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Message: 426 - Contents - Hide Contents Date: Wed, 27 Jun 2001 20:51:54 Subject: Re: ET's, unison vectors (and other equivalences) From: jpehrson@r... --- In tuning-math@y..., "monz" <joemonz@y...> wrote: Yahoo groups: /tuning-math/message/417 * [with cont.]> > ----- Original Message ----- > From: D.Stearns <STEARNS@C...> > To: <tuning-math@y...> > Sent: Tuesday, June 26, 2001 5:27 PM > Subject: Re: [tuning-math] Re: ET's, unison vectors (and other equivalences) > >>> Hi Paul and everyone, >> >> You can also use the 2d lattice as a basic model for plotting >> coordinates other that 3 and 5. >> >> Earlier I gave the [4,3] 7-tone, neutral third scale as an example >> with the unison vectors 52/49 >> = 2^2 * 7^-2 * 13^1 = [ 2 0 0 -2 0 1] > > >> and 28672/28561. >> = 2^12 * 7^1 * 13^-4 = [ 12 0 0 1 0 -4] > >>> This would be an example of plugging 13 and 7 into >> a 2D lattice space while retaining the diatonic matrix. >> As cab be seen at a glance in either of the prime-factor notations. >I can see a cab at a glance, but I don't get this vector notation... Could you please gently run it down again, or would that be for the *arithmetic* list?? Thanks! _________ ______ _____ Joseph Pehrson
Message: 427 - Contents - Hide Contents Date: Wed, 27 Jun 2001 21:03:39 Subject: Re: 41 "miracle" and 43 tone scales From: jpehrson@r... --- In tuning-math@y..., "M. Edward (Ed) Borasky" <znmeb@a...> wrote: Yahoo groups: /tuning-math/message/425 * [with cont.]> On Wed, 27 Jun 2001 jpehrson@r... wrote: > >> A question: >>>> In arithmetic and mathematics is the *numerator* of a fractionever considered "more important" than the *denominator?*> > Not that I know of -- see the definition of the rational numbers asequivalence classes of ordered pairs of integers. In an ordered pair, *somebody's* gotta be number one and somebody else's gotta be number two :-). Ya ain't got no ordered pair otherwise :-).>Got it! Thanks, Ed!>>> Just as in "otonal??" Hasn't the "otonal" series, on the overall, >> been considered *significantly* more important than the *utonal*over the years??> > Outside of Partch, yes -- Otonal/Major is *musically* moreimportant than Utonal/Minor *in common practice Western music*. One of the things Partch was trying to do, after having defined Otonal and Utonal to begin with, was to treat them equally in his music and right what he considered to be a wrong in this respect. I haven't heard enough of his music to know whether Otonal and Utonal are in fact equally respected in his works. Gee... this is an interesting question, but Jon Szanto isn't on this list... Maybe I'll post something to the "biggie...">>> Or am I just "out to lunch..." >> Are you buying? :-)Sure! But, unfortunately... you're in Oregon at the moment.... :) _________ _______ _____ Joseph Pehrson
Message: 428 - Contents - Hide Contents Date: Wed, 27 Jun 2001 23:37:11 Subject: Re: 41 "miracle" and 43 tone scales From: Dave Keenan --- In tuning-math@y..., jpehrson@r... wrote:> In arithmetic and mathematics is the *numerator* of a fraction ever > considered "more important" than the *denominator?*No. I don't think so. It's all completely dual.> Or is that a silly question...?No. Its a good question.> It seems to me in simple arithmetic, > the numerator seems more "impressive..." maybe because the numbers > are larger??In ordinary (non-musical) usage the numerator is just as likely to be smaller than the denominator.> Just as in "otonal??" Hasn't the "otonal" series, on the overall, > been considered *significantly* more important than the *utonal* over > the years??Yes. But this doesn't make the numerator or denominator special. It makes _the_smallest_of_the_two_numbers_ special. Several frequencies having their fundamentals ocurring as if they are the harmonics of a lower virtual fundamental, gives more consonance than several frequencies that each have one harmonic corresponding to a higher "guide-tone". In the case of octave-equivalent pitches we have a convention to put them in a form that is between 1/1 and 2/1 so they have positive logarithms. But for non octave-equivalent pitches we can have 2/3 different from 3/2. For intervals, octave equivalence doesn't matter. 2:3 describes exactly the same interval as 3:2. I have argued before for a convention of putting the small number first, as we do for "extended ratios" such as 4:5:6. But when we want to take its logarithm (to convert to cents) we will still enter it as 3/2, i.e. big number as numerator, so that we are dealing with positive logarithms. But remember these are only conventions or conveniences. The musical specialness is in "big number versus little number", not "numerator versus denominator". Regards, -- Dave Keenan
Message: 429 - Contents - Hide Contents Date: Thu, 28 Jun 2001 01:23:43 Subject: Re: Hypothesis revisited From: Paul Erlich --- In tuning-math@y..., Graham Breed <graham@m...> wrote:> Yes, they both give Miracle41, but a different Miracle41 each time/Can you explain what you mean by "different"? They're both centered around the 1/1, so it's not the mode that's different . . .> > If you invert and normalize the octave-invariant matrix, the left hand column > gives you the prime intervals in terms of generators.Well that sounds like it solves the Hypothesis in a demonstrative fashion, yes?
Message: 430 - Contents - Hide Contents Date: Thu, 28 Jun 2001 01:25:20 Subject: Re: ET's, unison vectors (and other equivalences) From: Paul Erlich --- In tuning-math@y..., "D.Stearns" <STEARNS@C...> wrote:> Hi Paul and everyone, > > You can also use the 2d lattice as a basic model for plotting > coordinates other that 3 and 5.We've done that before, for example Margo and my discussions (about 22-tET, among other things) in the (3,7) plane.
