Definitions of tuning terms
© 1998 by Joseph L. Monzo
All definitions by Joe Monzo unless otherwise cited
Euler Genus
Scales consisting of all the tones which are multiples of a
set of factors and divisors of the the product of those same factors
were discovered by Leonhard Euler and called by the term Genus Musicum
(Euler, 1739; Fokker, 1966; Rasch, 1987) .
[Chalmers gives an example here, but it really illustrates
specifically an
Euler-Fokker genus. -Monzo]
[from John Chalmers, Divisions of the Tetrachord]
When used specifically to differentiate from
Euler-Fokker genera, Euler genus refers to a 2-dimensional
5-limit system.
For example, the factors 2n*3*5 generate the tones
1/1, 5/4, 3/2, 15/8, 2/1, an Euler genus that can be diagrammed
on a lattice as follows, using the 'triangular'
convention:
Another example: the factors sn*32*52
generate the tones 1/1, 9/8, 75/64, 5/4, 45/32, 3/2, 25/16, 225/128, 15/8,
2/1, which can be shown on the lattice as:
The lattice of a 2-dimensional (i.e.,
5-limit) Euler genus
will always bound a square or parallelogram structure.
Because of this square structure, an Euler genus can
be reckoned as either a
harmonic series or
otonality, with the
numerary nexus in the denominators
of the
fractions which describe the
ratios, and the 'root' known as
the fundamental; or as a
subharmonic series or
utonality, with the
numerary nexus in the numerators of the rational fractions;
Fokker called the 'root' of the latter the
guide-tone.
The 'fundamental' of both of these examples is 1/1;
the 'guide-tone' of the first is 15/8, and of the second,
225/128.
[from Joe Monzo,
JustMusic: A New Harmony]
See also
Euler-Fokker genus,
and my translation of Patrice Bailhache's
Music and Mathematics: Leonhard Euler.
5:4 ---15:8
/ \ /
/ \ /
/ \ /
1:1 --- 3:2
25:16---75:64---225:128
/ \ / \ /
/ \ / \ /
/ \ / \ /
5:4 ---15:8 --- 45:32
/ \ / \ /
/ \ / \ /
/ \ / \ /
1:1 --- 3:2 --- 9:8
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