19-Tone Theory and Applications

Hubert S.  Howe  Jr.

doi:10.33779/2658-4824.2020.4.081-099

Hubert S. Howe Jr. Princeton University https://orcid.org/0000-0001-5822-8364

DOI: https://doi.org/10.33779/2658-4824.2020.4.081-099

Keywords: equal temperament, pitch, pitch class, numerical representation of pitch classes, pitch structures, normal form representation, of pitch structures, intervals, interval content, multiplicative operations, complementary operations, number of distinct forms of a set, total chromatic, trichords, tetrachords, pentachords, hexachords, septachords, arrays, weighted pitch classes, array inclusions.

Abstract

Microtonal music is one of those subjects
that has always been around, but few people
have ever had the will to investigate
it thoroughly. The main reason why more
people have not dealt with microtonal music
is that there are almost no instruments that
allow composers to experiment with it.
In spite of all this, the music of many cultures
even at the present time employs non-equaltempered
scales, and even Western music
did until the eighteenth century,
when mathematicians worked out
the logarithmic basis of equal temperament.
In this article, the author explains how
he became interested in 19-tone equal
temperament and how he explored the possible
resources available in such a system.
This involves creating a chord grammar based
on similarity relationships, similar to what
he has used in his music written
in 12-tone equal temperament. Through these
considerations he discovered a particular
set of chords that have special properties
in terms of their interval content, number
of transpositions, and relationships
to other chords. Finally, the author explains
how he used these properties in the music
he composed in this temperament.