Message: 431 - Contents - Hide Contents Date: Thu, 28 Jun 2001 01:34:29 Subject: Re: 41 "miracle" and 43 tone scales From: Paul Erlich --- In tuning-math@y..., jpehrson@r... wrote:> --- In tuning-math@y..., "Dave Keenan" <D.KEENAN@U...> wrote: > > Yahoo groups: /tuning-math/message/407 * [with cont.] >>>> I'm interested now more than ever in knowing some of Daniel >>> Wolf's knowledge and opinions on this subject. A full-scale >>> analysis of the *non*-JI harmonies in Partch's compositions >>> would reveal a ton of information. >>>> Yes indeed. We might be able to better answer the "schismic vs. >> miracle" question based on that. >> >> -- Dave Keenan >> Doesn't this imply that, somehow, Partch was using the "non-JI" > harmonies in a different way than his "JI" harmonies??Well a question can't imply a fact. But if you mean, doesn't it _assume_ that, then no. In fact, the more Partch used them in the same way, the easier it will be to decide which unison vectors he may have accepted.> > Personally, I would doubt that. Once he had his scale, he probably > just used it "as is" regardless of the derivation of the notes.. > > ?? >Unfortunately, that may be a bit too much to hope for. Partch devised an involved compositional apparatus in _Genesis_ based on JI harmonies, and I would be shocked if this didn't still guide his later works somewhat.
Message: 432 - Contents - Hide Contents Date: Thu, 28 Jun 2001 01:37:51 Subject: Re: 41 "miracle" and 43 tone scales From: Paul Erlich> --- In tuning-math@y..., "M. Edward (Ed) Borasky" <znmeb@a...> wrote: >>>> Outside of Partch, yes -- Otonal/Major is *musically* more> important than Utonal/Minor *in common practice Western music*.On what basis do you make that claim? They seem to be equal enough in importance in this music to "fool" Riemann, Partch, and many other theorists to give them equal footing a priori.
Message: 434 - Contents - Hide Contents Date: Thu, 28 Jun 2001 02:36:46 Subject: Re: 41 "miracle" and 43 tone scales From: Paul Erlich --- In tuning-math@y..., "M. Edward Borasky" <znmeb@a...> wrote:> I was paraphrasing Partch ... I can probably find the line in _Genesis_, but > one of his goals was to restore Untonality to equal footing with Otonality, > thus implying an existing *in*equality.In musical _theories_ -- not in any of the musical _practice_ that he liked, as he understood it.
Message: 435 - Contents - Hide Contents Date: Thu, 28 Jun 2001 03:52:15 Subject: Re: 41 "miracle" and 43 tone scales From: jpehrson@r... --- In tuning-math@y..., "Dave Keenan" <D.KEENAN@U...> wrote: Yahoo groups: /tuning-math/message/428 * [with cont.]> > But remember these are only conventions or conveniences. Themusical specialness is in "big number versus little number", not "numerator versus denominator".> > Regards, > -- Dave KeenanGot it! Thanks, Dave! ________ _______ ______ Joseph Pehrson
Message: 437 - Contents - Hide Contents Date: Thu, 28 Jun 2001 12:00 +0 Subject: Re: questions about Graham's matrices (was: 13-limit mappin From: graham@m... In-Reply-To: <002c01c0ff33$9751a860$4448620c@a...> I'm replying to this here, as we've had complaints about over-precise cents values. Perhaps Monz could post back to TBL (The Big List) when he understands it.>>>> mapping by period and generator: >>>> ([1, 0], ([0, 2, -1], [5, 1, 12])) >>>> The first two-element list shows the mapping of the octave. The >> second >> element is always zero for both my scripts, as the period is always a >> fraction of an octave. >> So in other words you always use the nearest integer here? > I'm still confused about that "0".It isn't the nearest integer to anything: these lists are the definition of the temperament. All the temperaments I'm currently considering use a period that's an equal division of the octave. So you never need the generator to get the octave, and that parameter's always zero.>> So the first number tells you how many equal >> parts the octave is being divided into. Here it's 1 which is the >> simplest case. >> Confused about this too... I thought this example divided the > octave into 41 parts? Again, is this pair of numbers expressing > the nearest integer fraction of an octave?The octave is divided into 41 *unequal* parts. There are no equal divisions of the octave. Compare with this 13-limit temperament I gave before: 9/52, 103.897 cent generator basis: (0.5, 0.086580634742799478) mapping by period and generator: ([2, 0], ([3, 5, 7, 9, 10], [1, -2, -8, -12, -15])) mapping by steps: [(58, 46), (92, 73), (135, 107), (163, 129), (201, 159), (215, 170)] unison vectors: [[1, 2, -3, 1, 0, 0], [-4, 0, 2, 1, -1, 0], [1, -3, 2, 1, 0, -1], [2, -1, 0, 1, -2, 1]] highest interval width: 17 complexity measure: 34 (46 for smallest MOS) highest error: 0.004911 (5.893 cents) unique and diaschismic temperament: 2/11, 105.214 cent generator basis: (0.5, 0.087678135277931377) mapping by period and generator: ([2, 0], ([3, 5], [1, -2])) mapping by steps: [(12, 10), (19, 16), (28, 23)] unison vectors: [[11, -4, -2]] highest interval width: 3 complexity measure: 6 (8 for smallest MOS) highest error: 0.002716 (3.259 cents) unique Both of them divide the octave in 2 equal parts all the time.> Also, I think it's confusing the way you give the "octave correction" > first and the "number of generators" second in this line, but > it's reversed in all the following lines, generator first and > octave second.Octave is always first in the printouts. I explained the generator mapping first because it makes more sense that way round. As another thread shows, you can ignore octaves completely and define the scale using the generator mapping and number of equal divisions of the octave. You don't know how many notes the MOS will contain, or whether it's unique assuming octave invariance, but you can still work out this information.>> In more familiar terms, the generator is a 5:4 major third. 5 major >> thirds are a 3:1 perfect twelfth. >> (2^(380.391/1200))^5 does indeed equal exactly 3. > > Following you so far... Oh good! > So I follow this too. Now comes the tricky part... >>> (2*(5,0) - (12,-1) = (-2, 1)) >> OK, so as I said above, ((2^(380.391/1200))^5) * (2^0) = 3 . > The "2*" means that we square that, and so the first group > stands for 3^2 = 9 .Oh, that is putting the generator first, isn't it? That's wrong, you're right to pull me up on it. So it should be (2*(0, 5) - (12, -1)) = (1, -2)) I think you're complicating it by bringing ratios and exact pitches back into it. A 9:1 is two 3:1 steps, hence 2*(0, 5). Any ratio can be prime factorized, and worked out in terms of octaves and generators using the conversion matrix.