hunting for a well temperament

Is there a 12-note scale with (very nearly) the TOP meantone
octave, 2:3:4-chord brats of 2 (or 3:2 if necessary), and fifths
between the 69- and 22-equal fifths?

Alaska-fried has a slightly flatter octave than the above, and
some brats of 1 (which I don't think are desirable).

Heya Gene- I notice this page is missing...

http://66.98.148.43/~xenharmo/meanwil.html

-Carl

I wrote...
>Is there a 12-note scale with (very nearly) the TOP meantone
>octave, 2:3:4-chord brats of 2 (or 3:2 if necessary), and fifths
>between the 69- and 22-equal fifths?
>
>Alaska-fried has a slightly flatter octave than the above, and
>some brats of 1 (which I don't think are desirable).
>
>Heya Gene- I notice this page is missing...
>
>http://66.98.148.43/~xenharmo/meanwil.html
>
>-Carl

Using Gene's...

>If we define the octave-fifth beat ratio as
>
>(2f - 3)/(o - 2)

...and the below, I find two 5ths give a 2:1 brat with an
octave of 1197 cents (577/289)...

697.956768 (865/578)
and
700.956308 (1733/1156)

One approach would be to use as many of the smaller of these
as possible (for good major 3rds) and then close the circle
with big ones. It happens this can be done with only 2 big ones.
Or, one can get even closer to an 1197-cent octave with 1 just
5th.

Of course the 1197 number isn't sacrosanct. Here are the
octaves of TOP-tuned 12-equal tunings...

5-limit - 1197.67 cents
7-limit - 1195.08 cents

...going beyond consistency (which seems to change TOP's
agreement with Gene's Zeta and Gram-point tunings)...

11-limit - 1193.61
13-limit - 1198.44

...and that's pretty much where it stays forever.

When I was doing the Alaska tunings (lots of playing on my
keyboard and tunings on my piano), I found 1197 to be a good
number. And for 'authentic' pre-20th-century music, it seems
the 5-limit result is most pertinent.

-Carl

At 02:11 PM 8/25/2005, I wrote:
>>This has probably been discussed here before... I'm looking
>>for a method to find the rational with the lowest Tenney height
>>within n cents of an irrational target. My recollection is
>>that Brun's algorithm does this, but that Gene said there
>>are better ways...
>
>Here's some scheme that attempts this by interating mediants.
>I think this is the same as traversing the Stern-Brocott tree.
>Is this the same as Brun's algorithm? Does it give correct answers?
>
>-Carl
>
>;; Returns the mediant of two fractions.
>
>(define middle
> (lambda (left right)
> (if (not (and (exact? left) (exact? right)))
> 'inputerror
> (/ (+ (numerator left) (numerator right))
> (+ (denominator left) (denominator right))))))
>
>;; Returns the simplest rational within a given factor (range) of
>;; target, where complexity is defined as numerator*denominator
>;; and 1 < range < target.
>
>(define gear
> (lambda (target range)
> (letrec
> ((loop (lambda (left right target range)
> (let ((mediant (middle left right)))
> (if (<= (/ (max mediant target)
> (min mediant target)) range)
> mediant
> (loop
> (if (< mediant target) mediant left)
> (if (> mediant target) mediant right)
> target
> range))))))
> (if (or (<= range 1) (>= range target))
> 'inpur
> (loop
> (inexact->exact (floor (/ target range)))
> (inexact->exact (ceiling (* target range)))
> target
> range)))))

>Using Gene's...
>
>>If we define the octave-fifth beat ratio as
>>
>>(2f - 3)/(o - 2)
>
>...and the below, I find two 5ths give a 2:1 brat with an
>octave of 1197 cents (577/289)...
>
>697.956768 (865/578)
>and
>700.956308 (1733/1156)
>
>One approach would be to use as many of the smaller of these
>as possible (for good major 3rds) and then close the circle
>with big ones. It happens this can be done with only 2 big ones.
>Or, one can get even closer to an 1197-cent octave with 1 just
>5th.

Sorry: using one 700.956308 5th gets closer than using two.

The actual brat target octave used was 1197.00219 cents. Using
one of these large tempered 5ths or one just 5th gets you .07
cents flat or sharp of it, respectively. The just 5th gets you
four thousandths of a cent closer, which is possibly within my
margin of error.

-Carl

At 03:06 PM 8/25/2005, you wrote:
>>Using Gene's...
>>
>>>If we define the octave-fifth beat ratio as
>>>
>>>(2f - 3)/(o - 2)
>>
>>...and the below, I find two 5ths give a 2:1 brat with an
>>octave of 1197 cents (577/289)...
>>
>>697.956768 (865/578)
>>and
>>700.956308 (1733/1156)
>>
>>One approach would be to use as many of the smaller of these
>>as possible (for good major 3rds) and then close the circle
>>with big ones. It happens this can be done with only 2 big ones.
>>Or, one can get even closer to an 1197-cent octave with 1 just
>>5th.
>
>Sorry: using one 700.956308 5th gets closer than using two.

Of course this is a very 'equal' well temperament. So I looked
for fifths with a 3:2 and 2:3 brat against the 577/289 octave.
Unfortunately the results are even more mild...

1731/1156 - 698.957192
2599/1734 - 700.623282

Gene once said one can get as close to just as one likes with
any desired brat. That makes sense. So holding one interval (in
this case, the octave) still and solving for the other (5ths)
isn't a good approach.

Anybody have a better one?

-Carl

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> I wrote...
> >Is there a 12-note scale with (very nearly) the TOP meantone
> >octave, 2:3:4-chord brats of 2 (or 3:2 if necessary), and fifths
> >between the 69- and 22-equal fifths?
> >
> >Alaska-fried has a slightly flatter octave than the above, and
> >some brats of 1 (which I don't think are desirable).
> >
> >Heya Gene- I notice this page is missing...
> >
> >http://66.98.148.43/~xenharmo/meanwil.html
> >
> >-Carl
>
> Using Gene's...
>
> >If we define the octave-fifth beat ratio as
> >
> >(2f - 3)/(o - 2)
>
> ...and the below, I find two 5ths give a 2:1 brat with an
> octave of 1197 cents (577/289)...

I thought you wanted to have octaves near the TOP meantone octave
(see above)?

>
> 697.956768 (865/578)
> and
> 700.956308 (1733/1156)
>
> One approach would be to use as many of the smaller of these
> as possible (for good major 3rds) and then close the circle
> with big ones. It happens this can be done with only 2 big ones.
> Or, one can get even closer to an 1197-cent octave with 1 just
> 5th.
>
> Of course the 1197 number isn't sacrosanct. Here are the
> octaves of TOP-tuned 12-equal tunings...
>
> 5-limit - 1197.67 cents
> 7-limit - 1195.08 cents
>
> ...going beyond consistency (which seems to change TOP's
> agreement with Gene's Zeta and Gram-point tunings)...
>
> 11-limit - 1193.61
> 13-limit - 1198.44

Consistency is meaningless in this context. For one thing,
consistency is defined with respect to odd limits, while TOP tunings
are defined with respect to prime limits. But certainly it's far from
clear which mapping (or set of vanishing ratios)is implied when you
say 13-limit 12-equal, and supporting more than one seems like a good
idea.

> ...and that's pretty much where it stays forever.

