On the theory of the art of singing 1)

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Definitions

1st Definition

Step is the next subsequent ascent which one rises in natural singing, of which the smaller variety is called minor step, the larger, major step.

2nd Definition 2)

Natural singing is that which by an orderly ascent takes place as follows: two major steps, one minor, three major steps, one minor, two major steps, one minor, three major steps, one minor, and so on gradually, in orderly sequence.

Explanation

Since lay people, not knowing the difference between half steps and whole steps, use this progression by natural inclination, it is with good reason called natural singing, because singing two or three half steps, or four or five whole steps in succession not only is difficult to accomplish, but also is unpleasant to hear as well as unnatural.

3rd Definition

These seven steps, ascending in an orderly way according to natural singing, make one round of singing.

Explanation

When one rises seven steps above a given note in an orderly ascent, the last sound is so similar to the first that it seems as if one had made a round and arrived again at where one began, so that, because of that similarity, this is called a round. This is rather similar to what happens in astronomy with the helices described daily by the moon owing to its daily motion, which, though not being properly parallel circles, are so called because of this similarity.

4th Definition

Those seven steps are called as follows: ut, re, mi, fa, sol, la, si, among them the steps from mi to fa and from si to ut are 3) minor steps and all the others major steps.

5th Definition

Two sounds having the same pitch, their relation is called selftone. But differing by a minor step, semitone. Differing by a major step, a whole tone. Differing by one major and one minor step, one-tone-and-half. Differing by two major steps, ditone. And so forth in an orderly way.

1) The selected pages which follow have been taken from the part that is in Stevin's own handwriting, with the exception of the chapter on the modes.
2) The second definition corresponds to the scale, sung to the vocables ut, re, mi, fa, sol, la, si, ut, re, etc.
3) The fourth definition was in contradiction to the second definition, in that the step la : si was here called a minor step, which does not fit in with the sequence major - major - minor - major - major - major - minor - major - etc. We therefore put: si to ut. If si is flattened by a semitone, the resulting note is called sa. Thus la to sa and si to ut are semitones; la to si and sa to ut whole tones.

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6th Definition

Two sounds having the same pitch, their relation is also called a first (or prime), but when differing by one step, it is called a second, and when this step is a minor step, it is properly called a minor second, when it is a major step, a major second. Likewise, when the sounds differ by two steps, their relation is called a third, and when one of them is a minor step, it is called a minor third, but when both are major steps, it is called a major third, and so on to the seventh, the steps following it being called double-first, double-second, and so on in a regular way like the simple first, second, and the following.

Explanation

Singable sounds receive names of two different kinds, as described in the foregoing 5th and 6th definitions, each of which has its special use. For when the ratios of sounds have to be added or subtracted, they are more conveniently referred to by the names of the tones, since, the tritone added to a ditone, their sum is a five-tone; the ditone being subtracted from the three-tone-and-half, the remainder is the one-tone-and-half, in such a way that sums and remainders receive names in conformity with the numbers to which they correspond.
The names of firsts, seconds, thirds, etc. are more convenient for practical use in the composition *) of songs. For it is easier and more convenient to count the difference between two sounds by steps ascending or descending according to natural singing than by whole tones and semitones, since the number of the steps does not agree with the number of tones.

Postulate

We postulate that as one part of a string is to another, so is the coarseness of the sound of the one to that of the other.

Explanation

When two persons sing together a double-first, the coarser voice of the lower has an appearance of doubleness with respect to the sharp voice of the higher, i.e. as 2 yards are double to 1 yard, so this lower voice in coarseness seems to be double to the higher. It is true that this doubleness does not present itself quite so clearly and intelligibly in sound as it does in size, number, weight, time, motion, and otherwise; yet the stretched string itself induces us to grant this, since if its parts are in the double ratio as to size, the same sounds ring that we say to be in the double ratio as to coarseness. For the whole string, when played against its half, together make us hear the aforesaid double-first. Further, as it has here been said that, the whole string being to its half in double ratio as to size, its sound is in double ratio as to coarseness, so also it is to be understood that, the whole string being to its quarter in a certain ratio as to size, its sound has the same ratio as to coarseness, and so on for all other cases, parts of the string against each other as well as parts against the whole string.
Now if anyone should wish to deny one half of the string to sound against the whole string as a double-first, and on this account should not admit the coarseness of sounds to be in the ratio of the parts of the string, what is described above has been put as a postulate, since such a postulate in principle does not require any proof. Anyhow, as experience teaches, the contrary does not happen unless owing to false strings or some other mishap.

*)Compositione cantus.

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2nd Postulate

All whole tones to be equal and likewise all semitones to be equal.

Explanation

The meaning is this: that one rises as much from ut to re as from re to mi, and also from fa to sol, from sol to la, and from sa 1) to ut. That likewise one rises as much from mi to fa as from la to sa.

Theorem. 2) Proposition
1 : 1 Selftone First (prime)
1 : (12) 1/2 Semitone Minor second
1 : (6) 1/2 Whole tone Major second
1 : (4) 1/2 One-tone-and-half Minor third
1 : (3) 1/2 Ditone Major third
1 : (12) 1/32 Two-tone-and-half Fourth
1 : 1/2 Tritone {Bad major fourth or bad minor fifth}
1 : (12) 1/128 Three-tone-and-half Fifth
1 : (3) 1/4 Four-tone Minor sixth
1 : (4) 1/8 Four-tone-and-half Major sixth
1 : (6) 1/32 Five-tone Minor seventh
1 : (12) 1/2048 Five-tone-and-half Major seventh
1 : 1/2 Six-tone Double-first (octave)

Proof (abest)
On Ratio in General

Because ratios in the field of sound are not as manifestly known as in other fields where we meet with them, for the sake of greater clarity we shall first speak about ratios and equirationality 3) in general; subsequently about the aspect of ratio in singing by comparison with the familiar ratio in geometry. And finally of the ratios proper to musical sounds.
Ratio as defined generally is the relation according to quantity between things of the same nature. As in number, size, weight, time: 6, 6 feet, 6 pounds, 6 hours are in the double ratio to 3, 3 feet, 3 pounds, 3 hours. Equirationality is the relation of two equal ratios. For instance, 6 to 3 is a double ratio, so also is 8 to 4, therefore the ratio of 6 to 3 is equal to the ratio of 8 to 4. They are thus equal ratios, and their relation, i.e. saying that as 6 is to 3, so is 8 to 4, is an equirationality, or 6, 3, 8, 4 are terms of an equirationality.
Look here: Dutch words, easy to understand and modest in appearance, but in reality of an infinite power. For if one conciders the thing defined, viz. equirationality, it is like a definition of its substance, the mere ring of which, at first hearing, brings home to us and shows us that the very thorough understanding of

1) In the manuscript, erroneously, si. This same error has been pointed out in note 3 p. 423. Sometimes Stevin writes the scale as fa sol la sa ut re mi fa. Cf. figure 1 p. 436.
2) The notation (n) a is currently used by Stevin for na
3) In Stevin's Dutch: everedenheijt.

