This is an Opt In Archive . We would like to hear from you if you want your posts included. For the contact address see About this archive. All posts are copyright (c).
- Contents - Hide Contents - Home - Section 21000 1050 1100 1150 1200 1250 1300 1350 1400 1450 1500 1550 1600 1650 1700 1750 1800 1850 1900 1950
1300 - 1325 -
Message: 1325 - Contents - Hide Contents Date: Tue, 21 Aug 2001 01:09:57 Subject: Re: Hypothesis From: Paul Erlich --- In tuning-math@y..., genewardsmith@j... wrote:> --- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote: >>> 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! >> It seems to me the comma problem is less of a problem if you are only > interested in melody, and this whole business is justified in terms > of melody.In terms of harmony?> Is it really true that a pentatonic or diatonic melody > sounds better in a meantone tuning than it does in just tuning?Probably Pythagorean is everyone's favorite melodic tuning. And yes, I do dislike the melodic jaggedness of just scales . . . but why don't we just assume harmony _is_ important for the purposes of the Hypothesis. Let's assume that the only reason for tempering is to tame those nasty wolves.> Moreover, the smaller the scale steps the harder it becomes to tell > the difference between them. If hearing the difference between 9/8 > and 10/9 is hard, hearing the difference between 16/15 and 15/14 will > certainly be harder.Probably . . . let's just say that tempering out the 225:224 is more of a harmonic, than a melodic, consideration.
Message: 1326 - Contents - Hide Contents Date: Tue, 21 Aug 2001 18:55:06 Subject: Re: Tetrachordal alterations (was: Hi gang.) From: Paul Erlich --- In tuning-math@y..., "Dave Keenan" <D.KEENAN@U...> wrote:>> --- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote: >> What we need is a really user-friendly, _practical_ guide to a > bunch>> of the new temperaments and their MOSs (and ideally, tetrachordal >> alterations of those MOSs in cases like 10-of-22 and 22-of-46). >> Are "tetrachordal alterations" only possible when the interval of > repetition is some whole-number fraction of octave?In an MOS, the interval of repetition is _always_ some whole-number fraction of the interval of equivalence.> > How do you do them, in general?I don't know if there's a general way, but you understand what omnitetrachorality is, right? "Alteration" simply means re-shuffling the step sizes in an MOS or hyper-MOS.> > What would be a "tetrachordal alteration" of Blackjack?Don't know if there is one! Can you make a blackjack-like scale omnitetrachordal?
Message: 1327 - Contents - Hide Contents Date: Tue, 21 Aug 2001 03:18:23 Subject: Re: The hypothesis From: Dave Keenan --- In tuning-math@y..., genewardsmith@j... wrote:> 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. ...There's no confusion over whether 2 is a prime. We understand quite well, all that you wrote. Each of us has probably railed against it at some time. But it would be too confusing to change it now. You'd best just learn to accept it. Rank 1 = equal temperament Rank 2 = linear temperament Rank 3 = planar temperament Regards, -- Dave Keenan
Message: 1328 - Contents - Hide Contents Date: Tue, 21 Aug 2001 19:03:00 Subject: Re: Hypothesis From: Paul Erlich --- In tuning-math@y..., genewardsmith@j... wrote:> --- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote: >>>> Is it really true that a pentatonic or diatonic melody >>> sounds better in a meantone tuning than it does in just tuning? >>> Probably Pythagorean is everyone's favorite melodic tuning. >> I don't know--to my ears, melodically Pythagorean is brighter and > more aggressive, (and actually not too much different from 12 ET), > but JI diatonic melody is smooth and refined, so to speak. Maybe my > ears are no good. :)You like JI diatonic melody even when there is a direct leap of 40:27?> > And yes,>> I do dislike the melodic jaggedness of just scales . . . but why >> don't we just assume harmony _is_ important for the purposes of the >> Hypothesis. Let's assume that the only reason for tempering is to >> tame those nasty wolves. >> As you can see, "jagged" is not how JI diatonic melodies strike me at > all.Even when there is a direct leap of 40:27? P.S. The only culture where the major scale is tuned in JI is in Indian music. But they tune it 1/1 9/8 5/4 4/3 3/2 27/16 15/8 2/1, which has two identical tetrachords, and avoid direct leaps between 5/4 and 27/16.> If you are tempering merely to tame wolves, why does this WF > stuff concern you, however?I don't understand this question. If we prove the hypothesis, we'll essentially be saying that in a sense, the most harmonically interesting fixed-pitch scales are all MOS scales. Think of MOS as a nice, abstract property, and please disregard all the recent music- theory literature! I think it's extremely interesting if we can determine a nice, simple property that will have to hold for any scale that comes out of the PB-tempering process.
