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Message: 3625 - Contents - Hide Contents Date: Thu, 31 Jan 2002 04:24:16 Subject: Re: new cylindrical meantone lattice From: paulerlich --- In tuning-math@y..., "monz" <joemonz@y...> wrote:>> Somewhere a long time ago, perhaps in the Mills times, I posted a >> proposed formula for these lengths. But I'm not too picky about it. >> Why not show mistuning as a tiny "break" in the consonant connections? > >> Hmm ... sounds interesting. You mean like on the recent Blackjack > lattice you posted?Well, sort of. Except that all the consonant intervals would be broken.>Please elaborate.Well, for example, you could do it like in Hall's hexagonal lattices, where he puts a little number representing the mistuning of each interval, but you could do it visually, like representing each consonant interval as a twig and putting a physical "break", the size of the mistuning, into it.>>>>> And I *still* don't understand how a note that I factor as, >>>>> for example, 3^(2/3) * 5^(1/3) (ignoring 2), which is the >>>>> 1/6-comma meantone "whole tone", can be represented as >>>>> anything else. There is no other combination of exponents >>>>> for 3 and 5 which will plot that point in exactly that spot. >>>>>>>> Sure there are -- they'd just be irrational exponents. But when >>>> some meantones, like LucyTuning, will require irrational exponents >>>> anyway in order to get a "spiral" happening for them, that >>>> implies to me that irrational exponents are just as meaningful >>>> as rational ones. . . . > >> I'm still not getting this. It seems to me that you're thinking > in terms of something other than the 3x5 plane within which I'm working.No I'm not :)> But I can certainly buy what you're saying about irrational exponents. > As I said, if I could figure out *how* to lattice them, I would. > (See the bit above about the xenharmonic bridge.)You'll have to determine what the "spiral" means for some hypothetical musician who might want to see it, if anything. Once you define the problem, it can be solved.>>> Hmmm ... yes, I'm thinking that the meantone-spiral thing really >>> might be a useful new contribution to tuning theory. Thanks for >>> the acknowledgement! >>>> You got it, though I'm not banking on the "useful" part :) >> Particularly irksome is that you must choose a 1/1, such as C, which >> flies in the face of the true nature of meantone as a transposible >> system. > >> Ah! ... Paul, this is where you'd understand my ideas a little > better if you finally succumb to my always begging you to join > my justmusic group! <Yahoo groups: /justmusic * [with cont.] > > > > The idea is that the user could: > > 1) Choose meantone as the type of tuning desired: JustMusic > then draws the syntonic-comma based cylinder on the lattice.Right . . .> 2) Define which meantone by fraction-of-a-comma (or Lucy or Golden, > if I ever figure out how): JustMusic draws the spiral around > the cylinder.And the spiral helps me, as a musician, do . . . ?> 3) Use the mouse to roll the cylinder around on the lattice > to get whichever key-center is desired, or simply input the > key and let JustMusic do the rolling.Well this I was expecting anyway.> That's just a brief outline.I'm afraid you haven't told me anything new :) :)
Message: 3626 - Contents - Hide Contents Date: Thu, 31 Jan 2002 13:28 +0 Subject: Re: interval of equivalence, unison-vector, period From: graham@xxxxxxxxxx.xx.xx In-Reply-To: <a3am2v+eddv@xxxxxxx.xxx> paulerlich wrote:> The period is often 1/2-octave, 1/3-octave, 1/4-octave, 1/9- > octave, . . . so that's clearly not a "unison-vector". > > The "interval of equivalence" is a unison vector in Graham's system, > but Graham's system seems more limited than Gene's. Gene treats it as > only one of the "constructing" consonant intervals, and then > somehow "sticks it back in at the end" with some LLL reduction of > something.What distinction are you seeing between us? The octave is mathematically like a chromatic unison vector. Like any other chromatic unison vector, you "stick it back in at the end" to get your linear temperament. LLL reduction has nothing to do with this. Divisions of the period are torsion relating the the equivalence interval as a unison vector. For example, if you define 25:24 as a unison vector in Miracle temperament, you get a decimal scale with torsion. That means, when you stop tempering out the 25:24 you find the approximation to it is always divided into two equal steps (the quommas). If the octave is a chromatic unison vector in twintone, it also gets divided into two equal parts because of torsion. The only thing stopping an octave being a real unison vector is that it isn't like a unison. Graham
Message: 3627 - Contents - Hide Contents Date: Thu, 31 Jan 2002 23:49:55 Subject: Re: interval of equivalence, unison-vector, period From: monz> From: paulerlich <paul@xxxxxxxxxxxxx.xxx> > To: <tuning-math@xxxxxxxxxxx.xxx> > Sent: Thursday, January 31, 2002 9:48 PM > Subject: [tuning-math] Re: interval of equivalence, unison-vector, period > > > --- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:>> --- In tuning-math@y..., "paulerlich" <paul@s...> wrote: >>>>>> If the octave is a chromatic unison vector in twintone, >>>> it also gets divided into two equal parts because of torsion. >>>>>> Warning -- this does not agree with the definition of >>> torsion that Gene was talking about. >>>> If you take a set of unison vectors defining an equal temperament, >> as for instance {50/49, 64/63, 245/243} and now add 2 to the set, >> then {2, 50/49, 64/63, 245/243} generates a kernel K such that N7/K >> = C22--we have a map of the 7-limit to a cyclic group of order 22-- >> This, I think, corresponds to how Graham thinks of things, and how I > _used_ to think of things, before I understood torsion in the period- > is-1/2-or-1/9-or-1/N-octave sense.Paul, you're really good at explaining things. Please elaborate on this until I understand it. :)>> which is a torsion group, since everything has finite order. All >> elements are torsion elements, and we have a finite group, >> And this could happen just as well for a group with a prime number of > elements, such as {2, 25/24, 81/80} -> C7. >>> so this is rather different than a block with torsion elements. >> Yes it is. Now we really need to revise the definition of torsion :(, > and think of different names for these two things.I'm really interested in this difference between "a block with torsion elements" and the finite group where "all elements are torsion elements". I don't recall anyone ever responding to the lattice diagram I made for the torsion definition: Definitions of tuning terms: torsion, (c) 2002... * [with cont.] (Wayb.) I thought that showing the pairs of pitches that are separated by two unison-vector candidates that are smaller than the actual unison-vectors defining the torsional-block might have been saying something significant about what a torsional-block is, or maybe at least something about this particular example. Any thoughts?>> If we take 50/49^64/63, the wedgie for twintone, and wedge it with >> 245/245 we get the 7-limit val h22 of the 22-et, which of course >> defines a temperament. If we wedge the twintone wedgie with 2 >> instead, we also get a val--the mapping to generators of the non- >> octave generator of twintone. >> Can you go into this in more detail, pretty please with sugar on top?And I second that request. The fog has still not cleared about the three items in the subject line. -monz _________________________________________________________ Do You Yahoo!? Get your free @yahoo.com address at Yahoo! Mail - The best web-based email! * [with cont.] (Wayb.)