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equirationality was not found among the Greeks and their successors. For (leaving aside many other things, to be discussed elsewhere) from saying that 6, 4, 3 of three sounds make a musical equirationality endless vanities follow and are concluded.1)
The answer, called forth by such a saying is: these form no equal ratios, and consequently there is no equirationality either. The cause of these errors is that the language of the Greeks could not interpret this term together with all other mathematical terms as properly as the Dutch language does. Therefore, if one wished to speak with good discrimination of the suitability of languages, one might say that knowledge of Greek is useful in order to bring to light various discoveries of the Greek, which in their days were the most important, by translation from the Greek into other languages; as is done daily. Latin (because it has happened to become the common language of the world) serves to be understood in all countries, and also to study all sciences about all subjects, which have been described in it by various kinds of people. French, Italian, Spanish, Polish, etc. serve to carry on trade, everybody according to his situation. But DUTCH serves to teach the liberal arts, to fathom the hidden secrets of nature, and to prove that miracle is no miracle.2) Therefore, whoso should be minded to seek after the great Wisdom, of which the knowledge of the Chaldees and the Egyptians formerly was a remnant, would find it useful to go to this source or first origin from which they had got it, learning diligently in Dutch, among other things, what is the aforesaid equirationality. For this word depicts the character of this great matter properly. Other words, such as Proportio, Analogia, are unable to do so, as is clearly shown by practice in several places.

Comparison of Geometrical Ratio mith Musical Ratio

So far we have spoken of ratios in general, but in order to explain now, as intended, ratio in singing by comparison with geometrical ratio, we are to know that, as the geometrical ratio consists in the largeness and smallness of figures, which is measured by length, so ratio in singing consists in the coarseness and sharpness of sounds, which is measured by height or lowness. Thus when two persons sing a double-first, it is said, in view of this difference in lowness of one below the other, that the coarser voice is double below the sharper one. And all the other greater or smaller singing ratios are made of the same stuff as the stuff this doubleness is made of. Again, just as all the ratios of two given rectilinear plane figures or solids cannot be recognized by sight, but obey geometrical rules teaching us how to find them, so all the ratios of two given sounds cannot be judged by hearing, but they are revealed by means of the musical rules governing them, which we now have to discuss.

1) Between two numbers p and q one can have an arithmetic mean (a), a harmonic mean (h), and a geometric mean (g). These are defined by p - a = a - q; 1/p - 1/h = 1/h - 1/q, and p : g = g : q. Obviously the numbers 6, 4, and 3 quoted by Stevin show the harmonic mean 4 of the outer terms 6 and 3, 4 being 1/3 more than 3 and 1/3 less than 6 so that 1/3 - 1/4 = 1/4 - 1/6. Accordingly the note, given by a length of string 4, is the harmonic mean of the notes given by the lengths 6 and 3. The latter make an octave. The harmonic mean gives a fifth against the lower, a fourth against the higher note. Stevin has in mind geometrical ratio only, and he objects to equating two musical (singconstighe) ratios to be construed from the three numbers in question. Obviously he refuses to admit the harmonic ratio to be called a ratio.
2) "Wonder en is gheen wonder" was the motto on the frontispiece of Stevin's treatise on statics, De beghinselen der Weeghconst.

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On the Ratios of Singable Sounds According to the Opinion of the Greeks

Experience shows that a stretched string on some instrument, such as a lute, a cither, a violin, or the like, produces against its half a sound so similar to it that it has a semblance of identity, the coarseness ratio of which, by some natural inclination, we understand to be double, but not with the same evidence as the doubleness with which we meet in other matters, as has been said before; but this is confirmed more perceptibly by the bodies producing these sounds, such as the whole string and its half, which are also in the double ratio. The same also applies to half the string and its quarter, its eighth, its sixteenth part, etc. in this sequence, for all these sounds have the aforesaid semblance of identity, with the understandable form of the fourfold, eightfold, sixteenfold ratio of coarseness. The same is also obvious in all the ratios besides the above-mentioned sequence. For if we take a part of the string whose ratio to the whole string is one half of the ratio of the double part,1) its sound will also have dropped halfway by properly estimated lowness. But because this known dropping is the measure of the coarseness, as we have stated above, here the ratio of coarseness is known to us, and the same also applies to other similar sounds, from which it is concluded that as this part of the string is to that part of the string, so also is the sound of this to the sound of that, i.e. the parts of the string produce sounds in the ratio of their sizes.
This being noted in former times, the impulse to the true division of the string was such that it should comprise the notes proper which we sing in natural singing. Which the Greeks, to distinguish it from what they called Chromaticum and Enharmonicum genus,2) called Diatonicum genus, in order that thus natural singing might be hit off flawlessly on musical instruments.
To effect this, one merely requires some interval 3) containing a semitone, such as a one-tone-and-half, a two-tone-and-half, a three-tone-and-half, etc., because from these, by way of addition and subtraction of the ratios, one can accurately find all the rest without hearing any further sounds. For this purpose they took the fifth, i.e. the three-tone-and-half, and found the true ratio of the length of the string and its part producing this three-tone-and-half to be very close to the ratio of 3 to 2; they estimated that the ratio 3 : 2 was the true one. Proceeding therewith as if it were the true ratio, they subtracted it from the ratio 2 : 1 of the six-tone, the remainder being the ratio 4 : 3 of the two-tone-and-half; the latter being subtracted from the ratio 3 : 2 of the three-tone-and-half, the remainder is the ratio 9 : 8 for the whole tone. When another ratio 9 : 8 is added thereto, this gives the ratio 81 : 64 of the ditone. When this is subtracted from the ratio 4 : 3 of the two-tone-and-half, the remainder for the semitone is 256 : 243, etc. But if the melodic line, or to speak more concretely: the neck of a lute or cither, is divided according to the above-mentioned ratios, experience shows patently by hearing that this is not the semitone, because it is much too small. Therefore the natural notes are not correctly hit off by such a division. And although the Ancients perceived this fact, nevertheless they took this division to be correct and perfect, and preferred to think that the defect was in our singing

1) Stevin means. the square root of 1/2.
2) The manuscript, erroneously, has harmonicum.
3) Stevin writes: toon.