> And ((2^(380.391/1200))^12) * (2^-1) = ~6.983305074 , > which agrees with your definition above as ~7. > > The minus sign means we divide the terms, and... > Voilą! ... ~9/7 .7:1 is written as (2,-1). You read that straight off. So 9:7 or (0 2 0 -1)H is 2*(0,5)-(-1,12). You could write that (0 2 0 -1)( 1 0) ( 0 5) ( 2 1) (-1 12)> And checking the answer: > ((2^(380.391/1200))^-2) * (2^1) does indeed equal the ~9/7. > > So you're putting an equivalence relationship in here.That's a check, yes, but a simpler one would be to use the relationship at the top of the printout, that the generator is 13 steps from 41. 41-2*13=41-26=15. And 9:7 does approximate to 15 steps from 41.> That was confusing... I had a hard time understanding how > ~9/7 = "An octave less two major thirds". Now it's clear.It means you can construct an augmented triad with two 5-limit and one 9-identity thirds.>>>> mapping by steps: >>>> [(22, 19), (35, 30), (51, 44), (62, 53)] >>>> Each pair shows the size of a prime interval in terms of scale steps. >> Call the steps x and y. An octave is 22x+19y. For the case where >> x=y, >> you have 41-equal. Where x=0, you have 19-equal. Where y=0, you have >> 22-equal. So 19, 22 and 41-equal are all members of this temperament >> family. >> >> 3:1 is 35x+30y, 5:1 is 51x+44y and 7:1 is 62x+53y. You can get any >> 7-prime limit interval in terms of x and y by combining these. >> OK, I understand all the math here, but I'm not quite following the > logic which deterimines that they are "all members of this temperament > family". How does your program find the 19, 22 and 41 in this example?My program *starts* with 19 and 22, along with the prime intervals and odd limit, and works everything out from them. It gets 41 by adding 19 and 22. My other program starts with the unison vectors, but that's more complex. A third program could go from unison vectors to a mapping in terms of generators within a period that's a fraction of an octave. In that case, it'd have to get the equal temperaments by optimizing for the best generator/period ratio, and walking the scale tree.>> and use simpler coordinates. Here, q=x+y and p=x >> >> 3:2 is 11q + 2p >> 5:4 is 6q + p >> 8:7 is 4q >> Oops... now you lost me.Substitute in for p and q: 11q+2p = 11(x+y) + 2x = 13x + 11y 6q+p = 6(x+y) + x = 7x + 6y 4q = 4(x+y) = 4x + 4y>> So q is 2 steps in 41-equal, or 1 step in 22- or 19-equal >> and p is 1 step in 41-or 22-equal, and no steps in 19-equal. > > Getting foggier...| 19= | 22= | 41= --------------------------------- x | 0 | 1 | 1 y | 1 | 0 | 1 p | 0 | 1 | 1 q | 1 | 1 | 2>>>> unison vectors: >>>> [[-10, -1, 5, 0], [5, -12, 0, 5]] > >> So these are the ratios 3125/3072 and 537824/531441 ?The might well be, you're as capable as me of working them out. I missed off the bottom of the file before:>> highest interval width: 12This is the maximum number of generators needed for an interval within the consonance limit.>> complexity measure: 12 (13 for smallest MOS)The complexity measure is the previous number multiplied by the number of equal divisions of the octave. The number of otonal or utonal chords is the number of notes in the scale minus this measure. Roughly. In this case there's a 13 note MOS that can hold that complete otonality.>> highest error: 0.004936 (5.923 cents)This comes from the minimax optimisation. It's a bit over-precise so it can be checked against another program, and because it's possible for the program to throw out *very* accurate temperaments. The temperaments are sorted assuming the fewer notes needed and the smaller the error the better.>> uniqueThis is either there or not. I added it to be able to assess keyboard mappings as well as temperaments. I have lists of keyboard mappings, not uploaded yet, that rate a mapping twice as badly if it isn't unique, and ignore the accuracy. Graham
Message: 438 - Contents - Hide Contents Date: Thu, 28 Jun 2001 12:49 +0 Subject: Re: Hypothesis revisited From: graham@m... In-Reply-To: <9he0uv+d2ll@e...> Paul wrote:>> Yes, they both give Miracle41, but a different Miracle41 each time/ >> Can you explain what you mean > by "different"? They're both > centered around the 1/1, so it's > not the mode that's different . . .One is 10+41n, the other 31+41n. The mapping by period and generator is the same both times. So they're both aspects of the same temperament. It depends on whether you take this "set of MOS scales" result seriously. It doesn't come out of the octave invariant method discussed below.>> If you invert and normalize the octave-invariant matrix, the left >> hand column >> gives you the prime intervals in terms of generators. >> Well that sounds like it solves the > Hypothesis in a demonstrative > fashion, yes?If you can prove it will always work. I can't, but am pleased it does. You can certainly always define the scale in terms of some kind of octave-invariant interval, and call that the generator. Perhaps that's all it comes down to. But I've always said this was obvious from the matrix technique. But showing that the unison vectors lead to a linear temperament is different from showing they give a CS periodicity block, or whatever it is you asked. The octave-specific method doesn't always give a result. It fails with the unison vectors I'm using for the multiple-29 temperament. But you can always define a temperament in terms of a pair of intervals, even if they aren't the ones you want for the MOS. The octave-invariant result for multiple-29, BTW, is this mapping: [0, 707281, 707281, 707281, 707281] when I wanted [0, 29, 29, 29, 29] Incidentally, an alternative octave-specific case would be to define an extra chromatic unison vector instead of the octave. The the two left hand columns of the inverse will be the mapping by scale steps. Graham
