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Message: 9475 - Contents - Hide Contents Date: Wed, 28 Jan 2004 09:46:59 Subject: 1705229 equal From: Gene Ward Smith On this web page Tabulating values of the Riemann-Siegel Z func... * [with cont.] (Wayb.) we find that this was the highest value (converted to an octave division) the author found for Z(t) after going out well past 2000000. While integrating Z works better than merely taking the highest value, I note that 1705229-et is very nice if you have godlike hearing and enjoy the 41 through 57 limits.

Message: 9476 - Contents - Hide Contents Date: Wed, 28 Jan 2004 17:23:00 Subject: Re: 114 7-limit temperaments From: Carl Lumma>> >art of the problem is these contours represent musical values, so >> they're ultimately a matter of opinion. >>Not the problem here, as Dave simply wanted a graph,I was referring here to lists, not the graph. I was glad to see Dave ask for that graph by the way, because I think it would help clarify my perception of the lists I've seen. -C.

Message: 9480 - Contents - Hide Contents Date: Wed, 28 Jan 2004 17:32:37 Subject: Re: rank complexity explanation updated From: Carl Lumma>>>>>> >00.0 : 2 4 5 7 9 11 12 >>>>>> 200.0 : 2 3 5 7 9 10 12 >>>>>> 400.0 : 1 3 5 7 8 10 12 >>>>>> 500.0 : 2 4 6 7 9 11 12 >>>>>> 700.0 : 2 4 5 7 9 10 12 >>>>>> 900.0 : 2 3 5 7 8 10 12 >>>>>> 1100.0: 1 3 5 6 8 10 12 >>>>>>>>>> Q: Shouldn't the first row be 000.0? >>>>>>>> It is quite safe to ignore the values before the colon, as they >>>> are merely an artifact of scala's output. >>>>>> What kind of artifact are they, if not an error? >>>> They indicate the interval between 1/1 and the degree of the >> original 'native' mode, on which the mode shown on the particular >> line is based. >>In that case, shouldn't the first entry be 000.0?It should be "1/1". Looks like this was a typo on my part.>> In this case they are artifactal only because they are given in >> different units than is the interval matrix itself. >>OK, but more serious would seem to be the actual error.Maybe I clipped this bit when I copied and pasted, and then filled in by hand (incorrectly) in subconscious mode. -Carl

Message: 9481 - Contents - Hide Contents Date: Wed, 28 Jan 2004 12:08:32 Subject: Re: Gene's Improved notation for prime-form reduction From: Carl Lumma>Carl, See 8118: (quoting Gene) > >It seems to me that the set of prime-form reductions of n things >taken m at a time modulo some permutation group of degree G could be >given a name--"(n,m) reduced G" or "G{n,m}" or something--and then >we'd really be cooking. The prime form in question could be defined >as the least base-2 number in the orbit. You might also want a name >for the function which takes a chord or PC or whatever you wish to >call it to its G-reduction. Something like "Pfred(s, G)" where s is >the PC set and G is the permuation group. Thanks! -C. ________________________________________________________________________ ________________________________________________________________________ ------------------------------------------------------------------------Yahoo! Groups Links To visit your group on the web, go to: Yahoo groups: /tuning-math/ * [with cont.] To unsubscribe from this group, send an email to: tuning-math-unsubscribe@xxxxxxxxxxx.xxx Your use of Yahoo! Groups is subject to: Yahoo! Terms of Service * [with cont.] (Wayb.)