Message: 1329 - Contents - Hide Contents Date: Tue, 21 Aug 2001 20:26:30 Subject: Re: Tetrachordal alterations From: David C Keenan I've ignore tetrachordality for too long, it's time I figured out how to make it happen in any linear temperament. Paul, I had hoped you had more of a handle on it than I did. It seems not. --- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:> --- In tuning-math@y..., "Dave Keenan" <D.KEENAN@U...> wrote:>> Are "tetrachordal alterations" only possible when the interval of >> repetition is some whole-number fraction of octave? >> In an MOS, the interval of repetition is _always_ some whole-number > fraction of the interval of equivalence.I meant "fraction" in the popular sense, as not including the whole. But no, I now believe tetrachordal (not necc. omnitetrachordal) alterations are always possible, it's just a question of finding the minimal alteration that makes it tetrachordal and just how big that alteration turns out to be in each case. It's not clear to me why a tetrachordal scale in some linear temperament, must have the same number of notes as a MOS in that temperament, but I'll assume it for now. Know any good reason(s)? I'd measure the magnitude of a particular alteration (the alteredness?) by how many notes would have to be added to the original MOS, by consecutive generators, keeping all chains the same, before you can contain the tetrachordal scale. e.g. Your tetrachordal decatonic has an alteredness of 2, the minimum possible with a half-octave period.>> How do you do them, in general? >> I don't know if there's a general way, but you understand what > omnitetrachorality is, right?Yes. But I wasn't considering _omni_tetrachordality. That looks too hard for now. Do any popular or historical scales have it?> "Alteration" simply means re-shuffling > the step sizes in an MOS or hyper-MOS. >>>> What would be a "tetrachordal alteration" of Blackjack? >> Don't know if there is one! Can you make a blackjack-like scale > omnitetrachordal? God knows!But I had previously failed to appreciate that Blackjack (a 21 note chain of Miracle generators = secors = 116.7c) is tetrachordal. duh! You get disjunct 9-step tetrachords when you start from the point of symmetry. Ok. Lets look at the 10 note Miracle MOS. What's the minimum alteration to make it tetrachordal. This just amounts to asking: What is the 10 note tetrachordal Miracle scale that spans the fewest secors. I'll notate my tetrachords as the conventional C...F and G...C. Since the tetrachords must share exactly one note (C) and must be melodically identical, a 10-note tetrachordal scale must have 5 notes in each tetrachord and 1 note outside the tetrachords. (The possibility of 3 notes outside the tetrachord can be ignored because even Blackjack has only 2 and we expect to get out of it with fewer generators than Blackjack). Here's what we start with 5 1 C> D[ Eb< Ev F Gb^ G> A[ Bb< Bv C Db^ D> E[ F< F#v G Ab^ A> B[ C< 5 1 That's a long enough chain of secors (Blackjack) where we will number the notes of each tetrachord of our 10-note scale in pitch order. One tetrachord above and one below. We'll then mark the note that's outside the tetrachords with an "X". We'll try to stay as close to the center of the chain as possible. Note that X can go on either Gb^ or F#v. So here are eight minimal possibilities. They all span 16 notes in the chain. 5 1 2 3 4 C> D[ Eb< Ev F Gb^ G> A[ Bb< Bv C Db^ D> E[ F< F#v G Ab^ A> B[ C< X 5 X 1 2 3 4 4 5 1 2 3 C> D[ Eb< Ev F Gb^ G> A[ Bb< Bv C Db^ D> E[ F< F#v G Ab^ A> B[ C< X 4 5 X 1 2 3 3 4 5 1 2 C> D[ Eb< Ev F Gb^ G> A[ Bb< Bv C Db^ D> E[ F< F#v G Ab^ A> B[ C< X 3 4 5 X 1 2 2 3 4 5 1 C> D[ Eb< Ev F Gb^ G> A[ Bb< Bv C Db^ D> E[ F< F#v G Ab^ A> B[ C< X 2 3 4 5 X 1 Actually, not all eight are distinct. Some are simply transposed. There are really only 5. 3 of which are disjunct-tetrachordal in 2 positions. Here's what they look like melodically (in steps of 72-tET). Vertical bar "|" is used to show tetrachords. 