A treatise of the natural grounds and principles of harmony by William Holder ...

CHAP. V. Of Proportion; and Applyed to Harmony.

WHereas it hath been said be∣fore, That Harmonick Bo∣dies and Motions fall under Nu∣merical Calculations, and the Rati∣ons of Concords have been already assign'd: It may seem necessary here (before we proceed to speak of Dis∣cords) to shew the manner how to calculate the Proportions appertaining to Harmonick Sounds: And for this, I shall better prepare the Reader, by premising something concerning Pro∣portion in General.

We may compare (i. e. amongst themselves) either (1.) Magnitudes, (so they be of the same kind;) Or (2.) the Gravitations, Motions, Velocities,

Durations, Sounds, &c. from thence a∣rising; or further, if you please, the Numbers themselves, by which the things Compared, are Explicated. And if these shall be Unequal, we may then consider, either, First, How much one of them Exceeds the other; or Se∣condly, After what manner one of them stands related to the other, as to the Quotient of the Antecedent (or former Term) divided by the Consequent (or latter Term:) Which Quotient doth Expound, Denominate, or shew, how many times, or how much of a time, or times, one of them doth contain the other. And this by the Greeks is called 〈 in non-Latin alphabet 〉〈 in non-Latin alphabet 〉, Ratio; as they are wont to call the Similitude, or Equality of Ratio's, 〈 in non-Latin alphabet 〉〈 in non-Latin alphabet 〉, Analogic, Proportion, or Proportionality. But Custome, and the Sense assisting, will render any over-curious Application of these Terms unnecessary.

From these two Considerations last 〈…〉〈…〉 there are wont to be de∣〈…〉〈…〉 sorts of Proportion, Arith∣•…•…l, Geometrical, and a mixt Pro∣•…•…tion, resulting from these two, cal∣led Harmonical.

1. Arithmetical, When three or more Numbers in Progression, have the same Difference; as, 2, 4, 6, 8, &c. or discontinued, as 2, 4, 6; 14, 16, 18.

2. Geometrical, When three or more Numbers have the same Ration; as 2, 4, 8, 16, 32; or Discontinued; 2, 4; 64, 128.

Lastly, Harmonical, (partaking of both the other) When three Numbers are so ordered, that there be the same Ration of the Greatest to the Least; as there is of the Difference of the two Greater, to the Difference of the two less Numbers. As in these three Terms; 3, 4, 6; the Ration of 6 to 3 (being the greatest and least

Terms) is Duple. So is 2, the Diffe∣rence of 6 and 4 (the two greater Numbers) to 1. the Difference of 4 and 3 (the two less Numbers) Duple also. This is Proportion Harmoni∣cal, which Diapason 6 to 3, bears to Diapente 6 to 4, and Diatessaron 4 to 3; as its mean Proportionals.

Now for the kinds of Rations most properly so called; i. e. Geometrical; first observe, that in all Rations, the former Term or Number (whether greater or less) is always called the An∣tecedent; and the other following Number, is called the Consequent. If therefore the Antecedent be the greater Term; then the Ration is ei∣ther Multiplex, Superparticular, Super∣partient, or (what is compounded of these) Multiplex Superparticular, or Multiplex Superpartient.

1. Multiplex; as Duple, 4 to 2; Triple, 6 to 2; Quadruple, 8 to 2.

2. Superparticular; as 3 to 2, 4 to 3, 5 to 4; Exceeding but by one aliquot part, and in their Radical, or least Numbers, always but by one; and these Rations are termed Sesquial∣tera, Sesquitertia, (or Supertertia) Ses∣quiquarta (or Superquarta) &c. Note, that Numbers exceeding more than by one, and but by one aliquot part, may yet be Superparticular, if they be not expressed in their Radical, i. e. least Numbers; as 12 to 8 hath the same Ration as 3 to 2; i. e. Superpar∣ticular; though it seem not so, till it be reduced by the greatest Common Divi∣sor to its Radical Numbers 3 to 2. And the Common Divisor (i. e. the Num∣ber by which both the Terms may se∣verally be divided) is often the Diffe∣rence between the two Numbers; as in 12 to 8, the Difference is 4, which is the Common Divisor. Divide 12 by 4, the Quotient is 3; Divide 8 by 4, the Quotient 2; so the Radical

is 3 to 2. Thus also 15 to 10, di∣vided by the difference 5, gives 3 to 2; yet in 16 to 10, 2 is the com∣mon Divisor, and gives 8 to 5; be∣ing Superpartient. But in all Superpar∣ticular Rations, whose Terms are thus made larger by being Multiplied: the Difference between the Terms is always the greatest common Divisor; as in the foregoing Examples.

