A proposal to perform musick in perfect and mathematical proportions containing I. the state of musick in general, II. the principles of present practice ..., III. the tables of proportions, calculated for the viol ... / by Thomas Salmon ... ; with large remarks upon this whole treatise by the reverend and learned John Wallis ...

A PROPOSAL To perform Musick, IN Perfect and Mathematical Proportions.

CHAPTER I. Of the State of Musick in General.

WHEN the great Empires of the World, were in the height of their Glory, espe∣cially the Grecian and Roman, (whose Authors have left us lasting Monuments of their Excellency) Then did all sorts of Learning flourish in the greatest Per∣fection: The Arms of the Conquerors ever carrying a∣long with them Arts and Civility.

But to bring about a fatal Period, did the North swarm with barbarous Multitudes, who came down like a mighty Torrent, and subdued the best Nations of the World; which were forc'd to become Rude and Illiterate, because their New Masters and Inhabitants were such.

Amidst these Calamities, no wonder that Musick pe∣rished: All Learning lay in the Dust, especially that which was proper to the Times of Peace.

But this Darkness was not perpetual; The Ages at last clear'd up; and from the Ruines of Antiquity, brought forth some broken Pieces, which were by de∣grees set together; and by this time of day are Arriv'd near their ancient Glory. Guido has been Refining a∣bove Six Hundred Years.

Two things are chiefly conducing to this Restoration: The great Genius of the Age we live in, and the great Diligence in searching after Antiquity: The excellent Editions of the best Authors, and the most laborious Comments upon them, abundantly testifie the Truth of this.

In both these Felicities, Musick has had as great a share as any; Aristoxenus, Euclide, Nichomachus, Aly∣pius, Bacchius, Gaudentius, Aristides, Martian, have been with a great deal of Diligence, set forth by Mar∣cus Meibomius, at Amsterdam, in the Year 1652.

And above all, is, Claudius Ptolomaeus, who Correct∣ed and Reconciled the Pythagoreans, and Aristoxeneans, the speculative and practical Parties: This Author was Published by Dr. Wallis, at Oxford, in the Year 1682, who added an Appendix, comparing the Ancient and Modern Harmony; which is as the Key to all our Spe∣culations, and without which the former Authors were hardly Intelligible.

Nor are we less beholding to the Excellent Genius of our Modern Musicians: There are, indeed, only two Fragments (as I know of) remaining of the an∣cient Grecian Compositions; One of Pindar's, found

by Kircher, at Messana in Sicily; the other of Dionysi∣us's, rescued by Dr. Bernard, from lying hid amongst some Papers of Arch-Bishop Vsher's; both Published with Chilmeads Notes in the end of Aratus, at the Ox∣ford Theatre: These are very short, and very imper∣fect, and therefore we cannot make any Judgment of their Songs or Lessons.

But by all that we can discern from their Harmoni∣cal Treatises, There never was such regularity in the de∣signing of Keys, such a pleasing sweetness of Air, such a various contexture of Chords, as the Practical Musi∣cians are at this day Masters of.

It may seem now, that there remains nothing to be added, or to be learn'd out of those Eminent Authors I have here Recited; And the mighty Power of Musick, Recorded by the most Grave and Authentick Historians, may be lookt upon as Romance, since all the Excellen∣cies now perform'd, cannot conquer the Soul, and sub∣due the Passions as has been done of Old.

But before we quit the Testimonies of what Musick has done, and despair of any further Advancement; Let us enquire whether there be not something very Consi∣derable still wanting, something Fundamental very much amiss, even that which the forementioned Philosophers were likely to be most Excellent at, when the Learn∣ed and Practical Part were met in the same Persons: Whether this be not the Accurate Observation of Pro∣portions, which the Soul is from Heaven inform'd to Judge of, and the Body in Union with it, must Sub∣mit to.

Surely, I need not prove, that all Musick consists in Proportion; that the more exact the Proportions,

the more Excellent the Musick: This is that, all the World is agreed in. For this, I have every Man of my side, that except the Voice, the Instrument be well in Tune, the best Composition that was ever made, will never please; And what is it to be in Tune, but for every Note to bear a due Proportion to one ano∣ther?

Indeed, the Proportions of Musick are twofold; First, In respect of Tune, and Second, In respect of Time: The latter of these, which Dr. Vossius contends so much about, is certainly very considerable; that the Musick should agree with the Poetical Prosodia; that all the Variety of Rythmical Feet should have their pro∣per Movements.