7 7 7 9|5 7|7 7 7 9 7 7 9 7|5 7|7 7 9 7 7 7 7 9|7 5|7 7 7 9 7 9 7 7|5 7|7 9 7 7 7 7 9 7|7 5|7 7 9 7 9 7 7 7|5 7|9 7 7 7 7 9 7 7|7 5|7 9 7 7 9 7 7 7|7 5|9 7 7 7 The middle one (the most even) looks like a very interesting detempering of your tetrachordal (pentachordal) decatonic. It has 2 of 4:5:6:7 2 of 1/(7:6:5:4) 2 of 4:5:6 2 of 1/(6:5:4) 2 of 3:7:9:21 ASS 1 of 3:9:11:33 ASS and probably some necessarily-tempered ASSes. But I don't see any sense in which any of these tetrachordal scales are an "alteration" of the 10 note MOS. It seems to me that tetrachordal scale creation (in a given temperament) is not related to MOS scale creation (in the same temperament) in any way. Here's the most compact (on the chain) inversionally-symmetric 7-note tetrachordal scale in Miracle. 3 4 1 2 C> D[ Eb< Ev F Gb^ G> A[ Bb< Bv C Db^ D> E[ F< F#v G Ab^ A> B[ C< 3 4 1 2 Steps 7 16 7|12|7 16 7 Here's the most even one that fits in Blackjack. 2 4 1 3 C> D[ Eb< Ev F Gb^ G> A[ Bb< Bv C Db^ D> E[ F< F#v G Ab^ A> B[ C< 2 4 1 3 Steps 9 12 9|12|9 12 9 Which is of course the neutral thirds MOS, or Mohajira. These are even less of an alteration of a MOS. -- Dave Keenan Brisbane, Australia Dave Keenan's Home Page * [with cont.] (Wayb.)
Message: 1330 - Contents - Hide Contents Date: Tue, 21 Aug 2001 03:38:23 Subject: Re: Microtemperament and scale structure From: Dave Keenan --- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:> 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.Shouldn't all those "L"s be "s"s and vice versa?
Message: 1331 - Contents - Hide Contents Date: Tue, 21 Aug 2001 19:05:24 Subject: Re: Hypothesis From: Paul Erlich --- In tuning-math@y..., genewardsmith@j... wrote:> but JI diatonic melody is smooth and refined, so to speak. Maybe my > ears are no good. :)Do you have an example of an actual diatonic melody that sounds good to you in JI? Most classical melodies, say Mozart for example, sound bad to me in JI -- the motivic unity between statements is disturbed by the variation in the size of the whole tone.
Message: 1332 - Contents - Hide Contents Date: Tue, 21 Aug 2001 03:50:10 Subject: Tetrachordal alterations (was: Hi gang.) From: Dave Keenan> --- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote: > What we need is a really user-friendly, _practical_ guide to a bunch > of the new temperaments and their MOSs (and ideally, tetrachordal > alterations of those MOSs in cases like 10-of-22 and 22-of-46).Are "tetrachordal alterations" only possible when the interval of repetition is some whole-number fraction of octave? How do you do them, in general? What would be a "tetrachordal alteration" of Blackjack? -- Dave Keenan
Message: 1333 - Contents - Hide Contents Date: Tue, 21 Aug 2001 19:08:23 Subject: Re: The hypothesis From: Paul Erlich --- In tuning-math@y..., genewardsmith@j... wrote:> --- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote: >>>> There is no >>> special treatment given to the octave in this method, but I see > no>>> reason in terms of psychological acoustics why there needs to be. >>> Right -- so mathematically, why don't we just call the octave (or > in>> some cases, like the BP scale, another simple interval) the >> equivalence interval, and deal with ETs as cyclic groups, etc., >> ignoring the question of whether the octaves are slightly tempered > or >> not? >> There are two distinct questions involved--tuning, and scale > construction. If you are discussing tuning, the octave is an interval > and needs to be tuned--even leaving it a 2 is after all a choice of > tuning. If you are constructing scales which repeat a particular > pattern of steps, the psycoacoustic properties of the octave make it > by far the most interesting choice.So I ask you again -- why not leave the question of how to tune the octave as an outside question, and deal with scales as if they exist in a cyclic continuum, modulo the octave, in the majority of our manipulations?
