from linear to equal

Hi,

Linear temperaments (or 2-dimensional tunings) are infinitely
extendable. Once you extend a linear temperament eonugh you'll start
getting different pitches that nevertheless are more or less
indistinguishable from each other. Even before that you'll get
approximations that are better than those the linear temperament is
supposed to give.

So what would be a good place to close the circle and go from linear
to equal?

For TOP tempered linear temperaments I suggest closing the circle
when you start getting better approximations to the primes for which
the tuning is optimized.

What are your thoughts about this?

Kalle

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

> So what would be a good place to close the circle and go from linear
> to equal?
>
> For TOP tempered linear temperaments I suggest closing the circle
> when you start getting better approximations to the primes for which
> the tuning is optimized.
>
> What are your thoughts about this?

For pure octave tunings, a system I sometimes use is to close at a
"poptimal" generator. A generator is "poptimal" for a certain set of
octave-eqivalent consonances if there is some exponent p, 2 <= p <=
infinity, such that the sum of the pth powers of the absolute value of
the errors over the set of consonances is minimal. This is convenient
for Scala score files, since the notes are now represented by
(reasonably small) integers. I also sometimes use it when cooking up a
Scala scl file (just did, in fact, over on the tuning list) though in
that case it makes little difference.

If you follow this system, 5-limit meantone closes for 81, 7-limit
meantone for 31, and 11-limit meantone for 31. 5 and 7 taken together
are 1/4-comma exactly, which doesn't close; 5 and 11 taken together
closes at 112, and 7 and 11 of course also at 31. One rarely
encounters problems; even a microtemperament like ennealimmal closes
at 1053, which is perfectly reasonable for Scala applications; one
does, however, need to ensure the division is divisible by 9.

A different naming convention than using TOP tuning would be to give
the same name iff the poptimal ranges intersect. This isn't very
convenient in practice, due to the difficulty of computing the
poptimal range, but clearly it leads to quite different results.
Miracle, for instance, has the same TOP tuning in the 5, 7 and 11
limits, but while the 5 and 7 limit poptimal ranges intersect, the 5
and 11 or 7 and 11 ranges apparently do not, though as I say computing
these is a pain, so I may have the range too small. In any case,
miracle closes at 175 in the 5 and 7 limits, and at 401 in the 11-limit.

Kalle Aho wrote:

> For TOP tempered linear temperaments I suggest closing the circle > when you start getting better approximations to the primes for which > the tuning is optimized. > > What are your thoughts about this?

That's pretty much my opinion, although you can go some ways beyond that point if you try to avoid the better approximations, and there may be other reasons to stick with the temperaments in particular cases. TOP father (g = 447.3863410, p = 1185.869125) has a better approximation of 5 after only 7 iterations of the generator, so you might want to switch to 8-ET. This works out fine for the 5-limit, since TOP 5-limit 8-ET <1185.032536, 1925.677871, 2814.452272] is pretty close to TOP 5-limit father <1185.869125, 1924.351908, 2819.124590]. But 8-ET isn't 7-limit consistent, so if you're using 7-limit father temperament <1185.869125, 1924.351908, 2819.124589, 3401.317477], you're probably better off sticking with the 8-note father MOS rather than going to one of the versions of TOP 8-ET.

--- In tuning-math@yahoogroups.com, "Kalle Aho" <kalleaho@m...> wrote:
> Hi,
>
> Linear temperaments (or 2-dimensional tunings) are infinitely
> extendable. Once you extend a linear temperament eonugh you'll
start
> getting different pitches that nevertheless are more or less
> indistinguishable from each other. Even before that you'll get
> approximations that are better than those the linear temperament is
> supposed to give.
>
> So what would be a good place to close the circle and go from
linear
> to equal?
>
> For TOP tempered linear temperaments I suggest closing the circle
> when you start getting better approximations to the primes for
which
> the tuning is optimized.

Not a bad idea. I don't think any of my horagrams go further than
this, although 5:4 is slightly better in TOP Catler, and maybe
there's another similar example somewhere.

You'd have to make your criterion a little more precise -- are you
assuming that the scales grow in one direction, or in both
directions, as you apply the generator more and more times?

9-limit should also be considered when you're going "poptimal".

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, "Kalle Aho" <kalleaho@m...>
wrote:
>
> > So what would be a good place to close the circle and go from
linear
> > to equal?
> >
> > For TOP tempered linear temperaments I suggest closing the circle
> > when you start getting better approximations to the primes for
which
> > the tuning is optimized.
> >
> > What are your thoughts about this?
>
> For pure octave tunings, a system I sometimes use is to close at a
> "poptimal" generator. A generator is "poptimal" for a certain set of
> octave-eqivalent consonances if there is some exponent p, 2 <= p <=
> infinity, such that the sum of the pth powers of the absolute value
of
> the errors over the set of consonances is minimal. This is
convenient
> for Scala score files, since the notes are now represented by
> (reasonably small) integers. I also sometimes use it when cooking
up a
> Scala scl file (just did, in fact, over on the tuning list) though
in
> that case it makes little difference.
>
> If you follow this system, 5-limit meantone closes for 81, 7-limit
> meantone for 31, and 11-limit meantone for 31. 5 and 7 taken
together
> are 1/4-comma exactly, which doesn't close; 5 and 11 taken together
> closes at 112, and 7 and 11 of course also at 31. One rarely
> encounters problems; even a microtemperament like ennealimmal closes
> at 1053, which is perfectly reasonable for Scala applications; one
> does, however, need to ensure the division is divisible by 9.
>
> A different naming convention than using TOP tuning would be to give
> the same name iff the poptimal ranges intersect. This isn't very
> convenient in practice, due to the difficulty of computing the
> poptimal range, but clearly it leads to quite different results.
> Miracle, for instance, has the same TOP tuning in the 5, 7 and 11
> limits, but while the 5 and 7 limit poptimal ranges intersect, the 5
> and 11 or 7 and 11 ranges apparently do not, though as I say
computing
> these is a pain, so I may have the range too small. In any case,
> miracle closes at 175 in the 5 and 7 limits, and at 401 in the 11-
limit.

