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Message: 750 - Contents - Hide Contents Date: Tue, 21 Aug 2001 15:38 +0 Subject: Re: The hypothesis From: graham@m... In-Reply-To: <9ls5st+gcan@e...> In article <9ls5st+gcan@e...>, genewardsmith@j... () wrote:>> 2 is certainly prime, but most of the time we consider octave- > invariant >> scales. >> Considering scales is another level of generality altogether--first > we have approximations (kernels, unison vectors, and so forth) then > we have tuning, and finally we select a subset and have scales.Oh, right, but whatever they are, they're octave invariant. CLAMPITT.PDF is a relevant place for definitions, as it's already been referenced: "By /scale/ we refer to a set of pitches ordered according to ascending frequencies (pitch height) bounded by an interval of periodicity."> Of course, unless you have an infinite number of notes in your scale, > which you may have conceptually but not in practice, you don't have > octave invariance anyway.Then we'll add a new category: 1) just intonation 2) approximations 3) tuning 4) scale 5) scale in practice In general, just intonation will have an infinite number of notes in multiple dimensions. Approximations will have an infinite number of notes in fewer dimensions. A tuning is a special case of either a just intonation of approximation. A scale is a subset of a tuning that has a finite number of notes, either overall or to the interval of repetition. A scale in practice will have a finite total number of notes. I don't agree with this anyway. You can have a scale without a tuning. For example, a C major scale can be tuned to either 12- or 19-equal.>> So the kernel has dimension 1 because it contains 1 unison vector? >> 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.>> So in octave-invariant terms, 5-limit is rank 2, but cyclic about > the >> octave. >> If we consider equivalence classes modulo octaves, the 5-limit is > free of rank two, but I don't know what you mean by "cyclic around > the octave".You describe an ET as cyclic below.>> An ET would be rank 0 I suppose, but you've already given the >> real name for that case. >> An ET would be free of rank 1, or "cyclic of infinite order". If we > mod out by octaves, it would no longer be free but would (still) be > cyclic, which implies one generator.But if we "mod out" by one scale step, it would still have rank 1? In fact, there are two different ways of treating octave equivalence: 1) Consider the scale repeating about the octave. 2) Consider the scale as only existing within the octave. And (2) is actually closer to the way octave-invariant matrices work.>> In the octave-invariant case the octave lies outside the system, so > you>> can't say anything about it. >> Whether to tune the octave exactly or not is a question which lies at > a more specific, less abstract level than that created by defining > certain things to be unison vectors. As a general rule, you only > confuse things by insisting on concrete particulars when they are not > required. About all one can say for certain is that you can't toss 2 > out of a discussion involving unison vectors, because without 2 we > can't tell what is a small interval and what is not.You certainly can tell a small interval if 2 is taken out. Of course, you need a way of calculating interval size. As I see it, that lies outside of the group theory we've been discussing so far. There are two ways of doing it, corresponding to the two interpretations above. For (1), you calculate the pitch, and allow an arbitrary number of octaves to be added or subtracted. For (2), you calculate the pitch modulo the octave. If you want a small interval, you take the smallest option for (1). That is actually more liberal than (2). [4 -1] will give 81:80 either way. But [-4 1] could be 80:81 for (1) but has to be 160:81 for (2). My program works with (2). 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. It happens that in the octave-specific case, the metric is itself a column matrix, so that is a simplification. It means you can define the tuning within the system. With octave invariant matrices, you can only comment on intervals within the octave. So now we come to consider unison vectors. It sort of looks like unison vectors have to be small intervals. But I haven't seen a definition of *how* small. I'm hoping that it doesn't matter at all for the octave invariant case, so that the tuning of that octave becomes irrelevant. If it doesn't work out that way perhaps you, as the mathematician, can tell us the size constraints on the unison vectors.>> For the octave invariant case, the fourth or fifth is the > generator, which>> I think agrees with both meanings of "generator". >> You can generate by fifths and octaves if you want, but you don't > need to.You do need to if you want to enforce octave equivalence. For other approximations, other generators will be needed. I've even done calculations with a different interval of equivalence. See <import temper, string * [with cont.] (Wayb.)> <2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 ... * [with cont.] (Wayb.)> <import temper, string * [with cont.] (Wayb.)> <2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 ... * [with cont.] (Wayb.)> I haven't considered any systems without an interval of equivalence. Your idea of using a maximal number of ETs may be a way forward.> You have octaves on the brain, which is the usual situation in music > theory; however when discussing tuning and temperment it really is > just another interval.Yes, it is the usual situation in music theory.> Suppose I decide to have a mean-tone system, so that 81/80 is a > unison vector. I could tune things so that octaves were pure 2's, but > I don't have to. Suppose instead I decide that I want the major sixth > to be exact. Now I can look at the circle of octaves, and notice that > it approximately returns after 14 octaves--14 octaves is almost the > same as 19 major sixths; 2^14 = (5/3)^18.9968... Suppose I decide to > tune octaves so that I represent 2 by (5/3)^(19/14); this is equal to > 2.000232... and is sharp by about 1/5 of a cent. Since I have fixed > two values and I am making 81/80 a unison vector, major thirds are > now determined also. Since 2 and 5/3 are not now incommensurable, I > actually have a rank 1 group. It is the 14 equal division of the > major sixth, with a very slightly sharp octave; it is in practice > more or less indistinguishable from the 19 equal division of the > octave, with very slightly flat major sixths.Yes, I think that would work.