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) .
[from Monzo: Chalmers gives an example here, but it really illustrates specifically an euler-fokker genus.]
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 thus:
5:4 ---15:8 / \ / / \ / / \ / 1:1 --- 3:2
Another example: the factors 2n * 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.
There's an error in the Euler genus def. page.
At the bottom: The fundamental of the first is 1/8 and the guide tone 15. If you'd draw a lattice without the factors of 2, you'd get a fundamental of 1/1 and a guide tone of 15/1. For the second likewise it's 1/1 and 225 if you leave out factors of 2.
So if you have the chord 1:3:5:15 you can't add any tones without increasing the guide tone. But you have 8:10:12:15 and I can add 1, 3 and 5 for example without increasing the guide tone (120).
If you include the mathematical definition in the page it will make things much more comprehensible.
So the fundamental is the greatest common divisor of the frequency ratios, the guide tone is the lowest common multiple.
So if you take only two tones, 1 and 3, chord is 1:3, then the GCD is 1 and the LCM is 3.
If you take the tones 1 and 3/2, chord is 2:3, then the GCD is 1 and the LCM is 6.
That makes it clear that factors of 2 make a difference.
For the chord 1:3:5:15 it's 1 and 15. This is also a "complete chord". I think you could write a definition page for this too (or combine it with the genus page).
Euler also defined "Exponens consonantiae" which is the quotient of the LCM and GCD of the chord. A complete chord is a chord to which no tones can be added without increasing the EC. This is an Euler-Fokker genus by definition. The fundamental and guide tone are in the chord and are diametrically opposed in the lattice.
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