Tuning-Math Archive Section 1: 702

Message: 702 - Contents - Hide Contents Date: Mon, 20 Aug 2001 04:26:27 Subject: Re: Hypothesis From: carl@l...

> I looked at CLAMPITT.pdf, and it seems to me the argument that > there is something interesting about WF scales is extremely > unconvincing. Can anyone actually *hear* this? I notice that > when you talk about periodiciy blocks, you ignore this stuff > yourself, as well you might so far as I can see. > > What gives? Am I missing something? Howdy, Gene!

I doubt the "synechdochic property" (the "self-similarity" at the center of the Carey and Clampitt article) is significant, except maybe in very special kinds of musical examples and with a lot of training. In my opinion the Carey and Clampitt article amounts to some interesting ideas for algorithmic composition. I don't think MOS itself means much for the perception of melody. Rather, I think it works together, or is often confounded with other properties: () Symmetry at the 3:2. The idea is that the 3:2 is a special interval, a sort of 2nd-order octave. When a scale's generator is 3:2, MOS means that a given pattern can more often be repeated a 3:2 away. Chains of 5, 7, and 12 "fifths" are historically favored, but where are all the MOS chains of 5:4, 7:4, etc.? In my experience, MOS chains of non-fifth generators can be special too, but we should be careful not to give MOS credit for symmetry at the 3:2. () Myhill's property -- every scale interval comes in exactly two acoustic sizes. This may make it easier for listeners to track scale intervals. Consider a musical phrase that is transposed to a different mode of the diatonic scale -- it is changed with respect to acoustic intervals but unchanged with respect to scalar intervals. I think this is an important musical device that is only possible with certain kinds of scales. Myhill's property may make it easier for the listener to access such a device, but probably doesn't mean much if the scale can't support the device in the first place. Here, I believe a property called "stability" comes into play.[1] Fortunately, we can test this by listening to un-stable MOS scales. I've done some of this listening informally. -Carl [1] Rothenberg, David. "A Model for Pattern Perception with Musical Applications. Part I: Pitch Structures as Order-Preserving Maps", Mathematical Systems Theory vol. 11, 1978, pp. 199-234. Rothenberg, David. "A Model for Pattern Perception with Musical Applications Part II: The Information Content of Pitch structures", Mathematical Systems Theory vol. 11, 1978, pp. 353-372. Rothenberg, David. "A Model for Pattern Perception with Musical Applications Part III: The Graph Embedding of Pitch Structures", Mathematical Systems Theory vol. 12, 1978, pp. 73-101.

Message: 703 - Contents - Hide Contents Date: Mon, 20 Aug 2001 04:45:13 Subject: Re: Mea culpa From: carl@l... Forgive me for stepping in here guys, but I'm online and figure that sooner is better...

> I must say I am surprised and pleased with the attitude around > here. The one time I tried to publish about music, the Computer > Music Journal turned it down as "too mathematical", so I thought > people were a little allergic. I would like a copy of that paper > now, and I could put it up on a web page--I think I sent a copy > to some just intonation library in San Francisco--does that ring > any bells?

The Just Intonation Network is here in SF: The Just Intonation Network * [with cont.] (Wayb.)

>PB I presume means periodicity block, and MOS is some kind of >jumped-up well-formed scale, I understand. Could you similarly >define MOS (and WF while you are at it?)

MOS, WF, and Myhill's property are all equivalent. They are usually given as something like: MOS or WF- any pythagorean-type scale in which the generating interval always spans the same number of scale degrees. While strict pythagorean scales are usually generated with 3:2's against 2:1's, MOS and WF allow any generator, and sometimes the interval of equivalence is allowed to be non-2:1. Myhill's property- all generic scale intervals have exactly two specific sizes. -Carl

Message: 705 - Contents - Hide Contents Date: Mon, 20 Aug 2001 07:03:57 Subject: Re: Mea culpa From: carl@l...

>> >he Just Intonation Network is here in SF: >> >> The Just Intonation Network * [with cont.] (Wayb.) >

> Thanks. Do you know if it has a library and if it would still > have a paper I sent to it back in the mid-80's? People have been > getting copies somehow, I've heard, and I suspect it comes from > there.

