TITLE:
Music as Mathematics of Senses
AUTHORS:
Hailong Li, Kalyan Chakraborty, Shigeru Kanemitsu
KEYWORDS:
Pythagorean Scale, Just Intonation, Temperament, Beat, Pythagoras’ Law of Small Numbers, Law of Cyclotomic Numbers
JOURNAL NAME:
Advances in Pure Mathematics,
Vol.8 No.12,
December
21,
2018
ABSTRACT: It is often said that music has
reached its supreme and highest level in the 18th and 19th centuries. One of
the main reasons for this achievement seems to be the robust structure of
compositions of music, somewhat remindful of robust structure of mathematics.
One is reminded of the words of Goethe: Geometry is frozen music. Here, we may extend geometry
to mathematics. For the Middle Age in Europe, there were seven main subjects in the universities or in higher education.
They were grammar, logic and rhetoric—these three (tri) were regarded as more
standard and called trivia (trivium), the origin of the word trivial.
And the remaining four were arithmetic, geometry, astronomy and music—these
four (quadrus) were regarded as more advanced subjects and were called
quadrivia (quadrivium). Thus for Goethe, geometry and mathematics seem to be
equivocal. G. Leibniz expresses more in detail in his letter to C. Goldbach in
1712 (April 17): Musica est exercitium arithmeticae occultum nescientis se
numerari animi (Music is a hidden arithmetic exercise of the soul, which doesn’t
know that it is counting). Or in other respects, J. Sylvester expresses more in
detail: Music is mathematics of senses. Mathematics is music of reasons. Thus, the title arises. This paper is a sequel to [1] and examines
mathematical structure of musical scales entailing their harmony on expanding
and elaborating material in [2] [3] [4] [5], etc. In statistics, the strong law of large numbers is well-known which claims
that This means that the relative frequency of
occurrences of an event A tends to
the true probability p of the
occurrences of A with probability 1.
In music, harmony is achieved according to Pythagoras’ law of small numbers,
which claims that only the small integer multiples of the fundamental notes can
create harmony and consonance. We shall also mention the law of cyclotomic
numbers according to Coxeter, which elaborates Pythagoras’ law and suggests a
connection with construction of n-gons
by ruler and compass. In the case of natural scales (just intonation), musical
notes appear in the form 2p3q5r (multiples of
the basic note), where p∈Z,q=-3, -2, -1, 0, 1, 2, 3 and r=-1, 0, 1. We shall give mathematical details of the
structure of various scales.