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Message: 1300 - Contents - Hide Contents Date: Mon, 20 Aug 2001 08:25:16 Subject: The hypothesis From: genewardsmith@xxxx.xxx I found a posting by Paul over on the tuning group, and it seems I may be closing on a statement of the Paul Hypothesis. "In fact, a few months ago I posted my Hypothesis, which states that if you temper out all but one of the unison vectors of a Fokker periodicity block, you end up with an MOS scale. We're discussing this Hypothesis on tuning-math@y..." Sounds like we may be getting there, but there seems to be some confusion as to whether 2 counts as a prime, and so whether for instance the 5-limit is 2D or 3D. Most of the time it makes sense to treat 2 like any other prime. "A temperament can be constructed by tempering out anywhere from 1 to n unison vectors. If you temper out n (and do it uniformly), you have an ET. If you temper out n-1, you have a linear temperament. If you temper out n-2, you have a planar temperament (Dave Keenan has created some examples of those)." From my point of view, the 5-limit is rank (dimension) 3, and the 7- limit 4, and so forth. If you temper out n-1 unison vectors which generate a well-behaved kernel, then you map onto a rank-1 group, and get an equal temperment. So "codimension" 1 (one less than the full number of dimensions) leads to a rank-1 group. In the same way, codimension 2 for the kernel leads to a rank 2 group, etc. If for instance you temper out 81/80 in the 5 limit, the kernel has dimension 1 and codimension 2, and leads to a rank 2 image group. We can tune the rank 1 group any way we like so long as the steps are of the same size, which means that our ET can have stretched or squashed octaves if we so choose. In the same way, we can tune the rank 2 group any way we like, except that we need to retain incommensurability of two generators (or at least to ignore the fact if they are not.) If we make the octaves pure in our example where the kernel is generated by a comma, we could for instance make the fifths pure also, leading to Pythagorean tuning. Alternatively, we could make the major thirds pure, leading to 1/4 comma mean tone temperment. (Pythagorean tuning is not considered a temperment, since the fifth isn't tempered, but it is the same sort of thing mathematically as 1/4 comma mean-tone temperment.) Other choices lead to other results, and all we need to do is to ensure the circle of fifths does not close--or at least to pretend otherwise it if it does. A rank 3 image group, coming from a kernel of codimension 3, is what people have been calling a 2D temperment. I hope that clarifies things (as it does for me) rather than further confuses them!
Message: 1301 - Contents - Hide Contents Date: Mon, 20 Aug 2001 20:33:30 Subject: Re: Hypothesis From: Paul Erlich --- In tuning-math@y..., carl@l... wrote:> I mean, I caught that they are non-parallelpiped PBs, but not > why this should translate into fewer harmonic structuresSee the last post.> (do > you mean only complete chords? total consonant dyads?).I'm thinking both, but I suppose the latter might do if we're trying to mathematize this.
Message: 1302 - Contents - Hide Contents Date: Mon, 20 Aug 2001 11:15 +0 Subject: Re: The hypothesis From: graham@xxxxxxxxxx.xx.xx In-Reply-To: <9lqhhc+s6ru@xxxxxxx.xxx> In article <9lqhhc+s6ru@xxxxxxx.xxx>, genewardsmith@xxxx.xxx () 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. Most of the time it makes sense to > treat 2 like any other prime.2 is certainly prime, but most of the time we consider octave-invariant scales.> "A temperament can be > constructed by tempering out anywhere from 1 to n unison vectors. If > you temper out n (and do it uniformly), you have an ET. If you temper > out n-1, you have a linear temperament. If you temper out n-2, you > have a planar temperament (Dave Keenan has created some examples of > those)."So this is all octave invariant.> From my point of view, the 5-limit is rank (dimension) 3, and the 7- > limit 4, and so forth. If you temper out n-1 unison vectors which > generate a well-behaved kernel, then you map onto a rank-1 group, and > get an equal temperment. So "codimension" 1 (one less than the full > number of dimensions) leads to a rank-1 group. In the same way, > codimension 2 for the kernel leads to a rank 2 group, etc. If for > instance you temper out 81/80 in the 5 limit, the kernel has > dimension 1 and codimension 2, and leads to a rank 2 image group.So the kernel has dimension 1 because it contains 1 unison vector? I think I see the codimension. So in octave-invariant terms, 5-limit is rank 2, but cyclic about the octave. An ET would be rank 0 I suppose, but you've already given the real name for that case. Tempering out 81/80 would than be rank-1, hence "linear" temperament. I think that terminology goes back to either Ellis or Bosanquet.