Some 13-limit stuff

I did a 13-limit survey on tuning-math recently, and thought I would pull out the results of possible interest to Margo-style practice. Here I first give the mapping and corresponding generators, then the mean error, two kinds of complexity and the corresponding badness measures, and finally MOS scales which might be used.

[[1, 0, -31, -21, -14, -9], [0, 1, 21, 15, 11, 8]] [1200., 1904.391710]

rms 6.209025908 comp 11.46080276 graham 21

bad 240.9051197 grabad 597.5199555

mos [12, 17, 29, 46, 63]

This has a fifth of 704.39, which is right in Margo's range.

[[1, 0, 15, 25, -33, -28], [0, 1, -8, -14, 23, 20]] [1200., 1902.127698]

rms 2.806870405 comp 19.72435043 graham 37

bad 245.8818240 grabad 631.7204666

mos [12, 17, 29, 41, 53, 94]

This is one version of 13-limit schismic, with a fifth of 702.13. Both of these look pretty good.

[[2, 0, 11, 31, 45, 55], [0, 1, -2, -8, -12, -15]] [600.0000000, 1903.786589]

rms 2.880707578 comp 19.68247952 graham 34

bad 251.5468524 grabad 571.1070885

mos [10, 12, 22, 34, 46, 58]

This has two chains of fifths of size 703.79 cents, separated by a half-octave, so it is readily adaptable to a double keyboard setup.

[[29, 46, 0, 14, 33, 40], [0, 0, 1, 1, 1, 1]] [41.37931034, 2788.239580]

rms 2.277983567 comp 22.46330341 graham 29

bad 242.5275176 grabad 355.7521911

mos [58, 87]

We can also give the mapping as

[29, 46, 67, 81, 100, 107]
[0, 0, 1, 1, 1, 1]

In this case the generators are 1/29 of an octave, and a comma of size
15.82578774 cents. Two 29-et keyboards separated by this comma should work very nicely.

--- In tuning@y..., "Gene Ward Smith" <genewardsmith@j...>
wrote:
> I did a 13-limit survey on tuning-math recently, and thought I
>would pull out the results of possible interest to Margo-style
>practice.

wouldn't that mean *not* including prime 5?

anyway, most, if not all of these results have been discussed by
graham breed -- and a few of them have been named after
corresponding keyboard mappings of erv wilson (such as
cassandra 1 and cassandra 2) . . .

--- In tuning@y..., "wallyesterpaulrus" <wallyesterpaulrus@y...> wrote:
> --- In tuning@y..., "Gene Ward Smith" <genewardsmith@j...>
> wrote:
> > I did a 13-limit survey on tuning-math recently, and thought I
> >would pull out the results of possible interest to Margo-style
> >practice.
>
> wouldn't that mean *not* including prime 5?

Could be, but I didn't think Margo *always* ignores 5.

Gene Ward Smith wrote:

> [[1, 0, -31, -21, -14, -9], [0, 1, 21, 15, 11, 8]] [1200., 1904.391710]
> > rms 6.209025908 comp 11.46080276 graham 21
> > bad 240.9051197 grabad 597.5199555
> > mos [12, 17, 29, 46, 63]
> > This has a fifth of 704.39, which is right in Margo's range.

What's this "grabad" supposed to be? My default figure of demerit for this is 3.89. The worst error is 10.6 cents. Doesn't seem to have a name. It's consistent with the prime mappings of 29- and 46-equal.

> [[1, 0, 15, 25, -33, -28], [0, 1, -8, -14, 23, 20]] [1200., 1902.127698]
> > rms 2.806870405 comp 19.72435043 graham 37
> > bad 245.8818240 grabad 631.7204666
> > mos [12, 17, 29, 41, 53, 94]
> > This is one version of 13-limit schismic, with a fifth of 702.13. Both of these look pretty good.

This is "Cassandra 1" and is probably the mapping he intended to call Cassandra. "Cassandra 2" is simpler, but with a worst error of 10.0 cents. Which still makes it comparable to the first on your list here.

