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.
144 128 120 112 108 96 80 72 64 48 36 32 24 16
For shorter parts of the string we get higher notes.
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.
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
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.
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.
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.
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
so as to have perfect consonance between 480 and 96, because
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):
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
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
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.
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
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
|Simon Stevin This is the introduction to Vande spiegheling der singconst ("On the theory of the art of singing") in Principal Works of Simon Stevin vol. 5, A.D. Fokker (ed.), Amsterdam, 1955-1966, pp. 413-464.|