
 their Vibrations, though Pendulums are 
 thought to move flower in their leaft 'Ran- 
 ges ; yet, as to Strings, in the very fmall 
 Ranges which they make, (which are much 
 lefs in other Inftruments, or founding Bo- 
 dies) I need add no more than this, that 
 the Continuance of the fame Tune to the 
 laft, after a Chord is ftruck, and the con- 
 tinned Motion in lefs Vibrations of a fyrii? 
 pathizing String, during the Continuance 
 of greater Vibrations of the String which 
 is ftruck, do either of them fufficiently de- 
 monftrate, that thofe greater or lefs Vibra- 
 tions, are both made in the fame Meafure 
 pf Time, according to their Proportions, 
 keeping exaft Pace with each other. O- 
 therwife ; In the former, the Tune would 
 feniibly alter, and in the latter, the fym- 
 •pathizing String could not be continued 
 
 in 
 
in its Motion. This was not fo well con* 
 eluded, till the late Difcoveries of the Pen* 
 tm gave light to it. 
 
 There is one thing more which . 1 
 nmft not omit. That the Motions of a 
 Pendulum, may feem not fo proper to ex-* 
 plicate the Motions of a String; becaufe 
 the faid Motions depend upon differing 
 Principles, viz. thofe of a Pendulum upon 
 Gravity ; thofe of a String upon Elafticky* 
 1 jfhall therefore endeavour to fhew, how 
 the Motions of a Tendulum, agree with 
 thofe of a Spring, and how properly the 
 Explication of the Vibrations of a String, 
 is deduced from the Properties of a Pen* 
 dulum. 
 
 The Elaftick power of a Spring, in a 
 Body indued with Elafticity, feems to be 
 nothing elfe, but a natural Propenfion and 
 Endeavour of that Body, forced out of its 
 own Place, or Poflure, to reftore it {elf a- 
 gain into its former, more eafie and natu- 
 ral Pofture of Reft. And this is found in 
 feveral Sorts of Bodies, and makes differ 
 rent Cafes, of which I (ball mention feme, 
 
 I f the Violence be by Compreffion, for-* 
 cing a Body into lefs room than it natu-* 
 rally requires ^ then the Endeavour of Re- 
 
 Q 4 fti^b 
 
%% Jff BKDIX* 
 
 ftitution, is by Dilatation to gain room e* 
 pough. Thus Air may be compreffed in- 
 to lefs Space, and then will have a great 
 Elafticity, and ftruggle to gain its room. 
 Thus, if you fqueeze a dry Sponge, it will 
 paturaliy, when you take off the Force^ 
 ipread it felf, and fill its former Place* So f 
 if you prefs with your Finger a blown 
 Bladder, it will fpriog and rife again to its 
 Place* And to this may be reduced the 
 Springs of a Watch, and of a Spiral Wire, 
 
 Again, aftiff, but pliable Body, fatten- 
 ed at one End, and drawn afide a.t the o- 
 ther, will fpring back to its former Place ; 
 this is the Cafe of Steel-fprings of Locks ? 
 Snap-haunces, £jjV. and Branches of Trees, 
 when fhaken with the Wind, or pulled a- 
 Iide, return to their former PoiTure : As is 
 faid of the Palm, DepreJJa Refurgo. And 
 there are innumerable inftances of this kind, 
 where the force is by bending, and the 
 Reftitution by unbending or returning. 
 
 This kind is refembled by a Pendulum? 
 or Plummet hanging on a String, whofe 
 gravity, like the Spring in thofe other Bo- 
 dies, naturally carries it to its place, which 
 liere is downward ; all heavy Bodies na« 
 lurally defcending till tliey meet with fome 
 
 Ob- 
 
'APPENDIX. 
 
 3 
 
 Obftacle to reft upon. And the loweft that 
 the Plummet can defcend in its Reftraint 
 by the String, is, when it hangs perpendi- 
 cular, as to A B ; where it is neareft to the 
 Horizontal Plane G H, and therefore low- 
 eft. Now, if you force the Plummet up- 
 ward (held at length upon the String) from 
 B to C, and let it go ; it will, by a Spon- 
 taneous Motion, endeavour its Reftitution 
 to B : But, having nothing to ftop it but 
 Air, the Impulfe of its own Velocity will 
 carry it beyond B, towards D ; and fo 
 backward and forward, decreafing at every 
 Range, till it come to reft at S. 
 
 G B H 
 
 Thus the Pendulum and Spring agree 
 In Nature, if you confider the Force a- 
 gainft them, and their Endeavour of Re- 
 $itutiono 
 
 But 
 
24 J PP END IX. 
 
 But further, if you take a thin ftiff La^ 
 mina of Steel, like a Piece of Two-penny 
 Riband of fome length, and naii it fail: at 
 one End, (the remainder of it being free 
 in the Air) then force the other End afide 
 and let it go j it will make Vibrations back- 
 ward and forward, perfectly anfwering 
 tliofe of a Pendulum. And much rnore^ 
 if you contrive it with a little Steel But- 
 ton at the End of it, both to help the Mo- 
 tion when once fet on foot, and to bear 
 it better againft the Refiftance of the Air. 
 There will be no difference between the 
 Vibrations of this Spring, and of a Tendu** 
 lum-> which in both, will be alike increafed 
 or decreafed in Proportion to their Lengths. 
 The fame End (viz. Reft) being, in the 
 fame manner, obtained by Gravity in one*, 
 and Elafticity in the other. 
 
 Further yet, if you nail the Spring 
 above, and let it hang down perpendicu- 
 iar, with a heavier Weight at the lower 
 End, and then fet it on moving, the Vi- 
 brations will be continued and carried on 
 both by Gravity and Elafticity, the Ten- 
 dulum and the Spring will be moft friend- 
 ly joyned to caufe a fimple equal Motion 
 of Librations, I mean, an equal Meafure 
 of Time in the Recourfes ; only the Spring 
 anlwerabl? to its Strength., xxiay caufe the 
 
APPENDIX. 2 j 
 
 Libratlons to be fome what fwifter, as an. 
 Addition of Tenfion does to a String con- 
 tinued in the fame length. 
 
 I come now to confider a String of an 
 Inftrument, which is a Spring faifned at 
 both Ends. It acquired! a double Elafti- 
 city. The firft by Tenfion, and the Spring 
 is ftronger or weaker, according as the 
 Tenfion is greater or lefs. And by how 
 much ftronger the Spring is, fo much more 
 frequent are the Vibrations, and by this 
 Tenfion therefore, the Strings of an Inftru- 
 ment keeping the fame length are put in 
 Tune, and this Spring draws length- ways, 
 endeavouring a Relaxation of the Tenfion, 
 
 But then, Secondly , the String being 
 under a ftated Tenfion, hath another E- 
 laftick Power fide-ways, depending upon 
 the former, by which it endeavours, if it 
 be drawn afide, to reftore it felf to the 
 eafieft Tenfion, in the jQiorteft, viz. ftreight- 
 eft line. 
 
 I n the former Cafe, Tenfion doth the 
 fame with abatement of length, and affe£is 
 the String properly as a Spring, in that 
 the String being forcibly ftretched, as for- 
 cibly draws back to regain the remifs Pof- 
 ture in which it was before: And bears 
 
 < -little 
 
26 APPENDIX, 
 
 little Analogy with the Pendulum, except 
 in general, in their fpontaneous Motions 
 m order to their RefKtution. 
 
 But there is great Correfpondence in 
 the fecond Cafe, between the Librations 
 of a Pendulum and the Vibrations of a 
 String (for fo, for diftinftions fake, I will 
 now call them) in that they are both pro- 
 portioned to their length, as has been 
 ihewn ; and between the Elafticity which 
 moves the String, and Gravity which moves 
 the Pendulum, bqth of them having the 
 fame Tendency to Reftitution, and by the 
 fame Method. As the Declivity of the 
 Motion of a Pendulum, and consequently 
 the Impuife of its Gravity is ftill leffened 
 m the Arch of its Range from a Semi- 
 Circle, till it come to reft perpendicular ; 
 the Defcent thereof being more downright 
 at the firft and greatcft Ranges, and more 
 Horizontal at the [aft and fhorteft Ranges, 
 as may be feen in the preceding Figure C I 
 IE E ( B ; fo the Impuife of Spring is ftill 
 gradually lelfened as the Ranges ihorten, 
 and as it gains of relaxation, till it come 
 to be tailored to reft in its fhorteft Line. 
 And this may be the Caufe of the Equality 
 of Time of the Librations of a "Pendulum 7 
 and alio of the Vibrations of a String. 
 Now, the Proportions of i-ength^ to the 
 
 VeioK 
 
APPENDIX. sy 
 
 Velocity of Vibrations in one, and of Li- 
 forations in the other, we are fure of, and 
 find by manifeft Experience to be qua- 
 druple in one, and duple in the other. 
 
 Now tack two equal Pendulums toge- 
 ther (as before) being faftned at both Ends, 
 take away the Plumbets, and you make it 
 a String, retaining till the fime Properties 
 of Motion, only what was faid before to 
 be caufed by Gravity, muft now be faid 
 to be done by Elafticity* You fee what 
 an eaiie Step here is out of one into the 
 other, and what Agreement there is be- 
 tween them. The Phenomena are the 
 fame, but difficultly experimented in a 
 String, where the Vibrations are too fwift 
 to fall under each exa£t Meafure ; but moft 
 eafie in a Pendulum^ whofe flow Libra- 
 tions may be meafured at pleafure, and 
 numbered by diftant Moments of Time. 
 
 T o bring it nearer, make your Tenfion 
 of the String by Gravity, inftead of fcrew- 
 ing it up with a Pegg or Pin : Hang weight 
 upon a Pulley at oae End of the String, and 
 as you increafe the Weight, fo you do 
 increafe the Tenfion, and as you increafe 
 the Tenfion^ fo you increafe the Velocity 
 of Vibrations, So the Vibrations are pro- 
 portionably.regulated immediately by Ten- 
 fion 
 
28 APPENDIX. 
 
 fion, and mediately by Gravity. So that 
 Gravity may claim a fhare in the Meafares 
 of thefe Harmonick Motions. 
 
 But to come ftill nearer, and home to 
 our purpofe. Faften a Gut or Wire-firing 
 above, and hang a heavy Weight on it 
 below, as the Weight is more or lefs, fo 
 will be the Tenfion, and confequently the 
 Vibrations. But let the fame Weight con- 
 tinue, and the String will have a ftated 
 fetled Tenfion* Here you have -both in 
 one, a Pendulum^ and the Spring of a Strings 
 which refembles a double TendtiJum ; Draw 
 the Weight afide, and let it range, and 
 It is properly a *Pendulum y librating after 
 the Nature of a f&ndulum. Again, when 
 the Weight is at reft, ftrike the String with 
 a gentle PleBrum made of a Quill, on the 
 tipper part, fo as not to make the Weight 
 move, and the String will vibrate, and give 
 Its Tune, like other Strings fattened at 
 both Ends, as this is alfo, in this Cafe. 
 So you have here both a Tendulum and a 
 String, or either, which you pleafe. And 
 (the Tenfion being fuppofed to be fettled 
 under the fame Weighr) the common Mea- 
 fure and Regulator of the Proportions of 
 them is the Length, and as you alter the 
 Length, fo proportionably you alter at once 
 the Velocity in the Recourfes of the Vi- 
 
 bra- 
 
"APPENDIX. 2 9 
 
 brations of the String, and of the Libra- 
 tions of the Tenduhm. And though the 
 Vibrations be fo much fwifter, and more 
 frequent than the Librations, yet the Ra- 
 tions are altered alike. If you fubduple the 
 Length of the String, then the Vibrations 
 will be dupled. And if you fubquadruple 
 it, then the Librations will be alfo dupled,, 
 allowing for fo much of the Body of the 
 Weight as muft be taken in, to determine 
 the Length of the Pendulum. 
 
 The Vibrations are altered in duple 
 Proportion to the Librations, becaufe (as 
 lias been fhewn) the String is as a double 
 Tendulum^ either one of which fuppofed 
 Pendulums is but half fo long as the Stringy 
 and is quadrupled by dupling the whole 
 String. Still therefore the Proportion of 
 their Alterations holds fo certainly and 
 regularly with the Proportion of every 
 Change of their common Length, that ? 
 if you have the Comparative Ration of 
 either of thefe two, viz.. Vibrations or Li- 
 brations to the Length, you have them 
 both : The increafe of the Velocity of Li- 
 brations being fubduple to the increafe of 
 the Velocity of Vibrations. And thus the 
 Motions of a "Pendulum do fully and pro- 
 perly difcover to us, the Motions of a 
 String, by the manifeft Correfpondence of 
 
 their 
 
3o APP ENBIX. 
 
 their Properties and Nature. The Texdih 
 lum^s Motion of Gravity, and the Strings 
 of Elafticity bearing fo certain Proportions 
 according to Length, that the Principles 
 of Harmony, may be very properly made 
 out, a*nd moft eafily comprehended, as ex* 
 plained by the Pendulum. And we find, 
 that in all Ages, this part of Harmony was 
 never fo cheerfully underftood, as fince the 
 late Difcoveries about the Pendulnnu 
 
 And I chufe to make this Illuftration 
 by the Pendulum, becaufe it is fo eafie for 
 Experiment, and for our Comprehenfion ; 
 and the Elaftick Power fo difficult* 
 
 Having [qqr the Origine of Tuneable 
 or Harmonick Sounds, and of their Dif* 
 ference in refpeft of Acutenefs and Gra- 
 vity : It is next to be confidered, how 
 they come to be affeQ;ed with Confonancj 
 and DifTonancy, and what thefe are* 
 
 CHAE 
 
j* 
 
 CHAP. III. 
 Of Confonancy and Diffonancy* 
 
 C^Onfonancy and DiiTonancy are the Re- 
 jj fult of the Agreement, mixture or u- 
 niting (or the contrary) of the undulated 
 Motions of the Air or Medium, caufed by 
 the Vibrations by which the Sounds of 
 diftinft Tunes are made. And thofe are 
 more or lefs capable of fuch Mixture or 
 Coincidence according to the Proportion 
 of the Meafures of Velocity in which they 
 are made, /. e . according as they are more 
 or lefs commenfurate. This I might well 
 fet down as a Fofhdatum. But I fhall by 
 feveral Inftances endeavour to illuftrate the 
 undulating Motions or Undulations of the 
 Air ; and confirm what is faid of their 
 Agreements and Difagreements. And firft 
 the Undulations, by fomewhat we fee in 
 other Liquids. 
 
 L e t a Stone drop into the Middle of 
 a fmall Pond of (landing Water when it is 
 quiet, you (hall fee a Motion forthwith im- 
 prefled upon the Water^ paffing and dila- 
 ting from that Center where the Stone 
 M\ in circular Waves one within an other ? 
 
 B iili 
 
3 2 Of Conionancy 
 
 ftill propagated from tlie Center, fp read- 
 ing till they reach and dafh againft the 
 Banks, and then returning, if the force of 
 the Motion be fufficient, and meeting thofe 
 inner Circles which purfue the fame Courfe, 
 without giving them any Check. 
 
 And if you drop a Stone in another 
 place, from that Centre will likewife fpread 
 round Waves ; which meeting the other, 
 will quickly pafs them, each moving for- 
 wards in its own proper Figure. 
 
 The like is better experimented in 
 Quick-filver, which being a more denfe 
 Body, continues its Motions longer, and 
 may be placed nearer your Eye. If you 
 try it in a pretty large round Veffel, flip- 
 pole of a Foot Diameter, the Waves will 
 keep their own Motion forward and back- 
 ward, and quietly pafs by one another as 
 they meet. Something of this may be feeri 
 in a long narrow Paffage, where there is 
 not room to advance in Circles* 
 
 Make a wooden Trough or long Box, 
 fuppofe of two Inches broad, and two 
 deep, and twenty long. Fill in three Quar- 
 ters or half full of quick-filver, and place 
 it Horizontally, when it is at quiet, give 
 it with your Finger a little patt at one End ? 
 
 and 
 
and Difforiancy* { 3 1 
 
 and it will imprefs a Motion of a ridged 
 Wave a crofs, which will pafs on to the 
 other End, and dafhing againft it, return 
 in the fame Manner, and dafh againft the 
 hether End, and go back again, and thus 
 backward and forward, till the Motion 
 ceafe. Now if after yau have fet this 
 Motion on foot, you caufe fuch -another, 
 you fhall fee each Wave keep its regular 
 Courfe ; and when they meet one another* 
 pafs on without any Reiu£tancy. 
 
 I do not fay thefe Experiments are full 
 to my purpofe, becaule thefe being upon 
 fingle Bodies, are not fuiHcient to exprefs 
 the Difagreements of Difproportionate Mo- 
 tions caufed by different Vibrations of 
 different founding Bodies ; but thefe may 
 ferve to illuftrate thofe invifible Undula- 
 tions of Air. And how a Voice reflected 
 by the Walls of a Room, or by Eccho be- 
 ing of adequate Vibrations, returns from 
 the Wall, and meets the commenfurate 
 Undulations paffing forwards, without hin- 
 dering one another. 
 
 But there are Inftances which further 
 confirm the Reafons of Confonancy and 
 Diffonancy, by the manifeft agreeing or 
 difagreeing Meafures of Motions already 
 fpoken of. 
 
 D % 1 T 
 
34 Of Confonarcy 
 
 It hatli been a common Practice to 
 imitate a Tabour and Pipe upon an Or- 
 gan. Sound together two difcording Keys 
 (the bafe Keys will fhew it belt, becaufe 
 their Vibrations are flower) let them, for 
 Example, be Gamut with Gamut fharp, 
 or F Faut (harp, or all three together. 
 Though thefe of themfelves fhould be ex- 
 ceeding fmooth and well voyced Pipes ; 
 yet, when ftruck together, there will be 
 iiich a Battel in the Air between their 
 difproportioned Motions, fuch a Clatter 
 and Thumping, that it will be like the 
 beating of a Drum, while a Jigg is play- 
 ed to it with the other hand If you 
 ceafe this, and found a full Clofe of Con- 
 cords, it will appear furprizingly fmooth 
 and fweet, which fhews how Difcords 
 well placed, fet off Concords in Compo- 
 iition. But I bring this Inftance to fhew, 
 how ftrong and vehement thefe undula- 
 ting Motions are, and how they corref- 
 pond with the Vibrations by which they 
 are made. 
 
 I t may be worth the while, to relate 
 an Experiment upon which I happened. 
 Being in an Arched founding Room near 
 a fhrill Bell of a Houfe Clock, when the 
 Alarm ftruck, I whittled to it, which I 
 did with eafe in the fame Tune with the 
 Bell, but, endeavouring to whiftle a Note 
 
 higher 
 
and Diflonancy f$ 
 
 higher or lower, the Sound of the Bell 
 and its crofs Motions were fo predominant, 
 that my Breath and Lips were checkM fa, 
 that I could not whittle at alienor make 
 any Sound of it in that difcording Tune. 
 After, I founded a fhrill whiffling Pipe, 
 which was out of Tune to the Bell, and 
 their Motions fo clafhed, that they feemed to 
 found like (witching one another in the Air. 
 
 GALILEO, from this Doftrine of 
 Tendulums , eafily and naturally explains 
 the fo much admired fympathy of Con- 
 fonant firings ; one (though untouched) 
 moving when the other is ftruck. It is 
 perceptible in Strings of the fame, or a- 
 nother Inftrumerit, by trembling fo as to 
 lhake off a Straw laid upon the other 
 String : But in the fame Inftrument, it 
 may be made very vifible, as in a Bafs- 
 Viol. Strike one of the lower Strings 
 with the Bow, hard and ftrong, and if any 
 of the other Strings be Unifon or Q£tave 
 to it, you ftiall plainly fee it vibrate, and 
 continue to do fo, as long as you continue 
 the Stroke of your Bow, and, all the while, 
 the other Strings which are diffonant, reft 
 quiet. 
 
 The Reafon hereof is this. When you 
 
 flrike your String, the Progreflive found 
 
 • D $ of 
 
%6 Of Confonancy 
 
 of it ftrikes and ftarts all the other Strings, 
 and every of them makes a Move- 
 ment In its own proper Vibration . The 
 Confonant firing, keeping meafure in its 
 Vibrations with thofe of the founding String 
 hath its Motion continued, and propaga- 
 ted by continual agreeing Pulfes or Stokes 
 of the other. Whereas the Remainder of 
 the Diffonant firings having no help, but 
 being checked by the Crofs Motions of the 
 founding String, are constrained to remain 
 ftill and quiet. Like as, if you ftand be- 
 fore a Pendulum, and blow gently upon 
 it as it paffeth from you, and fo again in 
 its next Courfes keeping exaft time with 
 it, it is moft eafily continued in its Mo- 
 tion, But if you blow irregularly in Mea- 
 sures different from the Meafure of the 
 Motion of the Pendulum, and fo moft fre- 
 quently blow againft it, the Motion of it 
 will be fo checked, that it moil quickly 
 ceafe* 
 
 'And here we may take Notice, (as has 
 been hinted before) that this alfo confirms 
 
 the aforefaid Equality of the Time of Vi- 
 brations to the laft, for that the fmall and 
 weak Vibrations of the fympathizing String 
 are regulated ^and continued by the Pulfes 
 ■of the greater and ftronger Vibrations of 
 the (bunding String, which proves, that 
 
 not- 
 
and Oifionancy* 37 
 
 notwithftanding that Difparity of Ranges 
 they are commenfurate in the Time ,01 
 their Motion. 
 
 This Experiment is ancient : I find it 
 in oAri Hides Quint Hi anus & Greek Author, 
 who is fuppofed to have been contempo- 
 rary with Tint arch. But the Reafon of it 
 deduced from the Pendulum, is m\v ? and 
 firft difcovered by Galileo. 
 
 I t is an ordinary Trial, to find out 
 the Tune of a Beer-g!afs without finking 
 it, by holding it near your Mouth, a i 
 humming loud to it, in feveral fingle Tunc 
 and when you at laft hitt upon the Tune 
 of the Glafs, it will tremble and Eccho to 
 you. Which fhews the Confent and Uni- 
 formity of Vibrations of the fame Tune, 
 though in feveral Bodies. 
 
 A^t. 
 
 To ciofe this Chapter. I may conclude 
 that Confonancy is the Paffage of feveral 
 Tuneable founds through the Medium, fre- 
 quently mixing and uniting in their undu- 
 lated Motions, caufed by the well pro- 
 portioned commenfurate Vibrations of the 
 ionorous Bodies, and confequently arriving 
 fmooth, and fweet, and pleafant to the Ear. 
 On the contrary, DiiTonancy is from dif- 
 proportionate Motions of Sounds, not mix- 
 D 4 ing 
 
3 8 Of Confonancy 
 
 kg, but jarring and chilling as they pafs, 
 and arriving to the Ear Harfh, and Gra- 
 ting, and Offenfive. And this, in the next 
 Chapter fhall be more amply explained. 
 
 Now, what Concords and Difcords are 
 thus produced, andinufe, in order to Har- 
 mony, I fhall next conilder* 
 
 C H A R IV. 
 
 Of Concords. 
 
 COncords are Harmonick founds, which 
 being joined pleafe and delight the 
 Ear ; and Difcords the Contrary. So that 
 It is indeed the judgment of the Ear that 
 determines which are Concords and which 
 are Difcords. And to that we muft firft 
 refort to find out their Number. And then 
 we may after fearch and examine how the 
 natural Produftion of thofe Sounds, dif- 
 pofeth tl em to be pleafing or unpleafant* 
 Like as the Palate is ablolute Judge of 
 Tafts, what is fweet, and what is bitter, 
 or iowr, £jfc- though there may be alfo 
 found out fome natural Caufes of thofe 
 Qualities. But the Ear being entertained 
 tyath Motions which fall under exaft De- 
 mon- 
 
Of Concords- 55) 
 
 monftrations of their Meafures, the Do- 
 £hine hereof is capable of being more ac- 
 curately difcovered. 
 
 First then, (fetting afide the Unifdn 
 Concord, which is no Space nor Interval, but 
 an Indentity of Tune) the Ear allows and 
 approves thefe following Intervals, and on- 
 ly thefe for Concords to any given Note, 
 -viz. the O&ave or Eighth, the Fifth, then 
 the Fourth, (though by later Mafter^of 
 Mufick degraded from his Place) then the 
 Third Major, the Third Minor, the Sixth 
 Major, and the Sixth Minor. And alio 
 fuchj as in the Compafs of any Voice or 
 Inftrument beyond the O&ave, may be 
 compounded of thefe, for fuch thofe are, 
 I mean compounded, and only the for- 
 mer feven are fimple Concords ; not but 
 that they may feem to be compounded, 
 viz. the greater of the lefs within an O- 
 £tave, and therefore may be called Syftems, 
 but they are Originals. Whereas beyond 
 an Oftave, all is but Repetition of thefe 
 in Compound with the Eighth, as a Tenth 
 is an Eighth and a Third ; a Twelfth is 
 an Eighth and a Fifth ; a Fifteenth is Dii- 
 diapaion, u e. two O&aves, &c. 
 
 But notwithstanding this DifUn&ion 
 of Original and Compound Concords j and 
 
 tha? 
 
40 Of Coneorek 
 
 tho' thefe Compounded Concords are founds 
 and difcerned by their Habitude to the 0~ 
 riginal Concords comprehended in the Sy« 
 ftem of Diapafon ; (as a Tenth afcending 
 is an Offcave above the Third, or a Third 
 above the Odave ; a Twelfth is an 0£tave 
 to the Fifth, or a Fifth to the Eighth, a 
 Fifteenth is an Eighth above the Odave, 
 n e« Ditdiapafon two Eighths, iafcj yet 
 they muft be ownM, and are to be efteemM 
 good and true Concords, and equally ufe- 
 till in Melody, efpecially in that of Con- 
 fort. 
 
 T h b Syftem of an Eighth, containing 
 feven Intervals, or Spaces, or Degrees, and 
 eight Notes reckoned inclufively, as ex- 
 preffed by eight Chords, is called Diapa- 
 ibn, L e» a Syftem of all intermediate Con* 
 cords, which were anciently reputed to 
 be only the Fifth and the Fourth, and it 
 comprehends them both, as being com- 
 pounded of them both ; And now, that 
 the Thirds and Sixths are admitted for 
 Concords, the Eighth contains them alfo : 
 Viz. a Third Major and Sixth Minor, and 
 again a Third Minor and Sixth Major* 
 The Odave being but a Replication of 
 the Unifon, or given Note below it, and 
 the fame, as it were in Miniature, it ck> 
 feth and terminates the firft perfed Syftem,, 
 
 and 
 
Of Concords. 41 
 
 and the next O&ave above it afcends by 
 the fame Intervals, and is in like manner 
 compounded of them, and fo on, as far as 
 you can proceed upwards or downwards 
 with Voices or Inliruments, as may be 
 feen in an Organ, or Harpftchord. It is 
 therefore moft juftly judged by the Ear f 
 to be the chief of all Concords, and Is 
 the only Confonant Syftem, which h® 
 added to it felf, ftill makes Concorde, 
 
 And to it all other Concords agree*, 
 and are Confonant, though they do no 
 agree to each other ; nor any of them naakfe 
 a Concord if added to it felf, and the com- 
 plement or Refidue of any Concord to 
 Diapafon, is alfo Concord. 
 
 The next in Dignity is the Fifth, then 
 the Fourth, Third Mstjor, Third Minor^ 
 Sixth Major, and laftly Sixth Mifior \ all 
 taken by Afcent from the Unifon or given 
 Note, 
 
 By Unifon is meant, fometimes the Ha- 
 bitude or Ration of Equality of two Notes 
 compared together, being of the very fame 
 Tune. Sometimes (as here) for the given 
 fingle Note to which the Diftance, or the 
 Rations of other Intervals are compared* 
 As, if we confider the Relations to Gamut \ 
 
 to 
 
42 Of Concords. 
 
 to which Are is a Tone or Second, B ml 
 a Third, C a Fourth, D a Fifth, &fr . We 
 call Gamut the Unifon, for want of a more 
 proper Word. Thus C faut, or any other 
 Note to which other Intervals are taken, 
 may be called the Unifon* 
 
 And the Reader may eafily difcern, in 
 which Senfe it is taken all along by the 
 Coherence of the Difcourfe* 
 
 I come now to confider the natural Rea~ 
 fons, why Concords pleafe the Ear, by exa- 
 mining the Motions by which all Con- 
 cords are made, which having been gene- 
 rally alledged in the beginning of the third 
 Chapter, fhall now more particularly be 
 difcuffed 
 
 And here I hope the Reader will par- 
 don fome Repetition in a SubjeQ: that ftands 
 in need of all Light that may be, if, for 
 his eafe and more Heady Progrefs, before I 
 proceed, I call him back to a Review and 
 brief Summary of fome of thofe Notions, 
 which have been premised and confider- 
 ed more at large, I have fhewed, 
 
 i. That Harmonick Sound or Tune 
 is made by equal Vibrations or Tremblings 
 of a Body fitly conftituted* 
 
 2* Th AT 
 
Of Concords. 4$ 
 
 2. That thofe Vibrations make their 
 Courfes and Recourfes in the fame Mea- 
 fure of Time ; from the greateft Range 
 to the -leffer, till they come to reft. 
 
 3. That thofe Vibrations are under 
 a certain Meafure of Frequency of Courfes 
 and Recourfes in a given Space of Time* 
 
 4. That if the Vibrations be more 
 frequent, the Tune will be proportionably 
 more Acute ; if lefs frequent, more Grave* 
 
 5. That the Librations of a "Pendulum 
 become doubly frequent, if the Pendulum be 
 made four times fhorter ; and twice flower, 
 if the Pendulum be four times longer* 
 
 6. That a Chord, or String of a Mu- 
 fical Inftrument, is as a double Pendulum, 
 or two "Pendulums tacked together at length, 
 and therefore hath the fame Effects by 
 dupling ; as a "Pendulum by quadrupling, 
 /. e. by dupling the Length of the Chord, 
 the Vibrations will be fubdupled, /. e. 
 be halt fo many in a given Time. And 
 by fubdupling the length of the Chord, 
 the Vibrations will be dupled, and propor- 
 tionably fo in all other Meafures of Length, 
 the Vibrations bearing a Reciprocal pro- 
 portion to the Length. 
 
 7. T H A T 
 
^4 Of Concords. 
 
 j* That thefe Vibrations imprefs a 
 Motion of Undulation or Trembling in the 
 Medium (as far as the Motion extends) of 
 the fame Meafure with the Vibrations. 
 