Message: 9482 - Contents - Hide Contents Date: Wed, 28 Jan 2004 17:36:58 Subject: Re: rank complexity explanation updated From: Carl Lumma>>> >he interval matrix often, if not typically, has all >>> unisons/octaves along the diagonal. This one is merely >>> a reshuffling so that the diagonal becomes a right vertical. >>>> No, the interval matrix is always written as above (the values >> after the colons, at least). >>Where do you get *always*??I have never seen it printed this way. Where do you get "often, if not typically"? -Carl

Message: 9483 - Contents - Hide Contents Date: Thu, 29 Jan 2004 01:54:48 Subject: Re: 114 7-limit temperaments From: Carl Lumma>So we're not goint to talk about how to obtain a finite scale from a >temperament. Now that the octave is just another generator (and the >pair of generators is no longer unique), don't we have some explaining >to do, about why we should iterate one particular generator modulo the >other?No generator is privileged in the mod relationship (any may be taken as mod any other). And since the pair of generators was never unique, we have no more explaining to do now than we ever did. Though I am interested in any relationship between these two ways of looking at temperament.>>> even if it only works for linear temperaments. >>>> The whole point is that it works for equal, linear, planar . . . etc. >> >> Oh, you said _such_ . . . Sorry. Well, I suspect that's still >> possible, especially since Gene himself explained the lower >> complexity of schismic vs. miracle in terms of generators-per-prime. >>Yes. I meant "even if some particular _interpretation_ only works for >linear". I understand the _complexity_measure_ is intended to work for >all, and as such it sounds brilliant.Graham saw fit to report the size of the smallest MOS, which seems founded on the noble goal of trying to measure complexity in terms of number of notes, across temperaments. However it seems likely that size of smallest MOS is just an artifact of the particular choice of generators, which we now know is not unique. I've suggested consonance/notes graphs, but again: how to choose the notes? I've also suggested number of notes in a corresponding Fokker block, but I'll have to defer to Paul on the status of that suggestion. -Carl

Message: 9484 - Contents - Hide Contents Date: Thu, 29 Jan 2004 01:57:06 Subject: Re: rank complexity explanation updated From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:>>>> The interval matrix often, if not typically, has all >>>> unisons/octaves along the diagonal. This one is merely >>>> a reshuffling so that the diagonal becomes a right vertical. >>>>>> No, the interval matrix is always written as above (the values >>> after the colons, at least). >>>> Where do you get *always*?? >> I have never seen it printed this way. Where do you get > "often, if not typically"? > > -CarlHere's just one example: Definitions of tuning terms: interval matrix, ... * [with cont.] (Wayb.)

Message: 9485 - Contents - Hide Contents Date: Thu, 29 Jan 2004 21:19:14 Subject: Re: 114 7-limit temperaments From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" <d.keenan@b...> wrote:> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:>> --- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" <d.keenan@b...> >> wrote: >>>>> That sounds pretty good, in a hand-waving sort of way, but when I >> try>>> to pin down what it actually means in terms of how big a scale we >>> might need to make good use of some temperament, my brain just keeps >>> sliding off it. >>>> Might that be, in a way, a necessary result of dropping octave- >> equivalence? >> It might be, but in any case I'd still like to get on top of it. > > Without octave equivalence we can still make it finite by limiting it > to the range of human hearing, say 10 octaves.Sure, but then we won't have a clear choice of generator even for linear temperaments. I think *scale* (finite non-ET pitch set) questions would require a separate paper, beginning from periodicity blocks . . .