Message: 439 - Contents - Hide Contents Date: Thu, 28 Jun 2001 18:29:07 Subject: Re: Hypothesis revisited From: Paul Erlich --- In tuning-math@y..., graham@m... wrote:> In-Reply-To: <9he0uv+d2ll@e...> > Paul wrote: >>>> Yes, they both give Miracle41, but a different Miracle41 each time/ >>>> Can you explain what you mean >> by "different"? They're both >> centered around the 1/1, so it's >> not the mode that's different . . . >> One is 10+41n, the other 31+41n.What do you mean by this notation?> The mapping by period and generator is > the same both times. So they're both aspects of the same temperament. > It depends on whether you take this "set of MOS scales" result seriously.I'm not following you.> It doesn't come out of the octave invariant method discussed below.What's "It" in this sentence?>>>> If you invert and normalize the octave-invariant matrix, the left >>> hand column >>> gives you the prime intervals in terms of generators. >>>> Well that sounds like it solves the >> Hypothesis in a demonstrative >> fashion, yes? >> If you can prove it will always work. I can't, but am pleased it does. > You can certainly always define the scale in terms of some kind of > octave-invariant interval, and call that the generator. Perhaps that's > all it comes down to.Yes, but this choice should be unique . . . there should only be one (octave-invariant) generator.> But I've always said this was obvious from the > matrix technique. But showing that the unison vectors lead to a linear > temperament is different from showing they give a CS periodicity block, > or whatever it is you asked.Well there may be some differences in our understanding of this, as the above (different miracle-41s) may be indicating. But I think we're on the right track . . . ?> > The octave-specific method doesn't always give a result.Uh-oh. So maybe I can convince you to switch over to octave-invariant?> It fails with > the unison vectors I'm using for the multiple-29 temperament. But you > can always define a temperament in terms of a pair of intervals, even if > they aren't the ones you want for the MOS.Don't they _have_ to be the generator and the interval of repetition?> > The octave-invariant result for multiple-29, BTW, is this mapping: > > [0, 707281, 707281, 707281, 707281] > > when I wanted > > [0, 29, 29, 29, 29]Can you explain how the number 707281 comes about?
Message: 440 - Contents - Hide Contents Date: Thu, 28 Jun 2001 12:01:49 Subject: Re: questions about Graham's matrices (was: 13-limit mappin From: monz Hi Graham,> From: <graham@m...> > To: <tuning-math@xxxxxxxxxxx.xxx> > Sent: Thursday, June 28, 2001 4:00 AM > Subject: [tuning-math] Re: questions about Graham's matrices (was: 13-limit mappin > > > I'm replying to this here, as we've had complaints about over-precise > cents values. Perhaps Monz could post back to TBL (The Big List) when he > understands it.Yup - I should have posted the other one here only, and not also on TBL. (Thanks for that convenient abbreviation).>>>>> [Graham] >>>>> mapping by period and generator: >>>>> ([1, 0], ([0, 2, -1], [5, 1, 12])) >>> >>> [Graham]>>> The first two-element list shows the mapping of the octave. >>> The second element is always zero for both my scripts, as >>> the period is always a fraction of an octave. >> >> [me, monz]>> So in other words you always use the nearest integer here? >> I'm still confused about that "0". > > [Graham]> It isn't the nearest integer to anything: these lists are the definition > of the temperament. > > All the temperaments I'm currently considering use a period that's an > equal division of the octave. So you never need the generator to get the > octave, and that parameter's always zero.OK, I think I understand that now. A counter-illustration using a period that's an unequal division, which would "need the generator to get the octave", would help.>>> So the first number tells you how many equal >>> parts the octave is being divided into. Here it's 1 which is the >>> simplest case. >>>> Confused about this too... I thought this example divided the >> octave into 41 parts? Again, is this pair of numbers expressing >> the nearest integer fraction of an octave? >> The octave is divided into 41 *unequal* parts. There are no equal > divisions of the octave.This was a bit confusing, because I got it tangled with what you said above about "a period that's an equal division of the octave". Now I see the difference. The scale under consideration is an temperament which is an *unequal* division of the octave (the period of equivalence), because it's a result of multiples of the generator and not of an equal division of anything. So all thru the rest of the explanation when you refer to "steps of 19=, 22=, 41="... they're all approximations to the generated scale. Right? I think what's been confusing me is that you refer to both ratios and EDOs as approximations of the scale resulting from your generator, and perhaps I've been taking them more literally than I should have been. I realize now that every interval is to be understood in terms of this temperament's approximations to the basic prime intervals 2, 3, 5, 7. So your matrices are presenting a set of transformations.>>> (2*(5,0) - (12,-1) = (-2, 1)) >>>> OK, so as I said above, ((2^(380.391/1200))^5) * (2^0) = 3 . >> The "2*" means that we square that, and so the first group >> stands for 3^2 = 9 . >> Oh, that is putting the generator first, isn't it? That's wrong, you're > right to pull me up on it. So it should be > > (2*(0, 5) - (12, -1)) = (1, -2))Oops!... your bad. You didn't reverse (12, -1) into (-1, 12) as you meant to do. I really think it's much more intuitive to have it the other way around (your mistake here shows the persistence of that way of thinking). Put the generator first and the octave second, consistently. I agree with you that the number of generators is the more important figure, and to me it makes sense to *see* that number first. (I think your unconscious switch in the original post shows that.) So reverse the "correction" you made here and put it back like it was, and reverse the *other* lines to agree with these. So your illustrated calcuation (call it "two fifths less a seventh") translates into approximate ratios as ~(3:2)^2 / ~7:4 = ~9:7 , and would look like: (2*(5, 0) - (12, -1)) = (-2, 1)) . And the "octave less two major thirds" translates into approximate ratios as ~2:1 / ~(5/4)^2 = ~32/25 . When I did the matrix calculation I got (0, 1) - 2*(1, 2) = (-2, -3) . Hmmm... the important number, the generator, works out to be the same -2, which is correct. But why is the period calculation not working out when the octave is included? Is is because there is no zero period? So your opening lines, with extra labels, would look like: basis (generator, period) as fraction of octave: (0.31699250014423125, 1.0) mapping by [generator], [period] (~2:1, (~3:2, ~5:4, ~7:4)) : ([0, 1], ([5, 1, 12], [0, 2, -1])) Actually, I think that second line would be better rearranged to agree with the octave notation: mapping by (generator, period) [~2:1, ~3:2, ~5:4, ~7:4]: [(0, 1), (5, 0), (1, 2), (12, -1)] To me, that's as plain as day. The octave can still be seen as set apart by virtue of being first/leftmost on the list.