Message: 3628 - Contents - Hide Contents Date: Thu, 31 Jan 2002 20:45:41 Subject: Re: interval of equivalence, unison-vector, period From: paulerlich --- In tuning-math@y..., graham@m... wrote:> In-Reply-To: <a3am2v+eddv@e...> > paulerlich wrote: >>> The period is often 1/2-octave, 1/3-octave, 1/4-octave, 1/9- >> octave, . . . so that's clearly not a "unison-vector". >> >> The "interval of equivalence" is a unison vector in Graham's system, >> but Graham's system seems more limited than Gene's. Gene treats it as >> only one of the "constructing" consonant intervals, and then >> somehow "sticks it back in at the end" with some LLL reduction of >> something. >> What distinction are you seeing between us? The octave is mathematically > like a chromatic unison vector.Only in your system.> Like any other chromatic unison vector, > you "stick it back in at the end" to get your linear temperament.You're meaning something very different from what I meant by "sticking it back in at the end".> LLL > reduction has nothing to do with this.Take a look again at the way Gene does it. He initially comes out with a scale with two generators, neither of which is the octave. This bears no resemblance to the way you do it.> If the octave is a chromatic unison vector in twintone,How odd to call an octave a chromatic unison vector. When a note is altered by a chromatic unison vector, it is supposed to undergo a small but nonzero change in pitch. In neither the octave-invariant nor the octave-specific case is this true for the octave!> it also gets > divided into two equal parts because of torsion.Warning -- this does not agree with the definition of torsion that Gene was talking about.
Message: 3629 - Contents - Hide Contents Date: Thu, 31 Jan 2002 20:53:39 Subject: Re: new cylindrical meantone lattice From: paulerlich --- In tuning-math@y..., "monz" <joemonz@y...> wrote:>> You're depriving it of much-needed meaning. You'll have to revise >> your definition, then. > >> How so? You were the one who pointed out to me that my xenharmonic > bridge concept, while very similar, was not identical to Fokker's > unison-vector concept.Yes, but at least before, you could still use xenharmonic bridges to imply finity.>>> I'm saying that these two tunings *are* different (altho >>> the auditory system can't hear the difference), so the >>> xenharmonic bridge here *is* allowing the listener to accept >>> 2^(2/10) * 3^(-2/10) * 5^(3/10) to be the same as 2^ (2pi+1)/4pi . >>>> Now you're claiming that our auditory system cares about these >> irrational ratios??? > >> Huh? No, I'm not claiming that. As always, I believe that our > auditory system cares about fairly-low-integer/prime ratios.So what kind of "acceptance" are you talking about above???> I'm stating that LucyTuning is not the same as 3/10-comma > meantone. Sure, it sounds the same.Well, it's _slightly_ different. But then why didn't Harrison> and Lucy simply write about 3/10-comma instead?, which I think would > be a lot easier to understand mathematically.Because they liked pi!> There is a difference, and there's a little tiny xenharmonic bridge > in effect which blurs that difference and allows us to accept them > as being exactly the same. *This* kind of fudging and blurring > is what I originally was trying to express with the xenharmonic bridge > concept.But it doesn't induce *finity* unless the fudging is between two different constructions from the same set of basis intervals (usually primes).>>> Am I missing your point simply because pi is a number that cannot >>> be finitely quantized? >> >> No. > >> So then please try to elaborate more ... I still don't get > what you're saying.You claimed this whole xenharmonic bridge stuff was relevant to finity, and I'm arguing that here, it's not.>>>> ... -- why not let the meantones look pretty much the >>>> same -- compositionally, they pretty much are, wouldn't you say? >>> >>>>>> Yes, I think I can agree with that. >>> >>> But then again, why would a composer choose one particular >>> meantone over any other? >>>> It would depend on the criteria they chose to determine their >> meantone. Are they concentrating on the fifths? Do they care about >> the thirds only? Do they minimize maximum error, or total error? See >> my table in your meantone definition page for examples of what >> meantones different desiderata can lead to. > >> Well there you go: there's the answer as to why the spirals > are useful! > > By seeing how a particular meantone spirals around the > cylindrical lattice -- which still has the JI points and > distances marked on it, albeit warped a bit so as to fit around > the circle -- one can see which JI intervals it favors. > > > Thus, it can be seen from my 1/4-comma lattice at the bottom of > Internet Express - Quality, Affordable Dial Up... * [with cont.] (Wayb.) > > that 1/4-comma meantone gives all the "major 3rds" and "minor 6ths" > exactly. > > > If you refer to the dotted lines on these graphs, > Internet Express - Quality, Affordable Dial Up... * [with cont.] (Wayb.) > > which show the fraction-of-a-comma tempering of each meantone note, > and imagine that these lattices are wrapped around a cylinder, > then for the "tonic" chord: > > - it could be seen from the spiral of 1/3-comma meantone that > it gives the "minor 3rd" and "major 6th" exactly; > > - it could be seen from the 2/7-comma spiral that both the "major" > and "minor" "3rd" and "6th" all have an equal amount of error. > > - 1/5-comma favors the "5ths/4ths" and "major 3rd / minor 6th" > (which have the same error) over the "minor 3rd / major 6th"; > > Etc.So the _angle_ is meaningful. But I still have problems with (a) the density of points along the line, which doesn't appear to be meaningful; (b) the fact that you have to pin the spiral to a particular "1/1" origin, which ruins the rotational symmetry of the cylindrical meantone lattice (c) the fact that you can't yet plot non-rational-fraction-of-a-comma meantones this way, though the _angle_ part should be just as meaningful for those.
