RE: 11-limit, 31 tones, 9 hexads within 2.7c of just

Congratulations on this latest tuning, Dave! This has got to be the
pinnacle of the journey into periodicity-block-like JI which started almost
a year ago.

>Why would anyone bother with strict JI beyond the 5-limit? For the cost of
>only a 2.7 c error in any 11-limit interval you can have a LOT more
>consonances.

Let's take a look at the error of the otonal 7-limit tetrad...

Just Wafso-Lumma Wafso-31
---------------------------------------
3/2 702.0 700.0 [1.9] 700.6 [1.4]
5/4 386.3 384.4 [1.9] 383.6 [2.7]
6/5 315.6 315.6 [0.0] 317.0 [1.4]
7/4 968.8 968.8 [0.0] 968.4 [0.4]
7/5 582.5 584.4 [1.9] 584.8 [2.3]
7/6 266.9 268.8 [1.9] 267.8 [0.9]
---------------------------------------
Total
error 0.0 7.6 9.1
---------------------------------------

I'm sorry to report that I could hear, without too much trouble, the
mistuning of the tetrad in the wafso-Lumma scale. I'm assuming that I
could also hear it in wafso-31. Not that such an error isn't worth the
extra intervals -- not that it isn't _well worth_ the extra intervals. But
if you can hear it... Composers who do not require more intervals or
economy of buttons would have reason to use JI, and in fact _no reason_ to
use the wafso version.

-Carl

Carl wrote,

>Composers who do not require more intervals or
>economy of buttons would have reason to use JI, and in fact _no reason_ to
>use the wafso version.

Disagree -- it is much as with the major 6/9 chord, where I would use
meantone even if I had an infinite number of pitches easily available.

Carl Lumma wrote,

>Congratulations on this latest tuning, Dave! This has got to be the
>pinnacle of the journey into periodicity-block-like JI which started almost
>a year ago.

You mean JI-like periodicity blocks?

Anyway, now that we're dealing with 4-d (11-prime limit) ones, a natural
question would be, can we represent Partch's 43-tone scale with a
periodicity block? Following Wilson, we'll identify 11/10 and 10/9 as an
equivalent pair (and their inversions as another) so we are dealing with a
41-tone periodicity block, one of whose unison vectors is 100:99. Unlike the
19- and 31-tone periodicity blocks found so far, this 41-tone one will not
use 81:80 as a unison vector, as several 81:80 pairs appear in Partch's
scale. . . .

[Paul Erlich]
>>Composers who do not require more intervals or economy of buttons would
>>have reason to use JI, and in fact _no reason_ to use the wafso version.
>
>Disagree -- it is much as with the major 6/9 chord, where I would use
>meantone even if I had an infinite number of pitches easily available.

True, I forgot that some chords are temperament-only. But a composer
demanding the sort of accuracy we're talking about wouldn't want them.

[Paul Erlich]
>Anyway, now that we're dealing with 4-d (11-prime limit) ones, a natural
>question would be, can we represent Partch's 43-tone scale with a
>periodicity block? Following Wilson, we'll identify 11/10 and 10/9 as an
>equivalent pair (and their inversions as another) so we are dealing with a
>41-tone periodicity block, one of whose unison vectors is 100:99. Unlike the
>19- and 31-tone periodicity blocks found so far, this 41-tone one will not
>use 81:80 as a unison vector, as several 81:80 pairs appear in Partch's
>scale. . . .

I think in some respects this over-plays 43. Partch didn't consider it
that special. But, the idea is intersting. The question is, why fuse
11/10 and 10/9? Wilson did it to commensurate it with modulus 41. Is that
important to us?

[Daniel Wolf]
>Wilson's mapping had 12/11 and 11/10 (and their inversions) sharing a key,
>unison vector 121:120.

No, 12/11 and 11/10 were on seperate keys. Paul is correct.

-Carl

>True, I forgot that some chords are temperament-only. But a composer
>demanding the sort of accuracy we're talking about wouldn't want them.

You mean a composer satisfied with a 2.7� accuracy would want chords only
possible with 2.7� errors?

>I think in some respects this over-plays 43. Partch didn't consider it
>that special.

When he wrote Genesis, he adopted it and mapped its resources in great
detail.

>But, the idea is intersting. The question is, why fuse
>11/10 and 10/9? Wilson did it to commensurate it with modulus 41. Is that
>important to us?

