Tuning-Math Digests messages 11127 - 11151

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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.


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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.


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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


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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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