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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
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
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.
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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