A.D. Fokker about Stevin

On request by Joe Monzo who liked to know more about Simon Stevin's
thinking about 12-tone e.t., here is the scanned introductory article
by Adriaan Fokker in his edition of Stevin's "On the theory of the art
of singing" (Vande spiegheling der singconst). I will put this text on
the Huygens-Fokker website later.

INTRODUCTION

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
defini-
tions. 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 pro-
pagating 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 pro-
ducing 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.
Table
144 128 120 108 96 80 72 64 48 36 32 24 16
108 96 90 81 72 6O 54 48 36 27 24 18 12
81 72 67.5 6O.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' = 6O 3/4 clashing with c' = 6O and c = 120 on the second
and first strings. There are notes g' = 40 1/2 and g" = 20 1/4 on the third
string
clashing with g = 80 on che 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
satis-
factory 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 6O 54 48 40 36
Gassani adds some more, filling up gaps
72 68 64 6O 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 : 6O = 18 : 17 : 16 : 15
6O : 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)
disclaim-
ed 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 18^7 : 17^7, about 6.12 : 4.10 (canceling 10^8),
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 18^4 : 17^4 = about
10.50 : 8.35 (canceling 10^4). 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
18^12 : 17^12, 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 3/4 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 2/3 : 143 5/9 : 96.
In order to have perfect major thirds (as between 480 = 5 X 96 and 384 =
4 X 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 occa-
sionally 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 Zar-
lino.
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 10 000 and 5 000, 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 10 000 : 9 438 = 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

(footnote)
1) The present editor believes that Stevin's duodecimal division of the octave
i5
now going to be superseded by the division into 31 steps, advocated by Nicola
Vicen-
tino (1588) and Christiaan Huygens (1691).
irregular, inexplicable, or surd numbers" (see this edition, VoI.II B, p. 532,
also
Vol. I, p. 23).

For him a number like 2^7/12 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 compli-
cated 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-Yu". He
points out that at the end of the l6th century there was a great flow of Chinese
information into Europe. He urges the probability of some idea of Chu Tsai-Y'u'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
semi-
tones, ascribing his success to the wonderful semantic power of his Dutch lan-
guage. 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-Yu 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-Yu: "Stevin him-
self 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 con-
sciousness of the frequenry of vibrations. He speaks of "coarseness" or
"fineness'
determining pitch, and postulates a proportionaliry 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
veri-
fied 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!
Never-
theless 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
mathemat-
ical axiom.

Manuel Op de Coul coul@ezh.nl