twintone, paultone (was something about schoenberg

> > > The 34-tET version of twintone? It's better than 12, but much
> worse
> > > than 22.
> >
> > Depends on what you are using the 7-limit stuff to do, I would
> think--it is sweeter so far as the 5-limit goes.
>
> Then we're talking diaschismic, not twintone. Or if they're the same
> thing, we need another word for "paultone".
>

More quick definitions needed : twintone, paultone and quick
comparison with diaschismic. Any additional commentary regarding
34 welcome too.

thanks (and thanks Paul for the quick tetrachord answer, in my MOS
musings I seem to have been finding the phenomena without really
pinning a name on it).

Bob

In-Reply-To: <200201231200.OAA67852@ius578.iil.intel.com>
Robert C Valentine wrote:

> More quick definitions needed : twintone, paultone and quick
> comparison with diaschismic. Any additional commentary regarding
> 34 welcome too.

I'm trying to keep track of these at
<http://x31eq.com/catalog.htm>. I do need some updates,
especially for this Pelog thing.

Diaschismic is any temperament with a period of a half octave, and a
mapping of [1, -2]. That is, you have a semitone generator. A perfect
fifth is a tritone plus a semitone. A major third is a tritone minus two
semitones.

Paultone, which now seems to be being called twintone, is a particular
7-limit diaschismic consistent with 22-equal. The mapping is [1, -2, -2]
so a 7:4 is an octave minus two semitones.

34-equal is an accurate, 5-limit diaschismic. But it isn't consistent in
the 7-limit. There are two 7-limit mappings which converge at 34, and
give good results for different tunings. Neither of them is paultone.

The typical diaschismic mapping, for me at least, is that consistent with
46- and 58-equal. That's [1, -2, -8] and it's accurate to about 6 cents.

A script for finding the linear temperament consistent with (the prime
mappings of) a pair of equal temperaments can be found at
<http://x31eq.com/temper/>

Graham

>I'm trying to keep track of these at
><http://x31eq.com/catalog.htm>.

I'd seen this before, but only now taken
the time to read it thoroughly. All I
can say is, "Thank you, thank you, thank
you, thank you, thank you, Graham!".

-Carl

--- In tuning-math@y..., graham@m... wrote:

> 34-equal is an accurate, 5-limit diaschismic. But it isn't consistent in
> the 7-limit. There are two 7-limit mappings which converge at 34, and
> give good results for different tunings. Neither of them is paultone.

This is getting ridiculous.

If h22 is the map [22, 35, 51, 62] and g34 the map [34, 54l 79, 96]
then:

Proof 1: h22 ^ g34 = [-2, 4, 4, -2, -12, 11] = h10 ^ h12 = twintone
wedgie.

Proof 2: h22(50/49) = g34(50/49) = 0; h22(64/63) = g34(64/63) = 0.
In other words, yes it *is* twintone and *no*, no new word is required.

Here is a comparison of twintone as tuned in h22 and g34, which I think shows g34 is perfectly practical and arguably preferable:

3: 7.13 3.93
5: -4.50 1.92
7: 12.99 19.41
5/3: -11.63 -2.01
7/3: 5.86 15.48
7/5: 17.49 17.49

The 22-et version has better 7/4s and 7/6s, and the 34-et version has
better 3/2s, 5/4s, and 5/3s; they share the twintone 7/5 of sqrt(2).

In-Reply-To: <a2n9fh+hqeh@eGroups.com>
Me:
> > 34-equal is an accurate, 5-limit diaschismic. But it isn't
> > consistent in the 7-limit. There are two 7-limit mappings which
> > converge at 34, and give good results for different tunings. Neither
> > of them is paultone.

Gene:
> This is getting ridiculous.

Why so? You haven't addressed anything I said there, so I'll have to do
it myself.

> > 34-equal is an accurate, 5-limit diaschismic.

The worst-tuned 5-limit interval in 34-equal is 5:4, which is 4 cents
sharp, or about 1/9 steps of 34. That's pretty good.

> > But it isn't consistent in the 7-limit.

The worst-tuned 7-limit interval in h34 (34-equal using the best
approximations to ratios of prime numbers) is 7:4, which is 0.56 steps
flat. Hence 34-equal is not 7-limit consistent.

