Euler-Fokker Genus

An Euler Genus which contains factors of not only 3 and 5, but also 7. It is thus represented by a 3-dimensional (rather than 2-dimensional) lattice.

John Chalmers gave the following example in his definition of Euler genera, but strictly speaking, since we are noting a difference here, it is an Euler-Fokker genus, because it is 3-dimensional:

[For example,] The factors 2n*3*5*7 generate the tones 1/1 35/32 5/4 21/16 3/2 105/64 7/4 15/8 and 2/1 which is also an Octony.

[from John Chalmers, Divisions of the Tetrachord]



Here is a lattice diagram of the above Euler-Fokker genus, using the 'triangular' convention:

            35:32 ---- 105:64
           .'/ \'.   .'/
        5:4 /---\15:8 /
        /:\/     \/: /
       / :/\     /\:/
      / 7:4 ---- 21:16
     /.'   '.\ /.'
   1:1 ----- 3:2

The lattice of an Euler-Fokker genus will always bound a cubic or parellelepiped structure.

See Euler Genus, and also my translation of Patrice Bailhache's Music and Mathematics: Leonhard Euler; also Manuel Op de Coul's page re: Euler-Fokker genus.

[from Joe Monzo, JustMusic: A New Harmony]

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