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(as if one should say: the sun may lie, but the clock cannot). They even considered the sweet and lovely sounds of the minor and the major third and sixth, which sounded unpleasant in their misdivided melodic line, to be wrong, the more so because a dislike for inappropriate numbers moved them to do so. But when Ptolemy afterwards wanted to amend this imperfection, he divided the aforesaid genus diatonicum in a different way, making a distinction between a major whole tone in the ratio 9 : 8 and a minor whole tone in the ratio 10 : 9, a difference that does not exist in nature, for it is obvious that all whole tones are sung equal. Since these tones of Pythagoras and of Ptolemy displeased Zarlino, he made yet another division, distributing a certain comma (which remained in Ptolemy's division) over one tone and another, where it seemed appropriate to him, but tentatively.1)
All these misconceptions have originated from the fact that fundamentally the nature of equirationality was not understood, which was not due to a lack of brains, for their acts as have come down to us show sufficiently that the Greek were of the most intelligent that Nature produces, but they lacked a good tool, viz. the Dutch language, without which in the most profound matters one can accomplish as little as a skilled carpenter without good tempered tools can carry out his trade. For just as one cannot grasp the geometrical properties of a square by means of an unsuitable curvilinear figure so well as by means of a proper square figure according to Proposition 4, sub 5,2) the continual sight of which continually strengthens one's insight, so it was not possible to penetrate into the most profound secrets of Nature as thoroughly by means of the unsuitable Greek language (unsuitable as compared to Dutch) as by means of this eminently suitable and most perfect language of languages, whose characteristic designation pictures the matter designated so clearly for us that the matter itself thus seems to be continually before our eyes, whilst in other languages it remains incomprehensible and obscure, as experience amply proves, among other things in the present matter. For whoso puts the ratio 3 : 2 for the fifth, proceeding therewith five times, and in the end not coming out right, still holds that the ratio 3 : 2 is the actual one, he in truth ignores the essential character of addition and subtraction of ratios.
But in order to show up this misapprehension as to the misunderstood character of ratios by means of an example in intelligible words with numbers, let us take some number, such as 110, as representing the double-first, or the six-tone, and five persons A, B, C, D, E, representing in this order the three-tone-and-half, the two-tone-and-half, the whole tone, the ditone, and the semitone, putting therewith the requirement similar to the preceding Pythagorean operation with the ratio, as follows: Subtract from 110 what A should have, the remainder is for B; subtract B from A, the remainder is for C; add to this the same amount, then the sum is for D; after subtraction of the latter from B, the remainder should be 35 for E. To arrive at this result, somebody takes for A a number which as a superficial guess appears to him close enough, for instance 60. Proceeding with this as if it were the true number, he subtracts it from 110, there remains 50 for B;

1) For this question, see note A, page 460.
2) Reference to an item of Stevin's treatise De Meetdaet, Work XI.

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this being subtracted from 60, there remains 10 for C; adding 10 again to this, he finds 20 for D; this being subtracted from the 50 of B, there remains 30 for E. But E was to have 35, so he sees patently that E has not received his due. But not perceiving that this follows from the fact that the first number for A, i.e. 60, had not been put right, he considers the last mishap to be the secret of Nature and looks upon his former supposition as correct. But what will the experienced arithmetician say to this? Certainly, and with good reason, that such a person is insufficiently acquainted with the properties of arithmetic, for he knows that the correct portion for A is 59; when this is subtracted from 110, there remains 51 for B; this from 59, there remains 8 for C; adding another 8, that makes 16 for D; this subtracted from 51 of B, there remains 35 for E, as required.
The same thing happened in the calculation of the ratios of sounds, for a ratio had to be put for the fifth and this had to be subtracted and added according to a given rule, in such a way that finally the true ratio of the semitone should remain. In fact this was found not to come right if the ratio was put to be 3 : 2, and people continually went on maintaining this ratio 3 : 2 to be the true one. Truly, as patently as the arithmetician above saw that the man who put 60 for A did not sufficiently understand arithmetic, feeling at once the cause of his error, so clearly the expert in equirationality sees that the supporters of the ratio 3 : 2 for the fifth have not essentially understood the fundamental character of ratios and equirationalities, which was due, as said above, to the fact that they had no words which could designate mathematical matters as adequately as DUTCH.

Of the True Ratios of Natural Tones

But, to come to the point and to describe the proper ratios of natural intervals 1), I say that the true ratio of the fifth or the three-tone-and-half is 1 to (12) 1/128, i.e. 1 to the twelfth root of 1/128. When this is subtracted from the ratio 2 : 1 of the sixtone, there remains the ratio of 1 to (12) 1/32 for the two-tone-and-half. When this again is subtracted from the aforesaid ratio of the three-tone-and-half, there remains the ratio of 1 to (6) 1/2 for the whole tone. Addition to this of the same ratio makes the ratio of 1 to (3) 1/2 for the ditone. This being subtracted from the above-mentioned ratio of the two-tone-and-half, there remains the ratio of 1 to (12) 1/2 for the semitone. To prove this, let A, B, C, D, E, F, G, a, b, c, d, e, f, g designate the keys of an organ or a harpsichord, and H, I, K, L, M, N, O, P, Q, R the intermediate keys, which are called slit keys. Because this instrument is more convenient for the present purpose than the monochord, let it be tuned with the perfect natural tones, as follows:

Above F the double-first f with the fifth c between them
Below c the double-first C with the fifth G between them
Above G the double-first g with the fifth d between them
Below d the double-first D with the fifth a between them
Below a the double-first A with the fifth E between them
Above E the double-first e with the fifth b between them
Below b the double-first B with the fifth L between them
Above L the double-first Q with the fifth O between them

1) Stevin writes: toonen (= tones).

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Below O the double-first I with the fifth M between them
Above M the double-first R with the fifth P between them
Below P the double-first K with the fifth N between them
Below N the double-first H

Figure 1. Plan of a harpsichord's keyboard. Between the keys, semitones supposed minor have been marked c by Stevin (from cleen = small); major semitones by g (from groot = large). Starting the scale with ut on G, Stevin in f has sa, on a white key.

This being so, experience shows that H and F make a perfect fifth, and although this is considered a common and certain rule by all those who are skilled in this matter, yet to convince those who should doubt it I thought fit to use the authority of .....1)

Thus from H to F being a perfect fifth, all semitones must needs be equal and an exact half of a whole tone, which is proved as follows.
Let the semitone from B to C and from E to F be smaller or larger, if possible, than a right half tone; I assume, according to the Pythagorean view, that it is smaller, which we therefore (as also b : c, and e : f) designate by c 2), meaning small; the letter g must be taken to stand for a major semitone. To proceed, L to B, by tuning, is a fifth, consisting of three whole tones and a minor semitone or, which is the same, of two whole tones, two minor semitones, and one major semitone. This being so, from F to L is a major semitone, which is proved as follows. BC is a minor semitone, CD and DE each a whole tone, EF a minor semitone, making together two whole tones and two minor semitones; therefore FL, to make up the fifth BL, must be a major semitone, and consequently from L to G is a minor semitone, for from F to G is a whole tone; subtracting from this the major semitone from F to L, from L to G must be a minor semitone. But f, Q, g make double-firsts with F, L, G, therefore from f to Q is a major semitone

1) The same diagram and the method of tuning, in which Stevin uses the expressions doctaaf (the octave) and de quinte (the fifth) was shown on a separate leaflet. A foot-note also was on a separate slip of paper. In the note Stevin supposes that in the tuning experiment one has started from E-flat (the keys K and P in the diagram). In that case the last step leads to G-sharp (gis, the keys M and R). Stevin argued that the people quoted by him proclaim that they find the starting note P to be identical with the perfect fifth (d-sharp or dis) above M. Hence, he says, F too is a perfect fifth above H.
2) c, taken from Dutch cleen = small. The letter r in earlier publications seems to be corrupt. The letter g is taken from Dutch groot = large.