Message: 1334 - Contents - Hide Contents Date: Tue, 21 Aug 2001 19:11:09 Subject: Re: Mea culpa From: Paul Erlich --- In tuning-math@y..., "D.Stearns" <STEARNS@C...> wrote:> <<Myhill's property isn't just about the step sizes. Recall the > melodic minor scale, which has two step sizes but isn't WF. Myhill's > property says it has two sizes of _every_ generic interval size. But > in the case of LssssLssss, all "sixths" are the same size: L+4*s. > There's only one size of "sixth" -- so Myhill fails.>> > > > Hi Paul (and everyone), > > Rather than thinking of Myhill's property, MOS and WF as one big > group, perhaps it's better to pair WF and Myhill's property on one > side and MOS and maximal evenness on the other -- here's the idea: > > all MOS scales will have a ME rotationWhat do you mean, an ME rotation? Surely you don't mean maximal evenness -- which is independent of rotation. Many MOS scales are not ME in any reasonable embedding -- the Blackjack scale, for instance. But basically, you're right -- there is very little difference between MOS and WF and Myhill's property.
Message: 1335 - Contents - Hide Contents Date: Tue, 21 Aug 2001 04:16:36 Subject: Re: Hypothesis From: genewardsmith@xxxx.xxx --- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:>> Is it really true that a pentatonic or diatonic melody >> sounds better in a meantone tuning than it does in just tuning?> Probably Pythagorean is everyone's favorite melodic tuning.I don't know--to my ears, melodically Pythagorean is brighter and more aggressive, (and actually not too much different from 12 ET), but JI diatonic melody is smooth and refined, so to speak. Maybe my ears are no good. :) And yes,> I do dislike the melodic jaggedness of just scales . . . but why > don't we just assume harmony _is_ important for the purposes of the > Hypothesis. Let's assume that the only reason for tempering is to > tame those nasty wolves.As you can see, "jagged" is not how JI diatonic melodies strike me at all. If you are tempering merely to tame wolves, why does this WF stuff concern you, however?
Message: 1337 - Contents - Hide Contents Date: Tue, 21 Aug 2001 04:22:49 Subject: Re: The hypothesis From: genewardsmith@xxxx.xxx --- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:>> There is no >> special treatment given to the octave in this method, but I see no >> reason in terms of psychological acoustics why there needs to be.> Right -- so mathematically, why don't we just call the octave (or in > some cases, like the BP scale, another simple interval) the > equivalence interval, and deal with ETs as cyclic groups, etc., > ignoring the question of whether the octaves are slightly tempered or > not?There are two distinct questions involved--tuning, and scale construction. If you are discussing tuning, the octave is an interval and needs to be tuned--even leaving it a 2 is after all a choice of tuning. If you are constructing scales which repeat a particular pattern of steps, the psycoacoustic properties of the octave make it by far the most interesting choice.
Message: 1338 - Contents - Hide Contents Date: Tue, 21 Aug 2001 19:27:09 Subject: Re: Chromatic = commatic? From: Paul Erlich --- In tuning-math@y..., genewardsmith@j... wrote:>> 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. >> Both of these are elements of the kernel of the 7-homomorphism > [7, 11, 16] and together they generate it. There really is no > distinction to be drawn beyond the obvious fact that 25/24 is bigger > than 81/80.There is an additional, formal distinction drawn, which you're not taking into account in your formalism.> You could always call the biggest element in your > generating set the chromatic unison and the rest commatic unison > vectors,It's not always done that way.> but I don't see the point.The point is that, for example, if only one of the unison vectors is chromatic and the rest are commatic, you end up with an MOS scale, which has up to two specific sizes for each generic interval. 7-tET has only one specific size for each generic interval. Did you read my "proof" of the Hypothesis?> Anyway, what is chromatic for one > set will end up being commatic for another!Sure -- but within a given complete set of unison vectors, choosing one of the unison vectors to be chromatic and the rest to be commatic leads uniquely to an MOS scale -- that's what we're trying to prove here.>>>> 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). >> Now I translate this to saying that if the rank of the kernel is n, > then we get a linear temperament. Since the rank of the set of notes > is n+1, this means the codimension is 1 and hence the rank of the > homomorphic image is 1, meaning we have an et--We most definitely do not have an ET. There are two step sizes, and in general, up to two sizes for each generic interval.> which is precisely > what we did get in the case where we had the 7-et.In 5-limit space, if 81/80 and 25/24 are both commatic unison vectors, then you essentially have 7-et. But if 81/80 is commatic while 25/24 is chromatic, you get a diatonic scale, where the cycle of steps is sLLLsLL, the cycle of thirds is sLssLsL, the cycle of fourths is sssssLs, etc.> Why do you say > linear temperament, which we've just determined means rank 2?Because the diatonic scale described above has a single generator, namely s+2*L (or any octave-equivalent, such as s+3*L).>>> 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. >> Aha! So perhaps what you are saying is if the codimension is 2, then > the rank of the homomorphic image is 2, and we have a linear > temperament.Perhaps -- if I count your way.>>> No -- you did that with n unison vectors -- I'm not counting the 2 >> axis as a "dimension" here. >> Not a good idea in this context--you should.OK -- I'll try to do it your way from now on.>>> MOS means that there is an interval of repetition >> What do the letters of the acronym stand for?You don't want to know. It's Moment of Symmetry -- I know, that means something completely different in other contexts. But I think the person who came up with it was a non-mathematician looking at the results of iterating a fixed generator. Imagine that the generator keeps building on top of itself, around and around the cycle that is the octave. At stage N in this process, we have a scale with N notes. If the scale has only two step sizes, then this "moment" in the process has a special "symmetry" -- thus the name.
