Lenses and Waves

98 CHAPTER 3
fourth which implied that “… the number 7, …, is not incapable of
producing consonance …”, a conclusion that ran in the face of all previous
musical theory. 202 At that time – around 1661 – Huygens decided not to
accept the consonance of intervals with 7 because they had no regular place
in the scale.
Next, Huygens addressed a problem in tuning. When keyboards are tuned
according to then customary mean tone temperament, the question was how
the fifths employed ought to be adjusted with respect to pure fifths. 203 In
order to determine a mathematical solution, Huygens started by deriving the
ratios of all twelve tones in terms of the string lengths of the octave and the
fifth. In the course of his investigation, Huygens found a new property of
mean tone temperament. It concerned the quantitative difference between
the diatonic and the chromatic semitones. 204 Calculating the ratio of both
kinds of semitones, he concluded that C-D can be divided into 5 equal parts
and, consequently, the octave into 31 equal parts. Thus Huygens arrived at
the 31-tone division of the octave he had found discussed by Mersenne and
Salinas. In a letter published 30 years later in Histoire des Ouvrages des Sçavans
(October 1691), Huygens elaborately explained how he calculated the various
string lengths and pointed out advantages of his 31-tone division. 205 The
paper did not contain a further consequence Huygens had drawn in his
private notes: the consonance of intervals based on the number 7. Thus,
Huygens’ ‘most original contribution to the science of music’ remained
unknown to the world until this century. 206
Huygens’ studies of consonance show, once more, his dexterity in
exploring and elaborating the mathematics of a topic. He added rigor and
precision to Mersenne’s science of music, using Galileo’s approach and
202 OC20, 37. Translation: Cohen, Quantifying music, 214.
203 The tones of the octave are found using the consonances; this is called the division of the octave. The
3
seven tones of the diatonic scale are found by means of the fifth (
2
) and its complement, the fourth
4
(
7
). Likewise the chromatic tones are found by addition of fifths. A problem arises, however, because a
complete octave cannot be reached again by continuous addition of fifths. A small difference, called the
3 12
2 7
Pythagorean comma, exists between 12 fifths ( 2)
and 7 octaves ( 1)
. As a result, the tones of the
octave ought to be tempered in musical practice, which means that the purity of some consonances is
sacrificed. In mean tone temperament most major thirds are pure and the fifths are made a bit too large;
in equal temperament all consonances save the octave are a bit impure. Huygens preferred the former, the
latter has become standard tuning in Western music since the early nineteenth century.
204 The diatonic semitone is the difference between E and F, B and c, etc.; the chromatic semitone is the
difference between, for example, C and C . The chromatically sharpened C and flattened D – C # and D b –
differ, whereby C-D b and C #-D have the size of a diatonic semitone. The difference between C-C # and C-
D b is the difference between both kinds of semitones.
205 Most of Huygens’ musical studies is reproduced in OC20, 1-173. The French and Latin versions of the
letter have been reprinted with Dutch and English translations by Rasch in: Huygens, Le cycle harmonique.
206 Cohen, Quantifying music, 225-226.