> When I was doing the Alaska tunings (lots of playing on my
> keyboard and tunings on my piano), I found 1197 to be a good
> number. And for 'authentic' pre-20th-century music, it seems
> the 5-limit result is most pertinent.
>
> -Carl
>
> At 02:11 PM 8/25/2005, I wrote:
> >>This has probably been discussed here before... I'm looking
> >>for a method to find the rational with the lowest Tenney height
> >>within n cents of an irrational target. My recollection is
> >>that Brun's algorithm does this, but that Gene said there
> >>are better ways...
> >
> >Here's some scheme that attempts this by interating mediants.
> >I think this is the same as traversing the Stern-Brocott tree.
> >Is this the same as Brun's algorithm? Does it give correct
answers?
> >
> >-Carl
> >
> >;; Returns the mediant of two fractions.
> >
> >(define middle
> > (lambda (left right)
> > (if (not (and (exact? left) (exact? right)))
> > 'inputerror
> > (/ (+ (numerator left) (numerator right))
> > (+ (denominator left) (denominator right))))))
> >
> >;; Returns the simplest rational within a given factor (range) of
> >;; target, where complexity is defined as numerator*denominator
> >;; and 1 < range < target.
> >
> >(define gear
> > (lambda (target range)
> > (letrec
> > ((loop (lambda (left right target range)
> > (let ((mediant (middle left right)))
> > (if (<= (/ (max mediant target)
> > (min mediant target)) range)
> > mediant
> > (loop
> > (if (< mediant target) mediant left)
> > (if (> mediant target) mediant right)
> > target
> > range))))))
> > (if (or (<= range 1) (>= range target))
> > 'inpur
> > (loop
> > (inexact->exact (floor (/ target range)))
> > (inexact->exact (ceiling (* target range)))
> > target
> > range)))))

--- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
wrote:
> --- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> > I wrote...
> > >Is there a 12-note scale with (very nearly) the TOP meantone
> > >octave, 2:3:4-chord brats of 2 (or 3:2 if necessary), and fifths
> > >between the 69- and 22-equal fifths?
> > >
> > >Alaska-fried has a slightly flatter octave than the above, and
> > >some brats of 1 (which I don't think are desirable).
> > >
> > >Heya Gene- I notice this page is missing...
> > >
> > >http://66.98.148.43/~xenharmo/meanwil.html
> > >
> > >-Carl
> >
> > Using Gene's...
> >
> > >If we define the octave-fifth beat ratio as
> > >
> > >(2f - 3)/(o - 2)
> >
> > ...and the below, I find two 5ths give a 2:1 brat with an
> > octave of 1197 cents (577/289)...
>
> I thought you wanted to have octaves near the TOP meantone octave
> (see above)?
>
> >
> > 697.956768 (865/578)
> > and
> > 700.956308 (1733/1156)
> >
> > One approach would be to use as many of the smaller of these
> > as possible (for good major 3rds) and then close the circle
> > with big ones. It happens this can be done with only 2 big ones.
> > Or, one can get even closer to an 1197-cent octave with 1 just
> > 5th.
> >
> > Of course the 1197 number isn't sacrosanct. Here are the
> > octaves of TOP-tuned 12-equal tunings...
> >
> > 5-limit - 1197.67 cents
> > 7-limit - 1195.08 cents
> >
> > ...going beyond consistency (which seems to change TOP's
> > agreement with Gene's Zeta and Gram-point tunings)...
> >
> > 11-limit - 1193.61
> > 13-limit - 1198.44
>
> Consistency is meaningless in this context. For one thing,
> consistency is defined with respect to odd limits, while TOP
tunings
> are defined with respect to prime limits. But certainly it's far
from
> clear which mapping (or set of vanishing ratios)is implied when you
> say 13-limit 12-equal, and supporting more than one seems like a
good
> idea.
>
> > ...and that's pretty much where it stays forever.

Where TOP stays forever? As you go to higher and higher prime limits
(P), there are more and more plausible mappings (or groups of
vanishing ratios), so there are more and more plausible answers for
the question "what is the TOP P-limit 12-equal tuning?".

>> ...and the below, I find two 5ths give a 2:1 brat with an
>> octave of 1197 cents (577/289)...

>> Of course the 1197 number isn't sacrosanct. Here are the
>> octaves of TOP-tuned 12-equal tunings...
>>
>> 5-limit - 1197.67 cents
>> 7-limit - 1195.08 cents

>> for 'authentic' pre-20th-century music, it seems
>> the 5-limit result is most pertinent.

>I thought you wanted to have octaves near the TOP meantone octave
>(see above)?

I did -- the quote above shows that could be anywhere from 1197.67
to 1195.08 cents with the former considered (by me) more authentic
for old music.

>> ...going beyond consistency (which seems to change TOP's
>> agreement with Gene's Zeta and Gram-point tunings)...
>>
>> 11-limit - 1193.61
>> 13-limit - 1198.44
>
>Consistency is meaningless in this context. For one thing,
>consistency is defined with respect to odd limits, while TOP tunings
>are defined with respect to prime limits. But certainly it's far from
>clear which mapping (or set of vanishing ratios)is implied when you
>say 13-limit 12-equal, and supporting more than one seems like a good
>idea.

All true, but my comparison of TOP and zeta equal tunings seemed
to show that they diverged when consistency broke (though the
sample was small). I never understood the zeta tunings, but I
thought any correspondence (or not) with TOP tunings might be of
number-theoretic significance.

-Carl

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> >> ...and the below, I find two 5ths give a 2:1 brat with an
> >> octave of 1197 cents (577/289)...
>
> >> Of course the 1197 number isn't sacrosanct. Here are the
> >> octaves of TOP-tuned 12-equal tunings...
> >>
> >> 5-limit - 1197.67 cents
> >> 7-limit - 1195.08 cents
>
> >> for 'authentic' pre-20th-century music, it seems
> >> the 5-limit result is most pertinent.
>
> >I thought you wanted to have octaves near the TOP meantone octave
> >(see above)?
>
> I did -- the quote above shows that could be anywhere from 1197.67
> to 1195.08 cents with the former considered (by me) more authentic
> for old music.

So you abandoned the desire for octaves near the TOP meantone octave?

> All true, but my comparison of TOP and zeta equal tunings seemed
> to show that they diverged when consistency broke

Again, "consistency broke" is meaningless in this context.

>>> ...and that's pretty much where it stays forever.
>
>Where TOP stays forever? As you go to higher and higher prime
>limits (P), there are more and more plausible mappings (or groups
>of vanishing ratios), so there are more and more plausible answers
>for the question "what is the TOP P-limit 12-equal tuning?".

Huh. Using your algorithm the value stopped changing there up
to the 4000-some limit. Though I see your algorithm was based
on the standard val.

-Carl

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> >>> ...and that's pretty much where it stays forever.
> >
> >Where TOP stays forever? As you go to higher and higher prime
> >limits (P), there are more and more plausible mappings (or groups
> >of vanishing ratios), so there are more and more plausible answers
> >for the question "what is the TOP P-limit 12-equal tuning?".
>
> Huh. Using your algorithm the value stopped changing there up
> to the 4000-some limit.

What algorithm?

> Though I see your algorithm was based
> on the standard val.

I really don't know what you could be talking about.

--- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
wrote:
> --- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> > >> ...and the below, I find two 5ths give a 2:1 brat with an
> > >> octave of 1197 cents (577/289)...
> >
> > >> Of course the 1197 number isn't sacrosanct. Here are the
> > >> octaves of TOP-tuned 12-equal tunings...
> > >>
> > >> 5-limit - 1197.67 cents
> > >> 7-limit - 1195.08 cents
> >
> > >> for 'authentic' pre-20th-century music, it seems
> > >> the 5-limit result is most pertinent.
> >
> > >I thought you wanted to have octaves near the TOP meantone
octave
> > >(see above)?
> >
> > I did -- the quote above shows that could be anywhere from 1197.67
> > to 1195.08 cents with the former considered (by me) more authentic
> > for old music.
>
> So you abandoned the desire for octaves near the TOP meantone
octave?

Maybe I misunderstood you, but the TOP meantone octave is 1201.7
cents, so I really don't know what you could mean.

>
> > All true, but my comparison of TOP and zeta equal tunings seemed
> > to show that they diverged when consistency broke
>
> Again, "consistency broke" is meaningless in this context.

>> >> ...and the below, I find two 5ths give a 2:1 brat with an
>> >> octave of 1197 cents (577/289)...
>>
>> >> Of course the 1197 number isn't sacrosanct. Here are the
>> >> octaves of TOP-tuned 12-equal tunings...
>> >>
>> >> 5-limit - 1197.67 cents
>> >> 7-limit - 1195.08 cents
>>
>> >> for 'authentic' pre-20th-century music, it seems
>> >> the 5-limit result is most pertinent.
>>
>> >I thought you wanted to have octaves near the TOP meantone octave
>> >(see above)?
>>
>> I did -- the quote above shows that could be anywhere from 1197.67
>> to 1195.08 cents with the former considered (by me) more authentic
>> for old music.
>
>So you abandoned the desire for octaves near the TOP meantone octave?

I meant the TOP-12-equal octave according to your algorithm. Of
course there are difference "meantone" mappings... most if not all
of them will give a flat octave. Or what's your point?

>> All true, but my comparison of TOP and zeta equal tunings seemed
>> to show that they diverged when consistency broke
>
>Again, "consistency broke" is meaningless in this context.

It's an observation and therefore not meaningless. It's further
not meaningless because it means the standard val is no longer the
best one. Meanwhile your top-et algorithm does just fine with
odd limits (it gives same values as at the next-lowest prime).

