Tuning-Math Archive Section 1: 125

Message: 125 - Contents - Hide Contents Date: Mon, 04 Jun 2001 03:46:22 Subject: Re: Constant Structure (was: Temperament program issues) From: Paul Erlich --- In tuning-math@y..., carl@l... wrote:

>> It's sounds to me as if you're trying to define a property other >> than Wilson's CS. As I understand it, Wilson uses CS to describe >> a pattern shared by a group of tunings that can be mapped onto a >> single scale tree pattern,even if the generator size is only a >> average value. For example, he maps the 3(1,3,7,9,11,15 Eikosany >> (plus two "pigtails"), a scale based on an Indian Sruti model, and >> 22tet onto a 22-tone keyboard and notation. The subsets described >> by the scale tree then becomes useful paths for orientation in >> ´the larger system. >

> Amazing, Daniel! This is just what I guessed CS was, before > the correspondence from Kraig. See monz's web site reffed earlier > in this thread.

Kraig also thought that CS meant something different before Wilson gave him the definition on Monz' web page.

>> (Writing the above, I've got the strong suspicion that >> constructing a formula that will predict whether and where in >> the scale tree a given scale will find a CS is probably very hard >> to construct.) >

> On the contrary, unless I mis-understood you, the scale tree > shows the generator ranges itself. I know you know this, so > maybe I _did_ mis-read you. Can I get you to check monz's > web page? > > Definitions of tuning terms: constant structur... * [with cont.] (Wayb.) > > -Carl

Daniel Wolf didn't post directly to this list. So make sure someone forwards this to him. Also, maybe he wrote Dave privately because the idea was that this list should cease to exist? Perhaps we should be posting all this to the tuning list?

Message: 126 - Contents - Hide Contents Date: Tue, 05 Jun 2001 03:20:06 Subject: Math models (about Hypothesis) From: Pierre Lamothe I was not available at moment the thread Hypothesis was active. I had begun to write this post but I had urgent tasks forcing me to stop. It is not complete but since the List will probably terminate and since I miss yet time I post that unfinished. I would like to use the Paul's conjecture as starting point to show the necessity for fine tuned definitions if we want to progress in mathematical way. I would have difficulty to express mathematically the given conjecture for I don't know the mathematical definition used for almost all the terms. ----------------------- distributional evenness ----------------------- I begin with this definition appearing in the Monzo site. << distributional evenness The scale has no more than two sizes of interval in each interval class. >> What follows is not a criticism of the Monzo's definition but only some arguments to show that the conjecture has not really a mathematical form. There exist maybe a light ambiguity with << no more than two >> but the hard problem would be rather with the link between _scale_ and _class_ terms. Forgetting for a moment the term scale, let us look at << no more than two