> > There are two 7-limit mappings which
> > converge at 34, and give good results for different tunings.

h34&h46 (the temperament consistent with 46-equal and the prime mapping of
34-equal) is accurate to 5.9 cents in the 7-limit with a generator of
103.9 cents. h34&h22 is accurate to 6.9 cents in the 7-limit with a
generator of 107.5 cents. These are the most accurate 7-limit
diaschismics I know of. (You could probably improve on them with much
more complex mappings, and there's also Shrutar and the like.) They both
involve the prime mapping of 34-equal, hence converge at 34-equal.

> > Neither
> > of them is paultone.

h34&h22 has a period-generator mapping of

[(2, 0), (3, 1), (5, -2), (4, 9)]

h34&h46 has a period-generator mapping of

[(2, 0), (3, 1), (5, -2), (7, -8)]

paultone has a period-generator mapping of

[(2, 0), (3, 1), (5, -2), (6, -2)]

These are clearly three different temperaments.

So what's the problem?

> Here is a comparison of twintone as tuned in h22 and g34, which I think
> shows g34 is perfectly practical and arguably preferable:

Preferable to what? All this seems to show is that 34-equal is
inconsistent in the 7-limit, as I stated above.

> 3: 7.13 3.93
> 5: -4.50 1.92
> 7: 12.99 19.41
> 5/3: -11.63 -2.01
> 7/3: 5.86 15.48
> 7/5: 17.49 17.49
>
> The 22-et version has better 7/4s and 7/6s, and the 34-et version has
> better 3/2s, 5/4s, and 5/3s; they share the twintone 7/5 of sqrt(2).

Of course g34 has good 5-limit intervals, they're the same as those from
h34 ! The 7-limit as a whole is much worse, and most errors are in the
same direction, so a compromise temperament won't improve matters.

As it happens, my optimum for paultone/twintone is a 109.4 cent generator,
with a worst 7-limit error of 17 cents. 3/34 octaves are 105.9 cents and
2/22 octaves are 109.1 cents. So the twintone optimum is not only closer
to 22 than 34, but falls the other side of 22 than 34. That makes
34-equal far less characteristic of paultone/twintone than the other two
diaschismics I mention above.

So again, what's the problem?

Graham

--- In tuning-math@y..., graham@m... wrote:
> In-Reply-To: <a2n9fh+hqeh@e...>

> > > 34-equal is an accurate, 5-limit diaschismic. But it isn't
> > > consistent in the 7-limit. There are two 7-limit mappings which
> > > converge at 34, and give good results for different tunings. Neither
> > > of them is paultone.

> Gene:
> > This is getting ridiculous.
>
> Why so? You haven't addressed anything I said there, so I'll have to do
> it myself.

You say, correctly, that there are two reasonable 34-et 7-limit mappings. You claim, incorrectly, that neither gives twintone.

> > > But it isn't consistent in the 7-limit.
>
> The worst-tuned 7-limit interval in h34 (34-equal using the best
> approximations to ratios of prime numbers) is 7:4, which is 0.56 steps
> flat. Hence 34-equal is not 7-limit consistent.

You can't hear consistency, so why is this relevant?

> > > There are two 7-limit mappings which
> > > converge at 34, and give good results for different tunings.
>
> h34&h46 (the temperament consistent with 46-equal and the prime mapping of
> 34-equal) is accurate to 5.9 cents in the 7-limit with a generator of
> 103.9 cents. h34&h22 is accurate to 6.9 cents in the 7-limit with a
> generator of 107.5 cents. These are the most accurate 7-limit
> diaschismics I know of.

I'm talking about twintone, which means a generator of a fourth or fifth, or of a half-octave translate. The fifth of 34-et is 705.9
cents, and taking 600 from this gives 105.9 cents.

> So what's the problem?

The problem is that I'm trying to clear up a misconception, that you can't do twintone with 34 equal. You can, it works and it makes good sense.

> > Here is a comparison of twintone as tuned in h22 and g34, which I think
> > shows g34 is perfectly practical and arguably preferable:
>
> Preferable to what? All this seems to show is that 34-equal is
> inconsistent in the 7-limit, as I stated above.

It's much better for the 5-limit than 22-et, and the 7-limit of twintone can never be improved beyond the 7/5~10/7 compromise tritone anyway. If you look at the numbers, the 7-limit is worse, but the 5-limit is much better--so the question is, which do you want?

