Mike's Scale in Other EDOs and Temperaments

For those just tuning in: Mike B. found a scale he liked in 12-EDO and wants to find a better harmonic tuning for it. His scale in 12-EDO is 1 2 1 2 2 1 2 1, which I pointed out has 4 large steps and 4 small steps, so could be a permutation of the diminished scale, and thus would work in any EDO divisible by 4. I looked at a 36-EDO version that gives excellent 7-limit harmonies, but it drastically worsens both 3-limit and 5-limit harmony to do this.

Since Mike found the scale in 12-tET, we can also look at his scale as being a subset of other scales. For starters, let's look at it as the subset of a 9-note scale of 1 2 1 2 1 1 1 2 1, a scale of 6 small steps and 3 large steps, which results in breaking down one of the large steps to two small steps. When made distributionally-even, it looks like 1 1 2 1 1 2 1 1 2. This scale will work in any EDO divisible by 3, so we know that 2s+L=400 cents. So we know that some degrees of the scale will be constant, regardless of the EDO we choose, and others will be variable:
I: 0 cents
II: s cents
III: 400-s cents
IV: 400 cents
V: 400+L cents
VI: 800-s cents
VII: 800 cents
VIII: 800+s cents
IX: 1200-s cents

Mike wants some stronger 7-limit harmony, but probably not at the expense of the 3-limit. This means we need the VI to be around 700 cents; this forces us to have an s not too far from 100 cents. If we want a strong 7th harmonic, we'd have to look at another mode, because there's no room in either the VIII or IX degree to have an approximate 7th harmonic. If we make VIII 800+L instead, giving enough room to approximate 7/4, then we'd want an L between 160 or 175 cents or so to get an approximate 7/4; since 2s+L=400, we'd get an s between 120 and 112.5; the former gives 680 cents on the VI degree, and suggests 30-EDO. The latter gives 687.5 cents on the VI degree, and suggests 96-EDO.

OTOH, we can try making IX be 1200-L cents to approximate 7/4, suggesting an L between 240 and 225 cents or so, giving an s between 80 cents and 87.5. The former gives 720 cents on the VI degree, and suggests 15-EDO; the latter gives 712.5 cents on the VI degree, and suggests 96-EDO as well. So regardless of how we approach it, 96-EDO seems to be a good option. However, this is also suggestive of Augene temperament, so looking at TOP Augene may be the best solution.

Next approach: back in 12-EDO, we can take that 9-note scale apart even further: from 1 2 1 2 1 1 1 2 1, we can get 1 1 1 1 2 1 1 1 2 1, which is 2L+8s. This suggests even-numbered EDOs, or a period of 1/2 octave (L+4s=600 cents, IOW). To get the best 7-limit harmony from this scale, I'll just cut the crap and say we're looking at Pajara temperament, since I know Pajara is optimized for 7-limit harmony AND gives a scale of this configuration. Since 22-EDO is pretty good for Pajara, let's take a look at how it renders Mike's scale: we've got an s of 2 steps and an L of 3; how do we get Mike's 8-note scale out of this? We add some small steps together and maybe move the large steps around, to get: 2 4 2 3 4 2 3 2, 2 4 2 4 3 2 3 2, 2 3 2 4 4 2 3 2, or 2 4 2 3 3 2 4 2. I leave it to Mike to find out which one he likes best.

But why stop at a 10-note scale? Let's go back to 12-tET and break it down further, to an 11-note superscale, 1 1 1 1 1 2 1 1 1 1 1. This gives 10s+1L, which is not very helpful in finding a pattern of EDOs. Technically, any EDO greater than 12 will work for this MOS shape. However, we want our 8-note subset to have some specific harmonic properties. Our subset could be something like s 2s s 2s L s 2s s, or s 2s s 2s 2s s L s, or s L s 2s 2s s 2s s. We probably want L to be not too far from 2s in size; maybe it should be bigger, maybe smaller. This is where the guess-work starts. For simplicity's sake, let's say (5s/4)</=L</=(5s). This gives us 45-EDO, 35-EDO, 34-EDO, 23-EDO, 12-EDO, 13-EDO, 14-EDO, and 15-EDO. None of these produce scales with any accuracy in the 3-limit except for 12-EDO, and none of them produce scales with strong 7-limit harmony, either. So the 11-note superscale is probably a bad choice.

Hope this helps, Mike!

-Igs

--- In tuning@yahoogroups.com, "cityoftheasleep" <igliashon@...> wrote:
So the 11-note superscale is probably a bad choice.
>
> Hope this helps, Mike!

You never got to the 12-note superscale. That means asking for temperaments supported by 12et. You've mentioned some already, but an obvious omission is meantone. There's also schismatic/Helmholtz and dominant.

--- In tuning@yahoogroups.com, "genewardsmith" <genewardsmith@...> wrote:

> You never got to the 12-note superscale.

That would mean dividing Mike's scale into a superscale of 1 1 1 1 1 1 1 1 1 1 1 1 in 12-tET, which implies 12-note MOS scales of ssssssssssss or LLLLLLLLLLLL, i.e. L=s. The point of my exercise was trying to first find MOS superscales that could contain Mike's scale based on its step-configuration in 12-tET, and then to find some good tunings for each scale given Mike's harmonic wish-list. There's only one tuning for an MOS of ssssssssssss or LLLLLLLLLLLL, and that's 12-tET, assuming octaves are pure.

