Tuning-Math Archive Section 1: 327

Message: 327 - Contents - Hide Contents Date: Mon, 25 Jun 2001 03:53:27 Subject: Re: 41 "miracle" and 43 tone scales From: Dave Keenan --- In tuning-math@y..., jpehrson@r... wrote:

>> almost identical to several of Partch's scales. I can't help seeing >> Partch's various scales as gropings towards either Canasta or >> MIRACLE-41. I expect Partch would not have been able to distinguish >> his scales from the corresponding MIRACLE-temperament of them, > since I

>> understand someone said he couldn't distinguish one of them from >> 41-EDO. I think the fact that Partch, doing it mostly by ear, and > we,

>> doing it mostly by math, (and George Secor doing it by ???), >> essentially converged on the same thing, is no accident. >> >

> I'm getting a little confused here... Did Harry Partch use a 41-tone > scale in addition to his 43-tone scale?? He never actually > used "Miracle 41" did he?? > > _________ _______ _______ > Joseph Pehrson

Message: 328 - Contents - Hide Contents Date: Mon, 25 Jun 2001 03:56:41 Subject: Re: Hypothesis revisited From: Dave Keenan --- In tuning-math@y..., "Dave Keenan" <D.KEENAN@U...> wrote:

> I wrote:

>> Because there are zillions of MOS scales that have no relationship >> with small unison vectors. Sure you could probably always find a >> corresponding periodicity block, but these will have "unison > vectors"

>> so large as not to merit the name. >

> Try this one: A chain of 10, 369c generators, octave period.

Sorry. That should have been "A 10 note chain of 369c generators..."

Message: 329 - Contents - Hide Contents Date: Mon, 25 Jun 2001 04:04:46 Subject: Re: 41 "miracle" and 43 tone scales From: Dave Keenan Sorry about the previous message, sent by mistake. --- In tuning-math@y..., jpehrson@r... wrote:

> I'm getting a little confused here... Did Harry Partch use a > 41-tone scale in addition to his 43-tone scale??

Sure. There are two in the Scala archive, But that's not what I meant.

> He never actually used "Miracle 41" did he??

No. I didn't say that either. But he might not have noticed if someone had substituted a scale which was MIRACLE-41 plus a couple of extra notes from MIRACLE-45. Read: http://www.anaphoria.com/secor.PDF - Type Ok * [with cont.] (Wayb.) and Lattices with Decimal Notation * [with cont.] (Wayb.) and then tell me what you don't understand. Regards, -- Dave Keenan

Message: 331 - Contents - Hide Contents Date: Sun, 24 Jun 2001 22:10:45 Subject: pairwise entropy minimizer From: Carl Lumma Back in the day, Paul Erlich was working on finding scales for which the sum of the harmonic entropy of their dyads was low. He found that this was a hard problem. He had a tool which would relax a scale to a dyadic entropy minimum, but which could not find a global minimum for a given cardinality. He attempted to seed this program with random scales, hoping to find global minima by the Monte Carlo method. Last I heard, he believed that the global minimum for 5-tone scales was the usual meantone pentatonic. I can not remember if... (1) There were ever results for other cardinalities. (2) If there were significant runners-up for the 5-tone case. ...does anyone have information on this? Paul, do you prefer if this is posted to the harmonic entropy list? -Carl

Message: 334 - Contents - Hide Contents Date: Sun, 24 Jun 2001 23:33:30 Subject: Re: 41 "miracle" and 43 tone scales From: monz

> ----- Original Message ----- > From: M. Edward Borasky <znmeb@a...> > To: <tuning-math@xxxxxxxxxxx.xxx> > Sent: Sunday, June 24, 2001 8:42 PM > Subject: RE: [tuning-math] 41 "miracle" and 43 tone scales > > > Hmmm ... nowhere in Genesis of a Music do I see any reference to 41 EDO. > There are references to 19 EDO and 53 EDO IIRC, but I don't remember any > 41s.

Hi Ed. Other than 12-EDO, the others Partch discusses are 19, 36, and 53, in connection with Yasser, Busoni, and Mercator, respectively. He never hints that he would consider using any of them himself. And you're right, he says nothing about 41. 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).

> As far as I can tell, Partch started with the 28 tonalities -- 12 > primary and 16 secondary. That makes a 29-note scale from 1/1 through 2/1. > Then he filled in some of the larger the gaps in this scale with notes from > the secondary tonalities. It's never been clear to me why he stopped at 43, > though, rather than completing all 28 tonalities.