> However, there is nothing in the nature of the problem to suggest I > need to make any interval exact. One obvious way to decide would be > to pick a set of intervals {t1, ... , tn} which I want to be well > approximated, and a corresponding set of weights {w1, ... , wn} > defining how important I think it is to have that interval > approximate nicely. Perhaps I could do this using harmonic entropy? > In any case, having done this I now have an optimization problem > which I can decide using the method of least squares. If I have two > generators, which I have in the case of the 5-limit with 81/80 a > unison vector, then solving this will give me tunings for the > generators and hence tunings for the entire system. 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.Hold on, you need more than that. You need a harmonic metric, so that you can decide how well approximated a given interval is. That needs to include the rule for adding together a number of approximations. And you need to decide the ideal tuning for each interval. Here are the choices I made: The set of intervals is an odd limit. That means, all ratios within in octave that don't contain and odd number larger than the chosen limit. All are weighted equally. The closeness of approximation is measured in cents relative to just intonation. This assumes octaves are already just. The rule for adding together approximations is to take the poorest alternative. The "method of least squares" is totally irrelevant here. Indeed, it seems to be part of the harmonic metric, and not a method at all. You still need a minimisation algorithm. In this case, it happens that the simplest tuning will always have one interval in the set under consideration just. As a mathematician, perhaps you can prove this. But it means all I have to do is consider each interval in term, which is a very simple method. So the nature of this problem is such that I do need to make two intervals exact. There are plenty of other problems that have different natures, but I'm puzzled as to why you insist on bringing them up when we're discussing octave-invariant systems. Graham
Message: 751 - Contents - Hide Contents Date: Tue, 21 Aug 2001 15:38 +0 Subject: Re: Microtemperament and scale structure From: graham@m... 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.>> If you want to push the definition and make a third a >> unison vector, you can define quarter comma meantone by setting it > just. >> Now I think you're pushing definitions too far. Let's not forget the > strong form of the hypothesis!I'd be quite happy to forget the strong form of the hypothesis.>> So the commatic UVs define the temperament class and the chromatic > UV is>> used to define the specific tuning. >> Hmm . . . perhaps one _can_ define things this way, but it's by no > means universal. How would one define LucyTuning in this way??( 1 0 0) (1 ) (-2 0 1)H' =~ (1/pi) Oct (-4 4 -1) (0 ) or (0 1)h' = (1/pi) Oct (4 -1) (0 )>> Whatever they mean, MOS and WF are the same thing: a generated > scale with>> only two step sizes. >> Not the same thing. Clampitt lists all the WFs in 12-tET, and there > is no sign of the diminished (octatonic) scale, or any other scale > with an interval of repetition that is a fraction of an octave. These > are all MOS scales, though.But Carey & Clampitt also say the two concepts are identical. Don't they? Yes, note 1 says they are "equivalent". 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. 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. Graham
Message: 752 - Contents - Hide Contents Date: Tue, 21 Aug 2001 15:38 +0 Subject: Re: Mea culpa From: graham@m... In-Reply-To: <009701c12a17$911e9700$5340d63f@s...> 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 propertySo WF/Myhill assumes an octave of 2:1 but MOS/ME can be any period? That would make sense, and I agree with you that MOS/ME is better. But I wonder if WF/Myhill was ever intended to be so restricted. Graham
Message: 753 - Contents - Hide Contents Date: Tue, 21 Aug 2001 17:29 +0 Subject: Re: Chromatic = commatic? From: graham@m... In-Reply-To: <9lt3q7+gtig@e...> In article <9lt3q7+gtig@e...>, genewardsmith@j... () wrote:> --- 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.If the octave counts as a "unison vector" then the determinant *does* come out right. If not, there's only a determinant for the octave-invariant matrix anyway! You seem to have a low view of algebra. Paul doesn't mention equal temperament. Only "the number of notes in the scale". This is correct. For example, here's that pair of octave-invariant unison vectors: [ 4 -1] [ 0 -3] Invert it to get [-3 1] [ 0 4] Each column is a mapping of generators. The left hand column is what you get by making [0 -3] the chromatic UV. As there's a common factor of 3, it means the octave is divided into three equal parts. A fifth is (-)1 generators, and a third is an exact octave division. The right hand column is the usual meantone mapping. The fifth is the generator, and a third is four generators. This is the full octave-invariant homomorphism. Equal temperaments are tricky when they're octave invariant. Effectively, it means the new period is a scale step. Hence all notes are identical. That would make equal temperaments zero-dimensional in octave invariant terms, which happens to agree with the Hausdorff dimension. To get the rest of the information out, I agree it makes sense to consider octave-specific matrices, and I have argued that before. (Check the archives.) However, as Carey and Clampitt show, you can convert between the MOS generator and scale steps, so the homomorphism is still there.>> 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!That's all true, but missing the point. Commas and chromatic semitones are certainly treated differently in diatonic music.> 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?Because the rest of the world says "linear temperament" and has done so for longer than we've been alive.>> 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.How about you stop telling us what we should be doing, and start listening to what we're trying to say?>> MOS means that there is an interval of repetition >> What do the letters of the acronym stand for?Moments of Symmetry. Graham
Message: 754 - Contents - Hide Contents Date: Tue, 21 Aug 2001 18:52:41 Subject: Re: Microtemperament and scale structure From: Paul Erlich --- In tuning-math@y..., "Dave Keenan" <D.KEENAN@U...> wrote:> --- 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?Whoops! Of course.
