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 ...

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.