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Message: 3450 - Contents - Hide Contents Date: Tue, 22 Jan 2002 12:32:17 Subject: Re: Minkowski reduction (was: ...Schoenberg's rational implications) From: paulerlich --- In tuning-math@y..., "monz" <joemonz@y...> wrote:> >> Message 2850>> From: paulerlich <paul@s...> >> Date: Sun Jan 20, 2002 10:48pm >> Subject: Re: lattices of Schoenberg's rational implications >> >> >> --- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:>>> --- In tuning-math@y..., "paulerlich" <paul@s...> wrote: >>>>>>> However, I think the only reality for Schoenberg's system >>>> is a tuning where there is ambiguity, as defined by the >>>> kernel <33/32, 64/63, 81/80, 225/224>. > >> Ah ... so then, Paul, you agreed with me that this PB is > a valid one for p 1-184 of _Harmonielehre_?I don't know . . . I'll give you the benefit of the doubt.> > >>>>> BTW, is this Minkowski-reduced? >>>>> Nope. The honor belongs to <22/21, 33/32, 36/35, 50/49>. >>>> Awesome. So this suggests a more compact Fokker parallelepiped >> as "Schoenberg PB" -- here are the results of placing it in different >> positions in the lattice (you should treat the inversions of these as >> implied): >> >> >> 0 1 1 >> 84.467 21 20 >> 203.91 9 8 >> 315.64 6 5 >> 386.31 5 4 >> 470.78 21 16 >> 617.49 10 7 >> 701.96 3 2 >> 786.42 63 40 >> 933.13 12 7 >> 968.83 7 4 >> 1088.3 15 8 >> >> >> 0 1 1 >> 119.44 15 14 >> 203.91 9 8 >> 315.64 6 5 >> 386.31 5 4 >> 470.78 21 16 >> 617.49 10 7 >> 701.96 3 2 >> 786.42 63 40 >> 933.13 12 7 >> 968.83 7 4 >> 1088.3 15 8 >> >> >> 0 1 1 >> 119.44 15 14 >> 155.14 35 32 >> 301.85 25 21 >> 386.31 5 4 >> 470.78 21 16 >> 617.49 10 7 >> 701.96 3 2 >> 772.63 25 16 >> 884.36 5 3 >> 968.83 7 4 >> 1088.3 15 8 >> >> >> 0 1 1 >> 84.467 21 20 >> 155.14 35 32 >> 266.87 7 6 >> 386.31 5 4 >> 470.78 21 16 >> 582.51 7 5 >> 701.96 3 2 >> 737.65 49 32 >> 884.36 5 3 >> 968.83 7 4 >> 1053.3 147 80 > > >> With variant alternate pitches written on the same line > -- and thus with invariant ones on a line by themselves -- > these scales are combined into: > > 1/1 > 21/20 15/14 > 35/32 9/8 > 7/6 25/21 6/5 > 5/4 > 21/16 > 7/5 10/7 > 3/2 > 49/32 25/16 63/40 > 5/3 12/7 > 7/4 > 147/80 15/8 > > > > My first question is: this is a 7-limit periodicity-block, > so can you explain how the two 11-limit unison-vectors disappeared?They didn't disappear! It's just that in these particular positions, the parallelepiped all lies within one "power of 11" plane. I'm sure Gene could produce an example that wouldn't.> I've been trying to figure it out but don't see it. > > One thing I did notice in connection with this, is that > 147/80 is only a little less than 4 cents wider than 11/6, > which is one of the pitches implied in Schoenberg's overtone > diagram (p 23 of _Harmonielehre_) : > > vector ratio ~cents > > [ -4 1 -1 2 0 ] = 147/80 1053.2931 > - [ -1 -1 0 0 1 ] = 11/6 1049.362941 > -------------------- > [ -3 2 -1 2 -1 ] = 441/440 3.930158439 > > > So I know that 441/440 is tempered out.NO IT ISN'T! I believe it maps to 1 semitone given the set of unison vectors you've put forward.> But I don't see > how to get this as a combination of two of the other > unison-vectors. YOU CAN'T! > Then, I reasoned that since all of these pitches are separated > by one or two of the unison vectors which define this set of PBs, > the lattice could be further reduced to a 12-tone set, one that > can still "define the same temperament": > > Monzo lattice of Monzo's ultimate reduction of Paul Erlich's > 4 variant Minkowski-reduced 7-limit PBs for p 1-184 of > Schoenberg's _Harmonielehre_, to one 12-tone PB : > > Yahoo groups: /tuning-math/files/monz/ult-red.gif * [with cont.] > > > Correct?What's so "ultimate" about it?

