Topic: Epimorphic genera

1 scales

Thread (4 messages)

From: Gene Ward Smith (2004-08-28)
Subject: Epimorphic genera

Genus(15^4) turns out to be permutation epimorphic; Scala cleverly
discovered this interesting fact once again. My Maple code only finds
if a scale is epimorphic according to the ordering I feed to it, so it
would be interesting to know what Scala does. Simply trying on
permutations gets rapidly out of hand.

Genus(15^4), like the rest of the Genus(15^i), has the nifty property
that under 3<-->5 it maps to itself, and 225, representing 7, is left
fixed, which could be interesting.

I ran through genus(3^a 5^b) under 2^50; a lot of the ones on this
list which were epimorphic had the a or b 0; eliminating those led to
the following list of nine epimorphic genera, of which many were
discouragingly rectangular. Below I give n, the monzo for n, and the
epimorph val; I haven't fiddled with the file to clarify notation but
it should be clear which is the val and which is the monzo.

15 [0, 1, 1] [4, 6, 9]
75 [0, 1, 2] [6, 9, 14]
225 [0, 2, 2] [9, 14, 21]
405 [0, 4, 1] [10, 16, 23]
675 [0, 3, 2] [12, 19, 28]
1875 [0, 1, 4] [10, 15, 23]
234375 [0, 1, 7] [16, 24, 37]
215233605 [0, 16, 1] [34, 54, 79]
114383962274805 [0, 28, 1] [58, 92, 135]

From: Gene Ward Smith (2004-08-28)
Subject: Re: Epimorphic genera

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:

> Genus(15^4) turns out to be permutation epimorphic; Scala cleverly
> discovered this interesting fact once again.

In a quest for similar facts, I did a search for cases where the
standard n-val distinguised every element of an n-note genus, once
again ignoring the pure 3 and 5 power cases. The results were more
encouraging than the search for strictly epimorphic scales. Below I
give m, the number of notes n of genus(m), and the monzo for m. Note
that 50625, with monzo |0 4 4> does appear and is joined by many other
genera, many not so depressingly skinny as before. Any monzo with the
second and third coefficient both greater than 1 is enough for some
marvel septimal harmony. Anything with the 3 coefficient > 1 and the 5
coefficient > 3 will have mivera harmonies, meaning 
{99/98, 176/175}-planar, involving both 225/224 and 5632/5625.

15 4 |0 1 1>
225 9 |0 2 2>
405 10 |0 4 1>
675 12 |0 3 2>
2025 15 |0 4 2>
3645 14 |0 6 1>
6075 18 |0 5 2>
18225 21 |0 6 2>
50625 25 |0 4 4>
91125 28 |0 6 3>
164025 27 |0 8 2>
455625 35 |0 6 4>
1476225 33 |0 10 2>
13286025 39 |0 12 2>
22143375 48 |0 11 3>
39858075 42 |0 13 2>
110716875 60 |0 11 4>
215233605 34 |0 16 1>
553584375 72 |0 11 5>
732421875 26 |0 1 12>
1076168025 51 |0 16 2>
2767921875 84 |0 11 6>
13839609375 96 |0 11 7>
26904200625 85 |0 16 4>
69198046875 108 |0 11 8>
91552734375 32 |0 1 15>
274658203125 48 |0 2 15>
345990234375 120 |0 11 9>
672605015625 119 |0 16 6>
1729951171875 132 |0 11 10>
4236443047215 52 |0 25 1>
12709329141645 54 |0 26 1>
43248779296875 156 |0 11 12>
114383962274805 58 |0 28 1>
571919811374025 87 |0 28 2>
1081219482421875 180 |0 11 14>

From: Gene Ward Smith (2004-08-28)
Subject: Re: Epimorphic genera

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:

> In a quest for similar facts, I did a search for cases where the
> standard n-val distinguised every element of an n-note genus, once
> again ignoring the pure 3 and 5 power cases. 

Below I do almost the same, except now I use the standard (n-1)-val.
This leads to situations where you must choose which note to drop off
the scale; the result is a scale which will be permuation epimorphic,
and can be strictly epimorphic. To show what sort of things can arise,
I give an 11-note permutation epimorphic scale, which marvelizes to
two major tetrads and two minor tetrads, extending to one major and
two minor pentads in the 9-limit. This seems pretty good. The Scott
scale turns up on this list as well.

45 5 [0, 2, 1]
75 5 [0, 1, 2]
135 7 [0, 3, 1]
1125 11 [0, 2, 3]
3645 13 [0, 6, 1]
16875 19 [0, 3, 4]
32805 17 [0, 8, 1]
703125 23 [0, 2, 7]
2109375 31 [0, 3, 7]
512578125 71 [0, 8, 7]
8968066875 79 [0, 15, 4]
32958984375 55 [0, 3, 13]
1121008359375 127 [0, 15, 7]

! genum1125.scl
Transposed genus(1125) minus a note; permutation epimorphic
11
!
9/8
6/5
5/4
32/25
45/32
3/2
8/5
9/5
15/8
48/25
2

From: Manuel Op de Coul (2004-08-30)
Subject: Re: [tuning-math] Epimorphic genera

Gene wrote:
>Genus(15^4) turns out to be permutation epimorphic; Scala cleverly
>discovered this interesting fact once again. My Maple code only finds
>if a scale is epimorphic according to the ordering I feed to it, so it
>would be interesting to know what Scala does.

The next version will indicate if it's epimorphic with non-monotonic
ordering. I've also improved it for that 5-limit 7-note scale you
mentioned.
So now it does the following steps. In each step checking epimorphism
is done by testing whether a prime-degree mapping visits each degree,
regardless of order by maintaining an array of booleans.
- check the standard val
- check the second best val, and two more if there are more than 2 primes.
- try finding a val by solving a set of linear equations, based on
degrees 1 .. m where m is the number of primes, check it, if not, 2 .. m+1,
etc.

Manuel