Tuning-Math messages 353 - 377

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Message: 353

Date: Sun, 25 Jun 2000 08:41:33

Subject: [tuning-math] better than ratios (was: Re: Hypothesis revisited)

From: monz

> ----- Original Message ----- 
> From: Paul Erlich <paul@s...>
> To: <tuning-math@xxxxxxxxxxx.xxx>
> Sent: Monday, June 25, 2001 4:27 AM
> Subject: [tuning-math] Re: Hypothesis revisited
>
>
> --- In tuning-math@y..., "D.Stearns" <STEARNS@C...> wrote:
>
> > (Incidentally, I think the 135/128 "major chroma" is a
> > chromatic unison vector of the so-called miracle generator at
> > two-dimensions; with 34171875/33554432 being the commatic unison
> > vector if the generator is taken to a 10- or 11-tone MOS.)
> 
> I see the MIRACLE scales as needing three or four unison vectors
> each, since they live in a 7- or 11-limit lattice (i.e., they're
> 3D or 4D).


For the benefit of anyone else who (like me) got lost in this
thread, Dan's commatic unison vector is a 5-limit one, expressible
in prime-factor notation (much better, I think) as 2^-25 * 3^7 * 5^6,
or [-25 7 6] (which is how Graham writes it).

Dan's chromatic unison vector (also 5-limit) is 2^-7 * 3^2 * 5^1
= [-7 2 1].

So much better than ratios...


-monz
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Message: 354

Date: Sun, 25 Jun 2000 08:45:13

Subject: Re: 41 "miracle" and 43 tone scales

From: monz

> ----- Original Message ----- 
> From: Paul Erlich <paul@s...>
> To: <tuning-math@xxxxxxxxxxx.xxx>
> Sent: Monday, June 25, 2001 4:38 AM
> Subject: [tuning-math] Re: 41 "miracle" and 43 tone scales
> 
>
> --- In tuning-math@y..., "monz" <joemonz@y...> wrote:
> 
> > It was Erv Wilson who hypothesized that Partch was intuitively
> > "feeling out" a version of 41-EDO where two of the pitches could
> > imply either of a pair of ratios (12/11 and 11/10, and their
> > "octave"-complements).
> 
> Actually, the pair was 11/10 and 10/9 . . . you don't get a
> PB or CS the other way.


OK, I understand that *theoretically* this is the elegant comparison.

But we had a discussion about this around two years ago...

Didn't Daniel Wolf present cases in Partch's actual compositions
where either pair could be interchangeable?  That's what I remember.

(I should have mentioned it the first time around... my bad.)


-monz
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Message: 355

Date: Sun, 25 Jun 2000 08:49:31

Subject: Re: pairwise entropy minimizer

From: monz

> ----- Original Message ----- 
> From: Paul Erlich <paul@s...>
> To: <tuning-math@xxxxxxxxxxx.xxx>
> Sent: Monday, June 25, 2001 4:35 AM
> Subject: [tuning-math] Re: pairwise entropy minimizer
>
>
> --- In tuning-math@y..., Carl Lumma <carl@l...> wrote:
> > Back in the day, Paul Erlich was working on finding scales for
> > which the sum of the harmonic entropy of their dyads was low.
> > ...
> > (1) There were ever results for other cardinalities.
> 
> Oh yes . . . by the time I got to 12 notes, I was finding that
> the program was getting "stuck" in some kind of higher-dimensional
> "crevices" leading to curious 12-tone well-temperaments which 
> were not even local minima . . . they could be nudged closer
> to 12-tET without ever increasing the total dyadic harmonic
> entropy at any stage. Monz made a webpage of these well-temperaments.
> This was all posted to the tuning list . . . you'll have to dig
> through the archives.


Uh-oh... apparently my webpages must be getting "stuck in some kind
of higher-dimensional crevices"!!!  This seems to be another of the
"lost Monzo webpages".

Again, Paul, if you can remember any text from this page I can
probably find it and make a prominent link.


-monz
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Message: 357

Date: Mon, 25 Jun 2001 16:39:33

Subject: Re: Hypothesis revisited

From: carl@l...

--- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:
> Progress seems to have halted on the paper that was to introduce 
> MIRACLE . . .
/.../
> If we can do the following math problem, we'll be fine:
> 
> Given a k-by-k matrix, containing k-1 commatic unison vectors and 1 
> chromatic unison vector, delimiting a periodicity block, find:
> 
> (a) the generator of the resulting WF (MOS) scale;
> 
> (b) the integer N such that the interval of repetition is 1/N
> octaves.

Can somebody fill me in on what is meant by "interval of
repetition" here?

-C.


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Message: 358

Date: Mon, 25 Jun 2001 16:45:16

Subject: Re: pairwise entropy minimizer

From: carl@l...

--- In tuning-math@y..., "M. Edward Borasky" <znmeb@a...> wrote:
> Hmmm ... multi-dimensional optimization isn't a particularly
> difficult problem, as long as the function to be optimized is
> reasonably well behaved.

IIRC, that's the problem with harmonic entropy.

-Carl


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Message: 359

Date: Mon, 25 Jun 2001 16:50:30

Subject: Re: Hypothesis revisited

From: Dave Keenan

--- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:
> > > > 2. Masses of people over centuries have effectively given us a 
> > > short 
> > > > list of those they found useful.
...
> I am [objecting to the above sentence].

I mean the ancient scales that are still in popular use today in 
various cultures. eg. "meantone" diatonic. Arabic scales. Various 
pentatonics. Gamelan scales.

> > I'll adress the 
> > second. Graham Breed (and George Secor) have shown that MIRACLE_41 
is 
> > almost identical to several of Partch's scales.
> 
> Eh . . . not quite.

Err Paul, "almost" is a synonym for "not quite". See my post to the 
tuning list entitled "Partch's scales on the Miracle keyboard".

> > I can't help seeing 
> > Partch's various scales as gropings towards either Canasta
> 
> Don't see it.

No. I was wrong there.

> > or 
> > MIRACLE-41.
> 
> Toward modulus-41, yes . . . with many other generators functioning 
as well as, if not better than, 
> the 4/41 (MIRACLE) generator.

No. I'm talking about Miracle-41 and the 7/72 oct generator. 4/41 oct 
is only borderline Miracle.

> Yes, Dave, we both want to "rule out" the MOSs with no 
approximations to any JI 
> intervals/chords (if such a thing is possible). That is where we 
(the originators of "MIRACLE") 
> differ from Dan Stearns (at least in the viewpoint that goes behind 
this paper we're 
> contemplating). But that still leaves a great number of 
possibilities, as Robert Valentine, for 
> example, has been finding.

Oh sure. I was assuming you had read Dan's post and my response to it, 
and were referring to that. Sorry.

-- Dave Keenan


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Message: 361

Date: Mon, 25 Jun 2001 16:58:55

Subject: Re: 41 "miracle" and 43 tone scales

From: Dave Keenan

--- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:
> He stopped at 43 in order to make a melodically fairly even scale. 
With 10/9 and 11/10 seen as 
> a commatic pair (the unison vector involved is 100:99), and their 
octave complements another 
> such pair, Partch's scale is a 41-tone periodicity block -- or what 
Wilson calls a "Constant 
> Structure".

I think George Secor, Graham Breed and Dave Keenan disagree with this 
analysis, preferring one based on filling in the the diamond gaps 
using rationalised Miracle generators. See
Yahoo groups: /tuning/message/25575 *

Does anyone know if Partch regularly used any of the many approximate 
JI intervals in his scale such as those with only a 224:225 or 384:385 
error (less than 8 cents)?

-- Dave Keenan


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Message: 362

Date: Mon, 25 Jun 2001 17:09:27

Subject: Re: Hypothesis revisited

From: Dave Keenan

--- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:
> --- In tuning-math@y..., "Dave Keenan" <D.KEENAN@U...> wrote:
> > Actually, with the 10-note 369c MOS, I was looking for a MOS scale 
> > that Paul would have difficulty finding unison-vectors for, that 
are 
> > anything like unisons. i.e. This one was meant to have _big_ UVs, 
and 
> > not to contain any good approximations to SWNRs.
> > 
> > Are you asking us to find a linear temperament that treats those 
> > unison vectors (49/40 and 4375/4096) as commas, and to tell you 
how 
> > "good" it is relative to the usual JI criteria.
> 
> I think Dan just found unison vectors for your example, Dave!