Message: 3451 - Contents - Hide Contents Date: Tue, 22 Jan 2002 22:39:55 Subject: Re: Heuristic? From: genewardsmith --- In tuning-math@y..., "paulerlich" <paul@s...> wrote:> Wait a minute -- this isn't even correct! > > n=64, d=63 -> (n-d)/(d*log(d)) = 0.0038312 > > n=160000, d=151263 -> (n-d)/(d*log(d)) = 0.0048429 (WORSE THAN 64/63!) > > n=204800, d=194481 -> (n-d)/(d*log(d)) = 0.0043569 (WORSE THAN 64/63!)What's going on here is that what you are calling bad, I was calling good. The reason for that is that I was seeing if one could reduce a lattice basis wrt your heuristic, and to do that 50/49 should be better than 64/63, etc, or at least better than something much smaller. It remains the case that no matter what we call "bad" or "good" the intervals I gave are in the range between 50/49 and 64/63, which means I take it that they define an amountof tempering between that defined by 50/49 and 64/63? I'm not convinced this is making sense.

Message: 3452 - Contents - Hide Contents Date: Tue, 22 Jan 2002 08:53:41 Subject: Heuristic? From: genewardsmith --- In tuning-math@y..., paul@s... wrote:> (n-d)/(d*log(d))I get stuff like 160000/151263 and 204800/194481 when I look at twintone commas with a better heuristic than 64/63--something which apparently is not hard to achieve. This does not seem right.

Message: 3453 - Contents - Hide Contents Date: Tue, 22 Jan 2002 12:35:35 Subject: Re: the Lattice Theory Homepage From: paulerlich --- In tuning-math@y..., "monz" <joemonz@y...> wrote:>> I'm afraid that's the wrong kind of lattice--as came up >> before, there are two different things called "lattice" >> in English-language mathematics. This kind is a kind of >> partial ordering, which is important in universal algebra >> among other things, which is why the univeral algebraists >> in Hawaii care about it. > >> Are there any other types of lattices or just these two? > (not counting the kind which hold up rose-bushes, etc., of course!) > > While I hardly understand it, I'm surprised to see that > Minkowski reduction applies to "our" lattices as well as > the regular mathematical kind, since I knew they are different.This is not true, Monz. Both kinds of lattice are "regular mathematical" kinds. Minkowski reduction only applies to "our" definition.

Message: 3454 - Contents - Hide Contents Date: Tue, 22 Jan 2002 22:44:06 Subject: Re: Heuristics (Was: Hi Dave K.) From: genewardsmith --- In tuning-math@y..., "paulerlich" <paul@s...> wrote:> Maybe we each have to write our own paper, then. I'm hoping someone > will help me with the math for mine . . .We could make a deal--we each help the other guy with what they need help on the most. :) So> why not look for a mathematically pretty way to find good > temperaments? That's something I'm interested in, at any rate.I'd like to see a quick and easy estimate which inputs a wedgie and outputsa badness measure; a second pass could then refine that.

Message: 3455 - Contents - Hide Contents Date: Tue, 22 Jan 2002 12:39:19 Subject: Re: A top 20 11-limit superparticularly generated linear temperament list From: paulerlich --- In tuning-math@y..., "monz" <joemonz@y...> wrote:> > Hi Dave, > >>> From: dkeenanuqnetau <d.keenan@u...> >> To: <tuning-math@y...> >> Sent: Monday, January 21, 2002 2:35 PM >> Subject: Re: A top 20 11-limit superparticularly generated linear > temperament list >> >>>> ... One might just as easily say "This subset of >> hemiennealimmal is so close to RI we might as well try >> to tune the instruments to the RI scale since that's >> easier to calculate and tunable by ear (assuming >> harmonic timbres). > >> Hmmm ... given remarks on the tuning list in the past by Daniel > Wolf about the multiple senses (other than the two obvious > ones) or other ratios implied by Partch's use of his 43-tone > scale, now *that* sounds like something pretty close to the mark! > > Partch apparently wove harmonic structures into his compositions > which sometimes require the listener to infer different rational > implications from his scale than the obvious ones. Without > examining the actual mathematics of it, your revised statement > here seems to me to be a good way to model that aspect of Partch's > compositional practice. > > > > -monzWell, Monz, it would be good to know if Partch exploited _only_ the hemiennealimmal equivalencies, or _only_ the MIRACLE equivalencies, or what. Wilson seems to have felt that he exploited enough equivalencies that a closed 41-tone system (as in 41-tET) was actually implied. But Wilson never seems to have thought much about MIRACLE, let along hemiennealimmal.

Message: 3456 - Contents - Hide Contents Date: Tue, 22 Jan 2002 23:00:17 Subject: Re: Heuristics From: dkeenanuqnetau Can you guys please drop the "(Was: Hi Dave K.)" from the title of this thread. I have so little time to spend on tuning at the moment that I'm only reading posts that have my name in them. (in the body text is fine). But the search finds it whether in the body or the title and this thread is driving me crazy. Thanks. By the way, I didn't have a clue what "taxicab error" was. I'm glad if Gene found a way to give it meaning. :-)

Message: 3457 - Contents - Hide Contents Date: Tue, 22 Jan 2002 12:50:57 Subject: Re: Heuristic? From: paulerlich --- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:> --- In tuning-math@y..., paul@s... wrote: > >> (n-d)/(d*log(d)) >> I get stuff like 160000/151263 and 204800/194481 when I look at >twintone commas with a better heuristic than 64/63--something which >apparently is not hard to achieve. This does not seem right.What exactly do you mean, Gene? This heuristic is intended to work when _only one_ comma is tempered out -- but aren't you hiding a 50:49 up your sleeve? If you're trying to apply the heuristic to different ways of defining a two-comma temperament, then please recall I did say the UVs should be as near orthogonal as possible -- therefore, if 50:49 is already tempered out, then you have to express the other UV of twintone as some JI vector orthogonal to 50:49. If all these alternatives are nearly orthogonal to 50:49, then perhaps it _is_ right -- can we figure out what versions of twintone are actually "implied" (hopefully in the sense of the correct metric)?

