Optimal Pajara Tuning (Was: [MMM] Re: 34-et in the 7-limit

On MMM, Carl Lumma wrote:

> These days, I suspect Paul would search not ETs, but linear
> temperaments, and see that pajara wins on 7-limit badness in
> many formulations. I suspect he'd find 34-tET a valid pajara
> tuning. Whether it's actually better than 22 is a question.
> The mad error of 34 is slightly better, and the rms slightly
> worse. However, according to his calculation on page 21,
> 22 is closer to the optimal decatonic tuning than 34.

There is a difference, but it comes about from Tenney weighting. The minimax 7-limit pajara tuning has a generator of 109.4 cents. The 7-limit optimal RMS tuning is 108.8 cents. 22-equal tuning gives a generator of 109.1 cents. So you can see that, as far as one measure agrees with the other, 22-equal is the optimal tuning.

34-equal's pajara-consistent mapping gives a generator of 105.9 cents. The 7-limit RMS is a compromise, but one that's way closer to 22 than 34. So by these calculations, 34 may be a valid tuning, but it's an eccentric one.

However, 34 is mostly penalized for it's poor mapping of 7:4 and this doesn't look as bad with Tenney weighting. The unstretched TOP generator is 106.8 cents, and the equivalent RMS generator is 107.0 cents. Looked at this way, pajara is a true linear temperament, and shouldn't be monopolized by a single equal tuning. 56-equal, with a generator of 107.1 cents, is close but complex.

This would be a good test for the validity of Tenney weighting. Does decatonic music really sound better in 22-equal or a 22/34 compromise?

34-equal's worst 7-limit error is 19.4 cents, compared to 17.5 for 22-equal. So even here, the difference isn't that great. The TOP error is 3.45 cents/octave for 34 and 3.28 for 22. The equivalent RMS is 2.66 for 34 and 2.85 for 22. Hence it's only the weighted RMS where 34 is better, which seems to contradict what Carl wrote. But still, it is better, and that's worth noting.

Graham

>> The mad error of 34 is slightly better, and the rms slightly
>> worse.

>it's only the weighted RMS where 34 is
>better, which seems to contradict what Carl wrote.

According to my code, the unweighted 7-limit error of
22 is

(9.933221735441748 mad)
(10.918951869200367 rms))

and of 34

(9.451546186426318 mad)
(11.697443921018829 rms))

>This would be a good test for the validity of Tenney weighting.
>Does decatonic music really sound better in 22-equal or a 22/34
>compromise?

A synthetic test with greater contrast would probably be better.
One idea would be to come up with triples of chords with the same
total error, but one chord piles it on the higher identities, one
on the lower, and one spreads it evenly. If someone up with a
such things, I promise to listen and report back.

-Carl

Carl Lumma wrote:
>>>The mad error of 34 is slightly better, and the rms slightly
>>>worse.
> > >>it's only the weighted RMS where 34 is >>better, which seems to contradict what Carl wrote.
> > > According to my code, the unweighted 7-limit error of
> 22 is
> > (9.933221735441748 mad)
> (10.918951869200367 rms))

I agree with you about the RMS in cents, except for the last two decimal places. But what's this mad? It can't be maximum absolute deviation in cents because then the maximum would be less than the mean. Mean absolute deviation?

> and of 34
> > (9.451546186426318 mad)
> (11.697443921018829 rms))

Whatever the mad is, it's still worse than 22. I get an RMS of 12.552 cents.

>>This would be a good test for the validity of Tenney weighting.
>>Does decatonic music really sound better in 22-equal or a 22/34
>>compromise?
> > > A synthetic test with greater contrast would probably be better.
> One idea would be to come up with triples of chords with the same
> total error, but one chord piles it on the higher identities, one
> on the lower, and one spreads it evenly. If someone up with a
> such things, I promise to listen and report back.

But what are we supposed to test with these chords?