 & That if the Motions made by dif- 
 ferent Chords be fo commenfurate, that 
 they mix and unite ; bear the fame Courfe 
 either altogether, or alternately, or fre- 
 quently : Then the Sounds of thofe diffe- 
 rent Chords, thus mixing, will calmly pafs 
 the Medium ) and arrive at the Ear as one 
 Sound, or near the fame, and fo will fmooth- 
 ly and evenly ftrike the Ear with Plea&re, 
 and this is Confonancy, and from the want 
 of fuch Mixture is Diffonancy. I may 
 add, that as the more frequent Mixture 
 or Coinfidence of Vibrations, render the 
 Concords generally fo much the more per- 
 fed; So, the lefs there is of Mixture, the 
 greater and more harlli will be the Dif- 
 cord* 
 
 From the Premifes, it will be eafie 
 to comprehend the natural Reafon, why 
 the Ear is delighted with thofe forenamed 
 Concords; and that is, becaufe they all 
 unite in their Motions often, and at the 
 leaft at every fixth Courfe of Vibration, 
 which appears from the Rations by which 
 they are conftitutedy which are all contain- 
 ed 
 
Of Concords. 45 
 
 ed within that Number, and all Rations 
 contained within that Space of Six, make 
 Concords, becaufe the Mixture of their 
 Motions is anfwerabJe to the Ration of 
 them, and are made at or before every 
 Sixth Courfe. This will appear if we exa- 
 mine their Motions. Firft, how and why 
 the Unifons agree fo perfeftly ; and then, 
 finding the Reafon of an O&ave, and fixing 
 that, all the reft will follow* 
 
 T o this purpofe, ftrike a Chord of a 
 founding Xnitrument, and at the fame 
 Time, another Chord fuppofed to be in 
 all refpe&s Equal, /; e. in Length, Matter, 
 Thicknefs and Tenfion. Here then, both 
 the Strings give their Sound ; each Sound 
 is a certain Tune •, each Tune is made by 
 a certain Meafure of Vibrations ; the fame 
 Vibrations are impreffed upon, and carried 
 every way along the Medium, in Undula- 
 tions of the fame Meafure with them, un- 
 til the Sounds arrive at the Ear. Now the 
 Chords being fuppofed to be equal in all 
 refpefts ; it follows, that their Vibrations 
 muft be alfo equal, and confequentiy move 
 in the fame Meafure, ioyning and uniting 
 in every Courfe and Recourfe, and keep- 
 ing ltill the fame Equality* and Mixture of 
 Motions of the String, and in the Medium. 
 Therefore the Habitude of thefe two 
 
 Strings 
 
4^ Of Concords. 
 
 Strings is called Unifon, and is fo perfe<9> 
 ly Confonant, that it is an Identity of 
 Tune, there being no Interval or Space 
 between them. And the Ear can hardly 
 judge, whether the Sound be made by two 
 Strings, or by one. ( 
 
 But Confonancy is more properly con- 
 sidered, as an Interval, or Space between 
 Tones of different Acutenefs or Gravity. 
 And amongft them 3 the mod perfefl: is that 
 which comes neareft to Unifon, (I do not 
 mean betwixt which there is the leaft Dif- 
 ference of Interval • but, in whofe Motions 
 there is the greateft Mixture and Agree- 
 ment next to Unifon. The Motions of 
 two Unifons are in Ration of i to i, or 
 of Equality. The next Ration in whole 
 Numbers is 2 to 1, Duple. Divide a Mo- 
 nochord in two Equal parts, half the 
 Length compared to the whole, being in 
 Subduple Ration, will make double Vi- 
 brations, making two Recourfes in the 
 fame time that the other makes one, and 
 fo uniting and mixing alternately, u e. eve- 
 ry other Motion. Then comparing the 
 Sounds of thefe two, and the half will be 
 found to found an Q&ave to the whole 
 Chord. Now the O&ave (afcending from 
 the Unifon) being thus found and fixed 
 tq be in duple Proportion of Vibrations, 
 
 and 
 
Of Concords. 47 
 
 and fubduple of Length ; confequently the 
 Proportions of all other Intervals are eafily 
 found out. 
 
 ^ Th ey are found out by refolving or di- 
 viding the OQrave into the Mean Rations 
 which are contained in it. Euclia y in his 
 Seffio QanonU{ Theorem 6 y gives two De- 
 monftrations to prove, that Duple Ration; 
 contains, and is composed of the two next 
 Rations, viz. Se [qui alt era and Sefquittriia* 
 Therefore an 0£tave which is in Duple Ra-, 
 tion 2 to 1 is divided into, and composed of 
 a Fifth, whofe Ration is found to be $?fi 
 quzaltera 3 to z ; and a Fourth, whofe Ra- 
 tion is Sefquitertia 4 to 3. In like manner 
 Sefqui altera is composed of Sefqmqtiarta and 
 Sefquiquinta ; that is* a Fifth, 3 to 2, may, 
 be divided into a Third Major, 5 to 4 • and 
 a Third Minor, 6 to 5, &a 
 
 There isan eafie Way to take a view 
 of the Mean Rations, which may be con- 
 tained in any Ration given, by transferring 
 - the Prime or Radical Numbers of the givea 
 Ration into greater Numbers of the lame 
 Ration, as 2 to 1 into 4 to 2, or 6 to &&€£ 
 which have the fame Ration of Duple. 
 Again, 3 to 2 into 6 to 4, which is ft ill <SV/» 
 quialierd* Now in 4 to 2 the Mediety is 3* 
 %q that 4 to 3^ and 3 to i } are comprea-^n-; 
 
4B Of Concords. 
 
 ^d in 4 to 2 ; that is, a Fourth and a Fifth 
 ar e comprehended in an Eighth. In 6 to 4 
 the Mediety is 5, fo 6 to 4 contains 6 to 5, 
 and 5 to 4 ; /. e. a Fifth contains the two 
 Thirds. Let 6 to 3 be the O&ave, and it 
 contains 6 to 5 Third &/}, 5 to 4 Third ma- 
 jor, and 4 to 3 a Fourth, and hath two Me- 
 dieties, 5 and 4. Of this I fliall fay more 
 in the xiQxt Chapter. 
 
 Th e se Rations exprefs. the Difference 
 of Length in feveral Strings which make 
 the Concords ; and confequently the Diffe- 
 rence of their Vibrations. Take two Strings 
 Afl, in all other refpe&s equal, and com- 
 pare their Lengths, which, if 'equal, make 
 Uaifon, or the iame Tune. If A be double 
 in Length to B, i. e. 2 to 1, the Vibrations 
 of B will be duple to thofe of A, and unite 
 alternately, viz. at every Courfe, eroding 
 at the Recourfe*, and give the Sound of an 
 Odave to A. 
 
 I f the Length of A be to that of B as 
 3 to 2, and confequently the Vibrations as 
 2 to 3, their Sounds will confort in a Fifth, 
 and their Motions unite after every fecond 
 Recourfe, u e* at every other or third 
 Courfe. 
 
 If 
 
Of Concords, 49 
 
 it A to B be as 4. to j 9 thev found a 
 Fourth, their Motions uniting after every 
 third Recourfe, viz. at every fourth Co urfe. 
 
 I f A to B be as 5 to 4, they found a 
 Ditone, or Third Major, and unite after 
 every fourth Recourfe, /"• e. every fifth 
 Courfe. 
 
 If A to B be as 6 to £, they found a 
 Trihemitone, or Third Mi nor i uniting af- 
 ter every fifth Recourfe, at every flxtli 
 Courfe, 
 
 Th u s» by the frequency of their being 
 mix'd and united, the Harmony of joyird 
 Concords is found fo very fweet and plea- 
 fing ; the Remoter being alfo combined by 
 their relation to other Concords befides the 
 Unifon. The greater Sixth, 5 to 3, is 
 within the compafs of Rations between 
 1 and 6 • but, I confefs, the lefler Sixth, 
 8 to 5, is beyond it ; but is the Comple- 
 ment of 6 to 5 to an Oftave, and makes a 
 better Concord by its Combinations with 
 the Q£tave, and Fourth from the UnKon ; 
 having the Relation of a Third Minor to 
 One, and of a Third Major to the 'Other, 
 and their Motions uniting accordingly. 
 And the Sixth Mc]or hathVne fame Ad- 
 vantage. Of thefe Combinations I fhall 
 
5<o Of Concords. 
 
 have occafion to fay fomcwhat more, after 
 I have made the Subje£fc in hand as plain 
 as I can. 
 
 I propqsM the collating of two feveral 
 Strings, to exprefs the Confort which is 
 made by them ; but otherwife, theie Ra- 
 tions are more certainly found upon the 
 Meaibres of a Monochord, taken by being 
 applyM to the Section of a Canon, or a 
 Rule of the String's length divided into 
 Parts, as occafion requires ; becaufe there 
 is no need fo oiten to repeat Ceteris pari- 
 Im, as is when feveral Strings are collated. 
 And if you take the Rations as Fractions, 
 it will be more eafie to meafure out the 
 given Parts of a Monochord, or fingle 
 String extended on an Inftrument : Thole 
 Parts of the String divided by a moveable 
 Bridge or Fret put under, and made to 
 found ; That Sound, related to the Sound 
 of the Whole, will give the Interval fought 
 after. Ex.gr. ~ of the Chord gives an 
 Eighth, j give a Fifth, i found a Fourth, 
 r (bund a Third Major, | a Third Minor, 
 4 a Sixth Major, \ a Sixth Minor : Now we 
 thus exprefe thefe Concords. 
 
 IJ-mfon* 
 
Of Concords. 
 
 yt 
 
 Vulfon* 3d Mhu 3d Maj. 4^h 
 
 5th 
 
 e'~fi-t--'fe 6 .-*■ 
 
 ..... — e-*— 
 -« -4— i- M 3 — 
 
 6th M11. 6th Maj. 8th 3d & 5th. 4th & 6th. 
 
 
 AuthentiCi 
 
 —44 
 
 L.c.1 D_i , 
 
 Fla<ral. 
 
 I faid, that all Concords are in Rations 
 within the Number Six ; and I may add, 
 that all Rations within the Number Six 
 are Concords : Of which take the following 
 Scheme, 
 
 6 to <$ 
 to 4 
 to 3 
 to 2 
 to I 
 
 3d Minor J\ '4 to 3 4th | 6 to 5 
 
 5 th 
 8th 
 iath 
 1 9 th 
 
 to 2 3th I 5 to 4 
 to i 1 5th 1 4 to 3 
 
 I 3 to 2 
 
 3 to 2 
 to I 
 
 to 4 3d Major/" 2 to 1 
 
 to 3 6th Major. 
 
 to 2 loth Maj or. 
 
 to 1 1 yth Ma] or. 
 
 5 th 
 
 1 2th 
 
 8th 
 
 i to 1 
 
 3d Minor! 
 3 d Major. 
 Fourth 
 Fifth 
 Eighth 
 
 All that are Concords to the Unifctn, 
 
 are alio Concords to the 0£tave And all 
 
 .that are Difcqrds to the Uniion 3 are D|£ 
 
 £ 3 confo 
 
5^ Of Concords. 
 
 cords to the G£tave.. And fome of the In- 
 terfiled iate Concords, are Concords one to 
 another ; as, the two Thirds to the Fifth, 
 and the Fourth to the two Sixths. So that 
 the Unifon, Third, Fifth, and 0£tave ; or 
 the Uniibn, Fourth, Sixth, and 0£tave, may 
 be founded together to inake a compleat 
 Cloie of Harmony : I do not mean a Clofe 
 to conclude with, for the Plagal is not 
 fucli ; but a compleat Clofe, as it includes 
 all Concords within the compafs of D'iapa- 
 foh. A Scheme of which I have let down 
 at the end of the 'foregoing Staff of five 
 Lin^s, which containerh the Notes by 
 which the aforefaid Concords are exprefsVh 
 The former two, which afcend from the, 
 Unifon, Gamut , by Third Major (or Minor) 
 and Fifth, up to the Oftave, are ufually 
 calPd Authentic!*, as relating principally to 
 ,the Uniibn, and btfl: iatisfying the Ear to 
 reft upon : The other two, which afcend by 
 the Fourth and Sixth Minor (or Major) up 
 to the fame Octave, are caii'd Plagal) as 
 more combining with the Octave, feeming 
 to require a more proper Bafs Note, viz-.' 
 an Eighth below the Fourth, and therefore 
 not making a good concluding Clofe : And 
 on the continual Shifting thefe, or often ' 
 . changing them, depends the Variety of 
 Harmony ( as far as Co'nfonancy reacheth, 
 ■^Jjiqli'is but as the E^ody of Mufick) in 
 
 all 
 
0/ Concords. 55 
 
 all Contrapun£b chiefly, but indeed iri all 
 lands of Compofition. I do not exclude a 
 Sprinkling of Difcords, nor here meddle 
 with Air, Meafure, and Rythmus, which 
 are the Soul and Spirit of Mufick, and give 
 it fo great a commanding Power. The 
 Plagal Moods defcend by the fame Inter- 
 vals, by which the Authentick afcend ; 
 which is by Thirds and Fifths; and the 
 Authentick defcend the fame by which the 
 Plagal afcend, viz. by Fourths and Sixths ; 
 one chiefly relating to the Unifon, the other 
 to the Octave. 
 
 But that, for which I defcribM thefe 
 full Clofes, was chiefly to give ( as I pro- 
 mised) a larger account of the before- 
 mentioned Combinations of Concords, which 
 encreafe the Confonancies of each Note, 
 and make a wonderful Variegation and 
 pelightfulnefs of the Harmony. 
 
 Cast your Eye upon the firft of 
 them in the Authentick Scale, you will fee 
 t\ia£*Bmi hath three Relations of Confo- 
 nancy, viz. to the Unifon, or given Note 
 G ; to the Fifth, and to the 0<9:ave : To 
 the Unifon as a Third minor ; to the Fifth 
 as a Third major ; to the Ottave a Sixth 
 major ; fo that its Motions joyn after every 
 fifth Recourfe, i.e. at every fixth Courfe^ 
 
 E 4 ' wm 
 
14 Of Concords, 
 
 jyith the Unifon ; every fifth with the Dia- 
 peate or Fifth ; -every fixth Courfe with the 
 p£tave. Then confider the Diapente, 
 D fol re ; as a Fifth to the Unifon, it joyns. 
 with it every third Courfe ; and as a 
 Fourth to the Odave, they joyn every 
 fourth Courfe. Then, the Qciave with the 
 XLiiibn, joyns after every fecond Vibration, 
 *, e. at every Courfe, 
 
 Now take a Review of the Variety of 
 Gpnibnancies in thefe four Notes. Here 
 are mixM together in one Confort the Ra- 
 tions of 2 to i 3 to 2, 4 to 3, 5 to 4, 6 to 5*, 
 5 to 3. And juft fo It Is in the other Clo- 
 ies, only changing alternately the Sixths. 
 
 You may fee here, within the fpace of 
 three Intervals from the Unifon, viz. 3d, 
 jth, and 8th, what a Concourfe there is of 
 Confonant Rations, to variegate and give 
 (as 'twere) a pleafant Purling to the Har- 
 mony within that Space : For now, all this 
 Variety is fbrnfd within one Syftem of 
 Dj'apafov, juftly bearing that Name. But 
 then, think what it will be when the re- 
 mote Compounded Concords are joyn'd to 
 them ; as when we make a full Cloie with 
 both Hands upon an Organ, or Harpfichord, 
 or when the higher Part of a Confort of 
 Muiick is reconcile to the lower, by the " 
 
 middle 
 
Of Concords* 5 f 
 
 middle Parts, viz. the Treble to the Bafs, 
 by the Mean and Tenor : And all this, re- 
 frefh'd by the Interchanges made be- 
 tween the Plagal and Authentick Moods. 
 Add to all this the infinite Variety of Move- 
 ment of fome Parts, thro 5 all Spaces, while 
 fome Part moves flowly ; and (as in FugesJ 
 one Part chafing and purfuing another. 
 
 The whole Reafon of Confonancy be- 
 ing founded upon the Mixture and Uni- 
 ting of the Vibrating Motions of feveral 
 Chords, or founding Bodies, 'tis fit it fihould 
 here be better explained and confirmed. 
 That their Mixtures accord to their Ra- 
 tions, 'tis eufie to be computed ; but it 
 may be reprefented to your Eye. 
 

 
 Of Concords, 
 
 V< 
 
 A 
 
 B 
 
 -0- 
 
 B 
 
 :>V 
 
 0^7 — ^>o 
 
 "If" 
 
 -B 
 
 I VIA. B B A 
 j OS ABBA ABBA 
 
 ABB A] A B, &c 
 AB BA AB BAjAB BA. 
 
 
 1 VI AB BA ■ j AB BA~T AB & c 
 * Dj ABC GAB J BAG C^A j ABC. ' 
 
 hz 
 
Of Concords. 57 
 
 Let V Vbea Chord, and ftand for 
 the Unifon : Let O O be a Chord half fo 
 long, which will be an O&ave to the Uni- 
 fon, and the Vibrations double : Then, I 
 ■fay, they will alternately (/. e. at every 
 other Vibration) unite. Let from A to B 
 be the Courfe of the Vibration, and from 
 B to A the Recourfe ; obferving by the 
 way, that (in relation to the Figures men- 
 tioned in this Paragraph and the next, as 
 alfo in the former Diagram of the 'Pendu- 
 lum, cap. 2, pag. 9.) when 1 fay, [ fromB 
 to A], and [overtakes V in A, c5'c] I do 
 there endeavour to exprefs the Matter brief 
 and perfpicuous, without perplexing the 
 Figures with many Lines ; and avoiding 
 the Incumbrance of fo many Cautions, 
 whereby to diftra£fc the Reader: Yetlmuft 
 always be undertfood to acknowledge the 
 continual Decreafe of the Range of Vibra- 
 tions between A a,nd B, while the Motion 
 continues ; and by A and B mean only the 
 Extremities of the Range of all thofe Vi- 
 brations, both the Firil greateft, and alfo 
 the Succeflive leffen'd, and gradually con- 
 tracted Extremities of their Range. And 
 the following Demonftration proceeds and 
 holds equally in both, being apply'dtothe 
 Velocity of Recourfes, and not to the Com-* 
 pais of their Range, which is not at all 
 here confider'd. Such a kind of Equity, I 
 
 mult 
 
58 Of Concords* 
 
 muft fometimes, in other parts of this Dif~ 
 courfe, beg of the candid Reader. To 
 proceed therefore, I fay, whiift V being 
 Amok, makes his Courfe from A to B ; O 
 ( ftruck like wife) will have his Courfe from 
 A to B, and Recourfe from B to A, Next, 
 whiift V makes Recourfe from B to A ? 
 O is making its Courfe contrary, from A 
 to B, but recourfeth. and overtakes V in A, 
 and then they are united in A, and begin 
 their Courfe together. So you fee, that 
 the Vibrations of Diafafcm unite alternate- 
 ly, joyning at every Courfe of the Unifon, 
 and eroding at the Recourfe. 
 
 Thus alfo Diapente, or Fifth, having 
 the Ration of 3 to 2, unites in like manner 
 at every third Courfe of the U-nifon. Let 
 the Chord D D be Diapente to the Unifon 
 
 V ; whiift V courfeth from A to B, the 
 Chord D courfeth from A to B, and makes 
 half his Recourfe as far as C ; /'. e. % to 2. 
 Whiift V recourfech from B to A, D paf- 
 feth from C to A, and back from A to B* 
 Whiift V courfeth again from A to B, D 
 paffeth from B to A, and back to C. Whiift 
 
 V recourfeth from B to A, D paffeth from 
 C to B, and back to A ; and then they 
 unite in A, beginning their Courfes toge- 
 ther at every third Courfe of V, In like 
 manner the reft of the Coacords unite^ at 
 
 the 
 
Of Concords. 59 
 
 the 4th, 5 th, 6th Courfe, according to their 
 Rations, as might this fame way be fhewn, 
 but it would take up too much room, and is 
 needlefs, being made evident enough from 
 thefe Examples already given. 
 
 Thus far the Rates and Meafures of 
 Confonance lead us on, and give us the true 
 and demonftrable Grounds of Harmony : 
 But ftill 'tis not compleat without Difcords 
 and Degrees (of which 1 fhall treat ia 
 another Chapter) intermixed with the Con- 
 cords, to give them a Foyl, and fet them 
 oft the better. For (to ufe a homely Re- 
 femblance) that our Food, taken alone, 
 tho' proper, and wholfome, and natural., 
 may not cloy the Palate, and abate the 
 Appetite, the Cook finds fuch kinds and 
 varieties of Sawce, as quicken and pleafe 
 the Palate, and fharpen the Appetite, tho* 
 not feed the Stomach- as Vinegar,Muftard, 
 Pepper, £jjV. which nourifh not, nor are 
 taken alone, but carry down the Nourifh- 
 ment with better Relifh, and aflift it in 
 Digeftion. So the Practical Matters and 
 Skilful Compofers make ufe of Difcords, 
 judicioufly taken, to relifh the Con fort, 
 and make the Concords arrive much twee- 
 ter at the Ear, in all forts of Defcant, but 
 moft frequently in Cadence to a Clofe. In 
 all which, the chief Regard is to be had 
 
 to 
 
60 Of Concords. 
 
 to what the Ear may expe<Et in the Con* 
 duct of the Composition, and muft be per" 
 form'd with Moderation and Judgment 5 
 which I now only mention, not intending 
 to treat of Compofing, which is out of my 
 Defign and Sphere , and would be too 
 large ; but my Defign is, to make thefe 
 Grounds as plain as I can, thereby to gra- 
 tifie thofe whofe Philofophical Learning, 
 without previous Skill in Mufick, will ea« 
 lily render them capable of this Theory : 
 And alfo thofe Matters in Praftick Mufick^ 
 and Lovers of it, who, tho' wanting Phi- 
 lofophy, and the Latin and other Foreign 
 Tongues, to read better Authors ; yet, by 
 the. help of their Knowledge in Mufick, 
 may attain to underiland the depth of the 
 Grounds and Reafons of Harmony, for 
 whofe fakes it is done in this Lan- 
 guage. 
 
 I fliall conclude this Chapter with fome 
 Remarks, concerning the Names given to 
 the feveral Concords : We call them Thirds 
 Fourth, Fifth, Sixth, and Eighth. Of tli^fe 
 the Thirds being Two, and Sixths being 
 alfo Two, want better diitinguifhing Names. 
 To call them Flat and Sharp Thirds, and' 
 Fiat and Sharp Sixths, is not enough, and 
 lies under a Miftake j I mean, it is not a 
 fufficient DifiinStion* to call the greater 
 
 TJiird 
 
Of Concords. 61 
 
 Third and Sixth, Sharp Third and Sharp 
 Sixth, and the leffer, Flat. They are fo 
 indeed in afcending from the Unilon, but 
 in defcending they are contrary ; for to 
 the O&ave that greater Sixth is a leffer 
 Third, and the greater Third is a leffer 
 Sixth ; which leffer Third and Sixth can- 
 not well be calPd Flat, being in a Sharp 
 Key ; Flat and Sharp therefore do not well 
 diftinguiih them in general ; the leffer 
 Third from the O&ave being (harp, and 
 the greater Sixth flat. So, from the Fifth 
 defcending by Thirds, if the firft be a mi- 
 nor Third, it is fharp, and the other being 
 a major Third, cannot be faid to be flat* 
 
 Th e other Diftin£Uon of them, viz.bf 
 Major and Minor, is more proper, and 
 does well exprefs which of them we mean. 
 But ftill the common and confafed Name 
 of "Third, if the Diftin&ion of major and 
 minor be not always well remembered, is apt 
 to draw young Practitioners, who do not 
 well confider, into another Error. I would 
 therefore call the greater Third (as the 
 Greeks do ) Ditone, i. e. of two whole 
 Tones ; and the Third minor, Trihemitone 
 or Sef quit one, as confifting of three half- 
 Tones, (or rather of a Tone and half a 
 Tone) and this would avoid the mentioned 
 
 Error which I am going to defcribe. 
 
 It 
 
6% Of Concords, 
 
 It is a Rule in compofing ConfortMu-* 
 fick, that it is not lawful to make a Move-* 
 ment of two Unifons, or two Eighths, or 
 two Fifths together ; nor of two Fourths^ 
 unlefs made good by the addition of Thirds 
 in another Part : But we may move as 
 many Thirds or Sixths together as we 
 pleafe. Which laft is falfe, if we keep to 
 the fame fort of Thirds and Sixths ; for 
 the two Thirds differ one from another in 
 like manner as the Fourth differs from the 
 Fifth : For in the fame manner as the 
 Eighth is divided into a Fifth and Fourth, 
 fo is a Fifth into a Third major and Third 
 minor. Now call them by their right 
 Names, and, I fay, it is not lawful to 
 make a Movement of as many Ditones, 
 or of as many SeJ quit o?ies as you pleafe ; 
 And therefore when you take the Liberty 
 Ipokeiv of, under the general Names of 
 Thirds^ it will be found that you mix Di~ 
 tones and Trthemitones, and fo are not con- 
 cerned in the aforefaid Rule ; and fo the 
 Movements of Sixths will be made with 
 mixture and interchanges of 6th major and 
 6th minor ) which is fafe enough. 
 
 Y e t, I confefs, there is a little more 
 
 Liberty in moving Trihemitones and Di- 
 to ies\ as like wile either of the Sixths, than 
 there is in moving Fourths or Filths ; and 
 
 the 
 
Of Concords 6\ 
 
 the Ear will bear it better. Nay, there is 
 neceflity, in a gradual Movement of Thirds, 
 to make one Movement by two Tribemi- 
 tones together in every Fourth and Fifth, 
 or Fourth disjunft; that is, twice in Dia- 
 fafon, or, at leaft, in two Fifths; as in 
 Gamut Key proper. The natural Afcent 
 will be Ut Re^ Ml \ Fa Sol La : Now, to 
 thefe join Thirds in Natural Afcent, and 
 then they will be Mi Fa Sol La Fa SoL 
 
 J Mi Fa Sol La Fa Sol\ A t ^ • .„ , 
 
 yUt Re Ml Fa Sol La r A ? d thllS lC Will be 
 
 in other Cliffs, but with fome variation, 
 according to the Place of the Hemitone. 
 Here \f e } and {2} are two Tribe* 
 mitones fucceeding one another, and you 
 cannot well alter them without difor- 
 dering the Afcent, and diiTurbing the Har- 
 mony; becaufe, where there is a Hernia 
 tone^ the Tone below joinM to it, makes a 
 Trihemitone ; and the next Tone above k y 
 joined to it, makes the fame. Thus you 
 fee the neceflity of moving two Trihemi* 
 tones together, twice in c Diapafov 1 or a 
 5?th, in progreflion of Thirds, in Diatonic 
 Harmony, but you cannot well go fur- 
 ther. 
 
 Now, there is Reafon why two Tribe* 
 mitones will better bear it, becaufe of their 
 different Relations, by which one Trihemi^ 
 
 F tone 
 
6 4 Of Concords. 
 
 tone is better diftinguifh'd from another^ 
 than one Oftave, or one Fifth, or one 
 Fourth from another. 
 
 In a Third minor \ which hath two De- 
 grees or Intervals, confifting of a Tone and 
 Hemitone^ the Hemitone may be placed 
 either in the lower Space, and then gene- 
 rally is united to his Third major (which 
 makes the Complement of it to a Fifth) 
 downward, and makes a fharp Key ; or 
 elfe it may be placed in the upper Space, 
 and then generally takes his third major 
 above, to make up the fifth upward, and 
 conftkute a flat Key. And thus a Tritone 
 is avoided both ways. I lay, if the Hemi- 
 tone in the Third minor be below, then 
 the Third major lies below it, and the 
 Air is fharp. If the Hemitone be above, 
 then the Third nujor lies above, and the 
 Air is flat. And thus the two minor 
 Thirds join'd in confequence of Move- 
 ment, are differenced in their Relations, 
 coaiequent to the place of the Hemitone ; 
 which Variety takes off all Naufeoufnefs 
 from the Movement, and Tenders it fweet 
 and piealant. 
 
 You cannot fo well and regularly make 
 a Movement of Ditoues, tho' it may be 
 doac focnetimes, once or twice, or mora, 
 
 in 
 
Of Concords, -&fi 
 
 in a Bearing Paflage (in like manner as 
 you may fometimes ufe Difcords ) to give^ 
 after a little grating, a better Reiifh. The 
 Skilful Artift may go farther in the ufe of 
 Thirds and Difcords than is ordinarily al- 
 lowed. 
 
 I might enlarge this Chapter, by fet- 
 ting down Examples of the Lawful and 
 Unlawful Movements of Thirds major and 
 minor ^ and of the Ufe of Difcords ; but, 
 as I faid before, my Defign is not to treat 
 of Compofition : However, you may caft 
 your Eye upon theie following Inftances^ 
 and your own Obfervation from the belt 
 Matters will furnifh you with the reft. 
 
 Lawful Movement Unlawful Movement 
 
 of Thirds, Mix'd. of Thirds Major. 
 
 S&&—Z Egl£*lE?E5ELEE 
 
 1? 
 
 #r% i% «s «S § *>5 
 
 Tha 
 
66 Of Concords, 
 
 That the Reader may not incurr any 
 Miftake or Confufion, by feveral Names 
 of the fame Intervals, I have here fet 
 them down together, with their Rations, 
 
 8th 
 
 9th Major. 
 
 7th Minor. 
 
 6th Major, 
 
 6th Minor. 
 
 5th 
 
 Odave 3 Diapaforu 
 Hsptachord Major. 
 Heptachord Minon 
 Hexachord Major* 
 Hexachord Minor. 
 Diapente^ Pentachord* 
 phFalfe (mdc-l .,. 
 feci: ) j*Semidiapente. 
 
 4th Falfe (in exO m . 
 cefs ) j^Tntone. 
 
 4^ , , . Diateffaron, Tetrachord. 
 
 3 d Major DkonQt 
 
 5 ,.. K Sefqukone. 
 
 ^ Minor ^Trihemitone. 1 
 
 2d Maj. orWhole ; ~j m MaXt 
 
 Note Major Tone Major 
 
 2d Afiw, or Whole > 
 Note Minor 
 
 >Tone Miner ^ 
 
 Min. 
 
 2d LeafiyOt Half- \Hemi.l tone 
 
 Note Greater i Semi. | Afoj J g Afi*i*». 
 C Hemi-Y ... ^ 
 
 ^Semi-> toneM ^/ 
 y Diefis Chromatic. )" 
 t Diefis Major. j 
 
 JT Diefis Enharmonic. 1 
 t Diefis Minor. J 
 
 Half Note Ee/i 
 
 Quarter- Note 
 
 Difference be- 
 tween Tone 
 Major, and 
 Tone Minor. 
 
 Co 
 
 Tim a. Comma Maj us . 
 Schifm. 
 
 > 
 
 2 to 
 
 IS 
 
 9 
 
 8 
 
 3 
 64 
 
 4* 
 
 5 
 3 
 S 
 
 2 
 
 4$ 
 
 32 
 3 
 
 10 
 
 16 i«5 
 
 25 24 
 
 128 125 
 
 81 So 
 
 ■2ft/«, Whenever I mention Diefis without diftin&ion* 1 
 rnean Diefis Minor, or Enharmonic: And when I fa 
 mention Comma, I mean Comma M*jiij P orSchifm. 
 
Of Concords, 67 
 
 1 fhould next treat of T^if cords, but be- 
 caufe there will intervene fo much Ufe of 
 Calculation, it is needful that (before I 
 go further) I premife lome account o£ 
 Proportion in General, and apply it to 
 Harmony, 
 
 CHAR V. 
 
 Of Proportion 3 and apply d to Harmony. 
 
 WH E R E A S it hath been faid before, 
 That Harmonick Bodies and Mo- 
 tions fall under Numerical Calculations, 
 and the Rations of Concords have been 
 already affign'd ; it may feem neceffary 
 here (before we proceed to fpeak of Dif- 
 cords ) to fliew the Manner how to cal- 
 culate the Proportions appertaining to 
 Harmonick Sounds : And for this I fhall 
 better prepare the Reader, by premifing 
 fomething concerning Proportion in Ge- 
 neral. 
 