Message: 3458 - Contents - Hide Contents Date: Tue, 22 Jan 2002 23:00:34 Subject: Re: Heuristic? From: paulerlich --- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:> --- In tuning-math@y..., "paulerlich" <paul@s...> wrote: >>> Wait a minute -- this isn't even correct! >> >> n=64, d=63 -> (n-d)/(d*log(d)) = 0.0038312 >> >> n=160000, d=151263 -> (n-d)/(d*log(d)) = 0.0048429 (WORSE THAN 64/63!) >> >> n=204800, d=194481 -> (n-d)/(d*log(d)) = 0.0043569 (WORSE THAN 64/63!) >> What's going on here is that what you are calling bad, I was >calling good. The reason for that is that I was seeing if one could >reduce a lattice basis wrt your heuristic, and to do that 50/49 >should be better than 64/63, etc,Why? This is not the heuristic for complexity we're looking at here -- it's the heuristic for error! It seems like you're confused about this . . .>or at least better than something >much smaller.I'm not following. And Gene, so far there's a heuristic only for the case of one unison vector tempered out. Anything beyond that is your invention, not mine.>It remains the case that no matter what we call "bad" or "good" the >intervals I gave are in the range between 50/49 and 64/63, which >means I take it that they define an amount of tempering between that >defined by 50/49 and 64/63?Alone, when defining a planar temperament? Sure, why not?

Message: 3459 - Contents - Hide Contents Date: Tue, 22 Jan 2002 13:55:34 Subject: Re: Heuristic? From: paulerlich --- In tuning-math@y..., "paulerlich" <paul@s...> wrote:> --- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:>> --- In tuning-math@y..., paul@s... wrote: >> >>> (n-d)/(d*log(d)) >>>> I get stuff like 160000/151263 and 204800/194481 when I look at >> twintone commas with a better heuristic than 64/63Wait a minute -- this isn't even correct! n=64, d=63 -> (n-d)/(d*log(d)) = 0.0038312 n=160000, d=151263 -> (n-d)/(d*log(d)) = 0.0048429 (WORSE THAN 64/63!) n=204800, d=194481 -> (n-d)/(d*log(d)) = 0.0043569 (WORSE THAN 64/63!) Remember, (n-d)/(d*log(d)) is a heuristic for the _amount of tempering_ -- and I assume we agree that the less tempering, the better! So what are you on about, Gene?

Message: 3460 - Contents - Hide Contents Date: Tue, 22 Jan 2002 23:01:37 Subject: Re: Heuristics From: paulerlich --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:> I'm glad if > Gene found a way to give it meaning.Haven't understood it, as of yet . . . :-)

