Organizing subgroup temperaments into families

There's still one really crucial part of full-limit temperament theory
which has yet to be adequately generalized to handle subgroups, and that's
to find some meaningful and concrete way to organize them.

For full-limit temperaments, this problem has an existing solution: Gene's
"family" structure. This lays out specific rules for when you can say one
temperament is "an extension" of another; the "best" extension typically
shares the name of the original. If we could simply set up an analogous set
of rules for subgroup temperaments, we'd be sitting pretty.

Luckily, we already have an obvious analogous concept of "extension" for
subgroups, obtained by tweaking an earlier suggestion from Keenan Pepper.

Keenan's come up with a set of rules for when you can say temperament x is
a "subgroup restriction" of temperament y. Those rules are: x is a
restriction of y if x is not contorted, if it isn't "insane" (defined
elsewhere previously on this list), and if x, treated as a multilinear
function, equals the restriction of y to x's subgroup. Under his rules, we
can say that the 2.3.7 49/48 "semaphore" temperament is the 2.3.7
restriction of godzilla, or "2.3.7 godzilla."

If we apply his rules in reverse, we thus come up with a well-defined way
to say that a temperament is "an extension" of another; the 2.3.5.7
"godzilla" temperament is thus an "extension" of the 2.3.7 "semaphore"
temperament.

A cascade of further insights can be further derived from this reasoning:

- Any temperament in subgroup S can have a temperament extension in a
subgroup T if T contains S as a subgroup.
- The item above is backward-compatible with Gene's existing family
structure, which takes as its set of possible subgroups the full-limits
only.
- While the set of full-limits forms a total order under inclusion (2 < 2.3
< 2.3.5 < 2.3.5.7 < 2.3.5.7.11 < ...), the set of arbitrary JI subgroups
forms a partial order.
- Under this setup, 2.3.5 "meantone" is NOT a "restriction" of 2.3.5.7.11
"mohajira," thus 2.3.5.7.11 mohajira is NOT an "extension" of 2.3.5
meantone.
- This last point is particularly important, because using other notions of
"extension," 2.3.5.7.11 mohajira IS an extension of 2.3.5 meantone.

More to come...

Mike

--- In tuning-math@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> There's still one really crucial part of full-limit temperament theory
> which has yet to be adequately generalized to handle subgroups, and that's
> to find some meaningful and concrete way to organize them.
>
> For full-limit temperaments, this problem has an existing solution: Gene's
> "family" structure. This lays out specific rules for when you can say one
> temperament is "an extension" of another; the "best" extension typically
> shares the name of the original. If we could simply set up an analogous set
> of rules for subgroup temperaments, we'd be sitting pretty.
>
> Luckily, we already have an obvious analogous concept of "extension" for
> subgroups, obtained by tweaking an earlier suggestion from Keenan Pepper.
>
> Keenan's come up with a set of rules for when you can say temperament x is
> a "subgroup restriction" of temperament y. Those rules are: x is a
> restriction of y if x is not contorted, if it isn't "insane" (defined
> elsewhere previously on this list), and if x, treated as a multilinear
> function, equals the restriction of y to x's subgroup. Under his rules, we
> can say that the 2.3.7 49/48 "semaphore" temperament is the 2.3.7
> restriction of godzilla, or "2.3.7 godzilla."
>
> If we apply his rules in reverse, we thus come up with a well-defined way
> to say that a temperament is "an extension" of another; the 2.3.5.7
> "godzilla" temperament is thus an "extension" of the 2.3.7 "semaphore"
> temperament.
>
> A cascade of further insights can be further derived from this reasoning:
>
> - Any temperament in subgroup S can have a temperament extension in a
> subgroup T if T contains S as a subgroup.
> - The item above is backward-compatible with Gene's existing family
> structure, which takes as its set of possible subgroups the full-limits
> only.
> - While the set of full-limits forms a total order under inclusion (2 < 2.3
> < 2.3.5 < 2.3.5.7 < 2.3.5.7.11 < ...), the set of arbitrary JI subgroups
> forms a partial order.
> - Under this setup, 2.3.5 "meantone" is NOT a "restriction" of 2.3.5.7.11
> "mohajira," thus 2.3.5.7.11 mohajira is NOT an "extension" of 2.3.5
> meantone.
> - This last point is particularly important, because using other notions of
> "extension," 2.3.5.7.11 mohajira IS an extension of 2.3.5 meantone.

I'm Keenan Pepper and I approve this message.

Keenan