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and from Q to g a minor semitone. Further, O is a fifth above L, by tuning, therefore from c to O is a major semitone, which is proved as follows: LG is a minor semitone, Gb two whole tones, bc a minor semitone, making together two whole tones and two minor semitones. Therefore cO, to make up the fifth LO, must be a major semitone, and consequently Od is a minor semitone. But c, O, d make double-firsts with C, I, D, therefore from C to I is a major semitone and from I to D a minor semitone. Again, M is a fifth above I, by tuning; therefore from G to M is a major semitone, for ID is a minor semitone and DE a whole tone, EF a minor semitone, FG a whole tone, making together two whole tones and two minor semitones, so that GM, to make up the fifth IM, is a major semitone, and consequently Ma is a minor semitone. But g, R make double-firsts with G, M, therefore gR is a major semitone.
Further, P is a fifth above M, by tuning, therefore from d to P is a major semitone also, for from M to a is a minor semitone, from a to b a whole tone, from b to c a minor semitone, from c to d a whole tone, making together two whole tones and two minor semitones, in consequence of which dP, to make up the fifth LP, must needs be a major semitone, and consequently Pe must be a minor semitone.
But d, P, e make double-firsts with D, K, E; therefore DK too is a major semitone and KE a minor semitone. Further N is a fifth above K, by tuning, therefore from a to N is a major semitone. Because from K to E is a minor semitone and from E to F also a minor semitone, and from F to a two whole tones, this makes together two whole tones and two minor semitones, in consequense of which aN, to make up the fifth KN, makes a major semitone, and consequently Nb is a minor semitone; but a, N, b make double-firsts with A, H, B, therefore AH is a major semitone and HB a minor semitone. This being so, HF consists of two whole tones and three minor semitones. For HB is a minor semitone; such is BC too, and CE are two whole tones and EF is a minor semitone; these make together, as said above, two whole tones and three minor semitones. HF therefore is no fifth, which would be contrary to experience, contrary to authority, contrary to common opinion, and a denial of the principles. Note further that so much smaller as BC would be than a right semitone, so much larger AH must needs be, and consequently the difference between them twice as much, which is contrary to the common opinion. For just as in singing the ascent from mi to fa is equal to that from la to sa, so the ascent from B to C is equal to that from A to H. BC therefore is not less than the right half of a whole tone; likewise one will also show it not to be more. Therefore it must needs be the right half, and thus all the others, from A to H, from H to B, etc., are also equal. This being so, the double-first must needs consist of six whole tones which are all equal, or of twelve equal semitones; therefore the requirement will be satistied if, between the bounds of the double-first 1 and 1/2, designated below by A and B, there have been found eleven mean proportional numbers C, D, E, F, G, H, I, K, L, M, N 1), as follows:

1) Mean proportionals, forming a geometric progression.

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A. 1 Selftone First
C. (12) 1/2 Semitone Minor second
D. (6) 1/2 Whole tone Major second
E. (4) 1/2 One-tone-and-half Minor third
F. (3) 1/2 Ditone Major third
G. (12) 1/32 Two-tone-and-half Fourth
H. 1/2 Tritone Bad major fourth or bad minor fifth
I. (12) 1/128 Three-tone-and-half Fifth
K. (3) 1/4 Four-tone Minor sixth
L. (4) 1/8 Four-tone-and-half Major sixth
M. (6) 1/32 Five-tone Minor seventh
N. (12) 1/2048 Five-tone-and-half Major seventh
B. 1/2 Six-tone Double-first, eighth

Thus, A,1 to A,1 is the ratio of the selftone or first, but A,1 to C, (12) 1/2 the ratio of the semitone or minor second, and A,1 to D, (6) 1/2 the ratio of the whole tone or the major second, and so on with the remaining ratios, from which it appears that the fifth and the others are in such ratios as we had proposed to prove.
Now, someone might wonder, according to the ancient view, how the sweet sound af the fifth could consist in so *) unspeakable, irrational, and inappropriate a number 1). To this we might give a detailed answer. However, since it is not our intention to teach to the unspeakable irrationality and inappropriateness of such a misunderstanding the speakability, rationality, appropriateness, and natural wonderful perfection of these numbers, we shall leave it at that because we have proved it elsewhere.
But if one wished to express all the above mentioned ratios by twelfth roots, one would find the progression of the denominators of the fractions in a regular sequence, from which all the ratios above the six-tone or double-first become easily known, as the following example shows sufficiently clearly:

*) Inexplicabili irrationali absurdo numero.
1) Referring to his work on arithmetic, (Work V) Stevin in advance ridicules objections concerning the irrationality of his numbers.

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A to A | A (12) 1 Selftone First 1)
A to H | C (12) 1/2 Semitone Minor second
A to B | D (12) 1/4 Whole tone Major second
A to C | E (12) 1/8 One-tone-and-half Minor third
A to I | F (12) 1/16 Ditone Major third
A to D | G (12) 1/32 Two-tone-and-half Fourth
A to K | H (12) 1/64 Tritone Bad major fourth or bad minor fifth
A to E | I (12) 1/128 Three-tone-and-half Fifth
A to F | K (12) 1/256 Four-tone Minor sixth
A to L | L (12) 1/512 Four-tone-and-half Major sixth
A to G | M (12) 1/1024 Five-tone Minor seventh
A to M | N (12) 1/2048 Five-tone-and-half Major seventh
A to a | B (12) 1/4096 Six-tone Double-first eighth (octave)
| (12) 1/8192 Six-tone-and-half Double minor second
| (12) 1/16384 Seven-tone Double major second
| (12) 1/32768 Seven-tone-and-half Double minor third

Geometrical Division of the * Monochord

Now in order to divide the monochord geometrically in such a way that we have the true perfect sounds of natural singing, i.e. in the above mentioned ratios, let AB designate the monochord, whose centre is C. Divide this in D, E, F, G, H, I, K, L, M, N, O in such a way that GB and LB are two mean proportional lines between AB and CB 2), which may be found in practice (for the mathematical method is still unknown) by different methods, but in my opinion most conveniently in the manner of ........