The Third kind of Ration, is Su∣perpartient, exceeding by more than One; as 5 to 3, which is called, Su∣perbipartiens Tertias (or Tria) contain∣ing 3 and ⅔; 8 to 5, Supertripartiens Quintas, 5 and ⅗.

The Fourth is Multiplex Superparti∣cular, as 9 to 4, which is Duple, and Sesquiquarta; 13 to 4, which is Tri∣ple, and Sesquiquarta.

The Fifth and last is Multiplex Su∣perpartient, as 11 to 4; Duple, and Supertripartiens Quartas.

When the Antecedent is less than the Consequent; viz. when a less is compared to a greater; then the same Terms serve to express the Rations, only prefixing Sub to them, as Sub∣multiplex, Subsuperparticular (or Sub∣particular) Subsuperpartient (or Sub∣partient) &c. 4 to 2 is Duple: 2 to 4 is Subduple. 4 to 3 is Sesquitertia; 3 to 4 is Subsesquitertia; 5 to 3 is Superbipartiens Tertias; 3 to 5 is Sub∣superbipartiens Tertias, &c.

This short account of Proportion was necessary, because almost all the Philosophy of Harmony consists in Rations, Of the Bodies; Of the Mo∣tions; and of the Intervals of Sound; by which Harmony is made.

And in searching, stating, and comparing the Rations of these, there is found so much Variety, and Cer∣tainty, and Facility of Calculation, that the Contemplation of them may seem not much less delightful, than

the very hearing the good Musick it self, which springs from this Foun∣tain. And those who have already an affection for Musick, cannot but find it improv'd and much inhansed by this pleasant recreating Chase (as I may call it) in the Large Field of Harmonic Rations and Proportions. where they will find, to their great Pleasure and Satisfaction, the hidden causes of Harmony (hidden to most, even to Practitioners themselves) so amply discovered and laid plain be∣fore them.

All the Habitudes of Rations to each other, are sound by Multiplica∣tion or Division of their Terms; by which any Ration is Added to, or Substracted from another. And there may be use of Progression of 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 Sub∣straction

of Rations; which, though they are not performed like Addition and Substraction of Simple Numbers in Arithmetick; but upon Algebraic Grounds; yet the Praxis is most easie.

One Ration is added to another Ration, by Multiplying the two An∣tecedent Terms together, i. e. the An∣tecedent of one of the Rations, by the Antecedent of the other (for the more ease, they should be reduc'd into their least Numbers or Terms) And then the two Consequent Terms in like manner. The Ration of the Product of the Antecedents, to that of the Pro∣duct of the Consequents, is equal to the other Two added or joined together. Thus (for Example) Add the Ration of 8 to 6; i. e. (in Radical Numbers) 4 to 3, to the Ration of 12 to 10; i. e. 6. to 5; the Product will be 24 and 15; i. e. 8 to 5; You may set them thus, and multiply 4 by 6, they

make 24, which set at the Bottom; then multiply 3 by 5, they make 15; which likewise set under, and you have 24 to 15; which is a Ration com∣pounded of the other two, and Equal to them both. Reduce these Products, 24 and 15, to their least Radical Numbers, which is, by dividing as far as you can find a Common Divi∣sor to them both (which is here done by 3) and that brings them to the Ration of 8 to 5. By this you see, that a Third Minor, 6 to 5; added to a Fourth, 4 to 3; makes a Sixth Mi∣nor, 8 to 5. If more Rations are to be added, set them all under each other, and multiply the first Antece∣dent by the Second, and that Product by the Third; and again, that Pro∣duct by the Fourth, and so on; and in like manner the Consequents.

This Operation depends upon the Fifth Proposition of the Eighth Book of Euclid; where He shews, That the

Ration of plain Numbers is compound∣ed of their sides. See these Diagrams.

[illustration]

Now compound these Sides. Take for the Antecedents, 4 the greater Side of the greater Plane, and 3 the great∣er Side of the less Plane, and they multiply'd give 12; then take the remaining two Numbers 3 and 2, being the less Sides of the Planes (for Consequents) and they give 6. So, the Sides of 4 and 3, and of 3 and 2 compounded (by multiplying the Antecedent Terms by themselves, and the Consequents by themselves) make 12 to 6; i. e. 2 to 1. Which being

apply'd, amounts to this; Ratio Ses∣quialtera, 3 to 2, added to Ratio Ses∣quitertia 4 to 3; makes Duple Ration, 2 to 1. Therefore Diapente added to Diatessaron, makes Diapason.

