Yahoo Tuning Groups Ultimate Backup tuning-math Re: [tuning] Studloco vs Rodan the Flying Monster

No need to bother the main list with the details of this.

On 16/02/07, Gene Ward Smith <genewardsmith@coolgoose.com> wrote:
>
> "Studloco" is the somewhat whimsical name which has been attached to
> Miracle[41], 41 notes to the octave in a chain of 40 secors. "Rodan" is
> the 41&87 temperament, with truncated 13-limit wedgie <<3 17 -1 -13 -
> 22 ...||. From this you can see, if you read these things, that the
> generator is an 8/7 and the period an octave.
>
> It's been pointed out that Studloco contains the entire 11-limit
> tonality diamond with the exception of 11/10 and 20/11, which falls
> between -19 and +19 secors. Studloco we can take as between -20 and +20
> secors, and it has three copies of this truncated tonality diamond in
> it. Studloco has about the same number of notes as the Partch Genesis
> scale, and people have had fun compariing the two.
>
> I want to point out here that Rodan[41] is remarkably similar to
> Studloco, and in some ways better. Rodan[41] also has everything in the
> 11-limit diamond with the exception of 11/10 and 20/11 between -19 and
> +19 generators, and we similarly have three copies of this truncated
> diamond between -20 and +20 generators. The tuning accuracies are
> comparable--a good rodan tuning being 128-et. But rodan is a much
> better 13-limit temperament than miracle, and has many 13-limit
> intervals, in better tune than 72-et allows, inside of the compass of
> Rodan[41].

Which is better depends on what you're looking for. In my TOP-RMS
lists, two different kinds of miracle come out ahead of rodan. Rodan
is the simplest but only narrowly. It's more pronounced in the 13-odd
limit. Yes, it's better in tune than 72-et, but miracle isn't
constrained to 72-et.

> The complexity of 3/2 in rodan is 3 rather than 6, of 7/4 is 1 rather
> than 2, and of 7/6 is 4 rather than 8. 11/8 has a complexity of 13 as
> opposed to miracle's 15. Rodan is high-complexity for 5/4: 17 as
> opposed to 7. But the tuning world is overrun with people who seem to
> get along fine without much 5 in their music, and certainly you still
> get plenty of it in Rodan[41]. The complexity of 13/8 is 22 in rodan,
> which is pretty high, but better than the secor complexity of 34 (or
> 38, in the other direction) it has in 72-et. Rodan really does much
> better at working the 13-limit into the picture.

It's part of the near-5-equal family that works well in the no-fives
7-limit. Yes, it's still okay in the no-fives 11-limit, but 9 and 11
have opposite signs.

> Here's Rodan[41] in 128-et:
>
> ! rodan41.scl
> Rodan[41] in 128-et tuning
> 41
> !
> 28.125
> 56.250
> 84.375
> 112.500
> 150.000
> 178.125
> 206.250
> 234.375
> 262.500
> 290.625
> 318.750
> 346.875
> 384.375
> 412.500
> 440.625
> 468.750
> 496.875
> 525.000
> 553.125
> 581.250
> 618.750
> 646.875
> 675.000
> 703.125
> 731.250
> 759.375
> 787.500
> 815.625
> 853.125
> 881.250
> 909.375
> 937.500
> 965.625
> 993.750
> 1021.875
> 1050.000
> 1087.500
> 1115.625
> 1143.750
> 1171.875
> 1200.000

Seeing as this is tuning-math now, I can pump out more data. Here's Rodan:

46&41
1199.989 cents period
234.480 cents generator
234.482 cents generator for pure octaves

Period and Generator Mappings
2 3 5 7 11 13
< 1, 1, -1, 3, 6, 8 ]
< 0, 3, 17, -1, -13, -22 ]

Constituent Equal Temperaments
2 3 5 7 11 13
< 46, 73, 107, 129, 159, 170 ]
< 41, 65, 95, 115, 142, 152 ]

Complexity 13.267
RMS Weighted Error 0.625 cents/octave

Here are the two miracles that beat it:

72&31
1200.861 cents period
116.657 cents generator
116.574 cents generator for pure octaves

Period and Generator Mappings
2 3 5 7 11 13
< 1, 1, 3, 3, 2, 7 ]
< 0, 6, -7, -2, 15, -34 ]

Constituent Equal Temperaments
2 3 5 7 11 13
< 72, 114, 167, 202, 249, 266 ]
< 31, 49, 72, 87, 107, 115 ]