Message: 3461 - Contents - Hide Contents Date: Tue, 22 Jan 2002 16:43 +0 Subject: Re: Heuristics (Was: Hi Dave K.) From: graham@xxxxxxxxxx.xx.xx In-Reply-To: <a2jliq+9hrb@xxxxxxx.xxx> Me:>> My experience of generating and sorting linear temperaments from> the 5- to>> the 21-limit is that the "right" error metric for one can be wildly >> inappropriate for others. Paul:> Can you give an example?The first run through of my temperament generator, when I was using step-cents gave absurdly complex and accurate 5-limit temperaments. Using only consistent ETs works well enough up to the 15-limit, but beyond that optimal temperaments are missed. At least with the current metrics. Me:>> One assumption behind the heuristic is that the error is > proportional to>> the size/complexity of the unison vector. Paul:> You can call it an assumption, if you wish -- I've verified its > approximate correctness for all 10 (wildly different) temperaments > I've tried, against Gene's rms measures.How many dimensions? Me:>> If you measure complexity as >> the number of consonant intervals, that's the best case of > tempering it >> out. Paul:> What does that mean?It's the microtemperament formula. Last time I mentioned it, you pointed me to one of your own messages. For the minimax temperament, tempering out one unison vector, the error is the size of the comma divided by the number of consonances making it up. When you're tempering out more than one comma, the result will typically be worse than the best case for any of the commas on their own. But it can never be better than for only one comma. Paul:> Well, so far I've only considered the case where one unison vector is > tempered out.I'm only questioning size/complexity as a heuristic when you have more than one unison vector. It might still work then.>> The other assumption is that the octave-specific Tenney metric >> approximates the number of consonant intervals a comma's composed > of. I'm>> not sure how closely this holds. >> This is based on the Kees van Prooijen lattice metric, and again its > good approximation was verified relative to Gene's rms measure. >From the exposition I have, 'The "length" of a unison vector... in the Tenney lattice with taxicab metric ... is proportional to ... the "number" ... of consonant intervals making up that unison vector.' That's what I'm disagreeing with. Why does the KvP metric behave differently? Me:>> For example, 2401:2400 works well in the 7-limit because the > numerator>> only involves 7, so it has a complexity of 4 despite being fairly > complex >> and superparticular. Paul:> This is only one possible complexity measure, not the one Gene's > currently using, which already showed a good match with the > heuristic. A better one awaits . . .It's a complexity measure based on 1) The odd limit 2) Minimax tuning I thought we agreed that (1) was as good as any simple, all-purpose, numerical dissonance metric. Also that it gave the same results as the octave-specific Tenney metric (or product limit) for small intervals. I'm not prepared to abandon this solely in order to make your heuristic work. I use (2) because it's simple to find the rule, at least for only one commatic unison vector. I expect RMS optimisation would give similar results provided all consonances are treated equally. If this isn't the case, I want a good reason why. Me:>> Whereas a comma involving 11**4, or 14641, still >> only has a complexity of 4 in the 11-limit. So if you could get a >> superparticular like that, it'd lead to a much smaller error. Paul:> You're missing the lattice justification for the heuristic. No wonder > you're skeptical!I'm working with the precise theory I already have. And you're asking me to give it up for a heuristic? Me:>> It should follow that 5**4:(13*3*2**4) or 625:624 will be > particularly>> inefficient between the 13- and 23-limits relative to what the > heuristic >> would predict. Paul:> Why? Try a 2D system based on 5 and 13. The heuristics should work > fine, especially if you weight ratios of 13 as less important than > ratios of 5.Yes, it'll work fine as long as you fudge the metric to get it to work. Making ratios of 13 "less important" means allowing them to be more out of tune. My experience is that the more complex intervals get, the more accurately they have to be tuned to sound right. A planar temperament optimising the minimax would give exactly 1200*log2(625/624)/4 = 0.7 cents for the worst interval. In fact, I think that's with 13:8 just so 5:4 and 13:10 are both out by 0.7 cents. Other methods are hardly likely to make anything badly out of tune, but that's fourth-order, superparticular, planar temperaments for you. How is one temperament supposed to work or not work according to a heuristic that only states a proportionality? My rule gives exact results, and it works. But only when one unison vector is being tempered out. Actually, not always then. For example 9*3 would be two consonances in the 9-limit but should be weighted as 1.5. But the difference between 1.5 and 2 is less than that between 5 and 23. Graham

Message: 3462 - Contents - Hide Contents Date: Tue, 22 Jan 2002 17:05:34 Subject: Re: Heuristics (Was: Hi Dave K.) From: paulerlich --- In tuning-math@y..., graham@m... wrote:> In-Reply-To: <a2jliq+9hrb@e...> > Me:>>> My experience of generating and sorting linear temperaments from>> the 5- to>>> the 21-limit is that the "right" error metric for one can be wildly >>> inappropriate for others. > > Paul:>> Can you give an example? >> The first run through of my temperament generator, when I was using > step-cents gave absurdly complex and accurate 5-limit temperaments. Using > only consistent ETs works well enough up to the 15-limit, but beyond that > optimal temperaments are missed. At least with the current metrics.What does any of this have to do with the validity of the heuristics? You seem to be talking about goodness/badness, as well as the generating-from-ETs-missed-some issue, neither of which have anything to do with the validity of the heuristics. Of course, once you define a goodness/badness measure, you should be able to use the heuristic for step/complexity, combined with the heuristic for cents/error, to approximate that goodness/badness measure.> Me:>>> One assumption behind the heuristic is that the error is >> proportional to>>> the size/complexity of the unison vector. > > Paul:>> You can call it an assumption, if you wish -- I've verified its >> approximate correctness for all 10 (wildly different) temperaments >> I've tried, against Gene's rms measures. >> How many dimensions?These were all 5-limit linear temperaments.> > I'm only questioning size/complexity as a heuristicHmm . . . you may be misunderstanding something. Can you clarify what you mean by this?> when you have more > than one unison vector.More than one tempered out? Why don't we focus on the case of just one tempered out first.>>> The other assumption is that the octave-specific Tenney metric >>> approximates the number of consonant intervals a comma's composed >> of. I'm>>> not sure how closely this holds. >>>> This is based on the Kees van Prooijen lattice metric, and again its >> good approximation was verified relative to Gene's rms measure. > >> From the exposition I have, 'The "length" of a unison vector> ... in the Tenney lattice with taxicab metric ... is proportional to ... > the "number" ... of consonant intervals making > up that unison vector.' That's what I'm disagreeing with.Why are you disagreeing? Note that "number" is _weighted_ -- more complex consonances are longer and count as "fewer" consonances. ?> > Me:>>> For example, 2401:2400 works well in the 7-limit because the >> numerator>>> only involves 7, so it has a complexity of 4 despite being fairly >> complex >>> and superparticular. > > Paul:>> This is only one possible complexity measure, not the one Gene's >> currently using, which already showed a good match with the >> heuristic. A better one awaits . . . >> It's a complexity measure based on > > 1) The odd limit > > 2) Minimax tuning > > I thought we agreed that (1) was as good as any simple, all- purpose, > numerical dissonance metric. Also that it gave the same results as the > octave-specific Tenney metric (or product limit) for small intervals. I'm > not prepared to abandon this solely in order to make your heuristic work.My heuristic says the complexity is proportional to log(d), where d is either the numerator or denominator (since they're close). Since either n or d is the odd limit, my heuristic is equivalent to (1). So you wouldn't be abandoning anything.> I use (2) because it's simple to find the rule, at least for only one > commatic unison vector. I expect RMS optimisation would give similar > results provided all consonances are treated equally.RMS optimiziation gave similar results to my heuristic -- so what's the problem?> > Yes, it'll work fine as long as you fudge the metric to get it to work. > Making ratios of 13 "less important" means allowing them to be more out of > tune. My experience is that the more complex intervals get, the more > accurately they have to be tuned to sound right.Well, this is the age-old question. It depends what you mean by "sounds right". We've spent so much time discussing this in the past, how this could go either way . . . personally, I don't find ratios of 13 to be "meaningful" as isolated dyads, and in the context of big otonalities, you can notice mistuning in the most consonant ratios more easily than mistuning in the 13 identity.> How is one temperament supposed to work or not work according to a > heuristic that only states a proportionality?I calculated the constants of proportionality for both heuristics for 10 vastly different temperaments, using Gene's rms optimized results, and the constants of proportionality for each heuristic were all within a factor of 2 of one another. Did you miss that post?