Substraction of One Ration from a∣nother greater, is performed in like manner, by Multiplying the Terms; but this is done not Laterally, as in Ad∣dition, but Crosswise; by Multiplying the Antecedent of the Former (i. e. of the Greater) by the Consequent of the Latter, which produceth a new Antece∣dent; and the Consequent of the For∣mer by the Antecedent of the Latter; which gives a new Consequent. And therefore it is usually done by an Ob∣lique Decussation of the Lines. For Ex∣ample, If you would take 6 to 5 out of 4 to 3, you may set them down thus. Then 4 mul∣tiply'd by 5 makes 20; and 3 by 6 gives 18. So 20 to 18; i. e. 10 to 9, is the Remain∣der.

That is, Substract a Third Mi∣nor out of a Fourth, and there will re∣main a Tone Minor.

Multiplication of Rations is the same with their Addition; only it is not wont to be of divers Rations, but of the same, being taken twice, thrice, or oftener, as you please. And as before, in Addition, you added di∣vers Rations by Multiplying them: So here, in Multiplication, you add the same Ration to it self, after the same manner, viz. by Multiplying the Terms of the same Ratio by them∣selves; i. e. the Antecedent by it self, and the Consequent by it self (which in other words is to Multiply the same by 2) and will, in the Operation, be to Square the Ration first propounded (or give the Second Ordinal Power; the Ration first given being the First Power or Side) And to this Product, if the Simple Ration shall again be ad∣ded (after the same manner as before)

the Aggregate will be the Triple of the Ration first given; or the Pro∣duct of that Ration Multiply'd by 3; viz. the Cube, or Third Ordinal Power. Its Biquadrate, or Fourth Power, proceeds from Multiplying it by 4; and so successively in order as far as you please you may advance the Powers. For instance, The Du∣ple Ration 2 to 1, being added to it self, Dupled, or Multiply'd by 2, produceth 4 to 1 (the Ration Qua∣druple) and if to this, the first again be added (which is equivalent to Mul∣tiplying that said first by 3) there will arise the Ration Octuple, or 8 to 1. Whence the Ration 2 to 1, be∣ing taken for a Root, its Duple 4 to 1, will be the Square; its Triple 8 to 1, the Cube thereof, &c. as hath been said above. And to use another instance; To Duple the Ration of 3 to 2; it must be thus Squar'd; 3 by 3 gives 9; 2 by 2 gives 4.

So the Duple or Square of 3 to 2, is 9 to 4. Again, 9 by 3 is 27; and 4 by 2 is 8: So the Cubic Rati∣on of 3 to 2 is 27 to 8. Again, to find the Fourth Power, or Biquadrate; (i. e. Squar'd Square) 27 by 3 is 81; 8 by 2 is 16: So 81 to 16 is the Ration of 3 to 2 Quadrupled; as 'tis Dupled by the Square, Tripled by by the Cube, &c. To apply this In∣stance to our present purpose; 3 to 2 is the Ration of Diapente, or a Fifth in Harmony; 9 to 4 is the Ration of twice Diapente, or a Ninth (viz. Diapason with Tone Major) 27 to 8 is the Ration of thrice Diapente, or three Fifths; which is Diapason with Six Major (viz. 13th Major) The Ration of 81 to 16 makes four Fifths, i. e. Dis-diapason, with two Tones Major; i. e. a Seventeenth Major, and a Comma of 81 to 80.

To Divide any Ration, you must take the contrary way; And by Ex∣tracting of these Roots Respectively, Division by their Indices will be per∣formed. E. gr. To Divide 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 just half so much as 9 to 4.

From hence it will be obvious to any to make this Inference; That Ad∣dition and Multiplication of Rations are (in this Case) one and the same thing. And these Hints will be suffi∣cient to such as bend their Thoughts to these kinds of Speculations, and no great Trespass upon those that do not.