Figure 2. Division of the monochord in equal tones and semitones.

Further in such a way that IB be the mean proportional between AB and CB, which is found by the .. th Proposition of the .. th Book of Euclid. Likewise EB between AB and GB. Again DB between AB and EB. Further FB between AB and IB. Likewise HB between GB and IB; and KB between EB and CB. Again MB between IB and CB, and NB between LB and CB. Finally OB between NB and CB.
But if these divisions from C to B are required to be continued, this can easily be done as follows. Mark P in the middle of DB and Q in the middle of EB, and so on, because PB will make against AB the six-tone-and-half or double minor second, and QB against AB the seven-tone or double major second.

1) The letters in the first calumn of the following table correspond to the diagram, fig. I, p. 436. In the second column they refer to the preceding table on p. 441.
2) AB : GB = GB : LB = LB : CB.
*) Regula harmonices seu monocordus.

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Arithmetical Division of the Monochord

So far we have spoken of the geometrical division; in the following we shall explain the arithmetical division, i.e. by simple numbers, which in practice suffice, as follows. I divide the line AB into 10 000 equal parts. Now in order to know how many of these parts pertain to each tone 1), we begin with the three-tone-and-half, saying: 1 gives (12) 1/128, what does 10 000 give? This makes
(12) 7 812 500 000 000 000 000 000 000 000 000 000 000 000 000 000,
which comes very near to the integer 6 674. For 6 675 is too much, which may appear from the twelfth power of the one and of the other. But to furnish a proper test in each part of the operation of finding these toots, it is to be noted that the square root of the above written fourth power 2) is nearly (6) 88 388 347 648 318 440 550 105. Those who wish to check this may multiply this number by itself and add thereto the defect 3) 96 389 809 968 824 984 488 975. It also appears from the aforesaid defect that the true root exceeds the number by less than one unit, only by about 96 389 809 968 824 984 488 975 / 176 776 695 296 636 881 100 211. When from the aforesaid (6) 88 388 347 648 318 440 550 105 the square root is again extracted, this makes (3) 297 301 778 750. The proof is evident by multiplication of this number by itself and addition of the defect 404 488 987 605; it also appears from the aforesaid defect that the true root exceeds the number found by less than one unit, the excess being only about 404 488 987 605 / 594 603 557 501 *). From this finally extracting the cube root, we get 6 674, the proof of this being that when 6 674 is raised to the third power and the defect 26 628 726 is added thereto, this makes the first cube number. It also appears from the aforesaid defect that the true cube root exceeds the number found by less than one unit, only by about 26 628 726 / 133 646 851, so that 6 674 is the nearest integer number to the aforesaid root. The ratio of the three- tone-and-half has to be taken in simple numbers, as from 10 000 to 6 674. Then, this ratio 10 000 : 6 674 of the three-tone-and-half being subtracted from the ratio 2 : 1 of the six-tone, there remains the ratio 13348 / 10000 of the two-tone-and-half. But the numerator is not 10 000. In order to bring it to this value, I say: 13 348 gives 10 000, what does 10 000 give? This makes 7 491. Thus the two-tone-and-half has the ratio 10 000 / 7491. This being subtracted from the ratio 10 000 / 6674 of the three- tone-and-half, there remains (after conversion to the common numerator 10 000) the ratio 10 000 / 8909 of the whole tone. Adding to this the same again makes the ratio 10 000 / 7937 of the ditone. This same subtracted from ratio 10 000 / 7491 of the two-tone-and-half, there remains the ratio 10 000 / 9438 of the semitone. These tones being known, all the others are revealed by various methods of operation, for in order to have the ratio of the one-tone-and-half, one may subtract the whole tone from the two-tone-and-half, or the ditone from the three-tone-and-half, or add the whole tone to the semitone, and thus with all others.

1) Read: interval.
2) The twelfth root of a number is the sixth root of the square root of that number. Stevin is computing the twelfth root, R, of (104)12 / 27 = ((R3)2)2. First he takes this to be a fourth power ("the above written fourth power"), and he draws a square root, which makes 8.8 . . . . × 1022. He again draws a square root, to find R3 = 2.97 . . . × 1011. The last step is drawing the third root. Stevin gets 6674.
3) Stevin writes: remainder. In the actual operation of extracting the root this is in fact the remainder after one stops the calculation.
*) These numbers should be compared once more with the original computation.

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One might also perform the aforesaid division as follows. Having found the ratio of the three-tone-and-half as above, I get that of the tritone by saying: 1 gives 1/2, what does 10 000 give? This makes 7 071, so that the ratio 10 000 / 7071 is that of the tritone. This being subtracted from the ratio 10 000 / 6674 of the three-tone-and-half, there remains (after conversion to the common numerator 10 000) the ratio 10 000 / 9438 for the semitone. Adding the same ratio makes for the whole tone the ratio 10 000 / 8 908, whereas by the first method we got 10 000 / 8909, the cause of which small difference is evident. In the table below, for the sake of a more regular sequence, we shall stick to 8 909, and for the same reason to 7 936 for the ditone. The numbers above the six-tone are easily found by halving the preceding number; thus, to get the number of the six-tone-and-half, I take one half of 9 438, which makes 4 719, and for the seven-tone one half of 8 908 etc. A monochord, therefore, thus being divided into 10 000 equal parts, for every whole tone there will be so many parts, reckoning from B in the direction to A, as the following description shows.

10 000 Selftone First
9 438 Semitone Minor second
8 909 Whole tone Major second
8 409 1) One-tone-and-half Minor third
7 936 Ditone Major third
7 491 Two-tone-and-half Good fourth
7 071 Tritone Bad fourth
6 674 Three-tone-and-half Fifth
6 298 Four-tone Minor sixth
5 944 Four-tone-and-half Major sixth
5 611 Five-tone Minor seventh
5 296 Five-tone-and-half Major seventh
5 000 Six-tone Double-first
4 719 Six-tone-and-half Double minor second
4 454 Seven-tone Double major second

If one now wants to see how far amiss were the erroneous divisions of Pythagoras, BoŽthius, and Zarlino, this is readily possible by putting the largest number of their ratio also 10 000. I take the Pythagorean division, whose table being described up to the three-tone-and-half, runs as follows:

1) In the table a mistake, 8404, has been corrected to 8409. The correct numbers should read:

10 000.0
9 438.7 7.491.5 5 946.0
8 909.0 7 071.1 5 612.3
8 409.0 6 674.2 5 297.2
7 937.0 6 299.0 5 000.0

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10 000 First
9 492 Lesser minor second
9 364 Greater minor second
8 888 Major second
8 437 Minor third
7 901 Major third
7 500 Good fourth
7 023 Bad fourth
6 666 Fifth