>Linear temperaments (or 2-dimensional tunings) are infinitely
>extendable. Once you extend a linear temperament eonugh you'll start
>getting different pitches that nevertheless are more or less
>indistinguishable from each other. Even before that you'll get
>approximations that are better than those the linear temperament is
>supposed to give.
>
>So what would be a good place to close the circle and go from linear
>to equal?
>
>For TOP tempered linear temperaments I suggest closing the circle
>when you start getting better approximations to the primes for which
>the tuning is optimized.
>
>What are your thoughts about this?

Hi Kalle,

I wouldn't indicate such a hard-and-fast rule. If you reach those
notes (the better approximations) by modulating in a piece of music,
I'd say use them. If not, don't. Of course you're not allowed to
use them as direct approximations and still call it the same regular
temperament you started with. Maybe Gene will correct me but I
think changing the map in this fashion means you're using a different
temperament. There's nothing wrong with that of course -- or one
could remain faithful to the original map and keep the fine
distinctions of the extended progression -- or one could equalize.
All seem valid.

-Carl

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

> 9-limit should also be considered when you're going "poptimal".

True enough. Alas, even though we have the same wedgie, commas and
tuning map, the poptimal range need not even overlap. Orwell is a
typical example--there seems to be no overlap from 7 to 9, and none
between 11 and 9, but the others are OK. So, 5 and 9 overlap, and have
43/190 as a common generator, but 7 and 9, no.

>> 9-limit should also be considered when you're going "poptimal".
>
>True enough. Alas, even though we have the same wedgie, commas and
>tuning map, the poptimal range need not even overlap. Orwell is a
>typical example--there seems to be no overlap from 7 to 9, and none
>between 11 and 9, but the others are OK. So, 5 and 9 overlap, and
>have 43/190 as a common generator, but 7 and 9, no.

This is AWESOME. Seriously, if you had come to me in a past
life and asked me to imagine the most heinously interesting
thing ever, for torturing curious folks in purgatory or
something, I wouldn't have come up with the half of this
temperaments thing.

How annoying, that there doesn't seem to be any really good way
to famlify the temperaments.

By the way, Gene, how does poptimal relate to TOP? If the
commas dictate the TOP tuning, is there necessarily a generator/
period pair that give it?

And if there is, is it not specified exactly by the TOP tuning,
for a given map?

Maybe what I'm asking is, "could you walk me through the
functions you'd call to find a generator/period pair for a
7-limit linear temperament?". I can look in your code.
Though I guess what I have doesn't cover TOP...

-Carl

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> >> 9-limit should also be considered when you're going "poptimal".
> >
> >True enough. Alas, even though we have the same wedgie, commas and
> >tuning map, the poptimal range need not even overlap. Orwell is a
> >typical example--there seems to be no overlap from 7 to 9, and none
> >between 11 and 9, but the others are OK. So, 5 and 9 overlap, and
> >have 43/190 as a common generator, but 7 and 9, no.
>
> This is AWESOME. Seriously, if you had come to me in a past
> life and asked me to imagine the most heinously interesting
> thing ever, for torturing curious folks in purgatory or
> something, I wouldn't have come up with the half of this
> temperaments thing.

Har. You think that is bad, try this: two different 11-limit linear
temperaments are the meantone variants meantone or meanpop (sharing
the same TOP tuning with the 7-limit temperament) and huygens (sharing
the same NOT tuning with the 7-limit temperament.) The poptimal ranges
for these two theoretically distinct temperaments, one of which adds
385 to the 7-limit comma set {81/80, 126/125} and the other of which
adds 99/98, actually overlap. Unsurprisingly, 31 is poptimal for both,
and 31 is what you get by adding *both* 99.98 and 385/384 to 7-limit
meantone. However, 31 is not the only poptimal possibility; the
allegedly universal 198 will work also. This, of course, uses two
different possible versions of 11 for huygens and meantone, with the
huygens version being more accurate, since 198 has a meantone fifth
very slightly (0.1955 cents) sharper than 31, where the two
temperaments are identical. If we used 267 equal instead, also
poptimal for both temperaments, the 11 would be more accurate in the
meantone version rather than the huygens version.

What it really boils down to is that for the 11-limit we should just
give it up, and use 31 equal. That way we've got both 99/98 and
385/384 to play with, commas also happy with 11-limit orwell, by the way.

> By the way, Gene, how does poptimal relate to TOP?

Not very well, apparently.

If the
> commas dictate the TOP tuning, is there necessarily a generator/
> period pair that give it?

You've lost me.

> Maybe what I'm asking is, "could you walk me through the
> functions you'd call to find a generator/period pair for a
> 7-limit linear temperament?". I can look in your code.
> Though I guess what I have doesn't cover TOP...

Do you mean in terms of cents, or a generator period pair in terms of
p-limit intervals which temper to the generator and period whatever
tuning you use? In terms of cents, the easy ones to find are TOP, NOT,
rms, and minimax, but each of these is different; rms involves least
squares, and the rest I set up as a linear programming problem and
solve using Maple's simplex method implementation.

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

> For pure octave tunings, a system I sometimes use is to close at a
> "poptimal" generator. A generator is "poptimal" for a certain set of
> octave-eqivalent consonances if there is some exponent p, 2 <= p <=
> infinity, such that the sum of the pth powers of the absolute value
of
> the errors over the set of consonances is minimal.