-Igs

Igs,

This is awesome. Thanks for taking the time out to do it. The fact
that it works in Pajara as a subset of a 10-note permuted MOS scale is
basically the hunch that I had, although I couldn't see it at the
time. Hopefully it turns out to be a major/augmented^7 variant of the
decatonic scales. That is, mixolydian:ionian::this scale:spm.

I only have the time to scan this right now, but I'll delve more into
this in the morning.

-Mike

On Mon, Aug 30, 2010 at 4:01 PM, cityoftheasleep
<igliashon@...> wrote:
>
> For those just tuning in: Mike B. found a scale he liked in 12-EDO and wants to find a better harmonic tuning for it. His scale in 12-EDO is 1 2 1 2 2 1 2 1, which I pointed out has 4 large steps and 4 small steps, so could be a permutation of the diminished scale, and thus would work in any EDO divisible by 4. I looked at a 36-EDO version that gives excellent 7-limit harmonies, but it drastically worsens both 3-limit and 5-limit harmony to do this.
>
> Since Mike found the scale in 12-tET, we can also look at his scale as being a subset of other scales. For starters, let's look at it as the subset of a 9-note scale of 1 2 1 2 1 1 1 2 1, a scale of 6 small steps and 3 large steps, which results in breaking down one of the large steps to two small steps. When made distributionally-even, it looks like 1 1 2 1 1 2 1 1 2. This scale will work in any EDO divisible by 3, so we know that 2s+L=400 cents. So we know that some degrees of the scale will be constant, regardless of the EDO we choose, and others will be variable:
> I: 0 cents
> II: s cents
> III: 400-s cents
> IV: 400 cents
> V: 400+L cents
> VI: 800-s cents
> VII: 800 cents
> VIII: 800+s cents
> IX: 1200-s cents
>
> Mike wants some stronger 7-limit harmony, but probably not at the expense of the 3-limit. This means we need the VI to be around 700 cents; this forces us to have an s not too far from 100 cents. If we want a strong 7th harmonic, we'd have to look at another mode, because there's no room in either the VIII or IX degree to have an approximate 7th harmonic. If we make VIII 800+L instead, giving enough room to approximate 7/4, then we'd want an L between 160 or 175 cents or so to get an approximate 7/4; since 2s+L=400, we'd get an s between 120 and 112.5; the former gives 680 cents on the VI degree, and suggests 30-EDO. The latter gives 687.5 cents on the VI degree, and suggests 96-EDO.
>
> OTOH, we can try making IX be 1200-L cents to approximate 7/4, suggesting an L between 240 and 225 cents or so, giving an s between 80 cents and 87.5. The former gives 720 cents on the VI degree, and suggests 15-EDO; the latter gives 712.5 cents on the VI degree, and suggests 96-EDO as well. So regardless of how we approach it, 96-EDO seems to be a good option. However, this is also suggestive of Augene temperament, so looking at TOP Augene may be the best solution.
>
> Next approach: back in 12-EDO, we can take that 9-note scale apart even further: from 1 2 1 2 1 1 1 2 1, we can get 1 1 1 1 2 1 1 1 2 1, which is 2L+8s. This suggests even-numbered EDOs, or a period of 1/2 octave (L+4s=600 cents, IOW). To get the best 7-limit harmony from this scale, I'll just cut the crap and say we're looking at Pajara temperament, since I know Pajara is optimized for 7-limit harmony AND gives a scale of this configuration. Since 22-EDO is pretty good for Pajara, let's take a look at how it renders Mike's scale: we've got an s of 2 steps and an L of 3; how do we get Mike's 8-note scale out of this? We add some small steps together and maybe move the large steps around, to get: 2 4 2 3 4 2 3 2, 2 4 2 4 3 2 3 2, 2 3 2 4 4 2 3 2, or 2 4 2 3 3 2 4 2. I leave it to Mike to find out which one he likes best.
>
> But why stop at a 10-note scale? Let's go back to 12-tET and break it down further, to an 11-note superscale, 1 1 1 1 1 2 1 1 1 1 1. This gives 10s+1L, which is not very helpful in finding a pattern of EDOs. Technically, any EDO greater than 12 will work for this MOS shape. However, we want our 8-note subset to have some specific harmonic properties. Our subset could be something like s 2s s 2s L s 2s s, or s 2s s 2s 2s s L s, or s L s 2s 2s s 2s s. We probably want L to be not too far from 2s in size; maybe it should be bigger, maybe smaller. This is where the guess-work starts. For simplicity's sake, let's say (5s/4)</=L</=(5s). This gives us 45-EDO, 35-EDO, 34-EDO, 23-EDO, 12-EDO, 13-EDO, 14-EDO, and 15-EDO. None of these produce scales with any accuracy in the 3-limit except for 12-EDO, and none of them produce scales with strong 7-limit harmony, either. So the 11-note superscale is probably a bad choice.
>
> Hope this helps, Mike!
>
> -Igs