Complexity 13.524
RMS Weighted Error 0.478 cents/octave

41&31
1200.758 cents period
116.813 cents generator
116.739 cents generator for pure octaves

Period and Generator Mappings
2 3 5 7 11 13
< 1, 1, 3, 3, 2, 0 ]
< 0, 6, -7, -2, 15, 38 ]

Constituent Equal Temperaments
2 3 5 7 11 13
< 41, 65, 95, 115, 142, 152 ]
< 31, 49, 72, 87, 107, 114 ]

Complexity 13.284
RMS Weighted Error 0.555 cents/octave

Here's 13 odd-limit rodan:

17/87, 234.4 cent generator

basis:
(1.0, 0.195338760118)

mapping by period and generator:
[(1, 0), (1, 3), (-1, 17), (3, -1), (6, -13), (8, -22)]

mapping by steps:
[(46, 41), (73, 65), (107, 95), (129, 115), (159, 142), (170, 152)]

highest interval width: 39
complexity measure: 39 (41 for smallest MOS)
highest error: 0.004801 (5.761 cents)
unique

There are 3 better and similar temperament classes, none of them
miraculous, and the search may not be complete. But here are the
13-odd limit printouts for the two similar miracles:

10/103, 116.6 cent generator

basis:
(1.0, 0.09716272355)

mapping by period and generator:
[(1, 0), (1, 6), (3, -7), (3, -2), (2, 15), (7, -34)]

mapping by steps:
[(72, 31), (114, 49), (167, 72), (202, 87), (249, 107), (266, 115)]

highest interval width: 49
complexity measure: 49 (72 for smallest MOS)
highest error: 0.003972 (4.767 cents)
unique

11/113, 116.8 cent generator

basis:
(1.0, 0.0973002582945)

mapping by period and generator:
[(1, 0), (1, 6), (3, -7), (3, -2), (2, 15), (0, 38)]

mapping by steps:
[(72, 41), (114, 65), (167, 95), (202, 115), (249, 142), (266, 152)]

highest interval width: 45
complexity measure: 45 (72 for smallest MOS)
highest error: 0.003102 (3.723 cents)
unique

Neither of them give the full 13-limit in StudLoco, which counts for something.

Rodan does look better in the no-fives 13-limit. Here's the shortlist
(TOP-RMS) of two it belongs to:

38&17
1200.530 cents period
283.374 cents generator
283.248 cents generator for pure octaves

Period and Generator Mappings
2 3 7 11 13
< 1, 3, 8, 7, 7 ]
< 0, -6, -22, -15, -14 ]

Constituent Equal Temperaments
2 3 7 11 13
< 38, 60, 106, 131, 140 ]
< 17, 27, 48, 59, 63 ]

Complexity 7.837
RMS Weighted Error 0.644 cents/octave

41&5
1199.954 cents period
965.494 cents generator
965.532 cents generator for pure octaves

Period and Generator Mappings
2 3 7 11 13
< 1, 4, 2, -7, -14 ]
< 0, -3, 1, 13, 22 ]

Constituent Equal Temperaments
2 3 7 11 13
< 41, 65, 115, 142, 152 ]
< 5, 8, 14, 17, 18 ]

Complexity 7.838
RMS Weighted Error 0.683 cents/octave

(Yes, it happens to come out as 41&5 rather than 41&46 or something
more sensible. These are the best kinds of 5, 41, and 46 for this
error but maybe not for odd-limit errors.)

Relax the cutoffs a bit, and this comes out, looks Secorish:

24&17
1198.763 cents period
847.435 cents generator
848.309 cents generator for pure octaves

Period and Generator Mappings
2 3 7 11 13
< 1, 3, 12, 7, 3 ]
< 0, -2, -13, -5, 1 ]

Constituent Equal Temperaments
2 3 7 11 13
< 24, 38, 67, 83, 89 ]
< 17, 27, 48, 59, 63 ]

Complexity 4.901
RMS Weighted Error 0.785 cents/octave

Graham

🔗Gene Ward Smith <genewardsmith@coolgoose.com>

--- In tuning-math@yahoogroups.com, "Graham Breed" <gbreed@...> wrote:

> Which is better depends on what you're looking for. In my TOP-RMS
> lists, two different kinds of miracle come out ahead of rodan.

Which, I presume, depends heavily on your assumptions about
optimality. But remember--I was comparing Rodan[41] to Miracle[41].

Rodan
> is the simplest but only narrowly. It's more pronounced in the 13-
odd
> limit. Yes, it's better in tune than 72-et, but miracle isn't
> constrained to 72-et.