-Carl

>> >>> ...and that's pretty much where it stays forever.
>> >
>> >Where TOP stays forever? As you go to higher and higher prime
>> >limits (P), there are more and more plausible mappings (or groups
>> >of vanishing ratios), so there are more and more plausible answers
>> >for the question "what is the TOP P-limit 12-equal tuning?".
>>
>> Huh. Using your algorithm the value stopped changing there up
>> to the 4000-some limit.
>
>What algorithm?
>
>> Though I see your algorithm was based
>> on the standard val.
>
>I really don't know what you could be talking about.

The algorithm you gave on this list for TOP-tuning the standard
val in an n-et. You want me to look up the message?

-Carl

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> >> >> ...and the below, I find two 5ths give a 2:1 brat with an
> >> >> octave of 1197 cents (577/289)...
> >>
> >> >> Of course the 1197 number isn't sacrosanct. Here are the
> >> >> octaves of TOP-tuned 12-equal tunings...
> >> >>
> >> >> 5-limit - 1197.67 cents
> >> >> 7-limit - 1195.08 cents
> >>
> >> >> for 'authentic' pre-20th-century music, it seems
> >> >> the 5-limit result is most pertinent.
> >>
> >> >I thought you wanted to have octaves near the TOP meantone
octave
> >> >(see above)?
> >>
> >> I did -- the quote above shows that could be anywhere from
1197.67
> >> to 1195.08 cents with the former considered (by me) more
authentic
> >> for old music.
> >
> >So you abandoned the desire for octaves near the TOP meantone
octave?
>
> I meant the TOP-12-equal octave according to your algorithm. Of
> course there are difference "meantone" mappings...

Huh? There aren't different meantone mappings, at least not for 5-
limit . . .

> most if not all
> of them will give a flat octave.

???

> Or what's your point?

I don't know -- your original post referred to the TOP meantone
octave:

/tuning-math/message/12496

and you verified that you meant that when I asked you about it, but
are you now saying that both times you didn't mean meantone at all?

> >> All true, but my comparison of TOP and zeta equal tunings seemed
> >> to show that they diverged when consistency broke
> >
> >Again, "consistency broke" is meaningless in this context.
>
> It's an observation and therefore not meaningless.

Could you elaborate on your "observation" then? As you stated it, it
seems meaningless to me.

> It's further
> not meaningless because it means the standard val is no longer the
> best one.

Consistency never meant that the standard val is not the best one in
the first place, so I don't know what you mean. There are plenty of
ETs that are inconsistent in a particular odd limit but where the
standard val is the best one in that limit.

> Meanwhile your top-et algorithm does just fine with
> odd limits (it gives same values as at the next-lowest prime).

When the octaves are tempered, consistency within an odd limit is
impossible, strictly speaking.

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> >> >>> ...and that's pretty much where it stays forever.
> >> >
> >> >Where TOP stays forever? As you go to higher and higher prime
> >> >limits (P), there are more and more plausible mappings (or
groups
> >> >of vanishing ratios), so there are more and more plausible
answers
> >> >for the question "what is the TOP P-limit 12-equal tuning?".
> >>
> >> Huh. Using your algorithm the value stopped changing there up
> >> to the 4000-some limit.
> >
> >What algorithm?
> >
> >> Though I see your algorithm was based
> >> on the standard val.
> >
> >I really don't know what you could be talking about.
>
> The algorithm you gave on this list for TOP-tuning the standard
> val in an n-et. You want me to look up the message?

Sure . . . I'm claiming it doesn't make any reference to the standard
val at all.

>>> >>> ...and that's pretty much where it stays forever.
>>> >
>>> >Where TOP stays forever? As you go to higher and higher prime
>>> >limits (P), there are more and more plausible mappings (or groups
>>> >of vanishing ratios), so there are more and more plausible answers
>>> >for the question "what is the TOP P-limit 12-equal tuning?".
>>>
>>> Huh. Using your algorithm the value stopped changing there up
>>> to the 4000-some limit.
>>
>>What algorithm?
>>
>>> Though I see your algorithm was based
>>> on the standard val.
>>
>>I really don't know what you could be talking about.
>
>The algorithm you gave on this list for TOP-tuning the standard
>val in an n-et. You want me to look up the message?

That was...

/tuning-math/message/8512

And my original zeta comparison...

/tuning-math/message/8529
/tuning-math/message/8530

-Carl

>> I meant the TOP-12-equal octave according to your algorithm. Of
>> course there are difference "meantone" mappings...
>
>Huh? There aren't different meantone mappings, at least not for 5-
>limit . . .

True enough.

>> most if not all
>> of them will give a flat octave.
>
>???

Apparently the TOP meantone octave is stretched. Wild.

>> Or what's your point?
>
>I don't know -- your original post referred to the TOP meantone
>octave:

Yes, I meant the TOP 12-equal octave. But it's interesting
that TOP meantone has a stretched octave...

>> It's further
>> not meaningless because it means the standard val is no longer the
>> best one.
>
>Consistency never meant that the standard val is not the best one in
>the first place, so I don't know what you mean. There are plenty of
>ETs that are inconsistent in a particular odd limit but where the
>standard val is the best one in that limit.

Hm...

-Carl

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> >>> >>> ...and that's pretty much where it stays forever.
> >>> >
> >>> >Where TOP stays forever? As you go to higher and higher prime
> >>> >limits (P), there are more and more plausible mappings (or
groups
> >>> >of vanishing ratios), so there are more and more plausible
answers
> >>> >for the question "what is the TOP P-limit 12-equal tuning?".
> >>>
> >>> Huh. Using your algorithm the value stopped changing there up
> >>> to the 4000-some limit.
> >>
> >>What algorithm?
> >>
> >>> Though I see your algorithm was based
> >>> on the standard val.
> >>
> >>I really don't know what you could be talking about.
> >
> >The algorithm you gave on this list for TOP-tuning the standard
> >val in an n-et. You want me to look up the message?
>
> That was...
>
> /tuning-math/message/8512

Carl, is "standard val" stated or implied anywhere in that message?

>> >>> >>> ...and that's pretty much where it stays forever.
>> >>> >
>> >>> >Where TOP stays forever? As you go to higher and higher prime
>> >>> >limits (P), there are more and more plausible mappings (or
>> >>> >groups of vanishing ratios), so there are more and more
>> >>> >plausible answers for the question "what is the TOP P-limit
>> >>> >12-equal tuning?".
>> >>>
>> >>> Huh. Using your algorithm the value stopped changing there up
>> >>> to the 4000-some limit.
>> >>
>> >>What algorithm?
>> >>
>> >>> Though I see your algorithm was based
>> >>> on the standard val.
>> >>
>> >>I really don't know what you could be talking about.
>> >
>> >The algorithm you gave on this list for TOP-tuning the standard
>> >val in an n-et. You want me to look up the message?
>>
>> That was...
>>
>> /tuning-math/message/8512
>
>Carl, is "standard val" stated or implied anywhere in that message?

Yes, you use [12 19 28]

http://tonalsoft.com/enc/s/standard-val.aspx

-Carl

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> >> >>> >>> ...and that's pretty much where it stays forever.
> >> >>> >
> >> >>> >Where TOP stays forever? As you go to higher and higher
prime
> >> >>> >limits (P), there are more and more plausible mappings (or
> >> >>> >groups of vanishing ratios), so there are more and more
> >> >>> >plausible answers for the question "what is the TOP P-limit
> >> >>> >12-equal tuning?".
> >> >>>
> >> >>> Huh. Using your algorithm the value stopped changing there
up
> >> >>> to the 4000-some limit.
> >> >>
> >> >>What algorithm?
> >> >>
> >> >>> Though I see your algorithm was based
> >> >>> on the standard val.
> >> >>
> >> >>I really don't know what you could be talking about.
> >> >
> >> >The algorithm you gave on this list for TOP-tuning the standard
> >> >val in an n-et. You want me to look up the message?
> >>
> >> That was...
> >>
> >> /tuning-math/message/8512
> >
> >Carl, is "standard val" stated or implied anywhere in that message?
>
> Yes, you use [12 19 28]
>
> http://tonalsoft.com/enc/s/standard-val.aspx
>
> -Carl

So what?? That was just an example, and certainly the obvious one in
this case. It happens to be the best val according to many different
measures, not just the standard val. Why on earth would you assume
that I meant or implied the standard val here, after all my posts
criticizing the notion and term?