Graham

Carl Lumma wrote:
>>>The mad error of 34 is slightly better, and the rms slightly
>>>worse.
> > >>it's only the weighted RMS where 34 is >>better, which seems to contradict what Carl wrote.
> > > According to my code, the unweighted 7-limit error of
> 22 is
> > (9.933221735441748 mad)

okay, that's the mean absolute deviation in cents

> (10.918951869200367 rms))
> > and of 34
> > (9.451546186426318 mad)
> (11.697443921018829 rms))

and your mad is lower for 34.

I get

10.038989128090492 cents mad
12.552429031044609 cents rms

Graham

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

> A synthetic test with greater contrast would probably be better.
> One idea would be to come up with triples of chords with the same
> total error, but one chord piles it on the higher identities, one
> on the lower, and one spreads it evenly. If someone up with a
> such things, I promise to listen and report back.

I think what kind of music you are writing is going to make a big
difference. Triadic harmony in 34-et is really quite pleasing, and if
you were to to take harmonic textures from, say, the year 1800, with a
base of triadic harmony overlayed with dominant seventh chords,
diminished seventh chords, and diminished triads, then I think 34
would shine. I haven't tried writing any ersatz classical era music in
34, though, so this is just theory, but I recommend the idea for
anyone who is game to try. But just because 34 isn't a meantone system
doesn't mean it is devoid of familiar chordal landmarks, and it isn't
as strange sounding to unaccustomed ears as 22.

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

> (9.933221735441748 mad)
> (10.918951869200367 rms))

Mad error seems to be catching on around here. Is this Paul E's
influence? When the heck is he returning?

At 06:19 PM 5/19/2006, you wrote:
>Carl Lumma wrote:
>>>>The mad error of 34 is slightly better, and the rms slightly
>>>>worse.
>>
>>
>>>it's only the weighted RMS where 34 is
>>>better, which seems to contradict what Carl wrote.
>>
>>
>> According to my code, the unweighted 7-limit error of
>> 22 is
>>
>> (9.933221735441748 mad)
>> (10.918951869200367 rms))
>
>I agree with you about the RMS in cents, except for the last two decimal
>places.

Rounding errors on my part, no doubt.

>But what's this mad? ... Mean absolute deviation?

That's what MAD has always stood for in these parts, to my
(probably incomplete) knowledge.

>> and of 34
>>
>> (9.451546186426318 mad)
>> (11.697443921018829 rms))
>
>Whatever the mad is, it's still worse than 22.

Looks like less than mad(22) to me.

>I get an RMS of 12.552 cents.

What intervals are you checking?

>>>This would be a good test for the validity of Tenney weighting.
>>>Does decatonic music really sound better in 22-equal or a 22/34
>>>compromise?
>>
>> A synthetic test with greater contrast would probably be better.
>> One idea would be to come up with triples of chords with the same
>> total error, but one chord piles it on the higher identities, one
>> on the lower, and one spreads it evenly. If someone up with a
>> such things, I promise to listen and report back.
>
>But what are we supposed to test with these chords?

The sound of them, of course.

-Carl

>> and of 34
>>
>> (9.451546186426318 mad)
>> (11.697443921018829 rms))
>
>and your mad is lower for 34.
>
>I get
>
>10.038989128090492 cents mad
>12.552429031044609 cents rms

What intervals are you checking?

-Carl

>> A synthetic test with greater contrast would probably be better.
>> One idea would be to come up with triples of chords with the same
>> total error, but one chord piles it on the higher identities, one
>> on the lower, and one spreads it evenly. If someone up with a
>> such things, I promise to listen and report back.
>
>I think what kind of music you are writing is going to make a big
>difference. Triadic harmony in 34-et is really quite pleasing, and if
>you were to to take harmonic textures from, say, the year 1800, with a
>base of triadic harmony overlayed with dominant seventh chords,
>diminished seventh chords, and diminished triads, then I think 34
>would shine. I haven't tried writing any ersatz classical era music in
>34, though, so this is just theory, but I recommend the idea for
>anyone who is game to try. But just because 34 isn't a meantone system
>doesn't mean it is devoid of familiar chordal landmarks, and it isn't
>as strange sounding to unaccustomed ears as 22.