Message: 9486 - Contents - Hide Contents Date: Thu, 29 Jan 2004 11:20:37 Subject: What the numbers mean From: Gene Ward Smith Here is a no doubt long overdue discussion of what the numbers in the wedgie for a linear temperament mean. The wedgie for miracle is <<6 -7 -2 -25 -20 15||. Normally we would content ourselves with saying the mapping to primes is given by [<1 1 3 3|, <0 6 -7 -2|], but that is a 2-centric way of putting it; we have other maps for 3 and generator, 5 and generator, and 7 and generator: 3&g: [<1 1 3 3|, <-6 0 -25 -20|] g ~ 7/72 ~ 11/114 5&g: [<-2 -8 1 4|, <7 25 0 15|] g ~ 58/72 ~ 135/167 7&g: [<1 7 -4 1|, <-2 -20 15 0|] g ~ 65/72 ~ 182/202 The reason for the second way of putting the generator is that this is now in terms of 3, 5, or 7 of the <72 114 167 202| val. So, for instance, we can consider miracle in terms of a generator which is 11 steps out of 114 representing 3, or 135 out of 167 steps representing 5, or 182 out of 202 steps representing 7. If we look at the absolute values of the numbers in the miracle wedgie, 6 is the number of generators to get to a 3 in a 2&g system, or to get to a 2 in a 3&g system, or to get to 3/2 in either. Similarly, 7 is the number of generator steps to 5/2, 2 is the number of generator steps to 7/2, 25 is the number of generator steps to 5/3, 20 the number to 7/3 and 15 the number to 7/5, in the systems whose period is one of the two primes in the ratio. So, the last three numbers of the wedgie can be related to 5/3, 7/3, 7/5 just as the first three can to 3/2, 5/2, 7/2. If we take absolute values and normalize each by dividing by the log base 2 of the two primes involved, we get [6/p3 7/p5 2/p7 25/(p3p5) 20/(p3p7) 15/(p5p7)] = [3.785578521, 3.014735907, .7124143742, 6.793166368, 4.494834256, 2.301151281] The complexity is defined by the weighted value for 5/3, which is the worst case (as, in fact, it clearly is.)

Message: 9487 - Contents - Hide Contents Date: Thu, 29 Jan 2004 21:24:40 Subject: Re: 114 7-limit temperaments From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> > wrote: >>> No particular generator basis is assumed in the TOP complexity >> calculations. Instead, it's a direct measure of how much the >> tempering simplifies the lattice, and reduces (Gene seems to >> imply/agree) to the number of notes per acoustic whatever in the > case>> of equal (1-dimensional) temperaments. >> I thought equal temperaments were "0-dimensional"?We've always said linear temperaments were actually 2-dimensional, so equal temperaments are 1-dimensional, and are generated by their step. Remember we've dropped octave-equivalence to get TOP.> The complexity measure for an equal temperament pretty much reduces > to the division of the octave.More precisely, to the division of the *acoustical* octave, or any other fixed value in cents, apparently . . .> You could take the maximum of > |(mapping of p)/log2(p)| if you wanted to be precise about it.Could you take another look at the "Attn: Gene 2" post and explain what's going on there, mathematically? I didn't use the maximum but that was only 3-limit . . .