Message: 3463 - Contents - Hide Contents Date: Tue, 22 Jan 2002 17:06:37 Subject: Re: Heuristics (Was: Hi Dave K.) From: paulerlich --- In tuning-math@y..., graham@m... wrote:> My experience is that the more complex intervals get, the more > accurately they have to be tuned to sound right.So perhaps you'd like to temper the octaves most of all?

Message: 3464 - Contents - Hide Contents Date: Tue, 22 Jan 2002 17:34:32 Subject: Re: the Lattice Theory Homepage From: paulerlich Monz, If you're looking at MathWorld, the definition of lattice that we care about is found under "Point Lattice".

Message: 3465 - Contents - Hide Contents Date: Tue, 22 Jan 2002 17:44 +0 Subject: Re: Heuristics (Was: Hi Dave K.) From: graham@xxxxxxxxxx.xx.xx In-Reply-To: <a2k66t+5ccs@xxxxxxx.xxx> paulerlich wrote:> --- In tuning-math@y..., graham@m... wrote: >>> My experience is that the more complex intervals get, the more >> accurately they have to be tuned to sound right. >> So perhaps you'd like to temper the octaves most of all?I've mostly used octave-equivalent systems so far, so the opportunity doesn't present itself. Making the octaves worse would also make the most complex intervals worse, and some instruments don't allow you to do so anyway. Besides, I treat octaves and fifths as special cases. But this is certainly something to look at in the future. Oh, and if we're getting into specifics, a system with 11:8, 9:7, 9:8 and 11:7 tends to leave 8 on a par with 7, 11 and 9. So 2 would only end up with a third the error of 7 and 11. Graham

Message: 3466 - Contents - Hide Contents Date: Tue, 22 Jan 2002 17:23:00 Subject: new Dictionary entry: "torsion" From: monz new Dictionary entry: "torsion" Definitions of tuning terms: torsion, (c) 2002... * [with cont.] (Wayb.) Feedback appreciated. (and thanks for the helpful criticisms, Paul) -monz _________________________________________________________ Do You Yahoo!? Get your free @yahoo.com address at Yahoo! Mail - The best web-based email! * [with cont.] (Wayb.)