Then would the Sense be favoured by such Measures, as were most fit to Excite or Allay the Passions aim'd at; and the Words of a Song would be capable of a more easie and intelligible Pronunciation.

Since Musicians have not undert••ken to be Poets, and Poets have left off being Musicians; this now disjoynt∣ed Work, of making Words, and setting Tunes to them, has not been so exactly done as formerly, when the same Authour perform'd both.

But were it never so well done for Time, and the Proportions of Tune neglected; it could signifie nothing: None will pretend to make Musick by playing good Time, except the Instrument and Voice be in Tune.

However, till both these Fundamental Points be ob∣served with such Exactness and Excellency, as the An∣cients took care of; we must not say we do all they did, or that they could not prevail more than we can; all the Modern Excellencies may be rendred Ineffectual,

by tolerating so many unproportionate imperfections, as are every where found amongst us.

I shall not here give an account of all those accurate Proportions, which the Ancients contended for, nor their little enharmonical Distances, whereof their more curi∣ous Musick did consist; but only of what is now practi∣sed amongst us, that the certain Knowledge of our Fundamental Principles may produce Performances, much more exact and powerful.

CHAP. II. The present Practice of Musick.

THE Hours of Study are tedious to some and pre∣cious to others: I cannot therefore suppose any man will search into the demonstrative Reasons, or ac∣quaint himself with the Mathematical Operations be∣longing to this Proposal, till he be first assured of the truth and usefulness of it.

So that what is purely Speculative shall be reserv'd at present: This offers nothing but the Principles of conti∣nual Practice, whereby the Reader may be lead into the knowledge of what he is always to design; and taking the String of any Instrument, may give his Eye, and his Ear, and his Reason, an immediate satisfaction, in all that is here dictated to him.

Before we compose or perform any Musick, two things must be provided for.

I. That we have some little gradual Notes, which

may (whilst the Voice rises or falls) succeed one ano∣ther in the best Proportions possible; whereof (as of so many Alphabetical Elements) the whole Musick must consist.

II. These gradual Notes must be placed in such or∣der, that the greater Intervals (compounded of them) may in the best Proportions possible arise out of them, and be come at with the greatest conveniency: That in all the Points, where the single Notes determine, there the larger Chords may be exactly coincident; if it was not for this, there could be no Consort-Musick.

To set forth this, we may as well use the first seven Letters of the Alphabet, as all the hard Names of Gui∣do's Gamut; because they were framed long before Mu∣sick was brought into that good order wherein it now stands, and the first intension of them is not agreeable to the present practise.

Only this will be worth our Observation, that where∣as in the Scale of Musick, there are three Octaves, (be∣sides the double Notes and Notes in Alt) viz. the Base, Mean, and Treble, we may use Three Sizes of Letters in a greater, middle and lesser Character: as will be found in the Tables of Proportions.

For understanding the two things pre-required, we suppose the proportion of one gradual Note to be con∣tained between A and B, then between B and C the proportion of another gradual Note, though much les∣ser; these two single proportions, viz. that of A B, and that of B C being added together, must exactly consti∣tute a lesser Third; the proportions of the two gradu∣al Notes must determine in that point, where the com∣pounded interval may be coincident with them.

To proceed, if we add another gradual Proportion from C to D, then must arise the exact proportion of a Fourth, from the first given A to the Note D: if one more be added from D to E, there must be found the exact proportion of a Fifth, from A to E, and of a grea∣ter Third from C to E.

Thus must the Gradual Notes be contrived to be ex∣actly subservient to the greater intervals thorough all the Octaves: and if at any time this cannot be (as may happen in two or three instances) such particular Chords must be esteem'd inconcinnous and inconvenient, but they are very few, and lye much out of the way.

If we settle one Octave, the whole work is as good as done; all the rest is only repetition of the same Notes in a larger or more minute figure: for the eight Notes which are used in constant practise, proceeding gradual∣ly, take up just half the string, from the sound open to the middle of it: And if we have occasion to go further, 'tis but just the same over again.

The great concern is in what order our gradual Notes (which are of different sizes) must stand, from the Key or Sound given, till we arrive at the Octave; for there will be a great variety, according as the lesser gradual Notes are placed sooner or later: This must be lookt upon as the internal constitution of an Octave, which practical Musicians commonly understand by their Flat or Sharp, that is, their greater or lesser Third.