Message: 9488 - Contents - Hide Contents Date: Thu, 29 Jan 2004 02:46:55 Subject: Re: 114 7-limit temperaments From: Dave Keenan --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:> --- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" <d.keenan@b...> > wrote:>> Yes, It's very clever in that way. But still it flies in the face of >> Partch's "observation one" (or whatever it is called) that as the >> complexity of a frequency ratio increases it must be tuned more and >> more accurately to be perceived as just (or something), until it >> becomes too complex to hear that way even when tuned precisely. >> I don't think that's quite what Partch says.So what _does_ Partch say?> Manuel, at least, has > always insisted that simpler ratios need to be tuned more accurately, > and harmonic entropy and all the other discordance functions I've > seen show that the increase in discordance for a given amount of > mistuning is greatest for the simplest intervals.But surely it's obvious that beat rates go up as something like error _times_ complexity, not down as error _divided_by_ complexity, even though beat amplitudes do go down.>> The very fact that TOP cannot distinguish between a good 7-limit >> temperament and a good 9-limit temperament (nor 8 or 10 limit) > should>> make one suspicious. >> Such distinctions may be important for *scales*, but for > temperaments, I'm perfectly happy not to have to worry about them. > Any reasons I shouldn't be?Sure. It is often possible to change generators slightly to reduce errors in 7-odd-limit intervals at the expense of 9-odd-limit ones. We've always accepted this in the past.>> Paul, that's clearly not what Gene has done in this list, otherwise >> the error figures would be a factor of 1200 smaller than they are. >> You mean 1200/log(2), right? No. 1200.Start with the error in ratio n:d as another frequency ratio R (usually irrational). You claimed Gene was giving minimax of log(R)/log(n*d) which is the same as lg2(R)/lg2(n*d) in fact he is giving minimax of 1200*lg2(R)/lg2(n*d) where 1200*lg2(R) is the interval's error in cents.>> Gene has in fact already correctly normalised the errors and given >> them in cents. >> That's a weird interpretation of the units because it's really only > in cents for the octave, and often the octave doesn't even achieve > the reported error.But you agree it's a perfectly valid interpretation, due to the obvious generalisation of the normalising of weighted MAD, RMS etc. to the normalising of weighted minimax?>>>> Is the weighting the same for the complexity? Minimax where > gens per>>>> interval is divided by lg2(product_complexity(interval))? >>>>>> No particular generator basis is assumed in the TOP complexity >>> calculations. Instead, it's a direct measure of how much the >>> tempering simplifies the lattice, and reduces (Gene seems to >>> imply/agree) to the number of notes per acoustic [interval] in >>> the case of equal (1-dimensional) temperaments. ...>> OK. That sounds alright, but how do I relate it to upper limits (or >> whatever) on the numbers of notes before I get a particular >> interval? >> The word "before" implies some particular generating scheme.Yes. Presumably any reasonable generating scheme should have some such interprtetation in terms of the TOP complexity, otherwise what use is it?> In fact > it seems to imply a *single* generator, which is not even possible > for planar, etc. temperaments.I suspect one can still specify reasonable (maybe even optimal) generating schemes with multiple generators. In fact since we're tempering the octave, we apparently need to find such schemes already for so-called linear temperaments. Musicians will ultimately want finite scales (even finite compasses) from these temperaments. In any case, I'd be happy to hear _any_ such interpretation for this new complexity measure, even if it only works for linear temperaments.> However, I wouldn't be surprised if it > is indeed possible to view TOP complexity this way, due to some > amazing theorem to be proved by Gene.I don't require any amazing theorems to be proved. I'd be happy if such a concrete interpretation could just be shown to work for all temperaments on Gene's latest list, or even just the first 10 or so. If some complexity measure cannot be interpreted in concrete terms of numbers of notes for particular intervals, then why should I trust it as a means for comparing temperaments? I'm sure it can be, I'm just dying to know how, or be given enough information to figure it out for myself. Feel free to point me to important posts I may have missed.

Message: 9490 - Contents - Hide Contents Date: Thu, 29 Jan 2004 21:28:08 Subject: Re: 114 7-limit temperaments From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:>> For linear temperaments, I agree that we may still want to give the >> period/generator specification, but there are many ways to justify >> the attention on this. For example, though we didn't assume octave- >> equivalence, we may want to assume that the musician will generally >> be most interested in subsets of the temperament that repeat at the >> (tempered) octave. >> As long as the temperament maps the octave, will it not be periodic > at said octave? > > -CarlUnless it's an equal temperament, it will have an infinite density of notes, periodic at any conceivable interval in the temperament. Whether the *finite subsets* you choose repeat at the octave or not, though, is up to you.

Message: 9491 - Contents - Hide Contents Date: Thu, 29 Jan 2004 21:29:42 Subject: Re: Graef article on rationalization of scales From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:>>> That the diatonic scale is a 'good' PB seems like the best example. >>>> Is PB a synonym for Fokker block, or is it more general, and if so, >> how precisely is it defined? >> My impression was that PB is weaker than Fokker, the later requiring > epimorphism and monotonicity (neither of which I have a solid > understanding of) and that the former requires, well, nothing more > than the correct number of commas that, when all of them are tempered > out, gives an equal temperament. > > -CarlPretty much. We've been through this before, whereupon Gene defined "block" for convex PB and "semiblock" for something "not too concave" or something, if my vague recollection is reliable.