Message: 755 - 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: 756 - 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: 757 - 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: 758 - 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: 759 - 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: 761 - 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: 762 - 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: 763 - 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: 764 - 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: 765 - 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@s...> > 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: 766 - Contents - Hide Contents
Date: Tue, 21 Aug 2001 21:03 +0
Subject: Generators and unison vectors
From: graham@m...
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: 767 - 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: 768 - 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: 769 - Contents - Hide Contents Date: Tue, 21 Aug 2001 20:25:22 Subject: Re: Microtemperament and scale structure From: genewardsmith@j... --- 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!
Message: 770 - Contents - Hide Contents Date: Tue, 21 Aug 2001 20:27:27 Subject: Re: Generators and unison vectors From: Paul Erlich --- In tuning-math@y..., graham@m... wrote:> 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 sHow did you get this, without appealing to a parallelogram or some such construction?> > 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. By definition. > 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.If you mean, start with the unison vectors as non-zero intervals and then selectively bring some of them to zero, yes, that's how I think of it in the lattice/PB regime.> > 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, How so? > so the analogy breaks down. But whatever, I'll call > all the intervals unison vectors anyway._All_ the intervals?> 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?A unison vector has to come out to a musical _unison_! An augmented unison maybe, but still a unison.> 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. Don't follow. > 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.You've completely lost me.> > An alternative to "unison vector" would be "melodic generator".Ouch! Are you sure? The steps in the scale can't be unison vectors, nor can the generator of the scale be a unison vector.> > 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.Gene has referred to that concept, I believe.> > 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.Please, please, please don't refer to a fifth as a unison vector. Musical intervals go: unison, second, third, fourth, fifth. There's a long way between a unison and a fifth.> 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.They can't be unison vectors in any space! If the generator is a unison vector, you never get beyond the first note of the scale and its chromatic alterations.> > 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.I suppose -- but you got me doing it too :)
Message: 771 - Contents - Hide Contents Date: Tue, 21 Aug 2001 20:57:27 Subject: Re: The hypothesis From: genewardsmith@j... --- In tuning-math@y..., graham@m... wrote:>> Considering scales is another level of generality altogether-- first >> we have approximations (kernels, unison vectors, and so forth) then >> we have tuning, and finally we select a subset and have scales.> Oh, right, but whatever they are, they're octave invariant. CLAMPITT.PDF > is a relevant place for definitions, as it's already been referenced: "By > /scale/ we refer to a set of pitches ordered according to ascending > frequencies (pitch height) bounded by an interval of periodicity."This definition of a scale does not assume it repeats octaves, so it does not assume octave invariance.> I don't agree with this anyway. You can have a scale without a tuning. > For example, a C major scale can be tuned to either 12- or 19-equal.OK, we can consider a conceptual scale to be something defined on the level of the homomorphic image (including the identity map.) Then C major (or mean tone diatonic) is a conceptual scale in the mean-tone linear temperament, which becomes a tuned scale if you pick a tuning. How's that?> It looks like a commatic UV would be in the kernel, and a chromatic UV > would not.So far as I can see, they are both in the kernel and refer to the same thing.>> If we consider equivalence classes modulo octaves, the 5-limit is >> free of rank two, but I don't know what you mean by "cyclic around >> the octave".> You describe an ET as cyclic below.I said an ET modulo ocatves was cyclic. In the case above, where we are considering equivalence classes of notes with representative elements 3^a * 5^b, we have't modded out and we have something which is free, not cyclic. By setting h(3^a*5^b) = 7*a+2*b (mod 12) we do get a circle of note equivalence classes.>> An ET would be free of rank 1, or "cyclic of infinite order". If we >> mod out by octaves, it would no longer be free but would (still) be >> cyclic, which implies one generator.