>> >> That was...
>> >>
>> >> /tuning-math/message/8512
>> >
>> >Carl, is "standard val" stated or implied anywhere in that message?
>>
>> Yes, you use [12 19 28]
>>
>> http://tonalsoft.com/enc/s/standard-val.aspx
>>
>> -Carl
>
>So what?? That was just an example, and certainly the obvious one in
>this case. It happens to be the best val according to many different
>measures, not just the standard val. Why on earth would you assume
>that I meant or implied the standard val here, after all my posts
>criticizing the notion and term?

I don't remember your criticism of it (though I think I remember you
and Gene arguing over 64-tET meantone maps). My mistake. Well, my
results were based on using the standard val.

-Carl

Hi Paul,

--- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
wrote:

> --- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
>
> > > [Paul Erlich:]
> > > /tuning-math/message/8512
> > >
> > > Carl, is "standard val" stated or implied anywhere in
> > > that message?
> >
> > Yes, you use [12 19 28]
> >
> > http://tonalsoft.com/enc/s/standard-val.aspx
> >
> > -Carl
>
> So what?? That was just an example, and certainly the
> obvious one in this case. It happens to be the best val
> according to many different measures, not just the
> standard val. Why on earth would you assume that I meant
> or implied the standard val here, after all my posts
> criticizing the notion and term?

I'd be really happy if you would submit the list of
those posts ... i'll add the links to the Encyclopedia entry.

-monz
http://tonalsoft.com
Tonescape microtonal music software

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> >> >> That was...
> >> >>
> >> >> /tuning-math/message/8512
> >> >
> >> >Carl, is "standard val" stated or implied anywhere in that
message?
> >>
> >> Yes, you use [12 19 28]
> >>
> >> http://tonalsoft.com/enc/s/standard-val.aspx
> >>
> >> -Carl
> >
> >So what?? That was just an example, and certainly the obvious one
in
> >this case. It happens to be the best val according to many
different
> >measures, not just the standard val. Why on earth would you assume
> >that I meant or implied the standard val here, after all my posts
> >criticizing the notion and term?
>
> I don't remember your criticism of it (though I think I remember you
> and Gene arguing over 64-tET meantone maps).

You probably mean 5-limit maps?

> My mistake. Well, my
> results were based on using the standard val.

OK -- my algorithm isn't.

--- In tuning-math@yahoogroups.com, "monz" <monz@t...> wrote:
> Hi Paul,
>
>
> --- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
> wrote:
>
> > --- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> >
> > > > [Paul Erlich:]
> > > > /tuning-math/message/8512
> > > >
> > > > Carl, is "standard val" stated or implied anywhere in
> > > > that message?
> > >
> > > Yes, you use [12 19 28]
> > >
> > > http://tonalsoft.com/enc/s/standard-val.aspx
> > >
> > > -Carl
> >
> > So what?? That was just an example, and certainly the
> > obvious one in this case. It happens to be the best val
> > according to many different measures, not just the
> > standard val. Why on earth would you assume that I meant
> > or implied the standard val here, after all my posts
> > criticizing the notion and term?
>
>
> I'd be really happy if you would submit the list of
> those posts ... i'll add the links to the Encyclopedia entry.

Ouch . . . how do you suggest I go about finding them?

>> >So what?? That was just an example, and certainly the obvious
>> >one in this case. It happens to be the best val according to many
>> >different measures, not just the standard val. Why on earth would
>> >you assume that I meant or implied the standard val here, after
>> >all my posts criticizing the notion and term?
>>
>> I don't remember your criticism of it (though I think I remember
>> you and Gene arguing over 64-tET meantone maps).
>
>You probably mean 5-limit maps?

Yes, sorry.

>> My mistake. Well, my
>> results were based on using the standard val.
>
>OK -- my algorithm isn't.

So how do you choose them?

-Carl

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> >> >So what?? That was just an example, and certainly the obvious
> >> >one in this case. It happens to be the best val according to
many
> >> >different measures, not just the standard val. Why on earth
would
> >> >you assume that I meant or implied the standard val here, after
> >> >all my posts criticizing the notion and term?
> >>
> >> I don't remember your criticism of it (though I think I remember
> >> you and Gene arguing over 64-tET meantone maps).
> >
> >You probably mean 5-limit maps?
>
> Yes, sorry.
>
> >> My mistake. Well, my
> >> results were based on using the standard val.
> >
> >OK -- my algorithm isn't.
>
> So how do you choose them?

The algorithm works correctly for *any* val. If you need to choose
one, clearly a sensible choice is the val which, when TOP-tempered,
has the lowest TOP damage.

>> >> My mistake. Well, my
>> >> results were based on using the standard val.
>> >
>> >OK -- my algorithm isn't.
>>
>> So how do you choose them?
>
>The algorithm works correctly for *any* val. If you need to choose
>one, clearly a sensible choice is the val which, when TOP-tempered,
>has the lowest TOP damage.

Ok! How might one automate this?

-Carl

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> >> >> My mistake. Well, my
> >> >> results were based on using the standard val.
> >> >
> >> >OK -- my algorithm isn't.
> >>
> >> So how do you choose them?
> >
> >The algorithm works correctly for *any* val. If you need to choose
> >one, clearly a sensible choice is the val which, when TOP-
tempered,
> >has the lowest TOP damage.
>
> Ok! How might one automate this?

I think this would work:

Start with the JI val. Multiply it by every possible positive real
number (in practice you just need to find a small enough step size to
increment this constant through) and round the result to the nearest
integers. Use the TOP-et algorithm on all of these, and calculate
their TOP damage. Then group the results by the octave part of the
val and pick out the lowest-damage one in each group.

I think any val that doesn't arise through this process can't be the
minimum-TOP-damage one (when TOP-tuned) for a given octave
cardinality.

Hi Paul,

--- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
wrote:

> --- In tuning-math@yahoogroups.com, "monz" <monz@t...> wrote:

> > I'd be really happy if you would submit the list of
> > those posts ... i'll add the links to the Encyclopedia entry.
>
> Ouch . . . how do you suggest I go about finding them?

Umm ... the same way i do? -- painfully sifting thru
the sorry-ass Yahoo archive search, one page at a time.

If it's too much trouble, then maybe you could just
sum up your critiques for inclusion on the Encyclopedia
page?

-monz
http://tonalsoft.com
Tonescape microtonal music theory

>> >> >> My mistake. Well, my
>> >> >> results were based on using the standard val.
>> >> >
>> >> >OK -- my algorithm isn't.
>> >>
>> >> So how do you choose them?
>> >
>> >The algorithm works correctly for *any* val. If you need to
>> >choose one, clearly a sensible choice is the val which, when
>> >TOP-tempered, has the lowest TOP damage.
>>
>> Ok! How might one automate this?
>
>I think this would work:
>
>Start with the JI val. Multiply it by every possible positive real
>number (in practice you just need to find a small enough step size
>to increment this constant through) and round the result to the
>nearest integers. Use the TOP-et algorithm on all of these, and
>calculate their TOP damage. Then group the results by the octave
>part of the val and pick out the lowest-damage one in each group.
>
>I think any val that doesn't arise through this process can't
>be the minimum-TOP-damage one (when TOP-tuned) for a given octave
>cardinality.

This sounds expensive. What if we assume that an interesting val
for prime-limit p and scale size s will be the 'standard val' of a
basis for p, given as a list primes or ratios-of-primes?

-Carl

--- In tuning-math@yahoogroups.com, "monz" <monz@t...> wrote:
> Hi Paul,
>
> --- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
> wrote:
>
> > --- In tuning-math@yahoogroups.com, "monz" <monz@t...> wrote:
>
> > > I'd be really happy if you would submit the list of
> > > those posts ... i'll add the links to the Encyclopedia entry.
> >
> > Ouch . . . how do you suggest I go about finding them?
>
>
>
> Umm ... the same way i do? -- painfully sifting thru
> the sorry-ass Yahoo archive search, one page at a time.
>
> If it's too much trouble, then maybe you could just
> sum up your critiques for inclusion on the Encyclopedia
> page?

Simple -- what Gene calls the "standard val" is often not the "best
val" according to any of the various criteria for assessing the
severity of mistuning that Gene and/or I use. Why "standardize"
something that we know will sometimes not be "best"?

I had no idea you were accepting amendments to your dictionary pages -
- last time I asked you if you were you didn't answer, and before
that you weren't accepting them because all the pages were being
reformatted -- and of course I've proposed countless amendments that
have not been implemented yet, but I feel are very important.