 W e may compare f i. e. amongft them- 
 felves) either (1.) Magnitudes, (fo they 
 be of the fame kind;) or (2*) the Gravi- 
 tations , Motions , Velocities , 'Duratio't , 
 Sounds, &c» from tl;ence arifing j or fur- 
 
 F 3 tber, 
 
68 Of Proportion. 
 
 ther, if you pleafe, the Numbers them- 
 felves, by which the Things compared are 
 explicated. And if thefe fliall be unequal, 
 we may then confide?, either, Firft h how 
 much one of them exceeds the other ; or, 
 Secondly, after what manner one of them 
 {lands related to the other, as to the Quo- 
 tient of the Antecedent (or former Term) 
 divided by the Confequent (or latter Term :) 
 Which Quotient doth expound, denomi- 
 nate, or (hew, how many times, or how 
 much of a time or times, one of them 
 doth contain the other. And this by the 
 Greeks is call'd hoy©-, Ratio ; as they are 
 wont to call the Similitude, or Equality of 
 Ratio's ivdhoyU^ Analogie, Proportion, or 
 Proportionality : But Cuftom, and theSenfe 
 afTifting, will render any over-curious Ap- 
 plication of thefe Terms unnecelTary* 
 
 From thefe two Confiderations laft 
 mentioned there are wont to be deduced 
 three forts of Proportion, Arithmetical, Geo-* 
 metrical, and a mix'd Proportion refulting 
 from thefe two, call'd Harmonicah 
 
 i. (Arithmetical, when three or more 
 Numbers in Progreflion have the fame Dif- 
 ference • as, 2, 4, 6, 8, f$ij m or difcontinued^ 
 as 2,4,6; 14, 16,18, 
 
 2 e Geo- 
 
Of Proportion. 6? 
 
 2. Geometrical, when three or more 
 Numbers have the fame Ration, as 2,4,8, 
 i<5, 32; or difcontinited, 2^) 64,128. 
 
 Laftly, Harmonica! j ( partaking of both 
 the other) when three Numbers are fo or- 
 dered that there be the fame Ration of the 
 Greateft to the Leaft, as there is of the 
 Difference of the two Greater to the Diffe- 
 rence of the two Lefs Numbers : As in 
 thefe three Terms, 3, 4, 6, the Ration of 
 6 to 3 (being the greateft and leaft Terms) 
 is Duple* So is 2, the Difference of 6 and 4 
 (the two greater Numbers) to I, the Dif- 
 ference of 4 and 3 (the two lefs Numbers) 
 Duple alfo. This is Proportion Harmonf- 
 cal, which Diapafon, 6 to 3, bears to Dia- 
 pente 6 to 4, and Diateifaron 4 to 3, as its 
 mean Proportionals. 
 
 Now for the Kinds of Rations moil 
 properly fo calPd, /. e. Geometrical : Firft 
 obferve, that in all Rations the former 
 Term or Number (whether greater or lefs) 
 is always cali'd the Antecedent, and the 
 other following Number is calPd the CW- 
 fequeht. If therefore the Antecedent be 
 the greater Term, then the Ration is either 
 Multiplex, Stiperp articular, Svperf art tent y 
 or (what is compounded of tlieiej MPiti- 
 
 F 4 pkx 
 
jo Of Proportion. 
 
 flex Superfi articular, or Multiplex Super* 
 fartient. 
 
 i. Multiplex ; as Duple, 4 to 2 ; Triple, 
 6 to* 2 ; Quadruple, 8 to 2. 
 
 2. Stit erf articular ; as, 3 to 2, 4 to 3, 
 5 to 4, exceeding but by one aliquot part, 
 and in their Radical or leaft Numbers, al- 
 ways but by one ; and thefe Rations are 
 term'd Stfqmaltera, Sefquitertia (or Super- 
 iertia) Sejquiquarta, (or Superquarta) &x* 
 Note j that Numbers exceeding more than 
 by one, and but by one aliquot part, may 
 yet be Superf articular, if they be not ex- 
 prefsM in their Radical, i. e. leaft Num- 
 bers ; as 1 2 to 8 hath the fame Ration as 
 3 to 2 • /. e. Superp articular, tho' it feem 
 not fo till it be reduced by the greatefl 
 Common Divifor to its Radical Numbers 
 3 to 2. And the Common Divifor (i. e. the 
 Number by which both the Terms may 
 ieverally be divided) is often the Diffe- 
 rence between the two Numbers ; as in 
 12 to 8, the Difference is 4, which is the 
 Common Divifor. Divide 12 by 4, the 
 Quotient is 3 ; divide 8 by 4, the Quotient 
 is 2 ; fo the Radical is 3 to 2. Thusalfo 
 15 to 10 divided by the Difference 5, gives 
 3 to 2 ; yet, in 16 to 10,. 2 is the Common 
 Divifor, and gives 8 to 5, being Superpar-* 
 
 tknt* 
 
Of Proportion. 71 
 
 tient. But in all Superp articular Rations, 
 whofe Terms are thus made larger by be- 
 ing multiply^}, the Difference between the 
 Terms is always the greateft Common Di~ 
 vifor ; as in the 'foregoing Examples. 
 
 The third kind of Ration Is Superp ar- 
 tient) exceeding by more than One, as % 
 to 3, which is calPd SuperbipartiensTertias 
 (or Tria) containing 3 and f- 8 to ^ 
 Sup ertripartiens Quint as,) 5 audi* 
 
 The fourth is Multiplex Superp art icular^ 
 as 9 to 4, which is duple, and Sefquiquarta^ 
 1 3 to 4, which is triple, and Sefquiquarta. 
 
 The fifth and laft is Multiplex Superp ar- 
 tient^ as 1 1 to 4 ; duple, and Supertripar- 
 tiens Quartos. 
 
 When the Antecedent is lefs than the 
 Confequent, viz. when a lefs is compared 
 to a greater, then the fame Terms ferve 
 to exprefs the Rations, only prefixing Suh 
 to them; as, Submultiplex y Subfuperp arti- 
 cular ( or Subp articular ) Subfuperpartient 
 (or Subpartienf) &c, 410 2 is Duple y 2 to 
 4 is Subduple. 4 to 3 is Sefquitertia ; 3 to 
 4 is Subfefquitertia \ 5 to 3 is Superbipar- 
 tiens Tertias ; 3 to 5 is Subfuperbipartiens 
 TertiaS) &c. 
 
 This 
 
yi Of Proportion; 
 
 Th is fhort Account of Proportion; was 
 necefTary,becaufe almoft all the Philofophy 
 of Harmony confifts in Rations, of the 
 Bodies, of the Motions, and of the In- 
 tervals of Sound, by which Harmony is 
 made. 
 
 A n d in fearching, ftating, and compa- 
 ring the Rations of thefe, there is found 
 fo much Variety, and Certainty, and Faci- 
 lity of Calculation, that the Contempla- 
 tion of them may feem not much fefs de- 
 lightful than the very Hearing the good 
 Mufick it felf, which fprings from this 
 Fountain. And thofe who have already 
 an affeftion for Mufick cannot but find it 
 improv'd and much enhaunc'd by this plea- 
 fant recreating Chace ( as I may call it ) 
 in the large Field of Harmonic Rations and 
 TrofQYtiom, where they will find, to their 
 great Pleafure and Satisfaction, the hidden 
 Caufes of Harmony ( hidden to moft, even 
 to Practitioners themfelvesj fo amply dif- 
 coverM and laid plain before them. 
 
 All the Habitudes of Rations to each 
 other are found by Multiplication or Divi- 
 fion of their Terms ; by which any Ration 
 is added to, or fubilra&ed from another : 
 And there may be ufe of Progreffion of 
 
 Ra« 
 
Of Proportion. 7 5 
 
 Rations, or Proportions, and of finding a 
 Medium or Mediety between the Terms of 
 any Ration: But "the main Work is done 
 by Addition and Subftra&ion of Rations ; 
 which, tho' they are not performed like 
 Addition and Subftra&ion of Simple Num- 
 bers in Arithmetic!*, but upon Algebraick 
 Grounds, yet the Praxis is moil eafie. 
 
 O nt e Ration is added to another Ra~ 
 tion, by multiplying the two antecedent 
 Terms together ; i. e. the Antecedent of 
 one of the Rations by the Antecedent of 
 the other (for the more eafe they fhould 
 be reduced into their leaft Numbers or 
 Terms) and then the two Confequent 
 Terms in like manner. The Ration of the 
 ProduQ: of the Antecedents, to that of the 
 ProduQ: of the Confequents, is equal to the 
 other two added or join'd together. Thus 
 (for Example) add the Ration of 8 to 6; 
 i.e. (m Radical Numbers) 4 to 3, to the 
 Ration of 12 to 10 ; i. e. 6 to 5, 
 the ProduQ: will be 24 and 15*; 4 J 3 
 /. e. 8 to 5. You may fet 'em thus, 5 j~ y 
 and multiply 4 by 6, they make . 
 
 24, which fet at the bottom; 2 ^' 
 then multiply 3 by ?, they make 
 15, which likewiie let under, and you have 
 24 to 15; which is a Ration compounded 
 
 of 
 
74 
 
 Of Proportion. 
 
 of the other two, and equal to them both. 
 Reduce thefe Produ&s, 24 and 15, to their 
 lead Radical Numbers, which is, by divi- 
 ding as far as you can find a common Di- 
 vifor to them both (which is here done 
 by 3 ) and that brings them to the Ration 
 of 8 to 5. By this you fee, that a Third 
 minor j 6 to 5, added to a Fourth, 4 to 3, 
 makes a Sixth minor ^ 8 to 5* If more Ra- 
 tions are to be added, fet them all under 
 each other, and multiply the firft Antece- 
 dent by the fecond, and that Produd by 
 the third, and again that Product by the 
 Fourth, and fo on ; and fo in like manner 
 the Confequents. 
 
 This Operation depends upon the Fifth 
 Propofition of the Eighth Book of Euclid r ; 
 where he fhews, that the Ration of Plain 
 Numbers is compounded of their Sides. 
 See thefe Diagrams : 
 
 1 
 
 
 
 12 
 
 
 
 1 
 
 
 
 1 
 
 
 Now 
 
Of Proportion- 75 
 
 Now compound thefe Sides. Take 
 for the Antecedents, 4 the greater Side 
 of the greater Plane, and 3 the grea- 
 ter Side of the lefs Plane, and they mul- 
 tiply'd give 1 z : Then take the remaining 
 two Numbers 3 and 2, being the lefs Sides 
 of the Planes (for Confequents) and they 
 give 6. So the Sides of 4 and 3, and of 
 3 and 2, compounded (by multiplying the 
 Antecedent Terms by themfelves, and the 
 Confequents by themfelves) make 12 to 6 r 
 i. e. 2 to 1 ; which being apply'd, amounts 
 to this ; Ratio Sej "qui 'altera, 3 to 2, added 
 to Ratio Sefquitertia 4 to 3, makes Duple 
 Ration, 2 to 1. Therefore T^i agent e added 
 to Diateffaron, makes Diagafon. 
 
 Substraction of one Ration from 
 another greater is performed in like man- 
 ner by multiplying the Terms ; but this is 
 done not Laterally, as in Addition, but 
 Crojjwife ; by multiplying the Antecedent 
 of the former (/'. e. of the greater) by the 
 Confequent of the latter, which produced! 
 a new Antecedent ; and the Coniequent of 
 the former by the Antecedent of the latter, 
 which gives a new Confequent. And 
 therefore it is ufually done by an Oblique 
 DecuiTation of the Lines. For Example, 
 If you would take 6 to 5 out of 4 to 3, 
 
 YOU 
 
y6 Of Proportion. 
 
 you may fet them down as in the Mar- 
 gin : Then 4 multiply'd by 5^ 
 4- 3 makes 20, and 3 by 6 gives 18 : 
 
 XSo 20 to 18, /'. e. 10 to 9, is the 
 Remainder. That is 5 fubftra£t a 
 6 5 Third Minor out of a Fourth, 
 20. 1 8. and there will remain a Tone 
 1.0. 9. Minor? 
 
 Multiplication of Rations is the 
 fame with their Addition, only 'tis not wont 
 to be of divers Rations, but of the fame, 
 being taken twice, thrice, or oftener, as you 
 pleafe. And as before in Addition you added 
 divers Rations by multiplying them, fo here 
 in Multiplication you add the fame Ration 
 to it felf, after the fame manner, viz. by 
 iTiultiplying the Terms of the fame Ration 
 by themfelves ; /. e. the Antecedent by it 
 felf, and the Confequent by it felf, (which 
 in other Words is to multiply the fame by 2) 
 and will, in the Operation, be to fquare 
 the Ration firft propounded ( or give the 
 Second Ordinal Power, the Ration firft gi- 
 ven being the Firft Power or Side.) And 
 to this Product, if the Simple Ration fhall 
 again be added (after the fame manner as 
 before) the Aggregate will be triple of the 
 Ration firft given j or the Produft of that 
 Ration multiply'd by 3, viz. the Cube, or 
 % bird Ordinal Power. Its Biquadrate, or 
 
 Fourth, 
 
Of Proportion. J? 
 
 Fourth Power, proceeds from multiplying 
 it by 4, and fo fucceffively in order as far 
 as you pleafe you may advance the Powers. 
 For inftance, the Duple Ration, 2 to 1, be- 
 ing added to it felf, dupled, or muitiply'd 
 by 2, produceth 4 to*i, (rhe Ration Qua- 
 druple) : And if to this, the firft again be 
 added, which is equivalent to multiply- 
 ing that faid firft by 3) there will arife the 
 Ration Octuple, or 8 to 1. Whence the 
 Ration 2 to 1 being taken for a Root, its 
 Duple 4 to 1 will be the Square, its Triple 
 8 to 1 the Cube thereof, £s;c. as hath been 
 faid above. And, to ufe another Inftance, 
 To duple the Radon of 3 to 1, it muft be 
 thus fquarM ; 3 by 3 gives 9 : 2 by 2 gives 
 4 ; fq the Duple or Square of 3 to 2 is 9 
 to 4. Again, 9 by 3 is 27, and 4 by 2 is &, 
 fo the Cubic Ration of * to 2 is 27 to 8. 
 Again, to find the Fourth Power, or Biqua- 
 drate, (./". e. fquarM Square) 27 by 3 is 8 x, 
 8 by 2 is 16 ; fo 81 to 16 is the Ration of 
 3 to 2 quadrupled, as 'tis dupled by the 
 Square, tripled by the Cube, &c. To ap- 
 ply this Inftance to our prefent purpofe, 
 3 to 2 is the Ration of Diafente, or a Fifth 
 in Harmony ; 9 to 4 is the Ration of twice 
 Diapente, or a Ninth (viz. Diapafon with 
 Tone Major) ; 27 to 8 is the Ration of 
 thrice Diapente, or three Fifths, which is 
 %)ia$afon with Six Mqjflt {viz. 13^ Major) 
 
 The 
 
y% Of Proportion^ 
 
 The Ration of 81 to x<5 makes four Fifths, 
 i.e. Dif-diafafon, with two Tones Major ', 
 i. e. a Seventeenth Major ^ and a Comma of 
 8x to 8o» 
 
 To divide any Ration, you muft take 
 the contrary Way, and by extra&ing of 
 thefe Roots refpe&ively, Divifion by their 
 Indices will be performed. E. gn To di- 
 vide it by 2, is to take the Square Root of 
 it ; by 3, the Cubic Root ; by 4, the Biqua* 
 dratick, &c* Thus to divide 9 to 4 by 2, 
 the Square Root of 9 is 3, the Square Root 
 of 4 is 2 • then 3 to 2 is a Ration juft half 
 fo much as 9 to 4* 
 
 From hence it will be obvious to any 
 to make this Inference ; That Addition and 
 Multiplication of Rations are (in this cafe) 
 one and the fame thing. And thefe Hints 
 will be fufficient to fuch as bend their 
 Thoughts to thefe kinds of Speculations, 
 and no great Trefpafs upon thofe that do 
 not. 
 
 The Advantage of proceeding by the 
 Ordinal Powers, Square, Cube, &c. (as is 
 before mentionM) may be very ufeful where 
 there is Occaiion of large Progreffions ; as, 
 to find (for Example) how many Com- 
 ma's are contained la a Tone Major, or other 
 
 Interval 
 
Of Proportion! T9 
 
 Interval ; let it be, How many are in Dta- 
 fafon ? Which muft be done by multiply- 
 ing Comma's, l. e. adding them, tijl you 
 arrive at a Ration equal to OBave, (if that 
 be fought) viz. Duple : Or elfe by dividing 
 the Ration of Diafafon by that of a Com- 
 ma, and finding the Quotient; which may 
 be done by Logarithms. And herein I 
 meet with fome Differences of Calcula- 
 tions. 
 
 Mersennus findsj by his Calculation, 
 5 8^ Comma's, and fome what more, in an 
 Qffiave : But the late Nicholas Mercator, 
 aModeft Perfon, and a Learned and Judi- 
 cious Mathematician, in a Manufcript of 
 his, of which I have had a Sight, makes 
 this Remark upon it ; In folvendo hoc Pro- 
 hlemate aberrat Merfennus ; And he, work- 
 ing by ahe Logarithms, finds out but 55, 
 and a little more; and from thence has de- 
 duced an ingenious Invention of finding 
 and applying a leaft Common Meafure to 
 all Harmonic Intervals, not precifeiy per- 
 fed, but very near it. 
 
 Supposing a Comma to be T y part of 
 Diapafon; for better Accommodation ra- 
 ther than according to the true Partition 
 rr ? which T \ he calls an Artificial Comma, 
 Hot exa^ but differing from the true Na- 
 
 G tural 
 
go 
 
 Of Proportion. 
 
 tural Comma about T V part of a Comma, 
 and ttW of Diapafon (which is a Diffe- 
 rence imperceptible) then the Intervals 
 within Diapafon will be meafurM by Com- 
 ma's according to the following Table ; 
 which you may prove by adding two, or 
 three, or more of thefe Numbers of Com- 
 ma's, to fee how they agree to conftitute 
 thofe Intervals, which they ought to make • 
 and the like by fubftra&ing* 
 
 Intervals 
 
 Comma 
 Die/is 
 
 Semit. Minm 
 Semit* Medium 
 Semit* Majw 
 Semit* Maximm 
 Tone Minor 
 Tone Major 
 3 d Minor 
 3 d Major 
 
 r\ Intervals 
 
 2 
 3 
 4 
 5 
 6 
 8 
 
 9 
 
 14 
 
 17 
 
 4 th 
 Tritone 
 
 Semidiagente 
 
 rtfa 
 
 6 th Minor 
 6 th Major 
 
 7 th Minor 
 7 tlx Major 
 
 OBave 
 
 22 
 26 
 
 2 7 
 
 31 
 
 3^ 
 19 
 
 45 
 48 
 
 53 
 
 This I thought fit, on this Occafion, 
 to impart to the Reader, having Leave fo 
 
 to do fromMr* Mercator\FiicRd 7 to whom 
 he prefented the faid Manufcript. ■ 
 
 HereI may advertife the Reader, that 
 
 k is indifferent whether you compare the 
 
 V greater 
 
Of Proportion. 8 i 
 
 greater Term of an Harmonic Ration to 
 the'lefs, or the lefs to the greater ; /, e. 
 whether of them you place as Antecedent, 
 e. gr> 3 to 2, or z to 3 ; becaufe in Har- 
 monies the Proportions of Lengths of 
 Chords, and of their Vibrations, are reci- 
 procal or counter-chang'd : As the Length 
 is encreas'd, fo the Vibrations are in the 
 fame proportion decreased ; £j? e contra. 
 If therefore (as in. Diafente) the length of 
 the Unifon String be 3^ then the length 
 (cateris paribus-) of the String which in 
 afcent makes Diapente to that Unifon muft 
 be 2, or — : Thus the Ration of Diatente 
 
 is 2 to 3 in refpeft of the Length of it, 
 comparM to the Length of the Uniibn 
 String. 
 
 Again, the String 2 vibrates thrice in 
 the fame Time that the String 3 vibrates 
 twice ; and thus the Ration of Diapente ^ in 
 refped of Vibrations, is 3 to 2 : So that 
 where you find in Authors fometimes the 
 greater Number in the Rations fet before 
 and made the Antecedent, fometimes let 
 after and made the Confequent, you muft 
 underftand in the former, the Ration of 
 their Vibrations ; and in the latter, the 
 Ration of their Lengths ; which comes all 
 to one* 
 
 G % O e, 
 
8 1 Of Proportion. 
 
 6 r, yon may underftand the Unifon to 
 be compared to Diafiente above it, and 
 the Ration of Lengths is 3 to 2, of Vibra- 
 tions 2 to 3, or elfe Diapente to be com- 
 pared to the Unifon, and then the Ration 
 of Lengths is 2 to 3, of Vibrations 3 to 2. 
 This is true in fingle Rations, or if one 
 Ration be compared to another ; then the 
 two greater Terms muff: be rankM as An- 
 tecedents; or otherwife, the two leffer 
 Terms. 
 
 The Difference between Arithmetical 
 and Geometrical Proportion is to be well 
 heeded. An Arithmetical mean Proportion 
 is that which has equal Difference to the 
 Antecedent and Coniequent Terms of thofe 
 Numbers to which it is the Mediety, and 
 is found by adding the Terms, and taking 
 half the SuAi. Thus between 9 and i, 
 tvhich added together make 10, the Me- 
 diety is 5; being Equi.different fiom 9 and 
 from 1 ; which Difference is 4: As 5 ex- 
 ceeds 1 by 4; fo like wife 9 exceeds 5 by 4. 
 And thus in Arithmetical Progreffion 2, 4, 
 6, 8; where the Difference is only confi- 
 der'd, there is the fame Arithmetical Pro- 
 portion between 2 and 4, 4 and 6, 6 and 8; 
 and between 2 and 6, and 4 and 8 : But in 
 Geometrical Proportion, where is confider'd 
 not the Numerical Difference, but another 
 Habitude of the Terms, viz. how many 
 
 times, 
 
Of Proportion- 8$ 
 
 times, or how much of a time or times, 
 one of them cloth contain the other (as 
 hath been explained at large in the begin- 
 ning of this Chapter.) There the Mean 
 Proportional is not the fame with Arithme- 
 tical, but found another way j and equidif. 
 ferent Progreffioris make different Rations- 
 The Rations (taking them all in their lead 
 Terms) cxprefsM by lefs Numbers, being 
 greater than thofe oi greater Numbers, I 
 mean in Proportions fyf>er Particular^ &C 
 where the Antecedents are greater than the 
 Confequents, (as, on the contrary, where 
 the Antecedents are lefs than the Conic- 
 quents, the Ratio's of left Numbers are 
 lefs than the Rations of greater.) The Me- 
 diety of 9 to 1 is not now y, but 3 ; 3 ha- 
 ving the fame Ration to 1 as 9 has to 3, 
 (as 9 to 3, fo 3 to 1) viz. triple. And lb 
 in Progreflion Arithmetical, of Terms ha- 
 ving the lame Differences ; if confidcr'd 
 Geometrically, the Terms will all be com- 
 prehended by unequal Rations. The Dif- 
 ferences of x to 4, 4 to 6, 6 to 8, arc equal, 
 but the Rations are unequal ; 2 to 4 is lefs 
 than 4 to 6, and 4 to 6 lefs than 6 to 8. 
 As on the contrary, 4 to 2 is greater than 
 6 to 4, and 6 to 4 greater than 8 to 6: 
 For 4 to 2 is duple, 6 to 4 but Sefquialtera 
 (one and a half only, or ' ) and 8 to 6 is 
 no more than Sefquitertia^ (one and a third 
 
 G 3 part, 
 
84 Of Proportion. 
 
 part, or y) which fliews a confiderable in- 
 equality of their Rations. In like manner 
 6 to 2 is triple ; 8 to 4 is but duple, and 
 yet their Differences equal. Thus the 
 mean Rations comprehended in any grea- 
 ter Ration divided Arithmetically, i. e. by 
 equal Differences, are unequal to one ano- 
 ther, confider'd Geometrically. Thus 2,3, 
 a ) 5, 6, if you confider the Numbers, make 
 an Arithmetical Progreflion : But if you 
 confider the Rations of thofe Numbers, as 
 is done in Harmony, then thev are unequal, 
 every one being greater or lefs (according 
 as you proceed by Afcent or Defcent) than 
 the next to it. Thus, in this Progreflion, 
 (underilanding, together with the Ratio's, 
 the Intervals themfelves, as is before pre- 
 mifed) 2 to 3 Is the greateft, being Dia- 
 f ente ; 3 to 4 the next^ Diateffaron ; 4 to 5 
 ftill. lefs, viz. Bit one ; 5 to 6 the leaft, be- 
 ing SefqmtqMe. Or, i^i you defcend, 6 to 5 
 lead ; 5 to 4 next, i$c. Thefe are the mean 
 Rations comprehended in the Ration of 
 6 to 2 , by which Diafafon cum Diafeitte^ 
 era i2 th us divided into the aforefaid In- 
 tervals, and meafured by them, viz* as 
 is 6 to 2, {ziz. triple) fo'is the Aggregate 
 of all the mean Rations within that Num- 
 ber, 6 to 5, § to 4, 4 to 3 5 and 3 to 2 : Or 
 6 to J, 5 to 2 ; or 6 to 4, 4 to z ; or 6 to 3, 
 3 to 2. The Aggregates of thefe are equal 
 to 6 to 2, viz* triple. This 
 
Of Proportion^ 8j 
 
 Th i s is premifed in order to proceed 
 to what was intimated in the 'foregoing 
 Chapter. 
 
 Ta king notice firft of this Procedure, 
 peculiar to Harmonics, viz. to make Pro- 
 greflion or Divifion in Arithmetical Propor- 
 tion in refpefl: of the Numbers ; but to 
 confider the things numbered according to 
 their Rations Geometrical. And thus Har- 
 monic Proportion is faid to be compounded 
 of Arithmetical and Geometrical. 
 
 You may find them all in the Divifion 
 of the Syftem of Diapafon into *Dia$ente 
 and Diatejfaron, i. e. 5 th and 4 th , aicending 
 from the Unifon. 
 
 If by Diapente firft, then by 2, 3, 4, 
 Arithmetically. If firft by Diatejjaron, then 
 by 3, 4, 6, Harmonically. And thefe Ra- 
 tions confiderM Geometrically, in relation 
 to Sound, there is likewife found Geome- 
 trical Proportions between the Numbers 
 6, 3 to 4, 2 • and <5, 4 to 3, 2, 
 
 The Ancients therefore owning only 
 gth, jth, anc j ^th, f or fimple Conformant Inter- 
 vals, concluded them all within the Num- 
 bers of 12, 9, 8, 6, which contained them 
 
 G 4 allj 
 
S£ Of Proportion. 
 
 all : viz. i: to 5, Diatcfon ; n to 8, D/£- 
 f£nte\ 12 to >. Diatejfaroki 9 to 8, 7h/:cn 
 
 And whi:h iz-r:' ; i to exprefs the three kinds 
 cf Proportion, one Harmonica!, between 
 1: to S, and 8 to 6 ; Arithmetical, between 
 12 to <?, and 9 to 6; and Geometrical, be« 
 tween 12 to 9 and S to 6: and between 
 1: t: 8. and 9 to 6. It was laid therefore, 
 tha: Me -gttriu^s Lyre was lining with four 
 Chords, having thole Proportions, 6, 8,9,12. 
 
 I intimated, that I would here more 
 largely explain that ready and eaiie Way 
 of finding and measuring the mean Ra- 
 tions contained in any of thole Harmonic 
 Rations given, by transferring them out 
 of their Prime or Radical Numbers into 
 i ibers of the lame Ration. By 
 :: the Ration, but the Terms of 
 it; mil continuing the lame Ration) you 
 will .-.•'": we Mediety ; as, 2 ro 1 dupied 
 is 4 to 2; ahc >u have 3 the Mediety. 
 5v tripling _ v. have two Means : 
 2 to 1 tripled is 5 to 3, containing 3 Ra~ 
 is ; 6 to 5^ j> to 4, 4 to 3 ; and io ftill 
 mare, the more : ou multiply it. 
 
 X j w obferve, firfir, that any Ration 
 Multiplex or 5 1 par^, :r by ::anu 
 
 ferring it o..v:v- Radical Numbers made 
 
Of Proportion. 87 
 
 like Superpartie?if) contains fo many Super- 
 particular Rations, as there are Units in 
 the Difference between the Antecedent 
 and the Confequent. Thus in 8 to 4 
 (being z to 1 transferred by quadrupling) 
 the Difference is 4, and it contains 4 Su- 
 perparticular Rations, viz. 8 to 7, 7 to 5, 
 
 6 to y, and 5 to 4 ; where tho' the Pro* 
 greffion of Numbers is Arithmetical, yet 
 the Proportions of Excefs are Geometrical 
 and Unequal. The Super particular Ra- 
 tions exprefs'd by lefs Numbers being grea- 
 ter (as hath been laid) than thofe that con- 
 fift of greater Numbers; 5 to 4 is a greater 
 Ration than 6 to 5, and 6 to 5 greater 
 than 7 to 6, and 7 to 6 than 8 to 7- as a 
 Fourth part is greater than a Fifth, and a 
 Fifth greater than a Sixth, &c But in 
 this Inftance there are two Rations not ap- 
 pertaining to Harmonics, viz. 8 to 7, and 
 
 7 t0 *• 
 
 Secondly therefore, you may make un~ 
 equal Steps, and take none but Harmonic 
 Rations, by fele&ing greater and fewer 
 intermediate Rations, tho' fome of them 
 composed of feveral Superparticulars; pro- 
 vided you do not difcontinue the Rational 
 ProgretTion, but that you repeat ilill the 
 laft Confequent, making it the next Ante- 
 cedent ) as if you meafure the Ration of 
 
 8t04 
 
88 Of Proportion, 
 
 8 to 4, by 8 to 6 and 6 to 4, or by 8 to 5 
 and 5 to 4 ? or by 8 to 6, and 6 to 5, and 
 
 5 to 4 ; in thefe three ways the Rations 
 are all Harmonlcal, and are refpe&ively 
 contained in, and make up the Ration of 
 8 to 4. Thus you may meafure f and di- 
 vide, and compound mod harmonic Rations 
 without your Pen* 
 
 T o that End I would have my Reader 
 to be very perfeft in the Radical Numbers 
 which exprefs the Rations of the feven firft 
 (or uncompounded) Confonants, viz. Dia- 
 fafon, 2toi; 'Diafiente, 3 to 2 ; Diateffa-* 
 ron, 4 to 3 ; Ditone, 5 to 4 ; Trihemitone, 
 
 6 to 5 ; Hexachordon Majm, 5 to 3 ; Hexa- 
 chorion Minm, 8 to 5 ; and likewife of the 
 Degrees in Diatonick Harmony f viz. Tone 
 Major ', 9 to 8 ; Tone Minor \ 10 to 9 ; He* 
 mitone Major, 1 6 to 1 5 ; and the Differen- 
 ces of thofe Degrees ; Hemitone Great 'eft \ 
 27 to 25 ; and Hemitone Minor, 25 to 24 ; 
 £ omnia, or Schifm, 81 to 80 •, Diefis Enhar- 
 monic, 1 a, 8 to 125. 
 
 Of other Hemitones I fliall treat in the 
 
 Eighth Chapter. 
 
