King Orwell

If we take the 11-limit diamond, sort it by size, and look at the
interval differences, we get

121/120, 100/99, 99/98, 81/80, 64/63, 56/55, 55/54, 50/49, 49/48,
45/44, 36/35, 33/32, 28/27, 12/11

A tuned temperament, to be consistent with the diamond, needs to send
each of the above to a positive value in terms of cents. I took a list
I have of 100 11-limit rank two temperaments, and found 51 of them
were consistent with the diamond. The minimum size of continguous
generators needed to get the whole diamond is 2G+1, where G is the
Graham complexity. The winner of this process turned out, as you've
already guessed, to be Orwell, which can do it in 35 notes. For a
scale you might take the 4mos of 36 notes or the 3mos of 39. You could
also take the 53 mos, but of course not with the 53 tuning!

Next up are two temperaments which work for a scale of 45 steps,
miracle and tritonic, the 29&31 temperament. For 51 steps, we have
nonkleismic, the 31&58 temperament, and for 53 steps the 22&58
temperament, which I think does not yet have a name.

So far as miracle foes, I don't think it much matters if 45 notes is
not a mos, it is pretty even anyway, especially if the tuning is
optimized for the 11-limit and not the 7-limit. For a pretty decent
quasi-Partch experience I think you could use this scale.

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

> A tuned temperament, to be consistent with the diamond, needs to send
> each of the above to a positive value in terms of cents. I took a list
> I have of 100 11-limit rank two temperaments, and found 51 of them
> were consistent with the diamond.

I should add that for the sake of specificity I used TOP tuning. Of
course orwell in 53-equal will *not* work!

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

> Next up are two temperaments which work for a scale of 45 steps,
> miracle and tritonic, the 29&31 temperament. For 51 steps, we have
> nonkleismic, the 31&58 temperament, and for 53 steps the 22&58
> temperament, which I think does not yet have a name.

The 22&58 should really be 54 steps, since it has two periods to an
octave. The correct formula in general is 2G+P, where P is the number
of periods to an octave, and this complexity measure (diamond
complexity?) is something to consider as an alternative to Graham
complexity. The diamond complexity of ennealimmal is therefore 2*27+9
= 63, which may explain why I went as far as 54 when I composed in it.
The diamond complexity of hemiennealimmal is 162, if someone is brave
enough to want to try that. The diamond complexity of octoid is even
worse, at 180; I guess you may as well just use 270 for both.

Of course this all depends on what limit you are in; the diamond
complexity of miracle for instance in the 7 limit is only 27, not 45,
and Canasta works fine.

>If we take the 11-limit diamond, sort it by size, and look at the
>interval differences, we get
>
>121/120, 100/99, 99/98, 81/80, 64/63, 56/55, 55/54, 50/49, 49/48,
>45/44, 36/35, 33/32, 28/27, 12/11
>
>A tuned temperament, to be consistent with the diamond, needs to send
>each of the above to a positive value in terms of cents. I took a list
>I have of 100 11-limit rank two temperaments,

Are these linear or planar?

>and found 51 of them were consistent with the diamond. The minimum
>size of continguous generators needed to get the whole diamond is
>2G+1, where G is the Graham complexity. The winner of this process
>turned out, as you've already guessed, to be Orwell, which can do it
>in 35 notes.

Er, you only need 29 notes in JI.

-Carl

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:

> >A tuned temperament, to be consistent with the diamond, needs to send
> >each of the above to a positive value in terms of cents. I took a list
> >I have of 100 11-limit rank two temperaments,
>
> Are these linear or planar?

Rank two means two generators, and therefore linear; but Paul tells us
we can't say that anymore, which is fine by me.

> >and found 51 of them were consistent with the diamond. The minimum
> >size of continguous generators needed to get the whole diamond is
> >2G+1, where G is the Graham complexity. The winner of this process
> >turned out, as you've already guessed, to be Orwell, which can do it
> >in 35 notes.
>
> Er, you only need 29 notes in JI.

A scale with steps of sizes ranging from 12/11 to 121/120, and with a
lot of approximate consonances which suggest tempering by miracle. In
miracle it ranges from -22 to 22 secors, and at least some of the
holes could reasonably be filled in to make it a little more even. But
why not go all the way to 45?