But as much as I can observe from the Compositions of the most Eminent Masters for these last Twenty years, this internal constitution of an Octave is but twofold: either with a greater Third, Sixth and Seventh; or a Lesser Third, Six and Seventh: In the same com∣position all are lesser, or all greater.

There needs then only this twofold constitution of the Octave to be considered by us, the two Keys A and C: all the rest serve only to render the same series of Notes in different pitches; which is demonstrable by transposing Tunes from one Key to another: The Tune remains the same, only the compass of the Voice or Ins∣trument is better accommodated.

These two Keys A and C are called Natural, because the Proportions, originally assigned to each Letter, keep those proper places, which either Guido the first restorer or immemorial Custom hath allotted to them; Where∣as by taking other Keys, as suppose G for A, the pro∣portions or different sizes of the gradual Notes are for∣ced to shift their quarters, and by flats or sharps to straiten or widen their usual distances.

'Tis sufficient demonstration for all this, that when any Tune is transposed into A or C, it wants nor flats nor sharps, whatever it did before.

I shall in the first place give you the natural order of the gradual Notes as they stand in the Key A, where we have a lesser Third Sixth and Seventh, exactly coinci∣dent with the Third Sixth and Seventh gradual Note.

You have between every Letter, that part or pro∣portion of the string assigned which each gradual Note requires: underneath you have the proportion of each compounded interval, what part of the string it's stop must be when compared with the whole string open from the Nut to the Bridge.

The Constitution of the Key A.

	A.	9.B.	16.C.	10.D.	9.E.	16.F.	9.G.	10.a.
A Lesser Third			6
A Fourth				4
A Fifth					3
A Lesser Sixth						3/8
A Lesser Seventh							4/9
An Eighth								2

The Experiment must be thus: You are to take any one String, and suppose it to be the Key A, when it is open: then measure the 9th part of it, you will have B or one gradual Note. Not that the first Fret must stand there, but the second; for we are not reckning according to Tableture, but Notes specified by the first Letters of their hard names: the half Notes shall be considered afterwards.

From the place of B measure the 16th part of the remaining String, there will be C, the least gradual Note: And there you will arrive at the 6th part of the whole String, which is the proportion of the Lesser Third; and the Ear will acknowledge it to be so.

From the place of C take the 10th part of the re∣maining String, there will be D, another gradual Note, much wider than the last, between B and C, but some∣thing less than the first between A and B. At the place of D, you will arrive at the 4th part of the whole String, which is the proportion of a practical Fourth.

Here, to prevent all perplexity and mis-understand∣ing, the Reader must carefully distinguish in the Terms of Art: The Practical Musician reckons how many gra∣dual Notes he has gone over from his Key or Sound given, and accordingly calls his Intervals a Third, Fourth, and Fifth, as having so many gradual Notes contained in them; but the Mathematician regards only the parts of the String, what proportion the part stopped bears to the String open.

Here indeed the Practical and Mathematical Terms are the same, a fourth part of the String mathematically measured, is a practical fourth; but in all other Chords they differ, as we have seen a Lesser Third to be the sixth part of a String.

From the place of D take the 9th part of the remain∣ing String, (which is a gradual Note of the same pro∣portion with the first, between A and B;) here will be E: and here you will find you are arrived at the third part of a String, which is the grateful proportion of a Practical Fifth.

The proportions of the Lesser Sixth and Seventh, viz. ⅜ and 4/9, are of a different sort from the rest; the former Chords arise from the natural division of an O∣ctave or Duple proportion, there are formed by an arti∣ficial addition of a second or a third to the Fifth: The former proportions are called by Arithmeticians Super-particular, these are super-partient.

I believe the Reader will not desire to be troubled with the nature of them here, but only to be informed how to measure them for his present satisfaction: He is to know then, that he must not take the upper number 3 of 3/8 for the third part of the whole String, for then a

Lesser Sixth would be the same as a Fifth; but he is to devide the whole String into eight parts, as the lower number specifies, and then where three of those parts determine from the Nut, there will be a Lesser Sixth.

This is the addition of the 16th part of the remaining String from E to F: for A open to E was the third part of the String, that is a Practical Fifth; from A open to F will be three parts of the whole String divided into eight parts, which is a practical Lesser Sixth.

A Lesser Seventh is produced by taking a 9th part of the remaining String from F to G, which is a Lesser Third above E, this will be found to determine at 4 when the whole String is divided into 9 parts, and therefore is the proportion 4/9.