 Now if you would divide any of the 
 
 Confonants into two Parts, you may do it 
 by the Mean or Mediety of the two Radi- 
 cal 
 
Of Proportion^ 8pl 
 
 cal Numbers, if they have a Mean ; and 
 where they have not, (as when their Ra- 
 tio's are Suf erf articular) you need but 
 duple thofe Numbers, and you will have 
 a Mean (one or more.) Thus duple the 
 Numbers of the Ration of Diafafon^ 2 to i, 
 and you have 4 to 2 ; and then 3 is the 
 Mean by which it is divided into two un- 
 equal, but proper and harmonical parts^ 
 viz. 4 to 3, and 3 to 2. After this man- 
 ner Diafafon^ 4 to 2, comprehends 4 to 3, 
 and 3 to 2 : So "Diafente^ 6 to 4, is 6 to j, 
 and 5* to 4: "Ditone^ 10 to 8, is 10 to 9, 
 and 9 to 8 ; fo Sixth major ^ 5 to 3, is 5 104^ 
 and 4 to 3. 
 
 Tho' ? from what was now obferv'd, 
 you may divide any of the Confonants into 
 intermediate Parts, yet when you divide 
 -thefe three following, viz. Sixth minor , 
 € DiatejJaron, and Trihemitone, you will find 
 that thofe Parts into which they are divi- 
 ded, are not all fuch Intervals as are har- 
 monical. The Sixth minor^ whofe Ration 
 Is 8 to 5, contains in it three Means, viz. 
 8 to 7, 7 to 6, and 6 to 5 ; the laft where- 
 of only is one of the harmonick Intervals, 
 of which the Sixth minor confifts, viz. Tri- 
 hemitone ; and to make up the other Inter- 
 val, viz. Diatejfaron, you muft take the 
 other two, 8 to 7, and 7 to 6 j which be- 
 
$o Of Proportion, 
 
 ing added (or, which is the fame thing, 
 taking the Ratio of their two Extream 
 Terms, that being the Sum of all the in- 
 termediate ones added) you have 8 to <5, 
 or (in the leaft Terms) 4 to 3, Again, 
 Diatefjaro??, in Radical Numbers 4 to 3; 
 being (if thofe Numbers are dupled) 8 to 6, 
 gives for his Parts 8 to 7, and 7 to 6 ; which 
 Ratio s agree with no Intervals that are 
 Harmonick ; therefore you muft take the 
 Ration pf Diatefj 'arou in other Terms,which 
 may afford inch Harmonick Parts. And to 
 do this, you muft proceed farther than 
 dupling (or adding it once to it felf ) for 
 to this Duple you muft add the firft Radi- 
 cal Numbers once again (which in effect 
 IS the fame with tripling it at firft) viz. 
 4 and 3) to 8 and 6; and the Aggregate 
 will be a new, but equivalent, Ration of 
 Diate\\'aron\ viz. 12 to 9. And this gives 
 you three Means, ix ton, and 11 to 10; 
 both Unharmonical ; but which together 
 are, as was fhewM before, the fame with 
 1 2 to 10 ( or 6 to 5 ) Trjhemitone ; and 
 ic to 9 Tone minor ; and are the two Har- 
 mcnkal Intervals 01 \\'KichlJzateJaro?z con- 
 . and which divide it into the two 
 •eft equal Harmonick Parts, Lahly, 
 Trihemitone^ or Third minpr^ 6 to j\ or 
 (thofe Numbers being dupled) 12 to ic, 
 pYQS :2 to 11, a:cd 11 to 70, which are 
 
 Un- 
 
Of Proportion. c \ 
 
 Unharmonical Rari : n . ; . : I : J fser 
 
 the former manna r :: j •• - : :: '■-'■ 
 which divides it felf (as before) inn 1 1 : 
 1 6, Tone major; and 16 to 15, & 
 
 Thus, by a little Pra S : e . 1 fl 1 1 ~k(H 
 nick Intervals will be mofte: 
 by the lefTer Intervals compriz; . . 
 Now, for Exercife lake, ta \ 1 tfac Mc j fuic a 
 
 of a greater Ration : Suppoft thai : : 1 6 : ; 3 
 
 be given as an Harmonick S flem . T : %n 2 
 
 what it is, ar. : : ::..:/:- 
 
 firft find the grofc I uidther. ::.t .nore 
 
 minute. You will prefentlv . ;z.. . ; : r 
 
 to 8 (being a r : : : due R :.; 
 
 fajon\ and 8 to 4 : 
 
 Then 1 6 to 4 is £ 
 
 and the remaining _ I : 
 
 then 16 to 3 i i : . I - 
 
 ; i. e. an Y 
 and 4:: 
 
 5 
 
<?2 Of Proportion. 
 
 But, to find all the Harmonick Intervals 
 within that Ration ( for we now confider 
 Rations as relating to Harmony) take this 
 Scheme. 
 
 16 to 3 contains. 
 In Radicals. 
 
 16 to 15, 
 
 
 Hemitone* 
 
 
 15 tO 12, 
 
 5 to 4, 
 
 Ditone* 
 
 
 12 tO IO, 
 
 6 to jj 
 
 Trihemitone*. 
 
 ' 
 
 10 to 9, 
 
 
 Tone Minor. 
 
 
 9 to 8, 
 
 
 Tone Major. 
 
 
 8 to 6, 
 
 4 to 3, 
 
 Diatejjaron. 
 
 
 6 to 5, 
 
 ■ 
 
 Trihemitone* 
 
 
 5 to 4> 
 
 
 Ditone* 
 
 
 4 t0 3, 
 
 
 Diatejjaron. 
 
 
 \tot. 16 to 3 
 
 Difdiafafon cum "Diatejjaron 
 
 16 to IP, 
 10 to 6, 
 
 6 to 4, 
 
 4 tQ h 
 Tot. 1 6 to 3 
 
 Or thus, 
 
 j# Radicals. 
 8 to 5, 
 
 5 to 3, 
 3 to 2, 
 
 ) 
 
 6 th ftff*o% 
 
 6 th Major. 
 
 4 th 
 Eighteenth. 
 
 All 
 
Of Proportion. 95 
 
 All thefe Intervals thus put together 
 are comprehended in, and make up, the 
 Ration of 1 6 to 3,, being taken in a conjunct 
 Series of Rations. 
 
 But otherw?fe, within this compafs of 
 Numbers are contain'd many more Expref- 
 fions of Harmonick Ration. Ex. gr. 
 
 Radicals, 
 
 Radicals. 
 
 16 to 
 
 15, 
 
 
 
 
 12 to <5, 
 
 2 to 1. 
 
 16 to 
 
 ia , 
 
 4 
 
 to 
 
 3- 
 
 12 to 4, 
 
 3 to I. 
 
 16 to 
 
 10, 
 
 8 
 
 to 
 
 5. 
 
 12 to 3, 
 
 4 to 1. 
 
 16 to 
 
 *, 
 
 a 
 
 to 
 
 I. 
 
 10 to 9, 
 
 ,i 
 
 16 to 
 
 *> 
 
 8 
 
 to 
 
 3- 
 
 10 to 8, 
 
 5 to 4. 
 
 id tO 
 
 4> 
 
 4 
 
 to 
 
 1. 
 
 10 to 6y 
 
 5 to 3. 
 
 16 to 
 
 3- 
 
 
 
 
 10 to 5, 
 
 z to I. 
 
 15 to 
 
 12, 
 
 S 
 
 to 
 
 4- 
 
 9 to 8, 
 
 
 15 to 
 
 10, 
 
 3 
 
 to 
 
 2. 
 
 9 to 6, 
 
 3 to 2. 
 
 15 to 
 
 i» 
 
 3 
 
 to 
 
 1. 
 
 9 to 3, 
 
 3 to 1. 
 
 15 to 
 
 3, 
 
 5 
 
 to 
 
 1. 
 
 8 to 6, 
 
 4 to 3. 
 
 14 to 
 
 7> 
 
 2 
 
 to 
 
 1. 
 
 8 to 5, 
 
 
 12 tO 
 
 10, 
 
 6 
 
 to 
 
 5- 
 
 8 to 4, 
 
 2 to I. 
 
 IX to 
 
 I 12 to 
 
 % 
 
 4 
 
 to 
 
 3- 
 
 6 to 5, 
 
 gjfr. 
 
 8, 
 
 3 
 
 to 
 
 2. 
 
 ^zV. Pag. 
 
 67. 
 
 And now I fuppofe the Reader better 
 prepar'd to proceed in the remainder of this 
 Difcourfe, where we fliall treat of Z>/7 lords. 
 
 CHAP. 
 
94 Of Difcords and Degrees. 
 
 C H A P. VI. 
 
 Of Difcords and Degrees. 
 
 A LL Habitudes of one Chord to ano- 
 -"* ther, that are not Concords r ( fuch as 
 are before defcribM ) are Difcords ; which 
 are or may be innumerable, as are the mi- 
 nute Tenfions by which one Chord may 
 be made to vary from it felf, or from ano- 
 ther* But here we are to confider only 
 fuch Difcords as are ufeful ( and in truth 
 neceflary} to Harmony \ or at leaft relating 
 to it, as are the Differences found between 
 Marmonick Intervals. 
 
 And thefe apt and ufeful Difcords are 
 either fimple uncompounded Intervals, fuch 
 as immediately follow one another, amen- 
 ding or defcending in the Scale of Mufick ; 
 as, tit, Re 9 Mi) Fa 9 Sol^ La, Fa^ Sol, and are 
 calFd Degrees : Or elfe greater Spaces or 
 Intervals compounded or Degrees inclu- 
 ding or skipping over fome of them, as all 
 the Comords do, Ut Mi^Ut Fa^ Ut Sol, &c* 
 And fuch are the "Difcords of which we 
 
 now 
 
Of Dif cords and Degrees. $ *} 
 
 now treat, as principally the Tritone, Falf e 
 Fifth, and the two Sevenths, Major and 
 Minor, if they be not rather among the 
 Degrees, 65V. For more Perfpicuity, I 
 fliall treat of them feverally, viz. of De- 
 gree s i of c Difcords i and of Differences. 
 
 And firft of "Degrees. 
 
 Degrees are uncompoondsd Inter- 
 vals, which are found upon eight Chords, 
 and in feven Spaces, by which an imme- 
 diate Afcent or Defcent is made from the 
 Unifon to the Qffave or Diafafon^ and 
 by the fame Progreflion to as many Otfaves 
 as there may be Occafion. Thefe are dif- 
 ferent, according to the different Kinds of 
 Mufic, viz. Enharmonic, Chromatic, and 
 Diatonic^ and the feveral Colours of the 
 two latter: (all which I fliall more con- 
 veniently explain by and by ) ; but of 
 thefe now mentioned, the Diatonic is tliQ 
 mod proper and natural Way : The other 
 two, if for Curiofities fake we confider 
 them only by running the Notes of m 
 Offave up or down in thefe Scales, feem 
 rather a Force upon Nature ; yet here- 
 in probably might lie the Excellency 
 of the ancient Greeks : But we now 
 life only the Diatonic Kind , intermix- 
 ing here and there forac^ of the Chro- 
 
 H matic* 
 
p 6 Of Dif cords and Begreesl 
 
 mafic, (and more rarely fome of the En- 
 harmonic : ) And our Excellency feenis to 
 lie in moft artificial Conipofing, and join- 
 ing feveral Parts in Symphony or Confort ; 
 which they cannot be fuppos'd to have ef- 
 fe&ed, at lead: in lb many Parts as we or- 
 dinarily make, hecaufe ( as is generally 
 affiroi'd of them ) they owftM no Concords 
 befides Eighth, Fifth, and Fourth, and the 
 Compounds of thefe. 
 
 F. JQrcher (cited alfo by Gajjendm with- 
 out any Mark of Diffent ) is of Opinion, 
 that the ancient Greeks never ufed Con- 
 fort Mufic, i. e. of different Parts at once, 
 but only Solitary, for one fingle Voice or 
 Inftrument ; and, that Guido Aretinm firft 
 indented and brought in Mufic of Sym- 
 phony or Confort, both for the one and 
 the other. They apply'd Inftruments to 
 Voice, but how they were managed, he 
 muft be wifer than I that can tell. 
 
 This Way of theirs feems to be more 
 proper (by the elaborate Curiofity and 
 Nicety of Contrivance of Degrees, and by 
 Meafures rather than by harmonious Con- 
 fonancy, and by long-ftudied Performance) 
 to make great Impreffions upon the Fan- 
 cy, and operate accordingly, as fome Hifto 
 lies relate : Ours more fedately affeds the 
 
Of Difcords and agrees. <?/ 
 
 Underftanding and Judgment, from the 
 judicious Contrivance and happy Compo- 
 sition of Melodious Confort. The One 
 quietly, but powerfully, affe&s the Intel- 
 led by true Harmony ; the Other, chiefly 
 by the Rphmm, violently attacks and h.ur* 
 ries the Imagination, In fine, upon the. 
 natural Grounds of Harmony (of which I 
 have hitherto been treating) is founded 
 the Diatonic Mufic ; but not fo, or not fo 
 regularly, the Chromatic or Enharmonic 
 Kinds. " Take this following View of 
 them. 
 
 The Ancients afcended from the Uni- 
 [on to an OHave by two Syftemes of Te~ 
 trachords or Fourths. Thefe were either 
 Conjunct, when they began the Second 
 Tetrachord at the Fourth Chord, viz. with 
 the laft Note of the firft Tetrachord, and 
 which being fo joined, conilituted but at 
 Seventh ; and therefore they affurned a 
 Tone beneath the Unifon (which they there- 
 fore calPd Profiambanomenos) to make a full 
 Eighth. 
 
 O r. elfe the two Tetrachords were dif* 
 jun£fc, the Second taking its beginning at 
 the Fifth Chord, there being always a Tone 
 Major between the Fourth and Fifth Chords, 
 So the Degrees were immediately applyM 
 
 H 2 to 
 
9 8 Of 'Dif cords and degrees. 
 
 to the Fourths, and by them to the Q&ave ; 
 and were different according to the diffe- 
 rent Kinds of Mufic. In the common Dia- 
 tonic Genus the Degrees were Tone and 
 Semitone ; Intervals more Equal and Eafy 9 
 and Natural. In the common Chromatic, 
 where the Degrees were Hemitones and 
 Trihemitones, the Difference of fome of the 
 Intervals was greater: But the greateft 
 Difference, and confequently difficulty, was 
 in the Enharmonic Kind, ufing only Diefis, 
 or quarter of a Tone^ and Ditone, as the 
 Degrees whereby they made the Tetra- 
 chord. 
 
 And to conftitute thefe Degrees, fome 
 of them, viz. th% Followers of Arifioxenus, 
 divided a Tone M^jor into Twelve equal 
 Parts, i. e. fuppofed it fo divided \ Six of 
 which being the Hemitone, (viz. half of it) 
 made a Degree of Chromatic Toni&um \ and 
 Three of them, or a quarter, calPd Diejis, 
 a Degree Enharmonic. The Chromatic 
 Fourth rofe thus, viz. from the firft Chord 
 to the fecond was a Hemitone ; from the 
 fecond to the third, a Hemitone \ from the 
 Third to the Fourth, a Trihemitone ; or as 
 much as would make up a juft Fourth. 
 And this laft Space ( in this cafe ) was ac- 
 counted as well as either of the other, but 
 one Degree or undivided Interval. And 
 
 they 
 
Of Difcords and Degrees* 99 
 
 they call'd them Sfifs Intervals [ vvwi ] 
 
 when two of thofe other Degrees put to- 
 gether, made not fo great an Interval as 
 one of thefe ; as, in the Enharmonic Tetra- 
 chord, two Diefes were lefs than the re- 
 maining r Dito?ie, and in the common Chro- 
 matic, two Hemitone Degrees were lefs. 
 than the remaining Trihemitone Degree. 
 
 Then for the Enharmonic Fourth, the 
 firft Degree was a Die/is, or quarter of a 
 Tone% the fecond alfo Three of thofe 
 Twelve Parts, viz. a Diejis ; the third a 
 T>itone, fuch as made up a juft Fourth. 
 And this Ditone ( tho' fo large a Degree ) 
 being confider'd as thus placed (in the 
 Enharmonic Tetrachord) was likewife to 
 them but as one uncompounded or entire 
 Interval. 
 
 These were the Degrees Chromatic 
 and Enharmonic : Tho' they alfo might be 
 placed otherwife, /. e. the greater Degree 
 in thefe may change its place, as the Hemi- 
 tone (the lefs Degree) doth in the 'Diatonic 
 Genus ; and from this Change chiefly arofe 
 the feveral Moods, 'Dorian, hydidn y &c. 
 From all which, their Mufic no doubt 
 (tho' it be hard to us to conceive) muft 
 afford extraordinary Delight and Pleafure, 
 if it did bear but a reafonable Proportion 
 
 H 3 to 
 
I go Of Dif cords and Degrees. 
 
 to their infinite Curiofity and Labour. And 
 as we may fuppofe it to have differM very 
 much from that which now is, arid for fe« 
 veral Ages hath been ufed ; lb confequently 
 we may look upon it as in a manner loft 
 jtp us. 
 
 In profecution of my Defign, I am on- 
 ly, or chiefly, to infift on the other Kind 
 of Degrees, which are moft proper to the 
 .Natural Explanation of Harmony, viz. the 
 Degrees Diatonic ; which are fo call'd, not 
 becaufe they are all Tones \hv& becaufe moft 
 of 'em, as many as can be, are fuch ; vim. 
 in every Diapafon five Tones and two He- 
 miton.es. Upon thefe, I fay, I am to infift, 
 as being, of thofe before mentioned, the 
 moil; Natural and Rational 
 
 Digrejjion* 
 
 But before we proceed, it may perhaps 
 be a Satisfaction to the Reader, after what 
 lias been laid, to have a little better Pro- 
 Iped of the ancient Greek Mufic, by fome 
 general Account ; not of their whole Do- 
 ctrine, but of that which relates to our 
 prefent Subject, viz. their Degrees, and 
 Scales of. Harmony, and Notes. 
 
 First 
 
Of Dif cords and Degrees. i o i 
 
 First then, take out of Euclid the 
 Degrees according to the three Genera \ 
 which were, Enharmonic^ Chromatic ^ and 
 Diatonic ; which Kinds have fix Colours 
 (as they calPd them). Euclid^ Introd. Harm. 
 pag. io. 
 
 The Enharmonic Kind had but one 
 Colour, which made up its Tetrachord by 
 thefe Intervals ; a Die/is (or quarter of a 
 Tone) then fuch another Diejis 7 and alfo a 
 Ditone incompofit. 
 
 The Chromatic had three Colours, by 
 which it was divided into MoI/e 9 Sefcufhm y 
 and Toniaum. 
 
 x ft > Molk y in which the Tetrachord rofe 
 by a Triental Diejis ( four of thofe twelve 
 Parts mentioned before) or third part of a 
 Tone ; and another fuch T)iejis ; and an in- 
 compofit Interval, containing a Tone and 
 half, and third part of a Tone : And it was 
 calPd Molle^ becaufe it hath the leaf}, and 
 confequently moil enervated Sfifs Intervals 
 within the Chromatic Genm* 
 
 2 d 5 Sefcuflum^ by a Diejis which \§Sef- 
 quialiera to the Enharmonic Diejis y and 
 another fuch Die/is, and an Incompofit In- 
 terval of feven Diefes Quadrantal 3 viz. each 
 being three Duodecimals of a 71m*. 
 
 H 4 i^To- 
 
102 
 
 Of Dif cords and Degrees. 
 
 3 d > Tontdum 3 by a Hemitone^ and Hemi-* 
 tone and Trihemitone ; and is calPd Toniaum 
 becaufe the two Sfifs Intervals make ^Tone* 
 And this is the ordinary Chromatic. 
 
 The Diatoniek had two Colours; it 
 
 was M?//<? and Syntonum* 
 
 i ft > MoUe^hj 2. Hemitone^ andanlncom- 
 pofit Interval of three Quadrantal Diefes 
 and an Interval of five fuch Diefes. 
 
 2 d > Syntonum^ by a Hemitone and a 7##^ 
 and a 7W. And this is the common Dia* 
 
 ionic. 
 
 To underftand this better, I inuft re- 
 affume fomewhat which I mentioned, but 
 not fully enough before. A Tone is fup- 
 pos\i to be divided into twelve leaft parts, 
 and therefore a Hemitone contains fix of 
 thofe Duodecimal (or twelfth) parts of a 
 Tone \ a Dihfis Trientalu fy DiefisQuadran- 
 talis 3, the whole DiateJJaron 3 o. And the 
 "Diateffaron in each of the three Kinds was 
 made and performed upon four Chords, 
 having three mean Intervals of Degrees, 
 according to the following Numbers and 
 Proportions of thofe thirty Duodecimal 
 farts, 
 
 Enhar* 
 
Of Vif cords and "Degrees. i o 5 
 
 Enharmonic, by 3, and 3, and 24 
 
 * MoDe, by 4, and 4, and 22. 
 
 \Hemiolion y "7 
 Chromatic, J or > by 44, and 44, and 2 i* 
 jSefcupluni) 3 
 
 v Tonimm, by 6, and 6, and 18, 
 C M>&, by (5, and s>, and 1 5 . 
 Diatonic, ^ 
 
 CSymonum, by <5> and 12, and 12; 
 
 T o each of thefe Kinds, and the Moods 
 of them, they fitted a perfeft Syftem or 
 Scale of Degrees to T^ifdia^afon ; as in the 
 following Example taken out of Nichoma- 
 chm j to which I have prefixed' our modern 
 Letters. 
 
 A 
 G 
 
 F 
 E 
 
 D 
 
 C 
 
 B 
 
 £. Nichomacho, pag. i% % 
 
 Nete Hy^erlolaon. 
 Paranete Hyper-1 ^ 7 ^ -. 
 holaon. ^Enbam.Chro. ptat. 
 
 Trite Hyperbolaon. Enh. Chro. Diat* 
 Nete T)iezeugnie- 
 
 non. 
 
 Paranete Diezeug-7 _ 7 ^ 7 _ . 
 www. 5 £ ^' C ^- ^^ 
 
 Taramefe* 
 
 V>\Nete 
 
i ©4 Of Difcords and Degrees* 
 
 D | Nete Synemmenon. 
 
 C Taranete Synem-1 _ . „ t n 
 
 menon* ^ Enh. Chro. Dtat. 
 
 B Trite Synemmenon* Enh. Chro. Diat. 
 
 A Mefe. " 
 
 6 J Lichanos Me [on. Enh. Chro. Diat. 
 
 F • Parypate Me/on. Enh. Chro. Diat* 
 
 E Hyp ate Mefon. 
 
 D Lychanos Hvpaton. Enh. Chro. Diat. 
 
 C Parypate Hypaton. Enh. Chro. Diat. 
 
 B I Hypate Hypaton. 
 
 A j Trojlambanomenos. 
 
 In this Scale of Difdiapafon you fee the 
 Mefe is an OHave below the Nete Hyper- 
 holaon, and an OHave above the ProJIam- 
 hanomenos : And the Lichanos, Parypate, 
 Parenete, and Trite, are changeable •, as 
 upon our Inftruments are the Seconds, and 
 Thirds, and Sixths, and Sevenths : The 
 Proflambanomenos, Hypate, Mefe, Paramefe, 
 and Nete, are immutable ; as are the Uni» 
 £on, Fourths, Fifths, and 0£taves. 
 
 Now from the feveral Changes of thefe 
 Mutable Chords chiefly arife the feveral 
 Moods (fome call'd them Tones) of Mu- 
 fie, of which Euclid fets down Thirteen ; 
 to which were joyn'd two more, viz. By- 
 
 peraoJian 
 
Of Bif cords and Degrees. 
 
 i© 
 
 feraoUan and Hyferlydian j and afterwards 
 Six more were added. 
 
 I ftiall give you, for a Tafte, Euclid's 
 Thirteen Moods. 
 
 Euclid* p. 19* 
 
 Hyfertnixolydim, five Hjferphrygim* 
 
 Mixolydim acutior^ five Hyperiaftius. 
 
 Mixolydius gravior, five Hjferdorim. 
 
 Lydim acutior. 
 
 Lydias gravior^ five Moliw. 
 
 Phrygias acutior. 
 
 Thrygius gravior, five lafiius* 
 
 Dorius. 
 
 Hyfolydhis acutior. 
 
 Hyfolydius gravior^ five HygoaoTius. 
 
 Hyfofhrygius acutior. 
 
 Hyfofhrygius gravior^ five Hy^oiaftius. 
 
 Hjfodorius. 
 
 O f thefe the moft grave, or lowed:, was 
 the Hyperdorian Mood, the Vroflambano- 
 menos whereof was fix'd upon the lowed 
 clear and firm Note, of the Voice or In- 
 ftrument that was fupposM to be of the 
 deepeft fettled Pitch in Nature, and adap- 
 ted freely to exprefs it : And then ail-along 
 from Grave to Acute the Moods took their 
 Afcent by Hemitones^ each Mood being a 
 
 Hemu 
 
106 Of Dif cords and "Degrees] 
 
 Hemitone higher or more acute than the 
 next under it. So that the Vrojlambano- 
 menos of the Hygertnixolyiian Mood was 
 juft an Eighth higher than that of the Hy~ 
 fodorian y and the reft accordingly. 
 
 Now each particular Chord in the pre- 
 ceeding Scale had two Signs or Notes 
 [ Mima, ] by which it was chara&eriz'd or 
 deTcribM in every one of thefe Moods re- 
 fpe&ively ; and alfo for all the Moods in 
 the feveral Kinds of Mufic ; Enharmonic^ 
 Chromatic ^ and "Diatonic ; of which two 
 Notes, the upper was for reading [ k^ ] 
 the lower for percuffion [ xpftw ] one for 
 the Voice, the other for the Hand. Con- 
 fider then how many Notes they ufed ; 
 18 Chords feverally for 13 Moods (or ra- 
 ther 15, taking in the Hy^eraoUan and 
 Hyperlydiaft, which are all defcribM by 
 Alyfius) and thefe fuited to the three 
 Kinds of Mufic. So many Notes, and fo 
 appropriated, had the Scholar then to learn 
 and conn who ftudied Mufic. Of thefe I 
 will give you in part a View out of A- 
 lyfim* 
 
 Notes 
 
Of Dif cords and Degrees! 1 07; 
 
 Notes of the Lydian Mood in the 
 'Diatonic Genus. 
 
 7.1.E.0.C.P.M.I.0 
 
 P.r.L.F,c.u.ri.<.v. 
 
 123456789 
 
 r/ixz.E.u.-e-. x.M.i. 
 
 N.Z.C.lI.Z.^. r|'<f, 
 
 IO II 12 13 I4 15 16 17 l8 
 
 4 Hyp at on*!)! at onos* Phi^^ndDigamma. 
 
 5 Hyf ate Mefon, Sigma^ and Sigma. 
 
 6 Tarypate Mej "on.— Rhofe Sigma inverted 
 
 7 Mefon Diatonos. — ilfy, and Pi drawn out. 
 
 8 Mrfe. — —■- Iota,ZEL&jLamda jacene 
 
I o 8 Of DiJ cords and Degrees. 
 
 f \i.„.o„<. «1^«**. STheta* and Lamhda 
 9 Trite Synemmenon. ^ lifted. 
 
 io Synemmenon Diato- Gamma, andiVy. 
 
 C Q fquared, lying fin 
 
 I I Afcfc* Synemmenon.< pine upwards, 
 
 £ and 2£e ta. 
 
 12 Taramefe. « — - J^e^ and P/ jacent. 
 
 13 Wte **«*»»».{ E ft^ and K 
 
 14 Diezeugmenon Dia-$ a fquared, lupine^ 
 
 tones. X and ^fta. 
 
 S^Phi jacent, and a 
 
 1 5 Nete r Dzeze%gmenon. m S carelefs Eta (w) 
 
 £ drawn out* 
 ^ T looking down, 
 
 16 Trite HyferloUon. *s &^//Z?^lefthalf, 
 
 C looking upward. 
 n . 7 ; „ r^,. C Mr, and F/ length- 
 n.mf>^*&+\ "ned, withal? A- 
 ^ w - £ cute above. 
 
 Xlota-t and Lambda 
 £8 jVdte Hy^erlol&on. < jacent, with an 
 
 C Acute above. 
 
 The Numeral Figures I have added under the Signs (or 
 Marks) only for Reference to the Names of the Notes 
 %nified by them 3 to fave defcribing them twice. 
 
 Notes 
 
Of Dif cords and Degrees. i o <? 
 
 Notes of the JEolian Mood in the 
 Diatonic Genus. 
 
 rf.X.I.X.T.C.O.K.I. 
 
 1134 5 6 7 8 9 
 
 Z ♦ A.H.Z .AX &-.OK, 
 
 IO II 12 13 14 15 16 17 l8 
 
 r£/ta (H) imperfe£fc 
 i Proflamlanomenos.< averted, and Equa- 
 
 C drate averted. 
 r Delta inverted, and 
 1 Hypate Hypaton^c'.< Tau jacent,aver- 
 
 L ted, &&. 
 
 eAri/iides (Pag. 91.) enumerates and de- 
 fcribes all the Variations of every Letter in* 
 the Greek Alphabet ; by which the Signs 
 or Notes above mentioned, and thofe of 
 the other Moods, were contrived out of 
 
 them* 
 
iio OfDifcords and Degrees. 
 
 them. They are in all 91 ; including the 
 Proper Letters : I fhall not defcribe, but 
 only number them. 
 