Message: 3630 - Contents - Hide Contents Date: Thu, 31 Jan 2002 04:58:05 Subject: Re: Approximate consonances of Parch's 43 tone scale From: dkeenanuqnetau --- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:> I searched for all instances where one tone of this scale differedfrom another by an amount within 5 cents of an 11-limit consonance; this led to four commas: 385/384, 8019/8000, 441/440, and 540/539.>This is fascinating. How far out does the error have to go before another comma appears? How far in can you come before one of those commas disappears and which one is it?> These commas are linearly independent, and define an equal temperament(as well as a PB, incidentally.) The et they define is> (drum roll please) 72 et.Neato! But that's dependent on your choice of allowable error?> The linear temperaments obtained by leaving out one of the commas were > <385/384,441/440,540/539>, miracle; <540/539,8019/8000,385/384>, > catakleismic;What's the generator and period of catakleismic?> and two unnamed temperaments with half-octave period: > > #5 on my list, <540/539,441/440,8019/8000> with wedgie > [12,34,20,30,26,-2,6,-49,-48,15] and generators a=4.9919/72, b=1/2;An 83.2 cent generator, half-octave period.> <441/440,8019/8000,385/384> with wedgie > [12,22,-4,-6,7,-40,-51,-71-90,-3] and generators a=10.9910/72, b=1/2.An 183.2 cent generator, half-octave period How many notes in contiguous, (equal-length?) chains of generators does each of these need to encompass Partch's 'Genesis' scale?
Message: 3631 - Contents - Hide Contents Date: Thu, 31 Jan 2002 20:57:32 Subject: Re: new cylindrical meantone lattice From: paulerlich --- In tuning-math@y..., "monz" <joemonz@y...> wrote:> Yes, he did the hexagonal ones. Oh, good. > Does he have more articles that I'd find of interest?Sure -- also his book, _Musical Acoustics_.> What I meant was: perhaps you could make a quick-and-dirty graphic > illustrating that "twig" idea. That was what I found interesting.Use your imagination!>> Because the *spiral* will be *pinned* to 1/1 no matter how we roll >> the cylinder, correct? > >> Doesn't have to be. It could be shifted along either axis by any > fraction of 3 or 5 or both that you'd like. It doesn't affect the > *relative* relationships between the meantone and the JI,Well, then, I think it should be *optional*, because for me, it's more relevant to see the relationships between the notes on the rolled lattice, without any differentiation.> only the > specific relationships between specific pairs of notes.How do you mean?
Message: 3632 - Contents - Hide Contents Date: Thu, 31 Jan 2002 21:02:50 Subject: Fwd: Re: interval of equivalence, unison-vector, periodd From: paulerlich --- In tuning-math@y..., "monz" <joemonz@y...> wrote:>>> From: paulerlich <paul@s...> >> To: <tuning-math@y...> >> Sent: Wednesday, January 30, 2002 9:52 PM >> Subject: [tuning-math] Fwd: Re: interval of equivalence, unison- vector, > period >> >> >>>>>> I'm having a really hard time understanding the differences >>> between "interval of equivalence", "period", and "unison- vector". >>> >>> Why aren't they *all* unison-vectors? >>>> The period is often 1/2-octave, 1/3-octave, 1/4-octave, 1/9- >> octave, . . . so that's clearly not a "unison-vector". > >> But what *is* the period? I mean, not what interval or size, > but what is it? What significance does it have?It's the smallest interval at which a scale can be transposed without changing the scale at all. For example, the diminished (octatonic) scale in 12-tET has a period of 1/4-octave. For another, my symmetrical decatonic scale in 22-tET has a period of 1/2-octave.>>> And recall that the interval of equivalence is usually a _large_ >>> interval, usually an octave, so not really much like a _unison_ >at >>> all! > >>> But ... but ... > >> Say the syntonic comma is a unison-vector. So pick a reference note; >> the note a comma away is considered equivalent. But on an >> "8ve"-equivalent lattice, the note *an "8ve" and a comma away >> (~1222 cents) is also considered equivalent*!! So then why >> not the "8ve" itself?On an octave-equivalent lattice, yes, it would be considered equivalent. That's not enough to make it a unison vector, though. The way Gene does things, unison vectors are all _small intervals_ defined with specific ratios, for example 81:80 but not 81:40, and then Gene can construct temperaments or whatever, and then octave- equivalence can be stuck back in at the end, if desired. If you don't do it this way, you won't be able to deal with torsion properly.