From this it appears that the smallest term, of the three-tone-and-half, is 8 parts too short, for when 6666 is subtracted from 6 674, there remains 8, but the semitone is 54 parts too long. Now one might think: why is this difference so much greater in the semitone than in the three-tone-and-half 1)? I would say that the cause of this is obvious in the example given above in simple language, where we put 110 instead of the six-tone and where five persons, A .... E in due order stood for the three-tone-and-half, the two-tone-and-half, the whole tone, the ditone, and the semitone, where A in the wrong operation received only one too many and B one too few, C two too many, D four too many, but E five too few, so that E had too few five times more than A had too many. And likewise from the same cause the semitone here gets too few, five times more (with respect to the ratio of coarseness) than the three-tone-and-half has too many. From this it is also obvious that the difference between the lesser and the greater semitone is sixteen times 2) greater than the excess in the ratio of the three-tone-and-half, which is the cause of this error becoming so much more perceptible in the semitone than in the other tones.

Note

One must remember that the names of doubleness 3), triplicity, quadruplicity of firsts, seconds, thirds, etc. do not refer to the coarseness of the sounds, but to the rounds (taking eight 4) successive steps for a round), for just as one may say that two, three, or four turns of a helix are the double, triple, or quadruple of one circumference, not with respect to the unequal lengths of the lines, in which such a ratio does not consist, but with reference to the number of the turns, so these firsts, seconds, etc. are called double, triple, quadruple with respect to the rounds, without minding the coarseness of the sounds, according to which the constituent notes of the triple first are in the fourfold ratio, and those of the quadruple first in the eightfold ratio. With the double-first the matter happens to be different. For a first or selftone is a ratio 1 : 1. Doubling it means adding another ratio 1 : 1 to it; this makes the ratio 1 : 1 again. Therefore the name double-first refers to the rounds of sound only.

1) Stevin explains why the deviation between his scale and the Pythagorean scale is so much larger for the (minor) semitone (9492 - 9438) than for the fifth (6666 - 6674).
2) The Dutch text has: "ten times", obviously by mistake.
3) By definition Stevin defined his double-first, our octave. In a double-first (according to the 6th definition) the two notes are in the twofold ratio. This is different from the double of, i.e. twice the ratio of the first.
4) This should be: "seven". See the 3rd Definition.

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Appendix 1)

Preface / On the fourth / On la, si, ut / On the twelve modes 2) / In musical composition nature is not followed as it is in Rhetoric / The joint descent and ascent of sixths and thirds is allowed if minor ones alternate with major ones / Why not numerals in the long notes? / Bemollaris cantus is a useless distinction. / It is a common saying that whoso distinguishes well teaches well, but besides it must be known that whoso distinguishes improperly, teaches improperly. Species perfecta and imperfecta is an evil distinction.

Having above described the theory of music, I thought it useful to add, in brief words, an explanation of some obscurities and errors rooted in present-day musical practice.

Chapter on the Fourth *)

The fourth is considered a discord by present-day composers, so that in singing with three or more voices it must not be heard against the lowest part; nay, below two voices it is not suffered at all. But when one asks: why?, they answer: because it displeases our ears. Which I deny, and I shall also prove the contrary, first by argument, next - which is more - in practice. 3)
The argument is as follows. Two sounds making a double-first have so great a similitude that, two persons singing a song - I take an old person and a child, the latter singing a double-first higher than the former, but without knowing about double-firsts - usually for all they know they are singing the same note. Nay, we shall also prove here that this will sometimes happen even to the most experienced people. So great is this similitude, that it has an appearance of identity; therefore, when to the two above-mentioned voices producing the double-first is added a third voice, the same concord or discord that this third voice makes with one of them is also made with the other. If then, for instance, this third voice is a whole tone above the lowest voice, it produces therewith the discordant second and with the higher voice the discordant seventh of a similar character. But if the third voice is two whole tones above the lowest voice, it produces therewith the concord third, and with the highest voice not a discordant tone, but the concordant sixth of a similar character. And consequently, if the third voice produces with the highest voice a pleasant fifth, it cannot be unpleasant with the lowest voice, but produces therewith a pleasant concordant fourth. For this is a common rule that an identical tone is to the same in the ratio 1. To this might also be added the authority of the Greeks and their successors, who also regarded it in this way, but, leaving them alone, we shall come to the empirical proof.
If anyone considers the fourth a discord, saying that it is unpleasant to his hearing, the contrary and his error are proved to him as follows. Take any two different sounds, such as those of a string and a human voice, or a flute and a string, or a voice and a flute, producing therewith now a fourth, now a fifth, and

1) Only some of the announced items will be discussed in the following pages.
2) A mode consists in the manner in which tanes and semitones are distributed within the compass of an octave.
*) The diatessaron.
3) In a similar argument contained in the other draft Stevin quotes in his favour a treatise of Andreas Papius (1547-1581): De consonantiis, seu pro diatesseron libri duo, Antwerp, 1581, Chrstph. Plantinus.

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this several times and also several fourths and fifths higher and lower, each asking his partner what it is that he is singing or playing, then we shall find in practice that, judging thereof without certainty, he will frequently contradict himself, often regarding as a fifth what he previously stated to be a fourth, and again conversely judging that to be a fourth which he had previously stated to be a fifth. This being found in practice, what more words do we need? Who is so unreasonable as to disgrace himself by his own words, saying: the fourth displeases me and the fifth pleases me very well? But to set forth the cause of these things somewhat more amply, it must be known that when two such sounds produce together a first or a double-first, the very keenest hearing cannot tell for certain which of the two it is. To speak of this even more clearly by way of example, I suppose that there are two persons, one playing the flute and the other singing, each having sung his part of a given song as one would consider it should be. If this song is thereafter repeated again, but in such a way that the singer goes a double-first higher than before, everyone (for the aforesaid reason, to wit that but for a double-first there is no certainty) thinks it is all right. Nevertheless, at the note where the singer first was a fifth below the flutist, he now inevitably will be a fourth above him, in such a manner that, taking that the first time it was a fifth, one now will hear the fourth as a fifth. Therefore, as to those who still say that the fourth sounds unpleasant in their ears, but the fifth very pleasant, I cannot but conclude that the habit grown during the last century has bred a certain effeminacy in them.

On the twelve modes 1)

As I intended to speak with Master David about the twelve modes of Zarlino, I copied the notes in my own way, to show him that there were twelve. But though I further meant to prove that there could not be more, I found on the contrary that there were fourteen, the proof of which I am sending you herewith. Therefore, please undeceive me, or be deceived yourself. Zarlino's six modes (which would make twelve with their contraries 2), as you send them to me, are as follows:

Figure 3. The principal notes of the odd numbers of the modes in the Dodekachordon of Glareanus, given to Stevin by a friend who took them from Zarlino.