Message: 9492 - Contents - Hide Contents Date: Thu, 29 Jan 2004 21:31:49 Subject: Re: Graef article on rationalization of scales From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:>>> That the diatonic scale is a 'good' PB seems like the best example. >>>> Is PB a synonym for Fokker block, or is it more general, and if so, >> how precisely is it defined? >> My impression was that PB is weaker than Fokker, the later requiring > epimorphism and monotonicity (neither of which I have a solid > understanding of) and that the former requires, well, nothing more > than the correct number of commas that, when all of them are tempered > out, gives an equal temperament. > > -CarlActually, all PBs always have epimorphism, but I don't know what monotonicity is. Fokker blocks are parallelograms/parallelepipeds -- bounded exactly by the linearly independent set of unison vectors.

Message: 9493 - Contents - Hide Contents Date: Thu, 29 Jan 2004 03:15:17 Subject: Re: 114 7-limit temperaments From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" <d.keenan@b...> wrote:> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:>> --- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" <d.keenan@b...> >> wrote:>>> Yes, It's very clever in that way. But still it flies in the face of >>> Partch's "observation one" (or whatever it is called) that as the >>> complexity of a frequency ratio increases it must be tuned more and >>> more accurately to be perceived as just (or something), until it >>> becomes too complex to hear that way even when tuned precisely. >>>> I don't think that's quite what Partch says. >> So what _does_ Partch say?I'll have to go home and look.>> Manuel, at least, has >> always insisted that simpler ratios need to be tuned more accurately, >> and harmonic entropy and all the other discordance functions I've >> seen show that the increase in discordance for a given amount of >> mistuning is greatest for the simplest intervals. >> But surely it's obvious that beat rates go up as something like error > _times_ complexity, not down as error _divided_by_ complexity, even > though beat amplitudes do go down.Yes. But beating is virtually irrelevant for the perception of higher- limit harmonies. Such harmonies, it seems to most people, *must* be otonal in order to sound convincing at all, and this extreme otonal bias indicates that virtual pitch and/or combinational tones are the dominant factors -- beating awould predict equal or greater consonance for the utonal harmonies, yet high-limit utonalities sound very dissonant to many people.>>> The very fact that TOP cannot distinguish between a good 7-limit >>> temperament and a good 9-limit temperament (nor 8 or 10 limit) >> should>>> make one suspicious. >>>> Such distinctions may be important for *scales*, but for >> temperaments, I'm perfectly happy not to have to worry about them. >> Any reasons I shouldn't be? >> Sure. It is often possible to change generators slightly to reduce > errors in 7-odd-limit intervals at the expense of 9-odd-limit ones. > We've always accepted this in the past.Sure, but we were always assuming full octave-equivalence then, for one thing.>>>> Gene has in fact already correctly normalised the errors and given >>> them in cents. >>>> That's a weird interpretation of the units because it's really only >> in cents for the octave, and often the octave doesn't even achieve >> the reported error. >> But you agree it's a perfectly valid interpretation, due to the > obvious generalisation of the normalising of weighted MAD, RMS etc. to > the normalising of weighted minimax?It's hard to say, because of course it's inconceivable for all of the errors to be equal.>>>>> Is the weighting the same for the complexity? Minimax where >> gens per>>>>> interval is divided by lg2(product_complexity(interval))? >>>>>>>> No particular generator basis is assumed in the TOP complexity >>>> calculations. Instead, it's a direct measure of how much the >>>> tempering simplifies the lattice, and reduces (Gene seems to >>>> imply/agree) to the number of notes per acoustic [interval] in >>>> the case of equal (1-dimensional) temperaments. > ...>>> OK. That sounds alright, but how do I relate it to upper limits (or >>> whatever) on the numbers of notes before I get a particular >>> interval? >>>> The word "before" implies some particular generating scheme. >> Yes. Presumably any reasonable generating scheme should have some such > interprtetation in terms of the TOP complexity, Possibly. > otherwise what use is it?See above. This seems like a more direct definition of 'complexity' to me. If I take Gene's confirmation to heart, it's the affine- geometrical size measure (length, area, volume, etc.) of the portion of the lattice sufficient, under the relevant temperament, to represent the entire lattice.>> In fact >> it seems to imply a *single* generator, which is not even possible >> for planar, etc. temperaments. >> I suspect one can still specify reasonable (maybe even optimal) > generating schemes with multiple generators. In fact since we're > tempering the octave, we apparently need to find such schemes already > for so-called linear temperaments.I don't think we need to do any of this, nor should we want to for the purposes of a paper brief enough to be published.> In any case, I'd be happy to hear _any_ such interpretation for this > new complexity measure,See Yahoo groups: /tuning-math/message/8781 * [with cont.] and Yahoo groups: /tuning-math/message/8806 * [with cont.] . . .> even if it only works for linear >temperaments.The whole point is that it works for equal, linear, planar . . . etc. Oh, you said _such_ . . . Sorry. Well, I suspect that's still possible, especially since Gene himself explained the lower complexity of schismic vs. miracle in terms of generators-per-prime.> I'd be happy if > such a concrete interpretation could just be shown to work for all > temperaments on Gene's latest list, or even just the first 10 or so.Yes, I'm hoping Gene will help me to see this too. I've been groping here for a better understanding of the math . . .