Message: 3633 - Contents - Hide Contents Date: Thu, 31 Jan 2002 21:05:37 Subject: Re: new cylindrical meantone lattice From: paulerlich --- In tuning-math@y..., "monz" <joemonz@y...> wrote:>>> If one keeps the JI pitches in place, and moves the bounding >>> unison-vectors and the meantone 1/2-step to the right along >>> the 3-axis, so that the boundaries enclose only 12 >>> pitches (the minimal set for this pair of unison-vectors) >>> and the meantone is exactly centered and symmetrical to those >>> 12 pitches, the entire system is centered and symmetrical >>> around the ratio 3^(1/2), or in terms of "8ve"-equivalent >>> ratios, the square-root of 3/2. That's great. >>> In fact this structure more accurately portrays what the >>> meantone really represents:No way, Jose.>>> because of the disappearance >>> of the syntonic comma unison-vector, this flat lattice >>> should be imagined to wrap around as a cylinder, so that >>> the right and left edges connect. Thus the centered >>> meantone may imply either of any pair of pitches which >>> would be separated by a comma on the flat lattice, and >>> each of those pairs of points on the flat lattice map to >>> the same point on a cylinder.This is true also for an infinite number of pitches on the flat lattice that are even further from the line. So the centering of the meantone line between 1/1 and 3/2 is irrelevant. So is the use of 2/9- comma meantone as opposed to any other meantone, as far as I can tell.
Message: 3634 - Contents - Hide Contents Date: Thu, 31 Jan 2002 05:13:18 Subject: Re: ET that does adaptive-JI? From: paulerlich --- In tuning-math@y..., "monz" <joemonz@y...> wrote:>>> From: paulerlich <paul@s...> >> To: <tuning-math@y...> >> Sent: Wednesday, January 30, 2002 8:08 PM >> Subject: [tuning-math] Re: ET that does adaptive-JI? >> >> >> I have offered 152-tET as a Universal Tuning -- one reason for this >> is that it supports the wonderful adaptive JI system of two (or >> three, or more, if necessary) 1/3-comma meantone chains, tuned 1/3- >> comma apart. This gives you 5-limit adaptive JI with no drift >> problems, and the pitch shifts reduced to normally imperceptible >> levels. 1/152 oct. ~= 1/150 oct. = 8 cents. > >> Would those be equidistant chains of 1/3-comma MT?Yes, equidistant at intervals of about 1/3-comma -- 1/152 oct. ~= 1/150 oct. = 8 cents.> Or is > there some special interval between chains?Yes, about 1/3-comma -- 1/152 oct. ~= 1/150 oct. = 8 cents.> Does it also > work equivalently as chains of 19-EDO, since that's so > close to 1/3-comma MT?Right. Just like Vicentino's second tuning, where two 31-tET chains 1/4-comma apart could do all its tricks really well, here two (or three, if you need certain chords besides major and minor triads) 19- tET chains 1/3-comma apart do those same kinds of tricks.
Message: 3635 - Contents - Hide Contents Date: Thu, 31 Jan 2002 21:07:59 Subject: Re: 152-EDO as adaptive-JI (was: IM conversation with Monz) From: paulerlich --- In tuning-math@y..., "monz" <joemonz@y...> wrote:> Ah ... 152-EDO gets more and more interesting the more I look > at it! > > So, since 19-EDO is a meantone, the lattice wraps into a cylinder. > > And since 1 step of 152-EDO is ~1/3-comma, you only need *3* > chains of 19-EDO each separated by 1 step of 152-EDO, in order > to represent the entire infinite JI lattice! And you don't > get comma problems! *WAY* cool !!!!!!Umm, something slipped up somewhere. Each 19-tET chain itself represents the entire JI lattice already. But if you want larger and larger chords in JI, you need more and more of the 19-tET chains. Nothing special happens when you have 3 chains.> So the 57-tone subset of 152-EDO which is three 19-EDOs starting > on the 1st, 2nd, and 3rd 152-EDO degrees is really some sort of > magical tuning for us adaptive-JI fans!Nope. You need as many 19-tET chains as the piece dictates.
Message: 3636 - Contents - Hide Contents Date: Thu, 31 Jan 2002 05:15:33 Subject: Re: ET that does adaptive-JI? From: paulerlich --- In tuning-math@y..., "monz" <joemonz@y...> wrote:> Hmmm ... can you elaborate more on that bit about "exploiting > their inconsistency"? I don't quite get that.Well, in most of the systems I described, you're using the second- best or even third-best approximations to some consonant intervals, relative to what 152-tET affords.> Anyway, 152-EDO seems like a fantastic tuning to play around with! > Why have I read nothing about it before?Probably because such large tunings don't get mentioned much.