Message: 3467 - Contents - Hide Contents Date: Tue, 22 Jan 2002 17:44 +0 Subject: Re: Heuristics (Was: Hi Dave K.) From: graham@xxxxxxxxxx.xx.xx In-Reply-To: <a2k64u+qmc3@xxxxxxx.xxx> paulerlich wrote:> What does any of this have to do with the validity of the heuristics? > You seem to be talking about goodness/badness, as well as the > generating-from-ETs-missed-some issue, neither of which have anything > to do with the validity of the heuristics. Of course, once you define > a goodness/badness measure, you should be able to use the heuristic > for step/complexity, combined with the heuristic for cents/error, to > approximate that goodness/badness measure.I'm saying that one dimensional results don't usually generalise well to more complex cases. Me:>> I'm only questioning size/complexity as a heuristic Paul:> Hmm . . . you may be misunderstanding something. Can you clarify what > you mean by this? Me:>> when you have more >> than one unison vector. Paul:> More than one tempered out? Why don't we focus on the case of just > one tempered out first.Yes, that's fine, we agree on that case. That's why I said I wasn't questioning it. Me:>>> From the exposition I have, 'The "length" of a unison vector>> ... in the Tenney lattice with taxicab metric ... is proportional > to ...>> the "number" ... of consonant intervals making >> up that unison vector.' That's what I'm disagreeing with. Paul:> Why are you disagreeing? Note that "number" is _weighted_ -- more > complex consonances are longer and count as "fewer" consonances.I was assuming that numbers were numbers and didn't carry weights. I thought that was the difference between numbers and amounts.> My heuristic says the complexity is proportional to log(d), where d > is either the numerator or denominator (since they're close). Since > either n or d is the odd limit, my heuristic is equivalent to (1). So > you wouldn't be abandoning anything.The examples I gave before show that the Tenney length of a unison vector isn't a good predictor of the smallest number of intervals within a given odd limit that make it up. The numerator and denominator being close simply mean that the Tenney metric will be a predictor of the odd limit *for that interval*. It works differently when you look at combinations of intervals.> RMS optimiziation gave similar results to my heuristic -- so what's > the problem?I assume there's no problem with RMS as opposed to minimax. If your results agree, there's no problem.> Well, this is the age-old question. It depends what you mean > by "sounds right". We've spent so much time discussing this in the > past, how this could go either way . . . personally, I don't find > ratios of 13 to be "meaningful" as isolated dyads, and in the context > of big otonalities, you can notice mistuning in the most consonant > ratios more easily than mistuning in the 13 identity.Yes, it's not something I would normally pursue. I work on the simplest-case metric that all intervals deemed consonant are treated equally in tuning. I can then do the fine tuning by ear. But if you're suggesting something that will only work with the opposite weighting to what I now prefer, I'll disagree with it.>> How is one temperament supposed to work or not work according to a >> heuristic that only states a proportionality? >> I calculated the constants of proportionality for both heuristics for > 10 vastly different temperaments, using Gene's rms optimized results, > and the constants of proportionality for each heuristic were all > within a factor of 2 of one another. Did you miss that post?I obviously didn't pay much attention to it. Do you have a rough figure for "Erlich's constant" then? A factor of 2 still sounds a bit wayward. Graham

Message: 3468 - Contents - Hide Contents Date: Tue, 22 Jan 2002 17:47:04 Subject: Re: Heuristics (Was: Hi Dave K.) From: paulerlich --- In tuning-math@y..., graham@m... wrote:> In-Reply-To: <a2k66t+5ccs@e...> > paulerlich wrote: >>> --- In tuning-math@y..., graham@m... wrote: >>>>> My experience is that the more complex intervals get, the more >>> accurately they have to be tuned to sound right. >>>> So perhaps you'd like to temper the octaves most of all? >> I've mostly used octave-equivalent systems so far, so the opportunity > doesn't present itself. Making the octaves worse would also make the most > complex intervals worse,Huh? And if this is true, why wouldn't this be true of, say, ratios of 5?

Message: 3469 - Contents - Hide Contents Date: Tue, 22 Jan 2002 17:41:30 Subject: Re: the Lattice Theory Homepage From: monz> From: genewardsmith <genewardsmith@xxxx.xxx> > To: <tuning-math@xxxxxxxxxxx.xxx> > Sent: Tuesday, January 22, 2002 4:17 PM > Subject: [tuning-math] Re: the Lattice Theory Homepage > > > Blichfeldt's theorem, Minkowski's theorem, and the > Jarnick-Nosarzewska inequality are quite relevant, and > the Voronoi cell is a bit of terminology I've been > considering introducing.Paul used Voronoi cells in several diagrams he made a couple of years ago, illustrating harmonic entropy.> I keep thinking I'll apply Pick's theorem, but never have, > however I have had occasion to mention random lattice walks.Thanks for all of those, Gene! Lattice School is now in session! Looks to me like "Ehrhart Polynomial" is worth looking into as well.> If you want specific lattices to look at, the root lattice > An and its dual An* are the key ones, I think.I don't understand that. Where can these be found? -monz _________________________________________________________ Do You Yahoo!? Get your free @yahoo.com address at Yahoo! Mail - The best web-based email! * [with cont.] (Wayb.)