The test would go beyond the question of pajara/22/34, to the
question of whether weighted errors should be used in optimization
calculations, and which way they should be weighted (see a recent
thread between Paul and Graham on that, for instance).

-Carl

>> (9.933221735441748 mad)
>> (10.918951869200367 rms))
>
>Mad error seems to be catching on around here. Is this Paul E's
>influence? When the heck is he returning?

I haven't spoken with Paul in a while, but I bet he's enjoying
his break. MAD has been around these lists for a long time (I
wrote that code in 1999, I think). I've always liked RMS, but
I've never tested the alternatives in a satisfactory way.

-C.

Carl Lumma wrote:
>>>and of 34
>>>
>>> (9.451546186426318 mad)
>>> (11.697443921018829 rms))
>>
>>and your mad is lower for 34.
>>
>>I get
>>
>>10.038989128090492 cents mad
>>12.552429031044609 cents rms
> > What intervals are you checking?

The 7-limit, or half of it. In octave-equivalent vector form:

(1, 0, 0), (0, 1, 0), (-1, 1, 0), (0, 0, 1), (-1, 0, 1), (0, -1, 1)

Graham

Carl Lumma wrote:

>>But what's this mad? ... Mean absolute deviation?
> > That's what MAD has always stood for in these parts, to my
> (probably incomplete) knowledge.

Quite possibly, but my knowledge is sufficiently incomplete to not remember any of this. It's always a good idea to define non-standard terms.

>>>>This would be a good test for the validity of Tenney weighting.
>>>>Does decatonic music really sound better in 22-equal or a 22/34
>>>>compromise?
>>>
>>>A synthetic test with greater contrast would probably be better.
>>>One idea would be to come up with triples of chords with the same
>>>total error, but one chord piles it on the higher identities, one
>>>on the lower, and one spreads it evenly. If someone up with a
>>>such things, I promise to listen and report back.
>>
>>But what are we supposed to test with these chords?
> > The sound of them, of course.

What about the sound? I can't see it's any better than any other experiments about isolated chords. Complete pieces of decatonic music would test how palatable 7-limit chords are in a harmonic context and whether the 7:4 truly functions as a consonance, which is a lot more useful.

Graham

>>>>A synthetic test with greater contrast would probably be better.
>>>>One idea would be to come up with triples of chords with the same
>>>>total error, but one chord piles it on the higher identities, one
>>>>on the lower, and one spreads it evenly. If someone up with a
>>>>such things, I promise to listen and report back.
>>>
>>>But what are we supposed to test with these chords?
>>
>> The sound of them, of course.
>
>What about the sound? I can't see it's any better than any other
>experiments about isolated chords. Complete pieces of decatonic music
>would test how palatable 7-limit chords are in a harmonic context and
>whether the 7:4 truly functions as a consonance, which is a lot more
>useful.

That's again a separate question -- whether pajara works musically
has nothing to do with whether weighted or unweighted error gives
better discordance values.

-Carl

>>>>and of 34
>>>>
>>>> (9.451546186426318 mad)
>>>> (11.697443921018829 rms))
>>>
>>>and your mad is lower for 34.
>>>
>>>I get
>>>
>>>10.038989128090492 cents mad
>>>12.552429031044609 cents rms
>>
>> What intervals are you checking?
>
>The 7-limit, or half of it. In octave-equivalent vector form:
>(1, 0, 0), (0, 1, 0), (-1, 1, 0), (0, 0, 1), (-1, 0, 1), (0, -1, 1)

My procedure is meant to work for generic chords. I check all
2-combinations of the chords identities. For the 7-limit, I
feed in (1 3 5 7). I think that gives me half of the 7-limit,
just like you. I get the same answer I originally gave
whether I feed in (4 5 6 7) or (1/1 5/4 3/2 7/4) or (1/1 7/6 7/5 7/4).
I wonder how we're getting different answers.

-Carl

--- In tuning-math@yahoogroups.com, Graham Breed <gbreed@...> wrote:
>
> Carl Lumma wrote:
>
> >>But what's this mad? ... Mean absolute deviation?
> >
> > That's what MAD has always stood for in these parts, to my
> > (probably incomplete) knowledge.
>
> Quite possibly, but my knowledge is sufficiently incomplete to not
> remember any of this. It's always a good idea to define
non-standard terms.

