Tuning-Math Digests messages 6728 - 6752

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Message: 6728

Date: Sun, 06 Apr 2003 12:16:25

Subject: Re: The canonical homomorphism

From: Carl Lumma

>Since the ratio between TH(a^n)TH(b^n) and TH((ab)^n) is bounded, when
>we take the nth root they tend towards the same limit, and so TH is a
>homomorphic mapping--that is to say TH(1) = 1, TH(a)TH(b) = TH(ab).
>
>This is as slick as goose grease, and whether or not we want to use
>this exact tuning is not to me the real question. What puts a smile on
>my face is that in this way we can in a theoretical sense define the
>temperament as being the mapping. The definition has nothing whatever
>to do with octaves or octave equivalence, and in no way depends on the
>definition or choice of consonances. I propose to call this the
>canonical homomorphism.

Wow, sometimes you do get what you've always wanted.  Can you show
how this works with an example?

-Carl


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Message: 6729

Date: Sun, 06 Apr 2003 00:21:32

Subject: Re: Scales of Wizard

From: Joseph Pehrson

--- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> 

Yahoo groups: /tuning-math/message/6168 *

wrote:
> Here are scales for wizard. Since the main interest I presume is 
using
> it for 72-et, I've reduced everything using 72-et, which means
> including 243/242. A poptimal generator however is found in the
> 310-et, in case some one cares.
> 
> Scales of Wizard
> 
> Wizard commas [225/224, 243/242, 385/384, 4000/3993]
> 72 et commas [225/224, 243/242, 385/384, 4000/3993]
> Wedgie [12, -2, 20, -6, -31, -2, -51, 52, -7, -86]
> 
> 
> Wizard[28] 
> 
> Chromas 45/44, 49/48
> 
> 72 et period [1, 3, 3, 3, 1, 3, 3, 3, 3, 1, 3, 3, 3, 3]
> 
> Scale
> [1, 33/32, 35/33, 15/14, 11/10, 25/22, 7/6, 33/28, 40/33, 5/4, 9/7,
> 33/25, 4/3, 11/8, 99/70, 16/11, 3/2, 50/33, 14/9, 8/5, 33/20, 5/3,
> 12/7, 44/25, 20/11, 15/8, 66/35, 35/18]
> 
> [[35/33, 33/25, 33/20, 33/32, 9/7, 8/5, 1, 5/4, 14/9, 35/18, 40/33,
> 50/33, 66/35, 33/28], [3/2, 15/8, 7/6, 16/11, 20/11, 25/22, 99/70,
> 44/25, 11/10, 11/8, 12/7, 15/14, 4/3, 5/3]]
> 
> Circles of 5/4 generator
> 
> [1, 5/4, 22/15, 12/7] [1, 33/28, 22/15, 12/7]
> [1, 5/4, 22/15, 12/7] [1, 33/28, 22/15, 12/7]
> [1, 5/4, 22/15, 12/7] [1, 33/28, 22/15, 12/7]
> [1, 5/4, 22/15, 12/7] [1, 33/28, 22/15, 12/7]
> [1, 5/4, 22/15, 12/7] [1, 33/28, 22/15, 12/7]