 Out of 
 
 A 
 
 were made 7 
 
 N 
 
 were made 2 
 
 B 
 
 z 
 
 S 
 
 2, 
 
 r 
 
 7 
 
 O 
 
 2 
 
 A 
 
 4 
 
 n 
 
 1 
 
 E 
 
 5 
 
 p 
 
 z 
 
 Z 
 
 z 
 
 s 
 
 6 
 
 H 
 
 5 
 
 T 
 
 4 
 
 e 
 
 z 
 
 X 
 
 3 
 
 1 
 
 4 
 
 $ 
 
 4 
 
 K 
 
 3 
 
 X 
 
 4 
 
 A 
 
 5 
 
 llf* 
 
 2r 
 
 M 
 
 5 |& 
 
 4 
 
 
 - 
 
 «— 
 
 
 49 1 
 
 4 1 
 
 
 9i 
 
 
 I fhall only add a Word or two con- 
 cerning their ancient Ufe of the Words 
 Diaflem and Syslem* Diafiem fignilies an 
 Interval or Space; System, a Conjunction 
 or Compofition of Intervals. So that, ge- 
 nerally fpeaking, an Qttave, or any other 
 SyBem, mi^ht be truly calPd a Diafiem^ 
 and very frequently ufed to be fo calPd* 
 where there was no occafion of Diflin&ion. 
 Tho' a Tone^ or Hemitone, could not be 
 calPd a Sy&em ; for when they fpoke fidd- 
 ly, by a Diafiem they underftood only an 
 
 Incom- 
 
Of Difcords and Degrees. 1 1 1 
 
 Incompofit Degree, whether DieJis,Hemi~ 
 tone, Tone, Sefguitone, or Ditone ; for the 
 two laft were fometimes but Degrees, one 
 Enharmonic, the other Chromatic. By 
 SyHem they meant, a Comprehensive Inter- 
 val, compounded of Degrees, or of lefs 
 SyHems, or of both. Thus a Tone was a 
 Diaftem, and Diatejjaron was a System^ 
 compounded of Degrees, or of a 3 d and a 
 Degree. Diapafon was a Syflem, com- 
 pounded of the leffer SyHems, 4 th , and 5 th ; 
 or 3 d and 6 th - or of a Scale of Degrees: 
 And the Scale of Notes which they ufed, 
 was their Greateft, or Perfeft SyHem. Thus 
 with them, a 3 d Major, and a 3 d Minor, in 
 the Diatonic Genus ? were ( properly fpeak- 
 ing ) Syflems ; the former being compoun- 
 ded of two Tones, and the latter of three 
 He?nitones, or a Tone and Hemitone : But 
 In the Enharmonic Kind, a Ditone was not 
 a Syflem, but an Incompofit Degree ; which 
 added to two Diefes, made up the Diatef- 
 faron : And in the Chromatic Kind, a Tr/Z^- 
 mitone was the like ; being only an Incom* 
 pofit Diaftem, and not a Syjienu 
 
 But to return from this Digreflioiv 
 (which is not fo much to my Purpofe, as 
 to gratifie the Reader's Curio(ity) and con- 
 tinue our DHcourfe according to Nature's 
 Guidance, upon the Diato?iic Degrees* It 
 
 \ was 
 
iii Of Bij cords and Degrees] 
 
 was faid, that there are fiveTWi and two 
 Hemitones in every Diapafon* Now the 
 reafon why there muft be two Hemitones, 
 is, becaufe an 8 th is naturally compofed of f 
 and divided into 5 th and 4 th ; and a Fifth is 
 three Tones and a half; a Fourth twoTones 
 and a half ; and the Afcent, by Degrees, 
 muft pafs by Fourth and Fifth ; which are 
 always unchangeable, and keep the fame 
 diftance from Unifon ; and a juft Tone Ma- 
 jor of 9 to 8 always between them. There- 
 fore the Diapafon has not an Afcent of fix 
 Tones, but of five Tones and two Hemitones^ 
 one Hemitone being placed in each Fourth 
 Disjunct ; in either of which Fourths, the 
 Degrees may be alterM by placing thei&- 
 mitone in the Firft, or Second, or Third 
 Degree of either. As, MI y FA, Sol, La. 
 La, MI, FA, Sol Sol, La, MI, FA. If 
 this be done in the former Tetrachord, then 
 is chang'd the Second, or Third Chord ; if 
 in the other Disjun£l Tetrachord, then the 
 Sixth, or Seventh is changM : The Fourth 
 and Fifth being ftable and immutable, by 
 them we naturally divide the Diapafon : 
 The Second, Third, Sixth, and Seventh are 
 alterable, as Minor, and Major, according to 
 the Place of "the Hemitone. 
 
 These Tones and Hemitones thus pla- 
 ced, are the Degrees or Notes by which 
 
 m 
 
Of Dif cords and Degrees* i i j 
 
 an Afcent or Defcent is made from the Uni- 
 [on to the OHave, or thro 7 any other Syftem^ 
 giving all the Concords their juft Meafures 
 or Rations ; and without which, we could 
 neither Meafure, nor Divide, nor well 
 Pra£tife, to learn the greater Intervals or 
 Syfiems. 
 
 As we Naturally by the Judgment of 
 our Ear, own, and reft in the Octave, as the 
 chief Confonant ; fo we do as Naturally 
 (without Study or Skill in Mufic) meafure 
 the Syftem of a Diafafon by thefe Diatonic 
 Degrees ; and can do no other wife. We 
 cannot with our Voice, without infinite 
 Study, frame to run up or down eight 
 Notes, without fuch a Mixture of Tones 
 and Hemitones ; and we do it eafieft when 
 we avoid Tritones. We fee it in a Ring of 
 Bells, of which the compjeateft and mod 
 pleafant is a Peal of Six ; which are 
 beft foited to have the Hemitone in the 
 midft ; i. e. between the Third and Fourth, 
 both in Afcending and Defcending; and 
 then there will be no Tritone : Ex. gr. La, 
 Sol, Fa, Mi, Re, Ut. Where all Afcents 
 and Defcents are made by juft Diatejfa- 
 tens. Ut, Re, Mi, Fa. Re, Mi, Fa, Sol. 
 Mi, Fa, Sol, La. Or downwards ; La, 
 Sol, Fa, Mi. Sol, Fa, Mi, Re. Fa, Mi, 
 Re, Ut. 
 
 I \ And 
 
\ 1 4 Of Dif cords and Degrees. 
 
 And this is fo Natural that it pleafetlt 
 all Ears ; and if they fliouid be difpofed in 
 any other Order, it would be fo difagree- 
 able, that any Ruftick or unlearnM Ear, of 
 fuch as know not what a Tritone is, would 
 be able to judge, and find a Diflike of it* 
 But then, how much more, if the Ring 
 of Bells were difposM by Chromatic or En- 
 harmonic Degrees,conftituting the Diateffa- 
 tons ? how abfiird and uncouth it would 
 appear ! The pra&ife of thofe kinds there- 
 fore, and in fuch a manner, feems to be 
 ( as has been faid ) a Violence upon Na- 
 ture, and only for Curiofity. 
 
 I n Diatonic Mufic there Is but one fort 
 of Hemitone amongft the Degrees, calPd 
 Remit one Major, whofe Ration is 16 to 15 j 
 being the Difference, and making a Degree 
 between a Tone Major and Third Minor ; 
 or between a Third Major and a Fourth. 
 
 Th ere are two forts of Tones ; viz. 
 Major, and Minor. Tone Major ( 9 to 8 ) 
 being the Difference between a Fourth and 
 Fifth : And Tone Minor ( 10 to 9) which 
 is the Difference between Third Minor 
 and Fourth. But both the Tones arifing 
 (as hath been faid ) out of the Partition of 
 a Third Major, in like manner as 5 th and 4 th 
 do by the Partition of an 8 th : I may (with 
 
 fub- 
 
Of Vifcords and Degrees* 1 1 5 
 
 fubmiflioii ) maike the following Remark J 
 wherein, if I be too bold, or be miftaken 3 1 
 iliall beg the Reader's Pardon. 
 
 The ancient Greek Mailers found out 
 the Tone by the Difference of a Fourth and 
 Fifth, fubftra&ing one from the other : But 
 had they found it alfo (and that more Na- 
 turally) by the Divifion of a Fifth; firft 
 into a Dit one and SeJ quit one , and then by 
 the like proper Divifion of a true Ditone 
 ( or Third Major ) into its proper Parts :, 
 they muft have found both Tone Major and 
 Tone Minor. Euclid refts fatisfied, that In- 
 ter fu^er particular e non c adit Medium. A 
 fuper-particular Ration cannot have a Me- 
 diety ; viz. in whole Number : Which is 
 true in its Radical Numbers. But had he 
 doubled the Radical Terms of a Super- 
 particular, he might have found Mediums 
 nioft Naturally and Uniformly dividing the 
 Sy Items of Harmony ; ex. gr. The Duple 
 Ration 2 to 1, as the Excefs is but by an 
 Unity, has the Nature of Super-particular : 
 but 2 to 1, the Terms being dupled, is 4 
 to 2 ; where 3 is a Medium, which divides 
 it into 4 to 3 (4 th ) and 3 to 2 (5 th ) A- 
 gain, 3 to 2, dupling each Term, is 6 to 4; 
 and in the fame manner gives the two 
 Thirds, viz. 6 to 5, (3 d Minor) and 5104, 
 (3 d Major). Likewife the 3 d Major, 5 to 4, 
 
 I 3 dupled 
 
1 1 6 Of T)if cords and Degrees. 
 
 dupled as before, ioto 8, gives the two 
 Tones, i. e. 10 to 9, Tone Minor, and 9 to 8, 
 Tone Major* 
 
 And it feems to be a Reafon why the 
 Ancients did not difcover and ufe the Tone 
 Minor* and confequently not own the *D/- 
 tone for a Concord ; becaufe they did not 
 purfue this Way of dividing the Syftems. 
 Altho* Euclid had a fair Hint to fearch fur- 
 ther, when he meafured the Diafafon by 
 fix Tones [ Major'} and found them to ex- 
 ceed the Interval of Diapafon. 
 
 The Pythagoreans, not ufing Tone Mi- 
 nor, but two equal Tones Major, in a Fourth, 
 were forced to take a leffer Interval for the 
 Hemitone ; which is calPd their Limma, or 
 Pythagorean Hemitone ; and, which added 
 to thofe two Tones, makes up the Fourth : 
 Tis a Comma lefs than Hemitone Major, 
 (\6 to 15) and the Ration of it is z$6 to 
 243. 
 
 Yet we find the later Greek Matters, 
 Ptolemy, to take Notice of Tone Minor ; 
 and Jrifiides Quintiliantts, to divide a Sef- 
 qtiioBave Tone (9 to 8^ by dupling the 
 Terms of the Ration thereof into two He- 
 mitones ; 1 8 to 1 7, and 17 to 16. And thofe 
 again, by tfre lame Way, each into two 
 
 Diefes j 
 
Of Dif cords and Degrees'. 1 1 y 
 
 Diefes ; 36 to 35, 35 to 34 ; the Divifion of 
 18 to 17, the lefs He mi tone • And 34 to 33., 
 and 33 to j 2 ; the jParts of 17 to 16, the 
 greater Hemitone* But yet, none of thefe 
 were the Complement of two Sefquia&ave 
 Tones to Diatejjaron: but another Hemi* 
 tone, whofe Ratio is about 20 to 19 ; not 
 f xa&ly, but fo near it, that the Difference 
 is only i2i<6to 1215; both which together 
 make the Limma Pythagoricum. 
 
 But I no where find, that they thus 
 divided the Fifth and Third major, but ra- 
 ther feem'd to dillike this Way, becaufe of 
 the Inequality of the Hem? tones and Diefes 
 thus found out ; and chofe rather to eon* 
 ftitute their Degrees by the SefquioBave 
 Tone, and thofe Duodecimal fuppos'd-equal 
 Pi villous of it. But to return. 
 
 There are, you fee, three Degrees 
 Diatonic ; viz, Ht mi tone major, Tone minor^ 
 and Tone major. The firft of thefe fomc 
 call Degree minor ; the fecond, Degree ma- 
 jor ', the third, Degree maxim, Now thefe 
 three forts of Degrees are properly to be 
 intermixed, and ordered, in every Afcent to 
 im Eighth, in relation to the Key, or Uni- 
 fon given, and to the Affc&ions of that 
 Key, as to Flat and Sharp, in our Scale of 
 Muiic ; io, that the Concords may be all 
 
 I 4 true, 
 
11 8 Of D if cords and Degrees. 
 
 true, and ftand in their own fettled Ration* 
 Wherefore if you change the Key 7 they 
 mud be changed too ; which is the reafon 
 why a Harpfichord 5 whofe Degrees are 
 fixed ; or a fretted Inftrument, the Frets 
 remaining fix'd, cannot at once be fet in 
 Tune for all Keys : For, if you change the 
 Key, you withal change the Place of Tone 
 minor, and Tone major, and fall into other 
 Hemitcnes that are not proper 'Diatonic 
 Degrees, and confequently into falfe Inter- 
 vals* 
 
 You may fully fee this, if you draw 
 Scales of Afcent fitted to feveral Keys ( as 
 are here infetted J and compare them. 
 For an Example of this, Take the firft Scale 
 of Afcent to Diafafon [ I ] viz. upon C 
 Key Proper, by Diatonic Degrees ; (making 
 the firft to be Tone minor, as convenient 
 for this Inftance) intermixing the Chroma- 
 tic and other Hemitones, as they are ufual- 
 ly placed in the Keys of an Organ ; /, e. 
 run up an Eighth upon an Organ (tuned 
 as well as you can) by Half-Notes, begin- 
 ning at C Sol fa tit, and you will find thefe 
 Meafures. The Proper Degrees ftanding 
 right, as they ought to be, being defcrib'd 
 by Breves ; the other by Semi breves ; The 
 Breves reprefenting the Tones of the broad 
 Gradual Keys of an Organ ; the Semibreves 
 
 repre- 
 
<2* 
 
 recJ . / 
 
 3 
 
 1 
 
 I 
 
 
 A- 5 & 
 
 t ^„- 4 ^-^: 45^^ ff^ %. 
 
 ■ b'<? ^b , — . h ^ .r 
 
 " ^ 7 ^ - ^5 ^ • "3-7 ^ 5 : . 
 
 b£> »H,,,B^ 
 
 
 "BLust i/uJ aft^Fdf.-JlS. 
 
Of Vtjcords and f)cgrees. 1 1 p 
 
 reprefenting the narrow Upper Keys, which 
 are ufually call'd Mufics. And let this 
 be the firft Scale, and a Standard to the 
 reft. 
 
 Then draw a fecond Scale [II J run- 
 ning up an Eighth in like manner ; but let 
 the Key, or Firft Note be D Sol re, with a 
 Flat Sixth, on the fame Organ ftanding 
 tuned as before ; which Key is fet a Note 
 (or Tone Minor) higher than the former. 
 
 Draw alfo a third Scale [III] for 
 D Sol re Key with Sharps, viz. Third and 
 Seventh Major ; i. e, F, and C, fharp, 
 
 I n the Firft of thefe Scales, the Degrees 
 (exprefsM by "Breves) are fet in good and 
 natural Order. 
 
 I n the Second Scale (changing the Key 
 from C to DJ you will find the Second, 
 Fourth, and Sixth, a Comma (81 to 80) 
 too much ; but between the Fourth and 
 Fifth, a Tone Minor, which fhould be al- 
 ways a Tone Major, So, from the Fourth 
 to the Eighth, is a Comma fhort of T)ia- 
 fente , and from the Sixth, a Comma fhort 
 of Third Major. And this, becaufe in this 
 £cale the Degrees are mifplaced. 
 
 Ths 
 
1 10 Of Dif cords and Degrees. 
 
 The Third Scale makes the Second, 
 Fourth, and Sixth, from the Unifon, each 
 a Comma too much ; and from the OBave^ 
 as much too little. In it, the third De- 
 gree, between % F and G, is not the Pro- 
 per Hemitone^ but the Greateft Hemitone % 
 *7 to 25* Arid all this, becaufe in this 
 Scale alfo the Degrees are mifplaced ; and 
 there happen fas you may fee; three Tones 
 Minor., and but two Major, the deficient 
 Comma being added to the Hemitone. 
 
 I have added one Example more, of a 
 Fourth Scale j [ IV ] viz. beginning at the 
 Key $ C ; with the like Order of Degrees 
 as in the firft Scale ( from the Note C|) 
 upon the fame Instrument, as it ftands 
 tuned after the firft Scale : And this will 
 j-aife the firft Scale half a Note higher. 
 
 I n this Scale, all the Hemitones are of 
 the fame Meafure with thofe of the firft 
 Scale refpeftively* 
 
 And the Internals fhould be the fame 
 with thofe of the firft Scale ; which has 
 Third, Sixth, Seventh, Major. 
 
 But in this fourth Scale, the firft De- 
 gree, from $ C to h E ? is T$pe major, and 
 
 Die ft s ; 
 
Of Dif cords and Degrees. 121 
 
 Die/is; as being compounded of 16 to 15, 
 and 27 to 25. 
 
 Th e Second Degree from IE toF, is 
 Tone 'Minor ; therefore the Ditone, made 
 by thefe two Degrees, is too much by a 
 Die (is , (128 to 1 25) and as much too little 
 the Trihemitone, from the Ditone to the 
 Fifth. 
 
 The Third Degree, from F to $F, is a 
 Minor Hemit one, 25 to 24; which (tho' 
 a wrong Degree) fets the DiateJJ'aron 
 right. 
 
 The Fourth Degree; from | F to $ G, 
 is 33m«? Major, and makes a true Fifth. 
 
 The Fifth Degree, from f G to I B, is 
 7W major, and D*>/£r j fetting the Hexa- 
 chord (or Sixth) a *DieJis and Comma too 
 much, or too high. It ought to have been 
 !*<?#? minor. 
 
 The Sixth, from £ B to C, is Tone 
 minor ; too little in that place by a 
 Comma. 
 
 The Seventh, from C to i C, is Bern- 
 tone Minor ; too little by a Die/is. And 
 fo, thefe two laft Degrees are deficient by 
 
 a 
 
122 Of Dif cords and Degrees. 
 
 a Diefis and Comma ; which Diefis and 
 Comma being Redundant (as before) in the 
 fifth Degree, are balanced by the deficien- 
 cy of a Comma in the fixth Degree, and of 
 a Diefis in the feventh : And fo the OBave 
 is fet right. 
 
 These Difagreements may be better 
 view'd, if we fet together, and compare 
 the Degrees of this IV Scale, and thofe of 
 the I : Where we lhall find but one of all 
 the feven Degrees, to be the fame in both 
 Scales. 
 
 Scale I. Scale IV. 
 
 Degrees, 
 
 i ft* Tone minor. Tone ma), and Diefis* 
 
 2 d > Tone ma)or. Tone minor. 
 
 3 d > Hemit. major. Hemitone minor. 
 
 4 th > Tone major. Tone major. 
 
 5 th > Tone minor. Tone ma), and Diefis. 
 
 6 th ' Tone major. Tone minor* 
 
 7 th > Hemit. major. Hemitone minor* 
 
 And thus 'twill fucceed in all Inftru- 
 ments, tuned in order by Hemitones, which 
 are fix'd upon Strings; as Harp, &c. or 
 Strings with Keys ; as Organ, Harpfichord, 
 l£)C. or diftinguifh'd by Fretts ; as Lute, 
 Viol, fcffc. for which there is no Remedy, 
 but by fome alterations of the Tune of the 
 
 Strings 
 
Of Difcords and Degree?. 1 1 3 
 
 Strings in the two former ; and of the Space 
 of the Fretts in the latter ; as your prefent 
 Key will require, when you change from 
 one Key to another, in performing Mufical 
 Compofitions. 
 
 Tho' the Voice, in Singing, being freet 
 is naturally guided to avoid and. correct 
 thofe before defcrib'd Anomalies, and to 
 move in the true and proper Intervals : It 
 being much eafier with the Voice to hit 
 upon the right, than upon the anomalous or 
 wrong Spaces. 
 
 Much more of this Nature may be 
 found, if you make and compare more 
 Scales from other Keys. You will ftill find, 
 that, by changing the Key, you do withal 
 change and dilplace the Degrees, and make 
 ufe of Improper Degrees^ and produce In- 
 congruous Intervals. 
 
 For, inftead of the Proper Hemitone, 
 fome of the Degrees will be made of other 
 fort, of Hemitones ; amongft which chiefly 
 are thefe two : viz. Hemitone Maxim. 27 
 to 2 5 ; and Hemitone Minor, or Chromatic, 
 2 $ to 24. Which Hemitones conflitute and 
 divide the two Tones ; viz. Tone major, 
 9 to 8 : the Terms whereof tripled, are 
 27 to 24; and give 27 to z^ and 25 to 24. 
 
 The 
 
124 Of DiJ cords and Degrees. 
 
 The Tone mi?ior likewife is divided into 
 two Hemitones\ viz. Major ^ 16 to i^\ and 
 Minor <> z$ 1024. 
 
 These two ferve to meafure the Tones^ 
 and are ufed alfo when you Divert into 
 the Chromatic Kind. But the Hemitone 
 Degree in the Diatonic Genm, ought al- 
 ways to be Hemitone Major ^ 16 to 15 \ as 
 being the Proper Degree and Difference 
 between Tone major and Trihemitone^ be- 
 tween Ditone and a Fourth, between Fifth 
 and Sixth minor\> and alfo between Seventh 
 major and Offave. 
 
 Music would have feemM much eafier, 
 if the Progreflion of Dividing had reached 
 the Hemitones : I mean, if, as by dupling 
 the Terms of T)ia£afon^ 4 to 2 ; it divides 
 in 4 to 3, and 3 to 2 ; 'DiateJJaron, and 
 Diapente : And the Terms of Diapente 
 dupled, 6 to 4; fall into 6 to 5, and 5 104, 
 Third minor, and Third major ; and Ditone^ 
 or Third major, fo dupled, 10 to 8, falls 
 into 10 to 9, and 9 to 8, Tone minor and 
 Tone major : If, I fay, in like manner, the 
 dupled Terms of Tone major 18 to 16, thus 
 divided, had given Ufeful and Propet Hemi- 
 tones, 18 to 17, and 17 to i<5. But there 
 are no fuch Hemitones found in Harmony, 
 and we are put to feek the Hemitones out 
 
 ox 
 
Of Difcords and Degrees'. i z j 
 
 of the Differences of other Intervals ; as 
 we fhall have more Occafion to fee, when 
 I come to treat of Differences, in Chap. 8. 
 
 I may conclude this Chapter, by fhewing 
 how all Confonants, and other Concinnous 
 Intervals, are Compounded of thefe three 
 Degrees ; Tone major y Tone minor y and He* 
 mitone ma]or \ being feverally placed, as 
 the Key fhall require. 
 
 Tone Major, and \ joyn'd, \ , M , 
 Hemitone Major, $ make j3dM/w?, 
 
 Tone Major, and) joyn'd,"}., n/r . - 
 Tone Minor, J make ^j***$ 
 
 7o»^ Major, and* 
 7W Minor, and 
 Hemitone Major, 
 
 « MW 3 and^ j# °y n J dj C 4th. 
 
 make 
 
 2 To/w Major, 
 
 joyn'd, 
 
 1 Tow jtffttr, f J ^ a ke ?" 5*& 
 
 1 Hemitone Maj. 3 3 
 
 2 Tones Major, ~? • » , 
 
 1 tone Minor, > ^^ ?• tfth ilfw 
 
 2 Hemitones Maj. J 
 
 2 7Wx Major, 
 2 7om M«&r, 
 I Hemitone Maj 
 
 2 7W »r[ £ ^fcf'f tfth Af/^A 
 
 I Texts' 
 
Tl6 Of Difcords. 
 
 3 Tones Major, 7 : VQ *^ 7 
 
 i Towe Afc'wor, & > ma fc e ' > 7th Miwt. 
 
 i Hemitones Maj. j & 
 
 3 7*oh« Myw, 7 . , , 7 
 
 a Tones Minor, > •'■■■■■* , ' > yth Major. 
 
 \t HemttoneMaj.S G S 
 
 3 Tones Major, 7 . , 7 
 3 Hemitones Maj. \ j 
 
 2 Tow Major, \ joyn'd, \7ritone, or 
 !i lone Minor s j make J falfe 4th. 
 
 I Tew Major. J * >• 7 « • .,*, 
 
 1 7W Afaor V )o y n d * £ Semidtapente, 
 
 1 vltZ* \ mak ^ C or falfe S tfi. 
 
 2 Hemit. Major , 3 3 J 
 
 CHAP. VIL 
 
 , 6?/ Difcords. 
 
 BESIDES the Degrees, which, tho* 
 they conftitute and compound all Con- 
 cords, yet are reckoned amongft: Difcords ; 
 becaufe every Degree is Dxfcord to each 
 Chord, to, or from which it is a Degree, 
 either Afcending or Defcending, as being a 
 Second to it : Befides thefe, I fay, there 
 are other Difcords, fome greater, and fome 
 
 kfs* 
 
Of Vifcords. 127 
 
 lefs. The lefs will be found amongft the 
 Differences in the next Chapter ; and are 
 fit, rather to be known as Differences, than 
 to be ufed as Intervals. 
 
 Th e greater Difcords are generally made 
 of fuch Concords as, by reafon of mifpla- 
 ced Degrees happen to have a Comma, or 
 Die/is, or fometimes a Hemitone too much, 
 or too little ; and fo become Difcords, moft 
 of them being of little Ufe, only to know 
 them, for the better meafuring and recti- 
 fying the Syftems : Yet they are found 
 amongft the Scales of our Mufic. 
 
 Sometimes a Tone Mtjor being 
 where a Tone Minor fhould have been pla- 
 ced, or a Tone Minor inftead of a Tone Ma- 
 jor ; fometime other Hemitones, getting the 
 place of the "Diatonic Hemitone Major, and 
 ferving for a Degree, create unapt Difcor- 
 ding Intervals : amongft which may be 
 found at leaft two more Seconds, two more 
 Thirds, two more Sixths, and two more 
 Sevenths. In each of which, one is lefs^ 
 and the other greater, than the true legi- 
 timate Intervals, or Spaces of thofe Deno- 
 minations ; as will be more explained in the 
 enfuing Difcourfe. 
 
 K But 
 
tZi 
 
 OfDijcords. 
 
 But befides thefe ( or rather amongft 
 them, for I here treat of Degrees as Dif- 
 cords ) there ara two Difcords eminently 
 confiderable, viz. Tritone, and Semidia- 
 fente. The Tritone, ( or falfe Fourth ) 
 whofe Ration is 45 to 32, confifts of three 
 whole Notes ; viz. two Tones Major, and 
 one Minor* The Semidiapente (or falfe 
 Fifth) 641045 ; is compounded of a Fourth 
 and Remit one Major* 
 
 And thefe two divide Diafafon, 64 to 
 31, by the Mediety of 45; And they di- 
 vide it fo near to Equality, that in Practice 
 they are hardly to be diftinguifh'd, and 
 may almoft pafs for one and the fame : 
 but in Nature, they are fufficiently diftin- 
 guifliM ; as may be feen both by their fe- 
 veral Rations, and feveral Compounding 
 Parts. 
 
 I think we may reckon yths for Degrees^ 
 as well as among the greater Difcording In- 
 tervals ; becaufe they are but Seconds from 
 the Octave, and aro as truly Degrees De« 
 fcending, as the Seconds are in Afcent : tho' 
 they be great Intervals in refpe£fc of the 
 Unifon^ and fuch as may be here regarded. 
 
 These Difcords, thsTritone, and Semi* 
 diapente \ as alio, the Seconds, and Sevenths^ 
 
 are 
 
Of T)if cords. I 29 
 
 are of very great ufe in Mufic, and add a 
 wonderful Ornament and Pleafure to it, if 
 they be judicioufly managed- Without 
 them, Mufic would be much lefs grateful ; 
 like as Meat would be to the Palate with- 
 out Salt or Sawce. But, the farther Con- 
 fideration of this, and to give Dire&ions 
 when, and how to ufe 'em, is not my Task, 
 but muft be left to the Matters of Compo- 
 fition. 
 
 Discords then, fuch as are more apt 
 and ufe ful ( JntervaUa Concinna) are thefe 
 which follow. 
 
 2d Minor ; or, Hemit one Major, 16 to 15. 
 2d Major ; Tone Minor, 10 to 9. 
 
 2d Great eft ; Tone Major 9 to S. 
 
 7th Minor ; 5th & 3d Minor, 9 to 5. 
 ytfa Major; 5th & 3d Major, 1 5 to 8. 
 Tritone ; 3 d Maj. & Tone Maj. 45 to 3 x. 
 
 Semidiafente ', 4th 8c Remit. Maj. 64 to 4 j. 
 
 Th e s e are the Simple diffonant apt 
 Intervals within T)ia£afon; if you go a 
 further Compafs, you do but repeat the 
 fame Intervals added to Diapafon, or T)if- 
 diapajon, ovTrif-diafafon, &c. as, Ex.gr. 
 
 K % , A 
 
130 Of Dif cords* 
 
 A 9th is Diapafon with a 2d. 
 10th "Diapafon with a 3d 
 i ith 5 Diapafon with a 4th, or 
 
 {^Diapafon cum Diatejjaron* 
 f t f Diapafon with a 5 th, or 
 
 i Diapafon cum Diapente* 
 !i 5 th r Dif-diapafon» 
 19th T)if-diapafon cumDiapente* 
 2 2 th Trif-diapafon^ &c 
 
 HerEj by the way, the Reader may 
 take a little Diverfion, in praftifing to mea- 
 fure the Rations of fome of thofe Intervals 
 in the 'foregoing Catalogue of Difcords, by 
 comparing them with Diapafon \ as thofe of 
 the Sevenths ? which I felefl:, becaufe they 
 are the moft diftant Rations under 'Diapa- 
 fon ; viz. Seventh minor ^ 9 to 5 ; and Se- 
 venth major j 15 to 8. Now to find what 
 Degree or Interval lies between thefe and 
 Diapafon. 
 
 Firft, 9 to 5 is 10 to 5, wanting 10 to 9 
 {Tone minor J Next, 15 to 8 is 16 to 8, 
 wanting 16 to 15 {Hemitone major**) So 
 the Degree between Seventh minor and 
 Diapafon, is Tone minor ; and between Se- 
 venth major and Diapafon, is Hemitone 
 major* 
 
 Then 
 
Of Difcords. 1 1 1 
 
 Then he may exercife himfelf in a Sur- 
 vey of what Intervals are compriz'd in 
 thofe feveral Sevenths^ and of which they 
 are compounded. 
 
 First, 9 to $ comprizeth 9 to 8, and 
 8 to j : Or, 9 to 8, 8 to 6, and 6 to 5* 
 Next, 15 to 8 contain 15 to 12, 12 to 10, 
 10 to 9, and 9 to 8 : Or, 15 to 12, and 12 
 to 8 : Or, 1510 10, and 10 to 8, &c. I 
 fuppofe that the Reader, before this, is fo 
 perfeQ: in thefe Rations, that I need not 
 lofe Time to name the Intervals exprefs'd 
 by the Mean Rations, contained in the Yore- 
 going Rations of the Sevenths, which fhew 
 of what Intervals the feveral Sevenths are 
 compounded. 
 
 Besides thefe ( by reafon of Degrees 
 wrong placed) there are two more Sevenths ; 
 [falfe Sevenths'] one, lefs than the true ones, 
 and another greater, The leaft compoun- 
 ded of two Fourths, whofe Ration is 16 
 to 9, and wants a Comma oi Seventh minor^ 
 and a Tone ma]or of Diapafon : The other 
 is the greateft, calPd Sgmidiaf>afon y whofe 
 Ration is 48 to 25 j being 3. Diefis more 
 than Seventh major, and wanting Hemiton® 
 minor of Diagafqn. 
 
 K 3 N0W3 
 
i%z Of Dif cords* 
 
 Now, nrffy 16 to 9 Is 16 to 8 (2 to 1) 
 wanting 9 to 8 ; /. e. wanting Tone major 
 of Diafafon ; and contains 16 to 10 (8 to 5) 
 and 10 to 9 ; Or, 16 to 15, 15 to 12 (5 104) 
 12 to 10 (6 to 5) and 10 to 9. Next,Smi- 
 diafafon 48 to 25, is 50 to 25, wanting yo 
 to 48 ; /. f. 25 to 24 (viz. Heniit one minor) 
 of Diafafon. 
 
 And the like happens, as hath been 
 faid, to the other Intervals, which admit of 
 major and minor ; viz. Seconds y Thirds ^ and 
 Sixths. The Fourth^ and F//%, and Eighth 
 ought always to remain immutable ; tho^ 
 they may fuffer too fotiietimes, and incline 
 to Difeord, if we afcend to them by very 
 wrong Degrees ; as you may fee in the 
 II d Scale la the Yoregoing Chapter ; where 
 the Fourth having two Tones major ^ is a 
 Comma too much. 
 