Remember how I was saying that reasonable PBs are CS?

> [Carl Lumma, TD 463.12]
> Let's take a look at the error of the otonal 7-limit tetrad...
>
> Just Wafso-Lumma Wafso-31
> ---------------------------------------
> 3/2 702.0 700.0 [1.9] 700.6 [1.4]
> 5/4 386.3 384.4 [1.9] 383.6 [2.7]
> 6/5 315.6 315.6 [0.0] 317.0 [1.4]
> 7/4 968.8 968.8 [0.0] 968.4 [0.4]
> 7/5 582.5 584.4 [1.9] 584.8 [2.3]
> 7/6 266.9 268.8 [1.9] 267.8 [0.9]
> ---------------------------------------
> Total
> error 0.0 7.6 9.1
> ---------------------------------------
>
> I'm sorry to report that I could hear, without too much
> trouble, the mistuning of the tetrad in the wafso-Lumma scale.
> <... etc., snip>
>

Carl, I get the drift of what you're saying here, but
I'm at a loss to understand you *precisely*, because, if
one assumes (as probably everyone here would) that by
'*the* otonal 7-limit tetrad', you mean a 4-note chord
with the frequency proportions 4:5:6:7, there's not a
single one to be found among the pitches/intervals in
your list!

Here's a handy way to find all integer-ratios, reduced to
lowest terms, within a JI set (which I actually introduced
here last week discussing Ganassi). Simply divide the
proportions by each sucessive integer within the prime-limit
of the set, retaining only those which divide exactly and
discarding those with a decimal remainder.

7/6 6/5 5/4 7/5 3/2 7/4

/ 1 70 : 72 : 75 : 84 : 90 : 105
/ 2 35 36 42 45
/ 3 24 25 28 30 35
/ 4 18 21
/ 5 14 15 18 21
/ 6 12 14 15
/ 7 10 12 15
/ 8 9
/ 9 8 10
/10 7 9
/12 6 7
/14 5 6
/15 5 6 7
/16
/18 4 5
/20
/21 4 5
/24 3
/25 3
/27
/28 3
/30 3
/32
/35 2 3
/36 2

It's easy to see that there is no 4:5:6:7 tetrad in this set.

Of course, it's even easier to see on a lattice diagram.
Here's one which follows (as closely as ASCII allows) my
regular lattice formula. Notes not in Carl's set are indicated
by parenthesis (either empty or with the ratio where's it's
important for reference):

() ---------- (16/15) <- the otonal systemic 'fundamental'
/ '-._ / '-._
/ 7/6 -----/------'()
/ / / /
7/5 ------------ () /
'-._ / / '-._ /
7/4 -----/------'(1/1)
/ / '-._
6/5 / 5/4
'-._ /
'3/2

Can you explain? Did you mean 'hexad'? And what about 'otonal'?

-monz

Joseph L. Monzo Philadelphia monz@juno.com
http://www.ixpres.com/interval/monzo/homepage.html
|"...I had broken thru the lattice barrier..."|
| - Erv Wilson |
--------------------------------------------------

Joe, you misunderstood. Carl meant intervals, not ratios. If you start with
the otonal tetrad, you'll get the intervals in Carl's list.

Well, intervals are ratios. What I meant was, Carl's ratios meant intervals,
not pitches. He should have used ":" instead of "/" but I'm certainly not
consistent about that either.

In TD 464.3 Carl Lumma wrote (about H. Partch):

>I think in some respects this over-plays 43. Partch didn't consider it
>that special.

In TD 464.10 Paul Erlich replied:

>When he wrote Genesis, he adopted it and mapped its resources in great detail.

...and Partch was at the least curious (and, also at least, furious) with
those that latched onto 43 as a 'magic number'. After all, that's just
where *he* happened to stop. In a set of correspondences between himself
and John Cage in 1967 Partch wrote:

"Again however, if you dare to mention that number 43 you are
deliberately misrepresenting me. It is the one-half truth of the one-fourth
factor. And I shall *curse* you."

"You have been cursed before, but never by me, and if you are cursed
by me there will be a difference."

If nothing else, he was pretty clear in his determination. While the
theoretical underpinnings may have been delineated in Delusion (back in
1949 and the later edition in 1974), the years of _actual composition_ may
have pointed him to *his* best use of the tonality diamond / JI resources
he had set up, more than any conceptual materials the raw numbers would
have led to.