> [1, 5/4, 22/15, 12/7] [1, 33/28, 22/15, 12/7]
> [1, 5/4, 3/2, 12/7] [1, 33/28, 3/2, 12/7]
> [1, 5/4, 3/2, 12/7] [1, 6/5, 3/2, 12/7]
> [1, 5/4, 3/2, 12/7] [1, 6/5, 3/2, 12/7]
> [1, 5/4, 3/2, 12/7] [1, 6/5, 3/2, 12/7]
> [1, 5/4, 3/2, 7/4] [1, 6/5, 3/2, 7/4]
> [1, 5/4, 3/2, 7/4] [1, 6/5, 3/2, 7/4]
> [1, 5/4, 3/2, 7/4] [1, 6/5, 3/2, 7/4]
> [1, 14/11, 3/2, 7/4] [1, 6/5, 3/2, 7/4]
> 
> 
> [1, 44/35, 22/15, 44/25] [1, 8/7, 22/15, 5/3]
> [1, 44/35, 22/15, 44/25] [1, 8/7, 22/15, 5/3]
> [1, 9/7, 22/15, 44/25] [1, 8/7, 22/15, 5/3]
> [1, 9/7, 22/15, 44/25] [1, 8/7, 22/15, 5/3]
> [1, 9/7, 22/15, 44/25] [1, 7/6, 22/15, 5/3]
> [1, 9/7, 22/15, 44/25] [1, 7/6, 22/15, 5/3]
> [1, 9/7, 3/2, 44/25] [1, 7/6, 3/2, 5/3]
> [1, 9/7, 3/2, 44/25] [1, 7/6, 3/2, 56/33]
> [1, 9/7, 3/2, 44/25] [1, 7/6, 3/2, 56/33]
> [1, 9/7, 3/2, 44/25] [1, 7/6, 3/2, 56/33]
> [1, 9/7, 3/2, 44/25] [1, 7/6, 3/2, 56/33]
> [1, 9/7, 3/2, 44/25] [1, 7/6, 3/2, 56/33]
> [1, 9/7, 3/2, 44/25] [1, 7/6, 3/2, 56/33]
> [1, 9/7, 3/2, 9/5] [1, 7/6, 3/2, 56/33]
> 
> 
> [1, 8/7, 22/15, 12/7] [1, 11/8, 22/15, 12/7]
> [1, 8/7, 22/15, 12/7] [1, 11/8, 22/15, 12/7]
> [1, 8/7, 22/15, 12/7] [1, 11/8, 22/15, 12/7]
> [1, 8/7, 22/15, 12/7] [1, 11/8, 22/15, 12/7]
> [1, 7/6, 22/15, 12/7] [1, 11/8, 22/15, 12/7]
> [1, 7/6, 22/15, 12/7] [1, 11/8, 22/15, 12/7]
> [1, 7/6, 3/2, 12/7] [1, 11/8, 3/2, 12/7]
> [1, 7/6, 3/2, 12/7] [1, 11/8, 3/2, 12/7]
> [1, 7/6, 3/2, 12/7] [1, 11/8, 3/2, 12/7]
> [1, 7/6, 3/2, 12/7] [1, 11/8, 3/2, 12/7]
> [1, 7/6, 3/2, 7/4] [1, 11/8, 3/2, 7/4]
> [1, 7/6, 3/2, 7/4] [1, 7/5, 3/2, 7/4]
> [1, 7/6, 3/2, 7/4] [1, 7/5, 3/2, 7/4]
> [1, 7/6, 3/2, 7/4] [1, 7/5, 3/2, 7/4]
> 
> 
> 
> 
> Wizard[34]
> 
> Chroma 21/20
> 
> 72 et period
> [1, 3, 3, 1, 2, 1, 3, 3, 3, 1, 2, 1, 3, 3, 3, 1, 2]
> 
> Scale
> [1, 33/32, 25/24, 35/33, 15/14, 11/10, 25/22, 7/6, 33/28, 6/5, 
40/33,
> 5/4, 9/7, 33/25, 4/3, 15/11, 11/8, 99/70, 16/11, 22/15, 3/2, 50/33,
> 14/9, 8/5, 33/20, 5/3, 56/33, 12/7, 44/25, 20/11, 15/8, 66/35, 
27/14,
> 35/18]
> 
> [[15/11, 56/33, 35/33, 33/25, 33/20, 33/32, 9/7, 8/5, 1, 5/4, 14/9,
> 35/18, 40/33, 50/33, 66/35, 33/28, 22/15], [27/14, 6/5, 3/2, 15/8,
> 7/6, 16/11, 20/11, 25/22, 99/70, 44/25, 11/10, 11/8, 12/7, 15/14, 
4/3,
> 5/3, 25/24]]
> 
> Circles of 5/4 generator
> 
> [1, 5/4, 10/7, 5/3] [1, 8/7, 10/7, 12/7]
> [1, 5/4, 10/7, 5/3] [1, 8/7, 10/7, 12/7]
> [1, 5/4, 10/7, 5/3] [1, 8/7, 10/7, 12/7]
