136 Chapter 9
accounts did not seek to bridge these two; with few exceptions to this day, he was a lone
pioneer of mathematical aesthetics.^13 He founds his new theory on “ the exact knowledge
of sound, ” understood from the mechanics of waves, and on “ metaphysics ” : “ Led by reason
as well as experience, we attacked that problem and drew the conclusion that two or more
sounds are pleasing when the ratio, which exists between the numbers of vibrations pro-
duced at the same time, is understood; on the other hand, dissatisfaction is present when
either no order is felt or that order which it seems to have is suddenly confused. ” To make
this quantitative, “ we graded this perceptive ability in certain degrees, which are of greatest
importance in music and also may be found to be of great value in other arts and sciences
of which beauty is a part. Those degrees are arranged in accordance with the ease of
perceiving the ratios, and all those ratios that can be perceived with equal facility are
related to the same degree. ” He calls this their degree of agreeableness ( gradus suavitatis ),
using a Latin word that might also be translated as sweetness , charm, or tunefulness.^14
Though Euler ’ s exposition of these degrees may remind us of the ancient ordering based
on the perfection of intervals, his definition reminds us that he ranks “ how much agree-
ableness each consonance has in itself or, what amounts to the same thing, how much
facility is required for perceiving it. ” Where the ancients had placed the priority on the
intervals and ratios themselves, Euler now places it in the perceiving human subject. Still,
his prior mathematical sense of the relative simplicity of various ratios informs his ensuing
definitions (see box 9.1 for details).Box 9.1
How Euler constructed his degree of agreeablenessFirst, he assigned degree 1 to 1:1 and degree 2 to 1:2, which sets the basic pattern: “ by the
simple operation of halving or doubling, the degree of agreeableness is changed by unity. ”
Then to ratios of the form 1:2 n he assigned the degree ( n + 1) because “ the degrees progress
equally in ease of perception. Thus, the fifth degree is perceived with more difficulty than
the fourth, ” and so on. For ratios of the form 1: p , where p is prime, he assigns the degree p ;
thus, both 1:3 and 1:4 have degree 3, to accord with both principles he used thus far. He then
argues that 1: pq (where both p and q are prime) has degree p + q – 1. A few more steps led
him to the general conclusion that for any composite number m composed of n prime factors
whose sum is s , the ratio 1: m has the degree of agreeableness s – n + 1. Based on this, he
then argued that the degree of a series of proportions such as p:q or p : q : r (where p , q , r are
primes) is the same as that of 1: pq or 1: pqr respectively, where Euler calls the least common
multiple of these primes the exponent of that ratio. Hence, he assigned to 1: pqr and to p : q : r
the same degree, p + q + r – 2. Thus, the fifth (2:3) has degree (5 – 2) + 1 = 4, the same as
the degree of 1:6. A major triad C – E – G (4:5:6) has the same degree as 1:4 · 5 · 6 = 1:120, whose
exponent 60 = 2^2 · 3 · 5, s = 12 and n = 4, thus the degree is s – n + 1 = 9, the same as the
dissonant major seventh C – E – G – B, 8:10:12:15, whose exponent is also 60. A minor triad
like E – G – B (5:6:7) has exponent 210 = 5 · 6 · 7, so s = 18, n = 3, and degree s – n + 1 = 16.