Message: 1340 - Contents - Hide Contents Date: Tue, 21 Aug 2001 19:30:59 Subject: Re: Microtemperament and scale structure From: Paul Erlich --- In tuning-math@y..., genewardsmith@j... wrote:> --- In tuning-math@y..., graham@m... wrote: >>> It's what <Unison vector to MOS script * [with cont.] (Wayb.)> is all about. >> <Unison vectors * [with cont.] (Wayb.)> is a list of examples. >> It was pretty hard to figure out what they were examples of. > > Let me give an example matrix computation, and see if it looks > familiar. Let's take three et's in the 5-limit, for 12, 19, and 34. > If we make a matrix out of them, we have > > [12 19 34] > S = [19 30 54] > [28 44 79] > > Since this consists of three column vectors pointing in more or less > the same direction, the determinant is likely to be small; however > none of these three is a linear combination of the other two (as > often will happen--ets tend to be sums of other ets) the determinant > is nonzero--in this case, 1. If we invert it, we get > > [-6 -5 6] > S^(-1) = [11 -4 -2] > [-4 4 -1] > > The row vectors of S^(-1) are now 15625/15552, 2048/2025, and 81/80. > Taken in pairs, these give generators for the kernel of each of the > above systems, and hence good unison vectors for a PB. Each is a step > vector in one system, and a unison vector in the other two, in the > obvious way (given how matrix multiplication works.) > > In the same way, we could start with three linearly independent > unison vector candidates, and get a matrix of three ets by inverting. > > The single vectors generate the intersection of the kernels of a pair > of ets, and so define a linear temperament which factors through to > each of the ets. That is, 81/80 generates the intersection of the > kernel of the 12-system and the 19-system, and produces the mean tone > temperaments. Both 12 and 19 belong to this system--we can send it to > first the mean tone, then to either 12 or 19 (then to tuning as the > last step!) Similarly, 2048/2025 defines a temperament which is > common to both the 12 and the 34 system. It essentially defines what > they have in common.The latter are called diaschismic temperaments.> > There are other types of matrix computations we could make, but I'm > wondering if this seems familiar?Yes it does -- see Herman Miller's posts to this forum -- but thanks for this formalism, it's sure to be very valuable.
Message: 1341 - Contents - Hide Contents Date: Tue, 21 Aug 2001 07:49:27 Subject: Chromatic = commatic? From: genewardsmith@xxxx.xxx --- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:> 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.What you are calling the determinant is just the determinant of the minor you get by setting 2 aside--and there is 2 on the brain again. From the point of view of approximations and real life, your comment is true. From the point of view of pure algebra, it isn't. From an algebraic point of view, the 7-et might be [7, 11, 17] and not [7, 11, 16]--they are different homomorphisms. To recover the whole homomorphism, and not just the number of steps in an octave, we need all three minor determinants.> 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.Both of these are elements of the kernel of the 7-homomorphism [7, 11, 16] and together they generate it. There really is no distinction to be drawn beyond the obvious fact that 25/24 is bigger than 81/80. You could always call the biggest element in your generating set the chromatic unison and the rest commatic unison vectors, but I don't see the point. Anyway, what is chromatic for one set will end up being commatic for another!>>> 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).Now I translate this to saying that if the rank of the kernel is n, then we get a linear temperament. Since the rank of the set of notes is n+1, this means the codimension is 1 and hence the rank of the homomorphic image is 1, meaning we have an et--which is precisely what we did get in the case where we had the 7-et. Why do you say linear temperament, which we've just determined means rank 2?>> 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.Aha! So perhaps what you are saying is if the codimension is 2, then the rank of the homomorphic image is 2, and we have a linear temperament.> No -- you did that with n unison vectors -- I'm not counting the 2 > axis as a "dimension" here.Not a good idea in this context--you should.> MOS means that there is an interval of repetitionWhat do the letters of the acronym stand for?
