Marvel and musical genera

A musical genus is the set of all divisors of some positive integer n,
reduced to an octave. Since we plan on reducing to an octave, we can
restrict n to be odd, and then there is a 1-1 relationship between odd
integers and musical genera, which we can denote by the function
genus(n). The concept and terminology is (more or less) due to Euler.

For any odd n, the number of notes in genus(n) is d(n), where d is the
number of divisors function. d(n) is a multiplicative function; that
means it is a function f whose domain is the positive integers, and
such that f(a*b) = f(a)f(b) if gcd(a,b)=1. If the monzo for n is
|e2 e3 e5 ... ep> then d(n) = (e2+1)(e3+1) ... (ep+1); of course if n
is odd (e2+1) is just 1 and we can ignore it.

If m divides n, then the number of translates inside of genus(n) of
genus(m) is d(n/m), which is convenient for counting chords. For
example, genus(15), which consists of {1, 3/2, 5/4, 15/8}, contains
the major triad {1, 5/4, 3/2} and the minor triad {5/4, 3/2, 15/8}. We
can count the number of major and minor triads in a genus just by
counting the number of genus(15) translates in the genus, and this we
easily find from d(n/15). For example genus(675), the Duodene, has
d(675/15) = d(45) = 6 major triads and 6 minor triads.

We can also apply this method to tempered genera. The 5-limit marvel
equivalent of 224 = 32*7 is 225, and we can count marvel tetrads in a
5-limit genus, genus(n), by genus(n/225). The least common multiple of
225 and 9 is 225, so we also count the marvel pentads in this way--in
other words, every tetrad is already in a pentad. The marvel version
of 1/11 is 375, and 375*225 is 84375, so we can count marvel hexads
using d(n/84375). Another useful way to get 11-limit harmony is
{99/98, 176/175}-planar; this has 5632/5625 as a comma (size 2.153
cents) and putting it together with 225/224 gives {99/98, 176/175} as
a TM basis. Genus(5635) is a scale of 15 notes, having an otonal and
utonal hexad using 5625/512 for an 11 and 225/32 for a 7, and the
hexad count now is found using d(n/5625).

From the point of view of 7-limit marvel tempering, a genus of the
form genus(15^i) is especially nice; this makes a scale of size
(i+1)^2 with (i-1)^2 otonal and (i-1)^2 utonal pentads. Genus(15^4) is
particularly worthy of notice, as it is epimorphic. The Scala archives
calls this one "Genus bis-ultra-chromaticum [33335555]", which makes
me wonder if Euler talked about it; I recall him mentioning some
particular possibilities. 15^4/5625 = 9, and d(9) = 3, so this has
three otonal and three utonal hexads of the 5625 variety. It also has
3^2 = 9 otonal and utonal marvel pentads.

>We can also apply this method to tempered genera. The 5-limit marvel
>equivalent of 224 = 32*7 is 225, and we can count marvel tetrads in a
>5-limit genus, genus(n), by genus(n/225).

Swell.

-C.

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:

> >We can also apply this method to tempered genera. The 5-limit marvel
> >equivalent of 224 = 32*7 is 225, and we can count marvel tetrads in a
> >5-limit genus, genus(n), by genus(n/225).
>
> Swell.

Is this mild enthusiasm, or a mild lack of enthusism? :)

>> >We can also apply this method to tempered genera. The 5-limit marvel
>> >equivalent of 224 = 32*7 is 225, and we can count marvel tetrads in a
>> >5-limit genus, genus(n), by genus(n/225).
>>
>> Swell.
>
>Is this mild enthusiasm, or a mild lack of enthusism? :)

Sorry, it was meant to be high enthusiasm.

-Carl