> > 3: 7.13 3.93
> > 5: -4.50 1.92
> > 7: 12.99 19.41
> > 5/3: -11.63 -2.01
> > 7/3: 5.86 15.48
> > 7/5: 17.49 17.49

> As it happens, my optimum for paultone/twintone is a 109.4 cent generator,
> with a worst 7-limit error of 17 cents. 3/34 octaves are 105.9 cents and
> 2/22 octaves are 109.1 cents. So the twintone optimum is not only closer
> to 22 than 34, but falls the other side of 22 than 34.

This is a weak argument--the rms optimized generator of 708.8 cents falls between the 22-et fifth of 709.1 cents and the 34-et fifth of
705.9 cents, but of course the whole thing is a one-parameter family which will give you different optimum values depending on what you decide to optimize, and what type of optimization you use. Weighting the 5-limit more heavily will move things in the direction of 34-et.

>You can't hear consistency, so why is this relevant?

You can hear consistency, when neighboring chords involve
different approximations to the same interval.

-Carl

I wrote...

>>You can't hear consistency, so why is this relevant?
>
>You can hear consistency, when neighboring chords involve
>different approximations to the same interval.

Which is not to say that this is in any way "bad".

-Carl

--- In tuning-math@y..., "clumma" <carl@l...> wrote:

> >You can't hear consistency, so why is this relevant?
>
> You can hear consistency, when neighboring chords involve
> different approximations to the same interval.

That wouldn't happen in the case Gene is talking about.

>>>You can't hear consistency, so why is this relevant?
>>
>>You can hear consistency, when neighboring chords involve
>>different approximations to the same interval.
>
>That wouldn't happen in the case Gene is talking about.

I didn't say it would. Gene's was a general dismissal of
consistency.

-Carl

--- In tuning-math@y..., "clumma" <carl@l...> wrote:
> >>>You can't hear consistency, so why is this relevant?
> >>
> >>You can hear consistency, when neighboring chords involve
> >>different approximations to the same interval.
> >
> >That wouldn't happen in the case Gene is talking about.
>
> I didn't say it would. Gene's was a general dismissal of
> consistency.

You can have neighboring chords involve different approximations to
the same interval even in a consistent tuning. I see this happening
in 76-tET, where one could modulate between twintone, meantone,
double-diatonic, as well as other systems.

>You can have neighboring chords involve different approximations
>to the same interval even in a consistent tuning. I see this
>happening in 76-tET, where one could modulate between twintone,
>meantone, double-diatonic, as well as other systems.

Example? I don't see how this could happen, unless it involved:

() Switching between subsets of the ET, which is cheating.
() Invoking higher-order approximations (ie, 10:12:15->16:19:24).

-Carl

--- In tuning-math@y..., "clumma" <carl@l...> wrote:

> I don't see how this could happen, unless it involved:
>
> () Switching between subsets of the ET, which is cheating.

Cheating? Jeez, can't I modulate from diatonic to diminished to whole-
tone to augmented in 12-tET?

--- In tuning-math@y..., "clumma" <carl@l...> wrote:
> I wrote...
>
> >>You can't hear consistency, so why is this relevant?
> >
> >You can hear consistency, when neighboring chords involve
> >different approximations to the same interval.
>
> Which is not to say that this is in any way "bad".

I can approximate 1-5/4-3/2-7/4 by 0-18-32-44 or 0-18-32-45 in 55-et, and so I can claim to "hear" inconsistency. I can also approxmiate
it by 0-6-11-15 or 0-6-11-16 in 19-et; can I also claim to hear the
7-inconsistency of the 19-et? Why or why not? What about both
0-10-18-25 and 0-10-18-26 in the 31-et? Can you hear inconsistency here?

--- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:

> What about both
> 0-10-18-25 and 0-10-18-26 in the 31-et?

The first is a 7-limit tetrad or German augmented sixth, the second
is a classical dominant seventh (I know, we argued about this a while
ago, you do not believe this, but I sure do).

>>>You can't hear consistency, so why is this relevant?
>>
>>You can hear consistency, when neighboring chords involve
>>different approximations to the same interval.
>
>
>I can approximate 1-5/4-3/2-7/4 by 0-18-32-44 or 0-18-32-45 in
>55-et, and so I can claim to "hear" inconsistency.

