NOT tuning

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.

--- In tuning-math@yahoogroups.com, "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?

--- In tuning-math@yahoogroups.com, "Kalle Aho" <kalleaho@m...> wrote:
> --- In tuning-math@yahoogroups.com, "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.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, "Kalle Aho" <kalleaho@m...>
wrote:
> > --- In tuning-math@yahoogroups.com, "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.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:

> > 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 didn't mean to be rude. You can post anything you like. I just
wanted to understand the point of NOT.

Kalle

>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.

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> 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.
>
> Cool; I've been waiting for this.

Cool; let's look at some results.

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)

11-limit huygens: 697.6458 (same as 7-limit)

11-limit meantone (meanpop): 696.9010 (close to 31; the next
convergent is 169/291)

dominant sevenths: 702.1396 (12, 41, 94)

flattone: 693.9317 (19, 64, 83, 313)

Miracle

5-limit: 116.6593 (very close to 72; 31, 72, 2191, 2263)

7 and 11 limits are the same.

Diaschismic

5-limit: 104.7764 (126 is close)

7-limit: 104.5806 (46, then 218)

7-limit pajara: 109.1845 (22 is very close; 10, 22, 2230)

7-limit shrutar: 58.3882 (46, then 252)

11-limit pajara same as 7-limit pajara

11-limit shrutar same as 7-limit shrutar

Magic

5-limit: 381.1024 (continued fraction gives 19, 22, 63, 85, 148, 233,
381 with 2^(121/381) being very close. 41 isn't in there!)

7-limit: same as 5-limit

11-limit gives a family of temperaments

<<5 1 12 -8 -10 5 -30 25 -22 -64|| same as 5 and 7 limit magic

<<5 1 12 33 -10 5 35 25 73 51|| 380.6009 (19, 22, 41, 227, 268)

<<5 1 12 14 -10 5 5 25 29 -2|| 381.4284 (19, 22, 129, 280; 280 is very
close)

Orwell

5-limit: 271.5994 (22, 31, 53, 190, 243)

7-limit: 271.4707 (22, 31, 53, 84, 305)

11-limit: 271.8716 (22, 53, 128, 2741)

>> >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

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:

> Drat! I've lost Paul's comment on this. Did you see it? IIRC he
> accused you of measuring reciprocals differently.

He didn't like my definition of odd Tenney height. Too bad, it's the
only way this works, so we are stuck with it.

> >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.

It is in many cases. NOT meantone sneers at 6/5 and 5/3; and this is
how it works in general, it just plain likes odd integer numerators
better than ratios of two odd integers, which probably doesn't make a
lot of sense and has to do with why Paul is complaining.

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:

> >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

Carl, when Graham investigated this same question here a few months
ago, he concluded that pure-octaves TOP would be a uniform stretching
or compression of TOP, except where TOP already had pure octaves, in
which case it would actually change!

>> >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
>
>Carl, when Graham investigated this same question here a few months
>ago, he concluded that pure-octaves TOP would be a uniform stretching
>or compression of TOP,

That seems obvious for ETs....

>except where TOP already had pure octaves, in
>which case it would actually change!

That's impossible given the criterion of NOT.

Maybe I don't comprehend you.

-Carl

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:

> >except where TOP already had pure octaves, in
> >which case it would actually change!
>
> That's impossible given the criterion of NOT.
>
> Maybe I don't comprehend you.

Some examples of this method of tuning would be nice, and a definition
even better.

>> >except where TOP already had pure octaves, in
>> >which case it would actually change!
>>
>> That's impossible given the criterion of NOT.
>>
>> Maybe I don't comprehend you.
>
>Some examples of this method of tuning would be nice, and
>a definition even better.

Which method? Graham's? I think he gave examples.

Graham, what's a good word to search for? I know I have that
post. I think I replied to it.

-Carl

Carl Lumma wrote:
>>>>except where TOP already had pure octaves, in >>>>which case it would actually change!
>>>
>>>That's impossible given the criterion of NOT.
>>>
>>>Maybe I don't comprehend you.
>>
>>Some examples of this method of tuning would be nice, and
>>a definition even better.
> > > Which method? Graham's? I think he gave examples.
> > Graham, what's a good word to search for? I know I have that
> post. I think I replied to it.

I searched my local folder for "Kees metric" and found a post on 2nd Feb that you can work back from. I didn't originally know I was using a Kees metric, so you won't find that post.

I think it must be different to NOT, partly because Gene mentioned some problems that I'm sure I'd already solved.

Graham

Paul Erlich wrote:
> --- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> > >>>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
> > > Carl, when Graham investigated this same question here a few months > ago, he concluded that pure-octaves TOP would be a uniform stretching > or compression of TOP, except where TOP already had pure octaves, in > which case it would actually change!

You can always define the method to give the same answer for pure-odd ratios. But yes, for the 5-limit it should give quarter comma meantone, because the 81:80 is shared between the four factors of 3 in the numerator. It's clearly doing something different.