> > I think you're complicating it by bringing ratios and exact pitches back > into it.Agreed... but using the ratios allowed *me* to do the math in an Excel spreadsheet so that I could follow your reasoning. I went thru it step by step, looking at the cents values all along the way. If I understood better how to manipulate the matrices, I certainly would have done it that way too. I can see that it's *much* more elegant that way, even tho I've been having trouble understanding it. This is along the lines of what I was trying to get Paul to understand a couple of different times in the past. It's not necessary to always use prime-factors as the basis for lattice metrics... any numbers that give even, consistent divisions of the pitch-space *in SOME way* will do. The different ways of dividing (and multiplying) produce different kinds of lattices. (Of course Paul already understands all this, *and* the math to manipulate it, as do you. But I don't think he was following my reasoning when I was trying to make that point... probably because *I don't* understand the math! I'm not speaking the same language you guys are... altho I'm trying hard...)> A 9:1 is two 3:1 steps, hence 2*(0, 5).Yes, now that's *very* clear. I caught it, but certainly didn't explain it as elegantly as this.> Any ratio can be prime factorized, and worked out in terms of > octaves and generators using the conversion matrix.*That's* what's been giving me such trouble! Relating the calculations given in terms of this temperament to the approximate ratios really did confuse me, even tho in hindsight now I think it helped in the process of understanding.> >>> And ((2^(380.391/1200))^12) * (2^-1) = ~6.983305074 , >> which agrees with your definition above as ~7. >> >> The minus sign means we divide the terms, and... >> Voilą! ... ~9/7 . >> 7:1 is written as (2,-1).Oops! Your bad again... you meant (12, -1). ... well, actually you meant (-1, 12). (Boy, this interval sure keeps giving you the slip!)> You read that straight off. > So 9:7 or (0 2 0 -1)H is 2*(0,5)-(-1,12). You could write that > > (0 2 0 -1)( 1 0) > ( 0 5) > ( 2 1) > (-1 12) >Thanks, Graham. Seeing the matrix conversion broken down like this helps me a lot. So keeping to my (generator, period) reversal of your notation, (0 2 0 -1)H = 2*(5,0)-(12,-1) because ~3 = (5,0) and ~7 = (12,-1). Of course, in the case of an octave-invariant scale like this it's much simpler to just omit the period. So ignoring the first column in "H" because it's powers of 2, ratio prime vector 380.391-cent generators 2:1 = ( 1 0 0 0)H = ~ 0 3:2 = (-1 1 0 0)H = ~ 5 5:4 = (-2 0 1 0)H = ~ 1 7:4 = (-2 0 0 1)H = ~12 9:7 = ( 0 2 0 -1)H = ~(2*5)-12 = ~-2.>> And checking the answer: >> ((2^(380.391/1200))^-2) * (2^1) does indeed equal the ~9/7. >> >> So you're putting an equivalence relationship in here. >> That's a check, yes, but a simpler one would be to use the relationship > at the top of the printout, that the generator is 13 steps from 41. > 41-2*13=41-26=15. And 9:7 does approximate to 15 steps from 41.This helps a lot too. Thanks. So here you're back again to "an octave less two major thirds". ~2:1 / ~(5/4)^2 = ~32/25 or (0, 1) - 2*(1, 2) = (-2, -3) .>> That was confusing... I had a hard time understanding how >> ~9/7 = "An octave less two major thirds". Now it's clear. >> It means you can construct an augmented triad with two 5-limit and one > 9-identity thirds.Er... this is a little confusing, because a triad is constructed of only two intervals. You mean that if one measured all the intervals in an augmented triad *and its inversions*, the result would be two ~5:4s and one ~9:7.>>>>> mapping by steps: >>>>> [(22, 19), (35, 30), (51, 44), (62, 53)] >>>>>> Each pair shows the size of a prime interval in terms of scale steps. >>> Call the steps x and y. An octave is 22x+19y. For the case where >>> x=y, you have 41-equal. Where x=0, you have 19-equal. Where y=0, >>> you have 22-equal. So 19, 22 and 41-equal are all members of this >>> temperament family. >>> >>> 3:1 is 35x+30y, 5:1 is 51x+44y and 7:1 is 62x+53y. You can get any >>> 7-prime limit interval in terms of x and y by combining these. >>>> OK, I understand all the math here, but I'm not quite following the >> logic which deterimines that they are "all members of this temperament >> family". How does your program find the 19, 22 and 41 in this example? >> My program *starts* with 19 and 22, along with the prime intervals and > odd limit, and works everything out from them. It gets 41 by adding 19 > and 22.Ahhh!... that certainly explains *that*!> My other program starts with the unison vectors, but that's more complex. > A third program could go from unison vectors to a mapping in terms of > generators within a period that's a fraction of an octave. In that case, > it'd have to get the equal temperaments by optimizing for the best > generator/period ratio, and walking the scale tree.Hmmm.... that last algorithm sounds like a good one! Exactly the kind of thing I always wanted to include in my JustMusic software, applicable to rational systems as well as irrational.>>> and use simpler coordinates. Here, q=x+y and p=x >>> >>> 3:2 is 11q + 2p >>> 5:4 is 6q + p >>> 8:7 is 4q >>>> Oops... now you lost me. >> Substitute in for p and q: > > 11q+2p = 11(x+y) + 2x = 13x + 11y > 6q+p = 6(x+y) + x = 7x + 6y > 4q = 4(x+y) = 4x + 4yAh... if only I had been paying attention in algebra class... OK, now it's perfectly clear.