>> as if the property could be applied to a simple set. Supposing that,

<< . . . has no more than two sizes of interval in each interval class. >> might be mathematically written A partition of a set S by an equivalence relation R is _distributionally even_ if for any class x in S/R 0 < Card x < 3 where it is only assumed each class has either one or two elements. Thus the partition {{a},{b},{c}} of a set {a,b,c} would be distributionally even. For Paul, is that sense allowed? Besides, << no more than two >> implies that the case "for any x, Card x = 2" is only a possibility among others. So, if the partition {{a},{b},{c}} is not _distributionally even_, what rule is used to determine allowed combinations of classes with one or two elements? Is the form << no more than >> used only to allow the class of unison having only one interval while all others would have two? I imagine this light ambiguity is easy to clarified. The next one seems more serious. --------------- class and scale --------------- Is a _scale_ anything else, for Paul, than an ordered finite set of reals. If not so, what are the essential conditions? Has a _scale_ an essential link, or only an optional one, with periodicity block and also with properties often mentioned like consonance possibilities? Are interval classes in a scale something added to that scale by an external equivalence relation or are classes "sui generis" in sense that the classes follows (by a general principle used for any scale) from the intervals itself given as a whole? In the following set and partitions S = {1, 9/8, 32/27, 4/3, 3/2, 27/16, 16/9} P1 = {{1}, {9/8, 32/27}, {4/3}, {3/2}, {27/16, 16/9}} P2 = {{1}, {9/8}, {32/27}, {4/3}, {3/2}, {27/16}, {16/9}} P3 = {{1}, {4/3, 27/16}, {9/8, 16/9}, {32/27, 3/2}} how the definitions of scale, class and distributionally evenness should be applied? As you may see, I'm far from knowing clearly what this simple definition mathematically means, but it could be worst with other definitions. My discussions with Paul on the Tuning List shows we could differ on periodicity block and srutis and probably on unison vectors and steps. ----------------- periodicity block ----------------- *** TONE *** I will use here the term _tone_ in the strict sense of an interval of the first octave as representing its class modulo 2. For me, a normal set of unison vectors in a discrete tones Z-module determine a periodicity block refering to a canonical set of tones determined by an oriented object with origin : the hyperparallelopiped defined by the given unison vectors sharing a common origin. This canonical set of tones corresponds both to - tones inside the hyperparallelopiped having unison as origin and - tones on the unison vectors (*** so neither ambiguity nor double counting ***). Since the origin of the unison vectors is never inside the hyperparallelopiped, so keeping the same shape but changing the origin for another vertices don't correspond (most cases) to a simple translation of the block in the lattice. One tone, minimally, would be changed compared to the translated canonical set. Besides, since Paul and Dave seems to refer rather at periodical shapes, I would add that a same shape may correspond to distinct sets of unison vectors. If you know only the tile you don't know forcely the tiling. Another thing seems mysterious comparing our uses of periodicity block. For me, a system build from the periodicity block concept implies all modes in the system have the same amount of tones or degrees. If we use only one block we may have only one mode. Using N blocks surrounding unison we obtain for each D classes possibly N intervals. It remains possible after that to use appropriate temperament (removing commatic vectors or keeping integrally the structure). But it seems Paul and Monz would start with periodicity blocks having too tones, to be interpreted as degrees of a musical mode, in their approach of the Hindu system where the amount of degrees is 7 (SA RI GA ...) rather than 22, and it seems it is a temper process that serves to . . . (I don't understand, so I have some questions). If you use only one block, which have by definition only one element in each of the D classes determined by the periodicity, what is the sense of the presumed new classes which would appear after a temper treatment? Is it like that you obtain seven classes in the Hindu system from a block having a 22 or more periodicity? Do you have then an explicit epimorphism transforming an interval in its degree value? ----- As I said, I cut here for I have not much time to write. However, the questions already written may have interest. Pierre

Message: 132 - Contents - Hide Contents Date: Tue, 05 Jun 2001 18:32:54 Subject: Re: Refinement (?) of "true 5-limit" adaptive tuning From: Paul Erlich --- In tuning-math@y..., "John A. deLaubenfels" <jdl@a...> wrote:

>>> At first I thought a new set of tuning files would be necessary to >>> accomplish this, but then realized that a weak, but not negligible, >>> tritone spring would also address the goal. A tritone _must_ be >>> targeted to 600 cents, because we know nothing of inversion(s) in the >>> tuning-file free approach. > > [Paul E:]

>> I don't understand this . . . I thought your program recognized >> octave-equivalence, and hence inversions, independently of the tuning >> files. >

> Your statement is correct. But, for example, in 7-limit, a tritone is > NOT 600 cents, yet the program knows which side of the inversion it's > on because of pattern-matching between the notes on the the desirable > intervals specified by the tuning file.

This I understand . . .

>Am I being clear?

Still confused as to the original statement.

>> And by 'weak', do you mean 'strength approaches zero'? >

> That would be 'negligible'. Here, 'weak' means 1/8 of nominal. Do the > tables make sense? >

Very much so. I guess I was just wondering if your concept here was similar to those cases where you needed to put a negligible strength on some spring for computational reasons (i.e., the matrix wouldn't 'relax' correctly otherwise) but conceptually the strength was really zero.

Message: 133 - Contents - Hide Contents Date: Tue, 05 Jun 2001 19:14:07 Subject: Fwd: True nature of the blackjack scale (in 7-limit) . . . and more epimoress From: Paul Erlich --- In tuning@y..., "Paul Erlich" <paul@s...> wrote: Dave Keenan wrote (privately),

>What's the >set of 7-limit UV's for Blackjack again?