This is quite an interesting approach. What makes poptimal generators
good? And why can't p be 1?

> This is convenient
> for Scala score files, since the notes are now represented by
> (reasonably small) integers. I also sometimes use it when cooking
up a
> Scala scl file (just did, in fact, over on the tuning list) though
in
> that case it makes little difference.
>
> If you follow this system, 5-limit meantone closes for 81, 7-limit
> meantone for 31, and 11-limit meantone for 31. 5 and 7 taken
together
> are 1/4-comma exactly, which doesn't close; 5 and 11 taken together
> closes at 112, and 7 and 11 of course also at 31. One rarely
> encounters problems; even a microtemperament like ennealimmal closes
> at 1053, which is perfectly reasonable for Scala applications; one
> does, however, need to ensure the division is divisible by 9.

These results are interesting. Do these poptimal generators make
these linear temperaments close exactly at these ETs?!

> A different naming convention than using TOP tuning would be to give
> the same name iff the poptimal ranges intersect. This isn't very
> convenient in practice, due to the difficulty of computing the
> poptimal range, but clearly it leads to quite different results.
> Miracle, for instance, has the same TOP tuning in the 5, 7 and 11
> limits, but while the 5 and 7 limit poptimal ranges intersect, the 5
> and 11 or 7 and 11 ranges apparently do not, though as I say
computing
> these is a pain, so I may have the range too small. In any case,
> miracle closes at 175 in the 5 and 7 limits, and at 401 in the 11-
limit.

--- In tuning-math@yahoogroups.com, Herman Miller <hmiller@I...>
wrote:
> Kalle Aho wrote:
>
> > For TOP tempered linear temperaments I suggest closing the circle
> > when you start getting better approximations to the primes for
which
> > the tuning is optimized.
> >
> > What are your thoughts about this?
>
> That's pretty much my opinion, although you can go some ways beyond
that
> point if you try to avoid the better approximations, and there may
be
> other reasons to stick with the temperaments in particular cases.
TOP
> father (g = 447.3863410, p = 1185.869125) has a better
approximation of
> 5 after only 7 iterations of the generator, so you might want to
switch
> to 8-ET. This works out fine for the 5-limit, since TOP 5-limit 8-
ET
> <1185.032536, 1925.677871, 2814.452272] is pretty close to TOP 5-
limit
> father <1185.869125, 1924.351908, 2819.124590]. But 8-ET isn't 7-
limit
> consistent, so if you're using 7-limit father temperament
<1185.869125,
> 1924.351908, 2819.124589, 3401.317477], you're probably better off
> sticking with the 8-note father MOS rather than going to one of the
> versions of TOP 8-ET.

Wasn't consistency supposed to be a nonissue with TOP tunings? But I
understand why you say these things.

--- In tuning-math@yahoogroups.com, "Kalle Aho" <kalleaho@m...> wrote:
> --- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
> wrote:
>
> > For pure octave tunings, a system I sometimes use is to close at a
> > "poptimal" generator. A generator is "poptimal" for a certain set of
> > octave-eqivalent consonances if there is some exponent p, 2 <= p <=
> > infinity, such that the sum of the pth powers of the absolute value
> of
> > the errors over the set of consonances is minimal.
>
> This is quite an interesting approach. What makes poptimal generators
> good? And why can't p be 1?

It could be 1. In fact, Paul thinks it should be 1. My feeling is that
we accept 2 and infinity, so anything in between is OK, but I'm not
confident with 1. What makes them good is that they approximate a
given list of target consonances in an optimal way, for some sense of
optimal.

> These results are interesting. Do these poptimal generators make
> these linear temperaments close exactly at these ETs?!

Right. While I don't try to compute what exponent p they close for, I
can prove it must exist.

>> By the way, Gene, how does poptimal relate to TOP?
>
>Not very well, apparently.
>
> If the
>> commas dictate the TOP tuning, is there necessarily a
>> generator/period pair that give it?
>
>You've lost me.

I meant, for a given TOP-tuned linear temperament, does it not
stand to reason that there is at least one generator/period
pair (in cents) that produces scales in said tuning?

>> Maybe what I'm asking is, "could you walk me through the
>> functions you'd call to find a generator/period pair for a
>> 7-limit linear temperament?". I can look in your code.
>> Though I guess what I have doesn't cover TOP...
>
>Do you mean in terms of cents,

Yes.

>or a generator period pair in terms of
>p-limit intervals which temper to the generator and period whatever
>tuning you use?

You lost me here; sounds interesting but I don't think I meant this.

>In terms of cents, the easy ones to find are TOP, NOT,
>rms, and minimax, but each of these is different; rms involves least
>squares, and the rest I set up as a linear programming problem and
>solve using Maple's simplex method implementation.

Ok. I know roughly what this means. But it'd still be nice to
know what data you feed into which processes. You need a map at
some point, I'd think, so you can specify whether to find, say,
fifths or fourths for meantone... I'm trying to build a picture
of what kinds of things I need to know to get what kinds of
answers out.

-Carl

--- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
wrote:
> --- In tuning-math@yahoogroups.com, "Kalle Aho" <kalleaho@m...>
wrote:
> > Hi,
> >
> > Linear temperaments (or 2-dimensional tunings) are infinitely
> > extendable. Once you extend a linear temperament eonugh you'll
> start
> > getting different pitches that nevertheless are more or less
> > indistinguishable from each other. Even before that you'll get
> > approximations that are better than those the linear temperament
is
> > supposed to give.
> >
> > So what would be a good place to close the circle and go from
> linear
> > to equal?
> >
> > For TOP tempered linear temperaments I suggest closing the circle
> > when you start getting better approximations to the primes for
> which
> > the tuning is optimized.
>
> Not a bad idea. I don't think any of my horagrams go further than
> this, although 5:4 is slightly better in TOP Catler, and maybe
> there's another similar example somewhere.
>
> You'd have to make your criterion a little more precise -- are you
> assuming that the scales grow in one direction, or in both
> directions, as you apply the generator more and more times?