1) In the manuscript a chapter on the twelve modes, as given by Zarlino, is missing. We insert, as a substitute, a copy of a letter from Stevin to an unnamed person, preserved by his son Hendrick. Cf. Note B on p. 461.
2) Stevin refers to the plagal modes as contrary to the authentic modes. By the middle joint note the octave of the mode is divided into a fifth and a fourth. In the authentic modes the fifth is below the fourth, in the plagal modes the fourth is below the fifth. A pair of conjugated authentic and plagal modes, Stevin's contrary modes, share their fifths.

455

I arrange them in my own way (adding thereto in the seventh place the thirteenth mode), as follows:

Primus Tertius Quintus Septimus Nonus Undecimus
First Third Fifth Seventh Ninth Eleventh Thirteenth

Figure 4. The principal notes of the odd numbers of the modes, supplemented by a thirteenth of Stevin's invention. The small figures indicate the numbers of the ecclesiastical modes.

Now it is evident that it is by the different places of the semitones that the various kinds of music called different modes are distinguished. Therefore I make seven equal scales, one in each mode, as shown below. By the dotted steps I indicate, for greater clarity, the places which make semitones with their preceding notes.

Figure 5. Places of semitones in the various modes, indicated by the dotted lines.

I say that these are all different, which I prove as follows.

4th, 8th. The first mode has its fourth and eighth note making semitones.
3rd, 7th. The difference between the third mode and the first is that the former has its third and seventh notes making semitones, the latter has them making whole tones.
2nd, 6th. The difference between the fifth mode and the two preceding ones is that the former has its second and sixth notes making semitones, in the latter they make whole tones.
5th, 8th. The difference between the seventh mode and the three preceding ones is that the former has its fifth note making a semitone, the latter have it making a whole tone; further that the former has its eighth note making a semitone, but the third and fifth modes have it making a whole tone.
4th, 7th. The difference between the ninth mode and the third, fifth and seventh modes is that the former has its fourth note making a semitone, but the latter have it making a whole tone; further that the former has its seventh note making a semitone, but those of the first, fifth, and seventh modes make whole tones.
3rd, 6th. The difference between the eleventh mode and the first, fifth, seventh, and ninth modes is that the former has its third note making a semitone, the latter have it making a whole tone; further that the former has its sixth note making a semitone, but in the preceding first, third, seventh, and ninth modes it makes a whole tone.
2nd, 5th. The difference between the thirteenth mode and the first, third, seventh, ninth, and eleventh modes is that the former has its second note making a semitone, the other have it making a whole tone; further that the fifth note of the former makes a semitone, but the first, third, fifth, ninth, and eleventh modes have it making a whole tone.

457

These modes with their contraries (which contraries I leave aside for the sake of brevity) make fourteen modes, and there can be no more, for the next, which would be the fifteenth in the series, would be identical with the first, which I intended to prove.
The motive why Zarlino did not locate his first mode like the others has good reason, since the regular progress of the ascent from one mode to the other, i.e. from step to step would have been disturbed by this. For if he had taken the first mode of the Ancients 1) as his first, his eleventh mode 2) would not be one step higher than his preceding ninth 3), but five steps lower. And for the same reason his second tone must also be in the same place.
Note that our first mode corresponds to their tenth, because both are in fa.4) They only differ in that the latter makes its middle joint note one step lower than the former. Likewise the second corresponds to the eleventh, the third to the twelfth, the fifth to the seventh, the sixth to the eighth. As to the fourth and the ninth they correspond neither to one another nor to any of all the others.

On the Usual Distinction between the Singing that is called Bemollaris and Beduralis.5)

The usual distinction that is made between the music that is called Bemollaris and Beduralis is quite useless and is no real distinction, but they are the same thing. For if you give the C sol fa ut key, where B is flattened (Bemollaris), the name of G sol re ut, singing B natural thereto (Beduralis), you have entirely the same music that was written with B-flat. The same is found when the F fa ut key with flattened B is given the name of C sol fa ut with B natural, and likewise when the G sol re ut key with flattened B is given the name of D la sol re with B natural. Or if conversely you give all the latter the name of the former, you will have the same. It is thus no other kind of music than the other, and consequently it is a useless distinction.

Chapter in which is explained the Cause of the Imperfection that arises when Organs and Harpsichords are tuned

What we have said and concluded in a former chapter 6) about the fifth MP is the finding that this same fifth MP is good,*) but it happens in different

1) on d.
2) on c.
3) on a.
4) Stevin here writes fa for c. This agrees with his nomenclature in figure 1 (p. 436).
5) Cf. Note C on p. 463.
6) See note 1) on p. 437. - Cf. Note D on p. 464.
*) To prove this, it is to be noted that if one sings (as people sometimes call it) g-sharp against d-sharp upwards, as Orlando, among others, etc., we hear this to be a good fifth.

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instruments, as experience shows, that at one time it turns out slightly too large, at another time slightly too small, sometimes also good; but because in playing it is used very little, if at all, many masters who tune the instruments leave it to chance, since all the rest that is used is good enough as regards the ear. But to explain the unexplained cause why this fifth on the aforesaid instruments cannot be hit off as right as the natural singing of human voices testifies it should be, the common imperfection should be understood of practical operation in all matters, which cannot be performed as perfectly as mathematical operations. Thus, for instance, when a piece of linen of 50 yards is carefully measured by different people, one will find a strawbreadth, or an inch more than another. But it seldom happens, and then accidentally, that they all arrive altogether and quite alike at the same result. The same also applies in the measurement of surfaces, solids, and other things, such as time, motion, weight, and again in the matter of sound, which is our point in question. For one cannot match two sounds, such as make a fourth, a fifth or a sixth, etc. in such a way that these intervals are quite perfect, unless by chance; nor can they be proved to be so. But to show this practical imperfection clearly, place the fret of a lute in the place where you think its string will make a perfect fifth against another string. After this, shift this fret over the thickness of a hair only, upwards or downwards, and you will find that no appreciable change takes place, although, to be sure, some change does take place. But if you surmise and concede to yourself that you perceive this falseness of the fifth, let the fret be shifted by someone else, in such a way that you do not know whether he shifts it a hair's breadth upwards or downwards, or leaves it in the same place; when he several times thus inquires of you after the goodness of the fifth, in practice you will find your judgment uncertain, often saying that the good fifth is bad and that the bad fifth is good. It is therefore obvious that no human hearing, however keen it may be, is able to fit two tones quite surely in their perfection. From this it follows that many such mistakes, each of which in itself is inappreciable, yet in combination produce an appreciable error. For as in the aforesaid piece of linen measured by various people many small differences in every single yard, added together, finally make an appreciable difference, so here in the case of sounds too. For since this fifth MP is very rarely used, one let these tiny inappreciable errors drift, which together may finally be appreciable, sometimes also inappreciable, as the case may be. Therefore it is not astonishing that superior masters, tuning these instruments carefully, nevertheless in the end find bad notes which ought to be good; this is but natural. And whoso does not understand this, lacks discrimination between practical and mathematical operations, which we had to prove.