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> >> >> >> My mistake. Well, my
> >> >> >> results were based on using the standard val.
> >> >> >
> >> >> >OK -- my algorithm isn't.
> >> >>
> >> >> So how do you choose them?
> >> >
> >> >The algorithm works correctly for *any* val. If you need to
> >> >choose one, clearly a sensible choice is the val which, when
> >> >TOP-tempered, has the lowest TOP damage.
> >>
> >> Ok! How might one automate this?
> >
> >I think this would work:
> >
> >Start with the JI val. Multiply it by every possible positive real
> >number (in practice you just need to find a small enough step size
> >to increment this constant through) and round the result to the
> >nearest integers. Use the TOP-et algorithm on all of these, and
> >calculate their TOP damage. Then group the results by the octave
> >part of the val and pick out the lowest-damage one in each group.
> >
> >I think any val that doesn't arise through this process can't
> >be the minimum-TOP-damage one (when TOP-tuned) for a given octave
> >cardinality.
>
> This sounds expensive.

It isn't -- it's very cheap, as it just increments a single quantity.

> What if we assume that an interesting val
> for prime-limit p and scale size s will be the 'standard val' of a
> basis for p, given as a list primes or ratios-of-primes?

I have no idea what that means.

>> What if we assume that an interesting val
>> for prime-limit p and scale size s will be the 'standard val' of a
>> basis for p, given as a list primes or ratios-of-primes?
>
>I have no idea what that means.

Like, (2 3 5) and (2 5/3, 5/4) are both basises for 5-limit JI
(I didn't verify this)... if you find the 'standard val' for
all of these you'll get all of the interesting vals. Just an
idea.

-Carl

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> >> What if we assume that an interesting val
> >> for prime-limit p and scale size s will be the 'standard val' of a
> >> basis for p, given as a list primes or ratios-of-primes?
> >
> >I have no idea what that means.
>
> Like, (2 3 5) and (2 5/3, 5/4) are both basises

bases

> for 5-limit JI
> (I didn't verify this)... if you find the 'standard val' for
> all of these you'll get all of the interesting vals. Just an
> idea.

There are an infinite number of bases for 5-limit JI, so unless I'm
misunderstanding you, this seems infinitely more costly than my
proposal.

--- In tuning-math@yahoogroups.com, "monz" <monz@t...> wrote:

> I'd be really happy if you would submit the list of
> those posts ... i'll add the links to the Encyclopedia entry.

What's the entry on? By the way, I wonder what Paul thinks of
"semistandard" vals.

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

> This sounds expensive. What if we assume that an interesting val
> for prime-limit p and scale size s will be the 'standard val' of a
> basis for p, given as a list primes or ratios-of-primes?

I don't get what you mean, but the semistandard vals for limit p and
division n would be a good start.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, "monz" <monz@t...> wrote:
>
> > I'd be really happy if you would submit the list of
> > those posts ... i'll add the links to the Encyclopedia entry.
>
> What's the entry on? By the way, I wonder what Paul thinks of
> "semistandard" vals.

You'll have to refresh my memory. I vaguely remember that these are
ones which, when TOP-tuned, have the same TOP damage as the standard
vals. But since the standard vals themselves can often be beaten in
this regard, I'm probably remembering wrong.

At 05:53 PM 8/29/2005, you wrote:
>--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
>> >> What if we assume that an interesting val
>> >> for prime-limit p and scale size s will be the 'standard val' of a
>> >> basis for p, given as a list primes or ratios-of-primes?
>> >
>> >I have no idea what that means.
>>
>> Like, (2 3 5) and (2 5/3, 5/4) are both basises
>
>bases

I did that on purpose, to be funny. Along with the "Like,".

>> for 5-limit JI
>> (I didn't verify this)... if you find the 'standard val' for
>> all of these you'll get all of the interesting vals. Just an
>> idea.
>
>There are an infinite number of bases for 5-limit JI, so unless I'm
>misunderstanding you, this seems infinitely more costly than my
>proposal.

The original quote had the restriction that only primes or
fractions of primes could be used. Or something like that.

-Carl

>> This sounds expensive. What if we assume that an interesting val
>> for prime-limit p and scale size s will be the 'standard val' of a
>> basis for p, given as a list primes or ratios-of-primes?
>
>I don't get what you mean, but the semistandard vals for limit p and
>division n would be a good start.

Is there a reference on them somewhere?

-Carl

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> At 05:53 PM 8/29/2005, you wrote:
> >--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> >> >> What if we assume that an interesting val
> >> >> for prime-limit p and scale size s will be the 'standard val'
of a
> >> >> basis for p, given as a list primes or ratios-of-primes?
> >> >
> >> >I have no idea what that means.
> >>
> >> Like, (2 3 5) and (2 5/3, 5/4) are both basises
> >
> >bases
>
> I did that on purpose, to be funny. Along with the "Like,".
>
> >> for 5-limit JI
> >> (I didn't verify this)... if you find the 'standard val' for
> >> all of these you'll get all of the interesting vals. Just an
> >> idea.
> >
> >There are an infinite number of bases for 5-limit JI, so unless
I'm
> >misunderstanding you, this seems infinitely more costly than my
> >proposal.
>
> The original quote had the restriction that only primes or
> fractions of primes could be used. Or something like that.

This seems awfully complex to implement *and* arbitrary. My
suggestion, OTOH, is a very short computer program, whose results
I've already made plenty of graphs of.

--- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...> wrote:

> You'll have to refresh my memory. I vaguely remember that these are
> ones which, when TOP-tuned, have the same TOP damage as the standard
> vals. But since the standard vals themselves can often be beaten in
> this regard, I'm probably remembering wrong.

It depends on your definition of "TOP damage", I suppose. By the
definition I was using, a standard val does have minimal TOP damage,
and any val which does also is a semistandard val.

For example, in 64-et we have <64 101 149| as the standard val, and it
and <64 101 148| are the two 5-limit semistandard vals. Taking
64*<1 p3 p5|, where p3=log2(3) and p5=log2(5), and the difference to
the vals, and dividing by 1, p3, p5 respectively gives <0 .2761 .2599|
for <64 101 148| and <0 .2761 -.1708| for <0 101 149|. In the sense of
maximum weighted absolute value, the TOP damage is identical. To
choose one out of the list of semistandard vals as best, you could
first get the list, which isn't hard, and then choose the best
according to some other standard.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
wrote:
>
> > You'll have to refresh my memory. I vaguely remember that these
are
> > ones which, when TOP-tuned, have the same TOP damage as the
standard
> > vals. But since the standard vals themselves can often be beaten
in
> > this regard, I'm probably remembering wrong.
>
> It depends on your definition of "TOP damage", I suppose. By the
> definition I was using, a standard val does have minimal TOP damage,
> and any val which does also is a semistandard val.
>
> For example, in 64-et we have <64 101 149| as the standard val, and
it
> and <64 101 148| are the two 5-limit semistandard vals. Taking
> 64*<1 p3 p5|, where p3=log2(3) and p5=log2(5), and the difference to
> the vals, and dividing by 1, p3, p5 respectively gives
<0 .2761 .2599|
> for <64 101 148| and <0 .2761 -.1708| for <0 101 149|. In the sense
of
> maximum weighted absolute value, the TOP damage is identical. To
> choose one out of the list of semistandard vals as best, you could
> first get the list, which isn't hard, and then choose the best
> according to some other standard.

So you're not doing what I was talking about, which was TOP-tuning
first and then comparing the TOP damage. The maximum weighed absolute
value of these difference vectors (one element of which is
necessarily zero) seems rather uninteresting, since it doesn't
translate to a statement about all the intervals in the tuning the
way the TOP damage of a TOP-tuned temperament does.

My idea was that another way of defining the "standard val" would be
to choose the one which, when TOP-tuned, had the lowest TOP damage. I
was hoping that this would also be the one that when Kees tuned,
would have the lowest Kees damage, but you seem to be saying (in
another thread) that this isn't necessarily so beyond the 5-limit.

--- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...> wrote:

> So you're not doing what I was talking about, which was TOP-tuning
> first and then comparing the TOP damage.

No, but that would be one of the possible second steps in a decision
proceedure if you wanted to decide between semistandard vals.

The maximum weighed absolute
> value of these difference vectors (one element of which is
> necessarily zero) seems rather uninteresting, since it doesn't
> translate to a statement about all the intervals in the tuning the
> way the TOP damage of a TOP-tuned temperament does.

Sure it does; both of the semistandard 64 vals give the same maximum
for absolute error over Tenney distance, obtained for powers of 3, for
example.