I would look at all intervals of the chain so it doesn't matter which
way they grow. As soon as I get a better approximation to any one of
the primes I would choose the nearest (in terms of cardinality) equal
temperament that both supports the linear temperament and is better
than the ones preceding it.

Kalle

>> For pure octave tunings, a system I sometimes use is to close at a
>> "poptimal" generator. A generator is "poptimal" for a certain set of
>> octave-eqivalent consonances if there is some exponent p, 2 <= p <=
>> infinity, such that the sum of the pth powers of the absolute value
>> of the errors over the set of consonances is minimal.

I guess I never understood how poptimal is different than 'all
the error functions ever advocated here'.

>> A different naming convention than using TOP tuning would be to give
>> the same name iff the poptimal ranges intersect. This isn't very
>> convenient in practice, due to the difficulty of computing the
>> poptimal range, but clearly it leads to quite different results.
>> Miracle, for instance, has the same TOP tuning in the 5, 7 and 11
>> limits, but while the 5 and 7 limit poptimal ranges intersect,
>> the 5 and 11 or 7 and 11 ranges apparently do not, though as I say
>> computing these is a pain, so I may have the range too small. In
>> any case, miracle closes at 175 in the 5 and 7 limits, and at 401
>> in the 11-limit.

Isn't TOP a minimax (p=inf.) method? Oh, but in the definition above
you restrict poptimal to octaves. . . .

-Carl

Carl wrote:

> Hi Kalle,
>
> I wouldn't indicate such a hard-and-fast rule. If you reach those
> notes (the better approximations) by modulating in a piece of music,
> I'd say use them. If not, don't. Of course you're not allowed to
> use them as direct approximations and still call it the same regular
> temperament you started with. Maybe Gene will correct me but I
> think changing the map in this fashion means you're using a
different
> temperament. There's nothing wrong with that of course -- or one
> could remain faithful to the original map and keep the fine
> distinctions of the extended progression -- or one could equalize.
> All seem valid.

Right and true. But I think it would be nice to have some systematic
choice(s) when one wants to close the circle. For me the existence of
better approximations is somewhat disturbing. Note that we don't have
this "problem" in JI because every interval you reach will be worse
than the generating primes.

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

> Isn't TOP a minimax (p=inf.) method? Oh, but in the definition above
> you restrict poptimal to octaves. . . .

TOP is a weighted method, not restricted, which can be regarded as
minimax applied to just the primes.

>> Isn't TOP a minimax (p=inf.) method? Oh, but in the definition above
>> you restrict poptimal to octaves. . . .
>
>TOP is a weighted method, not restricted, which can be regarded as
>minimax applied to just the primes.

Oh, and why should I believe that p=inf gives minimax again?

-Carl

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> >> Isn't TOP a minimax (p=inf.) method? Oh, but in the definition above
> >> you restrict poptimal to octaves. . . .
> >
> >TOP is a weighted method, not restricted, which can be regarded as
> >minimax applied to just the primes.
>
> Oh, and why should I believe that p=inf gives minimax again?

If you have a>b, then a^p+b^p is dominated by a as p goes to infinity,
since (b/a)^p --> 0. Hence (|a|^p+|b|^p)^(1/p) --> max(|a|, |b|) as
p --> infinity. (If a and b are of the same size, the
doubling makes no difference, since 2^(1/p)-->1)

>> >TOP is a weighted method, not restricted, which can be regarded as
>> >minimax applied to just the primes.
>>
>> Oh, and why should I believe that p=inf gives minimax again?
>
>If you have a>b, then a^p+b^p is dominated by a as p goes to infinity,
>since (b/a)^p --> 0. Hence (|a|^p+|b|^p)^(1/p) --> max(|a|, |b|) as
>p --> infinity. (If a and b are of the same size, the
>doubling makes no difference, since 2^(1/p)-->1)

Of course. Thanks.

-Carl

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> > >> 9-limit should also be considered when you're going "poptimal".
> > >
> > >True enough. Alas, even though we have the same wedgie, commas
and
> > >tuning map, the poptimal range need not even overlap. Orwell is a
> > >typical example--there seems to be no overlap from 7 to 9, and
none
> > >between 11 and 9, but the others are OK. So, 5 and 9 overlap, and
> > >have 43/190 as a common generator, but 7 and 9, no.
> >
> > This is AWESOME. Seriously, if you had come to me in a past
> > life and asked me to imagine the most heinously interesting
> > thing ever, for torturing curious folks in purgatory or
> > something, I wouldn't have come up with the half of this
> > temperaments thing.
>
> Har. You think that is bad, try this: two different 11-limit linear
> temperaments are the meantone variants meantone or meanpop (sharing
> the same TOP tuning with the 7-limit temperament) and huygens
(sharing
> the same NOT tuning with the 7-limit temperament.)

Isn't that a ridiculous name for an 11-limit temperament?

--- In tuning-math@yahoogroups.com, "Kalle Aho" <kalleaho@m...> wrote:
> --- In tuning-math@yahoogroups.com, "Gene Ward Smith"
<gwsmith@s...>
> wrote:
>
> > For pure octave tunings, a system I sometimes use is to close at a
> > "poptimal" generator. A generator is "poptimal" for a certain set
of
> > octave-eqivalent consonances if there is some exponent p, 2 <= p
<=
> > infinity, such that the sum of the pth powers of the absolute
value
> of
> > the errors over the set of consonances is minimal.
>
> This is quite an interesting approach. What makes poptimal
>generators
> good?