From G take a 10th part of the remaining String, you will arrive at a, the precise middle of the whole String, so that an Octave is a duple proportion; the fullest and most perfect satisfaction that can be given to the Ear. And by this is the whole proceeding demonstra∣ted to be right, because not only by the way, every in∣terval was exactly found in its proper place, but at last this Chord, the sum total of all Musick, does just con∣tain all its Particulars.

After the same manner may the internal-constitution of an Octave in the Key C be demonstrated: I shall set it down without any explication, because the Experi∣ment and Reason of both are alike.

The Constitution of the Key C.

	C.	10.D.	9.E.	16.F.	9.g.	10.a.	9.b.	16.c.
A Greater Third			5
A Fourth				4
A Fifth					3
A Greater Sixth						2/5
A Greater Seventh							7/15
An Eighth								2

Though we have all along supposed a Monochord or single String, to make this demonstration more evident, and to shew that all the gradual Notes of an Octave put together, arrive just at the middle of the String; yet the progress of the Proportions is the same, when we take some of them upon one String, some upon ano∣ther. For each String is tuned Unison to some part of that which went before; so that 'tis all one whither the Proportions go along upon the same String, or go on to the next, when we come at the place of tuning Unisons.

As suppose upon the Viol the Fourth String be C open, when you come to E, or the Fourth Fret, you have a greater Third; then 'tis all one whether you take the 16th part of the same String to make F at the fifth Fret, or the 16th part of the next whole String to make F upon the first Fret: 'Tis all one, because the

third String open is tuned Unison to E, or the fourth Fret upon the fourth String.

As I would avoid troubling my Reader with need∣less difficulties, so I would not omit any thing of neces∣sary information; this last consideration makes me here add a discourse of Seconds, which is the name whereby the gradual Notes are commonly called: for reckning inclusively in Musick, one Interval, which must needs be contain'd between two Sounds, is term'd a Second.

It is best to treat of them in that method, which our Authors used in the Classical times, because 'tis their Perfection we are now aiming at: They divided their Musick into three sorts, Diatonick, Chromatick, Enhar∣monick, which was so diversified by those several sorts of Seconds or gradual Proportions they used therein.

1. In Diatonick Musick, the foregoing constitution of an Octave discovers three several sizes of Seconds, viz. the 9th part of a String, the 10th part, and the 16th part. I would satisfie the Reader in this variety, because he will think much to enter upon an Observa∣tion, not yet received, except he knows some necessity for it.

We must have the Proportion of the 16th part of the String between B and C, as also between E and F, or we cannot bring our gradual Notes into the form of an Octave, into the compass of a duple proportion; this is already acknowledged both by speculation and pra∣ctice: No one ever yet pretended to rise or fall eight Notes one after another, all of the same size.

To this 16th part we must a 9th part, or we can ne∣ver have an exact Lesser Third, which is the 6th part of the whole String; but if we add another Note of the

same size, viz. a 9th part to make up that Lesser Third a Fourth, we shall find that we have a great way over-shot the fourth part of the String, and without taking the 10th part, we can never hit it; as will appear by the former demonstrations upon the Monochord, in ma∣ny instances.

I must confess, this is so contrary to the common O∣pinion of Practical Musicians, that I would not insist up∣on it, did not necessity compel me, did not the greatest Reason and Authority assure me, that it will not be hereafter denied: Of these three sizes of Seconds does the whole progress, from the Key to the Octave, con∣sist in the forementioned order, being all along exactly coincident with the larger Intervals.

My Authorities are Cartes's Musick, Gassendus's In∣troduction, Wallis's Appendix, and all other Learned men, who have in this last Age reviewed the Harmoni∣cal concerns. 'Tis time certainly to receive into pra∣ctice those Improvements, which the greatest Modern Philosophers in the World have afforded Musick.

And indeed 'tis in vain to stand out, Nature always acknowledged and received them; a good Voice per∣forming by it self, a faithful Hand guided by a good Ear upon an unfretted unconfined Instrument, exactly ob∣serves them: All that I contend for is, that the Practi∣ser may know what he does, and may always make that his design, which is his excellency.

When we have thus much granted, then may the last Chapter of this Proposal be very acceptable; which puts into his hands the Tables of Proportions calculated for every Key, that he may perform them upon those Instruments, which have not hitherto been capable there∣of. But to pursue our present subject.

2. Chromatick Musick is that which ascends and de∣scends gradually by half Notes. I don't mean such as is commonly call'd the half Note in Diatonick Musick, the 16th part of the String, the proportion assigned be∣tween B and C, between E and F: These are self-sub∣sistent, and reckned as two compleat Steps, as well as any of the rest. And if we consider the value of their proportions, deserve rather to be reputed three-quarter than half-Notes.