> [1, 5/4, 10/7, 5/3] [1, 8/7, 10/7, 12/7]
> [1, 5/4, 10/7, 5/3] [1, 8/7, 10/7, 12/7]
> [1, 5/4, 10/7, 5/3] [1, 8/7, 10/7, 12/7]
> [1, 5/4, 3/2, 5/3] [1, 8/7, 3/2, 12/7]
> [1, 5/4, 3/2, 5/3] [1, 6/5, 3/2, 12/7]
> [1, 5/4, 3/2, 5/3] [1, 6/5, 3/2, 12/7]
> [1, 5/4, 3/2, 5/3] [1, 6/5, 3/2, 12/7]
> [1, 5/4, 3/2, 7/4] [1, 6/5, 3/2, 12/7]
> [1, 5/4, 3/2, 7/4] [1, 6/5, 3/2, 12/7]
> [1, 5/4, 3/2, 7/4] [1, 6/5, 3/2, 12/7]
> [1, 5/4, 3/2, 7/4] [1, 6/5, 3/2, 9/5]
> [1, 5/4, 3/2, 7/4] [1, 6/5, 3/2, 9/5]
> [1, 5/4, 3/2, 7/4] [1, 6/5, 3/2, 9/5]
> [1, 21/16, 3/2, 7/4] [1, 6/5, 3/2, 9/5]
> 
> 
> [1, 11/9, 10/7, 12/7] [1, 10/9, 10/7, 5/3] [1, 11/8, 10/7, 5/3]
> [1, 11/9, 10/7, 12/7] [1, 10/9, 10/7, 5/3] [1, 11/8, 10/7, 5/3]
> [1, 9/7, 10/7, 12/7] [1, 10/9, 10/7, 5/3] [1, 11/8, 10/7, 5/3]
> [1, 9/7, 10/7, 12/7] [1, 10/9, 10/7, 5/3] [1, 11/8, 10/7, 5/3]
> [1, 9/7, 10/7, 12/7] [1, 7/6, 10/7, 5/3] [1, 11/8, 10/7, 5/3]
> [1, 9/7, 10/7, 12/7] [1, 7/6, 10/7, 5/3] [1, 11/8, 10/7, 5/3]
> [1, 9/7, 3/2, 12/7] [1, 7/6, 3/2, 5/3] [1, 11/8, 3/2, 5/3]
> [1, 9/7, 3/2, 12/7] [1, 7/6, 3/2, 5/3] [1, 11/8, 3/2, 5/3]
> [1, 9/7, 3/2, 12/7] [1, 7/6, 3/2, 5/3] [1, 11/8, 3/2, 5/3]
> [1, 9/7, 3/2, 12/7] [1, 7/6, 3/2, 5/3] [1, 11/8, 3/2, 5/3]
> [1, 9/7, 3/2, 12/7] [1, 7/6, 3/2, 7/4] [1, 11/8, 3/2, 7/4]
> [1, 9/7, 3/2, 12/7] [1, 7/6, 3/2, 7/4] [1, 11/8, 3/2, 7/4]
> [1, 9/7, 3/2, 12/7] [1, 7/6, 3/2, 7/4] [1, 11/8, 3/2, 7/4]
> [1, 9/7, 3/2, 9/5] [1, 7/6, 3/2, 7/4] [1, 11/8, 3/2, 7/4]
> [1, 9/7, 3/2, 9/5] [1, 7/6, 3/2, 7/4] [1, 36/25, 3/2, 7/4]
> [1, 9/7, 3/2, 9/5] [1, 7/6, 3/2, 7/4] [1, 36/25, 3/2, 7/4]
> [1, 9/7, 3/2, 9/5] [1, 7/6, 3/2, 7/4] [1, 36/25, 3/2, 7/4]
> 
> 
> 
> 
> 
> Wizard[38]
> 
> Chroma 22/21
> 
> 72-et period 
> [1, 3, 3, 1, 2, 1, 3, 3, 1, 2, 1, 2, 1, 3, 3, 1, 2, 1, 2]
> Scale
> [1, 33/32, 25/24, 35/33, 15/14, 12/11, 11/10, 25/22, 7/6, 33/28, 
6/5,
> 40/33, 5/4, 9/7, 35/27, 33/25, 4/3, 15/11, 11/8, 99/70, 16/11, 
22/15,
> 3/2, 50/33, 54/35, 14/9, 8/5, 33/20, 5/3, 56/33, 12/7, 44/25, 20/11,
> 11/6, 15/8, 66/35, 27/14, 35/18]
> 
> [[12/11, 15/11, 56/33, 35/33, 33/25, 33/20, 33/32, 9/7, 8/5, 1, 5/4,
> 14/9, 35/18, 40/33, 50/33, 66/35, 33/28, 22/15, 11/6], [54/35, 
27/14,
> 6/5, 3/2, 15/8, 7/6, 16/11, 20/11, 25/22, 99/70, 44/25, 11/10, 11/8,
> 12/7, 15/14, 4/3, 5/3, 25/24, 35/27]]
> 
> Circles of 5/4 generator
> 
> [1, 5/4, 11/7, 11/6] [1, 44/35, 11/7, 12/7] [1, 5/4, 11/7, 11/6] [1,
> 44/35, 11/7, 12/7]