Not much, IMHO -- the "true" value of p in any situation will be some
number, not an infinite range of numbers.

> And why can't p be 1?

My graphs show p going even slightly below 1, and I think this is
more than appropriate when you look at the kinds of discordance
curves Bill Sethares predicts and George Secor prefers. Very sharp
spikes at the simple ratios.

> These results are interesting. Do these poptimal generators make
> these linear temperaments close exactly at these ETs?!

"Poptimal" doesn't imply uniqueness the way "optimal" does. Any
generator within a certain finite range will be poptimal for a given
situation. So you have to "feed in" ET generators at the beginning if
you want the circle(s) to close.

--- 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:
> >
> > > For pure octave tunings, a system I sometimes use is to close
at a
> > > "poptimal" generator. A generator is "poptimal" for a certain
set of
> > > octave-eqivalent consonances if there is some exponent p, 2 <=
p <=
> > > infinity, such that the sum of the pth powers of the absolute
value
> > of
> > > the errors over the set of consonances is minimal.
> >
> > This is quite an interesting approach. What makes poptimal
generators
> > good? And why can't p be 1?
>
> It could be 1. In fact, Paul thinks it should be 1.

No I don't -- TOP actually uses p=inf, in a sense -- I just think 1
should be included in the range if infinity is.

> What makes them good is that they approximate a
> given list of target consonances in an optimal way, for some sense
of
> optimal.

It's not a sense which gives a unique answer, which seems to have
tripped everyone up so far. All your logic is right there in your
math, but you have to understand that this is essentially a foreign
language for most of us.

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> >> By the way, Gene, how does poptimal relate to TOP?
> >
> >Not very well, apparently.
> >
> > If the
> >> commas dictate the TOP tuning, is there necessarily a
> >> generator/period pair that give it?
> >
> >You've lost me.
>
> I meant, for a given TOP-tuned linear temperament, does it not
> stand to reason that there is at least one generator/period
> pair (in cents) that produces scales in said tuning?

Yes, and there's exactly one if you fix that the octave is multiple
of the period, and the generator is within a prescribed 1/2-period
range (say, between 0 or 1/2 period, or between 1/2 and 1 period).

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

> > Har. You think that is bad, try this: two different 11-limit linear
> > temperaments are the meantone variants meantone or meanpop (sharing
> > the same TOP tuning with the 7-limit temperament) and huygens
> (sharing
> > the same NOT tuning with the 7-limit temperament.)
>
> Isn't that a ridiculous name for an 11-limit temperament?

You'd maybe prefer Fokker?

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

> > This is quite an interesting approach. What makes poptimal
> >generators
> > good?
>
> Not much, IMHO -- the "true" value of p in any situation will be some
> number, not an infinite range of numbers.

What in the world does this mean? What allegedly "true" value?

> > And why can't p be 1?
>
> My graphs show p going even slightly below 1, and I think this is
> more than appropriate when you look at the kinds of discordance
> curves Bill Sethares predicts and George Secor prefers. Very sharp
> spikes at the simple ratios.

If you go below 1 your don't even get a corresponding metric, but you
can go as far as 1 and have a metric. Let's at least keep the triangle
inequality, please.

As for 1, I think a lot of people would find the supposedly optimal
tunings not really very optimal in some cases.

> > These results are interesting. Do these poptimal generators make
> > these linear temperaments close exactly at these ETs?!
>
> "Poptimal" doesn't imply uniqueness the way "optimal" does. Any
> generator within a certain finite range will be poptimal for a given
> situation. So you have to "feed in" ET generators at the beginning if
> you want the circle(s) to close.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
wrote:
>
> > > Har. You think that is bad, try this: two different 11-limit
linear
> > > temperaments are the meantone variants meantone or meanpop
(sharing
> > > the same TOP tuning with the 7-limit temperament) and huygens
> > (sharing
> > > the same NOT tuning with the 7-limit temperament.)
> >
> > Isn't that a ridiculous name for an 11-limit temperament?
>
> You'd maybe prefer Fokker?

Did Fokker have a particular route along the circle of fifths that he
preferred to get 11?

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
wrote:
>
> > > This is quite an interesting approach. What makes poptimal
> > >generators
> > > good?
> >
> > Not much, IMHO -- the "true" value of p in any situation will be
some
> > number, not an infinite range of numbers.
>
> What in the world does this mean? What allegedly "true" value?

If you're using this p-norm model in the first place, it's probably
because you think it's true for some value of p. If you run over all
possible 'p's, you're violating the assumptions of any such model.

> > > And why can't p be 1?
> >
> > My graphs show p going even slightly below 1, and I think this is
> > more than appropriate when you look at the kinds of discordance
> > curves Bill Sethares predicts and George Secor prefers. Very
sharp
> > spikes at the simple ratios.
>
> If you go below 1 your don't even get a corresponding metric, but
you
> can go as far as 1 and have a metric. Let's at least keep the
triangle
> inequality, please.

The behavior below 1 reflects the meaningful result that temperaments
do not improve on JI tunings there. It's helpful to think of the
bigger picture.

> As for 1, I think a lot of people would find the supposedly optimal
> tunings not really very optimal in some cases.

And yet there is a significant bunch of composers who refuse to
temper, clinging to their JI scales. Might they be modelled too? (no
offense to them, of course.)

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

> > You'd maybe prefer Fokker?
>
> Did Fokker have a particular route along the circle of fifths that he
> preferred to get 11?