Message: 1342 - Contents - Hide Contents Date: Tue, 21 Aug 2001 19:42:38 Subject: Re: The hypothesis From: Paul Erlich --- In tuning-math@y..., graham@m... wrote:>> >> Because it is generated by one unison vector. I'm not clear yet if a >> unison vector is supposed to be an element of the kernel or a >> generator of the kernel, as I mentioned. >> It looks like a commatic UV would be in the kernel, and a chromatic UV > would not.Yes and no. I suppose here we've hit the limits of Gene's formalism. The chromatic unison vector is not a true equivalence, but is considered an interval too small (or too whatever) to keep in the resulting scale. It's an abrupt boundary in the lattice.> > You seem to be saying that specifying the tuning of the octave is > confusing when *you* are the one who wants to specify it! By taking it > out of the system, I don't care either way. The size of the octave > becomes a property of the metric, not the matrices. I do consider the > metric to be less abstract than the algebra.I agree with you here, Graham!
Message: 1343 - Contents - Hide Contents Date: Tue, 21 Aug 2001 08:10:08 Subject: Re: Microtemperament and scale structure From: genewardsmith@xxxx.xxx --- In tuning-math@y..., graham@m... wrote:> It's what <Unison vector to MOS script * [with cont.] (Wayb.)> is all about. > <Unison vectors * [with cont.] (Wayb.)> is a list of examples.It was pretty hard to figure out what they were examples of. Let me give an example matrix computation, and see if it looks familiar. Let's take three et's in the 5-limit, for 12, 19, and 34. If we make a matrix out of them, we have [12 19 34] S = [19 30 54] [28 44 79] Since this consists of three column vectors pointing in more or less the same direction, the determinant is likely to be small; however none of these three is a linear combination of the other two (as often will happen--ets tend to be sums of other ets) the determinant is nonzero--in this case, 1. If we invert it, we get [-6 -5 6] S^(-1) = [11 -4 -2] [-4 4 -1] The row vectors of S^(-1) are now 15625/15552, 2048/2025, and 81/80. Taken in pairs, these give generators for the kernel of each of the above systems, and hence good unison vectors for a PB. Each is a step vector in one system, and a unison vector in the other two, in the obvious way (given how matrix multiplication works.) In the same way, we could start with three linearly independent unison vector candidates, and get a matrix of three ets by inverting. The single vectors generate the intersection of the kernels of a pair of ets, and so define a linear temperament which factors through to each of the ets. That is, 81/80 generates the intersection of the kernel of the 12-system and the 19-system, and produces the mean tone temperaments. Both 12 and 19 belong to this system--we can send it to first the mean tone, then to either 12 or 19 (then to tuning as the last step!) Similarly, 2048/2025 defines a temperament which is common to both the 12 and the 34 system. It essentially defines what they have in common. There are other types of matrix computations we could make, but I'm wondering if this seems familiar?
Message: 1344 - Contents - Hide Contents Date: Tue, 21 Aug 2001 20:01:52 Subject: Re: Microtemperament and scale structure From: Paul Erlich --- In tuning-math@y..., graham@m... wrote:> In-Reply-To: <9lrmea+klq4@e...> > Paul) 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)? >> No, the commatic UVs define the approximation, the chromatic UVs are used > to define the tuning.I see what you mean by that now. By varying the interval that the chromatic unison vector is mapped to, you can get different variations on the tuning system which obeys the given "approximation" you refer to. But you're missing the point of the chromatic unison vector. You can take _any_ non-commatic vector, vary the interval that that vector is mapped to, and likewise get different variations on the tuning system which obeys the "approximation". The chromatic unison vector comes into play by _delimiting_ the number of notes in the scale. As you know, if there is only one chromatic unison vector, then ignoring it and just looking at the commatic unison vectors defines an "approximation" that implies a particular linear temperament. This linear temperament is potentially infinite in extent. The chromatic unison vector allows for a sort of imperfect closure of the system because, after a certain number of iterations of the generator of the linear temperament, you'll generate the chromatic unison vector (this should be easy to prove). And that number of iterations of the generator is where you stop adding notes to your scale.> I'd be quite happy to forget the strong form of the hypothesis.That's too bad. But OK, let's address the weak form first. Then we throw out the one chromatic unison vector, as it doesn't come into play here. We already have many arguments as to why it works. Let's get it on firm mathematical footing with Gene. Maybe Gene can prove the rule about how to calculate the generator of the linear temperament.>> >> 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. >> But Carey & Clampitt also say the two concepts are identical. Don't they? > Yes, note 1 says they are "equivalent".Well obviously they made a mistake either here or in their list!> > They list as Example 2 of CLAMPITT.PDF "the 21 nondegenerate well- formed > sets in the twelve-note universe" but say nothing about those 12 notes > defining an *octave*. The octatonic scale would belong to the 2 tone > universe.You mean the 3 tone universe.> Also, on page 2, "By /interval of periodicity/ we mean an > interval whose two boundary pitches are functionally equivalent. > Normally, the octave is the interval of periodicity." They don't define > Well Formedness, but refer to another paper. I'm guessing it depends on > the interval of periodicity, not the octave. The same seems to be true of > Myhill's Property.But in the symmetrical decatonic scale, the boundary pitches are _not_ functionally equivalent. They're a half-octave apart, and one only assumes that _octaves_ are functionally equivalent when constructing the periodicity block. The half-octave periodicity comes in as a new feature, not as a previously assumed equivalence relation.
