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:

               
           5:4 ---15:8
           / \     /
          /   \   /
         /     \ /
       1:1 --- 3:2
    
    

    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:

    
    
              25:16---75:64---225:128
               / \     / \     /
              /   \   /   \   /
             /     \ /     \ /
           5:4 ---15:8 --- 45:32
           / \     / \     /
          /   \   /   \   /
         /     \ /     \ /
       1:1 --- 3:2 --- 9:8
    
    
    

    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.


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