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Message: 11127 Date: Sun, 27 Jun 2004 18:29:09 Subject: Re: NOT tuning From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote: > --- In tuning-math@xxxxxxxxxxx.xxxx "Kalle Aho" <kalleaho@m...> wrote: > > --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> > > wrote: > > > NOT being an acronym for No Octave Tempering. NOT tuning is TOP > > tuning > > > with the added constraint that octaves must be pure. For example, > > the > > > 7-limit NOT tuning for meantone is very close to 1/5-comma; this > > makes > > > the error for 3, weighted by log(3), equal to with opposite sign > > from > > > the error for 7 weighted by log(7). > > > > > > More anon, I think. > > > > Why? > > Why not? This is the tuning math list, after all. Just as TOP tuning > bounds the ratio of absolute error over Tenney height, NOT does the > same for odd Tenney height, defined as the Tenney height of the odd > part of a positive rational number. In other words, take out the even > factor, so that the numerator and denominator are two odd integers > with GCD 1, and take the log of the product. > > For example, the 5-limit NOT meantone tuning has fifths of size > 697.0197, about 2/11 comma flat, and close to many people's favored > 55-equal tuning. The error in the fifth, divided by log2(3), is 2.4829 > cents, the error in the major third is sharp rather than flat, but > divided by log2(5) is again 2.4829. The error in the minor third is > in the flat direction by about 9.7 cents; dividing this by log2(15) > again gives 2.4829. The error in any 5-limit interval, divided by the > log base two of the product of the numerator and denominator of the > odd part, is bounded by 2.4829. It seems to me this is interesting > enough to justify posting about it. I'm not here. The odd Tenney height should truly be 5 for both the major third and the minor third. They're both ratios of 5 -- members of the 5-odd- limit. I'm not here.
Message: 11129 Date: Sun, 27 Jun 2004 11:57:36 Subject: Re: NOT tuning From: Carl Lumma >NOT being an acronym for No Octave Tempering. NOT tuning is TOP tuning >with the added constraint that octaves must be pure. For example, the >7-limit NOT tuning for meantone is very close to 1/5-comma; this makes >the error for 3, weighted by log(3), equal to with opposite sign from >the error for 7 weighted by log(7). > >More anon, I think. Cool; I've been waiting for this. -C.
Message: 11141 Date: Mon, 28 Jun 2004 01:59:45 Subject: Re: Absolute TOP error From: Carl Lumma >We've been doing the weighted TOP error as the error in cents divided >by the log base two of the product of the numerator and the >denominator. This is good for most purposes, but if we stick to the >same log for the ratio (both cents, or both log base two, etc.) then >we get something with a meaning independent of unit/log base choice. >It can be described as the logarithm, base N = the product of >numerator and denominator, of the error; Log_N(E). The reciprocal is >Log_E(N); it is how many steps of size E (the error) are required to >get to N (product of numerator and denominator.) In TOP tuning, there >is a minimum value for this which defines the relative error. The same >remarks apply for NOT tuning and the product of the numerator and >denominator of the odd part of the interval. > >If T is the TOP error by the definition we've been using, then 1200/T >is the minimum number of error-sized steps needed to get to the >product of numerator and denominator. For (5, 7, 11-limit) meantone >that would be 706.497 steps, for instance; miracle would be 1901.701 >and ennealimmal 32987.408. I think I understand some of this. How is it absolute? It still sounds weighted to me. -Carl
Message: 11142 Date: Mon, 28 Jun 2004 02:01:18 Subject: Re: NOT tuning From: Carl Lumma >> >NOT being an acronym for No Octave Tempering. NOT tuning is TOP tuning >> >with the added constraint that octaves must be pure. For example, the >> >7-limit NOT tuning for meantone is very close to 1/5-comma; this makes >> >the error for 3, weighted by log(3), equal to with opposite sign from >> >the error for 7 weighted by log(7). >> > >> >More anon, I think. >> >> Cool; I've been waiting for this. > >Cool; let's look at some results. Drat! I've lost Paul's comment on this. Did you see it? IIRC he accused you of measuring reciprocals differently. >Meantone > >5-limit: 698.0187 (43, 55, 98, 153, 251, 404) > >7-limit: 697.6458 (31, 43, 934, 977; 0.0286 cents flatter than 43) Hmm... I dunno, this seems a bit far from the old-style rms optimum. -Carl
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