Simon Stevin's views on music


Simon Stevin for some time seems to have contemplated writing a treatise on music. If ever he accomplished this design, the work must obviously have been lost. Only some fragments were discovered in 1884 by D. Bierens de Haan in a collection of miscellaneous manuscripts, which belonged to Constantijn Huygens (1596-1687), the well-known secretary to the Princes of Orange, who, at the same time, was a gifted poet and musician. (Cf. this edition, Vol. I, p. 33, work XV). This collection, now in the possession of the Koninklijke Nederlandse Akademie van Wetenschappen at Amsterdam, is preserved at the Koninklijke Bibliotheek at The Hague, The Netherlands.

Stevin often divided his books into a main part, containing the established doctrine, and an appendix dealing with controversial matters, in order "not to obscure the instruction by dispute", as he says. Accordingly we find two main parts, or sketches thereof, and two appendices. Neither of these treatises is complete, and in the plan they show an appreciable difference in the stress laid on certain points. One draft is in Stevin's own handwriting. It dwells rather more upon discussions than the other. This part will be reproduced hereafter. The other draft, which has been copied as if in preparation for print, indulges rather more in elementary definitions. It shows some gaps, presumably to be filled up later.
Stevin used to open his books with a summary. In one of the drafts the main part and the appendix contain pages bearing the title cort begrijp (i.e. summary), but one of these pages is blank, and the other contains a dedication to the "singing masters' of his time and the statement that he will give his critical remarks in an appendix.
Nowhere does Stevin use the word music. He always writes singing. Composers are called makers of singing. The stave is called singing ladder. Perhaps the word singconst, the art of singing, was the best translation into Dutch he could think of for musica. As is well known, Stevin was extremely keen in inventing and propagating vernacular translations of Latin words (see Vol. I, p. 6 and p. 58). It is to the semantic power of the Dutch language in making a word express its meaning properly by means of its components, and to the lack of this power of the Greek language, that he ascribes the fact that the clever Greeks failed to find the correct solution of how properly to divide the string to suit the true musical scale, whereas he himself was able to offer this solution.
For his reflections and conclusions Stevin based himself on "natural singing" (natuerlicke sanck), taking for granted that natural singing is an empirical fact liable to be observed with an amount of reasonable exactitude sufficient for all kinds of practical purposes. That, of course, is not rigid mathematical precision. The scale of natural singing shows five major steps and two minor steps. Stevin maintains that all major steps must be equal. So are the minor steps, each being one half of a major step. Thus, the sum of all steps in a "round" (ommeganck; we call it octave) amounts to six major steps. The major steps are whole tones, the minor steps are semitones.
With this statement Stevin took sides in the controversy on the problem of how to place frets on the fingerboard of a stringed instrument in order to ensure the correct intervals. The problem arises from the recognition that two fundamental intervals are equally important, but at the same time clash owing to their mutual incommensurability. These concordant intervals are commonly called the perfect fifth and the perfect major third. Though a certain amount of musical training will be helpful in understanding the nature of the difficulty, the general reader might grasp the point at issue as follows, if he does not mind being confronted with figures.
It is quite natural to divide a string of a lute, between neck and bridge, into two equal parts, and to listen ta the sounds produced. Again, it is natural to divide the halves into two. It is also natural to divide the string into three equal parts, and again each third part into two, and into three. Thus, taking the whole length to have 144 parts, we get a division at the numbers

      144  128  120  112  108  96  80  72  64  48  36  32  24  16

For shorter parts of the string we get higher notes.
We place frets on the fingerboard according to this division. By pressing down the string on these frets, we can easily produce the notes required. We may use the numbers given to designate the notes thus produced. Actually, however, people have agreed to call them by letters, e.g. the following

       A    B    c    .    d    e   g   a   b   e'  a'  b'  e"  b"

The note 112 was not included in the ancient lettered system. For our purpose we may ignore it.
The most important relation is the interval between the notes corresponding to a certain length of string and its half. Such notes are designated by the same letters, as A and a (144:72), or a and a' (72:36); as B and b (128:64), or b and b' (64:32) and b' and b" (32:16); again e and e' (96:48), or e' and e" (48:24). Their relation, their interval (2:1), is called an octave.
Next comes the so-called interval of the perfect fifth, 3:2, as between A and e, e and b, c and g, d and a, a and e'. Then follows the interval of the perfect fourth, 4:3, as between A and d, d and g, B and e, e and a, b and e', e' and a', b' and e". There are the intervals of the major third, 5:4, as between c and e, g and b.