Simon Stevin, 1585

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NOTE A, referring to pp. 431/432

In his loose reference to Zarlino, tainted with some derision, Stevin does not do justice to the problem Zarlino was trying to solve. Representing the pitches of the common scale by

c d e f g a b c' d' e' .. etc.,

according to the theory of Ptolemy adopted by Zarlino, in the triad c, e, g the note e was the harmonic mean between c and g. The latter making a perfect fifth, c and e make a perfect major third. The triads f, a, c' and g, b, d' were transpositions of the same triad. Thus the numbers assigned to these major chords are 24 : 30 : 36 = 32 : 40 : 48 = 36 : 45 : 54 (= 4 : 5 : 6) respectively. This procedure entails a difference of one comma in the whole tones (d : c) and (e : d), with the ratios 9/8 and 10/9 respectively; likewise in the whole tones (g : f ) and (a : g). It follows that (a : d) is one comma short of a fifth (ratio 3/2). For a transposition of a melody from c : d : e to the initial note g, one therefore needs a note a, one comma sharper than a. This note can be readily produced in singing, but once an organ pipe or a harpsichord string has been tuned to a, it cannot suddenly be brought to a. Zarlino sought for a compromise, by which he might make slight alterations in the pitches that would not disturb the harmonies too much. Suppose we strain the fifth a little bit, by an amount x, and the major third by y, then the pitches of the common scale will become c, (d + 2x), (e + y), (f - x), (g + x), (a - x + y), (b + x + y), c', (d' + 2x). We want the new fifths to be equal, therefore (g + x) - c = (a - x + y) - (d + 2x) (g - c) - (a - d) = -4x + y = comma. Likewise there is equality of the new seconds (d + 2x) - c = (e + y) - (d + 2x), and again (d - c) - (e - d) = y - 4x = comma. Now, if by a kind of equipartition, the fifth being nearly double the third, one chooses for the strains a relation x = 2y, then the solution is x = - 2/7 comma, y = -1/7 comma. This is the solution which Zarlino offered in his Institutioni 1558 (Parte II, cap. 43), and to which Stevin refers. Obviously less damage is done to the fundamental intervals if one puts y = 0, hence x = -1/4 comma. This solution was put forward by Zarlino in his Dimostrationi harmoniche in 1571 (Ragionamento quinto). Full credit for the first description of this solution is given by Barbour to Pietro Aron (Venice, 1523). Stevin does not mention it. For him the problem does not exist, or rather: he puts x = -1/12 comma, this being the difference between his fifth and the Pythagorean fifth of 3/2. Therefore the damage done by his equal temperament to the perfect third of 5/4 amounts to y = 2/3 comma. That solution of the tuning problem was discussed by Zarlino in 1588, in his Sopplementi musicali (Libro quarto, cap. 28). Stevin does not mention this at all.

461

NOTE B, referring to pp. 452/453

In this letter, which is not preserved in his own handwriting, Stevin tentatively enumerates two more, i.e. fourteen different modes. In the diagram he indicates by minims (half notes) the principal notes of the odd modes, the so-called authentic modes. These notes are the points of convergence for melodic lines in singing. We believe that Stevin calls tseamval (coincidence) the principal note in the middle which joins the fourth and the fifth that constitute the octave. The fact that, against his usual method, he nowhere gives a definition of this term, shows that his work is not complete. The final and initial note of an authentic, odd mode becomes the principle note in the middle of the following even mode, which is called a plagal mode. It turns out that the first and the eighth mode have the same division of the octave, but the difference lies in the positions of the principal notes. In the third diagram 1), by the added numbers Stevin

Figure 6. The twelve modes of Zarlino, supplemented with numbers 13 and 14 by Stevin. The latter contain in f : b and in b : f' an augmented fourth and a diminished fifth respectively.

1) Fig. 5 on p. 454.

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indicates the similar octaves. For the even mode in question the minim must be put one note lower than for the odd mode. Stevin's thirteenth and fourteenth modes show between the principal notes the discord of a diminished fifth or an augmented fourth. For this reason they had been rejected by Zarlino.
The numbers 1, 3, 5, 7 in Stevin's second diagram refer ta the ecclesiastical modes.

In the accompanying diagram the editor gives a full exposition of all these modes (met haer contrariŽn). The editor has also added the title of the chapter and the line beginning with 4e.8e on p. 454.
The last two paragraphs are in Stevin's own handwriting again. He compares his own and Zarlino's numbering of the modes with the numbering of the ecclesiastical modes first. It is not clear what numbering the manuscript refers to in the last paragraph. I keep the numbers as written by Stevin. The correlation between pairs of modes as pointed out by him is found in the pairs (1,8), (2,9), (3,10), (4,11), (5,12), with 6 and 7 solitary.

463

NOTE C, referring to pp. 456/457

Stevin remarks that one might do without a sign for flat at the beginning of the stave. By a simple transposition, choosing an appropriate clef, a tune can be written without such a flat. He says that if you prescribe a b-flat, having c on a certain line of the stave, then, by changing the clef you can place g on this line. By the same notes on the same lines, without a flat, the same melody comes out as before, written with a b-flat.
In a similar way, in figure 1, p. 436, placing ut on the key for g, sa came out on the key for f, and there was no need for a black key.
In Stevin's days no importance was attached to absolute pitch. That is the gist of his remark.
The same change of clef transforms the stave line for f into a line for c, and the line for g into a line for d.
An example is shown in figure 7.

Figure 7. A simple melody: re mi fa sol la sa - ut la sa sol fa -, where sa has been written as b-flat and as f, respectively.

464

NOTE D, referring to pp. 458/459

This discussion more or less completes the argument in the former chapter on the true ratios of natural tones, pp. 434-435, where the tuning was described of a harpsichord. There the tuning started from F and resulted in a tone H (our A-sharp or B flat) that by ear was to be judged a perfect fifth below F.
The present argument presumes that the tuning has started from K or F (see figure 1, our E-flat), and led to M (our G-sharp). Stevin's contention here is that M and P (G-sharp and e-flat) make a perfect fifth. In the note where he refers to other-people's meaning, M and P are called g-sharp and d-sharp. This is the only place where Stevin writes sharps.
It is quite remarkable that Stevin, discussing the possibility of perfect tuning, nowhere mentions that listening to beats of simultaneous sounds offers a powerful means for judging the accuracy of tuning.

Graphics created by Ad Davidse.