> [1, 5/4, 11/7, 11/6] [1, 44/35, 11/7, 12/7]
> [1, 5/4, 11/7, 11/6] [1, 44/35, 11/7, 12/7]
> [1, 5/4, 11/7, 11/6] [1, 44/35, 11/7, 12/7]
> [1, 5/4, 11/7, 11/6] [1, 44/35, 11/7, 12/7]
> [1, 5/4, 11/7, 11/6] [1, 44/35, 11/7, 12/7]
> [1, 5/4, 3/2, 11/6] [1, 44/35, 3/2, 12/7]
> [1, 5/4, 3/2, 11/6] [1, 6/5, 3/2, 12/7]
> [1, 5/4, 3/2, 11/6] [1, 6/5, 3/2, 12/7]
> [1, 5/4, 3/2, 11/6] [1, 6/5, 3/2, 12/7]
> [1, 5/4, 3/2, 7/4] [1, 6/5, 3/2, 12/7]
> [1, 5/4, 3/2, 7/4] [1, 6/5, 3/2, 12/7]
> [1, 5/4, 3/2, 7/4] [1, 6/5, 3/2, 12/7]
> [1, 5/4, 3/2, 7/4] [1, 6/5, 3/2, 12/7]
> [1, 5/4, 3/2, 7/4] [1, 6/5, 3/2, 12/7]
> [1, 5/4, 3/2, 7/4] [1, 6/5, 3/2, 18/11]
> [1, 5/4, 3/2, 7/4] [1, 6/5, 3/2, 18/11]
> [1, 5/4, 3/2, 7/4] [1, 6/5, 3/2, 18/11]
> [1, 25/21, 3/2, 7/4] [1, 6/5, 3/2, 18/11]
> 
> 
> [1, 27/20, 11/7, 66/35] [1, 11/9, 11/7, 5/3]
> [1, 27/20, 11/7, 66/35] [1, 11/9, 11/7, 5/3]
> [1, 9/7, 11/7, 66/35] [1, 11/9, 11/7, 5/3]
> [1, 9/7, 11/7, 66/35] [1, 11/9, 11/7, 5/3]
> [1, 9/7, 11/7, 66/35] [1, 7/6, 11/7, 5/3]
> [1, 9/7, 11/7, 66/35] [1, 7/6, 11/7, 5/3]
> [1, 9/7, 3/2, 66/35] [1, 7/6, 3/2, 5/3]
> [1, 9/7, 3/2, 66/35] [1, 7/6, 3/2, 5/3]
> [1, 9/7, 3/2, 66/35] [1, 7/6, 3/2, 5/3]
> [1, 9/7, 3/2, 66/35] [1, 7/6, 3/2, 5/3]
> [1, 9/7, 3/2, 66/35] [1, 7/6, 3/2, 5/3]
> [1, 9/7, 3/2, 66/35] [1, 7/6, 3/2, 5/3]
> [1, 9/7, 3/2, 66/35] [1, 7/6, 3/2, 35/22]
> [1, 9/7, 3/2, 9/5] [1, 7/6, 3/2, 35/22]
> [1, 9/7, 3/2, 9/5] [1, 7/6, 3/2, 35/22]
> [1, 9/7, 3/2, 9/5] [1, 7/6, 3/2, 35/22]
> [1, 9/7, 3/2, 9/5] [1, 7/6, 3/2, 35/22]
> [1, 9/7, 3/2, 9/5] [1, 7/6, 3/2, 35/22]
> [1, 9/7, 3/2, 9/5] [1, 7/6, 3/2, 35/22]
> 
> 
> [1, 11/9, 11/7, 11/6] [1, 27/20, 11/7, 12/7]
> [1, 11/9, 11/7, 11/6] [1, 27/20, 11/7, 12/7]
> [1, 11/9, 11/7, 11/6] [1, 9/7, 11/7, 12/7]
> [1, 11/9, 11/7, 11/6] [1, 9/7, 11/7, 12/7]
> [1, 7/6, 11/7, 11/6] [1, 9/7, 11/7, 12/7]
> [1, 7/6, 11/7, 11/6] [1, 9/7, 11/7, 12/7]
> [1, 7/6, 3/2, 11/6] [1, 9/7, 3/2, 12/7]
> [1, 7/6, 3/2, 11/6] [1, 9/7, 3/2, 12/7]
> [1, 7/6, 3/2, 11/6] [1, 9/7, 3/2, 12/7]
> [1, 7/6, 3/2, 11/6] [1, 9/7, 3/2, 12/7]
> [1, 7/6, 3/2, 7/4] [1, 9/7, 3/2, 12/7]
> [1, 7/6, 3/2, 7/4] [1, 9/7, 3/2, 12/7]
> [1, 7/6, 3/2, 7/4] [1, 9/7, 3/2, 12/7]
> [1, 7/6, 3/2, 7/4] [1, 9/7, 3/2, 12/7]
> [1, 7/6, 3/2, 7/4] [1, 9/7, 3/2, 12/7]
> [1, 7/6, 3/2, 7/4] [1, 9/7, 3/2, 18/11]
> [1, 7/6, 3/2, 7/4] [1, 9/7, 3/2, 18/11]
> [1, 7/6, 3/2, 7/4] [1, 9/7, 3/2, 18/11]
> [1, 7/6, 3/2, 7/4] [1, 9/7, 3/2, 18/11]