Message: 3637 - Contents - Hide Contents Date: Thu, 31 Jan 2002 21:11:16 Subject: Re: 217-EDO as adaptive-JI (was: 152-EDO as adaptive-JI) From: paulerlich --- In tuning-math@y..., "monz" <joemonz@y...> wrote:.> > So the 124-tone subset of 217-EDO which is four 31-EDOs starting > on the first four 217-EDO degrees also covers the whole meantone > lattice.Again, your reasoning is slipping up somewhere. Each 31-tET covers the whole meantone lattice, and the whole JI lattice. But depending on the complexity of the chords you're trying to tune in JI, you may need only two, or three, or perhaps five or six of the 31-EDOs. But Monz, remember that meantone tuning has some chords (like CEGAD) that simply _can't_ be rendered nicely in JI. So I would put good odds that Mahler would work out better in straight meantone than in 152-tET or 217-tET.
Message: 3638 - Contents - Hide Contents Date: Thu, 31 Jan 2002 05:23:52 Subject: Re: Approximate consonances of Parch's 43 tone scale From: paulerlich --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:> --- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:>> I searched for all instances where one tone of this scale differed> from another by an amount within 5 cents of an 11-limit consonance; > this led to four commas: 385/384, 8019/8000, 441/440, and 540/539. >> > > This is fascinating.Where was this posted originally?> How far out does the error have to go before another comma appears? > How far in can you come before one of those commas disappears and > which one is it? >>> These commas are linearly independent, and define an equal > temperament> (as well as a PB, incidentally.) The et they define is>> (drum roll please) 72 et. >> Neato! But that's dependent on your choice of allowable error?But of course. The commas are, in cents, 4.5026 cents, 4.1068 cents, 3.9302 cents, and 3.209. All about the same -- but you could draw a line between them, I guess.>> The linear temperaments obtained by leaving out one of the commas > were>> <385/384,441/440,540/539>, miracle; <540/539,8019/8000,385/384>, >> catakleismic; >> What's the generator and period of catakleismic? >>> and two unnamed temperaments with half-octave period: >> >> #5 on my list, <540/539,441/440,8019/8000> with wedgie >> [12,34,20,30,26,-2,6,-49,-48,15] and generators a=4.9919/72, b=1/2; >> An 83.2 cent generator, half-octave period. >>> <441/440,8019/8000,385/384> with wedgie >> [12,22,-4,-6,7,-40,-51,-71-90,-3] and generators a=10.9910/72, > b=1/2. >> An 183.2 cent generator, half-octave period > > How many notes in contiguous, (equal-length?) chains of generators > does each of these need to encompass Partch's 'Genesis' scale?What's the answer for MIRACLE? Wasn't it an non-'Genesis' 43-tone scale that MIRACLE comprised in 45 consecutive notes in a chain of generators? This is really interesting, as it makes one wonder, to what extend was Secor's original proposal "unique" or "best".
Message: 3639 - Contents - Hide Contents Date: Thu, 31 Jan 2002 21:12:37 Subject: Re: new cylindrical meantone lattice From: paulerlich --- In tuning-math@y..., "monz" <joemonz@y...> wrote:> > Paul, *you* were the one who kept nagging me about "... but how > are you going to lattice LucyTuning or Golden Meantone on this > kind of lattice?". > > So if your point is that LucyTuning and 3/10-comma meantone > are "functionally identical", then I can simply lattice > LucyTuning *as* 3/10-comma meantone and call it a day.Not quite -- it would have to be *very slightly different*.> > > > Now, about that bit where I wrote: "... finding some way to > represent pi in a universe where everything is factored by > 3 and 5", I had an idea for a simpler beginning approach. > > Let's start with meantone-like EDOs instead. As an example, > we want to lattice 1/3-comma meantone alongside 19-EDO. > > Can you devise some formula that would find the fractional > powers of 3 and 5 that would be needed to plot 19-EDO on a > trajectory that would follow closely alongside the 1/3-comma > trajectory? Now, I think *that* would be a meaningful lattice!OK, I'm in favor of thinking in this direction, as it addresses at least one of the three objections I raised in a post to you a few minutes ago.