The Advantage of proceeding by the Ordinal Powers, Square, Cube, &c. (as is before mentioned) may be ve∣ry usefull where there is occasion of

large Progressions. As to find (for Example) how many Comma's are contained in a Tone Major, or other Interval. Let it be, How many are in Diapason; Which must be done by Multiplying Comma's; i. e. Ad∣ding them, till you arrive at a Ration Equal to Octave (if that be sought) viz. Duple. Or else by Dividing the Ration of Diapason, by that of a Comma, and finding the Quotient; which may be done by Logarithms. And herein I meet with some Diffe∣rences of Calculations.

Mersennus finds, by his Calculation, 58 1/•• Comma's, and somewhat more in an Octave. But the late Nicholas Mercator, a Modest Person, and a Learned and Judicious Mathematici∣an, in a Manuscript of his, of which I have had a Sight; makes this Re∣mark upon it. In solvendo hoc Proble∣mate aberrat Mersennus. And He, working by the Logarithms, finds out

but 55, and a little more. And from thence has deduced an Ingenious Inven∣tion of finding and applying a least Common Measure to all Harmonic Intervals; not precisely perfect, but very near it.

Supposing a Comma to be 1/53 part of Diapason; for better Accommo∣dation rather than according to the true Partition 1/55; which 1/53 he calls an Artificial Comma, not exact, but dif∣fering from the true Natural Comma about 1/20 part of a Comma, and 1/1000 of Diapason (which is a Difference imperceptible) Then the Intervals within Diapason will be measured by Comma's according to the following Table. Which you may prove by adding two, or three, or more of these Numbers of Comma's, to see how they agree to constitute those Intervals, which they ought to make; and the like by substracting.

Intervals	••/53
Comma	1
Diesis	2
Semit. Minus	3
Semit. Medium	4
Semit. Majus	5
Semit. Max.	6
Tone Minor	8
Tone Major	9
3d Minor	14
3d Major	17
4th	22
Tritone	26
Semidiapente	27
5th	31
6th Minor	36
6th Major	39
7th Minor	45
7th Major	48
Octave	53

This I thought fit, on this occasi∣on, to impart to the Reader, having leave so to doe from Mr. Mercator's Friend, to whom He presented the said Manuscript.

Here I may advertise the Reader; that it is indifferent whether you com∣pare the greater Term of an Harmo∣nic Ration to the less, or the less to the

greater; i. e. whether of them you place as Antecedent. E.gr. 3 to 2, or 2 to 3. Because in Harmonics, the proporti∣ons of Lengths of Chords, and of their Vibrations are reciprocal or Counter-changed. As the Length is increased, so the Vibrations are in the same proportion decreased; & è con∣tra. If therefore (as in Diapente) the length of the Unison String be 3, then the length (caeteris paribus) of the String which in ascent makes Diapente to that Unison must be 2, or 2ds / 3. Thus the Ration of Diapente is 2 to 3 in respect of the length of it, compared to the length of the Unison String.

Again, the String 2 vibrates thrice, in the same time that the String 3 vi∣brates twice. And thus the Ration of Diapente in respect of Vibrations is 3 to 2. So that where you find in Au∣thors, sometimes the greater Number in the Rations set before and made the Antecedent, sometimes set after

and made the Consequent; You must understand in the former, the Ration of their Vibrations; and in the latter, the Ration of their Lengths; which comes all to one.

Or you may understand the Uni∣son to be compared to Diapente above it, and the Ration of Lengths is 3 to 2; of Vibrations 2 to 3: or else Di∣apente to be compared to the Unison, and then the Ration of Lengths is 2 to 3; of Vibrations 3 to 2. This is true in single Rations, or if one Ration be compared to another; then the two Greater Terms must be ranked as Antecedents: or otherwise, the two Less Terms.

The Difference between Arithme∣tical and Geometrical Proportion is to be well heeded. An Arithmetical mean Proportion is that which has Equal difference to the Antecedent and Consequent Terms of those Num∣bers to which it is the Mediety; and

is found by adding the Terms and ta∣king half the Sum. Thus between 9 and 1, which added together make 10, the Mediety is 5; being Equidif∣ferent from 9 and from 1; which Dif∣ference is 4: As 5 exceeds 1 by 4; so likewise 9 exceeds 5 by 4. And thus in Arithmetical Progression 2, 4, 6, 8; where the Difference is onely consider∣ed, there is the same 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 Proporti∣on where is considered, not the Nu∣merical Difference, but another Habi∣tude of the Terms, viz. How many times, or how much of a time, or times, one of them doth contain the other (as hath been explained at large in the beginning of this Chapter.) There the Mean Proportional is not the same with Arithmetical, but found another way; and Equidifferent Progressions make different Rations. The Rations