Message: 9494 - Contents - Hide Contents Date: Thu, 29 Jan 2004 21:33:12 Subject: Re: Maximum Error in TOP From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "Kalle Aho" <kalleaho@m...> wrote:> Hi! > > How do I get the maximum Tenney-weighted error in a TOP temperament? > Is it the same as the maximum weighted error of the primes Yes. > or some > other set of lattice-spanning intervals?That works too, I believe.

Message: 9495 - Contents - Hide Contents Date: Thu, 29 Jan 2004 03:19:32 Subject: Re: Graef article on rationalization of scales From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:>>> Since you practically single-handedly launched the 'popular scales >>> are good PBs' program, I find it highly unusual that you are now >>> asking me what it is. >>>> Now I understand you better. Yet, 'popular scales' will often have a >> large number of plausible derivations from a PB, >> You mean to a PB, I think?A gets derived *from* B, it doesn't get derived *to* B, right?> But if your 'the basis > doesn't matter' reasoning applied to blocks,It only applies to the *temperament kernel*, or to the commatic unison vectors -- not the chromatic ones.> If we could > then further show that historical scales correspond to gooder blocks > than random scales...Since Gene is done enumerating 12-note Fokker blocks, I hope he will treat us to a collection of 5-note ones soon . . .

Message: 9496 - Contents - Hide Contents Date: Thu, 29 Jan 2004 21:38:20 Subject: Re: 114 7-limit temperaments From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:>> So we're not goint to talk about how to obtain a finite scale from a >> temperament. Now that the octave is just another generator (and the >> pair of generators is no longer unique), don't we have some explaining >> to do, about why we should iterate one particular generator modulo the >> other? >> No generator is privileged in the mod relationship (any may be taken > as mod any other). And since the pair of generators was never unique, > we have no more explaining to do now than we ever did.Well, it was a lot more unique before, since one of the generators was totally unique -- the period -- and the other could be defined uniquely within any given half-period range (usually taken to be 0 to 1/2).> I've > also suggested number of notes in a corresponding Fokker block, but > I'll have to defer to Paul on the status of that suggestion.Again, this is exactly what I've been discussing in this thread and in "Attn: Gene 2", except that you have to replace "Fokker block" with "Fokker strip", "Fokker slice", or whatever is appropriate (block only appropriate for ET). In the latter cases, the number of notes is infinite, but the "size" of the relevant construct in the lattice is still ultra-meaningful, and what I've been itching about here.

Message: 9497 - Contents - Hide Contents Date: Thu, 29 Jan 2004 03:25:23 Subject: Re: Attn: Gene 2 From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote: > > '>> geometry, where angles are left undefined . . . Anyhow, since both of >> these methods could be used to address a 3-limit TOP temperament, in >> 5-limit could they be still both be expressible in a single form in a >> general enough framework, say exterior algebra? >> That was my suggestion. You normalize by dividing each coordinate by > log2(p) and take the wedge product up to the (normalized) multival, > and then measure complexity by taking the max of the absolute values > of the coefficients. If you start from the monzo side, you get the > same normalized coefficients up to a constant factor, but now you > might rather take the sum of the absolute values (L1 vs L infinity.)Sounds like there are still alternatives here (L1 vs L infinity), while the complexity definition I was sketching should not have this ambiguity. So I'm not sure if I'll end up agreeing with Gene's complexity calculations. Again, it would really help if he went through my original post and showed, step by step, how all the results there generalize to a single formula.