Chromatick half Notes arise by the division of Diato∣nick whole Notes into the two best proportions, so that they will follow one another, and be all along coinci∣dent with the greater Intervals. But those two vulgar half Notes in the Diatonick Scale will not do so; 'twould make mad work, to place two or three of these (viz. 16th parts) one after another, you would neither have true Thirds, Fourths, nor Fifths, in your whole Octave; you could not maintain any coincidence with other In∣tervals.

A Chromatick half Note is truly made by placing the Fret exactly in the middle between the two Frets of the Diatonick whole Note: This I first learn'd by the ma∣thematical division of an Octave or duple Proportion in∣to its natural parts; then I was confirmed in it by Ari∣stides, lib. 3. pag. 115. who requires such a Fund for the Enharmonical Dieses, and since upon tryal I find Practi∣cal Musicians very much satisfied in the Experiment of such a Division as fully answering their expectations.

I think only this last Age, ever since Musick has be∣gan to revive, has aspired after these Chromatick Hemi∣tones, and now they are used three, four, or five of them, in immediate sequence one after another; if their

proportions be truly given them, they are certainly the most charming Musick we have: but whereas a natu∣ral Genius easily runs into the Diatonick Intervals, these are not perform'd without a great deal of Art and Pra∣ctice.

3. Enharmonick Musick is that which ascends and de∣scends gradually by quarter Notes, which the Ancients called Dieses: I don't mean that the whole Octave, ei∣ther in this or the Chromatick Musick, did consist only of these; but after having used some of them, they took wider Steps and larger Intervals afterwards to compleat the Fourth and Fifth.

I could here add an account of the true Enharmonical quarter-Notes; the same Mathematical Operations pro∣duce their Proportions: The Grecian Authors (parti∣cularly Aristides in the fore-cited place) determine and record them, and they may become practical again; but I resolve to propose nothing here, but what is of present practice.

This I must say, that those invented for the Harpsi∣chord, are nothing to the ancient purpose: The Harp∣sichord quarter-Notes are designed only for playing more perfectly in several Keys, with lesser Bearings, which are never used in sequence, so as to hit four or five of them one after another; but the true Enharmo∣nick Scale offer'd its Dieses, as gradual Notes, whereby Musick stole into the Affections, and with little insinua∣ting Attempts got access, when the bold Diatonick would not be admitted.

CHAP. III. An Account of the Tables of Proportions.

IT is very possible, that those, who are devoted only to the Pleasures of Musick, may not care to trouble themselves with the foregoing Considerations: 'Tis not every mans delight to be diving into the Principles of a Science, and to be enquiring after those Causes which produce an Entertainment for his Senses; 'tis satisfacti∣on enough to the greatest part of the World, that they find them gratified.

And indeed the delights of Practical Musick enter the Ear, without acquainting the Understanding, from what Proportions they arise, or even so much, as that Proportion is the cause of them: this the Philosopher observes from Reason and Experience, and the Mecha∣nick must be taught, for the framing Instruments; but the Practiser has no necessity to study, except he desires the Learning as well as the Pleasure of his Art.

I have therefore so Calculated my Tables, that a man may without thinking perform his Musick in per∣fect Proportions; the Mechanical Workman shall make them ready to his hand, so that he need only shift the up∣per part of his Finger-board as the Key requires.

This I have tried, and found very convenient; I shall therefore give a Table of Proportions in every Key, that the Mechanick may accordingly make a sett of Finger-boards for each Instrument, according to its particular

length; the Proportions ever remaining the same, though the size be various.

It is evident that one Fret quite cross the neck of the Instrument, cannot render the Proportions perfect upon every String; because sometimes a greater Note is re∣quired from the Nut or String open, sometimes a lesser: if then the Fret stands true for one, 'twill be false for the other; if it stands between both, it will be perfect in neither.

As for example: Take the Viol tuned Note-ways, (which is ever the same) if you look back to the natural Constitution in the former Chapter, you will find that from the String D open, you must take the 9th part of the String; from the String G open, you must take the 10th ••art of the String: accordingly the first Fret from D (which is the Chromatick or just half the space of the whole Note) must be a great deal sharper, than the first Fret of the String G. And the first Fret of the String E being the least Diatonick Note to F, must be a great deal sharper than that which belonged to the String D, or G.