I doubt it. These two temperaments should both probably be melted down
into 31 equal, however, which of course makes them the same; hence
huygens or fokker might be good names.

--- 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:
> >
> > > > This is quite an interesting approach. What makes poptimal
> > > >generators
> > > > good?
> > >
> > > Not much, IMHO -- the "true" value of p in any situation will be
> some
> > > number, not an infinite range of numbers.
> >
> > What in the world does this mean? What allegedly "true" value?
>
> If you're using this p-norm model in the first place, it's probably
> because you think it's true for some value of p.

I have no idea how a norm can possibly be either true or false. My
assumption is that it is a valid definition of optimum for any p in
the range 2 to infinity, and that is simply because if you assume the
endpoints define a valid sense of optimum, so should all the
intermediate values.

> > As for 1, I think a lot of people would find the supposedly optimal
> > tunings not really very optimal in some cases.
>
> And yet there is a significant bunch of composers who refuse to
> temper, clinging to their JI scales. Might they be modelled too? (no
> offense to them, of course.)

If you refuse to temper at all, what in the world are you doing trying
to decide which tuning is best, on the assumption that a given
temperament will be used?

>> I meant, for a given TOP-tuned linear temperament, does it not
>> stand to reason that there is at least one generator/period
>> pair (in cents) that produces scales in said tuning?
>
>Yes, and there's exactly one if you fix that the octave is multiple
>of the period, and the generator is within a prescribed 1/2-period
>range (say, between 0 or 1/2 period, or between 1/2 and 1 period).

Great, thanks. That's as I thought, then.

-Carl

hi Gene and Paul,

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

> --- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...> wrote:
>
> > > You'd maybe prefer Fokker?
> >
> > Did Fokker have a particular route along the circle of fifths that he
> > preferred to get 11?
>
> I doubt it. These two temperaments should both probably be melted down
> into 31 equal, however, which of course makes them the same; hence
> huygens or fokker might be good names.

huygens or fokker are indeed the two most appropriate names
for 31edo.

but if your main criteria in naming is to honor someone
who advocated 11-limit, a good choice might be Ptolemy.
his "smooth (or 'equable' as quoted by Partch) diatonic"
and "syntonic chromatic" genera both used ratios of 11.

here are the tetrachord structures of those two tunings:

Ptolemy - _genos homalon diatonon_ = "even diatonic genus"
http://tonalsoft.com/enc/diatonic.htm#equable

>> string-length proportions: 9 : 10 : 11 : 12
>>
>>
>> note ...... ratio ... ~ cents
>>
>> mese ...... 1/1 ....... 0
>> .................................... > . 9:10 . ~ 182.4037121 cents
>> lichanos .. 9/10 .. - 182.4037121
>> .................................... >. 10:11 . ~ 165.0042285 cents
>> parhypate . 9/11 .. - 347.4079406
>> .................................... >. 11:12 . ~ 150.6370585 cents
>> hypate .... 3/4 ... - 498.0449991
>>
>> The string-length proportions of Ptolemy's "even diatonic"
>> have the smallest-number consecutive ratios which can describe
>> a four-fold division of the 4:3 "perfect-4th". The top interval
>> is thus the 5-limit 10:9 "lesser tone", the middle interval is
>> the 11:10 "undecimal tone", and the bottom interval is the
>> 12:11 "neutral 2nd" functioning as a very wide semitone.

Ptolemy - _genos syntonon chromatikon_ = "tense chromatic genus"
http://tonalsoft.com/enc/chromatic.htm#ptolemy-tense

>> string-length proportions: 66 : 77 : 84 : 88
>>
>>
>> note ...... ratio ... ~ cents
>>
>> mese ....... 1/1 ...... 0
>> .................................... >. 6:7 .. ~ 266.8709056 cents
>> lichanos ... 6/7 .. - 266.8709056
>> .................................... > 11:12 . ~ 150.6370585 cents
>> parhypate . 11/14 . - 417.5079641
>> .................................... > 21:22 . ~ 80.53703503 cents
>> hypate ..... 3/4 .. - 498.0449991
>>
>> In his "tense" version of the chromatic genus, Ptolemy used
>> the narrow 7:6 "septimal minor-3rd" as his "characteristic
>> interval" at the top of the tetrachord, thus simultaneously
>> placing his lichanos a wide 8:7 "septimal tone" above the
>> bottom note hypate. The middle interval is the 12:11
>>"undecimal neutral 2nd", functioning here as a very wide semitone.

in a later era, Partch might be the obvious choice
... but then again, he stressed JI so much that it
would never be a good idea to honor him by naming a
temperament after him. he was philosophically opposed
to temperament, you could almost say on moral grounds.
Ben Johnston speaks of temperament in much the same way.
and of course, here on the list Jon Szanto, Kraig Grady,
and David Beardsley and Pat Pagano would hold similar views.

-monz

--- 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:
> > >
> > > > > This is quite an interesting approach. What makes poptimal
> > > > >generators
> > > > > good?
> > > >
> > > > Not much, IMHO -- the "true" value of p in any situation will
be
> > some
> > > > number, not an infinite range of numbers.
> > >
> > > What in the world does this mean? What allegedly "true" value?
> >
> > If you're using this p-norm model in the first place, it's
probably
> > because you think it's true for some value of p.
>
> I have no idea how a norm can possibly be either true or false. My
> assumption is that it is a valid definition of optimum for any p in
> the range 2 to infinity, and that is simply because if you assume
the
> endpoints define a valid sense of optimum, so should all the
> intermediate values.

OK. And a lot of other values may define a valid sense of optimum, as
well, for example weightings, etc.