Message: 3640 - Contents - Hide Contents Date: Thu, 31 Jan 2002 05:33:26 Subject: Re: new cylindrical meantone lattice From: paulerlich --- In tuning-math@y..., "monz" <joemonz@y...> wrote:> > OK, in case there are other subtle distinctions between xenharmonic > bridges and unison-vectors, I'm going to stick with my terminology.You're depriving it of much-needed meaning. You'll have to revise your definition, then.> I'm saying that these two tunings *are* different (altho the auditory > system can't hear the difference), so the xenharmonic bridge here > *is* allowing the listener to accept 2^(2/10) * 3^(-2/10) * 5^(3/10) > to be the same as 2^(2pi+1)/4pi .Now you're claiming that our auditory system cares about these irrational ratios???> Am I missing your point simply because pi is a number that cannot > be finitely quantized? No.>>> If I could find some way to represent LucyTuning on the flat >>> lattice (which means finding some way to represent pi in a >>> universe where everything is factored by 3 and 5), then bend >>> the lattice into a meantone cylinder, then warp the cylinder >>> so that the LucyTuning "5th" occupies the same point as the >>> 3/10-comma meantone "5th", I'd have it. >>> >>> The fact that pi is transcendental, irrational, whatever, >>> makes it hard for me to figure out how to do this. >>>> If you want people to help you figure this out, you'll have to >> determine what the above construction is all about. What does it >> mean? Why do we want to see it on our perfectly good lattices- wrapped- >> onto-cylinders? > >> Sheesh, I don't know! I was hoping *you* could help me figure > that out! (OK, good questions from you *do* do that.) > >>> P.S. There are so many wonderful varieties of cylindrical tunings >> your JustMusic software could be helping people visualize -- they'll >> all look different -- >> Wow, Paul, I'm amazed that you wrote this just now. > > Another project I've been working on for a couple of months > is a MIDI-file of Beethoven's "Moonlight" Sonata, tuned in > Kirnberger III well-temperament. I chose Kirnberger simply > because it's quite likely to have been a tuning that Beethoven's > piano tuner might have used, and because it has a simplicity > and elegance (in terms of portraying it on my lattice) that > make it easy to lattice, where other WTs are more complicated. > > Anyway, one of the things that I found fascinating about > Kirnberger III (and this probably applies to many other WTs > as well ... haven't looked yet) is that the ends of the > tuning chain invoke the skhisma, which means in effect > (assuming the ~2-cent difference falls outside the capability > of a human tuner in Beethoven's day), that it's a closed tuning. > Thus, a nice cylinder, perpendicular to the skhisma, which is > *extremely* different from the meantone cylinder.Schismic tuning, gets lots of discussion, usually with reference to 17-tone, 29-tone, 41-tone, 53-tone, and similar MOSs of often 1/8- to 1/9-schisma tempered fifths. But once you've created a finite 12-tone periodicity block, such as a well-temperament, you can no longer attribute much importance to the schisma, which is one of only an infinite number of unison vectors which, in pairs, can produce the 12- tone periodicity block (or a torsional "multiple" if you're not careful).>> ... -- why not let the meantones look pretty much the >> same -- compositionally, they pretty much are, wouldn't you say? > >> Yes, I think I can agree with that. > > But then again, why would a composer choose one particular > meantone over any other?It would depend on the criteria they chose to determine their meantone. Are they concentrating on the fifths? Do they care about the thirds only? Do they minimize maximum error, or total error? See my table in your meantone definition page for examples of what meantones different desiderata can lead to.> There must be *some* reason/s > ... so what's wrong with making a visual model of those choices > too? > > Perhaps there are deeper things about the differences between > different meantones that we haven't noticed yet. Having nice > pictures of them would make it easier to find those as-yet > undiscovered aspects, I think.I could come up with lots of nice mathematical formulae for differentiating them, but if none of them have any meaning that we can understand, how can we choose one over another?
Message: 3641 - Contents - Hide Contents Date: Thu, 31 Jan 2002 23:58:45 Subject: Re: 217-EDO as adaptive-JI (was: 152-EDO as adaptive-JI) From: paulerlich --- In tuning-math@y..., "monz" <joemonz@y...> wrote:> And so, likewise, 217-EDO can perform this function for 31-EDO. > > 217 = 31 * 7. So 217-EDO is like 7 bicycle chains of 31-EDO. > 1 step of 217-EDO is ~1/4-comma. 1/4-comma meantone, represented > by 31-EDO, bends the JI lattice into a cylinder in which 4 chains > of the meantone can cover the whole lattice. > > So the 124-tone subset of 217-EDO which is four 31-EDOs starting > on the first four 217-EDO degrees also covers the whole meantone > lattice.Again, your reasoning is slipping up somewhere. Each 31-tET covers the whole meantone lattice, and the whole JI lattice. But depending on the complexity of the chords you're trying to tune in JI, you may need only two, or three, or perhaps five or six of the 31-EDOs.
Message: 3642 - Contents - Hide Contents Date: Thu, 31 Jan 2002 05:35:48 Subject: Re: new cylindrical meantone lattice From: paulerlich --- In tuning-math@y..., "monz" <joemonz@y...> wrote:>> From: paulerlich <paul@s...> >> To: <tuning-math@y...> >> Sent: Wednesday, January 30, 2002 8:24 PM >> Subject: [tuning-math] Re: new cylindrical meantone lattice >> >>>>>> Why not show mistuning as a tiny "break" in the consonant >>>> connections? >>> >>>>>> Hmm ... sounds interesting. You mean like on the recent Blackjack >>> lattice you posted? >>>> Well, sort of. Except that all the consonant intervals would be >> broken. >> >>> Please elaborate. >>>> Well, for example, you could do it like in Hall's hexagonal lattices, >> where he puts a little number representing the mistuning of each >> interval, but you could do it visually, like representing each >> consonant interval as a twig and putting a physical "break", the size >> of the mistuning, into it. > >> Hmmm ... this sounds *really* interesting! I'd like to see > more on this from you and the others who understand how to do it.Seems simple enough. What more do you want? Did Hall do the hexagonal ones in the article I sent you? Do I need to send you another Hall article?>>> 3) Use the mouse to roll the cylinder around on the lattice >>> to get whichever key-center is desired, or simply input the >>> key and let JustMusic do the rolling. >>>> Well this I was expecting anyway. > >> Really? Hmmm ... if you were expecting it, then why'd you ask > me the bit about choosing "a 1/1, such as C, which flies in the > face of the true nature of meantone as a transposible system" ? > > If the user can transpose/roll the cylinder at will, what > difference does it make which note is chosen as 1/1 ?Because the *spiral* will be *pinned* to 1/1 no matter how we roll the cylinder, correct?