The 29-tone scale comes directly from the 11-limit Tonality Diamond, and only involves the secondary tonalities in that they are *partially* present within that scale. Partch got those 29 pitches from the 12 primary tonalities: 6 otonal and 6 utonal hexads. Because 1/1 is represented 5 times, and 4/3 and 3/2 each represented twice, the potential (6*6) = 36 different pitches are reduced to 36 - 5 - 2 = 29. Partch was essentially satisfied with the harmonic possibilites of this 29-tone scale, since the formed his neat and compact Tonality Diamond. He "filled in the gaps" mainly because he wanted a certain measure of melodic evenness in the basic scale which formed essentially his full set of resources. (I qualify this with "essentially" because there are many, many other ratios which do in fact appear in Partch's compositions. As he himself emphatically reiterated, he considered the 43 pitches to be only a peripheral aspect of his whole technique, and it was a limitation which he often ignored.) Once he reached the point where the whole 2:1 was divided into approximately equal steps, he stopped. That division happened to be into 43 different degrees. You're correct that the notes filling the gaps were taken from expansion of the pitch-space into the secondary tonalities, so that the new pitches would form familiar harmonic relationships to the primary ones. But Partch's main consideration in choosing the new pitches was to divide the melodic gaps in the scale into the appropriately-spaced intervals in terms of *pitch-height*. So his goal was not to complete the secondary tonalities. If he had chosen more than the 14 secondary pitches he did choose, he would have ended up melodically with either less even spacing throughout one or more of the gaps, or only some of 43-tone steps divided in half and others not divided, which would give a scale still less even. And in definite answer to Joe Pehrson's question: NO, Partch *never* considered MIRACLE or any other temperament. One Partch made the break with 12-EDO around 1929 or so, he never wrote any other music in non-JI tunings, with the sole exception of the piano parts of _Bitter Music_ in the mid-1930s, and which "piece" was really a private journal and which he thought he had destroyed before he died. (_Bitter Music_ only exists now because a copy was stored on microfilm at a university and got past Partch.) *We* (Joe P., myself, Dave Keenan, Paul, Graham, Herman, and the others interested in MIRACLE) are the ones who like it's terrific emulation of Partch's scale. Hmmm... but George Secor knew Partch too. I wonder if Partch was familiar with Secor's discovery of MIRACLE...? -monz Yahoo! GeoCities * [with cont.] (Wayb.) "All roads lead to n^0" _________________________________________________________ Do You Yahoo!? Get your free @yahoo.com address at Yahoo! Mail - The best web-based email! * [with cont.] (Wayb.)

Message: 335 - Contents - Hide Contents Date: Mon, 25 Jun 2001 09:15:03 Subject: Re: Hypothesis revisited From: Dave Keenan --- In tuning-math@y..., "D.Stearns" <STEARNS@C...> wrote:

> Hi Dave K., > > <<Try this one: A chain of 10, 369c generators, octave period.>> > > Though this sounds more like a loaded mousetrap than a practical type > question... if small is all your really looking for, how about > something on the order of 49/40 and 4375/4096? > > --Dan Stearns

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 don't know how. But Graham or Paul may be able to soon.

Message: 336 - Contents - Hide Contents Date: Mon, 25 Jun 2001 10:53 +0 Subject: Re: Hypothesis revisited From: graham@m... In-Reply-To: <021301c0fd42$4989a440$77bcd33f@s...> Dan Stearns wrote:

> I think it was in Blackwood's book, I don't have right now so I can't > check, that I remember the 25/24 being called a "minor chroma" and the > 135/128 a "major chroma".

I did get hold of that book a while back, so it may be where I got the idea from. (I say "may be" because I don't consciously remember this, but as I read the book not long ago I can't claim it's a coincidence.)

> So generalizing commatic unison vectors in periodicity blocks as > chromas would seem at odds with this as the 25/24 "minor chroma" is a > chromatic unison vector in the two-dimensional diatonic periodicity > block.

Sure, but it's the *chromatic* unison vector I was going to call a "chroma" so no problem. The *commatic* unison vectors can easily enough be called "commas".

> (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.)

Don't know about this offhand. When I get home, I might plug it in. Graham

Message: 337 - Contents - Hide Contents Date: Mon, 25 Jun 2001 10:53 +0 Subject: Re: 41 "miracle" and 43 tone scales From: graham@m... In-Reply-To: <004601c0fd40$bfed2d20$4448620c@a...> monz wrote:

> Hmmm... but George Secor knew Partch too. I wonder if > Partch was familiar with Secor's discovery of MIRACLE...?