They do in fact have a tremendous library, mostly of stuff from the 80's, when the Network was at its peak. Unfortunately it is very disorganized, to the point where the chance they'll know if they have thing x is less than 50%, and it would take hours, even days to say for sure. Xeroxes, dot-matrix printouts abound, in boxes in Henry Rosenthal's basement. -Carl

Message: 706 - Contents - Hide Contents Date: Mon, 20 Aug 2001 08:25:16 Subject: The hypothesis From: genewardsmith@j... I found a posting by Paul over on the tuning group, and it seems I may be closing on a statement of the Paul Hypothesis. "In fact, a few months ago I posted my Hypothesis, which states that if you temper out all but one of the unison vectors of a Fokker periodicity block, you end up with an MOS scale. We're discussing this Hypothesis on tuning-math@y..." Sounds like we may be getting there, but there seems to be some confusion as to whether 2 counts as a prime, and so whether for instance the 5-limit is 2D or 3D. Most of the time it makes sense to treat 2 like any other prime. "A temperament can be constructed by tempering out anywhere from 1 to n unison vectors. If you temper out n (and do it uniformly), you have an ET. If you temper out n-1, you have a linear temperament. If you temper out n-2, you have a planar temperament (Dave Keenan has created some examples of those)." From my point of view, the 5-limit is rank (dimension) 3, and the 7- limit 4, and so forth. If you temper out n-1 unison vectors which generate a well-behaved kernel, then you map onto a rank-1 group, and get an equal temperment. So "codimension" 1 (one less than the full number of dimensions) leads to a rank-1 group. In the same way, codimension 2 for the kernel leads to a rank 2 group, etc. If for instance you temper out 81/80 in the 5 limit, the kernel has dimension 1 and codimension 2, and leads to a rank 2 image group. We can tune the rank 1 group any way we like so long as the steps are of the same size, which means that our ET can have stretched or squashed octaves if we so choose. In the same way, we can tune the rank 2 group any way we like, except that we need to retain incommensurability of two generators (or at least to ignore the fact if they are not.) If we make the octaves pure in our example where the kernel is generated by a comma, we could for instance make the fifths pure also, leading to Pythagorean tuning. Alternatively, we could make the major thirds pure, leading to 1/4 comma mean tone temperment. (Pythagorean tuning is not considered a temperment, since the fifth isn't tempered, but it is the same sort of thing mathematically as 1/4 comma mean-tone temperment.) Other choices lead to other results, and all we need to do is to ensure the circle of fifths does not close--or at least to pretend otherwise it if it does. A rank 3 image group, coming from a kernel of codimension 3, is what people have been calling a 2D temperment. I hope that clarifies things (as it does for me) rather than further confuses them!

Message: 708 - Contents - Hide Contents Date: Mon, 20 Aug 2001 11:45 +0 Subject: Re: Microtemperament and scale structure From: graham@m... In-Reply-To: <9lq0ik+d44m@e...> In article <9lq0ik+d44m@e...>, genewardsmith@j... () wrote:

> At last we are making progress! I don't see much role for > the "chromatic" element here, though. If the n-1 unison vectors are > linearly independent, we've already seen recently how to tell if they > generate a kernel of something mapping to Z: compute the gcd of the > determinant minors, and see if it is 1 or not. If they have no common > factor, then they define such a mapping, and the "chromatic vector" > will go to a certain number of steps in this mapping--hopefully 1, > but perhaps 2, 3, 4 ... etc.

Yes, the chromatic UV should be redundant. That reminds me of two hypotheses I didn't get round to implementing in Python code: 1) If you use an octave-invariant matrix, with the chromatic UV at the top, the left hand column of the adjoint matrix is the mapping by generator, like the second column was before. (May need to be divided through by the GCD.) That makes the adjoint matrix a list of generator mappings (the octave being a special case where it's specific). 2) Any simple interval will do for the chromatic UV, so long as all the UVs are linearly independent. I don't have strict criteria for "simple" here, any more than criteria for what work as unison vectors in the first place. But trying [1 0 0 ...], [0 1 0 ...], etc until something works (non-zero determinant) should do the trick. I don't think such things can be described as unison vectors, but I believe they do work in this context. So are you saying you have an equivalent to (2)? I don't think the GCD has to be 1.

> As for temperment, that has to do with tuning and you cannot draw any > conclusions about tuning unless you introduce it into your statement > somewhere--nothing in, nothing out.

I suppose it depends on how you define "temperament". Is "meantone" a temperament or a class of temperaments? The chromatic UV is used to define the tuning. If you want to push the definition and make a third a unison vector, you can define quarter comma meantone by setting it just. So the commatic UVs define the temperament class and the chromatic UV is used to define the specific tuning. I make the octave explicit for the same reason.