> My idea was that another way of defining the "standard val" would be
> to choose the one which, when TOP-tuned, had the lowest TOP damage. I
> was hoping that this would also be the one that when Kees tuned,
> would have the lowest Kees damage, but you seem to be saying (in
> another thread) that this isn't necessarily so beyond the 5-limit.

I think they should get less and less alike for higher prime limits.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
wrote:
>
> > So you're not doing what I was talking about, which was TOP-
tuning
> > first and then comparing the TOP damage.
>
> No, but that would be one of the possible second steps in a decision
> proceedure if you wanted to decide between semistandard vals.

Are you sure that the best val according to this procedure always
*is* a semistandard val?

> > The maximum weighed absolute
> > value of these difference vectors (one element of which is
> > necessarily zero) seems rather uninteresting, since it doesn't
> > translate to a statement about all the intervals in the tuning
the
> > way the TOP damage of a TOP-tuned temperament does.
>
> Sure it does; both of the semistandard 64 vals give the same maximum
> for absolute error over Tenney distance, obtained for powers of 3,
for
> example.

I was referring to properties such as the fact that optimality
remains even if you restrict your attention to intervals within 1
octave, or whatever bound on span you wish.

> > My idea was that another way of defining the "standard val" would
be
> > to choose the one which, when TOP-tuned, had the lowest TOP
damage. I
> > was hoping that this would also be the one that when Kees tuned,
> > would have the lowest Kees damage, but you seem to be saying (in
> > another thread) that this isn't necessarily so beyond the 5-limit.
>
> I think they should get less and less alike for higher prime limits.

I'm still waiting for an example.

P.S. Though I stand by my reply, I guess I shouldn't use the
word "val" at all. What I have in mind are *projective* "vals"
or "breeds", only defined up to an overall constant, and certainly
not carrying any implication whatsoever that the octave is to be
tuned pure. For ETs, the numbers in the projective breed will have a
integer relation with one another, since a bracket product with any
of the vanishing "monzos" will yield zero. But this does not depend
on the overall scaling of the projective breed, which should be taken
as undefined/arbitrary.

If we do want to talk about "vals" or whatever in an absolute sense,
where precise intervals are implied, it seems to me we should state
them in consistent units of interval measure, such as cents.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
wrote:
>
> > So you're not doing what I was talking about, which was TOP-
tuning
> > first and then comparing the TOP damage.
>
> No, but that would be one of the possible second steps in a decision
> proceedure if you wanted to decide between semistandard vals.
>
> The maximum weighed absolute
> > value of these difference vectors (one element of which is
> > necessarily zero) seems rather uninteresting, since it doesn't
> > translate to a statement about all the intervals in the tuning
the
> > way the TOP damage of a TOP-tuned temperament does.
>
> Sure it does; both of the semistandard 64 vals give the same maximum
> for absolute error over Tenney distance, obtained for powers of 3,
for
> example.
>
> > My idea was that another way of defining the "standard val" would
be
> > to choose the one which, when TOP-tuned, had the lowest TOP
damage. I
> > was hoping that this would also be the one that when Kees tuned,
> > would have the lowest Kees damage, but you seem to be saying (in
> > another thread) that this isn't necessarily so beyond the 5-limit.
>
> I think they should get less and less alike for higher prime limits.

--- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...> wrote:

> > No, but that would be one of the possible second steps in a decision
> > proceedure if you wanted to decide between semistandard vals.
>
> Are you sure that the best val according to this procedure always
> *is* a semistandard val?

Of my proceedure, of course. Of your proceedure, not necessarily, but
I'd hate to try to come up with a counterexample, as I suspect it
would be hard.

--- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...> wrote:
> P.S. Though I stand by my reply, I guess I shouldn't use the
> word "val" at all. What I have in mind are *projective* "vals"
> or "breeds", only defined up to an overall constant, and certainly
> not carrying any implication whatsoever that the octave is to be
> tuned pure.

The way I defined them, vals are not tunings and do not carry such an
implication. Looking at them projectively is something altogether
different; vals define points in projective space, but when you do
that you erase all the information which allows tuning to even be
discussed. They are, in particular, not points in Tenney space, and
don't carry a metric structure (though there is a topology.)

Projective vals can be wedged together, and projective multivals carry
all the information needed to define regular temperaments, without
regard to questions of tuning. Regular temperaments are in 1-1
correspondence with points on the associated Grassmannian:

http://en.wikipedia.org/wiki/Grassmanian

> If we do want to talk about "vals" or whatever in an absolute sense,
> where precise intervals are implied, it seems to me we should state
> them in consistent units of interval measure, such as cents.

Those would then be tuning maps.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
wrote:
> > P.S. Though I stand by my reply, I guess I shouldn't use the
> > word "val" at all. What I have in mind are *projective* "vals"
> > or "breeds", only defined up to an overall constant, and
certainly
> > not carrying any implication whatsoever that the octave is to be
> > tuned pure.
>
> The way I defined them, vals are not tunings and do not carry such
an
> implication. Looking at them projectively is something altogether
> different; vals define points in projective space, but when you do
> that you erase all the information which allows tuning to even be
> discussed.

But if they weren't tunings in the first place, what have we erased?

> They are, in particular, not points in Tenney space, and
> don't carry a metric structure (though there is a topology.)

Well, my hex chart is a chart of projective vals (in projective
Tenney space?), and there seem to be a lot of interesting things
going on; for example, the TOP error of the TOP-tuned incarnations of
these follow the hexagonal contours in this space. So it seems tuning
*can* be discussed, though perhaps indirectly. And I don't know what
you mean about metric structure; it seems perfectly possible to
define a 'distance' between two projective vals, say by just
measuring it on that chard.

> Projective vals can be wedged together, and projective multivals
carry
> all the information needed to define regular temperaments, without
> regard to questions of tuning. Regular temperaments are in 1-1
> correspondence with points on the associated Grassmannian:
>
> http://en.wikipedia.org/wiki/Grassmanian

I'll look at this and see if I can learn anything.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
wrote:
>
> > > No, but that would be one of the possible second steps in a
decision
> > > proceedure if you wanted to decide between semistandard vals.
> >
> > Are you sure that the best val according to this procedure always
> > *is* a semistandard val?
>
> Of my proceedure, of course. Of your proceedure, not necessarily, but
> I'd hate to try to come up with a counterexample, as I suspect it
> would be hard.

Is it that hard? Isn't [20 32 47] the val for 20-equal which, when TOP
tuned, has the lowest TOP damage? And am I correct that this is neither
the "standard" nor a "semistandard" val for 20-equal according to your
definitions?

--- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...> wrote:

> I'm still waiting for an example.

Hmmm. What about

<<3 -5 -6 -1 -15 -18 -12 0 15 18||

with TM basis {56/55, 64/63, 77/75} and mapping

[<1 3 0 0 3|, <0 -3 5 6 1|]

--- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...> wrote:

> Is it that hard? Isn't [20 32 47] the val for 20-equal which, when TOP
> tuned, has the lowest TOP damage? And am I correct that this is neither
> the "standard" nor a "semistandard" val for 20-equal according to your
> definitions?

OK, I guess it isn't hard.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
wrote:
>
> > I'm still waiting for an example.
>
> Hmmm. What about
>
> <<3 -5 -6 -1 -15 -18 -12 0 15 18||
>
> with TM basis {56/55, 64/63, 77/75} and mapping
>
> [<1 3 0 0 3|, <0 -3 5 6 1|]

You'll have to tell me what you think the TOP and Kees tunings for this
are.

--- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...> wrote:
> --- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
> wrote:
> > --- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
> wrote:
> >
> > > I'm still waiting for an example.
> >
> > Hmmm. What about
> >
> > <<3 -5 -6 -1 -15 -18 -12 0 15 18||
> >
> > with TM basis {56/55, 64/63, 77/75} and mapping
> >
> > [<1 3 0 0 3|, <0 -3 5 6 1|]
>
> You'll have to tell me what you think the TOP and Kees tunings for this
> are.

Great. Yahoo is again deleting my posts again and not telling me.

The stretched TOP tuning has pure 7s, and the Kees tuning has
the error of 3, over log(3), equal to the error of 5, over log(5).