If that's the case, then it makes my point quite well. Isn't it just a 
little ridiculous to refer to intervals of 351c and 114c as "unison" 
vectors or "commas"?

-- Dave Keenan


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Message: 363

Date: Mon, 25 Jun 2001 17:16:05

Subject: Re: 41 "miracle" and 43 tone scales

From: Dave Keenan

--- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:
> Graham and Dave, Wilson knew Partch, and his mappings for the 
[43-tone scale] to Modulus-41 and 
> Modulus-72 keyboards did not use the MIRACLE generator, but rather 
other generators.

Which ones?

> So I 
> don't see how one could say that Partch was using, or implying 
MIRACLE, in any way 
> whatsoever.

All that means is that Partch wasn't intentionally using Miracle and 
that Wilson missed the fact that Partch's scales imply it.

-- Dave Keenan


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Message: 364

Date: Mon, 25 Jun 2001 17:25:33

Subject: Re: 41 "miracle" and 43 tone scales

From: Dave Keenan

--- In tuning-math@y..., "monz" <joemonz@y...> wrote:
> > > It was Erv Wilson who hypothesized that Partch was intuitively
> > > "feeling out" a version of 41-EDO where two of the pitches could
> > > imply either of a pair of ratios (12/11 and 11/10, and their
> > > "octave"-complements).
> > 
> > Actually, the pair was 11/10 and 10/9 . . . you don't get a
> > PB or CS the other way.
> 
> 
> OK, I understand that *theoretically* this is the elegant 
comparison.
> 
> But we had a discussion about this around two years ago...
> 
> Didn't Daniel Wolf present cases in Partch's actual compositions
> where either pair could be interchangeable?  That's what I remember.

It's interesting that Miracle distinguishes all three of these ratios, 
as Partch did.

11:12 is  -9 generators
10:11 is  22 generators
 9:10 is -19 generators

-- Dave Keenan


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Message: 365

Date: Mon, 25 Jun 2001 17:38:34

Subject: Re: Hypothesis revisited

From: Dave Keenan

--- In tuning-math@y..., carl@l... wrote:
> Can somebody fill me in on what is meant by "interval of
> repetition" here?

It's just Paul inventing yet another term for what has been called 
(ill advisedly when relating to MOS)
  formal octave
  interval of equivalence
and more sensibly called
  period
  interval of periodicity

It gets a little ridiculous referring to 1/29 octave as a formal 
octave or an interval of equivalence, as in Graham's 15-limit 
temperament.

-- Dave Keenan


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Message: 366

Date: Sun, 25 Jun 2000 12:58:27

Subject: Re: 41 "miracle" and 43 tone scales

From: monz

I'm replying here to two of Graham's posts about Partch and MIRACLE.

> ----- Original Message -----
> From: <graham@m...>
> To: <tuning-math@xxxxxxxxxxx.xxx>
> Sent: Monday, June 25, 2001 2:53 AM
> Subject: [tuning-math] Re: 41 "miracle" and 43 tone scales
>
>
> One question is, how much did Partch know about Miracle when he drew up
> that original, unpublished scale?  It may be stretching credulity to
> suggest he worked it all out, and then pretended it was pure JI.  But the
> criteria he was using may well have matched those that are enshrined in
> Miracle.  Roughly equal melodic steps will of course favour an MOS.  And
> he would have been able to hear the intervals that were almost just by
> Miracle approximations.  And so he could have chosen the extra notes to
> maximise these consonances.
>
> In which case, why did he change his mind later?  I think it was to get
> more modulation by fifths in the 5-limit plane.  With experience, he
> decided this was more important than matching the consonances.
>
> The limitations on modulation by fifths is one of the problems with
> Miracle, at least in a traditional context.  Boomsliter and Creel's
> theories work very well with schismic, but not at all well with Miracle,
> temperament.
>


> ----- Original Message -----
> From: <graham@m...>
> To: <tuning-math@xxxxxxxxxxx.xxx>
> Cc: <gbreed@c...>
> Sent: Monday, June 25, 2001 7:03 AM
> Subject: [tuning-math] Re: 41 "miracle" and 43 tone scales
>
>
> Oh, come come.  If Partch was ever feeling towards Miracle he would have
> stopped doing so long before Wilson came up with his Modulus-41 ideas.
> That the scale works so well with 41 and 72 does imply Miracle.  Then
> again, simply using 11-limit JI implies Miracle.
>
> It is interesting that 31, 41 and 72 don't get a mention in Genesis.
> Deliberate avoidance of temperaments he can't dismiss so lightly?  You
> decide!