> We can tune the rank 1 group any way we like so long as the steps are > of the same size, which means that our ET can have stretched or > squashed octaves if we so choose.In the octave-invariant case the octave lies outside the system, so you can't say anything about it.> In the same way, we can tune the > rank 2 group any way we like, except that we need to retain > incommensurability of two generators (or at least to ignore the fact > if they are not.) If we make the octaves pure in our example where > the kernel is generated by a comma, we could for instance make the > fifths pure also, leading to Pythagorean tuning. Alternatively, we > could make the major thirds pure, leading to 1/4 comma mean tone > temperment. (Pythagorean tuning is not considered a temperment, since > the fifth isn't tempered, but it is the same sort of thing > mathematically as 1/4 comma mean-tone temperment.) Other choices lead > to other results, and all we need to do is to ensure the circle of > fifths does not close--or at least to pretend otherwise it if it does.For the octave invariant case, the fourth or fifth is the generator, which I think agrees with both meanings of "generator".> A rank 3 image group, coming from a kernel of codimension 3, is what > people have been calling a 2D temperment. I hope that clarifies > things (as it does for me) rather than further confuses them!It's also thought of as repeating every octave. You'll have to come up with the group theoretic description of that. Although the same tuning can be described the way you did, octave repetition is another constraint or simplification. Graham
Message: 1303 - Contents - Hide Contents Date: Mon, 20 Aug 2001 20:35:34 Subject: Re: Mea culpa From: Paul Erlich --- In tuning-math@y..., "D.Stearns" <STEARNS@C...> wrote:> Hi Carl, > > It's a 12-tet scale with a 5/12 generator.I'm not seeing the 12-tET-ness or the 5/12-ness of this at all:>>> This is not quite true -- for example, LssssLssss is MOS but not > WF>>> and doesn't have Myhill's property.
Message: 1304 - Contents - Hide Contents Date: Mon, 20 Aug 2001 20:51:12 Subject: Re: Mea culpa From: carl@xxxxx.xxx>>>> >OS, WF, and Myhill's property are all equivalent. >>>>>> This is not quite true -- for example, LssssLssss is MOS but not >>> WF and doesn't have Myhill's property. >>>> What single generator produces the scale? >> >> -Carl >> One possibility is s -- here the interval of repetition is the half- > octave.Then my reply is that the scale is MOS/WF at the half-octave. -Carl
Message: 1305 - Contents - Hide Contents Date: Mon, 20 Aug 2001 20:55:57 Subject: Re: Mea culpa From: Paul Erlich --- In tuning-math@y..., carl@l... wrote:> Then my reply is that the scale is MOS/WF at the half-octave.Well there's no point in going into a big debate on terminology here, but note that Clampitt' list of WFs in 12-tET is sorely incomplete if you allow this kind of construction. Let's just say "MOS" and forget about it.
Message: 1307 - Contents - Hide Contents Date: Mon, 20 Aug 2001 21:53:18 Subject: Re: Mea culpa From: Paul Erlich --- In tuning-math@y..., "D.Stearns" <STEARNS@C...> wrote:> Paul Erlich wrote, > > <<I'm not seeing the 12-tET-ness or the 5/12-ness of this at all>> > > Hmm, why not?Sorry, I hadn't seen your subsequent message where you said you were interpreting this scale as a 12-tET scale. As I'm sure you know, the discussion of this scale started with them in 22-tET or something close to it. There one could say the generator is 1/11 of an octave.
Message: 1308 - Contents - Hide Contents Date: Mon, 20 Aug 2001 23:18:53 Subject: Re: The hypothesis From: genewardsmith@xxxx.xxx --- In tuning-math@y..., graham@m... 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. 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.> 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.> 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".>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.> 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.> 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 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. 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. 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.