> [[2, 0, 11, 31, 45, 55], [0, 1, -2, -8, -12, -15]] [600.0000000, 1903.786589]
> > rms 2.880707578 comp 19.68247952 graham 34
> > bad 251.5468524 grabad 571.1070885
> > mos [10, 12, 22, 34, 46, 58]
> > This has two chains of fifths of size 703.79 cents, separated by a half-octave, so it is readily adaptable to a double keyboard setup.

That's the 13-limit diaschismic.

> [[29, 46, 0, 14, 33, 40], [0, 0, 1, 1, 1, 1]] [41.37931034, 2788.239580]
> > rms 2.277983567 comp 22.46330341 graham 29
> > bad 242.5275176 grabad 355.7521911
> > mos [58, 87]

That's Mystery.

> We can also give the mapping as
> > [29, 46, 67, 81, 100, 107]
> [0, 0, 1, 1, 1, 1]
> > In this case the generators are 1/29 of an octave, and a comma of size
> 15.82578774 cents. Two 29-et keyboards separated by this comma should work very nicely.

Yes! I've got it on my ZTar with 29-equal running down the neck, and the strings tuned to neutral thirds. It works, but it's cumbersome so I haven't got much to show for it yet.

The best reason I've found for using 13-limit harmony is to get 8:11:13 chords. They share some rare properties that may be important with 4:5:6. And they sound okay. The mapping can be optimised for them by having whole tones (5 steps of 29-equal) down the neck instead of single steps. That leaves neutral third scales as compact structures. It seems to work melodically, but needs some more tinkering to be useful, because the notes you want are only on one part of the neck. It still gives the 3-approximation, but 5 and 7 are more remote.

Graham

--- In tuning@y..., Graham Breed <graham@m...> wrote:
> Gene Ward Smith wrote:
>
> > [[1, 0, -31, -21, -14, -9], [0, 1, 21, 15, 11, 8]] [1200., 1904.391710]
> >
> > rms 6.209025908 comp 11.46080276 graham 21
> >
> > bad 240.9051197 grabad 597.5199555
> >
> > mos [12, 17, 29, 46, 63]
> >
> > This has a fifth of 704.39, which is right in Margo's range.
>
> What's this "grabad" supposed to be?

Err, would you believe rms error times 21^(3/2)?

> > [[2, 0, 11, 31, 45, 55], [0, 1, -2, -8, -12, -15]] [600.0000000, 1903.786589]
> >
> > rms 2.880707578 comp 19.68247952 graham 34
> >
> > bad 251.5468524 grabad 571.1070885
> >
> > mos [10, 12, 22, 34, 46, 58]
> >
> > This has two chains of fifths of size 703.79 cents, separated by a half-octave, so it is readily adaptable to a double keyboard setup.
>
> That's the 13-limit diaschismic.

That's what I'd call it.

> > [[29, 46, 0, 14, 33, 40], [0, 0, 1, 1, 1, 1]] [41.37931034, 2788.239580]
> >
> > rms 2.277983567 comp 22.46330341 graham 29
> >
> > bad 242.5275176 grabad 355.7521911
> >
> > mos [58, 87]
>
> That's Mystery.

Mystery? Have you been adding to your web page, or did I miss it? I know the idea has been discussed before, but I don't recall Mystery.

Gene Ward Smith wrote:

> Mystery? Have you been adding to your web page, or did I miss it? I know the idea has been discussed before, but I don't recall Mystery.

I suggested it here on the 19th of July, and nobody said anything.

Graham

--- In tuning@y..., Graham Breed <graham@m...> wrote:
> Gene Ward Smith wrote:
>
> > Mystery? Have you been adding to your web page, or did I miss it? I know the idea has been discussed before, but I don't recall Mystery.
>
> I suggested it here on the 19th of July, and nobody said anything.

I don't recall anyone noticing anything during July; it was awfully slow this summer around here. I mentioned the 11-limit version back on Feb 6, it appears, and no one commented then either I suppose.