Graham, you know that I also love speculation!
I'm very impressed by yours here.

John Chalmers is the subscriber on this list who can really
document the relationship between Secor and Partch.  (Perhaps
we should also post a query on another list for Kraig Grady?)

I do know, however, that their meeting ocurred quite late in
Partch's life.  Partch lamented that Secor's Scalatron was the
instrument he had always wanted, but it came along too late to
do him any good.  This was probably early 1970s, possibly late 1960s.


_Genesis_ was published in 1947 or 1949 [1] (1st ed.) and
1974 (2nd ed.), and the only substantial changes in the 2nd edition
concerned Partch's new instruments.  The theoretical and historical
sections of the book remained virtually intact.


So I'm certain beyond any doubt that Partch was not *consciously*
aware of MIRACLE before the late 1960s. (note my emphasis)

But Graham's speculations are intriguing, and I'm fairly convinced
by them that Partch *intuitively* understood the MIRACLE concept
and perhaps was indeed guided in constructing his 43-tone scale
by some of the additional "senses" in which the 14 new (and
original 29) pitches could be taken in MIRACLE.

Daniel Wolf, who has had the opportunity to study Partch's
scores in *much* greater depth than I have, has remarked on how
Partch did not always construct his harmonies according to the
lowest-odd-integer hexadic theory presented in _Genesis_.
So perhaps some of these "nonstandard" usages *do* conform
to MIRACLE-like approximations.


Partch's 14 additional pitches are, as Graham correctly states,
primarily an expansion of the Tonality Diamond in the prime-factor-3
dimension, which Graham notes is *not* a feature of MIRACLE.

I've noted before how I thought it was a paradox that for all
his vitriolic abrogation of Pythagoreanism, Partch took exactly
this route in expanding his pitch gamut.  It seems that he valued
*something* about traditional music-theory after all, and that
"something" is, again as Graham points out, modulation or
root-movement by 3:2s.


About the equal temperaments discussed in _Genesis_:

First of all, I should say that I was simply writing from memory
before.  Now I have the book in front of me, and there are indeed
some ETs that I left out.  I'll correct that omission abundantly
now.


Partch (1974, p 417) does make this interesting general observation:

> Fundamentally, equal temperaments are based upon and deduced
> from Pythagorean "cycles," in whole or part.

He opens his chapter on equal-temperaments with a long and
scathing diatribe against 12-EDO, which, by this point in the
book, should not surprise the reader.

Then he discusses the 'First Result of Expansion - "Quartertones"'.
Upon mentioning Carillo, Partch also thus mentions 48- and 96-EDO.

But he actually does go into a little detail about 24-EDO, and
he's even generous enough about its potential to say that
'As a temporary expedient, as an immediately feasible method
of creating new musical resources, "quartertones" are valuable'.
He mentions Haba [which should be spelled Hába], Hans Barth, and
Mildred Couper and their use of dual regular keyboards, and Meyer
and Moellendorf and their new keyboards.

Then Partch breifly discusses Busoni and 36-EDO, which he characterizes
as "another Polypythagoreanism in tempered expression".

In the middle of this text, on p 430, is Partch's comparative table
of tunings.  I will come back to say more about this table after
describing the rest of the text.

Next comes the discussion of Yasser's 19-EDO, then finally 53-EDO.

