King Orwell overthrown in coup, King Miracle takes the throne

I was counting orwell as working simply because of roundoff error from
the TOP tuning. In fact, it is not consistent with the diamond, which
we can easily learn by algebraic methods, or simply noting that 99/98
and 121/120 are commas of 11-limit orwell.

On the other hand, 45 notes of miracle, detempered, seems to be a
pretty solid alternative to Partch's famous 43:

1, 45/44, 33/32, 25/24, 21/20, 15/14, 12/11, 11/10, 10/9, 9/8, 8/7,
7/6, 25/21, 6/5, 11/9, 5/4, 14/11, 9/7, 21/16, 4/3, 15/11, 11/8, 7/5,
10/7, 16/11, 22/15, 3/2, 32/21, 14/9, 11/7, 8/5, 18/11, 5/3, 27/16,
12/7, 7/4, 16/9, 9/5, 20/11, 11/6, 15/8, 21/11, 27/14, 35/18, 49/25

The largest steps are 45/44, and the smallest, 126/125.

! mund45.scl
Tenney reduced 11-limit Miracle[45]
45
!
45/44
33/32
25/24
21/20
15/14
12/11
11/10
10/9
9/8
8/7
7/6
25/21
6/5
11/9
5/4
14/11
9/7
21/16
4/3
15/11
11/8
7/5
10/7
16/11
22/15
3/2
32/21
14/9
11/7
8/5
18/11
5/3
27/16
12/7
7/4
16/9
9/5
20/11
11/6
15/8
21/11
27/14
35/18
49/25
2

Gene Ward Smith wrote:
> > I was counting orwell as working simply because of roundoff error from
> the TOP tuning. In fact, it is not consistent with the diamond, which
> we can easily learn by algebraic methods, or simply noting that 99/98
> and 121/120 are commas of 11-limit orwell.

You mean it isn't unique? That's been the problem all along! 58 is the smallest constant structure (MOS/periodicity block) that uniquely represents the 11-limit diamond. You could try proving this. I think you need to assume that the pitches are monotonically increasing, but not that it's a remotely good approximation.

> On the other hand, 45 notes of miracle, detempered, seems to be a
> pretty solid alternative to Partch's famous 43:

Yes, you're now the third person to notice this! See George Secor's original XH3 paper. Also partch_43a.scl from Manuel's archive.

> 1, 45/44, 33/32, 25/24, 21/20, 15/14, 12/11, 11/10, 10/9, 9/8, 8/7, > 7/6, 25/21, 6/5, 11/9, 5/4, 14/11, 9/7, 21/16, 4/3, 15/11, 11/8, 7/5,
> 10/7, 16/11, 22/15, 3/2, 32/21, 14/9, 11/7, 8/5, 18/11, 5/3, 27/16,
> 12/7, 7/4, 16/9, 9/5, 20/11, 11/6, 15/8, 21/11, 27/14, 35/18, 49/25
> > The largest steps are 45/44, and the smallest, 126/125.

That's a new scale, anyway. It doesn't have the ascending/descending symmetry which explains some of the differences with Partch. Partch's first note is 49/48. I obviously missed this before because on my website I said 45/44. Otherwise, if Partch's note isn't in your scale, the octave complement is.

Graham

--- In tuning-math@yahoogroups.com, Graham Breed <gbreed@g...> wrote:
> Gene Ward Smith wrote:
>
> You mean it isn't unique?

Sorry, no. I was all excited by the idea we could go as low as 35, but
orwell will not take us there.

That's been the problem all along! 58 is the
> smallest constant structure (MOS/periodicity block) that uniquely
> represents the 11-limit diamond. You could try proving this.

It would be easy enough to get bounds on the ranges of values for
v/v(2) by solving a system of inequalities; but I think this wouldn't
make for an easy proof. Probably a brute force look at vals would be
better.

> > On the other hand, 45 notes of miracle, detempered, seems to be a
> > pretty solid alternative to Partch's famous 43:
>
> Yes, you're now the third person to notice this! See George Secor's
> original XH3 paper. Also partch_43a.scl from Manuel's archive.

Hmmm. If I ask Scala for what scales it contains, the largest it finds
is partch_29, so I think this scale is actually new. It should also be
good at taking advantage of miracle-type commas; that is, 225/224,
243/242, 385/384, 441/440, and 540/539.

Here's an alternative, Euclidean reduction. Once again partch_29 is
the largest scale it contains.

! mundeuc45.scl
Euclidean reduced detempered Miracle[45] with Tenney tie-breaker
45
!
45/44
33/32
28/27
21/20
15/14
12/11
11/10
10/9
9/8
8/7
7/6
32/27
6/5
11/9
5/4
14/11
9/7
21/16
4/3
15/11
11/8
7/5
10/7
16/11
22/15
3/2
32/21
14/9
11/7
8/5
18/11
5/3
27/16
12/7
7/4
16/9
9/5
20/11
11/6
28/15
21/11
27/14
64/33
55/28
2