 All thefe Intervals may be fubjeQ: to 
 more Mutations, by more abfurd placing 
 of Degrees, or of Differences of Degrees ; 
 but it is not worth the Curiofity to iearch 
 
 farther into them : The Reader may take 
 Fleafure, and fufficiently exercife himfelf, 
 
 in comparing and meaiuring thefe which 
 
 are already laid before him, 
 
 Bu T 
 
Of Dif cords, i 3 5 
 
 But to return from this Digreffion. 
 There are many unapt Difcords, which may 
 arife* by continual Progrefllon of the fame 
 Concords ; I. e* by adding (for Example) 
 a Fourth to a Fourth, a Fifth to a Fifth, &c« 
 for 'tis obfervable, That only Diapafon ad- 
 ded (as oft as you pleafe) to Diapajon, ftill 
 makes Concord : But any other Concord,, 
 added to it felf, makes Difcord* 
 
 You will fee the Reafon of it, when you 
 have confiderM well the Anatomy (as I 
 may call it) of the Conftitutive Parts of 
 'Diafafon ; which contains, and is composed 
 of feven Spaces of Degrees, or of Fourth and 
 Fifth, or of Thirds and Sixths, or of Seconds 
 and Sevenths ; which muft all keep their 
 true Meafures and Rations belonging to 
 them, and otherwife are eafily and often 
 diforderM. 
 
 Then, confider Diapafon as confHtuted 
 of two Fourths disjunct, and a Tone major 
 between 'em. And this laft is moll need- 
 ful to be very well confiderM ; as mod 
 plainly fhewing the Reafons of thofe Ano«? 
 malies, or irregular Intervals, which are 
 produced by changing the Key, and confe- 
 quently giving a new and wrong Place to 
 this odd Jon? major, which ftands in the 
 
 K 4 midil 
 
1 34 Of Difcords. 
 
 midft of DiaPafon, between the two Fourths 
 disjunct. 
 
 Every fourth muft confift of one Tone 
 major , one Tone minor, and one Hemitone 
 major , as its Degrees, placing them in what 
 Order you pleafe ; whofe Rations, added 
 together, make the Ration of Diatejjaron. 
 And of thefe fame Degrees contained in the 
 Fourth^ are made the two Thirds, which 
 conftitute the fifth. Tone major and Hemi- 
 tone major make the lefs Third, or Trihemi- 
 tone ; Tone major and Tone minor make the 
 greater Third, or Ditone ; Trihemitone and 
 Ditone make Diafente ; Trihemitone and 
 Tone Minor (as likewife Ditone and Hemi- 
 tone major) make Diafeffaron. 
 
 Now this Tone Major, that (lands in the 
 middle of Diafafon, between the two 
 fourths, which it disjoins ; and the Degrees 
 required to the Fourths, will not in a fixed 
 Scale ftand right, when you alter your Key, 
 and begin your Scale of ^Diafafon from 
 another Note : For that which w T as the Fifth, 
 will now be the Fourth, or Sixth, &c. and 
 then the Degrees will be diforder'd, and 
 create fome difcording Intervals. If you 
 continue conjunct Fourths, there will be a 
 Defefl: of Tones Major ; if you continue con- 
 juniV Fifths, there will be too many Tones 
 
 Major 
 
Of Dif cords. 13 j 
 
 Major in the Syftems produced. And if a 
 Tone Major be found, where it ought to 
 have been a Tone Minor ; or a Minor inftead 
 of a Major \ that Interval will have a Com- 
 ma too much, or too little. And fo like- 
 wife will from a wrong Hemitone be found 
 the Difference of a Diejis. And thefe two, 
 Comma and Diejis, are fo often redundant, 
 or deficient, according as the Degrees hap- 
 pen to be diforder'd or mifplaced; that 
 thereby the Difficulties of fixing half-Notes 
 of an Organ in tune for all Keys, or giving 
 the true Tune by Fretts, become fo infu- 
 perable. 
 
 You fee, that in every Space of an 
 Eighth, there are to be three Tones major % 
 and two Tones minor, and two Hemitones 
 major : One Tone major between the Dia- 
 tejJ : aron and Diafente, and a Tone major r 
 a Tone minor, and Hemitone major in each of 
 the disjun£t Fourths. 
 
 These Are the proper Degrees by 
 which you fhould always Afcend or De- 
 fcend t\iYQ r Diapafon, in the T>iatonicKind ; 
 which Diafafon being the compleat Syftem, 
 containing all primary Simple Harmonic In* 
 tervals that are ; (and for that reafon calPd 
 Diafaj'on ;J you may multiply it, or add it 
 |Q its felf as oft as you pleafe, as far as Voice 
 
 or 
 
%l6 Of D if cords* 
 
 or Inftrument can reach, and it will ftill be 
 Concord, and cannot be diforder.M by fuch 
 Addition ; becaufe every of them will con- 
 tain (however placed) juft three Tones 
 major ^ two Tones minor ^ and two Hemitones 
 major* 
 
 Whereas, if you add any^ other In- 
 terval to it felf, the Degrees will not fall 
 right, and it will be Difcord, becaufe all 
 Concords are compounded of unequal Parts, 
 as hath been fhewn before ; and if you car- 
 ry them in equal Progreffion, they will mix 
 with other Intervals by incongruous De- 
 grees, and thofe diforderM Degrees will 
 create a diffonant Interval See the follow- 
 ing Scheme of it. 
 
 * 5 th, wanting Hemit. min. 
 5th, and Hemit. minor. 
 
 2 3ds minor "J 
 a 3ds major 
 
 2 4ths j>^ j 8th, wanting Tone major, 
 
 £ ^ 8th, and Tone major. 
 8th, and Ditone^tk Die/is, 
 t 8th, and ^th^k Hem. min* 
 
 2 5ths 
 
 2 6ths minor 
 2 6th$~major„ 
 
 To which may be added, That 
 
 2 Tones min. \f%5 Dito0e 9 vr&nting a Comma, 
 1 Tones ma). J S 1 Ditone 7 and a Comma. 
 
 It was faid above, That Diafafon may 
 
 be added to it felf as oft as. you pleafe, and 
 
 i there 
 
Of f)if cords. 137 
 
 there will be no Diforder, becaufe every- 
 one of 'em will ftill retain the fame Degrees 
 of which the firft was composed : But it is 
 not fo in other Concords ; of which I will 
 add one more Example, becaufe of the Ufe 
 which may be made of it. 
 
 Make aProgreffion of four c Diapenfe\ 
 and, as was fhew'd in the Fifth Chapter, it 
 will produce "Difdiapafon, and two Tones 
 ma)or, which is a 17 th * with a Comma too 
 much ; becaufe in that Space there ought 
 to be jufl: feven Tones ma)or, and five Tones 
 minor ; whereas in four Fifths continued, 
 there will be found eight Tones mdpr, and 
 but four Tones minor : So that a Tone ma)or^ 
 getting the Place of a Tone minor, there will 
 be in the whole Syftem a Comma too much. 
 One of thefe maftr Tones fliould have been 
 a Tone minor, to make the Excefs above Dif 
 diapafon & juft T) it one. 
 
 On the other fide, if you continue the 
 Ration of four Diatejfarons, there will be a 
 Tone minor, inftead of a Tone ma)or ; and 
 confequently a Comma deficient in conftitu- 
 ting Diapafon and Sixth minor. For fince 
 every Fourth muft confift of the Degrees of 
 Tone 'mi7tor, one Tone ma\or, one Remit o7ie 
 ma)or ; it follows, that if you continue four 
 Fourths, there will be four Tones minor* four 
 
 Tones 
 
138 QfVijcords, 
 
 'Tones major, and four Hemitones major : 
 Whereas in the Interval of r Diapafon with 
 Sixth minor., there ought to be five Tones 
 major, and but three minor. 
 
 B y this you may fee the Reafon, why, 
 to put an Organ or Harpfichord into more 
 general ufeful Tune, you muft tune by 
 Eighths and Fifths; making the Eighths 
 perfect, and the Fifths a little bearing down- 
 ward ; /. e . as much as a quarter of a Com- 
 ma, which the Ear will bear with in a Fifth, 
 tho' not in an Eighth. For Example, be- 
 gin at C la ut ; make C Sol fa ut a perfeft 
 Eighth to it, and G Sol re ut a bearing Fifth \ 
 then tune a perfect Eighth to G, and a 
 bearing Fifth at D Lafolre-, and from 
 thence downwards (that you may keep to- 
 wards the middle of the Inflrument) a per- 
 fe& Eighth at D Sol re : And from thence 
 a bearing Fifth up at A ; and from A, a 
 perfect Eighth upwards, and bearing Fifth 
 at E La mi. From E an Eighth down- 
 wards ; and fo go on, as far as you are led 
 by this Method, to tune all the middle part 
 of the Inflrument ; and at laft fill up all 
 above, and below, by Eighths from thofe 
 which are fettled in Tune, according to the 
 Scheme annexed ; obferving (as was faid ) 
 tp tune the Eighths perfect, and the Fifths 
 a little bearing flat; except la the three 
 
 4aft 
 
Of D if cords. 1 1$ 
 
 laft Barrs of Fifths, where the Fifths begin 
 to be taken downward from C, as they 
 were upwards in all before : Therefore, as 
 before, the Fifth above bore downward ; 
 fo here, the Fifth below mufl bear upward, 
 to make a bearing Fifth : but that being 
 not fo eafie to be judg'd, alter the Note 
 below, till you judge the Note above to 
 be a bearing Fifth to it. This will reftifie 
 both thofe Anomalies of Fifths and Fourths; 
 For the Fifth to the Unifon, is a Fourth to 
 the Oflave ; and what the Fifth lofeth by 
 Abatement, the Fourth will gain : Which 
 doth in a good Degree reftifie the Scale of 
 the Inftrument. Taking Care withal, that 
 what Anomalies will ftill be found in this 
 Hemitonic Scale, may, by the Judgment 
 of your Ear, in tuning, be thrown upon 
 fuch Chords as are leait ufed for the Key ; 
 as $ G, b E, &c. even which the Ear will 
 bear with, as it doth with other Difcords 
 in binding PafTages *, if fo, you clofe not 
 upon them. But the other Difcords, fo 
 ufed, are moft Elegant ; thefe only more 
 Tolerable. 
 
 CHAP, 
 
Of "Differences. 
 
 CHAP. VIIL 
 
 Of Differences. 
 
 ALL Rations and Proportions of In- 
 equality^ have a Difference between 
 them, when compared to one another ; and 
 confequently the Intervals, exprefs'd by 
 thofe Rations, differ likewife. A Fifth is 
 different from a Fourth, by a Tone Major ; 
 from a Third Minor, by a Third Major j fo 
 an Eighth from a Fifth, by a Fourth. Of 
 the Compounding Parts of any Interval, one 
 of them is the Difference between the other 
 Part and the whole Interval. 
 
 ButI treat now of fuch Differences as 
 are generally lefs than a Tone, and create 
 the Difficulties and Anomalies occurring in 
 the two 'foregoing Chapters. I have the 
 lefs to fay of them apart, becaufe I could 
 not avoid touching upon them ail-along. 
 'Twill only therefore be needful, to fet be- 
 fore you an orderly View of them. And, 
 firft, taking an Account of the true Harmo- 
 nic Intervals, with their Differences, and the 
 Degrees by which they arife ; 'twill be ea- 
 fier to judge of the falfe Intervals, and of 
 what Concern they are to Harmony. 
 
 Table 
 
Of "Differences. 
 
 HI 
 
 Table of true Diatonic Intervals within 
 Diapafbn, with the Differences between 
 them. 
 
 TTheir 
 
 Hemitone Major, 
 Tone Minor* 
 Tone Major. 
 3d Minor > T 
 3d Major ' 3 
 4th. 
 
 5 th. 
 
 6th Minor \ 
 6th Major \ 
 7th Minor*, 
 
 7 th Major \ 
 Diapafon ; 
 
 Tritone \ 
 Semidiapente ; 
 
 (Rations 
 
 Differen- 
 
 ces. 
 
 } 
 
 1 G . 
 
 O 
 
 e 
 
 o 
 
 fTone Maj. ScHemit.Maj. 
 Tone Maj. 3c Tone Min 
 3d minor & Tone min* » 
 
 or 3d tfz*yw 6c He- > 
 
 mitone major* j 
 
 4th 3 and XW ^ztf/w ; *> 
 
 or of the two 3ds. f 
 5th 3 and Hemit. maj. 
 
 or 4th 3 and 3d min 
 $thj and Tone minor *S 
 
 or 4th and 3d maj. y 
 6 th maj. & Hem. max. 
 
 or 6th min. & 7W 
 
 major \ or 5 th and 
 
 3d minor. 
 6th ;##/. 8c TW ?## /. ") 
 
 or $ th, and 3d maj. I 
 7th, and 2d or 6thi ") 
 
 6c 3d or 5th,6c4th.j 
 
 16 to 
 
 10 to 
 
 9 to 
 6 to 
 5 to 
 
 *%5 to 24 
 *8i to 80 
 b 16 toi^ 
 5J25 to 24 
 
 10 to 9 
 
 4 to 3 
 
 3 to 
 8 to 
 
 5 to 
 
 9 to $ 
 
 IS to 
 
 2 to 
 
 3d maj. and 7W iw/y. 45 to 31 
 1 4th, anc * Hemit. major. J<54 to 45 
 
 9 to 8 
 
 16 to EC] 
 
 2 j to 2,4 
 
 27 to 25 
 
 25 to 24 
 6 to 15 
 
 2048 
 to 
 
 204$ j 
 
 Thofe which arife from the Differences of Confonant In ? 
 tervals, are call'd IntervaIJa Concinna, and properly apper- 
 tain to Harmony : The reft are neceflary to be known* for 
 making and undemanding the Scales of Mufick. 
 
 Table 
 
!4Z 
 
 Of Differences. 
 
 Table of faJfe Diatonic Intervals, caufed 
 hy Improper Degrees ; with their Rations 
 andT)ifferences from the true Intervals. 
 
 This Mark -J- fiands iotmove° s — forte,//. 
 
 rw^/A Leaft * T £f ^™ f > . and i> 
 
 *c Hemit. major* f 
 
 tone £ Grea£e fi . j^ tf ^ flfj anc i ^ 
 
 HemiUmax 9 f 
 
 Ditone *£ 
 
 Fourth 
 
 (Differences 
 from true. 
 
 C Leaft ; z Tones minor. 
 
 ^Greateft; 2 7W/ j»/y<?r. 
 
 ^ Lefs ; g Tones minor fie V 
 Hemit, major, y 
 
 l 
 
 Greater ; 2 7W/ /»/*)*. &7 
 Hemit* maj 9 y 
 
 c Lefs; Lefs 4th, and! 
 
 Fifth ^ 7fo* major. J 
 
 ^Greater ; Greater 4th 3 and! 
 
 yz to a7'8| to 80 — 
 
 100 to 81.81 to 80 — 
 
 81 to 6*4 81 to So -§~ 
 
 3^0 to 243 81 to 80 — 
 
 17 to 2,© 81 to 804* 
 
 Sixth 
 
 c Leaft 
 
 Strand Hemit.l 
 ■^ minor* y 
 
 £Greateft; $th, ^ and Tone 
 major* 
 
 } 
 
 C Leaft ; 6th major^ and 1 
 
 Seventh^ Hemit. major, y 
 
 £Greateft; 6th minor > and 1 
 
 3d minor. 3 
 
 40 to 27! 8 1 to 8o~*. 
 81 to 80 4- 
 
 81 to 80 — 
 81 to 80 -|- 
 
 81 to 80 — 
 u8toii$-f< 
 
 *5 
 
 to 
 
 16 
 
 *7 
 
 to 
 
 16 
 
 16 
 
 to 
 
 9 
 
 4 3 
 
 to 
 
 *5 
 
 Here 
 
Of Differences. 1 4 1 
 
 He r e in this Account may be feen,how 
 frequently the Comma , and the D ; efis, 
 Abounding or Deficient, by reafon of Mif- 
 placed Degrees, occafion Difcord in Har~ 
 monic Intervals* 
 
 The Comma, by reafon of a wrongTW, 
 /. e. too much, when a Tone Major hap- 
 pens where there ought to be a Tone Mi- 
 nor ; or too little, when the Tone Minor is 
 placed inftead of the Major. And the Diefis 
 is Redundant, or Deficient, by reafon of a 
 wrong Hemitone\ when the Major happens 
 inftead of the Minor, or the contrary : the 
 "Diefis being the Difference between them* 
 And if Hemitonium Maximum get in the 
 Place of Hemitonium Jtfajus, the Excefs will 
 be a Comma j if in the Place of Hemitone 
 Minor, the Excefs will be Comma and 
 Diefis. 
 
 And thefe Anomalies are not Imagina- ' 
 ry, or only Poffible, but are Real in an In- 
 ftrument nx'd in Tune by Hemitone s\ as, 
 Organ, Harpfichord, &c. And the Reader 
 may find fame of 'em amongft. thofe four 
 Scales of Diapafo??^ in the Sixth Chapter ; 
 to which alfo more may be added ; Out of 
 the Firft of which, I have fele&ed fome 
 Examples, ufing the common Marks, as he* 
 f&re ,viz 9 -J- for more • and— for left or wan- 
 ting. L From 
 
i:4i- Of Differences. 
 
 * ' ' C 2 d Af/#. — Hennt. 3 
 
 or, 
 
 
 3 d Af/ar. — Hemit. Min. 
 
 *r fn "P. S 3d Major, + D/V/w; or, 
 it ' t ' ' t 4th; — Hemit. Minor. 
 
 D, to G ; ■ 4th, 4- Comma. 
 
 in M.V S 3 d Min.—Dief.k Com. or 
 b E, to f F ; -£ ^ M># + ^ M;A 
 
 ^^toi^;t $dMaj.-\~ Hemit. Mix. 
 *T? id i idMaj.^Dtef.tkCom.Qr. 
 
 f t<, tO £ i3 j ^ 4thj __ ^r m> Su l minh)u 
 
 I F, to B ; 4th, -j" Comma. 
 J $ u, to J3 ; ^ ^ d M>> __ # m #. Min. 
 
 &<? tn C ■ $ 3d M?>r -f !>/<# ; or, 
 * b ' t0L ' 1 4 th, — Hemit. Minor. 
 
 i B, to D ; 3d M";/. — Comma. 
 
 Next, take account of fome Differences 
 which constitute feveral Hemito?ies> 
 
 Di£ 
 
Of Difference*. 143 
 
 r ':..'. ±Hemt.Maxw. 27 to 2">. 
 
 idMajor, and 4th. Hemit. Majus) 16 to 15, 
 
 Tone Major y and H;7/.'if. rHemt6ae\ . a 
 
 ZRemiton* 7 
 iTones Major y zv\A 4th <> (or Li//j?/:r^> 2^6 to 243, 
 £ Pythffgor.l 
 
 
 L 
 
 rgpr 
 
 C Afotoqte^ 21S7 to 2048 ; 
 
 57r.v /fcfc/. and Z;/;;//;.?,s or Hemit. /Ilea 7 , with 
 L Comma. 
 
 To' winch may be addecfout of Mefftrmus^ 
 
 r Hemit. Maxim, and tHemHomum} c , . 
 5.1 Bemt.Mmor. 1 M**^:)* 5 * 8 » 6 -v 
 
 CJ 
 
 £j ! Z0** Minor^ and -^ 
 
 "^ ' He mit one Ma \ im. / Hemito nium* ? 
 
 | Hemitone Alitor, V 
 ^ and Comma. J 
 
 
 Next, take a farther View of D'//y- 
 rences, moft of which arife out of die pre- 
 ceeding Differences^ by which you will bet- 
 ter fee how all Intervals arc Compounded 
 and Differenced, and more eafily lud^e of 
 their Meaiures. 
 
 t 2 Tab!- 
 
144 
 
 Of Differences. 
 
 a 
 
 a 
 
 Table of more Differences, 
 
 C Tone Maj. and To** M?tf. Cemma. 
 
 Tone Maj. and H<m Greateji. Bemitone Minor* 
 
 Tone Maj. and Hem. Medium. Hemitone Major, 
 
 Tone Maj. and Hem. Pythag. ^ Apotome. 
 
 Hem. Greateji, and HemMaj. Comma. 
 
 ]Hem.Greateft,8cHem.Min. ^mma, and Diefis; 
 I J Iviz.Hem. Minimum. 
 
 | Hemit . Major, and Minor. Diefu. 
 
 Ben*, Major, and Medium. { ^fZ mX "" 
 
 Hemit. Major, and Pythag. Comma 
 
 A P o, ml , and H»* «**{ bjfrhjj <g* 
 
 j Apoteme,3.nd Hemit. Med. Comma. 
 
 A j. a tt •+ z> *? (Command the zfote* 
 
 Afotome, and H^f. Pjf^.{ faid & l&mnc ^ 
 
 Apotome, and Hraif. Afe&r s. 2 Commas. 
 
 Hemit. Medium,zn&Pyttog. 4 D B^ ° f C ?T* 
 
 | y 6 I Majus&A Minus. 
 Hemit. Medium, and Minus. Comma. 
 m Hemit. Pytbrfg. and Minus. Comma Minus. 
 
 1 Somewhat 7 3125 
 Hemit Minus } and Die ft s. < more than > to 
 
 ( Comma jivz. 3 3072. 
 Hemit- Minus, and Comma. Bern. Submimmum* 
 
 Diefts, and C«*. iCmma Minus, viz, 
 
 J ' .1 2048 to 2025. 
 
 LCbw. Maju$ } and C^ 4 Minus. 3 2803 to 3 2768 
 
 These 
 
Of Differences. 1 4 j 
 
 These Differences ( with fome more ) 
 are found between feveral other Intervals ; 
 of which more Tables might be drawn, 
 but I fhall not trouble the Reader with 
 them. Having here fhewn what they are, 
 he may ( if he pleafe ) exercife himfelf to 
 examine Thefe by Numbers, and alfo find 
 out Them ; and to fome it may be plea- 
 fant and delightful : And, for thatReafon, 
 I have the more largely infilled on this part 
 of my Subject, which concerns the Mea- 
 furesj Habitudes^ and Differences of Har- 
 monic Intervals. 
 
 I fhall add one Table more, of the Parts 
 of which thefe leffer Intervals are com- 
 pounded ; which will ftili give more 
 Light to the former ; and is, in Efeff y the 
 fame* 
 
 L 3 Tone 
 
u 7i /• \ Hemit one Pytht-fforicum. 
 
 Hem t Med.< n < Difference between Comm&MoAuu 
 
 ^ Comma. ) , ... ■ • > 
 
 and Minus, viz. 3^805 to 3*768. 
 
 1 46 0/" Differences* 
 
 fTons Minor, and C Hemiione Maxim. 
 
 Tone Al^jor con-\ Comma. £ and Hemitone Min, 
 
 tains, &i f s com- < ^ . „, . _ T . jU He#z. A//w. 
 
 j a j c jHemitone Ma], $ Limma, ) ^. _ 
 
 pounded or / „ . .. •' < Ai x 7i Vie (is. 
 
 r C tiemitone Med. I Abotome ) ^ J 
 
 w v ^ / I Comma. 
 
 T /!/' J Sem. Maxim. J Hemit. Major, f 2 Hemit. Min 
 
 ■^ eJV ^ m lHem.Submin.\Hemit. Minor A 1 Diejis. 
 
 Tf 7[J 3 ^ Wi ^ <7 /' J H^/. ifc&flf. % Hem.'Pyth.J Hemit. Min, 
 
 nem ' Mrix ^ Comma ' \DieJls. \zComma 3 j\Bief.^Com. 
 
 %j ust - J Rem.Med.j HemYyth. J Hem. Mln. f Hem.Subm, 
 liem - M *J'\Co m . Min\Comma. \l)iefis.. \Vief.&C* m . 
 \ Be 
 
 ¥ 
 
 ( ai 
 
 tt t» r 5 Semitone Minus* 
 
 llenuI y ih '\Comma Minus. 
 
 TT . f Hemit. Submin. f Diejis, and 
 
 Ifc». tdto-^Comm*. 1?i25 to 307*. 
 
 {Comma, 
 
 hlc I ts ' \ Comma Minus, 
 
 _ (Comma Minus. 
 
 C omnia, a, n rtJ ,._ ,,,.o 
 1 J 32.00$ to 52700* 
 
 I think there fcarce needs an Apology for 
 feme of thefe Appellations, in refpect of 
 Grammar. That I call Hemitoninm, and 
 HexachcrdGsi) Mo]m and Minm \ fometimes 
 Memztone^zn^ Hexachord 7 major, and minor . 
 Thefe two laft Words are lb well adapted to 
 pur Language, that therms no Englifa-man 
 but knows them. Therefore when I make 
 Htnntone an Englifo Word, I take major and 
 !»/>tfr to be io too, and fitteft to be joined 
 with, it, without refpeci of 'Gender* 
 
 CHA.P. 
 
A? 
 
 CHAP. IX. 
 
 Conclusion. 
 
 TO conclude all. Bodies by Motion 
 make Sound ; Sound, of fitly-conftitu- 
 ted Bodies, makes Tune : Tune, by Swift- 
 iiefs of Motion is rendered more acute; 
 by SSownefs more grave : in proportion to 
 the Meafure of Courfes and Recourfes, of 
 Tremblings or Vibrations of Sonorous Bo- 
 dies. Thofe Proportions are found out by 
 the Quantity and Afte£Hons of Sounding 
 Bodies ; ex.gr. by the Length of Chords* 
 If the Proportion of Length (ceteris Pari* 
 bua) and confequently of Vibrations of fe- 
 veral Chords, be commenfurate within the 
 Number 6 ; then thofe Intervals of Tune 
 are Confonant, and make Concord, the 
 Motions mixing and uniting as they pafs ; 
 If incommenfurate, they make Difcord by 
 the jarring and clafhing of the Motions. 
 Concords are within a limited Number 
 Difcords innumerable. But of them, thofe 
 only here confider'd, which are ( as the 
 Greeks termM them) if*M«?, Goncinnow y 
 apt and ufeful in Harmony : Or which, at 
 kail, are neceffary to be known, as being 
 
 L 4 the 
 
148 Conclufton. 
 
 the Differences and Meafures of the other ; 
 and helping to difcover the Reafon of 
 Anomalies, found in the Degrees of Inftru- 
 ments tuned by Hemitones. 
 
 All thefe I have endeavoured to ex- 
 plain, with the manifeft Reafons of Con- 
 fer! aney and Diffonancy (the Properties of 
 a Tendulum giving much Light to it) lb as 
 to render them eafie to be understood by 
 aimoft all forts of Readers ; and to that 
 end have enlargM, and repeated, where I 
 might (to the more intelligent Reader) have 
 compriz'd it very much fhorter. But I 
 hope the Reader will pardon that, which 
 could not well be avoided, in order to a 
 full and clear Explanation of that, which 
 was my De£gn, viz. the Phlofophy of the 
 Natural Grounds of Harmony. 
 
 Upon the whole, you fee how Ratio- 
 nally, and Naturally, all the Simple Con- 
 cords, and the two Tones, are found and ' 
 demonftrated, by Subdivisions of Diapafon. 
 
 2 to i, i.e. 4 to 2 ; into 4 to 3, and 3 to 2. 
 2 to 1, i.e. 6 to 3 1 into 6 to 5, and 5 to 3. 
 2 to 1, i.e. 8 to 4; into 8 to 5, and 5 to 4, 
 2 to 1, i.e.ictoy^ into 10 to 9, $ to 8, and 
 S to £. 
 
 In 
 
Conclufion* 1 49 
 
 I n which are the Rations ( in Radical, 
 or Leaft Numbers) of the O&ave, Fifth, 
 Fourth, Third Major, Third Minor, Sixth 
 Major, Sixth Minor, and Tone Major, and 
 
 Tone Minor. 
 
 And then, all the Hemitones, zn&DieJis, 
 and Comma, are found by the Differences 
 of thefe, and of one another; as hath been 
 Jhewn at large. 
 
 Now, certainly, this is much to be pre- 
 ferred before any Irrational Contrivance of 
 expreffing the feveral Intervals. The Ari- 
 Stoocenian Way of dividing a Tone [ Major ] 
 into twelve Parts, of which 3 made a Die- 
 fis, 6 made Hemitone, 30 made Diatejja- 
 ron, (as hath been faid) might be ufeful, 
 as being eafier for Apprehenfion of the In- 
 tervals belonging to the three Kinds of 
 Mufick j and might ferve for a leaft com- 
 mon Meafure of all Intervals (like Mr. Mer-* 
 cators artificial Comma) 72 of them being 
 contained in Diagafon. 
 
 But this Way, and fome other Methods 
 of dividing Intervals equally, by Surd Num- 
 bers and FraBions, attempted by fome mo- 
 dern Authors ; could never conftitute true 
 Intervals upon the Strings of an Inftrument, 
 
 nor 
 
1 50 Condufion. 
 
 nor afford any Reafon for the Caufes of 
 Harmony, as is done by the Rational Way 5 
 explaining- Confonancy by united Motions, 
 and Coincidence of Vibrations. And tho* 
 they fupposM fuch Divifions of Intervals ; 
 yet we may well believe, that they could 
 not make them, nor apply 'em in tuning a 
 Mufical Inftrument-, and if they could, the 
 Intervals would not be true, nor exa£t. 
 But yet, the Voice offering at thofe, might 
 more eafily fall into the true Natural In- 
 tervals. Ex. gr. The Voice could hardly 
 exprefs the ancient Ditone of two femes 
 Major ; but, aiming at it, would readily 
 fall into the Rational Confbnant Ditone of 
 5 to 4, con fitting of Tone Major and Tore 
 Minor. It may well be rejected as unrea- 
 sonable, to meafure Intervals by Irrational 
 Numbers, when we can fo eafily difcover 
 and aflign their true Rations in Numbers, 
 that are minute enough, and eafie to be 
 underftood. 
 
 I did not intend to meddle with the Ar- 
 tificial Part of Mufick: The Art of Con> 
 poling, and the Metric and Rhythmical 
 Parts, which give the infinite Variety of 
 Air and Humour, and indeed the very Life 
 to Harmony • and which can make Mu- 
 Jick, without Intervals of Acutenefs and 
 Gravity, even upon a Drum •, and by which 
 
 chiefly 
 
Conclujion. i 5 l 
 
 chiefly the wonderful Effefts of Mufick 
 are performM, and the Kinds of Air diftm- 
 guifh'd • as, Almand, Corant, Jigg, &c. 
 which varioufly attack the Fancy of the 
 Hearers ; iome with Sprightfulnefs, fome 
 with Sadnefs, and others a middle^ Way : 
 Which is alfo improved by the Differences 
 of thofe we call Fiat, or Sharp Keys ; the 
 Sharp, which take the Greater Intervals 
 within Diafafon, as Thirds, Sixths, and 
 Sevenths Major ; are more brisk and airy; 
 and being affifted with Choice of Mea- 
 fures laft fpoken of, do dilate the Spirits, 
 and rouze 'em up to Gallantry and Magna- 
 nimity. The Flat, confuting of all the left 
 Intervals, contrad and damp the Spirits, 
 and produce Sadnefs and Melancholy. 
 Lailly, a mixture of thefe, with a fuitable 
 Rfyytbmw, gently fix the Spirits, and com- 
 pofe them in a middle Way : Wherefore 
 the Firft of thefe is eall'd by the Greeks 
 Diafialtic, Dilating ; the Second, Systaltic, 
 Contra&ing; the Laft, Hcftchiafiic, Ap- 
 peafing. 
 
 I have done what I defigrfd, fearch'd 
 into the Natural Reafons and Grounds, 
 the Materials of Harmony ; not pretend- 
 ing to teach the Art and Skill of Mu- 
 fick, but to difcover to the Reader the 
 foundations of it, and the Reafons of the 
 
i 5 1 Conctufion. 
 