About Yasser's proposal, Partch emphasizes that its goal is
not the betterment of intonation, but simply an expansion of
scalar resources.  He notes the improved approximations to
5- and 7-limit ratios, and also that "The ratios of 7 are somewhat
better also, but still with a maximum falsity of 21.4 cents
(33.1 cents in twelve-tone temperament).  The ratios of 11 are
not represented at all".  Actually, 19-EDO's closest approximations
to the 11-limit ratios are all between +/- 17.1 and 31.5 cents,
significantly better than 12-EDO's.

Partch had mentioned in "Chapter 15: A Thumbnail Sketch of the
History of Intonation" that King Fang (in China) and Mersenne,
Kircher, and Mercator (in Europe) all proposed this tuning.
In the middle of the discussion of 53-EDO is a digression
"On the Matter of Hearing a 2-Cents Falsity".

Partch notes that 53-EDO is indeed extremely close to 3- and
5-limit JI, but does not consider it suitable for his own use
as it offers little improvement in approximating the 7- and
11-limit ratios he wanted to use.

Finally he examines the keyboard proposals of Nicolaus Ramarinus
(1640) [2], Bosanquet (no date given by Partch, c. 1875?), and
Jas. Paul White (1883) [3].

And that wraps up Partch's "Chapter 17: Equal Temperaments".


Now, back to that comparative table...

Partch's table on p 430 compares his Monophonic 43-tone scale with,
in order:

- 12-EDO,
- 12-tone Pythagorean: a 3^(-6...+5) system,
- 16-tone Meantone: a cycle of implied "5ths" 3^(-5...10) tuned in
    1/4-comma meantone, the pair of notes at either end of the cycle
    being the additional notes on Handel's organ (according to Partch),
- 17-tone Arabic: a Pythagorean 3^(-12...+4) system,
- 19-EDO,
- 24-EDO,
- 31-EDO,
- 36-EDO,
- 53-EDO.

First, I should note that there are obviously tunings here (the
second, third, and fourth) which are not ETs.  Partch had already
discussed these in his "Chapter 16: Polypythagoreanism".

But - SURPRISE! - there's 31-EDO in the table, but
WITH NO MENTION WHATSOEVER IN THE TEXT!!

And I checked all the other chapters in _Genesis_... there's no
mention at all of Huyghens, Fokker, or anything else concerning
31-EDO.

Now THAT'S interesting!  ... And I never noticed it before,
having been duped by 31-EDO's appearance in that table into
thinking that Partch said something about it somewhere.

So Graham is right that, except for this inconspicuous little
tabulation, Partch does not mention 31-, 41- or 72-EDO.
Good detective work, Graham!!!


NOTES

[1]  I asked before (on the main list) about the actual publication
date. I don't remember now what the outcome was, but I've seen it
listed in catalogs under both dates.  The original Preface
is dated April 1947, but the copyright date is 1949.


[2] About Ramarinus, Partch says:
     > the "tone" (9/8) was divided into nine "commas",
     > according to Hawkins [_History of the Science and
     > Practice of Music, vol 1, p 396].  The fifty-third part
     > of 2/1 is approximately the width of the "comma" of
     > Didymus, 81/80 (21.5 cents; see table above), and since
     > six 9/8's are larger than a 2/1 by approximately this
     > interval (the "comma" of Pythagoras, 23.5 cents), this
     > procedure would result in a fifty-three-tone scale.

Of course, we are well aware that the 9-commas-per-tone
temperament works out to exactly 55-EDO, which is a meantone,
whereas 53-EDO is quasi-just.  This choice probably reflects
Partch's own bias; I'd bet that Ramarinus most likely meant
something more like 55-EDO.


[3]  Paul (or anyone else in Boston): It still says in the 1974
edition of _Genesis_ that White's harmonium was housed in a
practice room at New England Conservatory, and that Partch
examined it in 1943.  I've found page references in _Genesis_
that should have been renumbered from the 1st edition and weren't,
so perhaps this is a story that also should have been updated.
Please... go take a look and let us know!