Message: 3643 - Contents - Hide Contents Date: Thu, 31 Jan 2002 05:45:37 Subject: IM conversation with Monz From: paulerlich paulerlich: It can only work its magic as adaptive JI if the separation between the two chains is either 1/4-comma for 1/4-comma meantone, or 1/3-comma for 1/3-comma meantone. No other combination of system and separation would work. Monz: Hmmm ... can you quote some of this IM to tuning-math with a nice elaboration on how this works. I don't quite see it. paulerlich: Simple: paulerlich: in 1/4-comma meantone, the major thirds are just, and the fifth and minor third are 1/4-comma flat -- so you simply go to the 1/4-comma meantone chain 1/4-comma away to get your just minor third and just fifth. paulerlich: in 1/3-comma meantone, the minor thirds are just, and the fifth and major third are 1/3-comma flat -- so you simply go to the 1/3-comma meantone chain 1/3-comma away to get your just major third and just fifth.
Message: 3644 - Contents - Hide Contents Date: Fri, 01 Feb 2002 09:01:37 Subject: Fwd: Re: interval of equivalence, unison-vector, periodd From: paulerlich --- In tuning-math@y..., "monz" <joemonz@y...> wrote:>>> From: paulerlich <paul@s...> >> To: <tuning-math@y...> >> Sent: Thursday, January 31, 2002 1:02 PM >> Re: interval of equivalence, unison-vector, period >> >>>>> But what *is* the period? I mean, not what interval >>> or size, but what is it? What significance does it have? >>>> It's the smallest interval at which a scale can be >> transposed without changing the scale at all. For example, >> the diminished (octatonic) scale in 12-tET has a period >> of 1/4-octave. For another, my symmetrical decatonic scale >> in 22-tET has a period of 1/2-octave. > >> Ah ... OK, I can grasp that. > > But then what makes the "interval of equivalence" different > from that?Well, the interval of equivalence is what you explicitly decide you want to treat as a kind of "generator" in your scale, in that the scale will automatically repeat every interval of equivalence because we'll just be hearing the "same" scale again, only higher or lower in pitch. The period, however, comes in at 1/N octaves, where N is an integer (usually 1, but not always), just because of the way the unison vectors work out.>> The way Gene does things, unison vectors are all >> _small intervals_ defined with specific ratios, for >> example 81:80 but not 81:40, and then Gene can construct >> temperaments or whatever, and then octave-equivalence >> can be stuck back in at the end, if desired. If you don't >> do it this way, you won't be able to deal with torsion >> properly. > >> Can you explain why not?Recall that a particular unison vector (or product of unison vectors, etc.) is candidate for torsion if it's a power (square, cube, etc.) of some other interval. Let's say you don't keep track of the factors of 2 making up the unison vectors. Now let's say you notice that a particular unison vector (or product of unison vectors, etc.) has all its prime-factorization exponents as multiples of N. Then it appears to be an Nth power of some unison vector, right? Well, not necessarily. If the power of 2 that you threw away was also a multiple of N, then you're fine -- the Nth root of the small interval is some even smaller interval. But if it wasn't, then you're really taking the Nth root of something close to an octave, or to two octaves, etc. . . . which may not be a small interval at all! For example: 6561:6400 = 2^-8 * 3^8 * 5^-2 This is the square of 81:80 = 2^-4 * 3^4 * 5^-4 So any periodicity block where is 6561:6400 is a unison vector, or the product of the unison vectors, etc., will be torsional. HOWEVER: 50:49 = 2^1 * 5^2 * 7^-2 What if we ignore the factors of 2? 50:49 "=" 5^2 * 7^-2 This is the square of 5^1 * 7^-1 which is a tritone, or tritone plus octave, or . . . etc., depending on how many factors of two you put back in. But a tritone is no kind of unison vector! Instead, it (as 1/2 octave) becomes the _period_ for any system involving the 50:49 unison vector. Catchin' on?
Message: 3645 - Contents - Hide Contents Date: Fri, 1 Feb 2002 12:58 +00 Subject: Re: 152-EDO as adaptive-JI From: graham@xxxxxxxxxx.xx.xx In-Reply-To: <a3cbnf+5q2u@xxxxxxx.xxx> Monz:>> And since 1 step of 152-EDO is ~1/3-comma, you only need *3* >> chains of 19-EDO each separated by 1 step of 152-EDO, in order >> to represent the entire infinite JI lattice! And you don't >> get comma problems! *WAY* cool !!!!!! Paul:> Umm, something slipped up somewhere. Each 19-tET chain itself > represents the entire JI lattice already. But if you want larger and > larger chords in JI, you need more and more of the 19-tET chains. > Nothing special happens when you have 3 chains.One period/generator mapping is [(19, 0), (30, 1), (44, 1), (53, 3), (66, -2)] This is accurate to 1.7 cents of 11-limit JI. You need 5 non-consecutive chains to carry one 11-limit otonality. But that means you get 19 of them into the bargain, and you can do infinite modulations along 19-equal, but not an infinite lattice. You do need all 6 chains for that. Another mapping is [(19, 0), (30, 1), (44, 1), (53, 3), (65, 6)] Tuned to 152-equal, that's 2.2 cents out, but it does mean the 6 should map to 0. It now means the 11-limit otonality can be got from 4 consecutive chains, but isn't so close to JI.>> So the 57-tone subset of 152-EDO which is three 19-EDOs starting >> on the 1st, 2nd, and 3rd 152-EDO degrees is really some sort of >> magical tuning for us adaptive-JI fans! >> Nope. You need as many 19-tET chains as the piece dictates.You don't need more than 6 to cover 152-EDO. The CNMI connected to London Guildhall University have built a trumpet for 19-equal. I was told that it can handle 7-limit JI fairly easily by pitch bending, and the method is to think of multiple chains of third-comma meantone. That looks to me pretty much like what you're doing, but in the 11-limit. So this might all be more practical than you realized. Graham
Message: 3646 - Contents - Hide Contents Date: Fri, 01 Feb 2002 03:42:53 Subject: Re: any ideas? From: paulerlich --- In tuning-math@y..., "D.Stearns" <STEARNS@C...> wrote:> This is a question that keeps coming back and interesting me from time > to time, and it's one that I've never come up with a nice answer for > either--maybe someone here can? > > If you take any random scale, what type of a method could be applied > to the rotations so as to result in some pleasing "best" rotation? > > By "pleasing" I mean elegant in the math or aesthetic sense, and by > "best" I could mean just about anything depending upon how "pleasing" > is defined! > > Anyway, any ideas?I like to leave considerations of "rotation" and "choosing a tonic" free to be informed by the particular style in which a scale is to be used. For Indian-type styles, I tend to center around whatever is the closest thing I can find to a central 1/1-3/2 dyad. But it varies. Also, the question of "rotation" is somewhat ambiguous, as distinct "octave species", "finalis", "dominant", and other aspects of this question could be asked independently.