Message: 1345 - Contents - Hide Contents Date: Tue, 21 Aug 2001 20:03:46 Subject: Re: Mea culpa From: Paul Erlich --- In tuning-math@y..., graham@m... wrote:> In-Reply-To: <009701c12a17$911e9700$5340d63f@stearns> > Dan Stearns wrote: >>> Rather than thinking of Myhill's property, MOS and WF as one big >> group, perhaps it's better to pair WF and Myhill's property on one >> side and MOS and maximal evenness on the other -- here's the idea: >> >> all MOS scales will have a ME rotation >> >> all WF scales will have Myhill's property >> So WF/Myhill assumes an octave of 2:1 but MOS/ME can be any period?No, WF/Myhill only assumes that the interval of repetition is equal to the interval of equivalence. ME shouldn't be in this discussion -- it assumes an embedding universe and Blackjack isn't an ME set in 72- tET.
Message: 1346 - Contents - Hide Contents
Date: Tue, 21 Aug 2001 21:03 +0
Subject: Generators and unison vectors
From: graham@xxxxxxxxxx.xx.xx
I was thinking about what n "unison vectors" in n-dimensional space could
mean. In fact, it's what I've already called a "basis". See
<404 Not Found * [with cont.] Search for http://www.microtonal.co.uk/matrix.htm in Wayback Machine>. I assume that's analogous to
Gene's "kernel". Let's revisit the example there:
(t) ( 1 -2 1)
(s) = ( 4 -1 -1)H
(p) (-4 4 -1)
Which can be used to define this scale
C D E F G A B C
t+p t s t+p t p+t s
The inverse of the matrix is
( 5 2 3)
( 8 3 5)
(12 4 7)
It defines H in terms of t, s and p. That I already knew. Now, the
interesting thing is, taking p out of the scale
C D E F G A B C
t t s t t t s
is the same as making p a commatic unison vector. And we can define the
approximate H in terms of t and s
( 5 2)(t)
H' = ( 8 3)(s)
(12 4)
I already knew this, but I didn't think of it as "designating a commatic
unison vector" or whatever. Before we've been thinking about starting
with commatic unison vectors and making them chromatic. It actually makes
more sense the other way around.
Making (4 -1 -1)H or 16:15 another commatic unison vector gives us
5-equal:
( 5)(t)
H' = ( 8)
(12)
Gene's already explained this in algebraic terms, but I think the example
makes it clearer. It means I can now interpret
( 1 -2 1)
( 4 -1 -1)
(-4 4 -1)
as the octave-specific equivalent of a periodicity block. Unfortunately
it only has 1 note, so the analogy breaks down. But whatever, I'll call
all the intervals unison vectors anyway.
Now, we can also define meantone as
( 1 0 0) (1 0)
( 0 1 0)C = (0 1)
(-4 4 -1) (0 0)
Here, the octave and twelfth are taking the place of the chromatic unison
vectors. So, can we call them unison vectors? I think not, because C
evaluates to
( 1 0)
( 0 1)
(-4 4)
That has negative numbers in it, which means the generators (for so they
are) don't add up to give all the primary consonances. So now we have a
definition for what interval qualifies as a "unison vector".
Interestingly, a fifth would work as a generator here, but not for a
diatonic scale.
An alternative to "unison vector" would be "melodic generator".
Usually we call the octave the period and a fifth (which is therefore
equivalent to the twelfth) the generator. It isn't much of a stretch to
think of two generators.