Message: 1309 - Contents - Hide Contents Date: Tue, 21 Aug 2001 00:00:39 Subject: Re: Mea culpa From: genewardsmith@xxxx.xxx --- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:> --- In tuning-math@y..., carl@l... wrote:>> MOS, WF, and Myhill's property are all equivalent.> This is not quite true -- for example, LssssLssss is MOS but not WF > and doesn't have Myhill's property.If "L" is a large scale step and "s" is a small scale step, then this has two sizes of steps. If that is Myhill's property then it should have it, so why doesn't it?
Message: 1310 - Contents - Hide Contents Date: Tue, 21 Aug 2001 15:38 +0 Subject: Re: Microtemperament and scale structure From: graham@xxxxxxxxxx.xx.xx In-Reply-To: <9lt510+dcmg@xxxxxxx.xxx> In article <9lt510+dcmg@xxxxxxx.xxx>, genewardsmith@xxxx.xxx () 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.Um, yes, it would be. You'll find the discussion in the archives.> 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.)Aaaaaah! So they are! I hadn't thought of doing that. Suddenly everything is a lot clearer. The main difference with what I do is that I consider two ETs and perfect octaves instead of three ETs. Presumably, chromatizing a unison vector is the same as junking one of the three ETs you get from the inverse.> 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.> 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.Yes, that figures. Let's go back to the example.> [12 19 34] > S = [19 30 54] > [28 44 79] > [-6 -5 6] > S^(-1) = [11 -4 -2] > [-4 4 -1]Rows and columns are interchanged when you take the inverse. So the [-4 4 -1] corresponds to 34. This is the unison vector that results from taking 34 *out* of the system. Take out the bottom two, and you should end up with 12= |[ 1 0 0]| |[11 -4 -2]| = 12 |[-4 4 -1]| so that works. Generalizing to more dimensions, presumably considering n unison vectors, and taking out n-2 of them, will give you the linear temperament. So, as I already have a program for generating consistent ETs, I could use it to generate a list of candidate unison vectors. And then use them to go back to ETs. All of it without assuming octave equivalence anywhere.> There are other types of matrix computations we could make, but I'm > wondering if this seems familiar?Okay, I think I see the connection. What you're doing with octave *specific* matrices is analogous to what I did with octave *invariant* matrices. So you invert the matrix of unison vectors to get sets of generators, where the generator is a step size in an ET. I invert a matrix of one less unison vector to get sets of generators, where the generator is the interval so called in WF or MOS theory. Algebraically, it's exactly the same operation. What I do with octave specific matrices is a bit more complicated. Because I'm considering one fewer ET, and setting the octave just, that means the inverse contains a mixture of step-size and MOS generators. I really should be *reducing* the number of unison vectors, but it is very interesting to see your method that works with one more. Graham
Message: 1311 - Contents - Hide Contents Date: Tue, 21 Aug 2001 21:56:30 Subject: Re: Hypothesis From: genewardsmith@xxxx.xxx --- 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.
Message: 1312 - Contents - Hide Contents Date: Tue, 21 Aug 2001 00:08:59 Subject: Re: Microtemperament and scale structure From: genewardsmith@xxxx.xxx --- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:> Are you saying that both keyboards are tuned identically, or that > there may be an offset?It certainly would be more interesting musically with an offset, but it doesn't matter in the sense that this is a tuning question, not a structure question. Either way, a comma interval is represented by jumping from the green keyboard to the red keyboard or vice-versa-- therefore, there is no distinction between a comma up and a comma down, and two commas are a unison.> I was just asking what it stood for. "Just Tuning"?Sorry, just trying to be one of the boys in this alphabet soup of acronyms around here.
Message: 1313 - Contents - Hide Contents Date: Tue, 21 Aug 2001 15:38 +0 Subject: Re: The hypothesis From: graham@xxxxxxxxxx.xx.xx In-Reply-To: <9ls5st+gcan@xxxxxxx.xxx> In article <9ls5st+gcan@xxxxxxx.xxx>, genewardsmith@xxxx.xxx () 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: 1314 - Contents - Hide Contents Date: Tue, 21 Aug 2001 21:59:06 Subject: Re: The hypothesis From: genewardsmith@xxxx.xxx --- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:> 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?That's certainly the way to deal with scales--I was still considering tuning and temperament.