(taking them all in their least Terms) expressed by less Numbers, being grea∣ter than those of greater Numbers, I mean in Proportions Super-particu∣lar, &c. Where the Antecedents are Greater than the Consequents: (as on the Contrary, where the Antecedents are Less than the Consequents, the Ra∣tio's of Less Numbers are Less than the Ratio's of Greater.) The Mediety of 9 to 1, is not now 5, but 3; 3 having the same Ration to 1, as 9 has to 3 (as 9 to 3, so 3 to 1) viz. Triple. And so in Progression Arith∣metical, of Terms having the same Differences; if considered Geometrical∣ly, the Terms will all be comprehen∣ded by unequal Rations. The Diffe∣rences of 2 to 4, 4 to 6, 6 to 8, are Equal; but the Rations are unequal: 2 to 4 is less than 4 to 6, and 4 to 6 less than 6 to 8. As on the Contrary; 4 to 2, is greater than 6 to 4; and •• to 4 greater than 8 to 6. For 4 to 2

is Duple; 6 to 4 but Sesquialtera (one and a half onely, or 2/2) and 8 to 6 is no more than Sesquitertia, (one and a Third part, or 4/3) which shews a con∣siderable Inequality of their Rations. In like manner, 6 to 2 is Triple; 8 to 4 is but Duple; and yet their Differen∣ces Equal. Thus the mean Rations comprehended in any greater Ration divided Arichmetically; i. e. by Equal Differences; are unequal to one ano∣ther considered Geometrically. Thus 2, 3, 4, 5, 6; if you consider the Numbers, make an Arithmetical Pro∣gression: But if you consider the Rati∣ons of those Numbers, as is done in Harmony, then they are Unequal; eve∣ry one being greater or less (according as you proceed by Ascent or Descent) than the next to it. Thus in this pro∣gression, 2 to 3 is the greatest, being Diapente; 3 to 4 the next, Diatessaron; 4 to 5 still less, viz. Ditone; 5 to 6 the least, being Sesquitone. Or, if you des∣cend,

6 to 5 least; 5 to 4 next, &c. These are the mean Rations compre∣hended in the Ration of 6 to 2, by which Diapason cum Diapente, or a 12th, is divided into the aforesaid Intervals, and measured by them: viz. as is 6 to 2, (viz. Triple.) So is the Aggregate of all the mean Rations within that Num∣ber; 6 to 5, 5 to 4, 4 to 3, and 3 to 2. Or 6 to 5, 5 to 2; or 6 to 4, 4 to 2; or 6 to 3, 3 to 2. The Ag∣gregates of these are Equal to 6 to 2, viz. Triple.

This is premised in order to pro∣ceed to what was intimated in the foregoing Chapter.

Taking notice first of this procedure, peculiar to Harmonics; viz. To make Progression or Division in Arithmetical Proportion in respect of the Numbers; but to consider the things Numbred ac∣cording to their Rations Geometrical. And thus Harmonic proportion, is said to be compounded of Arithmetical and Geometrical.

You may find them all in the Divi∣sion of the Systeme of Diapason, into Diapente and Diatessaron, i. e. 5th and 4th; ascending from the Unison.

If by Diapente first, Then by 2, 3, 4, Arithmetically: If first by Diatessa∣ron, Then by 3, 4, 6, Harmonically. And these Rations considered Geome∣trically, in Relation to Sound; There is likewise found Geometrical Proporti∣ons between the Numbers 6, 3, to 4, 2; and 6, 4, to 3, 2.

The Antients therefore owning one∣ly 8th, 5th, and 4th, for Simple Con∣sonant Intervals; concluded them all within the Numbers of 12, 9, 8, 6, which contained them all: viz. 12 to 6, Diapason; 12 to 8, Diapente; 12 to 9, Diatessaron; 9 to 8, Tone. And which served to express the three Kinds of Pro∣portion; viz. Harmonical, between 12 to 8, and 8 to 6; Arithmetical, between 12 to 9, and 9 to 6; and Geometri∣cal, between 12 to 9, and 8 to 6;

and between 12 to 8, and 9 to 6. It was said therefore, That Mercu∣rius his Lyre was strung with four Chords, having those Proportions, 6, 8, 9, 12. Gassend.