Not really. For isolated chords, you just always use the best
approximation (say, minimum rms). As long as you're happy with
that approximation, and you've based it on the chords you
actually want to use, not just the dyads involved (as some
early investigators did), then you're golden.

The "problem" occurs when modulating from the best approx. of
one chord to the best approx. of another, and thereby creating
anomalous (as in, non-existent in JI) commas. Some people
think commas are a "feature not a bug", others prefer to temper
them out. Others (apparently both you and I) think both
approaches are valid. And, as I said...

>>Which is not to say that this is in any way "bad".

...even tempering some commas out while inventing news ones can
probably be interesting. But for me, as a composer, this is just
too confusing. Thus, I restrict myself to consistent ets.

Consistency is also useful as a "badness" measure. It may not
be ideal for looking at ets up to 10 million, as some optimum
"flat" measure may be, but for any kind of goodness per notes
you'd actually care about from a pragmatic standpoint, it is
more than adequate.

>I can also approxmiate it by 0-6-11-15 or 0-6-11-16 in 19-et;
>can I also claim to hear the 7-inconsistency of the 19-et? Why
>or why not?

Depending on the context, the former chord is more likely to
approximate 4:5:6:7 or 1/1-5/4-3/2-12/7, and the latter chord
1/1-5/4-3/2-9/5 or 12:15:18:22... in other words, I'd guess
these would normally sound like different chords when
juxtaposed.

>What about both 0-10-18-25 and 0-10-18-26 in the 31-et? Can
>you hear inconsistency here?

The former chord is clearly 4:5:6:7. The latter chord would
attract the same suspects as 0-6-11-16 in 19-tET, and as Paul
points out, may be tuned any number of ways in diatonic music
since it functions as a dissonance there (1/1-5/4-3/2-16/9
often works well).

-Carl

>>>You can't hear consistency, so why is this relevant?
>>
>>You can hear consistency, when neighboring chords involve
>>different approximations to the same interval.
>
>
>I can approximate 1-5/4-3/2-7/4 by 0-18-32-44 or 0-18-32-45 in
>55-et, and so I can claim to "hear" inconsistency.

Not really. For isolated chords, you just always use the best
approximation (say, minimum rms). As long as you're happy with
that approximation, and you've based it on the chords you
actually want to use, not just the dyads involved (as some
early investigators did), you're golden.

The "problem" occurs when modulating from the best approx. of
one chord to the best approx. of another, which sometimes creates
anomalous (as in, non-existent in JI) commas. Some people think
commas are a "feature not a bug", others prefer to temper them
out. Others (apparently both you and I) think both approaches
are valid. And, as I said...

>>Which is not to say that this is in any way "bad".

...even tempering some commas out while inventing news ones can
probably be interesting. But for me, as a composer, this is just
too confusing. Thus, I restrict myself to consistent ets.

Consistency is also useful as a "badness" measure. It may not be
ideal for looking at ets up to 10 million, as some optimum "flat"
measure may be, but for any kind of goodness per notes you'd
actually care about from a pragmatic standpoint, it is more than
adequate.

>I can also approxmiate it by 0-6-11-15 or 0-6-11-16 in 19-et;
>can I also claim to hear the 7-inconsistency of the 19-et? Why
>or why not?

Depending on the context, the former chord is more likely to
approximate 4:5:6:7 or 1/1-5/4-3/2-12/7, and the latter chord
1/1-5/4-3/2-9/5 or 12:15:18:22... in other words, I'd guess
these would normally sound like different chords when juxtaposed.

>What about both 0-10-18-25 and 0-10-18-26 in the 31-et? Can
>you hear inconsistency here?

The former chord is clearly 4:5:6:7. The latter chord would
attract the same suspects as 0-6-11-16 in 19-tET, and as Paul
points out, may be tuned any number of ways in diatonic music
since it functions as a dissonance there (1/1-5/4-3/2-16/9
often works well).

-Carl

--- In tuning-math@y..., "clumma" <carl@l...> wrote:

> The "problem" occurs when modulating from the best approx. of
> one chord to the best approx. of another, and thereby creating
> anomalous (as in, non-existent in JI) commas.