 Anomalous Thce?iomena, which occur in 
 the Scales of Degrees and Intervals ; which 
 tho' it be enough to my Purpofe, yet is 
 but a fmall (tho' indeed the moft cer- 
 tain, and, confequently, moft delightful ) 
 Part of the Philofophy of Mufick; in 
 which there remain infinite curious Dif- 
 quifitions, that may be made about it ; 
 as, what it is that makes Humane Voices, 
 even of the fame Pitch, fo much to differ 
 one from another? (For tho 7 the Diffe- 
 rences of Humane Countenances are vifi- 
 ble, yet we cannot fee the Differences of 
 Inftruments of Voice, nor confequently of 
 the Motions and Collifions of Air, by 
 which the Sound is made. ) What it is 
 that conftitutes the different Sounds of 
 the Sorts of Mufical Inftruments, and 
 even fingle Inftruments ? How the Trum- 
 pet, only by the Impulfe of Breath, falls 
 into fuch Variety of Notes, and in the 
 Lower Scale makes fuch Natural Leaps 
 into Confonant Intervals of Third, Fourth, 
 Fifth, and Eighth. But this, I find, is 
 very ingenioufly explicated by an honou- 
 rable Member of the R. S. and publiflhM 
 in the PhilofofhicalTranf actions, N Q 195. 
 Alfo how the Tufa-Marine, or Sea-Trum- 
 pet ( a Monochord) fo fully expreffeth the 
 Trumpet ; and is alfo made to render 
 other Varieties of Sounds ; as, of a Vio- 
 lin,. 
 
Conclufion. i 5 5 
 
 lin, and Flageolet, whereof I have been 
 an Ear-witnefs? How the Sounds of 
 Harmony are received by the Ear ; and 
 why fome Perfons do not love Mu- 
 fick? &c. 
 
 As to this laft • the incomparable 
 Dr. Willis mentions a certain Nerve in the 
 Brain, which fome Perfons have, and fome 
 have not. But further, it may be con- 
 fider'd, that all Nerves are compofed of 
 fmall Fibres ; Of fuch in the Guts of 
 Sheep, Cats, &c. are made Lute-Strings : 
 And of fuch are all the Nerves, and a- 
 mongft them, thofe of the Ear, compo- 
 fed. And, as fuch, the latter are affe&ed 
 with the regular Tremblings of Harmonic 
 Sounds. If a falfe String ( fuch as I have 
 before defcrib'd ) tranfmit its Sound to 
 the beft Ear, it difpleafeth. Now, if there 
 be found Falfenefs in thofe Fibres, of which 
 Strings are made, why not the like in 
 thofe of the Auditory Nerve in fome Per- 
 fons ? And then 'tis no Wonder if fuch 
 an Ear be not pleas'd with Mufick, whofe 
 Nerves are not fitted to correspond with 
 it, in commenfurate Impreflions and Mo- 
 tions. I gave an Inftance, in Chap. Ill, 
 how a Bell-Glafs will tremble and eccho 
 to its own Tune, if you hit upon it : And 
 I may add, That if the Glafs iliould be 
 
 irregu- 
 
i 54 Conclufion. 
 
 irregularly framed, and give an uncertain 
 Tune, it would not anfwer your Trial, 
 In line, Bodies mull be regularly framed 
 to make Harmonic Sounds,, and the Ear re- 
 gularly conftituted to receive them. But 
 this by the by ; and only for a Hint of 
 "Enquiry* 
 
 I was faying, That there remain infinite 
 Curiofities relating to the Nature of Har- 
 mony, which may give the molt Acute 
 Philosopher Bufinefs, more than enough, 
 to find out ; and which, perhaps, will not 
 appear fo eafie to demonftfate and explain, 
 as are the Natural Grounds of Confonancy 
 and <r Di$ona?icy. 
 
 After all therefore, and above all, hy 
 what is already difcovefd, and by what 
 yet remains to be found out; we cannot 
 but fee fuiBcient Caufe to rouze up our 
 beft Thoughts, to Admire and Adore the 
 Infinite Wifdom and Goodnefs of Almighty 
 G o d. His Wifdom, in ordering the Na- 
 ture of Harmony in fo wonderful a man- 
 ner, that it furpaffeth our Underftanding 
 to mate a through Search into k^ tho' (as 
 I faid) we find fo much by Searching, as 
 does recompenfe our Pains with Pleafure 
 and Admiration. 
 
 And 
 
Conclujion. 155 
 
 And his Goodnefs, in giving Mufiek 
 for the Refrefhings and Rejoycings of Man- 
 kind ; fo that it ought, even as it relates to 
 Common Ufe, to be an Inftrument of our 
 Great Creator's Praife, as H e is the Foun- 
 der and Donor of it. 
 
 Bu t much more, as 'tis advanced and 
 ordain'd to relate immediately to his Holy 
 Worfhip, when we Sing to the Honour 
 and Praife of God. It is fo Effential a 
 Part of our Homage to the Divine Majefty, 
 that there was never any Religion in the 
 World, Pagan, JewiJJ), Christian, or Me- 
 humetan, that did not mix fome Kind of 
 Mufiek with their Devotions ; and with 
 Divine Hymns, and Inftruments of Mufiek, 
 fet forth the Honour of God, and cele- 
 brate his Praife. Not only, Te decet Hym~ 
 nits 'Beus in Si on, (Pfal. 65.) but alio — ■ 
 Sing unto the Lord all the whole Earthy 
 (Pfal. 96.) 
 
 A n d it is that which is incelfantly per- 
 formed in Heaven, before the Throne of 
 God, by a General Confort of all the Holy 
 Angels and the BieiTed. 
 
 In fhort, we are in Duty and Gratitude 
 bound to blefs God, for our Delightful 
 
 Refrejh- 
 
i *)6 Conclufion* 
 
 Refrejhments by the Ufe of Mufick ; but 
 efpecially, in our Publick Devotions, we 
 are obliged by our Religion, with Sacred 
 Hymns and Anthems, to magnifie his Holy 
 Name ; that we may at laft find Admit- 
 tance above, to bear a Part in that Blefled 
 Confort, and eternally Sing Hallelujahs and 
 Trifagions in Heaven. 
 
 FINIS. 
 

 
 159 
 
 RULES 
 
 for 
 
 A 
 
 Playing 
 
 TROROW-BASS, 
 
 By the late Famous 
 
 Mr. Godfry Y^gller. 
 
 / 
 
 MUS ICK confifts In Concords, and D 1 fiords -, the Concords are Fdttr* 
 and of two kinds, viz,* Perfeft and Imperfett ; the Perfeft arc 
 the $th and %th ; the IrnperfecT: the id and 676. The Dlfcords are 
 Three, «u/£. the 2^, $th f and fr& ; the gth being the fame with the 2d, 
 bur differently accompany 'd. 
 
 The Flat IrnperfecT: $th is ufed, either as a Concord or Difcord, but mod 
 commonly like the latter. 
 
 The following Scheme, (the Treble afcending by Semitones) (hews all 
 Concords and Dijcords, as they ftand with regard to the Baft. 
 
 By Chords is meant either Concords or Dlfcords j by Semitones is meaitt half 
 Notes. 
 
 b* #^ b3 $3 b4 $4 S b6 #6 b7 $7 8 b? #9 
 
 r=dip.zp|zDzj 
 
 
 pzp::zi 
 
 Tt? — 
 
 There are other chords us'd fometimes, as the j&r* 5th, &>. but thdfc 
 Ihall be treated of hereafter. 
 
 In common Chords which are the id, Mh, and 8/& avoid the taking two 
 5^ or two Sths, together, not being allow'd either in Playing of Com* 
 pofition; and the beft way to do ic in pkying, is to friovc }6ur Hands 
 contrary one to the other. 
 
 When the firft common Chord you take is the 3d, the next : miift be tht 
 %$h and %th, and fo vise vcrfa, as the following Scheme will illuft'rate. 
 
l6o 
 
 Example of Common Chords C 8 I 5 j 3 
 different? taken* -s 5 1 5 
 
 j&g/ft /w ^ Tliorow-Bafs* 
 
 ill: 
 
 3-R-l-e-e- 
 
 
 __g O 1 jb> g {, 
 
 The Sixth may be taken With the third and C<5 J 3 
 
 eighth, ia full Playing the following feveral ■< 3 
 
 e 
 
 6 
 3 I 
 
 \ Example nf Common Chords ^^ Sixes, taken the feveral ways ahovt |( 
 
 mentioned 
 
 \ 
 
 _ M-4-ii t J^iiii 
 
 I! In 
 
 
 On any Note where nothing is marked, common Chords are playl|. 
 
 In Sixes moil be obferv'd that when the Bafs is low, and requires a nati 
 
 2fal flat 6ilh you moft play two fixes and one third ; if the Bafs is hii 
 
 snd requires a natural flat 6th, play two thirds and oqe fixth ; if 1 
 
 3d or 6th happens to be {harp inftead of two fixes or two third 
 
 * I Alfo in Divisions where a fixth is required, infte 
 ^ 1 1 1 of two thirdsj or two fixes* play the fame. 
 
 Aflat or fbarp maik'd over or under any Note in the Bafs, fignifiejj 
 flat or fbarp third tv be pky'd : A flat ox fbarp markM before a Note [ 
 figiare, figniiies that Not® or Figure to be play'd flat or fbarp 
 
 Flay 
 
 * oin 
 
Rules for a ' Thorow-Bafs, 
 
 161 
 
 ; Example. 
 
 It All Keys are known to be fiat or jharp, not by the fiats or /W^ plac'd 
 
 Tjthe beginning of a Leflon; but by the third above the Key, for if your 
 
 -tThird is Flat, the Key is Flat ; if your Third is Sharp, the Key is SA*r/>. 
 
 I All jharp Notes naturally require /**. Thirds, all flat Notes require 
 
 Marp Thirds \ the fame Rule hold as to Sixes. 
 
 I B, E, and A are naturally [harp Notes in an open Key ; F, C, and Q 
 
 ire naturally Hat Notes in an open Key. 
 
 K Difcords are prepared by Concords, and refolved into Concords, which 
 
 re brought in when a part lies {till, and are fometimes ufed in contrary 
 
 i lotion. 
 
 I There are three forts of Cadences, or full Clofes, as when the Bajs falls $ 
 
 jjth, crrifesa4th, *//« the Common C^w^ ; the 6th and 4th Cadence ; 
 
 fad the great (or fulleft) Cadence. Each of thefe may be accompany ? 4 
 
 fcdifferCBt wafs ; as will be feen by the following Examples. 
 
 The Common Cadence. 
 
 £ 
 
 b 7 
 
 #3 
 
 8 b? U#3 
 5 5 18 b 7 
 
 C.8 8 b7 
 he 6tb&xi&. tfh Cadence. <fi 5 .1 
 
l6i 
 
 Rules for a Thorow-Bafs* 
 
 5 
 
 ?S3 t 7 ir 3 44t3 
 
 The great tiuknti. 2 J 6 5 ,. "> 8 8 8 b 7 
 b )*3 44 f 3 7 6 5 5 
 
 €. ( 5 
 
 7 6 S J 
 
 5 44$! 
 #3 8 § b 7 
 
 8 
 
 — j. J J, J % 44t4-i 4d ^ ^4^- J 
 
 
 p. ».—*-. 
 
 -:|=— :z|:pt|zpzg:|:pri:: 
 — .— « -TT-Ktrzhzi 
 
 In al! Cadences whatfoever, where the Bafs rifesa 4th, or falls a 51 
 Obferve, that the 4th falls half a Note into a /harp Third* m& the 81 
 a whole Note into a j?*i? 7th. 
 
 There is another Cadence call'd the 7th and <5ch Cadence* which 
 counted but a half Clofe, and if the 6th is fiat , is never tifed for a fin 
 Clofe, becaufe it does not fadsfy the Ear, like as when the Bafs falls 
 5th, or rifes a 4th, 'tis often introduced in a piece of Muiick, as tl 
 Air may require ; and when it ends any one part of a piece, 'tis in ord 
 to begin a new Movement or Subjtft : The 7th and /harp 6th may be ui< 
 for a final Clofe, if the Bcfigti of the Compofer requires it* but 'tis vei 
 rarely done. 
 
 %he following Example mil ftsew how hoth the jtl and \>&th % and Jth, and ^o* 1 
 
 are us'd. 
 
 1 3284-' 
 
 |-^=|E^%=^pfei=^n=J 
 
 
 7#6 
 
 
 7b6 
 
 
 ^ipfzfcf-p^zlz^zzp 
 
 I ,,_^ . I 7#< 
 
Rules for a Thorow-Bafs* 
 
 16$ 
 
 Obferve when a Difccrd happens in the higher N«te, leave the Cno- 
 I cord out afeove it. 
 
 CThtFffi 
 I < ^th and 
 I ,/fecond. 
 
 \ 
 
 CThe Flat 
 «? $th and 
 [(jSth join'd. 
 
 6 b3lb5 
 
 b$i 6 3 
 
 3lb5! 6 
 
 here inftead of the 6th, the 
 %th may be added, but 
 then it ought to be mark'd- 
 
 here the Sth is o-C The perfeQ 
 mittedunlefsitbe^ <>th and 6tb 
 tin paffing Notes, cjoin'd. 
 
 The Sharp \ 6 J 2 
 4 f£and <# 4 6 
 fecond. 1 2 I $4 
 
 6 
 5 
 
 #4 
 2 
 
 6 
 
 here the 8/& 
 
 if one think 
 
 5 6 fictoplayfull 
 
 maybe added. 
 
 The perfeS fifth when joyn'd with a fixth is ufed like a Difcord. 
 
 Example. 
 
 4 * 
 
 
 
 4#3 _ 
 
 t 
 
 l#7 
 J 4 
 
 » 2 
 
 The 5£«rp 
 
 7^, when 
 the B^// 
 lies ftiil. 
 
 The gth re-\3 
 folving into \ 9 
 the 8*6. ( $ 
 
 2 
 
 #7 
 
 4Jhere to play f When the 3^ 7 
 2 full the Flat < and 4*& are 4 
 $7|67& may be (mark'd above 3 
 added. one another* 
 
 here inftead of 
 the Jth Sharp 6th 
 may be ufed, but 
 then it ought to 
 be mark'd. 
 
 9 8 The 6th and 4*JK6| a U 
 
 5 (when the B^/>de»< 4I6U 
 
 8' 3 [fcendsby degrees. C.2I4I6 
 
 The 6th and $th i 6rS 
 when the B#fs ) 46 ^ 
 skips or lies ftili. <! S^p 
 
 
 
 * *4 
 
 V 7 6 7 fi- ' f ..''-..■ > 
 
 
 
 
 Hv* 
 
 4-6-ilI 
 
 Tfif. 
 
■26$ 
 
 Rules for a Thorow-Bafs. 
 
 The ith and %th hap- r 7 1 3 
 pening juft before < 5 7 
 the Cadence Note. v. 3 \ 5 
 
 5 I here inftead of the third, the ninth if 
 3 prepaid may be ufed but then it ought 
 7 ! to be mark'd. 
 
 The extreme Flat feventh> ^ b? 3 j bs J the extreme FA*# feventh Is 
 and tfU* fifth happening jufU b$ b? 3 the fame with the Sharp 
 before the Cadence Note. V 3 | b5 | b; j fixth. 
 
 #2 
 
 4 1 the extreme Sharp fe- 
 I ^2 cond is the fame dis- 
 and fourth. ^ $2 f 4 | ' & j tance as the Flat third- 
 
 The extreme r ^ 
 
 ■&fcar£ fecond^ ti 4 j 6 J ^2 j cond is the fame'dis-^ Ex, as follows. 
 
 
 (S!« 
 
 f-v- 
 
 a=a 
 
 > 
 
 is.: 
 
 t— 
 
 4*3 __, 76 _*_« 
 
 ■_[ .*L_ft. 
 
 * 
 
 """"* — 'c_ — r" ~t ~ 1 
 
 _fe -'_4fe G _|_f -*S-w-4 Iff** • 
 
 I /is. PZ**. 
 
 DZOZ| 
 
 
 The 4 f & and 9^ 
 refolving into 
 the id and §th. 
 
 
 The $th ani jth C9 8 
 refolving into -s7 6 
 the 3^ and £*/&• £3 
 
 7<5 h 
 
 98)76 
 
 ttJ3:l^:=^™1^ 8 * 
 
 .4J «JS- 
 
 33 
 
 __j_ _. 
 
 .jizzzP 
 
 #;•"'.:.-■ 
 
 
 9 8 
 
 
 t-G-r 
 
 /||]fV *— 
 
 When the i*/> afcqnds or defc.ends one or two Notes, move your 
 Hands in playin" contrary one to the other. 
 
 After a fixth where ■ np eighth* fc concerned , pu nia ; y either afcend or 
 defend together. ' " 
 
Rules for a Thordw-Bafe. 
 
 16% 
 
 After a fixth where the eighth lies in the middle, you may either a£» 
 cend or defcend. 
 
 Example. 
 
 -B._-a._t: rpp-puf fS_ps— ' 
 
 6 6 
 
 l3 f^±te_J__£:li«- ^-ftft.l-f- — :z 
 
 - - ^-.-h-P-*— 
 
 
 _zbzczzt 
 
 __y_yy^yy_y_^_3=s 
 
 jpp-zlZszz- 
 ~:£z_z[:zp:: 
 
 6 4 3 
 
 ■■»— t— r — ■ r— t- •*— r*— ' -a — 
 
 6b* 
 
 -- 1— 
 
 0z 
 
 J — *P- 
 
 I 
 
 
 
 ittitS 
 
 ^-z**3ziZfcZiSzz(L^izi_a_ntn: 
 
 . _ Jll 4 # 3 _ 
 
 Example ofpajfmg Notes in Common-time. 
 
 ;-M-™ 
 
 P-r-f-6 
 
 ^pZzqddz-zzs:|:E^^ 
 
 — — — tra-ig^z._^_u -_:-j^~-wg~-c j^j_ 
 
 
 
 T", izziziici'-if tzzzi tzzrii HI " 
 
i66 
 
 Rules for a Thorow-Bafs. 
 
 t:£et:3^EEISEEd 
 f 
 
 E® ; 
 
 ^bfeaM 
 
 £EEztgb:}:t&2:_ 
 
 4 5 
 
 £S==Se| 
 
 6 
 
 "=tff 
 
 ^, 
 
 SKT 
 
 Ei 
 
 K*P 
 
 7 
 6 6 6 4 
 
 ± . .* . m i . J^fl #__ ( 
 
 _ V 9 
 
 = tttrfe 
 
 fM* 
 
 _S ^ _4^ _6 _ 
 
 i 
 
 pH-4- 
 
 6 
 
 4 
 
 - , 3zznt'"zzzt;zfc~izt" , ""~c~" 
 7 6 -#__*•; 
 
 
 I 
 
 
 tfLftZfE^.^^.. 
 
 4£3 
 
 <i~ ztpzC ccceKz ::EZZzziqzjz|ZEzzr "zQjzl'iff *Ci"l 
 
 -^btBzt^ 
 
 -^5i tptr* 5 Tar r-i- ~ tt-qi 
 
 
 :gzzz:gzz:[zzEZz:zjzz::zpz: 
 
 r- — r- — — — —— — > — ft-' 
 
 -9- 
 
 r^I 
 
 jqrtazzziz!: 
 
 -r 
 
 * j- « jlj1i 
 
 Ci 
 
 6 
 
 i-JZg: 
 
 i 
 
 6 6 6 6 
 
 1- 
 
Rules for a Thofow-Bals*' 
 
 iS-f 
 
 6 
 3 
 
 _*_J - i^ m 
 
 M-F- • 
 
 — 
 
 ; jfgg zg -rr 
 
 3 -p- , S> . *_3 ^— — 
 
 U # -3ji 
 
 Bxm^h of p offing Notes in Triple-time? 
 
 H*»fl 
 
 
 
 ¥*s- — n 
 
 gzF=:j : ~— f ~ 
 
 -# 
 
 *# 
 
 en— f— : 
 
 
 f~ 
 
 
 1— zfezziteCcrf^szpiEi 
 
 T -j- t«< 
 
 
 g 
 
 W t 
 
 _6 4J_ 
 
 
 bf 
 
 
 
168 
 
 Rules for a Thorow-Bafs, 
 
 
 __*__._«__ 
 
 #6 
 
 6 
 
 pf — i 
 
 
 6 < 
 
 L&2 
 
 _ 
 
 
 
 
 
 
 I 
 
 
 
 
 
 
 _|I«~ 
 
 E-— 
 
 ._" 
 
 -H„— ^ 
 
 — _.._ 
 
 * 
 
 i 
 
 , 
 
 
 '''ZiJz'ml^S 5 
 
 0/ Natural Sixes, 
 
 VUy common Chords en all Notes where the following Rules imit i 
 fe£t you otherwife. 
 
 The natural Sixes in a Sharp Key are on the half Note below the Ke 
 the third above the Key, and on all extraordinary Sharp Notes out of tl 
 Key, if not to the contrary mark'd, or prevented by Cadences* 
 
 The natural Sixes in a Flat Key are on the Note below the Key ; t 
 Note above .the Key ? and on all extraordinary Sharp Notes out of 1 
 Key, if not to the contrary mark'd, or prevented by Cadences. 
 
 When the Bafs either in a Flat, or a Sharp Key, afcends or defcem 
 half a Note, Sixes are proper on the xxrft Note 3 falling on the fecond, ui 
 lets prevented by a Cadence. 
 
 When the Bafi either in a Flat or Sharp Key„ defcends with a commc 
 Chord by thirds ; Sixes are proper on the falling thirds. 
 
 When the Bafs either in a Flat or a Sharp Key, afcends with a commi 
 Chord by thirds : Sixes are proper on the rifing thirds. In a Fiat Key tl 
 third above the Key generally requires a fixth to prepare the Cadence* tl 
 fifth being repugnant to tht half Note below the Key. 
 
 Seldom two ~ Notes aftend ©r defeejad but one of them hath i 
 
 .;£m 
 
 
 
Example of Natural Sixes and proper Cadences in a flwp 
 
 Key, 
 
 Rules fir a Thorow-Bafs. 
 
 169 
 
 J 6 S ~ 
 
 *~ 
 
 Z. |L 
 
 V£ 
 
 y =^zz:j " ^:j:=g===:D-^ I z jzzrDzzDzzrfc 
 si iz^zzis rj: zsidizzza zz £~: z^i3izi3 : 3:3z: 
 
 t -f-f - -Ff P *f— f '- F '-r-F # ? -f- 
 
 fcE=lfc=E= 
 
 ~Xx «~ — i."d — ■' 
 
 — s-- 
 -p- 
 
 
 ZZ3ZZiS? 
 
 
 
 ft- «JBSt.ft.^ 
 
 zzzfczEBrfz: 
 
 f o 
 
 * 3 
 
 -e- 
 -e- 
 
 :p: 
 
 u^ 
 
 1 .,,..- . . 
 
 *«*——— '§f-***«p«aw 
 
 C 3 
 
 Mxam* 
 
n? 
 
 Rules for a Thorow-Bafs. 
 
 
 Example of Natural Sixes and proper Cadences in a M 
 
 Key. 
 
 §£z-«zz:»zz"» zz?~ qtrrfz*3zriz a.*jirr? 1 
 
 
 Ig-Czpzpizt:: zp^ZEzzz z3Z£c:i:zzt:gzcz^:C:: 1 
 
 
 4 3 ->~ 
 
 
 **«-r*^p-^|-^ , | «— »• ■^^»^^^^'^ffl^^^*^^|^^ , * i a ' , ^ 4 "^ ^'**, 
 
 Nov 
 
Rules for a Thorow-Ba&v 
 
 m 
 
 Now all thcfe Sixes mentioned either in a Flat or Sharp Key, are not 
 
 only to be obferved in the Key you play in, but likewife in all other 
 
 Cadences you are going into : And for the time you keep in that Cadence^ 
 
 j obferve the Rules for Sixes as tho' you were in the Key your Leflon is 
 
 1 Compofed in. 
 
 Where the 5*/j afccnds a perfcft fourth, or defcends a perfect fifth, 
 ' ! Sixes arc generally left. 
 
 Other Rplcsfor Sixes are tvhere the Bsfs moves by degrees downwards* 
 | then thefe Sixes may be play'd on every other Note. 
 
 (Example. 
 
 The Compofer (efpecially in few parts) may Compofe as many Sixes 
 cither afcending or defcending by degrees as they think fit, but then 
 they ought to be marked. 
 
 ,-j 
 
 -i-L-A-A 
 
 Example. 
 
 6 
 ZflZ ft. J 5 «J 
 
 6-6 6 6 
 
 
 =Jt^b^^g 
 
 ■ *"""" *r m v- m :Z i ~~j~ i ^Jn'SZ. 'i V *-"" " "" " " "" """ ' •*" 
 
 6 .6 
 
 
 I 
 
 Now 
 
i 7 2 
 
 Rules for a Thorow-Bafs. 
 
 Now here follows an Example where two Sixes are abfolute aecejti 
 ty, and tnat defending befcaule they are fhort Cadencis inflaad of 7. 6 
 
 mark a. ' 
 
 In a fiat Key Defending. 
 
 
 
 Jfcendfag. 
 
 l=ii3lli^ilil|lllli 
 
 jfo a Jharp Key Defending. 
 
 
 U- 
 
 iHiiiiiiiliipipii 
 
 Afcen&hig. 
 
 
 ■**■ 
 
 6 6 
 
 a=p=Eli= 
 
 6 bS 
 
 H — — - ■ 
 
 w— '■ 
 
 When 
 
Rules for a Thorow-Bafs. 
 
 175 
 
 When the Bafs lies ftill, the Seventh is generally refolded Into tie 
 6th, and the 9th into the 8ch. The Example which follows, &ew$ 
 how Difcords may be Refolved feveral ways. 
 
 E xamfle. 
 
 z^i 
 
 jjfcj^ —$l •_!#? -j- 6 ~1 J L 7 L 7 1.1 J 7 
 
 
 nra 
 
 zffnrz 
 
 Several Examples of what may be done when the Bafs Defends 
 by degrees. 
 
 The Common so ay. 
 
 _p4— p p— i 
 
 I .z|i_p_j I " 
 
 Zcj. 
 
 76 
 
 *= 
 
 gz=::zd=:|| 
 
 J _. 
 
 Natural and Artificial, 
 
 16 76 
 
 
 -g- q 76^ 56 56 76 76 
 
 
 —pzzj 
 
 All 
 
*74 
 
 Rules for a Thordw-Bafs. 
 
 A% Artificial. 
 
 ^-rrE~:E==:l=zp-:p:-:}r=-6:z:^:=:fcz:p:z3g~J: 
 
 $6 76 
 
 
 • 
 
 When the Bafs Abends fy Degrees* 
 
 
 55 „ _!£„ _il;-._2 5 — -J 6 ™ _ f6 _ Q ~*S~ 
 
 
 76 6 98 7^ q ~P~- 
 
 Example of all forts of Difcords in a fiat Kcy a 
 
 l—j. — „ J-. . 
 
 [z:Czo:?izip;-L. 
 
 _ — B . 
 
 I—, 
 
 zzzuz 
 
 
 :cx 
 
 ■ _ 5 _ 6 — 7 *L i~t £fe; 
 
 6 S 
 
 4(54)4^ 
 B - 
 
 ™o~:; 
 
Rules for a Thorow-Bafs. 
 
 
 7 7^ 
 
 s 2 
 
 .zE=q: 
 
 54 « 
 
 6h? 
 
 zizizzzzzt 
 
 
 
 
 "l=izj:r:d=r- 
 
 ___4gjfe ffi3 4 6b5 | 4 6* J4 
 
 il^lillilfi 
 
 :qz — w 
 
 ~B~ 
 
 J> _ 4 jp 
 
 ^.— -jH^^n ~j^"iZ 
 
 
 
 r 
 
 •L&ra 
 
 rare: 
 
 _fe_t_L — t: 
 
 3fzi~ •ziLJ 
 
 I ? *4 „ 
 
 4 3 J -p- — #•-- 
 
 ?-f-| 
 
 6 6 
 
 i-f 
 
 
 
 -+ 
 
 _£_£-J#.JtL_ £_i_i#l. 
 
 ^r =: P- :1 ~Ft f r :| 
 
 O 
 
is© Rules for a Thorow-Bafe. 
 
 -r-r— f- rn-f 
 
 t-i— «■©— 
 
 Ef EF"f" f 
 
 35HJ#* 
 
 V 
 
 P "ft" 
 
 zacsz 
 
 f-p-f-f : t-t : "F-f-~-fc- 
 
 
 £ 
 
 - "« 6< .#5— i. 4 iL_-,_Ji 3 . 
 
 ***-*— H 
 
 #>" 
 
 ^ -f *£f 
 
 fE=EE:E==~ 
 
 tp- 
 
 87 6 
 #3 4(54) 
 
 <#3*>#3 
 
 
 |**» 
 
 Where the Figures are fet in Parent hefts* thofe I wou'd have only drop* 
 to fee off Playing* 
 
 Example of all forts of Difcords in a fiarp Kej« 
 
 ^-^ : Wi-ri^W 
 
 6 S 
 
 5454) 4(3*) 
 
 7 7 7? 
 
 a-Czz — 
 
 a: 
 
 
 43 4b3 
 
Rules for a Thorow-Bafs. 
 
 151 
 
 . L , 76 u.?** ,7^5 .,76 5- 
 
 J4 J5 J#j_ 
 
 -©- 
 
 -©- 
 
 
 7 6 S 76 y 5J6 4, 
 
 , ?B 4b i.-t, 54 _ 4 l_ -J ' gi it^t it? 4 r 62 
 
 rEzr:|z=-:=j:^=^=e|:^z£:3; 
 
 
 S 6 S 
 
 
 o 
 
itfci 
 
 Rules for a Thorow-Bafs. 
 
 fi^lE=lilii=fey: 
 
 
 fe#^jEfeJEEtSE=dEE 
 
 ^4r(^$=p5=E^=f=fj 
 
 < 
 
 6 7 6 
 
 7 6 9 6 7 6 6 45 
 
 ^ilPipgilpi 
 
 EEkiV^^ 
 
 ff 
 
 
 uzpr 
 
 6b? 
 
 I i-E K2 
 
 §g= 
 
 ISAfe 
 
 43 
 
 pi: 
 
 0— # 
 
 j-'**—" t^ 
 
 ^11 
 
 r^p^ 
 
 
 7 6 761 
 
 IS 
 
 Q:z-g: 
 
 II 
 
 1- B 
 
 ifrrr 
 
 -e- 
 
Rules for a Thorow-Bafs. 
 
 i8 3 
 
 To make feme Chords eafie to your memory, you may obferve as 
 follows: A common Chord to any Note makes a 3d* 6th.and 8th. to the 
 1 3d- above it, or 6th- below it» A common Chord makes a 4th. 6th. and 
 j&th. to the 5th- above It, or a 4th. below it. A common Chord makes 
 a 3d. 5th* and 7th. to the 6rh. above it, or a 3d. below it. A common 
 Chord makes a 4th. 6th. and 2d* to the 7th- above it, or a 2d. below 
 it. 
 