>> The strong form says that if you construct the >> Fokker (hyperparallelepiped) periodicity block >> from the n unison vectors, and again 1 is >> chromatic and n-1 are commatic, then the notes in >> the PB form an MOS scale. >

> PB I presume means periodicity block, and MOS is some kind of jumped- > up well-formed scale, I understand. Could you similarly define MOS > (and WF while you are at it?)

Whatever they mean, MOS and WF are the same thing: a generated scale with only two step sizes. It may be a useful property for alternative systems of tonality, but we don't have enough examples to pronounce on that yet. It's certainly useful to think about when designing generalized keyboards or alternative notation systems. (See <http://www.anaphoria.com/xen3b.PDF - Type Ok * [with cont.] (Wayb.)> if you haven't already.) Graham

Message: 709 - Contents - Hide Contents Date: Mon, 20 Aug 2001 11:45 +0 Subject: Re: Microtemperament and scale structure From: graham@m... In-Reply-To: <9lpte7+27ap@e...> Paul wrote:

>> The other columns happen to be the generator mappings for the >> equivalent column being a chromatic unison vector. >> I don't think there's a proof for >> this always working yet, but it does. >

> Can you show with examples?

It's what <Unison vector to MOS script * [with cont.] (Wayb.)> is all about. <Unison vectors * [with cont.] (Wayb.)> is a list of examples. Graham

Message: 711 - Contents - Hide Contents Date: Mon, 20 Aug 2001 18:28:26 Subject: Re: Microtemperament and scale structure From: Paul Erlich --- In tuning-math@y..., genewardsmith@j... wrote:

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

>> Gene -- first of all, start with a set of n unison >> vectors. The unison vectors that are tempered out >> or completely ignored are called "commatic unison >> vectors". The unison vectors that amount to a >> musically significant difference, but not (often) >> large enough to move you from one scale step to >> the next, are called "chromatic unison vectors". >

> Thanks! I'd guessed that was what it meant. I think you are adding to > the confusion by calling both of them "unison vectors", though--why > not unison and step vectors instead? Three reasons:

1) The number of notes in the scale should be (normally) the determinant of the matrix of unison vectors. One has to include both the chromatic and the commatic unison vectors in this calculation. 2) In the "prototypical" case, the commatic unison vector is "the comma", 81:80; and the chromatic unison vector is "the chromatic unison" or "augmented unison", 25:24. These define a 7-tone periodicity block: the diatonic scale. You see how the terminology is just a generalization of this case. 3) "Step vectors" would refer to something else. In the prototypical example, the step vectors would be 16:15, 10:9, and 9:8.

>> The weak form of the hypothesis simply says that >> if there is 1 chromatic unison vector, and n-1 >> commatic unison vectors, then what you have is a >> linear temperament, with some generator and >> interval of repetition (which is usually equal to the >> interval of equivalence, but sometimes turns out to >> be half, a third, a quarter . . . of it). >

> At last we are making progress! I don't see much role for > the "chromatic" element here, though.

You're right . . . it plays no role here.

> If the n-1 unison vectors are > linearly independent, we've already seen recently how to tell if they > generate a kernel of something mapping to Z:

No -- you did that with n unison vectors -- I'm not counting the 2 axis as a "dimension" here.

> PB I presume means periodicity block, and MOS is some kind of jumped- > up well-formed scale, I understand. Could you similarly define MOS > (and WF while you are at it?)

MOS means that there is an interval of repetition (normally equal to the interval of equivalence [usually octave], but sometimes it comes out as half, third, quarter . . . of the IE). The scale repeats itself exactly within each interval of repetition. Within each, there is a generating interval, which is iterated some number of times such that the scale has two step sizes. Examples: The diatonic scale (LsssLss) is MOS: the IoR is an octave, and the generator is L+s+s. The melodic minor scale (LssssLs) is not MOS: there is no generator that produces all the notes and no others.

Message: 712 - Contents - Hide Contents Date: Mon, 20 Aug 2001 18:39:04 Subject: Re: Hypothesis From: Paul Erlich --- In tuning-math@y..., genewardsmith@j... wrote:

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

>> There are a tremendous number of arguments as >> to why there is something interesting about WF or >> MOS scales in the literature. Personally, I buy very >> few of them, if any. But there are some very >> powerful WF/MOS scales around, especially, of >> course, the usual diatonic scale, and the usual >> pentatonic scale. >

> Unless I am missing something (highly likely at this point!) the > pentatonic and diatonic scales are WF in mean tone intonation but not > in just intonation. Is that right? If it is right, doesn't that serve > to make the whole idea seem fishy?