Paul, you've written at length many times on various matter regarding
acoustic properties, physical manifestations of sonic resources,
experiments, studies, test results. (See Ex. 1.)

[Ex. 1. (from the same digest): "This is very tricky -- it involves
segregating the noise and the amplitude fluctuations of the partials from
the signal to be Fourier transformed -- a process which involves some
subjective judgment (remember, a Fourier transform will give you a set of
pure tones that reproduce any original signal, but these pure tones always
have constant amplitude).]

Did your improvisations with that guy playing the garden hose shed any
light, in support or negation, regarding theories you might have had
previous to the garden hose performance?

Just curious.

Hey everyone, if for some reason the networks crash around the globe, have
a great 2K. OK?

Cheers,
Jon
`'`'`'`'`'`'`'`'`'`'`'`'`'`'`'`'`'`'`'`'`'`'`'`'`'`'`'`'`'`'`'`'`'`
Jonathan M. Szanto : Corporeal Meadows - Harry Partch, online.
jszanto@adnc.com : http://www.corporeal.com/
`'`'`'`'`'`'`'`'`'`'`'`'`'`'`'`'`'`'`'`'`'`'`'`'`'`'`'`'`'`'`'`'`'`

I'm certainly not putting any more importance on Partch's 43-tone scale than
Wilson did. But Partch did stop at 43 when he wrote Genesis, and if you
reduce the 43 to 41 the same way Wilson did, I've shown that this choice is
a periodicity block. By now I've shown that many, many important JI scales
are periodicity blocks.

Jon, all JI scales involve a sort-of-arbitrary decision of where to stop.
What I've shown is, this decision is not completely arbitrary, it almost
always seems to conform to a periodicity-block construction.

>Paul, you've written at length many times on various matter regarding
>acoustic properties, physical manifestations of sonic resources,
>experiments, studies, test results. (See Ex. 1.)

>[Ex. 1. (from the same digest): "This is very tricky -- it involves
>segregating the noise and the amplitude fluctuations of the partials from
>the signal to be Fourier transformed -- a process which involves some
>subjective judgment (remember, a Fourier transform will give you a set of
>pure tones that reproduce any original signal, but these pure tones always
>have constant amplitude).]

>Did your improvisations with that guy playing the garden hose shed any
>light, in support or negation, regarding theories you might have had
>previous to the garden hose performance?

The guy with the garden hose is one of the best microtonal composers I've
ever heard, Randy Winchester. Have you heard his music? Anyway, very funny,
so what's your point?

>>True, I forgot that some chords are temperament-only. But a composer
>>demanding the sort of accuracy we're talking about wouldn't want them.
>
>You mean a composer satisfied with a 2.7� accuracy would want chords only
>possible with 2.7� errors?

No, I mean a composer dis-satisfied with 2.7� accuracy would not want
chords with greater mistuning. All the "magic" chords depend on a good
deal of overall mistuning.

>>I think in some respects this over-plays 43. Partch didn't consider it
>>that special.
>
>When he wrote Genesis, he adopted it and mapped its resources in great
>detail.

When he was a composer, he denied it had any special properties.

>>11/10 and 10/9? Wilson did it to commensurate it with modulus 41. Is that
>>important to us?
>
>Remember how I was saying that reasonable PBs are CS?

Yes. What does CS have to do with 11/10 and 10/9?

-Carl

In TD 464.14, I presented a table which showed
how to find the small-integer ratios present within
any JI system. A few of the numbers in the table
were misplaced, so here's the correct version, with
dashes to indicate the non-integer ratios.

/ 1 70 : 72 : 75 : 84 : 90 : 105
/ 2 35 36 - 42 45 -
/ 3 - 24 25 28 30 5
/ 4 - 18 - 21 - -
/ 5 14 - 15 - 18 21
/ 6 - 12 - 14 15 -
/ 7 10 - - 12 - 15
/ 8 - 9 - - - -
/ 9 - 8 - - 10 -
/10 7 - - - 9 -
/12 - 6 - 7 - -
/14 5 - - 6 - -
/15 - - 5 - 6 7
/16 - - - - - -
/18 - 4 - - 5 -
/20 - - - - - -
/21 - - - 4 - 5
/24 - 3 - - - -
/25 - - 3 - - -
/27 - - - - - -
/28 - - - 3 - -
/30 - - - - 3 -
/32 - - - - - -
/35 2 - - - - 3
/36 - 2 - - - -

Paul Erlich responded to my posting:

> [Paul Erlich, TD 464.15]
> Joe, you misunderstood. Carl meant intervals, not ratios.
> If you start with the otonal tetrad, you'll get the intervals
> in Carl's list.