Without hard facts, all we have is speculation. Which is good, because it's much more fun that way. If Secor had shared this with Partch, I'm surprised he didn't find out about the earlier 43 note scale that fits Miracle better. I suspect if he knew Partch, he would also have known Wilson, hence learned of the 41- connection from him? 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. Graham

Message: 339 - Contents - Hide Contents Date: Mon, 25 Jun 2001 11:56 +0 Subject: Re: Hypothesis revisited From: graham@m... In-Reply-To: <9h68ft+386j@e...> Dave Keenan wrote:

> --- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:

>> Well, the picture is not that simple when you're talking about one >> of the unison vectors (the chromatic one) _not_ being tempered >> out. Then it _does_ matter which set you choose. >

> Yes. That's what I thought I said. It _does_ matter. But choosing one > to be chromatic, still doesn't uniquely determine the others does it? > (except in 5-limit). So how do you know which vectors to check angles > between?

Yes, it does matter, but the vectors aren't unique. Where you have more than one commatic vector you have a lot of freedom about which you choose. If you check the output file from my latest script, you should see this twice: mapping by steps: [[10 1] [16 1] [23 3] [28 3] [35 2]] It means there are two ways of defining 10 (or 1+10n) note Miracle. However, that may not be a good example because one of them is pathological: it actually gives a 20 note periodicity block, which is why I included it in the test. But it's the *chromatic* vector that differs, so there is more than one that works. 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]] 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. 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 left hand two columns should be [[ 41 0] [ 65 -6] [ 95 7] [115 2] [142 -15]] If they are, the two sets of unison vectors give exactly the same results. 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. The original matrix has an octave as the top row, the chroma as the next one down, and commas below that. Inverting it and taking the left hand two columns defines the prime intervals in terms of the octave and chroma. If the left hand column has a common factor, divide through by that factor. If the right hand column has a prime factor, that tells you how many equal parts you need to divide the octave into, but you don't need to worry about that yet. To get the MOS, you need to add a multiple of the left hand column to the right hand column so that it's divisible by the number of steps to the octave. This is what my program does. Use this as a new right-hand column and you have defined the octave in terms of two step sizes. You most certainly do need octave-specific matrices. Otherwise, that left-hand column won't be there. You also need to make sure the chroma is a small interval. 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. Graham

Message: 340 - Contents - Hide Contents Date: Mon, 25 Jun 2001 11:18:49 Subject: Re: Hypothesis revisited From: Paul Erlich --- In tuning-math@y..., "Dave Keenan" <D.KEENAN@U...> wrote:

>> Well, the picture is not that simple when you're talking about one >> of the unison vectors (the chromatic one) _not_ being tempered >> out. Then it _does_ matter which set you choose. >

You're right . . . the angle stuff only makes sense if two or more unison vectors are not being tempered out.

> That's a worthy aim, but it can be acheived by finding the > linear-temperaments by existing methods and working backwards to the > unison vectors. Correct. >

>>> If the question is not quite rhetorical: >>> 1. No one thinks that all linear temperaments are equally >> interesting. >> >> Erv Wilson? >

> Doesn't it seem to you, from his diagrams, that he at least considers > noble generators to be more interesting or useful or special in some > way?

We all know the special properties of noble generators . . . as far as Wilson being exclusive about them, Kraig has reported otherwise . . .

>>> 2. Masses of people over centuries have effectively given us a >> short

>>> list of those they found useful. (Popularity of Partch's scales >> would

>>> in effect tell us that MIRACLE is useful) >>

>> wha . . . wha . . . what?? >

> I assume you're not objecting to the first sentence? I am. > 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.

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

Don't see it.

> or > MIRACLE-41.

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

>> Yup! I just thought this paper would be better if it were capable of >> unifying different fields of tuning theory, and presenting a few >> new interesting scales with descriptions according to this new >> unified theory, than being some sort of attempt to crown a few >> scales with the title of "best". >

> Gimme a break Paul. Dan's already slapped me on the wrist for that. > The "political correctness police" are getting a little tedious.

I didn't see Dan's post on this, and believe me, the last thing I want to do is be politically correct.

>> >> Yes. So why don't they "flow naturally"? >

> Because there are zillions of MOS scales that have no relationship > with small unison vectors.

But the _whole idea_ of MOS -- where does that come from? Really just from looking at the diatonic scale and then generalizing. So perhaps I'm interested in showing _why_ the diatonic scale is MOS, and giving an _impetus_ for finding more MOSs . . . without taking it as an axiom that MOSs are special.

>> More than that -- I want to show that well-formedness should not >> be an "axiom" at all but could instead be derived from more >> "fundamental considerations". A JI-friendly underpinning to much >> modern scale theory. One might even include a case where >> _two_ of the unison vectors are not tempered out, and related >> this to a second-order ME scale, such as the Indian 7-out-of-22. >

> See response to previous paragraph. You can't derive MOS from JI or > vice versa. One is a horizontal melodic property, the other vertical > harmonic. Periodicity blocks may give you MOS approx-JI scales but > they won't give you the MOS non-approx-JI scales.