 A 3d. and 4th marked makes a common Chord to the Note above it, 
 
 obferving the <>th. perfect or imperfeft, according to the Key, as alio an 
 
 8th. 3d. and 6th. to the 4th. above it, or 5 ch- below it. A Sharp ych. 
 
 marked, where the Bafs lies ftill makes 8th. 3d, and Jharp 6th. to the 
 
 Note above it, and 5th. 7th. and jharp 3d. to the Note below it, An 
 
 .cxtream jharp 2d. and 4th. marked on a Flat Note, makes jharp 3d. 5 th- 
 
 ind 7th. on the half Note below it, as alfo a jharp 6th, 8th. and 3d. to 
 
 the jharp 4th above it, ot flat 5th below it ; the flat 5th. and extream/a* 
 
 7th. marked on a Sharp Note, makes 3d. fiat 5th. and 8ch. ro the 3d, 
 
 above it, or the 6th. below it, as alfo an 8th. 3d. and 6th- to the flat 
 
 5th. above it, or Sharp 4th. below it. The 4th or 9th, mark'd Is the 
 
 perfect 5 th. 6th. and 3d. on the whole Note below it, and the flat 5th. 
 
 6th ? and 3d, on the half Note below it, as alfo 3d. 7th and oth to the 
 
 3d. above it, or 6th. below it, the 9th. and 7th mark'd Is the 5th. oth* 
 
 and 4th. on the 3d. below it, and the 6th. 3d. and perfeft 5th- to the 
 
 perfeQ: 4th. below it, or the 5th. above it, and 6th. 3d. and flat 5th, 
 
 on the perfeft 4th. below it. 
 
 % > 8 7 6 8 8 
 
 6 6 S 4 5 6 f 
 
 3 14 J* p 4-3.4 J^J_. 
 
 ctoic, - 3:1 ^ tefcE3:s£ i! 
 
 Q-J3 — -ZiZXi- - li. :__'i_^.._: :il?:_ b _: :to LJb :zc__i ., b 
 
 L 9 , 
 
 7 #2.-0- b 7 I b 7 
 
 ■tr" 4 b L d L 
 
 ZIj5I||1qZZ-..| 3-. 
 
 9 M 
 4 
 
 _-PI 
 
 - ( — £- 
 
 — „^_U-4 
 
 9 4 9 6 9,6 
 
 7 9 . 7 S 7 _b£ 
 
 Thzflat *>th. and y&*r£ 4th. the extre&m jharp 2d. sad flat 3d. The 
 
 extream yto 7th, and 'jharp 6th- upon any fretted Inftruments, or Harp- 
 
 ficord. without quarter Notes, are the fame thing in diftance, yet th§ 
 
 diftin&ion is as follows 
 
 i } , 1 ■ ' . j 
 
 gzzp=jp 5 zzS^ 
 
 b$th 9 
 
 ^4^ 
 
 bsth. 
 
 i" 
 
 ^ai. 
 
 b 3 d. 
 
Rules for a Thorow-Bafs. 
 
 #6tb. 
 
 g:^ ~: ^eS-— q— j jffpJ 
 
 b;th. 
 
 r 
 
 #6tfa. 
 
 Hi- 
 
 There are fome other Chords of the fame kind, Viz. the extream flat jf 
 4th. being the fame as the flyarp 3d. and the extream fharp 5th. the fame '.! 
 as a /&** 6th- but thefe Chords are only ufeful in three Parts, and will 
 
 not admit a 4th. the dliiin£ilcn is as follows, 
 
 
 | — ~ 
 
 b4tb e iff^d. 
 
 b4.th. 
 
 
 * 
 
 d-h<L*d-!^::. 
 
 I 
 
 
 #5th. b^th. ^5th. b<5th. 
 
 This extream /^ 4th- admits a 6th- for the fecond Part and is re* 
 folv'd" into the third, and the 6th. into the flat $ch- The extream fharp. 
 5th. admits for the fecond. Part a third. 
 
 Of Trafifpofition. 
 
 Before any one can pretend to Tranfe-pofe from one Key into ano« 
 iher, it is neceiXkry they fhou'd know all the Flats and Sharps naturally 
 belonging to ai! ? at leaft the Practicable Keys. 
 
 Ab3 F#3 Db3 G#3 Eb$ B#3 
 
 
 c iB 
 
 vszzz zKzzzzi 
 
 
 ob 3 r>f 3 
 
 Bb^ 
 
 
 Ef 3 
 
 lfcn 
 
 Cb3 
 
 A#3 
 
 t^Elrfr:' 
 
 Fb3 A^.3 Fbj EJ:? Cbj. 
 
 X x x X X 
 
 Note : The Keys which are marked with a Crofs uncter them are feldom ufecL 
 
 ^ r i«- - J — i^JLj- — . Additional flats <w J (harps 111 CWer. 
 jjpIZlfeZ^ZZIgZZZ; fife, ~ The reafon why I call the F/^/ and 
 
 ZZ„rZZfci. Z-faZ|ZZ~II Sharps One, Two, Three, &c. is becaufe 
 
 123 4 s where B is flat E may not, but where 
 
 P. ^ ™ — j. — „:|_p— -j. £ i s ^ b muft, The fame reafon holdi 
 
 :^ZZZ?Z5gZiz|^._ good for jkarpu 
 
Next it is requifite to be acquainted with the feveral Cliffs and their 
 Removes, and iaft obferve, that all Flat Keys mud be Tr&nfpos'd into 
 *>*;Keys> and {harp into {bar p ones: Which Keys are known according 
 J b their gd« which is either Flap or Sharp. 
 
 Rules for a 1 norow-isai& 
 
 Wf 
 
 |W Cliffs. CfolfautClitts.GfolreutC. The Natural fharp Key. 
 
 6 7*6 6 
 
 ":f=g: 
 
 
 EK— 
 
 
 f»# 
 
 §1 
 
 d — id 
 
 43 f 
 
 A Note higher. 
 
 6 7#5 
 
 S 
 
 43 
 
 Afiarp 3d. higher. 
 
 6 7*6 6 
 
 % 
 
 l:*C: 
 
 -d:g: 
 
 aiisg 
 
 ::a5iC±H-d-3 
 
 A /^ 3d* higher. 
 
 6 7#6 
 
 43 
 
 A 4th» higher, 
 
 6 7§S6 6 
 
 S 
 43 
 
 fe c £J:^3E 
 
 
 A 5th. higher 
 
 4? 
 
 
 A Jfowp 6th. higher. 
 
 
 __6 7#6 6 *3 
 
 A flat 7th higher* 
 
 A flat 6th. higher. 
 
 6 7#6 ^ 
 
 5 
 
 45 
 
 
 uifl&C^zJ: 
 
 ! 7 # 5 
 
 l?lllliiP 
 
 43 
 
 In a /fc* Key, the Natural. 
 
 6 76 # 6 
 
 
 A 2d. lower. 
 
 ECztZC 
 
 5 
 
 *#3 
 
 
 A )b/ 3d. lower. A {harp 3d. lower. 
 
 if. 4fc 6 76 * 6 4fttl ^ 6 75 # 6 4#3 
 
 ^VH 
 
 ^-•"---tir-p 
 
 311 
 A 
 
^5^^^^ Rules for a Tnorow-Bafs. 
 
 A 4th. lower- 
 
 gin: 
 
 U t.-Q 6 '■■'• 76 -%*^-**&- 
 
 fc£±:i£dS|t| 
 
 I' 
 
 :dz: 
 
 _ Q_ 6 ?6 * 6 
 
 :fe C ;F; " 
 
 A $th» lower* 
 
 A /fe*r£ 6th, lower. A {tat 6th. lower. 
 
 ntr\ = , . - _""r t • . ^ r D" io " \ t ___■ " * 
 
 --■27,-2.: 
 
 
 d& *--— "t|t- t-t-P — t-+-i -.dftSZ— ^— - — tt-t_4_.fi- - H4— O 
 
 t 
 
 A jharp 7 th. lower. 
 
 IK 1 
 
 t«C: 
 
 6 76 # fi 4%3 
 
 fo-Pi 
 
 4?. 
 
 A fiat 7th. lower. 
 
 6 67 6 
 
 £c£E{S!:Ei{:pE3:!t|5; 
 
 4*3 
 
 -9 
 
 ]-_- 
 
 ?_sg^-±+4i:tJ-ro£--d^it= ! 
 
 You are to obferve what Flats or S&*r£, belong to all the Keys, and 
 imagine the Qliff that puts you in the Key you have a mind to Play in 5 
 and what you find too high or two low. according to the Compafs of 
 the Inftrument you play on, you moil Tranfpofe an 8th* higher, or low- 
 er which is eafle enough to be done* 
 
 Of Difccrds, how many ways they may be prepared and refoh*d. 
 
 The 4th. when joyn'd with a 5th. or 6th. and is generally refolv'd into 
 the third, may be prepar'd by a 3d. 5th. 6th- or 8th. 
 
 The 4th- preparM by a 3d* and / 
 S-eib'l.v'd into a 3d. 
 
 The 4th. prepar'd by a 5th and 
 JUfolv'd into a 3d. 
 
 ¥~f-f 
 
 1 
 
 4 
 
 — e— 
 -e- 
 
 l§L:Ez| 
 
 it: 
 
 -pan: 
 
Rules for a Tllofow-Bafs. 
 
 The 4th. prepir'd by a 6th/ 
 and Refolv'd into a 3d. 
 
 
 —6 f-e-W- 
 
 4 1 
 
 
 fz£ :: 3 zzszzJ :] izlzazifzzE — ■ 
 fc2 z: g zzz:^:zztZ^z{fzz : -zz 
 
 The 4 th. prepar'd by an'° I fr "P 
 
 8jh. and Refolv'd into a }d.\ ] 4 1 
 
 The 4 th. on occafioa' " I ' n ' • [ Y 
 
 Refolv'd into a 6th. \ .546 » 6 4 6 4 
 
 
 1 
 
 * 5 5 ' • j 
 
 The 4 th Refolv'd Into the* : Jzp t"jf;:3:tZ " -^:pQ:^Zi\ZZ: 
 3d. fevers! times before you^;$ = hx^^ 
 come to the Cadence. •—• — _E-.|~~~ — □-}-- P— £. — 4_^™~f j — - 
 
 The ?th. may be prepar'd by a id. $th. 6th- 7th. or 8th. The ytfr 
 when the Bafs lies ftiM Reefolves into the 6th- and when the Bafs falls 
 Five Notes? or rifes four Notes it Refolves. into the 3d. The 7th. 
 fome times Refolves into a 5th. and then it is in order to a Cadence % 
 fo that the Bafs rifes one Note. I have feen the 7th. Refolv'd into an 
 8th. but it founds fo like two 8ths> that it makes me utterly againft 
 it. 
 
 The 7 th- prepare 
 bv a 3d. and Re- 
 folv'd into a 6th. 
 
 
 The 7th.prep8r'dJ 
 -jby a 5th. and Re~\ 
 folv'd into a 6th. 
 
 ^iZzfcZttijz 
 
 . ;^P_p,}J«. 
 
1 88 
 
 Rules for a Thorow-Bafs. 
 
 The 7th.prepar , d]^:^zf;rp:: 
 
 by a 6th. and Re-^ 
 iolv'd into a 6th- 
 
 J°-ka folv'd into a 6th, 
 
 ; The 7th. prepaid W±Z fezpz: ; 
 by an 8th. and Re«<^ 1 r I 
 
 Example of the 7th.« 
 Refolvinginto the gd. or/ 
 the $th- fome times* 
 
 32.f__f:_u4— I — l — i— t-i — £.-&-- 
 
 =Q~:: 
 
 liili^i: 
 
 „J 
 
 _J„.J * — }_J_o-L-ij 
 
 There Is fome times two 7ths 
 pos'd one after another, but 
 a Licence in Mufick and co 
 In order to a Cadence. 
 
 :ommonln 6 7 7 4 3 
 
 gztz^zizi:izi:zqzdzj; 
 
 The 9th. is generally prepared by a Jd- or a $th- and it may be by a 6th- 
 €>r 8th. but not fo naturally < The 9th when the Bafs lies ftill, Refolves 
 Into the 8th. The 9th. when the Bafs fails a 3d. Refolves into a 3d- 
 The 9th. when the Bafs rifes a 3d. Refolves into a 6th. The 9th. may 
 Kefolve into a $th- but not fo naturally as the ©ther 3 and then the B0JS 
 tiles four Notes. 
 
 The 9th. prepaid 
 I>y a 3d. and Re- 
 fold into the 
 
 "^ZmZ {The 9th- prepaid 
 by the 5 th. and 
 Refol?'d into the 
 8the 
 
Rules for a Thorow-Bais." 
 
 w 
 
 |Tfie9fcfi- prepaid 
 by a 6th. and Re- 
 folv'd into the 
 8th. 
 
 \=mu 
 
 
 The 9th. prepaid! <&r2r2-ZQ-* 
 by the 8th. and V~r f i T 
 
 6 93 _ n Refolv'dinto the\ % _. 
 
 8 th. 
 
 I£p=i 
 
 0f the 9th. Itefolving into the 3d. and 6tk but rarely into a yt& 
 
 Example* 
 
 L_4^-i-+- 
 
 
 1696 969 #6 69 S 
 
 liiimtiiiil 
 
 41 
 
 The F/atf 4th. and 2d. and Sharp 4th." and 2d. is brought in 
 when the Bafsin a driving Note defcends a half or whole Note, the 
 Sharp 4th- always Refolvcs into the 6th. as does generally the Flat 4th» 
 but fome times with the Flat 5th. the ad. Refolves into the 3d. 
 
 Where the feyeral driving Notes defeend by degrees* 
 
 Example* 
 
 / , 4 4 I 
 
 \ m 1 J 6 24 4 4 4 
 
 \ J -g— »T^ 6 z 6/ "^ z 6 2 6/ — " Z 6 
 
 . ^*a 
 
 Another Examples—I 
 
 4 6 
 
 :#tz 
 
 gJF»~<PM.£<« #|'«««-««-»«t 
 
 P 2 
 
IOQ 
 
 Rides for a Thorow-Bafs. 
 
 The 9th. and 7th. mark'd above one another, may be prepar'd by 
 the 3d. and 8th. and Refolv'd into the Sth and 6th. the Bafs lying ftill 
 &nd fome times is artificially into the Flat 5th. and 3d. and the Bafs 
 falls a Fiat 5 th, 
 
 Example* 
 
 
 >±i.— f«i — 1*#— 1« — 1—— t—-.t— -I— —r- 1 — I t—-t--— -U— jg;i— — I— — f— 
 
 - 6 
 
 hezc: 
 
 l^-l — a 
 
 1 
 
 9 8 
 
 7 6 
 
 9 8 
 
 
 i~ 
 
 :zcz: 
 
 - P 
 
 
 ,Q— 1~~- 
 
 X-j 
 
 , 9 8 b* 
 6 7 6 3 
 
 
 :cc 
 
 The 4th. and 9th mark'd one above another is beR prepar'd by the 3d* 
 
 and 5th. and Reiblv'd into the 3d. and 8th* the Bafs lying ftill, fome 
 tiroes artificially into the Flat 5th. and 3d. the Bafs failing a 3d. and 
 fome times into a 7th. the Bafs rifing four Notes contrary to the Tn* 
 
 Example. 
 
 
 L J^ 9 - ! 
 
 The 4?h. and 3d. mark'd one above another is commonly us'd 
 uhen the Bafs afcends by degrees, the 6th. and 4th. with a id, is com«* 
 monly us'd whea the Bafs defcends by degrees* 
 
 Example* 
 1 ? 6 3 6 
 
 6 6 
 
 4 6 4 6 
 
 r 
 
 i 1 
 
 *~\ •~ t |~" J t- — ■ g— t— 
 
 Th« 
 
Rules for a Thorow-Bafs. 191 
 
 The dth* anc! 4th. where the Bth. is joyn'd is commonly sis'd when 
 the Bafs lies ftiil in a Sharp Key, or when the Bafs either dcfcends four 
 Notes, or afcends five Notes. Example. 
 
 ZZZZJZZJZZZZLzi! !ZZ|ZZ!Z ZZJZ! !~Z3— -IJ— --*- 1 — ^ZZ! !I^JZi~^Z3ZI.tzZZ 
 
 O -I 
 
 t 
 
 6 5 B ' 4* ' .6 
 
 5 \ 1 6. . 7 7 7_^ S . a 6 .5 _ 4 3 
 
 ZZZZJI 
 
 — 3— -H— 1h-€ — 4- 
 
 fri-- 
 
 
 jljzzczzz: 
 
 Efzhzfa 
 
 — — zp-fzn 
 
 ! 
 
 — 4 _ 6 „* JL iL_ 6 _ _ 
 
 -r-gr- 
 
 The sharp 7th. when accompany'd with a 2d* and 4th. is us'd, when 
 the ft*/} lies ftill in a Flat Key. Example. 
 
 p^ 
 
 D- 
 
 - D- 
 
 I — 0- 
 
 llz=fc 
 
 ftjj* 78 3 J _S__: 
 
 "*. a 3 1 I. _"j it!!""" __ 
 
 * 
 
 r— r— r~ z] 
 
 — p_p„p. t 
 
 :□: 
 
 The excream Sibrf 2d- and 4th. generally prepares a Cadence- The 5th. 
 Ind 7th- and the Flat 5th. and extream Flat 7th. are generally the fore 
 miners of a Cadence- Example* 
 
 
 zz3zczg^^ 
 
 
 fcfc -^«-„L„«-|l»,-_-.l — 1„ 
 
 rpziz|izz„zi?zr^ 
 
 : f:±~ 
 
 
 ^r_pppp:|z p.:zpp--; 
 
 
 frd-p- 
 
 
 b 7 « 
 
 
192 
 
 Rules for a Thorow-Bafs, 
 
 The perfefi 5 th. and 6th. joyn'd is commonly us'd as the 7th. and $th« 
 before a Cadence, as alio when the Bajs defcends by 3ds. 
 
 The Flat $t\\* may be joyn'd to any Sharp Note that requires a &t% 
 emkis it be contrary to the Key, or it be marked ctherwife. 
 
 -*-* 
 
 
 The €%ttz*m Sharp, and the extream Pto Notes belonging naturally to 
 either Flat or Sharp Key : The extream Sharp in a ftiarp Key, is the hall 
 Note below the Key : The extream Sharp in a flat Key, is the Note 
 $b©ve the Key, unlets uktn off by an additional sharp: The extream 
 Flat in a Sharp Key, is a 4th- above or the 1th- below the Key : The 
 extream Flat in a flat Key, is a 3d. below or a 6th» above the Key. 
 
 The extream Sharp being too harfh, and the extream Flat too lufcious 
 unlefs taken off by an additional Sharp or JFte, or what is excepted in the 
 following Rules ought to be doubled. 
 
 On either extream Sharp or Flat Note, or any extraordinary Sharp 01 
 Flat Note out of the Key, that requires a common Chord* you double 
 the 3th. in Compofkion, or Playing four Parts. If the extream Sham 
 or an extraordinary ftiarp Note requires a natural Flat 6th, you leave outi 
 the 8th* in four parts, and Compofe, or Play two Sixes and one third 
 or two thirds according as the Bafs is too high, or too low. 
 
 If the extream Flat or any extraordinary flat Note requires a 6th*infteac 
 of double Sixes, or double thirds, you may Compofe, or Play in foul 
 Farts. 
 
 Where the e& cream Flat, or an extraordinary flat i 8 j 3 1 6 
 Note happens to make a ^ch. to any Note, never <k 6 J 8 | 3 
 double that SiKth. * 3 J 
 
 
 Example in a Sharp Key, 
 
 za::l:p:3i::g 
 
 # 
 
 tCt 
 
 
 
 [. — ,p__p4-P~p — -5"*""p"~ — ■ 
 
 -R: 
 
 -4s^-i- 
 
 
Rules for a Thorow-Bafs* 
 Example in a Flat Key. 
 
 191 
 
 6 .# 
 
 6 
 
 * . m 
 
 _ELt±rE:J:^ 
 
 4#3 
 
 _ 
 
 -W— # 6 
 
 :Er^zrz±:ii 
 
 6 87 ^6 . 
 
 iff fU 6^5 
 
 2pfc ; j£- _ 5 Jt#i*---~4.J3--.- ni-- 6b5 4 J- 6 t .,- 4 i 
 
 djuV-* s^~ » Jl^t*?_i*i 
 
 *yf- 
 
 il«^ 
 
 l^^^^^^tt^lg^ 76 
 
 
 #69 
 
 6 9 6 
 
 Efc±4~**^- 
 
 E=c:z rzpziz: :±:„pztr: jztazzp:: 
 
 4 
 
 . $ ri 6 ^ * ,* , * 7 6 S - 
 
 6 9 J5 . 7 1.4 
 
 zipirdti— fc- 
 
 ezze-: 
 
 , 
 
 -isl.^ 
 
 fv* 
 
 #_-l 
 
 >*•; 
 
 ȣj 
 
 *g&m&$m*m 
 
 6 IL 6 
 
 6___4 #__J? * 
 
 <5 
 
 
 te±= 
 
 >*««*■■« * 
 
194 
 
 Rules for a Thorow-Bafs. 
 
 6 6 6 
 
 »#$P_E?Zi 
 
 i 4—1 
 
 ItnBSzS:?: 
 
 4 
 
 52- 6 
 
 
 ,#z:3: 
 
 6 ?.4*3 
 
 
 T 
 
 , 6 * 
 
 b5 43 93 
 
 1=] 
 
 7 6 
 
 S 4 43 
 
 
 
 -9—1 
 
 — p- 
 
 
 p._l.BL J___J__ IZL - >®u«_L-_„ 
 
 6 1 5 4*j 
 
 So^ze Leffom where the E. fftfi lira G Cliffs Interfere one 
 with the other. 
 
 6 & 6 
 
 #6 5 
 
 6 76 #69 
 
 :z%ErW 
 
 «V£ : : 
 
 ^'f 1 L$L --M- *I ~*~-lJ§4 ' ilT 
 
 EEf|H^ESEEf=z 
 
 
 
 - — t 
 
 &4 
 
 * 2 6 
 
 4 6 
 1 b5 
 
 "fa:: 
 
 ^\* ' tr j r * i?S~~- — . J. l~ t , - „ L , L - r '! ■- u-SLl-, ffe _.i.,.„r^ .1.J 
 
 4 4 6 67 
 ^ 62 | 4f 4$3 
 
•_ 6 9 1 $_^_ ^j> |T j | h £• *. 1 6 
 
 Rules for a Thorow-Ba.fs; 
 
 6 9 „ 6 6 
 
 'i$i 
 
 9 
 
 . 7 4<J 
 
 7 li 7 
 
 m f'bS 4#B 
 
 
 & # s 
 
 i^k 
 
 fai— 
 
 *,'.# ,4 
 
 6 
 
 4 _ 6 
 
 _^ ^ ^ j&_ffL 4-— jfc _ i in i p n 
 
 4 3 
 
 76 # 
 
 
 =izfe 
 
 9-— 
 
 IZK. 
 
 -gp. 
 
 
 i^EsEsiEErfE:: :EEE 
 
 — 9- Sj {_,. f— — p— f— •("- j— >f 
 
 £-1 
 
 * 
 
 -ft.-.*-* I — >• 
 
 6 
 
 1? 
 
 
 6 6 s£ 
 
 l_*.^ S3— J*3_ i 
 
 -"prp:|:p-_r:pzJ~p:=^t:a-}iL::2 
 
 :Z2lt3' 
 
 Q. 
 
19& Rules for a Thorow-Bafs. 
 
 In this LeJJon the G, C, and F. are all us*cl* 
 
 
 
 F y jj^4_j_j__ : 
 
 4 6 
 7* - '6 U bJ 76 
 
 6 54 6 
 76 bS 2. hS 43 75 
 
 fi 
 
 6 Ji 6 76 6 
 
 P 
 
 54 6 fr 
 
 a bf 6 75 6 4 
 
 
 *fc 
 
 
 $ 
 
 MM 
 
 ^ 
 
 EC 
 
 ES 
 
 4 6 
 
 f3=Kp 
 
 ! lie. ^iLa.zdzt ibzp I . _. 
 
 6 6 _ -T^ 4 6 „ 7 6 7 ; 
 
 UJ ~>-43 7* b 5 -f"^ 2 "fc r^ 6 - 34 4 * 
 
 gzEezt:} izziz! 
 
 
 6 6 7 6 7 4b3 * b5 * bT 6 76 # 6 
 
 5 7 i1 i*L* 
 
 b5 . 9 
 
 I . b* 1 
 
 jjj*_7 43_ 5b4 4b3_4#3 b «& _ 7_ 6 J ^ * 
 
Rules for a Thorow-Bafs. 
 
 IPX 
 
 on ,_„j I I-Zj— s ,—«u-^0-^I-,E^$£— . -I-^0u.I-E pu QE— 
 
 7 k 7 
 
 6_6_« # _b_5 4#3 _ ^ 
 
 ^jz:p:):p:^:::zQz~pzz|t||j^zz:^ziz::r~ 
 
 
 i*1 " 6 
 
 to 
 
 _ S P— 6 6 4 1 
 
 ^F^F^PFp-rs-p-rf rr - --*r=rj* 
 
 **&' 
 
 
 ::c 
 
 Tirt"Ti 
 
 . 4r|z.i" E : ! t b Vt ~ 6 --1-* 6j ^ m 
 
 * 6 # 
 
 f-4---t---- H^-- ----|i-r-^z- 
 
 4 4 
 
 t _ 6 Jp J >#■ 
 
 S 6 6 
 
 43 -p-4 6 4 6 7 , 4B 
 
 |jE:|||=|:i^z zazj j||f zz?E=EEE 
 
 5 6 h7 $ 7 $ O 8 
 
 H^ggigMga 
 
 b7 
 76S £ ,2t^ #-§- 7b5 #34 4#* 
 
 
 0,2 
 
'198 
 
 Rules for a Thorow-Bafe. 
 
 
 4?. 55 76 7$6 ..-0-^f 7« #1 
 
 h---;j j-ir._L u . ; |'.3:L. : j *ZE~ p.j_£_J y : 
 
 76 f „76 S „76 75 y 
 
 .^L--i.±-i?_# 3 t 4 * | 3 #1+ .*3 *3ftJ, *#=3 16. 9 6 9 6 
 
 
 
 # 
 
 6 4 4 7 b 7 
 
 b5 : <"~^a 6<~^a 6 1 b S 43 
 
 
 fa^^^S^j fe==E= 
 
 # 
 
 6 4#3 , 
 
 JX^-I*.-!*™..- • 
 
 43 
 
 *? 6 6 
 
 6 £ # 1 4# 3 f ^ fe| 4*3 b'J 4#3 
 
 
 4 6 5 * bf 5 4*9 S*b5 S 2 6""^a 6a 6 
 
 -— - -hi— — w" 
 
 P 
 
 P#S 
 
 E-#-^ 
 
 7 7 
 
 
 RtfWB 
 
 « 6 
 
 5 zzz-: |ffiis~~~zz: nqqq::^ 
 
 
 4#3 7 7 #4 6^ m 
 
 *2# 7 J.. ^l-^ 2 j!!llfl f _. 
 
 
 _3j 
 
 :co 
 
Rules for a Thorow-Bafs. 
 
 199 
 
 6 7f_!g__ 
 
 :zE:!:grgp£p 
 
 t— 
 
 4 6 
 
 bS 
 
 9 6 $8 4-|3 Sa 
 
 #4 
 
 4 6 4 6 4 6 
 
 _J 2 _M Ji b5 S*_ b5 
 
 ^£_6 ^J#5 
 
 4 4 
 
 f i 6 a 4 6 6 J6 S.. 
 
 efrcdr 
 
 4 
 
 4 
 3 ^ 
 
 GjjS3^t,| 
 
 4 . 4 6 
 
 3 6 I n - 5 6 4 6 6 
 
 
 iifez:: 
 
 6 6 
 
 4 6 4.3 
 
 
 4 
 
 JT^ZiL-7 6 7 z!£~L ft I %, -j _ !*-~3 4 1_ 9 % — ?? x 
 
 
 E~~E ETEzu~^3f" 
 
 7 4 6? 
 _6^3 6_ _3_ _74 H _ 
 
 4 fj 
 
 52 M 
 
 6 76 f54 4#3 
 
 acdBS 
 
 6. 76 76 76 #6 9 b5 
 
 - h*^Pf^-^ 
 
2©0 
 
 6 
 
 9 6.$ 
 
 Rules for a Thorow-Bafs, 
 
 5* 4 4 4 7 S„ 6 
 
 
 *« 6 6 #_S 4#|_ 
 
 ^|g^^^-^R£f-fcp^^ffle — 
 
 l fhall here add fome fliort Leflbns by way of Fugeing. to 
 make the whole work Compleat. 
 
 ^Pfp-:^|li||t 
 
 •.♦I 
 
 :& 
 
 4 4 
 
 6 ^ 4 6 $ 4 7 62. 6 24 
 
 4 5^7-a #^6,7 7 7 | b 7 ^4 3; ~P •'T> 6Z _ 6 
 
 
 i 
 
 ■ v .- 7 iJt: 
 
 6 4 
 
 z±ErJt~ t;t Ef"i 
 ::E:Ep E: 4 :Sti 
 
 :3;zi3:-: 
 
 4 
 
 •i-l 
 
 #6 93 41 
 
 ^&:gt^:ii:"zq5zl:d5:iz J iz pz 
 
 43 *? 4$3 7 _ 6 * 6 Ll #0—' *• j*1 4#3 
 
 : ,-~~F - n—P- 
 
 M*,§l = g:j-»-^| : |---j||«-^ r 
 
 ^SiS$ 
 
 . . 4 _j ,_ 4 _ . jt:j j .4 . „.*_ 4 #i«. 
 
 6 76 
 
 ISEZE ! 
 
 s 
 
 pi 
 
 76 
 
 -j r i 
 
 '( — 
 
 7 4 5 
 
 z:eij 
 
 
 1..3 6 6_ -•— 
 
 -F- #3 
 
 
Rules for a Thorow-Bafs. 
 
 201 
 
 L*L* 
 
 6 7 # 4b3 
 
 $p§ 
 
 J I ^x # jLi.,I--*--i -._ 
 
 5 _4*a #56 
 
 U~m .KL.-— 4 JL,,Jl — J*,— -J 
 
 ™ II ^ft-$^-J 
 
 ufc : -F*£ 
 
 #-t:=bf:F::=E=E 
 
 zm 
 
 4 6 7 
 
 
 » 
 
 6 4 6 4 6 
 
 9 ^,*| 4 3 56 76 7*6 ?C"^ b5 f "-^ bS 6 9 6 
 
 siiqd:: ± zZKZpa:§:i z:E:i:g::^:j»:z4:i.i:z:z:z}:zz"":r; 
 
 — *-* 
 
 '1' 
 
 
 6 6 6J, S !f£ #<5:*. 
 
 6 ? 
 
 ■« *$?-#-* 4-¥t--frB 4--fcf4--Y-- tf i— • 
 
 e - : EEEt : f 
 
 ^^^^^^^j^^S 
 
 
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44 Rides for a Tho row-Baft. ■ 
 
 (Rules for Tuning a Harpsicord or Spinet. 
 
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 will permit ^ and all Fifths as flat as the Ear will permit. 
 
 Now and then by way of Tryal f touch Unifon Third, Fifth 9 
 and Eights^ and afterward Unifon Fourth and Sixth. 
 
 i 
 
 F 1 N I 
 
This book must not 
 be taken from the 
 Library building.