It's actually a standard term of sorts, but much less well known than
rms, and definately worth a definition.

Carl Lumma wrote:
>>>>>and of 34
>>>>>
>>>>> (9.451546186426318 mad)
>>>>> (11.697443921018829 rms))
>>>>
>>>>and your mad is lower for 34.
>>>>
>>>>I get
>>>>
>>>>10.038989128090492 cents mad
>>>>12.552429031044609 cents rms
>>>
>>>What intervals are you checking?
>>
>>The 7-limit, or half of it. In octave-equivalent vector form:
>>(1, 0, 0), (0, 1, 0), (-1, 1, 0), (0, 0, 1), (-1, 0, 1), (0, -1, 1)
> > > My procedure is meant to work for generic chords. I check all
> 2-combinations of the chords identities. For the 7-limit, I
> feed in (1 3 5 7). I think that gives me half of the 7-limit,
> just like you. I get the same answer I originally gave
> whether I feed in (4 5 6 7) or (1/1 5/4 3/2 7/4) or (1/1 7/6 7/5 7/4).
> I wonder how we're getting different answers.

Your figures are for the nearest mapping of each interval to 34-equal. That is, you're ignoring its inhereng inconsistency. My figures are for the pajara-consistent mapping:

<34, 54, 79, 96]

Graham

>> My procedure is meant to work for generic chords. I check all
>> 2-combinations of the chords identities. For the 7-limit, I
>> feed in (1 3 5 7). I think that gives me half of the 7-limit,
>> just like you. I get the same answer I originally gave
>> whether I feed in (4 5 6 7) or (1/1 5/4 3/2 7/4) or (1/1 7/6 7/5 7/4).
>> I wonder how we're getting different answers.
>
>Your figures are for the nearest mapping of each interval to 34-equal.
>That is, you're ignoring its inhereng inconsistency. My figures are for
>the pajara-consistent mapping:
>
><34, 54, 79, 96]

Ah, of course. I should have mentioned I was using the patent
val.

-Carl

Carl Lumma wrote:

> Ah, of course. I should have mentioned I was using the patent
> val.

In that case, you should be getting a MAD of 10.23 cents and an RMS of 12.81 cents. That's compared to the pajara-consistent mapping, which has a MAD of 10.04 cents and an RMS of 12.55 cents. However, it looked suspiciously like you were getting the MAD of 9.45 cents and RMS of 11.70 cents for an inconsistent calculation. But perhaps that was all a bad dream ;)

Glossary:

MAD -- mean absolute deviation

RMS -- root mean square (deviation)

Pajara -- the 7-limit regular temperament combining the optimal mappings for 12- and 22-equal. The pajara-consistent mapping for 34-equal is <34, 54, 79, 96]

Patent val -- the mapping taking the nearest approximation to each prime interval. For 34-equal in the 7-limit, this is <34, 54, 79, 95]

Inconsistent -- an equal temperament where the mapping of each interval doesn't agree with interval arithmetic

7-limit -- intervals with a frequency ratio that doesn't contain a number larger than 7, and octave equivalents.

Graham

>> Ah, of course. I should have mentioned I was using the patent
>> val.
>
>In that case, you should be getting a MAD of 10.23 cents and an RMS of
>12.81 cents. That's compared to the pajara-consistent mapping, which
>has a MAD of 10.04 cents and an RMS of 12.55 cents. However, it looked
>suspiciously like you were getting the MAD of 9.45 cents and RMS of
>11.70 cents for an inconsistent calculation. But perhaps that was all
>a bad dream ;)

Huh. The val I'm using is (0 11 20 27). Waitaminute. No, I'm
tuning all the dyads individually as you seem to have figured
out. See, originally this code checked for consistency and
returned an error otherwise. I just ripped that part out to do
this. But I wrote it assuming consistency... so even though it
reports using (0 11 20 27) it isn't. Sneaky! Looks like I'll
need to write a bunch of stuff to handle vals.

-Carl