> > > As for 1, I think a lot of people would find the supposedly
optimal
> > > tunings not really very optimal in some cases.
> >
> > And yet there is a significant bunch of composers who refuse to
> > temper, clinging to their JI scales. Might they be modelled too?
(no
> > offense to them, of course.)
>
> If you refuse to temper at all, what in the world are you doing
trying
> to decide which tuning is best, on the assumption that a given
> temperament will be used?

Exactly. This corresponds to the p<1 area, or at least to some of it.

--- In tuning-math@yahoogroups.com, "monz" <monz@a...> wrote:
> hi Gene and Paul,
>
>
> --- In tuning-math@yahoogroups.com, "Gene Ward Smith"
<gwsmith@s...>
> wrote:
>
> > --- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
wrote:
> >
> > > > You'd maybe prefer Fokker?
> > >
> > > Did Fokker have a particular route along the circle of fifths
that he
> > > preferred to get 11?
> >
> > I doubt it. These two temperaments should both probably be melted
down
> > into 31 equal, however, which of course makes them the same; hence
> > huygens or fokker might be good names.
>
>
> huygens or fokker are indeed the two most appropriate names
> for 31edo.
>
> but if your main criteria in naming is to honor someone
> who advocated 11-limit,

It would have to be a particular mapping of the 11-limit along a
particular chain of fifths that would, for example, correspond to a
particular path in the 31-equal circle of fifths. Huygens never went
beyond 7-limit, and Fokker didn't prescribe one method of generating
(by fifths) 31-equal's approximation of 11 over another.

hi Paul (and Gene),

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

> --- In tuning-math@yahoogroups.com, "monz" <monz@a...> wrote:
> > hi Gene and Paul,
> >
> >
> > huygens or fokker are indeed the two most appropriate names
> > for 31edo.
> >
> > but if your main criteria in naming is to honor someone
> > who advocated 11-limit,
>
> It would have to be a particular mapping of the 11-limit
> along a particular chain of fifths that would, for example,
> correspond to a particular path in the 31-equal circle of
> fifths. Huygens never went beyond 7-limit, and Fokker didn't
> prescribe one method of generating (by fifths) 31-equal's
> approximation of 11 over another.

well, of course neither Ptolemy nor Partch advocated
equal-temperament ... but if it's any help, these are
three xenharmonic-bridges that i've posited for Ptolemy
(in my book):

monzo ratio ~cents
3 5 7 11 23

[-4, 0, 1,-1, 0 > 896 / 891 9.687960643
[-6, 1,-1, 0, 0 > 5120 / 5103 5.757802203
[ 2,-1, 2,-1, 0 > 441 / 440 3.930158439

in a sense, one could say that all the differences
between Ptolemy's genera for notes of the same name
(i.e., all the different _lichanoi_ etc.) are
unison-vectors, since the notes all do carry the
same name.

but within Ptolemy's diatonic genera alone,
the difference between two of the _parhypatai_
(the "even diatonic" and "tonic diatonic") is
as large as 81/77 = ~88 cents.

the entire range of _parhypatai_ and _tritai_ in
Ptolemy's genera is covered by the difference in size
between his "even diatonic" and his enharmonic, and is
3;5,7,11,23-monzo [3, 1, 0,-1,-1> ratio 270 / 253
= ~112.5864268 cents.

the difference between Ptolemy's _lichanoi_ and
_paranetai_ is even greater: the usual 9/8
= ~203.9100017 cents.

the three xenharmonic-bridges in my table above are
the ones that are smallest in size.

for more on Ptolemy's genera, see:

http://tonalsoft.com/enc/diatonic.htm#genus
http://www.tonalsoft.com/enc/chromatic.htm#genus
http://www.tonalsoft.com/enc/enharmonic.htm#genus

-monz

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

> well, of course neither Ptolemy nor Partch advocated
> equal-temperament ... but if it's any help, these are
> three xenharmonic-bridges that i've posited for Ptolemy
> (in my book):
>
> monzo ratio ~cents
> 3 5 7 11 23
>
> [-4, 0, 1,-1, 0 > 896 / 891 9.687960643
> [-6, 1,-1, 0, 0 > 5120 / 5103 5.757802203
> [ 2,-1, 2,-1, 0 > 441 / 440 3.930158439

those interested in pursuing this further will find
helpful a brand new webpage i just wrote. as i have
time, i'll add much more to this one.

http://tonalsoft.com/enc/ptolemy.htm

-monz

Hi Monz,

Why is it that you're always creating new webpages and ignoring
corrections to your old ones? This seems to be a pattern with you.

The latest correction you ignored:
/tuning/topicId_53712.html#53792

-Paul

--- In tuning-math@yahoogroups.com, "monz" <monz@a...> wrote:
> --- In tuning-math@yahoogroups.com, "monz" <monz@a...> wrote:
>
> > well, of course neither Ptolemy nor Partch advocated
> > equal-temperament ... but if it's any help, these are
> > three xenharmonic-bridges that i've posited for Ptolemy
> > (in my book):
> >
> > monzo ratio ~cents
> > 3 5 7 11 23
> >
> > [-4, 0, 1,-1, 0 > 896 / 891 9.687960643
> > [-6, 1,-1, 0, 0 > 5120 / 5103 5.757802203
> > [ 2,-1, 2,-1, 0 > 441 / 440 3.930158439
>
>
>
> those interested in pursuing this further will find
> helpful a brand new webpage i just wrote. as i have
> time, i'll add much more to this one.
>
> http://tonalsoft.com/enc/ptolemy.htm
>
>
>
> -monz

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

> well, of course neither Ptolemy nor Partch advocated
> equal-temperament ... but if it's any help, these are
> three xenharmonic-bridges that i've posited for Ptolemy
> (in my book):
>
> monzo ratio ~cents
> 3 5 7 11 23
>
> [-4, 0, 1,-1, 0 > 896 / 891 9.687960643
> [-6, 1,-1, 0, 0 > 5120 / 5103 5.757802203
> [ 2,-1, 2,-1, 0 > 441 / 440 3.930158439

If these three are related, so they define an 11-limit planar
temperament. If you add 3388/3375 to this, you get Graham's mystery,
with a 1/29 period; if you add 385/384, rodan; if 243/242 an 11-limit
hemififths; and if 100/99, an 11-limit version of garibaldi (schismic
family) which I have listed as "garybald".