>>> So q is 2 steps in 41-equal, or 1 step in 22- or 19-equal >>> and p is 1 step in 41-or 22-equal, and no steps in 19-equal. >> >> Getting foggier... >> | 19= | 22= | 41= > --------------------------------- > x | 0 | 1 | 1 > y | 1 | 0 | 1 > p | 0 | 1 | 1 > q | 1 | 1 | 2 >Uh-oh... still not getting this part. Please elaborate. For some reason I'm not seeing the connection between p and q and the steps of the EDOs. Too many layers of abstraction for me to follow ...>>>>> unison vectors: >>>>> [[-10, -1, 5, 0], [5, -12, 0, 5]] >> >>>> So these are the ratios 3125/3072 and 537824/531441 ? >> The[y] might well be, you're as capable as me of working them out.OK... I was simply double-checking with you that the numbers stand for exponents of the prime-factors 2, 3, 5, and 7. So I guess they do. So your parenthetical lists are elegant, but IMO could use a little bit more of a legend explaining what those lists represent. Otherwise one has to learn the sequences beforehand and keep them in mind. I suggest adding a label giving the parameter list before each line. -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: 441 - Contents - Hide Contents Date: Thu, 28 Jun 2001 19:31:55 Subject: Re: questions about Graham's matrices (was: 13-limit mappin From: Paul Erlich --- In tuning-math@y..., "monz" <joemonz@y...> wrote:> > Er... this is a little confusing, because a triad is constructed of > only two intervals.Last time I checked, a triad had three intervals, a tetrad six, a pentad ten, and a hexad fifteen.
Message: 442 - Contents - Hide Contents Date: Thu, 28 Jun 2001 20:00:54 Subject: Re: Hypothesis revisited From: Graham Breed>> >ne is 10+41n, the other 31+41n. >> What do you mean by this notation?Temperements including the ETs with 10+41n or 31+41n notes, where n is a non-negative integer.>> The mapping by period and generator is >> the same both times. So they're both aspects of the same > temperament.>> It depends on whether you take this "set of MOS scales" result > seriously. >> I'm not following you.I explained this before. When you generate the scales from a set of unison vectors, one of them chromatic, the natural result is something like 10+41n rather than a single MOS or the full range of temperaments defined by the commatic unison vectors.>> It doesn't come out of the octave invariant method discussed below. >> What's "It" in this sentence?The restricted set of temperaments. But in fact I was wrong there. In fact, the second column of the normalized octave-specific inverse is the same as the first column of the octave-invariant one, but with an extra zero. I didn't notice it was the generator mapping before, but managed to get the right results anyway :)>> If you can prove it will always work. I can't, but am pleased it > does.>> You can certainly always define the scale in terms of some kind of >> octave-invariant interval, and call that the generator. Perhaps > that's>> all it comes down to. >> Yes, but this choice should be unique . . . there should only be one > (octave-invariant) generator.This brings us back to """" 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! """There are aways two generators that will work. The minus sign differentiates them.>> But I've always said this was obvious from the >> matrix technique. But showing that the unison vectors lead to a > linear>> temperament is different from showing they give a CS periodicity > block,>> or whatever it is you asked. >> Well there may be some differences in our understanding of this, as > the above (different miracle-41s) may be indicating. But I think > we're on the right track . . . ? Oh, unquestionably.>> The octave-specific method doesn't always give a result. >> Uh-oh. So maybe I can convince you to switch over to octave-invariant?I think it would be worth writing a script that only uses them. It would mean altering the code in temper.py to accept a mapping by generators, so it's a bit of work.>> It fails with >> the unison vectors I'm using for the multiple-29 temperament. But > you>> can always define a temperament in terms of a pair of intervals, > even if>> they aren't the ones you want for the MOS. >> Don't they _have_ to be the generator and the interval of repetition?No. If you take this matrix at face value:> [[ 41 0] > [ 65 -6] > [ 95 7] > [115 2] > [142 -15]]/41it defines Miracle using one 41st part of an octave, and a 41st part of the usual generator. That works, but it isn't efficient.>> The octave-invariant result for multiple-29, BTW, is this mapping: >> >> [0, 707281, 707281, 707281, 707281] >> >> when I wanted >> >> [0, 29, 29, 29, 29] >> Can you explain how the number 707281 comes about?It's 29^4. I'm sure it means I chose the chromatic unison vector wrongly. The interesting thing is that the generator matrix is a multiple of what it should be. In fact, the whole matrix has a common factor, which may be the clue that something's wrong. Although dividing through by that common factor won't work. Also, this is a case where the inverse of the octave-specific matrix doesn't get the generator mapping right. If the method almost works with an arbitrary chroma, that means we're a step towards getting it to work with only commatic unison vectors, which should be possible. Graham "I toss therefore I am" -- Sartre
Message: 444 - Contents - Hide Contents Date: Thu, 28 Jun 2001 22:18 +0 Subject: Re: questions about Graham's matrices From: graham@m... monz wrote:>> All the temperaments I'm currently considering use a period that's an >> equal division of the octave. So you never need the generator to get >> the >> octave, and that parameter's always zero. >> OK, I think I understand that now. A counter-illustration > using a period that's an unequal division, which would "need the > generator to get the octave", would help.As you never need it, it's difficult to find a counter-example... My schismic fourth keyboard mapping would be an example, where the period is a fourth. So the octave would be two periods plus a generator I think. But I've never worked that out with matrices.>> The octave is divided into 41 *unequal* parts. There are no equal >> divisions of the octave. >> This was a bit confusing, because I got it tangled with what you > said above about "a period that's an equal division of the octave". > Now I see the difference. > > The scale under consideration is an temperament which is an > *unequal* division of the octave (the period of equivalence), > because it's a result of multiples of the generator and not of > an equal division of anything.It's an unequal division of the period, which is an equal division of the interval of equivalence (in this case the trivial division of one) which for these example is always an octave.