-monz
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"All roads lead to n^0"





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Message: 368

Date: Mon, 25 Jun 2001 21:46:57

Subject: Re: Hypothesis revisited

From: Paul Erlich

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

> The unison vectors I used for 31+41n are:
> 
> [[ 2 -2  2  0 -1]
>  [-7 -1  1  1  1]
>  [-1  5  0  0 -2]
>  [-5  2  2 -1  0]]
> 
Getting rid of the first column:

[-2  2  0 -1]
[-1  1  1  1]
[ 5  0  0 -2]
[ 2  2 -1  0]

the resulting FPB is

       cents         numerator   denominator
       38.906           45           44
       70.672           25           24
       79.965          288          275
       111.73           16           15
       150.64           12           11
        182.4           10            9
       203.91            9            8
       235.68           55           48
       262.37           64           55
       294.13           32           27
       315.64            6            5
       347.41           11            9
       386.31            5            4
       425.22          225          176
       427.37           32           25
       466.28           72           55
       498.04            4            3
       536.95           15           11
       551.32           11            8
       590.22           45           32
       609.78           64           45
       648.68           16           11
       663.05           22           15
       701.96            3            2
       733.72           55           36
       772.63           25           16
       774.78          352          225
       813.69            8            5
       852.59           18           11
       884.36            5            3
       905.87           27           16
       937.63           55           32
       964.32           96           55
       996.09           16            9
       1017.6            9            5
       1049.4           11            6
       1088.3           15            8
         1120          275          144
       1129.3           48           25
       1161.1           88           45
         1200            1            1

> 
> That uses 100:99 as the chromatic UV.  The more obvious choice 
would be a 
> schisma, so that
> 
> [[-15 8  1  0  0]
>  [-7 -1  1  1  1]
>  [-1  5  0  0 -2]
>  [-5  2  2 -1  0]]
> 
> would give the same results.

Again getting rid of the first column, this is

[ 8  1  0  0]
[-1  1  1  1]
[ 5  0  0 -2]
[ 2  2 -1  0]

giving the FPB

       cents         numerator   denominator
       31.767           55           54
       60.412          729          704
       92.179          135          128
       111.73           16           15
        143.5           88           81
       172.14          243          220
       203.91            9            8
       235.68           55           48
       262.37           64           55
       296.09         1215         1024
       315.64            6            5
       347.41           11            9
       386.31            5            4
       407.82           81           64
       439.59          165          128
       466.28           72           55
       498.04            4            3
       519.55           27           20
       558.46          243          176
       590.22           45           32
       609.78           64           45
        643.5         1485         1024
       670.19           81           55
       701.96            3            2
       733.72           55           36
       760.41          256          165
       794.13          405          256
       813.69            8            5
       845.45           44           27
       884.36            5            3
       905.87           27           16
       937.63           55           32
       964.32           96           55
       996.09           16            9
       1017.6            9            5
       1056.5           81           44
       1088.3           15            8
       1107.8          256          135
       1141.5          495          256
       1168.2          108           55
         1200            2            1

The difference between these two scales is

     numerator     denominator
         242         243
        2187        2200
        4125        4096
           1           1
         242         243
        2187        2200
           1           1
           1           1
           1           1
       32805       32768
           1           1
           1           1
           1           1
          99         100
        4125        4096
           1           1
           1           1
          99         100
         243         242
           1           1
           1           1
       16335       16384
         243         242
           1           1
           1           1
        4096        4125
       91125       90112
           1           1
         242         243
           1           1
           1           1
           1           1
           1           1
           1           1
           1           1
         243         242
           1           1
        4096        4125
        4125        4096
         243         242
           2           1

So if the schisma (32805:32768) is the _chromatic_ unison vector of 
one of these scales, the two scales are _not_ equivalent, even up to 
arbitrary transpositions by _commatic_ unison vectors.



> I can't check this now, as I don't have 
> Numerical Python installed, or even Excel.  But you may be able 
to.  Try 
> inverting this matrix, and multiplying it by its determinant:
> 
 [[ 1  0  0  0  0]
  [-15 8  1  0  0]
  [-7 -1  1  1  1]
  [-1  5  0  0 -2]
  [-5  2  2 -1  0]]

The determinant is -41, and the inverse is

[      1            0            0            0            0  ]    
[    65/41         6/41        -2/41        -1/41        -2/41]    
[    95/41        -7/41        16/41         8/41        16/41]    
[   115/41        -2/41        28/41        14/41       -13/41]    
[   142/41        15/41        -5/41       -23/41        -5/41]  

> The left hand two columns should be
> 
> [[ 41   0]
>  [ 65  -6]
>  [ 95   7]
>  [115   2]
>  [142 -15]]

Up to a minus sign, yes.
> 
> If they are, the two sets of unison vectors give exactly the same 
> results.