The blackjack scale is the result of forming a periodicity block with the unison vectors 2401:2400, 225:224, and 36:35; treating the 2401:2400 and 225:224 as commatic unison vectors and tempering them out; and treating 36:35 as a chromatic unison vector and not tempering it out. Confused? Maybe it would help to add that the good old diatonic scale in 5-limit is the result of forming a periodicity block with the unison vectors 81:80 and 25:24; treating 81:80 as a commatic unison vector and tempering it out; and treating 25:24 as a chromatic unison vector and not tempering it out. In other words, the diatonic scale is an infinite 'band' of the infinite 2D 5-limit lattice, and the thickness of the band is given by the 25:24 interval. This is clearly explained and depicted in my paper, _The Forms Of Tonality_. The blackjack lattice that I posted (and need to correct) shows that the blackjack scale is an infinite 'slice' of the infinite 3D 7-limit lattice, and the thickness of the slice is given by the 36:35 interval. You can see that there is no 36:35 interval, which would be formed by moving two red connectors to the right, one green connector to the lower-left, and one blue connector to the lower left, within the blackjack scale. If you were to transpose the blackjack scale by this interval, it would fit, with no gaps, on top of its transposed self . . . and an infinite number of such 'layers' would fill the infinite 3D 7-limit lattice. So the blackjack scale is a "Form Of Tonality" (perhaps "Form of Modality" would be better since no 'tonal center' is necessarily implicated) with commatic and chromatic unison vectors very much in accordance with the sizes of commatic and chromatic unison vectors in the scales I've already described in that paper. Funny how things that appear unrelated at first seem to 'fit together'! Canasta seems less interesting from this point of view because its chromatic unison vector is 81:80 . . . which is more like a commatic unison vector in size, begging to be tempered out . . . and if you do that, you get the wonderful 31-tET . . . Always using epimoric ratios for the unison vectors has one advantage . . . the size of the numbers in the ratio immediately tells you both the melodic smallness of the interval, and its taxicab distance in the triangular lattice (suitably constructed, as in the second-to-last lattice on Kees van Prooijen's page lattice orientation * [with cont.] (Wayb.)). So tempering out an epimoric unison vector that uses numbers N times smaller than another one means that the constituent consonances will have to be tempered N^2 times as much . . . am I on to something? --- End forwarded message ---

Message: 134 - Contents - Hide Contents Date: Tue, 05 Jun 2001 20:13:26 Subject: Re: Fwd: True nature of the blackjack scale (in 7-limit) . . . and more epimoress From: jpehrson@r... --- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote: Yahoo groups: /tuning-math/message/133 * [with cont.] Thanks for posting and cross-posting this message, Paul. I never even understood that blackjack is an infinite 7-limit lattice as compared with our traditional diatonic being an infinite 5-limit lattice... Somehow I never "got" that before from the previous dialogues... No wonder there are so many "just tetrads" in it! __________ ___________ _______ Joseph Pehrson

Message: 136 - Contents - Hide Contents Date: Tue, 05 Jun 2001 21:09:32 Subject: Re: Refinement (?) of "true 5-limit" adaptive tuning From: Paul Erlich --- In tuning-math@y..., "John A. deLaubenfels" <jdl@a...> wrote:

> In the er3/es3, all spring strengths are greater than negligible. And > in truth I could have done the er2/es2 with zero-strength springs on 1, > 2, and 6; that won't hurt the matrix (it's only infinite strength > springs that are trouble-makers!). I just left the 1% springs in place > to see those other intervals represented in the table. >

So what you're saying is, in practice, you could use zero-strength springs on 1, 2, and 6 for most MIDI files, since the size of the tritone would be determined by various horizontal & grounding considerations . . . but for this single-chord example, you needed to use a finite-strength spring because otherwise, there would be no solution . . . ? In that case, I would favor using zero-strength springs in practice . . . P.S. Why is this discussion on this list? I thought it was voted that this list should not remain separate . . . but it seems to be growing! A monster! Perhaps I'd feel like I was respecting the vote if I at least transferred ownership of this list to someone else . . .

Message: 137 - Contents - Hide Contents Date: Tue, 05 Jun 2001 21:14:17 Subject: Re: Fwd: True nature of the blackjack scale (in 7-limit) . . . and more epimoress From: Paul Erlich --- In tuning-math@y..., jpehrson@r... wrote:

> --- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote: > > Yahoo groups: /tuning-math/message/133 * [with cont.] > > Thanks for posting and cross-posting this message, Paul. I never > even understood that blackjack is an infinite 7-limit lattice as > compared with our traditional diatonic being an infinite 5-limit > lattice... Somehow I never "got" that before from the previous > dialogues... No wonder there are so many "just tetrads" in it! >

Hey Joseph . . . look at blackjack3.bmp in the new bjlatt.ZIP . . . then look at Figure 8 in my paper _The Forms of Tonality_ . . . same idea, different scale (and different lattice orientation) . . . You can think of them as infinite, as we are here, or you can think of them simply as 'wrapping around' to meet themselves. Remember the 'donut'?