Strict, fixed-pitch just intonation has almost never been used in actual music with these scales. This is because of the so- called "comma problem". Don't let the JI advocates fool you: Pythagorean tuning and various meantone-like temperaments have been far more important than fixed-pitch 5-limit just intonation for the actual performance of these scales -- even in China! One thing I forgot to mention about the hypothesis: if you don't use the parallelepiped, you might end up with a scale that is not MOS, but has the same number of notes as the MOS that comes from the parallelepiped. I conjecture that in some precise sense, the MOS has more consonant structures (intervals and/or chords) than the corresponding non-MOS. This is seen, for example, in the decatonic case, where the chromatic unison vector is one member of the set {25:24, 28:27, 49:48}, and the commatic unison vectors are two members of the set {50:49, 64:63, 225:224}. The MOS is LssssLssss, which has 8 consonant 7-limit tetrads (4 4:5:6:7s and 4 1/7:1/6:1/5:1/4s); while a melodically superior non-MOS scale, LsssssLsss, has only 6.

Message: 713 - Contents - Hide Contents Date: Mon, 20 Aug 2001 18:41:50 Subject: Re: Microtemperament and scale structure From: Paul Erlich --- In tuning-math@y..., genewardsmith@j... wrote:

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

>> But, in this case, if you temper out the schisma and >> the diesis, you're tempering out their sum, which >> means you're tempering out _two_ syntonic >> commas . . . which means that you're either >> tempering out the syntonic comma, or setting it to >> half an octave. >

> I'm afraid that is where the "torsion" I was talking about comes in. > Suppose you color all 5-limit notes either green or red, by making > [a,b,c] green if a+b+c is even, and red if it is odd. Then two reds > add up to a green, a green and a red to a red, and two greens a green. > > Your two generators are green, but the comma is red. The generators > generate only greens, but you need two reds to get a green. Hence the > image under the homomorphism goes to a 12 et note, but there is a red > keyboard and a green keyboard!

Are you saying that both keyboards are tuned identically, or that there may be an offset?

>> Tell me what JT means. >

> To me, something defined in terms of rational numbers. What does it > mean to you?

I was just asking what it stood for. "Just Tuning"?

Message: 714 - Contents - Hide Contents Date: Mon, 20 Aug 2001 18:45:05 Subject: Re: Hypothesis From: Paul Erlich --- In tuning-math@y..., carl@l... wrote:

> > I don't think MOS itself means much for the perception of melody. > Rather, I think it works together, or is often confounded with > other properties: > > () Symmetry at the 3:2. The idea is that the 3:2 is a special > interval, a sort of 2nd-order octave. When a scale's generator > is 3:2, MOS means that a given pattern can more often be repeated > a 3:2 away. Chains of 5, 7, and 12 "fifths" are historically > favored, but where are all the MOS chains of 5:4, 7:4, etc.? In > my experience, MOS chains of non-fifth generators can be special > too, but we should be careful not to give MOS credit for symmetry > at the 3:2.

Did you get this from me? 'Cause you know I agree. But see the message I just posted about why MOSs appear to be _harmonically_ special for the class of scales with given step sizes and number of notes.

Message: 715 - Contents - Hide Contents Date: Mon, 20 Aug 2001 18:46:18 Subject: Re: Mea culpa From: Paul Erlich --- In tuning-math@y..., carl@l... wrote:

> > MOS, WF, and Myhill's property are all equivalent.

This is not quite true -- for example, LssssLssss is MOS but not WF and doesn't have Myhill's property.

Message: 716 - Contents - Hide Contents Date: Mon, 20 Aug 2001 18:49:32 Subject: Re: The hypothesis From: Paul Erlich --- In tuning-math@y..., genewardsmith@j... wrote:

> I found a posting by Paul over on the tuning group, and it seems I > may be closing on a statement of the Paul Hypothesis. > > "In fact, a few months ago I posted my Hypothesis, which states that > if you temper out all but one of the unison vectors of a Fokker > periodicity block, you end up with an MOS scale. We're discussing > this Hypothesis on tuning-math@y..." > > Sounds like we may be getting there, but there seems to be some > confusion as to whether 2 counts as a prime, and so whether for > instance the 5-limit is 2D or 3D. Most of the time it makes sense to > treat 2 like any other prime.