> So all thru the rest of the explanation when you refer to > "steps of 19=, 22=, 41="... they're all approximations to > the generated scale. Right?There isn't "a generated scale". The generator can be used for a whole family of scales. 19, 22 and 41-equal are merely special cases of the scales it can generate.> I think what's been confusing me is that you refer to both ratios > and EDOs as approximations of the scale resulting from your generator, > and perhaps I've been taking them more literally than I should > have been. I realize now that every interval is to be understood > in terms of this temperament's approximations to the basic prime > intervals 2, 3, 5, 7. So your matrices are presenting a set of > transformations.Yes, matrices are all about transformations. In this case between the harmonic and melodic ways of looking at things. It's a one-way transformation because you loose information in going from the just to tempered mapping.>> (2*(0, 5) - (12, -1)) = (1, -2)) > >> Oops!... your bad. You didn't reverse (12, -1) into (-1, 12) > as you meant to do. > > I really think it's much more intuitive to have it the other way around > (your mistake here shows the persistence of that way of thinking). > > Put the generator first and the octave second, consistently. > I agree with you that the number of generators is the more important > figure, and to me it makes sense to *see* that number first. > (I think your unconscious switch in the original post shows that.)No, because that would contradict the usual way of writing vectors from low to high primes.> And the "octave less two major thirds" translates into > approximate ratios as ~2:1 / ~(5/4)^2 = ~32/25 . > When I did the matrix calculation I got > > (0, 1) - 2*(1, 2) = (-2, -3) . > > Hmmm... the important number, the generator, works out to be > the same -2, which is correct. But why is the period calculation > not working out when the octave is included? Is is because > there is no zero period?You're using 5:1 instead of 5:4.>> I think you're complicating it by bringing ratios and exact pitches >> back >> into it. >> Agreed... but using the ratios allowed *me* to do the math in > an Excel spreadsheet so that I could follow your reasoning. > I went thru it step by step, looking at the cents values all > along the way. > > If I understood better how to manipulate the matrices, I certainly > would have done it that way too. I can see that it's *much* more > elegant that way, even tho I've been having trouble understanding it.If you've got Excel, you can do that! I explain it on my website. You use MINVERSE, MDETERM and MMULT, pressing CTRL-SHIFT-RETURN to enter the formulae.> This is along the lines of what I was trying to get Paul to > understand a couple of different times in the past. It's not > necessary to always use prime-factors as the basis for lattice > metrics... any numbers that give even, consistent divisions > of the pitch-space *in SOME way* will do. The different ways > of dividing (and multiplying) produce different kinds of lattices.Yes, I think this is what Pierre Lamothe was trying to get across before he left the list as well.> Of course, in the case of an octave-invariant scale > like this it's much simpler to just omit the period. So > ignoring the first column in "H" because it's powers of 2, > > ratio prime vector 380.391-cent generators > > 2:1 = ( 1 0 0 0)H = ~ 0 > 3:2 = (-1 1 0 0)H = ~ 5 > 5:4 = (-2 0 1 0)H = ~ 1 > 7:4 = (-2 0 0 1)H = ~12 > 9:7 = ( 0 2 0 -1)H = ~(2*5)-12 = ~-2.If you're thinking octave invariantly, you can simplify it further. ratio prime vector 380.391-cent generators 3:2 = (1 0 0)H = ~ 5 5:4 = (0 1 0)H = ~ 1 7:4 = (0 0 1)H = ~12 9:7 = (2 0 -1)H = ~(2*5)-12 = ~-2.>>> That was confusing... I had a hard time understanding how >>> ~9/7 = "An octave less two major thirds". Now it's clear. >>>> It means you can construct an augmented triad with two 5-limit and one >> 9-identity thirds. >> Er... this is a little confusing, because a triad is constructed of > only two intervals. > > You mean that if one measured all the intervals in an augmented triad > *and its inversions*, the result would be two ~5:4s and one ~9:7.Oh, however you count it, I was thinking of the octave as part of the chord.>> My other program starts with the unison vectors, but that's more >> complex. >> A third program could go from unison vectors to a mapping in terms of >> generators within a period that's a fraction of an octave. In that >> case, >> it'd have to get the equal temperaments by optimizing for the best >> generator/period ratio, and walking the scale tree. >> Hmmm.... that last algorithm sounds like a good one! Exactly the > kind of thing I always wanted to include in my JustMusic software, > applicable to rational systems as well as irrational.Try it. Type the octave-invariant unison vectors into the spreadsheet as an array, with the chromatic one at the top. Then select an equal sized square, type "=minverse(?:?)*mdeterm(?:?)" where ?:? is that original array, and the left hand column will be the mapping by generators.>>>> So q is 2 steps in 41-equal, or 1 step in 22- or 19-equal >>>> and p is 1 step in 41-or 22-equal, and no steps in 19-equal. >>> >>> Getting foggier... >>>> | 19= | 22= | 41= >> --------------------------------- >> x | 0 | 1 | 1 >> y | 1 | 0 | 1 >> p | 0 | 1 | 1 >> q | 1 | 1 | 2 >> > >> Uh-oh... still not getting this part. Please elaborate. > For some reason I'm not seeing the connection between > p and q and the steps of the EDOs. > > Too many layers of abstraction for me to follow ...If you tune to a given ET, the table shows you how many steps will be in each interval.> So your parenthetical lists are elegant, but IMO could use > a little bit more of a legend explaining what those lists > represent. Otherwise one has to learn the sequences beforehand > and keep them in mind. I suggest adding a label giving the > parameter list before each line.The "parenthetical lists" are the stringifications of the objects the program uses. I could get it to write out an HTML file for each temperament, but for the moment it's cutting-edge data. Graham