They don't!

> I think they must be, because I remember checking the 
> determinant before, and any chroma that gives a determinant of 41 
when 
> placed with Miracle commas should give this result.

Something must be wrong with one of your assumptions.

> You most certainly do need octave-specific matrices.  Otherwise, 
that 
> left-hand column won't be there.

I see that as a good thing . . . don't you?

> There may be an algorithm that works with octave 
> invariant matrices, but it's easier to upgrade them to be 
> octave-specific, and use a common or garden inverse.

?


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Message: 369

Date: Mon, 25 Jun 2001 21:50:22

Subject: Re: 41 "miracle" and 43 tone scales

From: Paul Erlich

--- In tuning-math@y..., graham@m... wrote:
> In-Reply-To: <9h7845+e2fi@e...>
> Paul wrote:
> 
> > Graham and Dave, Wilson knew Partch, and his mappings for the 
Diamond 
> > to Modulus-41 and Modulus-72 keyboards did not use the MIRACLE 
> > generator, but rather other generators. So I don't see how one 
could 
> > say that Partch was using, or implying MIRACLE, in any way 
whatsoever.
> 
> Oh, come come.  If Partch was ever feeling towards Miracle he would 
have 
> stopped doing so long before Wilson came up with his Modulus-41 
ideas.

???

> That the scale works so well with 41 and 72 does imply Miracle.

Now you're stretching the meaning of the word "imply".

> Then 
> again, simply using 11-limit JI implies Miracle.

Now you're _really_ stretching the meaning of the word "imply"!!! :)

> It is interesting that 31, 41 and 72 don't get a mention in 
Genesis.  
> Deliberate avoidance of temperaments he can't dismiss so lightly?  
You 
> decide!

I think he was simply ignorant of these temperaments, in the 
literature he was familiar with (which concentrated on 19, 24, and 
53). Actually, 31 _is_ in his ET comparison table, isn't it?


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Message: 370

Date: Mon, 25 Jun 2001 21:55:13

Subject: Re: pairwise entropy minimizer

From: Paul Erlich

--- In tuning-math@y..., carl@l... wrote:
> --- In tuning-math@y..., "M. Edward Borasky" <znmeb@a...> wrote:
> > Hmmm ... multi-dimensional optimization isn't a particularly
> > difficult problem, as long as the function to be optimized is
> > reasonably well behaved.
> 
> IIRC, that's the problem with harmonic entropy.
> 
Huh?


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Message: 371

Date: Mon, 25 Jun 2001 22:46:03

Subject: Unison vectors and MOS

From: Graham Breed

My temperament finding program has been updated to find commatic unison vectors
as well.  I tried to get it to find the simplest ones, but it still needs some
work there.  See

<Unison vector to MOS script *>
and
<Automatically generated temperaments *>

From plugging in some of the unison vectors mentioned before, it's apparent
that we don't have as much choice in the chroma as I thought.  But it still
isn't unique for a temperament. 

-- 

             Graham

"I toss therefore I am" -- Sartre


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Message: 372

Date: Mon, 25 Jun 2001 22:36:28

Subject: Re: 41 "miracle" and 43 tone scales

From: Graham Breed

Monz wrote:

> _Genesis_ was published in 1947 or 1949 [1] (1st ed.) and
> 1974 (2nd ed.), and the only substantial changes in the 2nd edition
> concerned Partch's new instruments.  The theoretical and historical
> sections of the book remained virtually intact.

So, if "Exposition on Monophony" was1933, that's well in advance.

> But Graham's speculations are intriguing, and I'm fairly convinced
> by them that Partch *intuitively* understood the MIRACLE concept
> and perhaps was indeed guided in constructing his 43-tone scale
> by some of the additional "senses" in which the 14 new (and
> original 29) pitches could be taken in MIRACLE.