For 11-limit pure-octave tuning, 72 is about as good as it gets,
isn't it? In deference to Woolhouse, we could use 71deg730, I suppose.

> Complexity 13.267

Measured how? Your new ideas?

> RMS Weighted Error 0.625 cents/octave

Weighted how?

> Relax the cutoffs a bit, and this comes out, looks Secorish:
>
> 24&17
> 1198.763 cents period
> 847.435 cents generator
> 848.309 cents generator for pure octaves
>
>
> Period and Generator Mappings
> 2 3 7 11 13
> < 1, 3, 12, 7, 3 ]
> < 0, -2, -13, -5, 1 ]

Secorish temperament? One way to extend it to the 5-limit is as 13-
limit hemififths, 41&58:

<<2 25 13 5 -1 ... ||

[<1, 1, -5, -1, 2, 4|, <0, 2, 25, 13, 5, -1|]

Comma basis: {144/143,196/195,243/242,364/363}

This is an important 13-limit temperament and it seems to me this is
the obvious way to think of it.

🔗Graham Breed <gbreed@gmail.com>

On 17/02/07, Gene Ward Smith <genewardsmith@coolgoose.com> wrote:
> --- In tuning-math@yahoogroups.com, "Graham Breed" <gbreed@...> wrote:
>
> > Which is better depends on what you're looking for. In my TOP-RMS
> > lists, two different kinds of miracle come out ahead of rodan.
>
> Which, I presume, depends heavily on your assumptions about
> optimality. But remember--I was comparing Rodan[41] to Miracle[41].

No, whatever your assumptions about optimality it depends on what
you're looking for. Both low-error 12-limit miracles have 13-limit
intervals in the 41 note MOS.

> Rodan
> > is the simplest but only narrowly. It's more pronounced in the 13-
> odd
> > limit. Yes, it's better in tune than 72-et, but miracle isn't
> > constrained to 72-et.
>
> For 11-limit pure-octave tuning, 72 is about as good as it gets,
> isn't it? In deference to Woolhouse, we could use 71deg730, I suppose.

Yes, but we're talking about the 13-limit.

> > Complexity 13.267
>
> Measured how? Your new ideas?

No. The range of weighted generator steps, which you say is the same
as the max-Kees weighted complexity.

> > RMS Weighted Error 0.625 cents/octave
>
> Weighted how?

Tenney weighting, like it always is, and like I said in the original message.

> > Relax the cutoffs a bit, and this comes out, looks Secorish:
> >
> > 24&17
> > 1198.763 cents period
> > 847.435 cents generator
> > 848.309 cents generator for pure octaves
> >
> >
> > Period and Generator Mappings
> > 2 3 7 11 13
> > < 1, 3, 12, 7, 3 ]
> > < 0, -2, -13, -5, 1 ]
>
> Secorish temperament? One way to extend it to the 5-limit is as 13-
> limit hemififths, 41&58:

George's paper is about a 17 note no-fives scale IIRC, with links to
12 notes. This splits the 12.

> <<2 25 13 5 -1 ... ||
>
> [<1, 1, -5, -1, 2, 4|, <0, 2, 25, 13, 5, -1|]
>
> Comma basis: {144/143,196/195,243/242,364/363}
>
> This is an important 13-limit temperament and it seems to me this is
> the obvious way to think of it.

Important in the sense of important music being written in it? Here
it is, anyway.

58&41
1198.890 cents period
351.248 cents generator
351.573 cents generator for pure octaves

Period and Generator Mappings
2 3 5 7 11 13
< 1, 1, -5, -1, 2, 4 ]
< 0, 2, 25, 13, 5, -1 ]

Constituent Equal Temperaments
2 3 5 7 11 13
< 58, 92, 135, 163, 201, 215 ]
< 41, 65, 95, 115, 142, 152 ]

Complexity 11.037
RMS Weighted Error 0.745 cents/octave

This one's comparable although it's much more complex in the 15-limit.
Maybe it has a name -- it covers 72 as well.

46&26
600.383 cents period
417.068 cents generator
416.802 cents generator for pure octaves

Period and Generator Mappings
2 3 5 7 11 13
< 2, -1, -3, 7, 9, 6 ]
< 0, 6, 11, -2, -3, 2 ]

Constituent Equal Temperaments
2 3 5 7 11 13
< 46, 73, 107, 129, 159, 170 ]
< 26, 41, 60, 73, 90, 96 ]

Complexity 11.209
RMS Weighted Error 0.564 cents/octave

Graham