That won't happen if you confine yourself to a regular temperamemt, such as the twintone version of 34-et, so I don't think it is relevant.

Carl:
> > The "problem" occurs when modulating from the best approx. of
> > one chord to the best approx. of another, and thereby creating
> > anomalous (as in, non-existent in JI) commas.

Gene:
> That won't happen if you confine yourself to a regular temperamemt,
> such as the twintone version of 34-et, so I don't think it is relevant.

No. If you're using a regular temperament, you can't be using 34-et.
34-et is an inconsistent, equal temperament. If you're using one of the
other diaschismic mappings of 34-et, the inconsistent chords will be
simpler than the regular ones. So what are you going to do? Pretend they
aren't there? Pretend they're not really 7-limit?

If you're not going to make use of the inconsistency, I don't see the
point in using 34-equal at all.

Graham

--- In tuning-math@y..., graham@m... wrote:

> No. If you're using a regular temperament, you can't be using 34-et.

This is just wrong; the whole thing is beginning to seem like another of those "religion" deal.

> 34-et is an inconsistent, equal temperament.

34-et isn't a regular temperament at all until you define a mapping to primes according to my proposed definition, which I think would help clarify all of this confusion.

If you're using one of the
> other diaschismic mappings of 34-et, the inconsistent chords will be
> simpler than the regular ones. So what are you going to do? Pretend they
> aren't there? Pretend they're not really 7-limit?

If you are using a 10-tone subset of 34 et, then they won't be there. In any case, this is not a new "problem"; it arises in meantone, where you get augmeted sixth intervals which are much closer to 7/4 than the 64/63 approximation ones intrinsic to diatonic 7-limit harmony, and so one has a connitption fit about it.

In the 34-et twintone, 96 is mapped to 96/54 mod 17 = -2, the familiar
64/63 approximation, whereas 95 maps to 95/54 mod 17 = -8, which isn't even a part of the 10 or 12 note twintone scales. This is quite analogous to the situation with the diatonic and standard septimal versions of 7-limit meantone; if g31 is the map [31,49,72,88] instead of the usual h31 of [31,49,72,87], then h12^g31 gives the temperament
[-1,-4,2,16,-6,-4] rather than h12^h31 = [-1,-4,-10,-12,13,-4]. The first is the diatonic version of 7-limit meantone, and may be regarded as the standard Western temperament of the last few centuries; the second uses the much better version of 7/4 which the 31-et allows, but it makes no appearance on the diatonic scale, as a glance at the wedgie shows. 31 equal can deal with either.

> If you're not going to make use of the inconsistency, I don't see the
> point in using 34-equal at all.

The point would be to make use of the superior 5-limit harmonies--compare the major sixth/minor thirds of 34-et to those of 22-et, for instance. If we consider 12-et, with a fifth which is two cents *flat* to be capable of producing a sort of twintone, we can certainly accept 34-et. If you look at how the fifth is tempered in various ets, a whole range of possibilities emerge:

h12: -1.96
g34: 3.93
g56: 5.19
h22: 7.14
h54: 9.16

There should be something for everyone in there.

Me:
> > No. If you're using a regular temperament, you can't be using 34-et.

Gene:
> This is just wrong; the whole thing is beginning to seem like another
> of those "religion" deal.

Close. It's actually a question of terminology. Equal and regular
temperaments are different things.

Me:
> > 34-et is an inconsistent, equal temperament.

Gene:
> 34-et isn't a regular temperament at all until you define a mapping to
> primes according to my proposed definition, which I think would help
> clarify all of this confusion.

34-et can be used *as* a regular temperament, but 34-et *is not* a regular
temperament.

Me:
> If you're using one of the
> > other diaschismic mappings of 34-et, the inconsistent chords will be
> > simpler than the regular ones. So what are you going to do? Pretend
> > they aren't there? Pretend they're not really 7-limit?

Gene:
> If you are using a 10-tone subset of 34 et, then they won't be there.

Bzzt -- yes they will. Try reading that paragraph a bit more carefully.

This is also the first time you've mentioned a 10-note *subset* which
would obviously skew towards twintone.

> In any case, this is not a new "problem"; it arises in meantone, where
> you get augmeted sixth intervals which are much closer to 7/4 than the
> 64/63 approximation ones intrinsic to diatonic 7-limit harmony, and so
> one has a connitption fit about it.