Message: 3470 - Contents - Hide Contents Date: Tue, 22 Jan 2002 17:56:38 Subject: Re: Heuristics (Was: Hi Dave K.) From: paulerlich --- In tuning-math@y..., graham@m... wrote:> Me:>>>> From the exposition I have, 'The "length" of a unison vector>>> ... in the Tenney lattice with taxicab metric ... is proportional >> to ...>>> the "number" ... of consonant intervals making >>> up that unison vector.' That's what I'm disagreeing with. > > Paul:>> Why are you disagreeing? Note that "number" is _weighted_ -- more >> complex consonances are longer and count as "fewer" consonances. >> I was assuming that numbers were numbers and didn't carry weights.Note that I put "number" in quotes.>> My heuristic says the complexity is proportional to log(d), where d >> is either the numerator or denominator (since they're close). Since >> either n or d is the odd limit, my heuristic is equivalent to (1). So >> you wouldn't be abandoning anything. >> The examples I gave before show that the Tenney length of a unison vector > isn't a good predictor of the smallest number of intervals within a given > odd limit that make it up.But I've always argued that the complexity measure should be weighted. It's easier to hear progressions by 3/2 than progressions by 5/4 . . .> The numerator and denominator being close > simply mean that the Tenney metric will be a predictor of the odd limit > *for that interval*. It works differently when you look at combinations > of intervals.Again, just focusing on linear temperaments from a "two-dimensional" just lattice for now . . .> Yes, it's not something I would normally pursue. I work on the > simplest-case metric that all intervals deemed consonant are treated > equally in tuning. I can then do the fine tuning by ear. But if you're > suggesting something that will only work with the opposite weighting to > what I now prefer, I'll disagree with it.Maybe we each have to write our own paper, then. I'm hoping someone will help me with the math for mine . . .>>> How is one temperament supposed to work or not work according to a >>> heuristic that only states a proportionality? >>>> I calculated the constants of proportionality for both heuristics for >> 10 vastly different temperaments, using Gene's rms optimized results, >> and the constants of proportionality for each heuristic were all >> within a factor of 2 of one another. Did you miss that post? >> I obviously didn't pay much attention to it. Yahoo groups: /tuning-math/message/2491 * [with cont.]"Expand Messages" as usual.> A factor of 2 still sounds a bit wayward.Not bad at all considering the wide range of complexities of these temperaments . . . but I'm still hunting for the "natural" set of definitions of "error" and "complexity" that will make the heuristic work real real good. Once you've found a good temperament, changing its error function is not going to change its goodness very much. So why not look for a mathematically pretty way to find good temperaments? That's something I'm interested in, at any rate.

Message: 3471 - Contents - Hide Contents Date: Tue, 22 Jan 2002 18:00 +0 Subject: Re: Heuristics (Was: Hi Dave K.) From: graham@xxxxxxxxxx.xx.xx In-Reply-To: <a2k8io+j4al@xxxxxxx.xxx> Me:>> I've mostly used octave-equivalent systems so far, so the > opportunity>> doesn't present itself. Making the octaves worse would also make > the most>> complex intervals worse, Paul:> Huh? And if this is true, why wouldn't this be true of, say, ratios > of 5?It would be true in a 25-limit system. Also, where 3 is involved, in a 15-limit system, although I'm not sure how the maths work out for that. So far, I've only tuned up 11-limit systems.>> and some instruments don't allow you to do so >> anyway. > > ?I have a Korg X5D (currently being repaired) which only supports octave based tuning tables. If I want to use it, everything else has to fall in line. Graham

Message: 3472 - Contents - Hide Contents Date: Tue, 22 Jan 2002 13:24:45 Subject: Re: Minkowski reduction (was: ...Schoenberg's rational implications) From: monz> From: paulerlich <paul@xxxxxxxxxxxxx.xxx> > To: <tuning-math@xxxxxxxxxxx.xxx> > Sent: Tuesday, January 22, 2002 4:25 AM > Subject: [tuning-math] Re: Minkowski reduction (was: ...Schoenberg's rational implications) > > > --- In tuning-math@y..., "monz" <joemonz@y...> wrote: >>> I've always been careful to emphasize that our tuning-theory use >> of "lattice" is different from the mathematician's strictly define >> uses of the term. >> This is not correct, Monz. There are several mathematical definitions > of "lattice" -- the one we use is most certainly one of these, as > we've discussed numerous times on the tuning list, and applied for > example in crystallographic theory.Duh!, of course. My bad! I've mentioned the crystallography bit myself in my webpage (even in the Dictionary).> ... >>> What's the purpose of wanting to find the Minkowski-reduced >> version of the PB instead of the actual one defined by >> Schoenberg's ratios? >> There's no purpose, as Schoenberg clearly meant for all the unison > vectors to be tempered out, and thus for 12-tET rather than JI to be > used. Tempering out the original set of unison vectors you posted is > exactly equivalent to tempering out the Minkowski-reduced set.Thanks, Paul, this helps. Of course, since Schoenberg intended *all* of the unison-vectors to be tempered out, it doesn't make a difference which set you look at ... he meant to imply them *all* simultaneously. -monz _________________________________________________________ Do You Yahoo!? Get your free @yahoo.com address at Yahoo! Mail - The best web-based email! * [with cont.] (Wayb.)