If we add 121/120 we get the 11-limit reduction of what Herman
dubbed "leapday" in the 13 limit. The TOP tunings are not identical
but they are close, and I suggest giving them the same name, and
perhaps "ptolemy" could be that name. The 11 and 13 limit
temperaments also have a common poptimal generator of 19/46, which
again supports giving them the same name, and could be another naming
idea along the lines of 19/84 I suppose. The corresponding fifth is
27/46; 2.39 cents sharp. I figured a Hellenistic Greek might like a
fifth as a generator, if introduced to temperament; but more
importantly, this (and "garybald"), is a "brigable" temperament since
it has a 1 as the first wedgie element. The xenharmonic bridge to 5
is |31 -21 1>.

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

> The TOP tunings are not identical
> but they are close, and I suggest giving them the same name, and
> perhaps "ptolemy" could be that name.

Note that Monz is only listing a small percentage of the "ptolemy
commas" that he found. You guys are playing real fast and loose!
(Don't take that as a complaint.)

> is a "brigable" temperament since

Did you see my question about this definition of yours? You didn't
reply. (Deja vu, anyone?)

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

> Did you see my question about this definition of yours?

No.

You didn't
> reply. (Deja vu, anyone?)

See above.

hi Paul,

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

> Hi Monz,
>
> Why is it that you're always creating new webpages and ignoring
> corrections to your old ones? This seems to be a pattern with you.

believe me, i've been making absurd numbers of corrections.

also, i really try to put a webpage together on the spot if
i have a few hours available when the inspiration strikes.

this last one is a good example of that. i've done a lot
of research into Ptolemy's tuning treatise, and have still
presented only a half-baked version of my work on it in
my book. i got the idea to collect this data on a webpage
about Ptolemy, and as the years pass i'll be stuffing it
full of ridiculous amounts of tables, graphs, lattices,
and long rambling text connecting Ptolemy with the Sumerians
and modern San Diego new-age UFOlogists. :) for now,
those two simple little tables are but the seed from which
a large fruitful tree will grow.

> The latest correction you ignored:
> /tuning/topicId_53712.html#53792
>
> -Paul

i really am sorry about that. my computer was infected
with a virus and now i have extra work to do just to
keep webpages working properly and try to remember the
stuff i lost that wasn't backed up.

rest assured, soon enough i'll be asking you (and everyone else)
to please itemize every error and broken link that can
be found on the tonalsoft site. but i have a huge load
of work to do for about the next month.

-monz

hi Gene,

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

> --- In tuning-math@yahoogroups.com, "monz" <monz@a...> wrote:
>
> > well, of course neither Ptolemy nor Partch advocated
> > equal-temperament ... but if it's any help, these are
> > three xenharmonic-bridges that i've posited for Ptolemy
> > (in my book):
> >
> > monzo ratio ~cents
> > 3 5 7 11 23
> >
> > [-4, 0, 1,-1, 0 > 896 / 891 9.687960643
> > [-6, 1,-1, 0, 0 > 5120 / 5103 5.757802203
> > [ 2,-1, 2,-1, 0 > 441 / 440 3.930158439
>
> If these three are related, so they define an 11-limit
> planar temperament. If you add 3388/3375 to this, you get
> Graham's mystery, with a 1/29 period; if you add 385/384,
> rodan; if 243/242 an 11-limit hemififths; and if 100/99,
> an 11-limit version of garibaldi (schismic family) which
> I have listed as "garybald".
>
> If we add 121/120 we get the 11-limit reduction of what
> Herman dubbed "leapday" in the 13 limit. The TOP tunings
> are not identical but they are close, and I suggest giving
> them the same name, and perhaps "ptolemy" could be that name.
> The 11 and 13 limit temperaments also have a common poptimal
> generator of 19/46, which again supports giving them the
> same name, and could be another naming idea along the lines
> of 19/84 I suppose. The corresponding fifth is 27/46;
> 2.39 cents sharp. I figured a Hellenistic Greek might like
> a fifth as a generator, if introduced to temperament; but
> more importantly, this (and "garybald"), is a "brigable"
> temperament since it has a 1 as the first wedgie element.
> The xenharmonic bridge to 5 is |31 -21 1>.

awesome, Gene !!! thanks !!! this is exactly the kind of
thing i was hoping for when i put out my bait. :)

seriously, i'm having so much trouble following almost
everything you post here ... by relating some of your work
to some of my own (which is obviously nice and familiar
to me), it helps me at least get an idea where this train
is going.

-monz

hi Paul and Gene,

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

> --- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
> wrote:
>
> > The TOP tunings are not identical but they are close,
> > and I suggest giving them the same name, and perhaps
> > "ptolemy" could be that name.
>
> Note that Monz is only listing a small percentage of the
> "ptolemy commas" that he found.

*real* small percentage.

completing these tables is the first thing i'm going to
add to this page, so stay tuned. (he he)

> You guys are playing real fast and loose!
> (Don't take that as a complaint.)

i'm having fun at a party i probably shouldn't have
even crashed.

-monz

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
> wrote:
>
> > Did you see my question about this definition of yours?
>
> No.

/tuning/topicId_53869.html#54026