For the technique of playing it is desirable that the fingers should not have to reach out far from the neck to the smaller numbers. For that reason a second string will be provided, with the same length as the first, a lighter one which at full length produces the note 108, which we called d. The second string at full length producing the note 108, the frets will produce notes corresponding to numbers proportionally reduced in the ratio 3/4. Here they are.

      108  96  90  81  72  60  54  48  36  27  24  18  12

The names by letters will be

       d   e   f   g   a   c'  d'  e'  a'  d"  e"  a"  e'"

There is a gain. A new note represented by a new letter, f. But there is a clash also. The fret number 108, producing by the first string a note d, is now producing a note g = 81, which cannot be the same note as g = 80 played on the first string.
Using a third string, which at full length produces the note 81 = g, again there appears a new note, but we find more clashes, as is seen from the table below of the notes given by numbers and by letters.

      144  128   120   108   96   80   72   64   48   36    32   24   16
      108   96    90    81   72   60   54   48   36   27    24   18   12
       81   72   67.5  60.75 54   45   40.5 36   27   20.25 18   13.5  9
       A     B    c     d    e    g    a    b    e'   a'    b'   e"   b"
       d     e    f     g    a    c'   d'   e'   a'   d"    e"   a"   e'"
       g     a   b-flat c'   d'   f'   g'   a'   d"   g"    a"   d"'  a"'

There is a note c' = 60.75 clashing with c' = 60 and c = 120 on the second and first strings. There are notes g' = 40.5 and g" = 20.25 on the third string clashing with g = 80 on the first string.
These clashes constitute a very serious difficulty in playing on a lute tuned in this way. The error 81:80 is called a "comma".

J. Murray Barbour, in his book on Tuning and Temperament (Michigan State College Press, 1953), presents a historical survey of the attempts to find a satisfactory solution for the problem how to improve the placing of the frets. His book contains an ancient explanatory picture of a lute, by Gassani, indicating the places of the frets.
It shows a division equivalent to the division given above, making in numbers

         72    64 60    54    48       40    36

Gassani adds some more, filling up gaps

         72 68 64 60 57 54 51 48 45    40    36

We can fill up the whole tones 45:40:36 in this way:

                                 45 42 40 38 36

We now see three series of semitones

      72 : 68 : 64 : 60                  = 18 : 17 : 16 : 15
      60 : 57 : 54 : 51  : 48 : 45 : 42  = 20 : 19 : 18 : 17  : 16 : 15 : 14
      42 : 40 : 38 : 36 (: 34 : 32 : 30) = 21 : 20 : 19 : 18 (: 17 : 16 : 15)

I continued the last series beyond 36 as a repetition of the first, lower series. There is a continuous range of semitones with the values 14 : 15 : 16 : 17 : 18 : 19 : 20 : 21, from major semitones 14:15 to minor semitones 20:21.
Vincenzo Galilei (Dialogo della musica antica e moderna, Firenze, 1581) disclaimed such a variety of semitones. Before Stevin, he wanted them all to be equal, and he chose the value 17:18, which happens to be midway. For the fifth, seven such semitones added would make 187 : 177, about 6.12 : 4.10 (cancelling 108), a rather good approximation to 6.15 : 4.10 = 3 : 2. The defect is 3 in 615, or 1 in 205.
For the major third, four such semitones would make 184 : 174 = about 10.50 : 8.35 (cancelling 104). This is a poorer approximation to the accepted value 5:4 = 10.50 : 8.40. The excess is about 5 in 840, or 1 in 168.
Now the octave, when taken to consist of twelve such semitones, turns out to be 1812 : 1712, approximately 11.57 : 5.83 instead of 11.66 : 5.83 = 2 : 1. The defect is 9 in nearly 1170, all but 1 in 130. It is three fifths (and in the opposite sense) of the comma excess of 1 in 80 in the values 81 and 80, 60.75 and 60, noted earlier.
The equal temperament by semitones 17:18 distributes the error of the comma over the octave (-3/5 c), the fifth (-2/5 c) and the major third (+1/2 c).
Vincenzo Galilei's rule seems to have been commonly accepted at the end of the sixteenth century, as it is to this day, but only for the lute, the viol, and similar instruments.