So that every String must have its particular Fret, whose Proportions are here given to the Mechanick, and he is to make use of them to the best advantage: Not that I would confine him to the way of shifting Finger-boards; 'tis possible the Makers of Instruments may find out some other way much more convenient: Their great excellency and industry in making Organs and Harpsichords, proves them sufficient to accommo∣date the designs of Musick: I only proposed what I had made use of, to shew that the Experiment is practi∣cable, which is enough for a Scholar to do, whose Pro∣vince lies only in the Rational part.

As I here inform the Mechanick, what Proportions he is to set upon every String, so I must inform the Pra∣ctiser what Keys he may play in, which is absolutely ne∣cessary; for no man can set about performing any thing in Musick, without knowing his Key.

This deserves to be consider'd, that the Writers of Musick may more certainly know where to fix their Flats and Sharps at the beginning of a Lesson or Song, and the number of them that is requisite: for as in Vo∣cal Musick 'tis a vast trouble in Sol-fa-ing to put Mi in a wrong place, so it is in Instrumental Musick, to have an Information renewed in several places thorough the whole Lesson by a Flat or a Sharp, which might have been known at first, once for all.

As for instance, C Key is now often chosen for a Lesser Third; there is no doubt but the Composer would have a Lesser Sixth as well as a Lesser Third, (as appears by the interspersed Flats); if so, there ought to have been three Flats prefixed, that A might be flat as well as E.

I shall in this Catalogue of Keys offer you the variety of fourteen; seven with a Greater Third, Sixth, and Seventh, the other seven with all these Intervals Lesser.

But for these fourteen Keys, you need to have only seven Finger-boards; for when the Proportions are lodged between the same Letters, then there will need no shifting: so that though the Key be different, yet the Instrument must be disposed in the same manner.

As for instance, in the two Natural Keys A and C, the same Finger-board will serve; you begin indeed in two different places, the Key A is a Lesser Third before C, but the series of Proportions required, will be found

exactly the same for both, according to the foremen∣tioned Internal Constitution.

You may take this following Catalogue of Keys, with the due Proportions assigned between each Letter.

I.

A.9.B.16.
C.10.D.9.E.16.F.9.G.10.a.9. b.16.c.

II. One Flat.

D.9.E.16.
F.10.G.9.A.16.B♭ 9.C.10.d.9.e.16.f.

III. One Sharp.

E.9.F♯ 16.
G.10.A.9.B.16.C.9.D.10.e.9.f.♯ 16.g.

IV. Two Flats.

G.9.A.16.
B♭ 10.C.9.D.16.E♭ 9.F.10.g.9.a.16.b♭.

V. Two Sharps.

B.9.C♯ 16.
D.10.E.9.F♯. 16.G.9.A.10.b.9.c♯.16.d.

VI. Three Flats.

C.9.D.16.
E♭ 10.F.9.G.16.A♭ 9.B♭ 10.c.9.d.16.e♭.

VII. Three Sharps.

F♯ 9.G♯ 16.
A.10.B.9.C♯ 16.D.9.E.10.f.♯ 9. g♯ 16.a.

By this may we understand what a Key is, and observe a series of Notes in their just Proportions passing on from the sound first given to the Octave: The Keys with Lesser Thirds have always in the first place a 9th part of the String, then a 16th part, and so on till you come to the same Letter again in a lesser Character: The Keys with Greater Thirds have always in the first place a 10th part, then a 9th and so on till you come to the same Letter again; but the three last Letters are in a lesser Character, to shew, that as you began a Lesser Third short of the other, so you go a Lesser Third beyond it.

Thus you have as many Keys provided for you, as need be used; some things indeed have been set with four Flats, but they are very difficult to the Practiser, and I never saw any of them published; but if it were requisite, other Finger-boards might also be made for them, by the same Rule as these are calculated.

I know the Keys B and F♯ with Lesser Thirds are seldom used, but D and A with Greater Thirds are: Now because the same Finger-boards that serve for the two later, serve also for the two former, and the Practi∣ser may have them into the bargain, I thought it better to give these also, than to omit any thing that might easily be useful. When the Composer finds that the Instrument goes well in tune upon these Keys, he will not hereafter be so much afraid of them.

This Calculation in the Tables is but for one length, viz. of 28 Inches from the Nut to the Bridge, and but for one Tuning upon the Viol; but the Workman may be directed from these Proportions given to fit them to the length of any Instrument: and from the Key given

in any Lyra tuning, for any sort of fretted Instruments, he may find out what Proportions fall upon every String.