Message: 1315 - Contents - Hide Contents Date: Tue, 21 Aug 2001 00:08:57 Subject: Re: Microtemperament and scale structure From: genewardsmith@xxxx.xxx --- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:> Are you saying that both keyboards are tuned identically, or that > there may be an offset?It certainly would be more interesting musically with an offset, but it doesn't matter in the sense that this is a tuning question, not a structure question. Either way, a comma interval is represented by jumping from the green keyboard to the red keyboard or vice-versa-- therefore, there is no distinction between a comma up and a comma down, and two commas are a unison.> I was just asking what it stood for. "Just Tuning"?Sorry, just trying to be one of the boys in this alphabet soup of acronyms around here.
Message: 1316 - Contents - Hide Contents Date: Tue, 21 Aug 2001 15:38 +0 Subject: Re: Microtemperament and scale structure From: graham@xxxxxxxxxx.xx.xx In-Reply-To: <9lrmea+klq4@xxxxxxx.xxx> 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: 1317 - Contents - Hide Contents Date: Tue, 21 Aug 2001 22:25:51 Subject: Re: The hypothesis From: genewardsmith@xxxx.xxx --- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:> 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.The dawn breaks! In other words, the difference between chromatic and commatic unison vectors has nothing to do with the kernel, it's purely a matter of tuning!
Message: 1318 - Contents - Hide Contents Date: Tue, 21 Aug 2001 00:13:34 Subject: Re: The hypothesis From: Paul Erlich --- In tuning-math@y..., genewardsmith@j... wrote:> In any case, having done this I now have an optimization problem > which I can decide using the method of least squares.We've done these sorts of things many times.> 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.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?
Message: 1319 - Contents - Hide Contents Date: Tue, 21 Aug 2001 15:38 +0 Subject: Re: Mea culpa From: graham@xxxxxxxxxx.xx.xx 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 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: 1320 - Contents - Hide Contents Date: Tue, 21 Aug 2001 23:45:01 Subject: Scales From: genewardsmith@xxxx.xxx Let's see if I understand the situation. In a p-limit system of harmony G, containing n primes, we select n-1 unison vectors and the prime 2, which is a sort of special unison vector. This defines a kernel K and so congruence classes on G, and a homomorphic image H = G/K which under the best circumstances is cyclic but which in any case is a finite abelian group of order h. We now select a set of coset representatives in G for H by taking elements in one period of the period lattice defined by the n generators of K. Now we select out a certain number m+1 of the unison vectors, including 2, and call them (aside from the 2) chromatic. The remaining unison vectors now define a temperament whose codimension is m+1, and hence is an m-dimensional temperament. We now select a tuning for that temperament, and we have a scale with h tones in an octave. If m = 0 we have a just scale; if m = 1 we have the scales with one chromatic vector which people have been talking about. I'm not sure if this helps anyone but me, but I think I am getting it.
Message: 1321 - Contents - Hide Contents Date: Tue, 21 Aug 2001 00:15:47 Subject: Re: Mea culpa From: Paul Erlich --- In tuning-math@y..., genewardsmith@j... wrote:> --- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:>> --- In tuning-math@y..., carl@l... wrote: >>>> MOS, WF, and Myhill's property are all equivalent. >>> This is not quite true -- for example, LssssLssss is MOS but not WF >> and doesn't have Myhill's property. >> If "L" is a large scale step and "s" is a small scale step, then this > has two sizes of steps. If that is Myhill's property then it should > have it, so why doesn't it?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.
Message: 1322 - Contents - Hide Contents Date: Tue, 21 Aug 2001 17:29 +0 Subject: Re: Chromatic = commatic? From: graham@xxxxxxxxxx.xx.xx In-Reply-To: <9lt3q7+gtig@xxxxxxx.xxx> In article <9lt3q7+gtig@xxxxxxx.xxx>, genewardsmith@xxxx.xxx () 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: 1323 - Contents - Hide Contents Date: Tue, 21 Aug 2001 00:32:56 Subject: Re: Hypothesis From: genewardsmith@xxxx.xxx --- 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. Is it really true that a pentatonic or diatonic melody sounds better in a meantone tuning than it does in just tuning? 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.
Message: 1324 - 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.
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