For organs and for harpsichords no attempt was made to divide the octave into twelve equal steps. Organists tried to have pure octaves and perfect major thirds by a slight adjustment of the fifth. They corrected the sequence of four fifths

486 : 324 : 216 : 144 : 96

so as to have perfect consonance between 480 and 96, because

480 : 96 = 5 : 1.

The comma excess 486:480 = 81:80 is distributed over the four steps, each step losing one fourth of a comma, i.e. 1 in 320, as follows (approximately):

480 : 321 : 214.67 : 143.56 : 96.

In order to have perfect major thirds (as between 480 = 5 × 96 and 384 = 4 × 96), a small infringement is thereby made of the perfect fifths.
Barbour (l.c., p. 26) gives the credit for the first description of this method of tuning to Pietro Aron (Toscanello in musica, Venice, 1523). It is the mean-tone temperament, strongly advocated by Gioseffo Zarlino (1517-1590; Stevin occasionally calls him Meester Sarlijn) and by Francesco Salinas (1577), two great early legislators of music. It has been in use for three centuries.

Stevin boldly did away with all these subtleties. In his view, all semitones had to be equal. In this he agreed with Vincenzo Galilei, that dissenting pupil of Zarlino.
He rejects any relationship between concordant intervals and ratios of integer numbers. For him the numbers resulting from the division of the ratio 2:1 into twelve equal ratios, twelve times the twelfth root of 2, are the true numbers. Barbour's remark is very appropriate (l.o., p. 7) : "In his days only a mathematician (and perhaps only a mathematician not fully cognizant of contemporary musical practice) could have made such a statement." Barbour adds: "It is refreshingly modern, agreeing completely with the views of advanced theorists and composers of our day."1) It is Stevin's outstanding achievement that he produced the exact proportional numbers, between 10000 and 5000, in four figures, representing the steps of twelve semitones in the octave leading from 1 to 1/2. He was able to do so, referring to his French work on arithmetic (1585, this ed. Vol. II B) where he had shown that the requirement of twelve equal ratios leading from 2 to 1 involves the twelfth root of 2. By combining the operations of computing two square roots and subsequently a cube root, he finds for the twelfth root of 2 the ratio 10000 : 9438 = 1.0595 : 1. The more exact figure is 1.059463 . . . . . Stevin never mentions the approximate value 18/17, familiar to makers of lutes, who used it in fixing frets on the fingerboards.
He had no bump for the plain simplicity of small integer numbers. In his treatise on arithmetic (Work V) he had explained that there are "no absurd, irrational, irregular, inexplicable, or surd numbers" (see this edition, Vol. II B, p. 532, also Vol. I, p. 23).
For him a number like 27/12 is as good as any other, say 3/2. If anybody should doubt that the sweet consonance of the fifth could be compatible with so complicated a number, then, says Stevin, rather haughtily and aggressively, he is not going to take pains to correct the inexplicable irrationality and absurdity of such a misapprehension. He repudiates the Pythagorean values for the intervals (3/2 for the fifth, 9/8 for the second, 81/64 for the major third, 4/3 for the fourth) on the ground that they lead up to the ratio 256/243 for a semitone (the minor limma). This, when subtracted from a whole tone (9/8), leaves another semitone with a ratio very close to 256/240. Stevin remarks that this major semitone is all but a quarter larger than the previous minor semitone (the differences of 243 and 240 from 256 being 13 and 16 respectively). All semitones having to be equal, the initial assumption of 3/2 for the ratio of the fifth must be wrong. For Stevin the equality of the twelve semitones follows from the fact that in tuning a harpsichord one obtains a closed cycle of fifths and fourths. Strictly speaking, the excess of twelve fifths over seven octaves should be 1 part in 73 (comma of Pythagoras). Stevin, however, ascribes any small deviations from the perfect cycle to unavoidable experimental errors.

Joseph Needham, in Vol. 4, Part 1, of his Science and Civilisation in China, refers to the duodecimal equal temperament as "the princely gift of Chu Tsai-YŁ. He points out that at the end of the 16th century there was a great flow of Chinese information into Europe. He urges the probability of some idea of Chu Tsai-YŁ's solution having floated towards Stevin's mind. Stevin himself refers to Prop. 45 in his book on arithmetic as the source of his method for finding the 12 equal semitones, ascribing his success to the wonderful semantic power of his Dutch language. He could not have said so, if he had to admit that a Chinese had been able to find the formula without Dutch words. The book of Chu Tsai-YŁ quoted by Needham is dated 1584. Stevin's book on arithmetic appeared in 1585. We can agree with Needham saying "the name of the inventor is of less importance than the fact of invention." As far as we know Stevin, we can apply to him the very same words of praise which Needham gives to Chu Tsai-YŁ: "Stevin himself would certainly have been the first to give another investigator his due, and the last to quarrel over claims of precedence".