Indeed Harpsichords and Organs, and such Instru∣ments, where Frets are not used, cannot be accommo∣dated the same way; but the Proportions and order of the Notes, are the same in them: They have something that makes the different gravity and acuteness of their Sounds, which may be so rectified, as also to render their Musick in a Mathematical perfection; but this is left to the ingenuity of the Artificer.

I shall now observe something particularly of the Ta∣bles of Proportions, according to the numbers of the forementioned Keys, which you will find prefixed at the head of each of them, as they are annexed to the end of this Treatise.

You will find, Number I. That A and C will not al∣low the sixth Base and Treble-string to have their fifth Frets upon the fourth part of the String, which makes a true Practical Fourth to the String open: For besides the least Diatonick Note, there falling two greater Notes upon the Strings D open, the stop G at the fifth Fret falls a pretty deal sharper; and accordingly the Fifth and Sixth Bases will not be a good Fourth to one ano∣ther, but the Fifth Base must be tuned Unison at that place where the Table is marked.

I have upon every Plate marked where the exact 4th and 5th and 6th part of the String falls, that you may see when the gradual Notes are not coincident with those larger Intervals, as in the forementioned case. Old Mr. Theodore Stefkins, (though he knew not the Mathemati∣cal reason) yet to make some allowance for this, was wont to direct the tuning of those Strings sharper than

ordinary; by this Table you will find exactly how much sharper the tuning and the stops must be.

Numb. II. In D and F with one Flat, you will find the same accident upon the fifth Base, where the same care must be taken, and all the Proportions will fall per∣fect.

Numb. III. In G and B♭ with two Flats, you have another affair to be consider'd; which is the tuning the third String to the Chromatick Note at the fourth Fret of the fourth String, which causes those two Strings not to be a true greater Third to one another. The reason is, because E, to which the third String is commonly tuned, does not in these Keys (G and B♭ with two Flats) fall upon the fourth, but the third Fret of the fourth String, which is E flat; so that the fourth Fret is now the Chro∣matick division between E flat and F: hence it follows also, that the first Fret upon the third String, which is F, is not the 16th part of the String, but the 17th, viz. the later part of that divided Note.

These two Accidents are all that I think need be ta∣ken notice of in all seven of them, because though they do occur in the rest, yet being of the same nature, the Reader will know how to understand them.

This may seem a difficulty and inconveniency; that after all, the Intervals of Musick could not every-where be given in perfect Proportions: And I will confess that there are a few instances wherein they cannot, as the lesser Note being the 10th part of a String, and the least Note which is the 16th part, will not make a true Lesser Third, that is the 6th part of the whole String.

But this does not proceed from the defect of this Pro∣posal, Nature it self will have it so; Scholars are not

to alter Nature, but to discover her Constitutions, and to give opportunity for the best management of all her Intrigues: I still perform my design, because I maintain those perfect and Mathematical Proportions in every place, where demonstration either requires or permits them.

That the Reader may know how few, and how easie to be avoided, these inconcinnous Intervals are, I will give him an account of all and every one of them: There are three in each Constitution of an Octave, which are exact and necessary to carry on the progress of single gradual Notes, but they must not be allowed in the Composition of Parts.

Inconcinnous Intervals from the Key C.

• 1. A Lesser Third, from the Seventh to the Ninth a∣bove the Key.
• 2. A Fourth, from the Second to the Fifth above the Key.
• 3. A Fifth, from the Fifth to the Ninth above the Key.

Inconcinnous Intervals from the Key A.

• 1. A Lesser Third, from the Second to the Fourth above the Key.
• 2. A Fourth, from the Fourth to the Seventh above the Key.
• 3. A Fifth, from the Seventh to the Eleventh above the Key.

This is the exactness, which Reason, guided by Ma∣thematical Demonstration, requires of us; and this ex∣actness

is rewarded by a proportionable pleasure, that a∣rises from it. Indeed since Musical Ears, (especially where Sence has no great acuteness) are commonly de∣baucht with bearings and imperfections, they may not perhaps at first observe the advantage offer'd; but I am sure Nature desires it, and will rejoyce in those Propor∣tions, which by the Laws of Creation she is to be de∣lighted with.

Yet there may be many an one, who will not care ei∣ther for the trouble or charge of changing Finger-boards; if some little thing would mend their Musick, it might be acceptable: I shall therefore add one more Table, Number VIII. which any person that uses a fretted In∣strument, either Lute, Viol, or Gittar, may easily make use of, and find the benefit of it.