Not new at all. In both cases there's a simplified 7-limit mapping that
optimises close to one of the extreme ETs -- 12 in meantone, 22 in
diaschismic. The difference with diaschismic is that the "normal" range
is covered by two different more-complex mappings, so we can't talk about
a "typical" 7-limit diaschismic. But this isn't new either, it's been on
my website for a few years.

Me:
> > If you're not going to make use of the inconsistency, I don't see the
> > point in using 34-equal at all.

Gene:
> The point would be to make use of the superior 5-limit
> harmonies--compare the major sixth/minor thirds of 34-et to those of
> 22-et, for instance.

That's not a sufficient reason. You can get better 5-limit harmonies with
a 105.2 cent generator (worst error 3.3 cents compared with 3.9 for
34-equal). So again, why use 34-equal?

> If we consider 12-et, with a fifth which is two
> cents *flat* to be capable of producing a sort of twintone, we can
> certainly accept 34-et. If you look at how the fifth is tempered in
> various ets, a whole range of possibilities emerge:
>
> h12: -1.96
> g34: 3.93
> g56: 5.19
> h22: 7.14
> h54: 9.16
>
> There should be something for everyone in there.

What does this have to do with the price of eggs?

Graham

--- In tuning-math@y..., graham@m... wrote:

> 34-et can be used *as* a regular temperament, but 34-et *is not* a regular
> temperament.

What is your definition of "regular temperament"? I gave mine.

> > If you are using a 10-tone subset of 34 et, then they won't be there.
>
> Bzzt -- yes they will. Try reading that paragraph a bit more carefully.

Where?

> This is also the first time you've mentioned a 10-note *subset* which
> would obviously skew towards twintone.

Twintone is the subject of our discussion--how can we skew towards where we already are?
> > The point would be to make use of the superior 5-limit
> > harmonies--compare the major sixth/minor thirds of 34-et to those of
> > 22-et, for instance.
>
> That's not a sufficient reason. You can get better 5-limit harmonies with
> a 105.2 cent generator (worst error 3.3 cents compared with 3.9 for
> 34-equal). So again, why use 34-equal?

This is simpl an agument that we should never use equal divisions at all. Should I go into reasons why we might want to?

> > h12: -1.96
> > g34: 3.93
> > g56: 5.19
> > h22: 7.14
> > h54: 9.16
> >
> > There should be something for everyone in there.
>
> What does this have to do with the price of eggs?

It shows 34-et as a part of a range of twintone et possibilities.

--- In tuning-math@y..., "genewardsmith" <genewardsmith@j...>
wrote:
> --- In tuning-math@y..., graham@m... wrote:
>
> > 34-et can be used *as* a regular temperament, but 34-et *is
not* a regular
> > temperament.
>
> What is your definition of "regular temperament"? I gave mine.

Yours seems to fit perfectly with the existing literature. I have no
idea what definition Graham might be going by.

You may note that many of the decatonic "keys" in the 22-tone
well-temperament in my paper are quite similar to 34-tET in
intonation.

genewardsmith wrote:

> What is your definition of "regular temperament"? I gave mine.

In fact, I've noticed that I have no idea what a "regular temperament" is.
So, I'd better beat a hasty retreat. I can't find your definition
either, though ...

> > > If you are using a 10-tone subset of 34 et, then they won't be
> > > there.
> > Bzzt -- yes they will. Try reading that paragraph a bit more
> > carefully.
>
> Where?

The one you cut out:

> If you're using one of the
> > other diaschismic mappings of 34-et, the inconsistent chords will be
> > simpler than the regular ones. So what are you going to do? Pretend
> > they aren't there? Pretend they're not really 7-limit?

Using either of the other diaschismic mappings of 34-et, the inconsistent
chords are those of twintone. You certainly do get some of them within
the 10 note MOS.

> > This is also the first time you've mentioned a 10-note *subset* which
> > would obviously skew towards twintone.
>
> Twintone is the subject of our discussion--how can we skew towards
> where we already are?

What? At this time of night I can't even understand why that objection's
bogus. You're saying because we're discussing something it must be right?