Message: 445 - Contents - Hide Contents Date: Thu, 28 Jun 2001 14:24:15 Subject: Re: questions about Graham's matrices (was: 13-limit mappin From: monz> From: Paul Erlich <paul@s...> > To: <tuning-math@xxxxxxxxxxx.xxx> > Sent: Thursday, June 28, 2001 12:31 PM > Subject: [tuning-math] Re: questions about Graham's matrices (was: 13-limit mappin > > > --- In tuning-math@y..., "monz" <joemonz@y...> wrote: >>>> Er... this is a little confusing, because a triad is constructed of >> only two intervals. >> Last time I checked, a triad had three intervals, a tetrad six, a > pentad ten, and a hexad fifteen.Oops... my bad this time! I compounded my correction of Graham's not-entirely-correct statement by introducing an actual error. Yes, Paul, of course a triad *does* have three intervals. I was speaking specifically of the types of "3rds". Thanks. -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: 446 - Contents - Hide Contents Date: Thu, 28 Jun 2001 22:29 +0 Subject: Re: Hypothesis revisited From: graham@m... Paul Erlich wrote:> I'm still confused about how there can be two different MIRACLE-41s. > Are there two different Canastas too, or does the divergence only > happen at 41?There are two Canstas, 10+31n and 21+31n.> A single MOS is what I expect. The number of notes in that MOS > normally equals the determinant of the matrix of unison vectors, > including the chromatic one. So where are we disagreeing?It's not clear to me if the duality is real or not.>> There are aways two generators that will work. The minus sign > differentiates >> them. >> But if you center the resulting scale around 1/1, either the plus- > sign or the minus-sign generator should give the same results. So > that can't account for the difference we saw.Are the FPBs different in this sense? For the matrices, it's because the mapping to steps in the MOS is always the same.>> No. If you take this matrix at face value: >>>>> [[ 41 0] >>> [ 65 -6] >>> [ 95 7] >>> [115 2] >>> [142 -15]]/41 >>>> it defines Miracle using one 41st part of an octave, and a 41st> part of the>> usual generator. That works, but it isn't efficient. >> How does it work? Certainly the scale doesn't repeat itself every > 41st of an octave.Yes, it would do. If you try tuning a 12-note meantone in cents relative to 12-equal, you'll see the pattern.>> If the method almost works with an arbitrary chroma, that means> we're a step>> towards getting it to work with only commatic unison vectors, which > should be >> possible. >> Well you _should_ be able to find the generator without specifying > the chroma, but you need the chroma to select a particular MOS.Indeed so! But the octave invariant matrix doesn't give you that particular MOS. Although it gives you enough of a clue to work it out from the determinant, the main result is the mapping in terms of generators. Graham
Message: 447 - Contents - Hide Contents Date: Thu, 28 Jun 2001 21:47:35 Subject: Re: questions about Graham's matrices From: Paul Erlich --- In tuning-math@y..., graham@m... wrote:> monz wrote:>> This is along the lines of what I was trying to get Paul to >> understand a couple of different times in the past. It's not >> necessary to always use prime-factors as the basis for lattice >> metrics... any numbers that give even, consistent divisions >> of the pitch-space *in SOME way* will do. The different ways >> of dividing (and multiplying) produce different kinds of lattices.Funny -- I would have said that this is something I was trying to get Monz to understand, rather than something Monz was trying to get me to understand . . . Maybe Monz could restate the context in which he was trying to get me to understand this?
Message: 448 - Contents - Hide Contents Date: Thu, 28 Jun 2001 21:55:38 Subject: Re: Hypothesis revisited From: Paul Erlich --- In tuning-math@y..., graham@m... wrote:> Paul Erlich wrote: >>> I'm still confused about how there can be two different MIRACLE- 41s. >> Are there two different Canastas too, or does the divergence only >> happen at 41? >> There are two Canstas, 10+31n and 21+31n.Hmmm . . . what's the _real_ difference between these two?> > Are the FPBs different in this sense?Yes -- look back a few days -- I showed that there was a schisma difference between a few corresponding pitches in the two FPBs, even though you're claiming the schisma as a chromatic unison vector (hence one that isn't tempered out).>> How does it work? Certainly the scale doesn't repeat itself every >> 41st of an octave. >> Yes, it would do. If you try tuning a 12-note meantone in cents relative > to 12-equal, you'll see the pattern.I see the pattern, but that doesn't make 1/12 octave the period of a 12-note meantone . . . ?>> Well you _should_ be able to find the generator without specifying >> the chroma, but you need the chroma to select a particular MOS. >> Indeed so! But the octave invariant matrix doesn't give you that > particular MOS.Sure it does! Just take the determinant (usually)! (Assuming you already know the generator.)> Although it gives you enough of a clue to work it out > from the determinant,A big clue!> the main result is the mapping in terms of > generators.Well, that does seem to be something very interesting you've found. How can we get that without plugging in a chroma at all?
Message: 449 - Contents - Hide Contents Date: Thu, 28 Jun 2001 22:29:28 Subject: Re: Hypothesis revisited From: Dave Keenan --- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:> --- In tuning-math@y..., Graham Breed <graham@m...> wrote:>>>> One is 10+41n, the other 31+41n. >>>>>> What do you mean by this notation? >>>> Temperements including the ETs with 10+41n or 31+41n notes, where n > is a >> non-negative integer. Graham,I still don't understand this. So 10+41n includes ETs 10 51 61 71 ... and 31+41n includes 31 72 113 ... Only the second looks anything like Miracle to me. Paul Erlich wrote:> I'm still confused about how there can be two different MIRACLE-41s. > Are there two different Canastas too, or does the divergence only > happen at 41? Graham replied: > There are two Canstas, 10+31n and 21+31n.What could this mean when 31 and 72 aren't members of either of these series? I'm very confused. -- Dave Keenan
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