Message: 3473 - Contents - Hide Contents Date: Tue, 22 Jan 2002 13:34:15 Subject: Re: Minkowski reduction (was: ...Schoenberg's rational implications) From: monz> From: paulerlich <paul@xxxxxxxxxxxxx.xxx> > To: <tuning-math@xxxxxxxxxxx.xxx> > Sent: Tuesday, January 22, 2002 4:32 AM > Subject: [tuning-math] Re: Minkowski reduction (was: ...Schoenberg's rational implications) > > > --- In tuning-math@y..., "monz" <joemonz@y...> wrote: >> >>> Message 2850>>> From: paulerlich <paul@s...> >>> Date: Sun Jan 20, 2002 10:48pm >>> Subject: Re: lattices of Schoenberg's rational implications >>> >>> >>>>> [Paul]>>>>> BTW, is this Minkowski-reduced? >>> >>>> [Gene]>>>> Nope. The honor belongs to <22/21, 33/32, 36/35, 50/49>. >>> >>> [Paul]>>> Awesome. So this suggests a more compact Fokker parallelepiped >>> as "Schoenberg PB" -- here are the results of placing it in >>> different positions in the lattice (you should treat the >>> inversions of these as implied): >>> >>>> >> [monz] >> My first question is: this is a 7-limit periodicity-block, >> so can you explain how the two 11-limit unison-vectors disappeared? > > [Paul]> They didn't disappear! It's just that in these particular positions, > the parallelepiped all lies within one "power of 11" plane. I'm sure > Gene could produce an example that wouldn't.OK ... I understood that intuitively ... just couldn't wrap my brain around the math.>> [monz] >> With variant alternate pitches written on the same line >> -- and thus with invariant ones on a line by themselves -- >> these scales are combined into: >> >> 1/1 >> 21/20 15/14 >> 35/32 9/8 >> 7/6 25/21 6/5 >> 5/4 >> 21/16 >> 7/5 10/7 >> 3/2 >> 49/32 25/16 63/40 >> 5/3 12/7 >> 7/4 >> 147/80 15/8 >> >> ... >> >> One thing I did notice in connection with this, is that >> 147/80 is only a little less than 4 cents wider than 11/6, >> which is one of the pitches implied in Schoenberg's overtone >> diagram (p 23 of _Harmonielehre_) : >> >> vector ratio ~cents >> >> [ -4 1 -1 2 0 ] = 147/80 1053.2931 >> - [ -1 -1 0 0 1 ] = 11/6 1049.362941 >> -------------------- >> [ -3 2 -1 2 -1 ] = 441/440 3.930158439 >> >> >> So I know that 441/440 is tempered out. >> NO IT ISN'T! I believe it maps to 1 semitone given the set of unison > vectors you've put forward. >>> But I don't see >> how to get this as a combination of two of the other >> unison-vectors. > > YOU CAN'T!Oops... my bad. Thanks, Paul. I see it now. If "C" is Schoenberg's 1/1, the 147/80 is mapped to "B" but 11/6 is mapped to "Bb". This is precisely the note which was misprinted in the diagram in the English edition ... guess I accepted it for so long that I got confused.>>> Then, I reasoned that since all of these pitches are separated >> by one or two of the unison vectors which define this set of PBs, >> the lattice could be further reduced to a 12-tone set, one that >> can still "define the same temperament": >> >> Monzo lattice of Monzo's ultimate reduction of Paul Erlich's >> 4 variant Minkowski-reduced 7-limit PBs for p 1-184 of >> Schoenberg's _Harmonielehre_, to one 12-tone PB : >> >> Yahoo groups: /tuning-math/files/monz/ult-red.gif * [with cont.] >> >> >> Correct? >> What's so "ultimate" about it?Simply that now it actually finally is reduced to one 12-one set. The point of my question is: even tho the shape of this particular PB is different from the 4 identical ones from which it was coalesced, all the same UVs are in effect, so it's still identical to those 4, yes? -monz _________________________________________________________ Do You Yahoo!? Get your free @yahoo.com address at Yahoo! Mail - The best web-based email! * [with cont.] (Wayb.)

Message: 3474 - Contents - Hide Contents Date: Tue, 22 Jan 2002 13:48:28 Subject: Re: A top 20 11-limit superparticularly generated linear temperament list From: monz> From: paulerlich <paul@xxxxxxxxxxxxx.xxx> > To: <tuning-math@xxxxxxxxxxx.xxx> > Sent: Tuesday, January 22, 2002 4:39 AM > Subject: [tuning-math] Re: A top 20 11-limit superparticularly generatedlinear temperament list> >>> Partch apparently wove harmonic structures into his compositions >> which sometimes require the listener to infer different rational >> implications from his scale than the obvious ones. Without >> examining the actual mathematics of it, your [Dave Keenan's] >> revised statement here seems to me to be a good way to model >> that aspect of Partch's compositional practice. >> >> >> >> -monz >> Well, Monz, it would be good to know if Partch exploited _only_ the > hemiennealimmal equivalencies, or _only_ the MIRACLE equivalencies, > or what. Wilson seems to have felt that he exploited enough > equivalencies that a closed 41-tone system (as in 41-tET) was > actually implied. But Wilson never seems to have thought much about > MIRACLE, let along hemiennealimmal.All good points, Paul. I was careful to add "without examining the actual mathematics", because I don't even know what "hemiennealimmal" is!!!!! (... but I'm still studying!) Why don't you guys take a look at that? Sounds interesting. -monz _________________________________________________________ Do You Yahoo!? Get your free @yahoo.com address at Yahoo! Mail - The best web-based email! * [with cont.] (Wayb.)

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