> But if we "mod out" by one scale step, it would still have rank 1?It would have one generator, but the rank is the rank of the group modulo torsion, so in this case the rank is 0. There is a structure theorem for finitely generated abelian groups (which is what we have been considering) which gives a complete set of invariants specifying the group; the rank being one of them and perhaps the most important. In the case of C(12), it is in terms of groups of prime power order C(3) x C(4), which specifies it.> You certainly can tell a small interval if 2 is taken out. Of course, you > need a way of calculating interval size. As I see it, that lies outside > of the group theory we've been discussing so far. There are two ways of > doing it, corresponding to the two interpretations above.I think your two ways in effect put the 2 back in by calculating what it must be.>If > it doesn't work out that way perhaps you, as the mathematician, can tell > us the size constraints on the unison vectors.So far as I can see, there aren't any. A unison vector is simply anything you've decided to regard as a unison, and so lies in (or generates?) the kernel.>> You can generate by fifths and octaves if you want, but you don't >> need to.> You do need to if you want to enforce octave equivalence.I don't see why. Why not by octaves and major thirds instead?>> However, there is nothing in the nature of the problem to suggest I >> need to make any interval exact. One obvious way to decide would be >> to pick a set of intervals {t1, ... , tn} which I want to be well >> approximated, and a corresponding set of weights {w1, ... , wn} >> defining how important I think it is to have that interval >> approximate nicely. Perhaps I could do this using harmonic entropy? >> In any case, having done this I now have an optimization problem >> which I can decide using the method of least squares. If I have two >> generators, which I have in the case of the 5-limit with 81/80 a >> unison vector, then solving this will give me tunings for the >> generators and hence tunings for the entire system. 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.> Hold on, you need more than that.Some one else said people around here have done such computations many times, so there seems to be some confusion on that score. I could certainly do one using the method above, at any rate.> So the nature of this problem is such that I do need to make two intervals > exact. There are plenty of other problems that have different natures, > but I'm puzzled as to why you insist on bringing them up when we're > discussing octave-invariant systems.I was discussing, among other things, tuning and temperament, where the question of octave tuning is relevant. As I pointed out, tuning an octave as a 2 is a choice of tuning.
Message: 773 - Contents - Hide Contents Date: Tue, 21 Aug 2001 21:50:01 Subject: Re: Chromatic = commatic? From: genewardsmith@j... --- In tuning-math@y..., graham@m... wrote:>You seem to have a low view of algebra.Actually, I'm an algebraist and number theorist, and algebraists are notorious for wanting to solve a problem using only algebraic methods when possible, which is what I was doing. If you calculate the "7" using determinants, you can get the rest of the homomorphism by using floating point calculations: 7 log_2(3) = 11.09... and 7 log_2(5) = 16.25... As an algebraist, I naturally want to do the computation using only algebra. As an algebraist, I also want to say "Now hold on! What if I want to say a third has 3 steps, instead of two?" Therefore I distinguish the systems [7, 11, 16] and [7, 11, 17].> For example, here's that pair of octave-invariant unison vectors: > > [ 4 -1] > [ 0 -3] > > Invert it to get > > [-3 1] > [ 0 4] > > Each column is a mapping of generators.Each column is a mapping of everything in the 5-limit modulo octaves, but I don't see how you draw the conclusions you do about them.> That's all true, but missing the point. Commas and chromatic semitones > are certainly treated differently in diatonic music.If you are looking *only* at diatonic music and not allowing any accidentals, then neither one is a scale step; they seem therefore to be treated in the same way. The comma, being smaller, lends itself better to approximations, of course.>> 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?> Because the rest of the world says "linear temperament" and has done so > for longer than we've been alive.You're missing my point--the world does not call something we get from codimension 1 a linear temperament, it calls codimension 2 (with 2 generators) a linear temperament.> How about you stop telling us what we should be doing, and start listening > to what we're trying to say?In a mathematical subject, mathematics is your friend; which means potentially so is a friendly mathematician. Why not view me as an ally in this endeavor?
Message: 774 - Contents - Hide Contents Date: Tue, 21 Aug 2001 21:56:30 Subject: Re: Hypothesis From: genewardsmith@j... --- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:>>> 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?I see what you are saying--I was assuming the JI melody was something appropriate to the scale. Of course if you translate something to JI it's likely not to be.> 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.That sounds like an excellent plan.
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