Message: 3647 - Contents - Hide Contents Date: Fri, 01 Feb 2002 09:03:34 Subject: Re: new cylindrical meantone lattice From: paulerlich --- In tuning-math@y..., "monz" <joemonz@y...> wrote:> I don't know about that, Paul. My intuition tells me that > if composers choose particular flavors of meantone based on > criteria such as the amount-of-error-from-JI relationships > among the basic consonant intervals (M3/m6, m3/M6, p4/p5), > then they *do* intend to emphasize/deemphasize specific > JI intervals, because the meantone they choose will do that. > > The centering of the meantone line -- which I prefer to call > a spiral since it belongs on a cylindrical lattice -- within > the PB, distributes the amount of error from JI as evenly as > possible among the intervals closest to the 1/1. That seems > to me to be something with an actual musical application.So you're deriving 2/9-comma meantone as an optimal meantone? What weights are you using to do so? I showed the meantones that I could easily derive from various weightings of the basic consonant intervals (M3/m6, m3/M6, p4/p5), and 2/9-comma wasn't one of them . . .
Message: 3648 - Contents - Hide Contents Date: Fri, 1 Feb 2002 12:58 +00 Subject: Re: interval of equivalence, unison-vector, period From: graham@xxxxxxxxxx.xx.xx In-Reply-To: <a3dlmu+4ceg@xxxxxxx.xxx> genewardsmith wrote:> Vals are the dual concept to intervals. We have prime number intervals, > such as octave or twelvth, and the dual to those are the p-adic > valuations vp, which count the number of powers of p (positive or > negative) in the prime factorization of a rational number. An interval > is a finite Z-linear combination of primes; that is, it is p1^e1 * > p2^e2 ... pk^ek for certain primes pn and certain integers en. A val is > a finite Z-linear combination of p-adic valuations: e1 v1 + e2 v2 + ... > + ek vk. Dual to the comma, or small interval is, more or less, an et > val. Another type of val of interest are the maps of generators to > primes.Does this mean "dual" is the proper word for what I'm calling the "complement"? Graham
Message: 3649 - Contents - Hide Contents Date: Fri, 01 Feb 2002 09:04:30 Subject: Re: interval of equivalence, unison-vector, period From: genewardsmith --- In tuning-math@y..., "paulerlich" <paul@s...> wrote:>> 50/49^245/243 = [6,10,10,-5,1,2] with generators >> a = 3.0143/22 = 164.4176 cents and b = 1/2 > > GlassicGood name--where does it come from?>> 64/63^245/243 = [1,9,-2,-30,6,12] with generators a = 8.9763/22 = >> 489.6152 cents and b = 1 >> "Big fifth" -- a unique facet of 22Not really; I would say it is even more characteristic of 27 or 49.>> A triple wedge product of three intervals will be a val, but it >> doesn't have to be an equal temperament val.> What other kinds are there?Vals are the dual concept to intervals. We have prime number intervals, such as octave or twelvth, and the dual to those are the p-adic valuations vp, which count the number of powers of p (positive or negative) in the prime factorization of a rational number. An interval is a finite Z-linear combination of primes; that is, it is p1^e1 * p2^e2 ... pk^ek for certain primes pn and certain integers en. A val is a finite Z-linear combination of p-adic valuations: e1 v1 + e2 v2 + ... + ek vk. Dual to the comma, or small interval is, more or less, an et val. Another type of val of interest are the maps of generators to primes.>> Mapping 2 to 1, and both 5 and 7 to 1/9 does not strike me as much> of a temperament. > > Well . . . I'm lost . . . does this have anything to do with what you > were once showing about your process, where for a "linear" or 2D > temperament, you started off with two generators, but then found a > different generator basis pair where you forced one member to be an > octave?Right--this val would be the starting point for that process, not a temperament. I can do the same sort of thing starting from [-2,-2,-7,-8], where I end up with [-2 2] [-2 3] [-7 5] [-8 6] as a mapping from generators to primes; here "b" is a wide fifth and "a" is a tritone below that.
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