Euler genus of a comma

Jacques Dudon proposed an interesting method of obtaining a scale from
a comma, as the union of the Euler genus of the numerator (the
divisors of the numerator, reduced to an octave) with that of the
denominator. A related idea is to use a single Euler genus associated
to the comma, by
taking the least common multiple of the numerator and denominator, and
then the genus from that. Both of these methods result in scales
containing the comma itself as a scale step, and hence both lead to
scales which are logical targets for tempering by the comma in question.
We might call this Euler genus the associated genus; if we remove the
steps the size of a comma, or temper them out, or simply TM reduce it
with respect to the comma, we occasionally get epimorphic scales.

Gene Ward Smith wrote:
> Jacques Dudon proposed an interesting method of obtaining a scale
> from a comma, as the union of the Euler genus of the numerator (the
> divisors of the numerator, reduced to an octave) with that of the
> denominator. A related idea is to use a single Euler genus associated
> to the comma, by
> taking the least common multiple of the numerator and denominator, and
> then the genus from that. Both of these methods result in scales
> containing the comma itself as a scale step, and hence both lead to
> scales which are logical targets for tempering by the comma in question.
> We might call this Euler genus the associated genus; if we remove the
> steps the size of a comma, or temper them out, or simply TM reduce it
> with respect to the comma, we occasionally get epimorphic scales.

So, generalising further, is a comma special; what can we say of the
Euler genus of an arbitrary rational interval n/d? (Mathematically, a
comma is just such a ratio.)

Again, is the octave 1:2 special for this reduction; what happens if we
reduce the divisors instead to a range of 1:3? of 2:3? etc ...

A musical note:
Melodically, I feel that the Dudon scales would be interesting even
without tempering; maybe more so than when tempered. The smaller the
interval from leading-tone to octave, the more expressive, I think.

Regards,
Yahya

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