How to construct a lumma

If you start class, prism and centaur at the bottom of the meantone
chain, step through the circle of modmos fifths in the 5-limit
versions, and take the ratio of the approximate fifths you get in this
way with 3/2, you obtain the following:

class

[128/125, 80/81, 1, 1, 1, 80/81, 128/125, 125/128, 1, 2048/2025,
1, 125/128]

prism

[1, 1, 80/81, 1, 1, 1, 2048/2025, 1, 125/128, 1, 2048/2025, 1]

centaur

[1, 1, 1, 80/81, 1, 1, 2048/2025, 1, 1, 1, 80/81, 1]

These all follow the following rules:

(1) Use "interesting" commas, meaning commas of 12-et less than 50
cents in size and such that the superpyth Graham complexity is less
than 23. In fact, these all use an "interesting" short list,
consisting only of 81/80, 128/125, and 2048/2025.

(2) Circulate--the product of the commas should be 2^19/3^12. This
cancels out the Pythagorean comma of a circle of 12 pure fifths, and
returns us to our starting point, giving circulation.

(3) Alternate commas greater than and less than one enough to keep the
deviation from 12-et within bounds.

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

> These all follow the following rules:

Actually, 128/125 has a superpyth complexity of 27 so I'd need to
relax it at least that far. It's also not clear how important the
circulation is, or how to go about it; you presumably can fudge a
little. You can manage to keep all of these rules with just the
Pythagorean scale and just one Pythagorean comma, but more interesting
I think would be
81/80 and the reciprocal of Pythagorean times 81/80, or
41943040/43046721. The latter gives a flat fifth, but one extremely
close to 19/13; of course we probably plan to temper it, after which
it no longer is. The difference here brings in the ultra-ultra
19-limit comma 272629760/272629233, which is 0.0033 cents and an
epimericity of 0.323.

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

You can manage to keep all of these rules with just the
> Pythagorean scale and just one Pythagorean comma...

If anyone wants to know what the lumma you get that way is, it's
basically this interesting scale called 12-equal. In the 1/4-kleismic
tuning, the fifth is 700.027 cents, but 700 cents is an excellent
marvel tuning for the fifth, shared by 72 and 84, and by their sum 156
which makes for a good marvel tuning--and any of these will reduce the
scale to 12-equal. If you Hahn reduce using 225/224 you get the exact
same scale which I recently presented on this list called rat12, which
I obtained by Hahn reducing via the commas of 72-et; apparently
225/224 is all that matters. It can be added to the detempered lumma
list, and even in detempered form it still circulates. The main
problem with it is that there isn't much point to it, since it doesn't
deliver in the smooth thirds department. Tempered by marvel there is
even less point, since it mutates into 12-equal. However it may be
interesting for some theoretical purposes.

>If you start class, prism and centaur at the bottom of the meantone
>chain, step through the circle of modmos fifths in the 5-limit
>versions, and take the ratio of the approximate fifths you get in this
>way with 3/2, you obtain the following:
>
>class
>[128/125, 80/81, 1, 1, 1, 80/81, 128/125, 125/128, 1, 2048/2025,
>1, 125/128]
>
>prism
>[1, 1, 80/81, 1, 1, 1, 2048/2025, 1, 125/128, 1, 2048/2025, 1]
>
>centaur
>[1, 1, 1, 80/81, 1, 1, 2048/2025, 1, 1, 1, 80/81, 1]
>
>These all follow the following rules:
>
>(1) Use "interesting" commas, meaning commas of 12-et less than 50
>cents in size and such that the superpyth Graham complexity is less
>than 23. In fact, these all use an "interesting" short list,
>consisting only of 81/80, 128/125, and 2048/2025.
>
>(2) Circulate--the product of the commas should be 2^19/3^12. This
>cancels out the Pythagorean comma of a circle of 12 pure fifths, and
>returns us to our starting point, giving circulation.
>
>(3) Alternate commas greater than and less than one enough to keep
>the deviation from 12-et within bounds.

Kyool.

-C.