I intimated that I would here more largely explain that ready and ea∣sie way of finding and measuring the mean Rations contained in any of those Harmonick Rations given, by transfer∣ring them out of their Prime or Radi∣cal Numbers, into greater Numbers of the same Ration. By Dupling (not the Ration, but the Terms of it: still conti∣nuing the same Ration) you will have one Mediety: as 2 to 1 Dupled is 4 to 2; and you have 3 the Mediety. By Tripling you will have two Means; 2 to 1 Tripled is 6 to 3, containing 3 Rations; 6 to 5, 5 to 4, 4 to 3; and so still more, the more you multiply it.

Now observe, First, That any Ra∣tion Multiplex or Superpartient (or by

transferring it out of its Radical Num∣bers made like Superpartient) contains so many Superparticular Rations, as there are Units in the Difference be∣tween the Antecedent and the Conse∣quent. Thus in 8 to 4 (being 2 to 1 transferred by Quadrupling) the Difference is 4, and it contains 4 Su∣perparticular Rations; viz. 8 to 7, 7 to 6, 6 to 5, and 5 to 4. Where though the Progression of Numbers is Arith∣metical, yet the Proportions of excess are Geometrical and Unequal. The Superparticular Rations expressed by less Numbers, being Greater, as hath been said, than those that consist 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 Instance, there are two Ra∣tions not appertaining to Harmonics; viz. 8 to 7, and 7 to 6.

Secondly therefore, you may make unequal Steps, and take none but Har∣monick Rations, by Selecting Greater and fewer intermediate Rations, tho' some of them composed of several Su∣perparticulars; provided you do not discontinue the Rational Progression, but that you repeat still the last Conse∣quent, making it the next Antecedent. As if you measure the Ration of 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 these three ways the Rations are all Harmonical, and are respectively contained in, and make up the Ration of 8 to 4. Thus you may measure, and divide, and compound most Harmonick Rations without you Pen.

To that End, I would have my Reader to be very perfect in the Radical Numbers, which express the Rations of the Seven first (or uncompounded) Consonants: viz. Diapason, 2 to 1;

Diapente, 3 to 2; Diatessaron, 4 to 3; Ditone, 5 to 4; Trthemitone, 6 to 5; Hexachordon Majus, 5 to 3; Hexa∣chordon Minus, 8 to 5. And likewise of the Degrees in Diatonick Harmony, viz. Tone Major, 9 to 8; Tone Minor, 10 to 9; Hemitone Major, 16 to 15. And the Differences of those Degrees; Hemitone Greatest, 27 to 25; and He∣mitone Minor, 25 to 24; Comma, or Schism, 81 to 80; Diesis Enharmonic, 128 to 125.

Of other Hemitones, I shall treat in the Eighth Chapter.

Now if you would divide any of the Consonants into two Parts, you may do it by the Mean, or Mediety of the two Radical Numbers; if they have a Mean: And where they have not (as when their Ratio's are Super∣particular) you need but Duple those Numbers, and you will have a Mean (one or more.) Thus Duple the Num∣bers of the Ration of Diapason, 2 to 1;

and you have 4 to 2; and then 3 is the Mean by which it is divided into two Unequal, but Proper and Harmonical parts; viz. 4 to 3, and 3 to 2. After this manner Diapason, 4 to 2; com∣prehends 4 to 3, and 3 to 2. So Dia∣pente, 6 to 4; is 6 to 5, and 5 to 4. Ditone, 10 to 8; is 10 to 9, and 9 to 8. So Sixth Major, 5 to 3; is 5 to 4, and 4 to 3.

Though, from what was now ob∣serv'd, you may divile any of the Consonants into intermediate Parts; yet when you divide these three fol∣lowing, viz. Sixth Minor, Diatessaron, and Trihemitone; you will find that those Parts into which they are divided, are not all such Intervals as are Harmoni∣cal. The Sixth Minor, whose Ration is 8 to 5, contains in it three Means; viz. 8 to 7, 7 to 6, and 6 to 5; the last whereof onely is one of the Har∣monick Intervals, of which the Sixth Minor consists; viz. Tribemitone: and