See above. 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.

Message: 341 - Contents - Hide Contents Date: Mon, 25 Jun 2001 11:25:21 Subject: Re: 41 "miracle" and 43 tone scales From: Paul Erlich --- In tuning-math@y..., "M. Edward Borasky" <znmeb@a...> wrote:

> Hmmm ... nowhere in Genesis of a Music do I see any reference to 41 EDO. > There are references to 19 EDO and 53 EDO IIRC, but I don't remember any > 41s.

Genesis of a Music was written very early in Partch's career. Later, he met Erv Wilson, who played Partch 41-tET and Partch couldn't distinguish it from his scale.

> As far as I can tell, Partch started with the 28 tonalities -- 12 > primary and 16 secondary. That makes a 29-note scale from 1/1 through 2/1.

The original 29-note scale is simply the Diamond -- only the primary 11-limit ratios. It has to do only with 6 Otonalities and 6 Utonalities, all containing 1/1. No secondary tonalities are explicitly involved at this stage.

> Then he filled in some of the larger the gaps in this scale with notes from > the secondary tonalities. It's never been clear to me why he stopped at 43, > though, rather than completing all 28 tonalities.

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".

Message: 342 - Contents - Hide Contents Date: Mon, 25 Jun 2001 11:27:48 Subject: Re: Hypothesis revisited From: Paul Erlich --- In tuning-math@y..., "D.Stearns" <STEARNS@C...> wrote:

> Hi Joe, > > I think it was in Blackwood's book, I don't have right now so I can't > check, that I remember the 25/24 being called a "minor chroma" and the > 135/128 a "major chroma". > > So generalizing commatic unison vectors in periodicity blocks as > chromas would seem at odds with this as the 25/24 "minor chroma" is a > chromatic unison vector in the two-dimensional diatonic periodicity > block.

No one was proposing generalizing commatic unison vectors as "chromas". They were suggesting generalizing _chromatic_ unison vectors as "chromas".

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).

Message: 343 - Contents - Hide Contents Date: Mon, 25 Jun 2001 11:35:50 Subject: Re: pairwise entropy minimizer From: Paul Erlich --- 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. > He found that this was a hard problem. He had a tool which > would relax a scale to a dyadic entropy minimum, but which could > not find a global minimum for a given cardinality. He attempted > to seed this program with random scales, hoping to find global > minima by the Monte Carlo method. Last I heard, he believed > that the global minimum for 5-tone scales was the usual meantone > pentatonic. I can not remember if... > > (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.

> > (2) If there were significant runners-up for the 5-tone case.

Yes . . . I posted a long list of the number of occurences of, and the rating of, many different pentatonic scales, obtained by starting the local minimizations from many, many random points.

> > ...does anyone have information on this?

You'll have to dig through the archives of the tuning list. Search for "relaxed".

Message: 344 - Contents - Hide Contents Date: Mon, 25 Jun 2001 11:38:27 Subject: Re: 41 "miracle" and 43 tone scales From: Paul Erlich --- 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.

Message: 346 - Contents - Hide Contents Date: Mon, 25 Jun 2001 11:39:45 Subject: Re: Hypothesis revisited From: Paul Erlich --- In tuning-math@y..., "Dave Keenan" <D.KEENAN@U...> wrote:

> --- In tuning-math@y..., "D.Stearns" <STEARNS@C...> wrote:

>> Hi Dave K., >> >> <<Try this one: A chain of 10, 369c generators, octave period.>> >> >> Though this sounds more like a loaded mousetrap than a practical > type

>> question... if small is all your really looking for, how about >> something on the order of 49/40 and 4375/4096? >> >> --Dan Stearns >

> 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!

Message: 347 - Contents - Hide Contents Date: Mon, 25 Jun 2001 11:43:01 Subject: Re: 41 "miracle" and 43 tone scales From: Paul Erlich 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.

Message: 348 - Contents - Hide Contents Date: Mon, 25 Jun 2001 11:43:52 Subject: Re: 41 "miracle" and 43 tone scales From: Paul Erlich I wrote,

> Graham and Dave, Wilson knew Partch, and his mappings for the Diamond to Modulus-41 and > Modulus-72

Oops -- I meant the Partch 43-tone scale, not the diamond.

Message: 349 - Contents - Hide Contents Date: Mon, 25 Jun 2001 15:03 +0 Subject: Re: 41 "miracle" and 43 tone scales From: graham@m... 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. 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