***Thanks, Gene... This looks promising...

J. Pehrson


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Message: 6741

Date: Sun, 06 Apr 2003 04:12:10

Subject: Re: Limit generator for 5-limit meantone

From: Carl Lumma

>This turns out to be 1/4-comma meantone; the sequence of fifths falls
>into a regular pattern of three pure fifts followed by a 40/27.

It's not clear to me why big n should be special.  Just because it
causes convergence?  Why should I optimize my generator for whatever
a chain of 500 generators approximates, over what a chain of 5 of
them approximates?

>I should add for this and the miracle example that the limit octave is
>2. This is not always the case; Paul may or may not like to know that
>for Pajara, for instance, it is distinctly flat. Carl I hope is taking
>note of this, as it seems to be along the lines he has been agitating
>for.

Indeed.  -C.


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Message: 6742

Date: Sun, 06 Apr 2003 04:17:33

Subject: Re: The limit generator

From: Carl Lumma

>I'm trying to figure out what it does, but the mode has nothing to do
>with it. It averages the generator along a line of generators, and I
>see no reason to think that it is independent of generator-pair
>choice, so I suppose its canonicity is open to question.

If a "generator-pair" is what I think it is, doesn't this method claim
to give the optimum size(s) *given* a generator-pair?  Would we expect
two different gen-pair choices for a temperament to result in the same
tuning if carried out forever at their respective optimum sizes?

-Carl


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