Well I've been treating 5-limit as 2D, following Fokker. In many contexts, it's important to keep 2 as an additional dimension -- but not in this context.

> I hope that clarifies > things (as it does for me) rather than further confuses them!

Well it certainly seems that you understand what we're talking about!

Message: 717 - Contents - Hide Contents Date: Mon, 20 Aug 2001 18:55:06 Subject: Re: Microtemperament and scale structure From: Paul Erlich --- In tuning-math@y..., graham@m... wrote:

> > I suppose it depends on how you define "temperament".

Is "meantone" a

> temperament or a class of temperaments? The chromatic UV is used to > define the tuning.

You mean the commatic UVs (81:80 in the case of meantone)?

> If you want to push the definition and make a third a > unison vector, you can define quarter comma meantone by setting it just.

Now I think you're pushing definitions too far. Let's not forget the strong form of the hypothesis!

> So the commatic UVs define the temperament class and the chromatic UV is > used to define the specific tuning.

Hmm . . . perhaps one _can_ define things this way, but it's by no means universal. How would one define LucyTuning in this way??

> > Whatever they mean, MOS and WF are the same thing: a generated scale with > only two step sizes.

Not the same thing. Clampitt lists all the WFs in 12-tET, and there is no sign of the diminished (octatonic) scale, or any other scale with an interval of repetition that is a fraction of an octave. These are all MOS scales, though.

Message: 718 - Contents - Hide Contents Date: Mon, 20 Aug 2001 18:55:44 Subject: Re: Microtemperament and scale structure From: Paul Erlich --- In tuning-math@y..., graham@m... wrote:

> In-Reply-To: <9lpte7+27ap@e...> > Paul wrote: >

>>> The other columns happen to be the generator mappings for the >>> equivalent column being a chromatic unison vector. >>> I don't think there's a proof for >>> this always working yet, but it does. >>

>> Can you show with examples? >

> It's what <Unison vector to MOS script * [with cont.] (Wayb.)> is all about. > <Unison vectors * [with cont.] (Wayb.)> is a list of examples. > > Graham

I meant for the particular case which you erased above.

Message: 719 - Contents - Hide Contents Date: Mon, 20 Aug 2001 19:20:53 Subject: Re: Mea culpa From: carl@l...

>> >OS, WF, and Myhill's property are all equivalent. >

> This is not quite true -- for example, LssssLssss is MOS but not WF > and doesn't have Myhill's property.

What single generator produces the scale? -Carl

Message: 720 - Contents - Hide Contents Date: Mon, 20 Aug 2001 19:23:18 Subject: Re: Hypothesis From: carl@l...

>> >) Symmetry at the 3:2. The idea is that the 3:2 is a special >> interval, a sort of 2nd-order octave. When a scale's generator >> is 3:2, MOS means that a given pattern can more often be repeated >> a 3:2 away. Chains of 5, 7, and 12 "fifths" are historically >> favored, but where are all the MOS chains of 5:4, 7:4, etc.? In >> my experience, MOS chains of non-fifth generators can be special >> too, but we should be careful not to give MOS credit for symmetry >> at the 3:2. >

> Did you get this from me? 'Cause you know I agree.

Absolutely -- I've long credited you with it, even in a pre-send version of that post.

> But see the message I just posted about why MOSs appear to be > _harmonically_ special for the class of scales with given step > sizes and number of notes.

I didn't catch the why, but I am of course familiar with the example you gave. -Carl

Message: 721 - Contents - Hide Contents Date: Mon, 20 Aug 2001 19:26:19 Subject: Re: Hypothesis From: carl@l... I wrote...

>> But see the message I just posted about why MOSs appear to be >> _harmonically_ special for the class of scales with given step >> sizes and number of notes. >

> I didn't catch the why, but I am of course familiar with the > example you gave.

I mean, I caught that they are non-parallelpiped PBs, but not why this should translate into fewer harmonic structures (do you mean only complete chords? total consonant dyads?). -Carl

Message: 724 - Contents - Hide Contents Date: Mon, 20 Aug 2001 20:30:09 Subject: Re: Mea culpa From: Paul Erlich --- In tuning-math@y..., carl@l... wrote:

>>> MOS, WF, and Myhill's property are all equivalent. >>

>> This is not quite true -- for example, LssssLssss is MOS but not WF >> and doesn't have Myhill's property. >

> What single generator produces the scale? > > -Carl

One possibility is s -- here the interval of repetition is the half- octave.