Message: 138 - Contents - Hide Contents Date: Tue, 05 Jun 2001 21:43:20 Subject: Re: Fwd: True nature of the blackjack scale (in 7-limit) . . . and more epimoress From: jpehrson@r... --- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote: Yahoo groups: /tuning-math/message/137 * [with cont.]

> Hey Joseph . . . look at blackjack3.bmp in the new bjlatt.ZIP . . . > then look at Figure 8 in my paper _The Forms of Tonality_ . . . same > idea, different scale (and different lattice orientation) . . . > > You can think of them as infinite, as we are here, or you can think > of them simply as 'wrapping around' to meet themselves. Remember > the 'donut'?

Yes, indeed! It became patently obvious once I was SHOWN it! :) Regarding the list, I don't know. I'm reluctant to post over to the "big" list right now, just because it's so "big!" and there have been so many complaints. Maybe it is right to keep a list of more numerical or math items over here. I was even reluctant to post the recent question Monz had about fractional remainders after division... And probably that would have been of interest to more people than just Monz and myself.... Dunno. Maybe it's best to keep things running like they are until there are "complaints" even though the vote indicated otherwise. After all, there were lots of abstentions... For myself, I'm pretty much getting used to having so many lists now... and, since I'm still trying to read most all of them, MATHEMATICALLY, it really doesn't make any difference. See... I *was* going to get to the "math" part... _________ _________ _______ Joseph Pehrson

Message: 139 - Contents - Hide Contents Date: Tue, 05 Jun 2001 21:45:05 Subject: combine with Harmonic Entropy From: jpehrson@r... HOWEVER, (and I forgot to mention this), I personally *do* believe that the "Harmonic Entropy" list should be "nuked" (messages saved, of course-- we'll have to ask Robert Walker) and any further posts on that subject be presented on the "Tuning Math" list... ___________ _________ ________ Joseph Pehrson

Message: 141 - Contents - Hide Contents Date: Tue, 05 Jun 2001 23:54:04 Subject: Re: Refinement (?) of "true 5-limit" adaptive tuning From: jpehrson@r... --- In tuning-math@y..., "John A. deLaubenfels" <jdl@a...> wrote: Yahoo groups: /tuning-math/message/140 * [with cont.]

> > If you are determined not to host this list any more, then please do > hand it off. I'd rather keep my long, detailed posts off the fat

list if possible. Maybe I'll start a boring-details@y...

> > JdL Hi John...

What I was thinking is that this list could include *BOTH* the salient details and *ALSO* expressions of exuberance! Let's say, after a long detailed post, somebody *else* could just write, "Dig it!" Right now, one can't do that on the "big" list, since somebody is always counting the words so that people with slow connections don't have to download so much! ________ _______ ___ Joseph Pehrson

Message: 142 - Contents - Hide Contents Date: Tue, 5 Jun 2001 22:59:19 Subject: keep the tuning-math list From: monz

> [Paul:]

>> P.S. Why is this discussion on this list? I thought it was voted that >> this list should not remain separate . . . but it seems to be >> growing! A monster! Perhaps I'd feel like I was respecting the vote >> if I at least transferred ownership of this list to someone >> else . . . >

> If you are determined not to host this list any more, then please do > hand it off. I'd rather keep my long, detailed posts off the fat list > if possible. Maybe I'll start a boring-details@y...

I know I'm repeating myself... but I totally agree with John. I think the microtonal community really needs *precisely this* list. Please keep it alive. -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: 143 - Contents - Hide Contents Date: Wed, 6 Jun 2001 10:01 +01 Subject: Re: Fwd: True nature of the blackjack scale (in 7-limit) . . .. From: graham@m... In-Reply-To: <9fjjpo+t5e1@e...> Joseph Pehrson wrote:

> Dunno. Maybe it's best to keep things running like they are until > there are "complaints" even though the vote indicated otherwise. > After all, there were lots of abstentions...