> > > The point would be to make use of the superior 5-limit
> > > harmonies--compare the major sixth/minor thirds of 34-et to those
> > > of 22-et, for instance.
> >
> > That's not a sufficient reason. You can get better 5-limit harmonies
> > with a 105.2 cent generator (worst error 3.3 cents compared with 3.9
> > for 34-equal). So again, why use 34-equal?
>
> This is simpl an agument that we should never use equal divisions at
> all. Should I go into reasons why we might want to?

You could do.

> It shows 34-et as a part of a range of twintone et possibilities.

Well, that's a radical idea. I'm sure I'd never have worked that out
myself.

BTW, what do you make the LLL reduction of

[ 1 1 1]
[-1 0 2]
[ 3 5 6]

?

Graham

--- In tuning-math@y..., graham@m... wrote:

> BTW, what do you make the LLL reduction of
>
> [ 1 1 1]
> [-1 0 2]
> [ 3 5 6]
>
> ?

Is the first column 2 or 3?

How about the Minkowski reduction?

--- In tuning-math@y..., graham@m... wrote:

> Using either of the other diaschismic mappings of 34-et, the inconsistent
> chords are those of twintone. You certainly do get some of them within
> the 10 note MOS.

What's an inconsistent chord?

> > It shows 34-et as a part of a range of twintone et possibilities.
>
> Well, that's a radical idea. I'm sure I'd never have worked that out
> myself.

You seemed to be objecting to it strongly, so I don't know what your point is.

>
> BTW, what do you make the LLL reduction of

> [ 1 1 1]
> [-1 0 2]
> [ 3 5 6]

That might depend on what inner product I use, and I don't know what this is supposed to represent. If I use the standard dot product, I get

[ 0 1 0]
[ 1 0 1]
[-1 0 2]

>>The "problem" occurs when modulating from the best approx. of
>>one chord to the best approx. of another, and thereby creating
>>anomalous (as in, non-existent in JI) commas.
>
>That won't happen if you confine yourself to a regular
>temperamemt, such as the twintone version of 34-et, so
>I don't think it is relevant.

Let's take a look at what you wrote...

>>>You can't hear consistency, so why is this relevant?

Seemed relevant from here.

>>>A temperament /.../ which maps rational intonation of a given
>>>rank in a consistent way to an intonation of smaller rank.

So what does it mean to map rational intonation "in a consistent
way"? From the Onto page a mathworld, it looks like a regular
temperament is just a consistent temperament in the first place.

-Carl

--- In tuning-math@y..., "clumma" <carl@l...> wrote:

> So what does it mean to map rational intonation "in a consistent
> way"? From the Onto page a mathworld, it looks like a regular
> temperament is just a consistent temperament in the first place.

I'm sure Gene would be happy with this statement. Only thing, it's
not consistent in the 'TTTTTT, footnote 8' sense of necessarily using
the best available approximation to every 'consonant' interval.

>>So what does it mean to map rational intonation "in a consistent
>>way"? From the Onto page a mathworld, it looks like a regular
>>temperament is just a consistent temperament in the first place.
>
>I'm sure Gene would be happy with this statement.

So then what's the point of saying that inconsistency won't cause
any problems in a regular temperament?

-Carl

--- In tuning-math@y..., "clumma" <carl@l...> wrote:
> >>So what does it mean to map rational intonation "in a consistent
> >>way"? From the Onto page a mathworld, it looks like a regular
> >>temperament is just a consistent temperament in the first place.
> >
> >I'm sure Gene would be happy with this statement.
>
> So then what's the point of saying that inconsistency won't cause
> any problems in a regular temperament?

There's a point if 'inconsistency' is defined in the TTTTTT, footnote
8 sense.

>>So then what's the point of saying that inconsistency won't cause
>>any problems in a regular temperament?
>
>There's a point if 'inconsistency' is defined in the TTTTTT,
>footnote 8 sense.

I don't get it.

-Carl

I like to re-visit this:

>>>The "problem" occurs when modulating from the best approx. of
>>>one chord to the best approx. of another, and thereby creating
>>>anomalous (as in, non-existent in JI) commas.
>>
>>That won't happen if you confine yourself to a regular
>>temperamemt /.../

I don't understand. Let's call 24-et a linear temperament of a
chain of 24 near 7:4's (19 steps each). Now play the following:

5:7:8 -> 7:8:10

Ignoring consistency and using the lowest-rms approximations, that
will be:

0,12,16 -> 12,17,24

The comma from 16 to 17 steps doesn't exist in JI. So a chain of
950-cent intervals must not be a "regular" temperament. Why?