Stretched TOP:

<1200 1915.578 2807.355 3368.826 4161.472|

Kees:

<1200 1915.929 2806.785 3368.142 4161.357|

While the tunings are pretty close, they clearly are different.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
wrote:
> > --- In tuning-math@yahoogroups.com, "Gene Ward Smith"
<gwsmith@s...>
> > wrote:
> > > --- In tuning-math@yahoogroups.com, "Paul Erlich"
<perlich@a...>
> > wrote:
> > >
> > > > I'm still waiting for an example.
> > >
> > > Hmmm. What about
> > >
> > > <<3 -5 -6 -1 -15 -18 -12 0 15 18||
> > >
> > > with TM basis {56/55, 64/63, 77/75} and mapping
> > >
> > > [<1 3 0 0 3|, <0 -3 5 6 1|]
> >
> > You'll have to tell me what you think the TOP and Kees tunings
for this
> > are.
>
> Great. Yahoo is again deleting my posts again and not telling me.
>
> The stretched TOP tuning has pure 7s,

What about the unstretched TOP tuning? That's what I was asking
about. Recall that if *that* has pure 7s, it's not unique, and this
has to be taken into account.

> and the Kees tuning has
> the error of 3, over log(3), equal to the error of 5, over log(5).
>
> Stretched TOP:
>
> <1200 1915.578 2807.355 3368.826 4161.472|
>
> Kees:
>
> <1200 1915.929 2806.785 3368.142 4161.357|
>
> While the tunings are pretty close, they clearly are different.

--- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...> wrote:

> > The stretched TOP tuning has pure 7s,
>
> What about the unstretched TOP tuning? That's what I was asking
> about. Recall that if *that* has pure 7s, it's not unique, and this
> has to be taken into account.

TOP tuning:

<<1195.729 1909.109 2796.794 3356.154 4146.545||

No exact anything that I can see.

>> >> All true, but my comparison of TOP and zeta equal tunings seemed
>> >> to show that they diverged when consistency broke
>> >
>> >Again, "consistency broke" is meaningless in this context.
>>
>> It's an observation and therefore not meaningless.
>
>Could you elaborate on your "observation" then? As you stated it, it
>seems meaningless to me.
>
>> It's further
>> not meaningless because it means the standard val is no longer the
>> best one.
>
>Consistency never meant that the standard val is not the best one in
>the first place, so I don't know what you mean. There are plenty of
>ETs that are inconsistent in a particular odd limit but where the
>standard val is the best one in that limit.

Yes but is there an example of an n-limit consistent ET where
the standard val isn't the best in that limit?

-Carl

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> >> >> All true, but my comparison of TOP and zeta equal tunings
seemed
> >> >> to show that they diverged when consistency broke
> >> >
> >> >Again, "consistency broke" is meaningless in this context.
> >>
> >> It's an observation and therefore not meaningless.
> >
> >Could you elaborate on your "observation" then? As you stated it,
it
> >seems meaningless to me.
> >
> >> It's further
> >> not meaningless because it means the standard val is no longer
the
> >> best one.
> >
> >Consistency never meant that the standard val is not the best one
in
> >the first place, so I don't know what you mean. There are plenty
of
> >ETs that are inconsistent in a particular odd limit but where the
> >standard val is the best one in that limit.
>
> Yes but is there an example of an n-limit consistent ET where
> the standard val isn't the best in that limit?

No -- consistent means you get the same answer for 'which is best'
regardless of which set of consonant basis intervals you look at, so
since the standard val looks at one such set, it will coincide with
all the other 'best' vals.

Though after Gene's reply to my post on projective vals, I have to
say I don't quite know what a val is :(

>>> my comparison of TOP and zeta equal tunings
>>> seemed to show that they diverged when consistency broke

>>> "consistency broke" is meaningless in this context.

>>> it's not meaningless because it means the standard val is
>>> no longer the best one.

>>>Consistency never meant that the standard val is not the best
>>>one in the first place, so I don't know what you mean. There
>>>are plenty of ETs that are inconsistent in a particular odd
>>>limit but where the standard val is the best one in that limit.
>>
>> Yes but is there an example of an n-limit consistent ET where
>> the standard val isn't the best in that limit?
>
>No -- consistent means you get the same answer for 'which is best'
>regardless of which set of consonant basis intervals you look at, so
>since the standard val looks at one such set, it will coincide with
>all the other 'best' vals.

Very good. Now, what I was trying to say, if I'm running up the
limit on your TOP-ET algorithm with any given val, chances are
it'll stop being the best val some time after consistency breaks,
and my hunch was that this is when your algorithm diverges from
the zeta tuning (or maybe it's just how absolutely good the given
val is...). Otherwise I'd expect it to get closer to zeta tuning at
higher limits...

>Though after Gene's reply to my post on projective vals, I have to
>say I don't quite know what a val is :(

I'm trying to sort that lot out too.

-Carl

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

> I'm trying to sort that lot out too.

For some fixed p-limit, just take <...| and fill it in with integers.
I orginally defined it as <.... infinitely, with all but a finite
number zero, but most of the time we work in a fixed p-limit anyway.

>> I'm trying to sort that lot out too.
>
>For some fixed p-limit, just take <...| and fill it in with integers.
>I orginally defined it as <.... infinitely, with all but a finite
>number zero, but most of the time we work in a fixed p-limit anyway.

Right. And I don't understand your objection to Paul's procedure.
We want to know what val (as you define it) leads to a tuning (not
that the val becomes a tuning) that is TOP (Tenney Optimal). Standard
and semi-standard vals don't do this, correct? So what is their
utility?

-Carl

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> >>> my comparison of TOP and zeta equal tunings
> >>> seemed to show that they diverged when consistency broke
>
> >>> "consistency broke" is meaningless in this context.
>
> >>> it's not meaningless because it means the standard val is
> >>> no longer the best one.
>
> >>>Consistency never meant that the standard val is not the best
> >>>one in the first place, so I don't know what you mean. There
> >>>are plenty of ETs that are inconsistent in a particular odd
> >>>limit but where the standard val is the best one in that limit.
> >>
> >> Yes but is there an example of an n-limit consistent ET where
> >> the standard val isn't the best in that limit?
> >
> >No -- consistent means you get the same answer for 'which is best'
> >regardless of which set of consonant basis intervals you look at,
so
> >since the standard val looks at one such set, it will coincide
with
> >all the other 'best' vals.
>
> Very good. Now, what I was trying to say, if I'm running up the
> limit on your TOP-ET algorithm with any given val,

Can you elaborate on what this means?

> chances are
> it'll stop being the best val some time after consistency breaks,

Best defined how? And what "chances" are you referring to? Is this a
probabilistic setup? Also, recall that consistency ceases to make any
sense in a TOP paradigm.

> and my hunch was that this is when your algorithm diverges from
> the zeta tuning (or maybe it's just how absolutely good the given
> val is...).

You lost me.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
>
> > I'm trying to sort that lot out too.
>
> For some fixed p-limit, just take <...| and fill it in with integers.
> I orginally defined it as <.... infinitely, with all but a finite
> number zero, but most of the time we work in a fixed p-limit anyway.

Now what's wrong with taking this to represent a projective class of
tunings, such that when the bracket product of this val (or any of the
tuning vectors it can represent) with any relevant comma is taken, the
result will be zero?

>> Very good. Now, what I was trying to say, if I'm running up the
>> limit on your TOP-ET algorithm with any given val,
>
>Can you elaborate on what this means?

I did it in my zeta vs. TOP-ET posts, which I recently linked to
here.

>> chances are
>> it'll stop being the best val some time after consistency breaks,
>
>Best defined how?

Your way -- lowest TOP damage of the resulting tuning.

>And what "chances" are you referring to? Is this a probabilistic
>setup?

It's just a trend I noticed when doing some numbers.

>Also, recall that consistency ceases to make any sense in a TOP
>paradigm.

It still shows when the val used to generate the TOP-ET is
no longer necessarily the one corresponding to the tuning with the
lowest TOP damage.

>> and my hunch was that this is when your algorithm diverges from
>> the zeta tuning (or maybe it's just how absolutely good the given
>> val is...).
>
>You lost me.

TOP ETs based on standard vals seem to be close to zeta and/or gram
tunings at limits where the corresponding standard ET is consistent.

-Carl

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

> >> Very good. Now, what I was trying to say, if I'm running up the
> >> limit on your TOP-ET algorithm with any given val,
> >
> >Can you elaborate on what this means?
>
> I did it in my zeta vs. TOP-ET posts, which I recently linked to
> here.
>
> >> chances are
> >> it'll stop being the best val some time after consistency breaks,
> >
> >Best defined how?
>
> Your way -- lowest TOP damage of the resulting tuning.
>
> >And what "chances" are you referring to? Is this a probabilistic
> >setup?
>
> It's just a trend I noticed when doing some numbers.