Message: 9498 - Contents - Hide Contents Date: Thu, 29 Jan 2004 21:51:52 Subject: Re: What the numbers mean From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:> Here is a no doubt long overdue discussion of what the numbers in the > wedgie for a linear temperament mean.I've noticed that the matrix of linear temperament commas with 0s along the diagonal has both the upper triangle and lower triangle very similar to the corresponding numbers in the wedgie (the latter are divisible by the former, in fact).> The wedgie for miracle is <<6 -7 -2 -25 -20 15||. Normally we would > content ourselves with saying the mapping to primes is given by > [<1 1 3 3|, <0 6 -7 -2|], but that is a 2-centric way of putting it; > we have other maps for 3 and generator, 5 and generator, and 7 and > generator: > > 3&g: [<1 1 3 3|, <-6 0 -25 -20|] g ~ 7/72 ~ 11/114 > > 5&g: [<-2 -8 1 4|, <7 25 0 15|] g ~ 58/72 ~ 135/167 > > 7&g: [<1 7 -4 1|, <-2 -20 15 0|] g ~ 65/72 ~ 182/202 > > The reason for the second way of putting the generator is that this is > now in terms of 3, 5, or 7 of the <72 114 167 202| val. So, for > instance, we can consider miracle in terms of a generator which is 11 > steps out of 114 representing 3, or 135 out of 167 steps representing > 5, or 182 out of 202 steps representing 7. > > If we look at the absolute values of the numbers in the miracle > wedgie, 6 is the number of generators to get to a 3 in a 2&g system, > or to get to a 2 in a 3&g system, or to get to 3/2 in either. > Similarly, 7 is the number of generator steps to 5/2, 2 is the number > of generator steps to 7/2, 25 is the number of generator steps to 5/3, > 20 the number to 7/3 and 15 the number to 7/5, in the systems whose > period is one of the two primes in the ratio. So, the last three > numbers of the wedgie can be related to 5/3, 7/3, 7/5 just as the > first three can to 3/2, 5/2, 7/2. > > If we take absolute values and normalize each by dividing by the log > base 2 of the two primes involved, we get > > [6/p3 7/p5 2/p7 25/(p3p5) 20/(p3p7) 15/(p5p7)] = > > [3.785578521, 3.014735907, .7124143742, 6.793166368, 4.494834256, > 2.301151281] > > The complexity is defined by the weighted value for 5/3, which is the > worst case (as, in fact, it clearly is.)Meaning what, exactly? Can you show this process in action for the simpler, 3-limit and 5- limit cases? And why do we take the worst case, instead of some sort of product (which would appear to get you your spacial measure once you've orthogonalized)? Thanks for this post, though; I'll be referring to it in the future.

Message: 9499 - Contents - Hide Contents Date: Thu, 29 Jan 2004 21:52:49 Subject: Re: What the numbers mean From: Dave Keenan --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:> Here is a no doubt long overdue discussion of what the numbers in the > wedgie for a linear temperament mean. ...That was a good explanation. Thanks heaps Gene!> If we take absolute values and normalize each by dividing by the log > base 2 of the two primes involved, we get > > [6/p3 7/p5 2/p7 25/(p3p5) 20/(p3p7) 15/(p5p7)] = > > [3.785578521, 3.014735907, .7124143742, 6.793166368, 4.494834256, > 2.301151281] > > The complexity is defined by the weighted value for 5/3, which is the > worst case (as, in fact, it clearly is.)You are dividing by the _product_ of the two logs, shouldn't you be dividing by their _sum_ (the log of the product of the primes)?

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