-Carl

--- In tuning-math@y..., "clumma" <carl@l...> wrote:

> I don't understand. Let's call 24-et a linear temperament of a
> chain of 24 near 7:4's (19 steps each). Now play the following:

OK. You are now looking at the linear temperament "Lumma", with MT reduced basis <49/48, 81/80> and wedgie [2,8,1,-20,4,8]. If you care about the thirds, you probably want to do this in 19-et; otherwise
24-et is a good choice.

> 5:7:8 -> 7:8:10

"Lumma" maps this to [8,-4]:[1,2]:[0,3] -> [1,2]:[0,3]:[8,-3]

> Ignoring consistency and using the lowest-rms approximations, that
> will be:

You're not allowed to do this, since you are using the regular temperament Lumma and must use what it gives you.

--- In tuning-math@y..., "clumma" <carl@l...> wrote:
> I like to re-visit this:
>
> >>>The "problem" occurs when modulating from the best
approx. of
> >>>one chord to the best approx. of another, and thereby
creating
> >>>anomalous (as in, non-existent in JI) commas.
> >>
> >>That won't happen if you confine yourself to a regular
> >>temperamemt /.../
>
> I don't understand. Let's call 24-et a linear temperament of a
> chain of 24 near 7:4's (19 steps each). Now play the following:
>
> 5:7:8 -> 7:8:10
>
> Ignoring consistency and using the lowest-rms
approximations, that
> will be:
>
> 0,12,16 -> 12,17,24
>
> The comma from 16 to 17 steps doesn't exist in JI. So a chain
of
> 950-cent intervals must not be a "regular" temperament. Why?

a regular temperament includes, in its specification, the interval
that each ji consonance is mapped to.

>>5:7:8 -> 7:8:10
>
>"Lumma" maps this to [8,-4]:[1,2]:[0,3] -> [1,2]:[0,3]:[8,-3]
>
>>Ignoring consistency and using the lowest-rms approximations, that
>>will be:
>
>You're not allowed to do this, since you are using the regular
>temperament Lumma and must use what it gives you.

Okay, so what you're saying is, you prefer to measure the error
of mappings from just intonation, rather than of ets -- it is a
mistake to say, 'such-and-such equal temperament has such-and-such
an error in the such-limit'. Fine by me, but since mappings are
consistent by nature, your use of them to support your statment
that you "can't hear consistency" is fallacious -- in order to
speak of the accuracy of ets, the notion of 'switching mappings'
must be substituted for the notion of inconsistency.

-Carl

> Okay, so what you're saying is, you prefer to measure the error
> of mappings from just intonation, rather than of ets -- it is a
> mistake to say, 'such-and-such equal temperament has such-and-such
> an error in the such-limit'. Fine by me, but since mappings are
> consistent by nature, your use of them to support your statment
> that you "can't hear consistency" is fallacious -- in order to
> speak of the accuracy of ets, the notion of 'switching mappings'
> must be substituted for the notion of inconsistency.

In fairness to you, Gene, Graham did seem to be artificially
kicking out the 34-et tuning of the two 7-limit mappings it
contains, for the simple reason there are two of them. That
would be wrong. I did it too:

>...even tempering some commas out while inventing news ones can
>probably be interesting. But for me, as a composer, this is just
>too confusing. Thus, I restrict myself to consistent ets.

The anomalous comma problem wouldn't be grounds for this. But
the fact that consistency serves as a badness measure might be.

-Carl

--- In tuning-math@y..., "clumma" <carl@l...> wrote:

> The anomalous comma problem wouldn't be grounds for this. But
> the fact that consistency serves as a badness measure might be.

I still don't know what the problem is. If you go to a high enough n, any n-et will have more than one good mapping. Is that still a problem?

>>The anomalous comma problem wouldn't be grounds for this. But
>>the fact that consistency serves as a badness measure might be.
>
>I still don't know what the problem is. If you go to a high enough n,
>any n-et will have more than one good mapping. Is that still a problem?

It's not a problem unless you're counting on the best rms
approximations a given et has to offer. All consistency
means at high n is that the best approximations all in the
et all fall in the same mapping.

-Carl