24 & 38 temperament

A while ago I posted about temperaments that equate two consecutive
superparticular ratios. For example, meantone equates 9/8 and 10/9,
minortone equates 10/9 and 11/10, etc. A lot of these temperaments
often seem to give particularly "magical" results, as Gene put it,
although I don't know if it has anything to do with this property or
something else.

At the very least I think they are conceptually useful temperaments in
that they take common intervals and just bisect them. So when 81/80
disappears, a major third gets split into two equal "meantones," which
that section of the harmonic series pretty easy to sing. Likewise,
tempering out 121/120 bisects the minor third into two neutral
seconds, which also makes that section of the harmonic series pretty
easy to sing. And tempering out 225/224 equates 15/14 and 16/15, which
makes "A# B C"... easy to sing.

So I decided to screw around a bit and see what the 13-limit rank-1
temperament is that tempers out 81/80, 121/120, 169/168, and 225/224.
You end up getting this unnamed "24 & 38" temperament where the
generator is ~252 cents and the period is ~600 cents. 13/62, 16/76,
18/86 are some decent generators for this. It also has an error of
about 5 cents. Also, despite the fact that the generator is right between
7/6 and 8/7, 49/48 isn't tempered out (except in the case of 38-et).

See here: http://x31eq.com/cgi-bin/rt.cgi?ets=24_38&error=8.280&limit=13&invariant=2_8_20_5_11_2_0_-8_-26_-1_-10

Does this temperament have any interesting properties besides
just making it easier to sing the harmonic series? I'm trying to
find all of the MOS scales in it now but I'm not sure how to
use Scala to do that for a non-equal temperament.

-Mike

Actually, looking into it more, it seems to just be, basically, some
kind of half-octave variant of meantone. Instead of the generator
being a fourth, the generator is a half a fourth. And instead of the
octave being 2/1, it's sqrt(2). So all of the MOS's you would expect
still apply - you have an MOS at 5, 7, 12, 19, 31, etc notes, which
when you stretch it to the octave gives you 10, 14, 24, 38, 62 note
scales. So I suppose if we have a "diaschismatic" temperament, then
this is "diameantone" or something.

The 10 and 14 note scales sound awesomely sinister but don't do too
much for traditional 5-limit harmony, unless there's some way of
permuting them that I'm not seeing.

-Mike

On Fri, Jun 18, 2010 at 1:53 PM, Mike Battaglia <battaglia01@...> wrote:
> A while ago I posted about temperaments that equate two consecutive
> superparticular ratios. For example, meantone equates 9/8 and 10/9,
> minortone equates 10/9 and 11/10, etc. A lot of these temperaments
> often seem to give particularly "magical" results, as Gene put it,
> although I don't know if it has anything to do with this property or
> something else.
>
> At the very least I think they are conceptually useful temperaments in
> that they take common intervals and just bisect them. So when 81/80
> disappears, a major third gets split into two equal "meantones," which
> that section of the harmonic series pretty easy to sing. Likewise,
> tempering out 121/120 bisects the minor third into two neutral
> seconds, which also makes that section of the harmonic series pretty
> easy to sing. And tempering out 225/224 equates 15/14 and 16/15, which
> makes "A# B C"... easy to sing.
>
> So I decided to screw around a bit and see what the 13-limit rank-1
> temperament is that tempers out 81/80, 121/120, 169/168, and 225/224.
> You end up getting this unnamed "24 & 38" temperament where the
> generator is ~252 cents and the period is ~600 cents. 13/62, 16/76,
> 18/86 are some decent generators for this. It also has an error of
> about 5 cents. Also, despite the fact that the generator is right between
> 7/6 and 8/7, 49/48 isn't tempered out (except in the case of 38-et).
>
> See here: http://x31eq.com/cgi-bin/rt.cgi?ets=24_38&error=8.280&limit=13&invariant=2_8_20_5_11_2_0_-8_-26_-1_-10
>
> Does this temperament have any interesting properties besides
> just making it easier to sing the harmonic series? I'm trying to
> find all of the MOS scales in it now but I'm not sure how to
> use Scala to do that for a non-equal temperament.
>
> -Mike
>

On 19 June 2010 02:19, Mike Battaglia <battaglia01@...> wrote:
> Actually, looking into it more, it seems to just be, basically, some
> kind of half-octave variant of meantone. Instead of the generator
> being a fourth, the generator is a half a fourth. And instead of the
> octave being 2/1, it's sqrt(2). So all of the MOS's you would expect
> still apply - you have an MOS at 5, 7, 12, 19, 31, etc notes, which
> when you stretch it to the octave gives you 10, 14, 24, 38, 62 note
> scales. So I suppose if we have a "diaschismatic" temperament, then
> this is "diameantone" or something.

If it's produced by two ETs with an even number of steps to the
octave, and you embrace any contorsion, the MOS family will equally
divide the octave. But 12&38 would already do that. So here, you're
dividing the octave, and dividing the generator.

It must be some permutation of meantone because you've specified that
81:80 be tempered out. It could work with 4 independent meantone
keyboards.

An MOS with an odd number of notes won't cover all ratios. It'll have
the opposite of contorsion.

> The 10 and 14 note scales sound awesomely sinister but don't do too
> much for traditional 5-limit harmony, unless there's some way of
> permuting them that I'm not seeing.

14 notes isn't many for something 4 times as complex as meantone.

Graham

On Sat, Jun 19, 2010 at 1:05 AM, Graham Breed <gbreed@...> wrote:
>
> If it's produced by two ETs with an even number of steps to the
> octave, and you embrace any contorsion, the MOS family will equally
> divide the octave. But 12&38 would already do that. So here, you're
> dividing the octave, and dividing the generator.
>
> It must be some permutation of meantone because you've specified that
> 81:80 be tempered out. It could work with 4 independent meantone
> keyboards.
>
> An MOS with an odd number of notes won't cover all ratios. It'll have
> the opposite of contorsion.

Eh, this is a basic question, but what is the difference between
contorsion and torsion? Torsion is when some interval x isn't tempered
out, but x^2 is, right? Or something similar. Would the opposite be if
some interval x is tempered out, but x^2 isn't?

-Mike

On 21 June 2010 21:53, Mike Battaglia <battaglia01@...> wrote:

> Eh, this is a basic question, but what is the difference between
> contorsion and torsion? Torsion is when some interval x isn't tempered
> out, but x^2 is, right? Or something similar. Would the opposite be if
> some interval x is tempered out, but x^2 isn't?

Torsion is a property of periodicity blocks, contorsion is a property
of regular temperaments (or things like regular temperaments that
don't qualify as temperaments because they have contorsion).

Graham