to make up the other Interval, viz. Diatessaron; you must take the other two, 8 to 7, and 7 to 6; which be∣ing added (or, which is the same thing, taking the Ratio of their two Extream Terms, That being the Sum of all the Intermediate ones added) you have 8 to 6, or (in the least Terms) 4 to 3. Again Diatessaron, in Radical Numbers, 4 to 3; being (if those Numbers are dupled) 8 to 6, gives for his Parts, 8 to 7, and 7 to 6; which Ratio's agree with no Inter∣vals that are Harmonick. Therefore you must take the Ration of Diatessa∣ron in other Terms, which may afford such Harmonick Parts. And to do this, you must proceed farther than dupling (or adding it once to it self) for to this Duple you must add the first Radical Numbers once again (which in effect is the same with Tri∣pling it at first) viz. 4 and 3, to 8 and 6; and the Aggregate will be a

new, but Equivalent, Ration of Dia∣tessaron; viz. 12 to 9. And this gives you three Means, 12 to 11, and 11 to 10; both Unharmonical; but, which together are, as was shewed be∣fore, the same with 12 to 10 (or 6 to 5) Trihemitone; and 10 to 9, Tone Minor: and are the two Harmonical Intervals of which Diatessaron consists, and which divide it into the two near∣est Equal Harmonick Parts. Lastly Trihemitone, or Third Minor, 6 to 5; or (those Numbers being dupled) 12 to 10, gives 12 to 11, and 11 to 10; which are Unharmonical Rations: but Tripled (after the former manner) 6 to 5 gives 18 to 15; which divides it self (as before) into 18 to 16, Tone Major; and 16 to 15, Hemitone Ma∣jor.

Thus by a little Practice all Har∣monick Intervals will be most easily measured, by the lesser Intervals com∣prized in them. Now, (for exercise

sake) take the Measures of a greater Ration: Suppose that of 16 to 3 be gi∣ven as an Harmonick Systeme. To find what it is, and of what Parts it con∣sists: First find the gross Parts, and then the more Minute. You will pre∣sently judge, that 16 to 8 (being a Part of this Ration) is Diapason; and 8 to 4 is likewise Diapason: then 16 to 4 is Disdiapason, or a Fifteenth; and the remaining 4 to 3, is a Fourth. So then, 16 to 3, is Disdiapason, and Dia∣tessaron; i. e. an Eighteenth: 16 to 8, 8 to 4, and 4 to 3.

But to find all the Harmonick Inter∣vals within that Ration (for we now consider Rations as relating to Harmo∣ny) take this Scheme.

16 to 3 contains,

	In Radicals.
16 to 15,		Hemitone.
15 to 12,	5 to 4,	Ditone.
12 to 10,	6 to 5,	Trihemitone.
10 to 9,		Tone Minor.
9 to 8,		Tone Major.
8 to 6,	4 to 3,	Diatessaron
6 to 5,		Trihemitone.
5 to 4,		Ditone.
4 to 3,		Diatessaron.
Tot. 16 to 3.	Disdiapason cum Diatessaron.

Or Thus,

	In Radicals.
16 to 10,	8 to 5,	6th Minor.
10 to 6,	5 to 3,	6th Major.
6 to 4,	3 to 2,	5th
4 to 3,		4th
Tot. 16 to 3.		Eighteenth.

All these Invervals thus put toge∣ther are comprehended in, and make up the Ration of 16 to 3; being ta∣ken in a Conjunct Series of Rations.

But otherwise, within this Compass of Numbers are contained many more Expressions of Harmonick Ra∣tion. Ex. gr.

	Radicals.
16 to 15.
16 to 12,	4 to 3.
16 to 10,	8 to 5.
16 to 8,	2 to 1.
16 to 6,	8 to 3.
16 to 4,	4 to 1.
16 to 3.
15 to 12,	5 to 4.
15 to 10,	3 to 2.
15 to 5,	3 to 1.
15 to 3,	5 to 1.
14 to 7,	2 to 1.
12 to 10,	6 to 5.
12 to 9,	4 to 3.
12 to 8,	3 to 2.
12 to 6,	2 to 1.
12 to 4,	3 to 1.
12 to 3,	4 to 1.
10 to 9.
10 to 8,	5 to 4.
10 to 6,	5 to 3.
10 to 5,	2 to 1.
9 to 8.
9 to 6,	3 to 2.
9 to 3,	3 to 1.
8 to 6,	4 to 3.
8 to 5,
8 to 4,	2 to 1.
6 to 5, &c.
Vid. Pag. 67.

And now I suppose the Reader bet∣ter prepared to proceed in the remain∣der of this Discourse, where we shall treat of Discords.

Notes

• 4	3
  6	5
  24.	15.
• 4	3
  6	5
  20.	28.
  10.	9.