In octave-equivalent terms, a unison is:
v v v v v v v v v v v w
where v is s fifth and w is a wolf. The same algebra applies, but it's
harder to visualise. I think the fifth here does count as a unison
vector.
Defining an MOS in octave-equivalent terms is easy, once you realize that
the generators that pop out needn't be unison vectors in octave-specific
space. So is defining temperaments by chromatic unison vectors with
octave-specific matrices. The difficult part is when you make the octave
a generator in those octave-specific matrices. Usually the octave won't
be a unison vector. Obviously not if you want to look at intervals
smaller than an octave! We end up with generators that aren't unison
vectors, and it all gets complicated.
The hard part of my original MOS finding script was getting the second
scale step from the one chromatic unison vector. It's because I was
getting mixed results: one column showing scale steps, the other the
generator mapping. Keep to only unison vectors (in the relevant space)
and it's all nice and simple.
That's the way I see it now. I've also realized that I've been saying
"octave invariant" when "octave equivalent" would probably be more to the
point.
Graham
Message: 1347 - Contents - Hide Contents Date: Tue, 21 Aug 2001 20:12:37 Subject: Re: Chromatic = commatic? From: Paul Erlich --- In tuning-math@y..., graham@m... wrote:> For example, here's that pair of octave-invariant unison vectors: > > [ 4 -1] > [ 0 -3] > > Invert it to get > > [-3 1] > [ 0 4] >The inverse times the determinant.> Equal temperaments are tricky when they're octave invariant. Effectively, > it means the new period is a scale step. Hence all notes are identical.Here you're going with the WF-related view that the interval of repetition must be equal to the interval of equivalence. I suggest instead the alternate view that they be allowed to differ, but that the interval of equivalence is always an integer number of intervals of repetition.> How about you stop telling us what we should be doing, and start listening > to what we're trying to say?Graham, I agree with you but maybe we should be willing to meet Gene halfway? Dave Keenan expressed an opinion on this but I'm not sure what it was.
Message: 1348 - Contents - Hide Contents Date: Tue, 21 Aug 2001 20:15:42 Subject: Re: Mea culpa From: Paul Erlich --- In tuning-math@y..., "D.Stearns" <STEARNS@C...> wrote:> Paul Erlich wrote, > > <<What do you mean, an ME rotation?>> > > In the 12-tet diatonic, the II, or dorian rotation, is ME.That is not the accepted definition of ME, according to any of the papers defining it by Clough, et al. It's a rotationally invariant property of scales. It does assume an embedding universe of notes -- normally a 12-note universe, but this can vary.> > <<Surely you don't mean maximal evenness -- which is independent of > rotation. Many MOS scales are not ME in any reasonable embedding -- > the Blackjack scale, for instance.>> > > Yes, maximal evenness -- show me any single generator MOS scale and > I'll show you its ME rotation!The blackjack scale is not ME in 72. The diatonic scale is not ME in 31. We've been over this before.
Message: 1349 - Contents - Hide Contents Date: Tue, 21 Aug 2001 20:25:22 Subject: Re: Microtemperament and scale structure From: genewardsmith@xxxx.xxx --- In tuning-math@y..., graham@m... wrote:>> In the same way, we could start with three linearly independent >> unison vector candidates, and get a matrix of three ets by inverting.> So is that what I'm doing? Hmm. Actually, I'd take two UVs along with > the octave. Hmm.This is probably going to annoy you, but I think it is true so I'll say it anyway--the octave is merely adding to the confusion. Let's take an example consisting of 128/125, 25/24 and 81/80. We get the matrix [ 7 0 -3] M = [-3 -1 2] [-4 4 -1] Then [ 7 12 3] M^(-1) = [11 19 5] [16 28 7] The columns of which are the 7, 12 and 3 divisions. (Little known but in some contexts actually useful fact--in the 3 et, a comma equals a third.) You can recover the same information by replacing one row of M in succession by [1 0 0] and taking adjoints or inverses, which you did before in the case of the top row. When you take the three adjoint matricies in succession, you get: [ 7 0 0] -[11 1 2] [16 4 1], [ 0 12 0] -[-1 19 3] [-4 28 0], [ 0 0 3] -[-2 -3 5] [-1 0 7]. You indicated you had an interpretation of the other columns of these matricies, but I don't see anything interesting about them, and hence my comment. I'd like to know if I am wrong!
1000 1050 1100 1150 1200 1250 1300 1350 1400 1450 1500 1550 1600 1650 1700 1750 1800 1850 1900 1950
1300 - 1325 -