> [Paul Erlich, TD 464.16]
> Well, intervals are ratios. What I meant was, Carl's ratios
> meant intervals, not pitches. He should have used ":" instead
> of "/" but I'm certainly not consistent about that either.

Oops, my bad again. OK, I see what Carl meant now.

-monz

Joseph L. Monzo Philadelphia monz@juno.com
http://www.ixpres.com/interval/monzo/homepage.html
|"...I had broken thru the lattice barrier..."|
| - Erv Wilson |
--------------------------------------------------

Daniel Wolf wrote,

>No, we're all wrong. Looking through the "Andrus" through "Vath" set of
>keyboard mappings, you can see Wilson trying out all the possibilities,
>including 12/11+11/10, 11/10+10/9, and 10/9+9/8 (the last to add 13s to the

>diamond).

Where can one find these? Are they all 41 tones? Anyway, I think it telling
that Wilson settled on the middle one for Xenharmonikon.

>>>True, I forgot that some chords are temperament-only. But a composer
>>>demanding the sort of accuracy we're talking about wouldn't want them.
>
>>You mean a composer satisfied with a 2.7� accuracy would want chords only
>>possible with 2.7� errors?

>No, I mean a composer dis-satisfied with 2.7� accuracy would not want
>chords with greater mistuning. All the "magic" chords depend on a good
>deal of overall mistuning.

But I was only talking about chords possible in Keenan's tuning described in
the subject of this thread.

>Yes. What does CS have to do with 11/10 and 10/9?

By identifying 10/9 with 11/10, Wilson was able to render Partch's scale as
a linear series -- a modified MOS -- a CS.

Carl wrote,

>>>True, I forgot that some chords are temperament-only. But a composer
>>>demanding the sort of accuracy we're talking about wouldn't want them.

I wrote,

>>You mean a composer satisfied with a 2.7� accuracy would want chords only
>>possible with 2.7� errors?

I meant "wouldn't want".

>I'm certainly not putting any more importance on Partch's 43-tone scale than
>Wilson did.

Talking with him, he puts too much importance on it.

>But Partch did stop at 43 when he wrote Genesis, and if you reduce the 43 to
>41 the same way Wilson did, I've shown that this choice is a periodicity
>block. By now I've shown that many, many important JI scales are periodicity
>blocks.

You mean "reasonable" periodicity blocks? Can't everything be viewed as a
periodicity block?

-Carl

>>But Partch did stop at 43 when he wrote Genesis, and if you reduce the 43
to
>>41 the same way Wilson did, I've shown that this choice is a periodicity
>>block. By now I've shown that many, many important JI scales are
periodicity
>>blocks.

>You mean "reasonable" periodicity blocks? Can't everything be viewed as a
>periodicity block?

Yes, I should have said "reasonable", but I guess that should go without
saying: the unison vectors must be _small_ intervals (I like them to be
smaller than any intervals in the scale, but even Fokker wasn't quite that
strict).

>>No, I mean a composer dis-satisfied with 2.7� accuracy would not want
>>chords with greater mistuning. All the "magic" chords depend on a good
>>deal of overall mistuning.
>
>But I was only talking about chords possible in Keenan's tuning described in
>the subject of this thread.

Fine: Somebody dissatisfied with 2.7� mistuning would be dissatisfied with
a 2.7� mistuning.

But sorry I tried to weasel out. If everything was covered by their
interest in accurate JI, I wouldn't have had to list extra intervals or
fewer buttons. So:

"The composer interested in accurate JI, with no need of more intervals,
fewer buttons, or temperament-only chords, would have no reason to use
Keenan's scale."

I might not be one of these people, but they do exist, and their viewpoint
is defensible. That's all I was saying (in answer to Dave's question).

>>Yes. What does CS have to do with 11/10 and 10/9?
>
>By identifying 10/9 with 11/10, Wilson was able to render Partch's scale
>as a linear series -- a modified MOS -- a CS.

Right. Almost forgot. We decided that your "good" periodicity blocks are
CS, right?

-Carl

Carl Lumma wrote,

>Right. Almost forgot. We decided that your "good" periodicity blocks are
>CS, right?

So it seems -- and all of Fokker's were CS too.