There is the ancient problem, come down from the Greeks, as to whatsoever sounds may have to do with numbers. In Stevin's time people had no clear consciousness of the frequenry of vibrations. He speaks of "coarseness" or "fineness" determining pitch, and postulates a proportionality of this coarseness to the length of the sound-producing part of a string. By way of example, he refers to the half, to the quarter, and to the eighth part of a string only. He does not mention other aliquot parts, or 2/3, or 3/4 of a string as examples. In this he shows a bias against integer numbers. Two is the only integer admitted by him in music. One would not have expected such a bias in a mind which knew quite well that the regular solids exhibit only selected integers in the number of their faces, edges, and angular points. Perhaps he would have admitted that consonant intervals, and their beauty, primarily have to do with integer numbers if he could have seen Lissajous' delightful figures of interfering oscillations. He never mentions the phenomenon of beats, so essential for tuning perfect concords. Stevin never verified whether on a harpsichord tuned with a closed cycle of fifths and fourths the thirds and sixths would turn out to be concords. They certainly would not! Nevertheless he takes the consonance of these intervals for granted as an empirical fact. He decides rather by definition which intervals are good and which are bad.

As a practical rule, the "singing masters" condemned the interval of the fourth in polyphonic singing. This interdiction is not recognized by Stevin. He argues that very often, when one hears two instruments, a and b, playing in unison, it is very difficult to know whether they are playing at the same pitch or one octave apart. If a third instrument, c, plays in consonance with both, then of course it is in consonance with each of them. In case the concord seems to be that of a fifth it is difficult for the ear to decide whether c makes a fifth with both a and b, or with one of them only, making a fourth with the other. But, this being so, the fourth must be a good concord too.
Stevin refuses to recognize a difference in singing with a flat on the stave or without (mollaris and duralis). He says that by transposition every tune can be written on the stave without a flat. In this he is right. Of course this has nothing to do with the difference in mode, with minor and major scales. There is no chapter on this subject of modes, but we have collected some scattered data.
Sometimes, in the scale the note si is flattened by a semitone to sa. Stevin seems to have seen a reason for giving sa a place on the stave without the sign for a flat. It is curious to see that in certain diagrams he assigns the vocables

           ut  re  mi  fa  sol  la  sa  ut

to the letters

           g   a   b   c   d    e   f   g

If he had assimilated ut to c, as we do, of course sa would have meant b-flat, and si would have to be b-natural (the Germans would say b and h, respectively). In one place Stevin promises to return to this question of sa and si, but no chapter on this question is included. In the manuscript there is no consistent notation of sa and si on the stave.

We do not know whether Stevin ever considered his work to have been brought to a satisfactory conclusion, and whether he intended to publish it. It might well be that discussions with musicians made him change his mind in some respect. Among the manuscripts of Constantijn Huygens mentioned above, published as an appendix to Stevin's Singconst (listed as Work XV in Vol. I of this edition, p. 33), there is a letter to Stevin from Abraham Verheyen, organist at Nijmegen (Gelderland), who urges that experiment, in tuning a harpsichord, shows that the three major thirds, i.e. six whole tones, do not make an octave. He explains to Stevin the merits of the current mean-tone temperament, and how to compute the ratios involved. Verheyen also produces an example of a song in two parts, clearly showing the difference of major and minor semitones. We know that Isaac Beekman (1588-1637, Journal, ed. C. De Waard, The Hague, 1942, Vol. 4, p. 157) at first very much admired Stevin's proportional division of the octave. Later he rejected it.
Maybe the criticisms of very able friends shook Stevin's sturdy conviction a little, so that he abandoned the idea of making a full size treatise based on his mathematical axiom.

1) The present editor believes that Stevin's duodecimal division of the octave is now going to be superseded by the division into 31 steps, advocated by Nicola Vicentino (1588) and Christiaan Huygens (1691).

A. D. Fokker, 1966