Be careful you don't get carried away with these speculations.  It seems
plausible that he was feeling for something like 41-equal but with improved
11-limit harmony.  In that case, you'd expect the result to look something like
a 41-note MOS of a good 11-limit temperament.  The scale he ends up with does
fit schismic better than Miracle.

As mathematicians, we should be aware of the dangers of imposing patterns on
data.  For the rest, I think the discussion should be taken to the main list if
you think you have a case.  Dave Keenan has already come up with some new
arguments.

> Partch's 14 additional pitches are, as Graham correctly states,
> primarily an expansion of the Tonality Diamond in the prime-factor-3
> dimension, which Graham notes is *not* a feature of MIRACLE.
> 
> I've noted before how I thought it was a paradox that for all
> his vitriolic abrogation of Pythagoreanism, Partch took exactly
> this route in expanding his pitch gamut.  It seems that he valued
> *something* about traditional music-theory after all, and that
> "something" is, again as Graham points out, modulation or
> root-movement by 3:2s.

D'alessandro also ends up with a long chain of 3:2s, and so doesn't work so
well as Miracle.

> And I checked all the other chapters in _Genesis_... there's no
> mention at all of Huyghens, Fokker, or anything else concerning
> 31-EDO.

I thought Fokker did his music theory during the Nazi occupation, hence after
the original publication of Genesis.  And Huygens' music theory wouldn't have
been known until then either.

Yasser still suggested eventual evolution to 31 though.

> Now THAT'S interesting!  ... And I never noticed it before,
> having been duped by 31-EDO's appearance in that table into
> thinking that Partch said something about it somewhere.
> 
> So Graham is right that, except for this inconspicuous little
> tabulation, Partch does not mention 31-, 41- or 72-EDO.
> Good detective work, Graham!!!

With you're detective work we can now say that he avoided *all* consistent
11-limit temperaments!

             Graham

"I toss therefore I am" -- Sartre


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Message: 373

Date: Mon, 25 Jun 2001 22:00:13

Subject: Re: Hypothesis revisited

From: Paul Erlich

--- In tuning-math@y..., "Dave Keenan" <D.KEENAN@U...> wrote:
> --- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:
> > > > > 2. Masses of people over centuries have effectively given 
us a 
> > > > short 
> > > > > list of those they found useful.
> ...
> > I am [objecting to the above sentence].
> 
> I mean the ancient scales that are still in popular use today in 
> various cultures. eg. "meantone" diatonic. Arabic scales. Various 
> pentatonics. Gamelan scales.

There are a lot of cultural accidents that lead to "popular use". And 
those Gamelan scales . . . you'd need some large unison vectors for 
those, wouldn't you?
> 
> > > I can't help seeing 
> > > Partch's various scales as gropings towards either Canasta
> > 
> > Don't see it.
> 
> No. I was wrong there.
> 
> > > or 
> > > MIRACLE-41.
> > 
> > Toward modulus-41, yes . . . with many other generators 
functioning 
> as well as, if not better than, 
> > the 4/41 (MIRACLE) generator.
> 
> No. I'm talking about Miracle-41 and the 7/72 oct generator. 4/41 
oct 
> is only borderline Miracle.

I meant 4/41 in a modulus-41, not 41-tET, sense. Doesn't the 19/72 
generator work as well for Partch's scale as the 7/72 generator?


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Message: 374

Date: Mon, 25 Jun 2001 22:02:47

Subject: Re: 41 "miracle" and 43 tone scales

From: Paul Erlich

--- In tuning-math@y..., "Dave Keenan" <D.KEENAN@U...> wrote:
> --- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:
> > He stopped at 43 in order to make a melodically fairly even 
scale. 
> With 10/9 and 11/10 seen as 
> > a commatic pair (the unison vector involved is 100:99), and their 
> octave complements another 
> > such pair, Partch's scale is a 41-tone periodicity block -- or 
what 
> Wilson calls a "Constant 
> > Structure".
> 
> I think George Secor, Graham Breed and Dave Keenan disagree with 
this 
> analysis, preferring one based on filling in the the diamond gaps 
> using rationalised Miracle generators. See
> Yahoo groups: /tuning/message/25575 *

The analyses are not necessarily incompatible!!!


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