I really have little idea what you're doing. Are you seeding these
with random numbers?

> >Also, recall that consistency ceases to make any sense in a TOP
> >paradigm.
>
> It still shows when the val used to generate the TOP-ET is
> no longer necessarily the one corresponding to the tuning with the
> lowest TOP damage.

Again, no idea what you mean here. If you're talking about "standard
vals" anywhere in the above, you should say so.

> >> and my hunch was that this is when your algorithm diverges from
> >> the zeta tuning (or maybe it's just how absolutely good the given
> >> val is...).
> >
> >You lost me.
>
> TOP ETs based on standard vals

So you were talking about standard vals above? You should have said
so. Now help fill in all the other blanks for me.

> seem to be close to zeta and/or gram
> tunings at limits where the corresponding standard ET is consistent.

Again, consistency doesn't make sense applied to vals in the TOP
paradigm, but there's perhaps some warpage of this statement that
could be made to make sense.

So, does this observation of yours apply for more than just the
usual, well-known "good" ETs?

>> >> Very good. Now, what I was trying to say, if I'm running up the
>> >> limit on your TOP-ET algorithm with any given val,
>> >
>> >Can you elaborate on what this means?
>>
>> I did it in my zeta vs. TOP-ET posts, which I recently linked to
>> here.
>>
>> >> chances are
>> >> it'll stop being the best val some time after consistency breaks,
>> >
>> >Best defined how?
>>
>> Your way -- lowest TOP damage of the resulting tuning.
>>
>> >And what "chances" are you referring to? Is this a probabilistic
>> >setup?
>>
>> It's just a trend I noticed when doing some numbers.
>
>I really have little idea what you're doing. Are you seeding these
>with random numbers?

I'm just comparing the octave stretch at different limits.

>> >Also, recall that consistency ceases to make any sense in a TOP
>> >paradigm.
>>
>> It still shows when the val used to generate the TOP-ET is
>> no longer necessarily the one corresponding to the tuning with the
>> lowest TOP damage.
>
>Again, no idea what you mean here. If you're talking about "standard
>vals" anywhere in the above, you should say so.

I thought we'd already established that I've been using standard
vals for this.

>> >> and my hunch was that this is when your algorithm diverges from
>> >> the zeta tuning (or maybe it's just how absolutely good the given
>> >> val is...).
>> >
>> >You lost me.
>>
>> TOP ETs based on standard vals
>
>So you were talking about standard vals above? You should have said
>so. Now help fill in all the other blanks for me.
>
>> seem to be close to zeta and/or gram
>> tunings at limits where the corresponding standard ET is consistent.
>
>Again, consistency doesn't make sense applied to vals in the TOP
>paradigm, but there's perhaps some warpage of this statement that
>could be made to make sense.
>
>So, does this observation of yours apply for more than just the
>usual, well-known "good" ETs?

Aren't these posts clear?

/tuning-math/message/8529
/tuning-math/message/8530

-Carl

Below is a message from last summer. Unfortunately using
approximations like this, the circle of fifths won't close
exactly. One needs to use an equation that includes the
pythagorean comma.

1. Listening tests of brats haven't been very exciting.
Things may be different on a real piano, though.

2. A closed circle of low-brat trines on paper may not
close on a piano. Slight adjustments to the pitches
may be needed, depending on how inharmonic the loudest
beating partials happen to be, and there's no guarantee
those deviations will add up to zero (so whatever you
worked out *still* might not close).

One can't tell if one doesn't try, though.

12f = 7o
o = 12f/7
f = 7o/12

(2f - 3)/(o - 2) = 2
f = o - 1/2
o = f + 1/2

Problem is, the top set of equations is log-scale and the
bottom set ain't. And apparently taking the logarithm of
an addition is a bit messy, involving something called
tropical geometry according to Wikipedia. Addition is
nearly but not exactly converted into max(). Hrm...

-Carl

At 03:06 PM 8/25/2005, I wrote:
>>Using Gene's...
>>
>>>If we define the octave-fifth beat ratio as
>>>
>>>(2f - 3)/(o - 2)
>>
>>...and the below, I find two 5ths give a 2:1 brat with an
>>octave of 1197 cents (577/289)...
>>
>>697.956768 (865/578)
>>and
>>700.956308 (1733/1156)
>>
>>One approach would be to use as many of the smaller of these
>>as possible (for good major 3rds) and then close the circle
>>with big ones. It happens this can be done with only 2 big ones.
>>Or, one can get even closer to an 1197-cent octave with 1 just
>>5th.
>
>Sorry: using one 700.956308 5th gets closer than using two.
>
>The actual brat target octave used was 1197.00219 cents. Using
>one of these large tempered 5ths or one just 5th gets you .07
>cents flat or sharp of it, respectively. The just 5th gets you
>four thousandths of a cent closer, which is possibly within my
>margin of error.
>
>-Carl

By gods, using the power of Maple, I think I've found the
only fifth and octave which have a brat of +2 and make the
pythagorean comma vanish

1197.394
698.480

I did it by expressing the first equation (below) using roots,
rather than taking the log of the second equation. I wish I
could have simplified further symbolically than I did, but
Solve kept giving me _Z and some bizarre "index" in the roots
instead of the variable I told it to solve for. Anyway, here's
the Scala file:

! synchtrines+2.scl
!
The 12-tone equal temperament with 2:3:4 brats of +2.
12
!
99.784
199.566
299.350
399.132
498.916
598.698
698.480
798.264
898.046
997.830
1097.612
1197.394
!

And with a brat of -2, we get a stretched octave:

1201.020
700.595

This one's so close to ET I doubt the difference is practical.

I get a degenerate solution (unity for fifth and octave) when
solving for a +1 brat.

The tuning for the -1 brat is just slightly more stretched
than that for the -2 brat:

1201.563372967
700.91196792

For now, I'm recommending synchtrines+2.scl. Maybe I'll try
and think of a way to get well temperaments like this.

-Carl

At 02:12 AM 5/29/2006, you wrote:
>Below is a message from last summer. Unfortunately using
>approximations like this, the circle of fifths won't close
>exactly. One needs to use an equation that includes the
>pythagorean comma.
>
>1. Listening tests of brats haven't been very exciting.
>Things may be different on a real piano, though.
>
>2. A closed circle of low-brat trines on paper may not
>close on a piano. Slight adjustments to the pitches
>may be needed, depending on how inharmonic the loudest
>beating partials happen to be, and there's no guarantee
>those deviations will add up to zero (so whatever you
>worked out *still* might not close).
>
>One can't tell if one doesn't try, though.
>
>12f = 7o
>o = 12f/7
>f = 7o/12
>
>(2f - 3)/(o - 2) = 2
>f = o - 1/2
>o = f + 1/2
>
>Problem is, the top set of equations is log-scale and the
>bottom set ain't. And apparently taking the logarithm of
>an addition is a bit messy, involving something called
>tropical geometry according to Wikipedia. Addition is
>nearly but not exactly converted into max(). Hrm...
>
>-Carl
>
>At 03:06 PM 8/25/2005, I wrote:
>>>Using Gene's...
>>>
>>>>If we define the octave-fifth beat ratio as
>>>>
>>>>(2f - 3)/(o - 2)
>>>
>>>...and the below, I find two 5ths give a 2:1 brat with an
>>>octave of 1197 cents (577/289)...
>>>
>>>697.956768 (865/578)
>>>and
>>>700.956308 (1733/1156)
>>>
>>>One approach would be to use as many of the smaller of these
>>>as possible (for good major 3rds) and then close the circle
>>>with big ones. It happens this can be done with only 2 big ones.
>>>Or, one can get even closer to an 1197-cent octave with 1 just
>>>5th.
>>
>>Sorry: using one 700.956308 5th gets closer than using two.
>>
>>The actual brat target octave used was 1197.00219 cents. Using
>>one of these large tempered 5ths or one just 5th gets you .07
>>cents flat or sharp of it, respectively. The just 5th gets you
>>four thousandths of a cent closer, which is possibly within my
>>margin of error.
>>
>>-Carl

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@...> wrote:
>
> By gods, using the power of Maple...

I thought you scoffed at Maple.

> I did it by expressing the first equation (below) using roots,
> rather than taking the log of the second equation. I wish I
> could have simplified further symbolically than I did, but
> Solve kept giving me _Z and some bizarre "index" in the roots
> instead of the variable I told it to solve for.

Sounds like a RootOf. You can get it to look nicer by aliasing the
root to something of your choice.