I haven't defined the 7-limit result because I don't generally know how to do 7-limit linear TOP. What I do have is:

Minimax 696.58
RMS (7) 696.65
RMS (9) 696.44
PORMSWE 697.22

The last one you may recall is my alternative to TOP. Here, the octave is stretched by 1.24 cents. I can't generalize it to the octave-equivalent case (which is why I switched to odd limits in the first place). But you can always unstretch the octave, which here gives a fifth of 696.49 cents.

Either there's a systematic error in all my calculations, or Gene's result is perverse.

Graham

--- In tuning-math@yahoogroups.com, Graham Breed <graham@m...> wrote:

> You can always define the method to give the same answer for pure-odd
> ratios. But yes, for the 5-limit it should give quarter comma
meantone,
> because the 81:80 is shared between the four factors of 3 in the
> numerator. It's clearly doing something different.
>
> I haven't defined the 7-limit result because I don't generally know how
> to do 7-limit linear TOP. What I do have is:
>
> Minimax 696.58
> RMS (7) 696.65
> RMS (9) 696.44
> PORMSWE 697.22
>
> The last one you may recall is my alternative to TOP. Here, the octave
> is stretched by 1.24 cents. I can't generalize it to the
> octave-equivalent case (which is why I switched to odd limits in the
> first place). But you can always unstretch the octave, which here
gives
> a fifth of 696.49 cents.
>
> Either there's a systematic error in all my calculations, or Gene's
> result is perverse.

I don't see why you think that; my results are a consequence of my
definition, and your results I presume of yours. NOT tunings are

NOT5: 698.02
error in 3, over log 3, is equal to error in 5, over log 5, is equal
to error in 5/3, over 15

NOT7: 697.65
error in 3, over log 3, is error in 7, over log 7, is error in 7/3,
over log 21

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> >> >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
> >
> >Carl, when Graham investigated this same question here a few
months
> >ago, he concluded that pure-octaves TOP would be a uniform
stretching
> >or compression of TOP,
>
> That seems obvious for ETs....

But it's not what Gene's definition gives you.

>
> >except where TOP already had pure octaves, in
> >which case it would actually change!
>
> That's impossible given the criterion of NOT.
>
> Maybe I don't comprehend you.

I didn't say NOT, I said "Graham" and "pure-octaves TOP".

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
>
> > >except where TOP already had pure octaves, in
> > >which case it would actually change!
> >
> > That's impossible given the criterion of NOT.
> >
> > Maybe I don't comprehend you.
>
> Some examples of this method of tuning would be nice, and a
definition
> even better.

I looked again and Graham said Feb 2nd.

>> >except where TOP already had pure octaves, in
>> >which case it would actually change!
>>
>> That's impossible given the criterion of NOT.
>>
>> Maybe I don't comprehend you.
>
>I didn't say NOT, I said "Graham" and "pure-octaves TOP".

Ok, it would seem to violate the definition of
"pure-octaves TOP".

-Carl

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> >> >except where TOP already had pure octaves, in
> >> >which case it would actually change!
> >>
> >> That's impossible given the criterion of NOT.
> >>
> >> Maybe I don't comprehend you.
> >
> >I didn't say NOT, I said "Graham" and "pure-octaves TOP".
>
> Ok, it would seem to violate the definition of
> "pure-octaves TOP".

It doesn't. It still has pure octaves.

>> >> >except where TOP already had pure octaves, in
>> >> >which case it would actually change!
>> >>
>> >> That's impossible given the criterion of NOT.
>> >>
>> >> Maybe I don't comprehend you.
>> >
>> >I didn't say NOT, I said "Graham" and "pure-octaves TOP".
>>
>> Ok, it would seem to violate the definition of
>> "pure-octaves TOP".
>
>It doesn't. It still has pure octaves.

Oh, I thought you were saying the octaves changed. So in
fact I have no clue what you were trying to say.

-Carl

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> >> >> >except where TOP already had pure octaves, in
> >> >> >which case it would actually change!
> >> >>
> >> >> That's impossible given the criterion of NOT.
> >> >>
> >> >> Maybe I don't comprehend you.
> >> >
> >> >I didn't say NOT, I said "Graham" and "pure-octaves TOP".
> >>
> >> Ok, it would seem to violate the definition of
> >> "pure-octaves TOP".
> >
> >It doesn't. It still has pure octaves.
>
> Oh, I thought you were saying the octaves changed. So in
> fact I have no clue what you were trying to say.

Graham's "pure-octaves TOP" is just a uniform stretching or
compression of normal TOP -- except in those cases where normal TOP's
already got pure octaves, in which case the change is not a mere
uniform stretching or compression.

>Graham's "pure-octaves TOP" is just a uniform stretching or
>compression of normal TOP -- except in those cases where normal TOP's
>already got pure octaves, in which case the change is not a mere
>uniform stretching or compression.

Aha! Got'cha.

That *is* interesting, and a bit unsettling I suppose.

-Carl

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> >Graham's "pure-octaves TOP" is just a uniform stretching or
> >compression of normal TOP -- except in those cases where normal
TOP's
> >already got pure octaves, in which case the change is not a mere
> >uniform stretching or compression.
>
> Aha! Got'cha.
>
> That *is* interesting, and a bit unsettling I suppose.

Yet true nonetheless.