How about insisting that, as the list already exists, an absolute majority of the Tuning List is required to vote it away? Then, the overwhelming number of people who don't bother to vote are sure to carry the day! Graham

Message: 146 - Contents - Hide Contents Date: Wed, 06 Jun 2001 13:30:20 Subject: Re: Fwd: True nature of the blackjack scale (in 7-limit) . . .. From: jpehrson@r... --- In tuning-math@y..., graham@m... wrote: Yahoo groups: /tuning-math/message/143 * [with cont.]

> In-Reply-To: <9fjjpo+t5e1@e...> > Joseph Pehrson wrote: >

>> Dunno. Maybe it's best to keep things running like they are until >> there are "complaints" even though the vote indicated otherwise. >> After all, there were lots of abstentions... >

> How about insisting that, as the list already exists, an absolute majority > of the Tuning List is required to vote it away? Then, the overwhelming > number of people who don't bother to vote are sure to carry the day! > > Graham

That's pretty funny, Graham. I can see you're good at "politics" too... (!!) Looks like the "Math List" won! ___________ ________ ________ Joseph Pehrson

Message: 147 - Contents - Hide Contents Date: Wed, 06 Jun 2001 13:33:11 Subject: Re: Refinement (?) of "true 5-limit" adaptive tuning From: jpehrson@r... --- In tuning-math@y..., "John A. deLaubenfels" <jdl@a...> wrote: Yahoo groups: /tuning-math/message/145 * [with cont.]

> [I wrote:]

>>> If you are determined not to host this list any more, then please do >>> hand it off. I'd rather keep my long, detailed posts off the fat >>> list if possible. Maybe I'll start a boring-details@y... > > [Joseph Pehrson:]

>> What I was thinking is that this list could include *BOTH* the >> salient details and *ALSO* expressions of exuberance! > >> Let's say, after a long detailed post, somebody *else* could just >> write, "Dig it!" >

> Oh, I agree! When I say "boring details", I just mean that I know > _some_ will be bored, but I'm excited, and if others are as well, it's > great to hear it! >

>> Right now, one can't do that on the "big" list, since somebody is >> always counting the words so that people with slow connections don't >> have to download so much! >

> I suppose if this list ever gets real fat, we'll have to be careful > here as well. > > JdL

I suppose that's true, John... but at the moment it still isn't the case... so I'm for lots of big numbers and stuff that I can't understand (well I'll get PART of it!) and EXUBERANCE! _________ ________ ________ Joseph Pehrson

Message: 148 - Contents - Hide Contents Date: Wed, 06 Jun 2001 18:51:15 Subject: Re: Refinement (?) of "true 5-limit" adaptive tuning From: Paul Erlich --- In tuning-math@y..., "John A. deLaubenfels" <jdl@a...> wrote:

> [Paul:]

>> So what you're saying is, in practice, you could use zero-strength >> springs on 1, 2, and 6 for most MIDI files, since the size of the >> tritone would be determined by various horizontal & grounding >> considerations . . . but for this single-chord example, you needed to >> use a finite-strength spring because otherwise, there would be no >> solution . . . ? In that case, I would favor using zero-strength >> springs in practice . . . >

> Are we clear that the 0% and the 1% springs (the latter represented > by er2/es2 files) would be almost identical in result, and that both > will be different from the er3/es3 I've been describing? Sure. > Are you making > a statement about your preference as to one group vs. the other, or > commenting on how you would do the former group?

I'm asking a question, and you didn't answer (see above).

> > [Paul:]

> If you are determined not to host this list any more, then please do > hand it off.

Are you volunteering?

Message: 149 - Contents - Hide Contents Date: Wed, 06 Jun 2001 18:55:38 Subject: Re: linear approximation From: Paul Erlich --- In tuning-math@y..., <manuel.op.de.coul@e...> wrote:

> It's exactly the same as the best LS generator for > a Pythagorean scale, with the generator being one step here. >

The best LS generator . . . Manuel, there are several _strange_ features in this calculation. First of all, you're including _both_ the 1/1 _and_ the 2/1, while every other pitch class appears only once. Second, you're forcing the fit line to pass through the 1/1. These are very odd features and I'm not sure how you'd justify them. As far as I can tell, if you do it correctly, the best LS step has to equal the mean step.