I call it the Lyrick Harmony, because our Lyra-tu∣nings require all the Proportions to be most convenient∣ly accommodated to the Strings open: Now if the Frets be placed at the distance assigned in this Table, they will be generally perfect.

This Table is calculated like the rest for a String 28 inches long from the Nut to the Bridge; but whatever length your instrument be, keep the same Proportions, and you will be right: a fourth part of a String is a fourth part, and the same Proportion, whether the String be longer or shorter.

For the first Fret then, take the 16th part of the whole String from the Nut, which is the least Diatonick Se∣cond that lies between B and C, between E and F; so that this will be always right, except in the Chromatick half Notes, not much used in Lyra Musick; but if the excellency of the Chroma be desired, then must the Pra∣ctiser

put himself to the expences of what has been for∣merly proposed: Jewels can never be had cheap.

For the second Fret you have two Lines, the upper∣most is the 10th part, the lowermost is the 9th part of the whole String from the Nut; we use no Proportion in Musick between the 16th and the 10th, as will ap∣pear by speculative Demonstration, and practical Expe∣riment.

If the tuning be with a greater Third, then the se∣cond Fret had best stand upon the 10th part; if the tuning be with a lesser Third, this second Fret had best stand upon the 9th part; for in Lyra-tunings the Key is generally some String open, and you will find by the twofold constitution of an Octave in the former Chapter, that the lesser Third requires the greatest Second from the Key, which is the 9th part, as from A to B; but the greater Third requires the lesser Second from the Key, which is the 10th part from C to D.

This may not be always convenient, in respect of Composition, and therefore the Practiser may set his Fret where he pleases between these two strokes, accor∣ding as he desires his bearings: however it can't but be a very advantageous satisfaction, to know his Latitude within which he may be right, and above or below which he must be wrong: These are the bounds,

Quos ultra citraque nequit consistere rectum.

The third Fret must be the 6th part of the String from the Nut, which is the Proportion of a lesser Third to the String open; for 'tis demonstrable, that in Mu∣sick we use no Proportion between the 9th and the 6th part: but if you are not to have a true lesser Third to

the String open, as may sometimes happen, when the tuning does not well favour your design, you may then use what bearing you please.

The fourth Fret must be a 5th part of the whole String, as being the Proportion of a greater Third to the String open.

The fifth Fret must be a 4th part of the whole String, as the just Proportion of a Practical Fourth.

The length of the Plate would not suffer me to give the sixth and seventh Frets, which are upon Viols; but the direction is easie.

The seventh Fret must be just the third part of the whole String from the Nut, as being the grateful Pro∣portion of a Practical Fifth.

The sixth Fret standing between the 4th and third parts of the String, may be usually placed in the precise middle, where you may make a stroke the Finger-board of your Viol: but if the tuning requires any impo••••••nt Note to fall upon it, then may you tune your Fret by moving it higher or lower, as its Octave upon some of the higher Frets requires.

Thus you may keep your former Gut-frets, which are movable and tyed quite cross the Viol; the strokes made upon the Finger-board, being as so many Land∣marks, either to keep you just in the right, or else to give you aim in the Variation.

I acknowledge that this will not come near perfe∣ction in the Note-way, nor always do in the other; but 'tis an advantage to make a good guess, and not always to do things at random: If I travel without a certain Track, an Information that I must leave a Town a quar∣ter of a mile on the right hand, is a satisfactory direction, though I am not to go thorough it.

For a Conclusion of this Proposal, I need only add, that the truth of it is evident, both from Rational and Sensible Demonstrations; for the usefulness and necessi∣ty of it, every Man that wears a Musical Ear shall be Judge; the difference of Seconds, the greater and les∣ser Note, (which have hitherto been used without any regard) is so very considerable, that whoever takes but a transient view of them, will confess his Frets must be rectified, he cannot bear so great a deviation from what is truly in Tune; and accordingly the Practical Master does rectifie them, when he passes from one suit of Les∣sons to another. For assigning these particular Propor∣tions, and denying that others are now to be used (as was asserted in the Lyrick Harmony) the Author de∣sires no longer to be trusted, than there shall appear an inclination in any to study the Arithmetical and Geome∣trical parts of Musick, which are ready to be publish∣ed.

〈 in non-Latin alphabet 〉〈 in non-Latin alphabet 〉.