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Message: 3575 - Contents - Hide Contents Date: Tue, 29 Jan 2002 07:33:35 Subject: Re: question for Gene From: paulerlich I wrote,> without pushing the badness over the limit > you've computed?I meant, over the limit you've adopted (500 I think it was . . .)?

Message: 3576 - Contents - Hide Contents Date: Tue, 29 Jan 2002 22:30:46 Subject: Re: new cylindrical meantone lattice From: paulerlich --- In tuning-math@y..., "monz" <joemonz@y...> wrote:>> And what is it for meantones like 31-tET >> or 69-tET or LucyTuning? Does it still exist? > >> Well ... the lattice I'm using here is "8ve-equivalent", > so I can't put any EDOs on them.31-tET or 69-tET or LucyTuning are no more or less "8ve-equivalent" than fraction-of-a-comma meantones. So I'm not sure what you meant my this.> But I'm not sure how well it works, because in my lattice > formula, 2 is the smallest step in ratio-space, and so > the entire 69-EDO, for example, would be crammed into a > space smaller than that which separates 1:1 from 3:2, > making it hard to give any real visual relevance.If this whole business really had any acoustical meaning, wouldn't the 69-tET line simply be somewhere between the 1/3-comma meantone line and the 2/7-comma meantone line?> > But of course I could easily change the "step sizes" of > my lattice metrics too. Could even reverse it, and make > 2 the *largest* step. Hmmm.... I'm thinking that that > might be a really useful idea... then it'd be easy to show > EDOs. > > > As for LucyTuning, I don't know ... other than the fact > that it's audibly indistinguishable from 3/10-comma MT, > which I could easily plot. But as far as actually putting > pi on my lattice, I don't have a clue.I'm hoping this sort of consideration will give you a clue that what you're plotting isn't actually meaningful.

Message: 3577 - Contents - Hide Contents Date: Tue, 29 Jan 2002 08:06:19 Subject: Re: question for Gene From: genewardsmith --- In tuning-math@y..., "paulerlich" <paul@s...> wrote:> Is there any way to directly compare the badnesses of equal > temperaments and linear temperaments and meaningfully ask the > question: Which of the linear temperaments that you found (in the 5- > limit, and whatever other cases you've completed) could be expressed > by an equal temperament, without pushing the badness over the limit > you've computed?Certainly--just recalculate rms error for the new tuning. Complexity will never increase, and if you allow it to decrease (as for instance in the 12-et version of schismic) you recalculate that also, by reducing it mod n, I suppose.

Message: 3578 - Contents - Hide Contents Date: Tue, 29 Jan 2002 14:46:50 Subject: Re: new cylindrical meantone lattice From: monz> From: paulerlich <paul@xxxxxxxxxxxxx.xxx> > To: <tuning-math@xxxxxxxxxxx.xxx> > Sent: Tuesday, January 29, 2002 2:30 PM > Subject: [tuning-math] Re: new cylindrical meantone lattice > > > --- In tuning-math@y..., "monz" <joemonz@y...> wrote: >>>> And what is it for meantones like 31-tET >>> or 69-tET or LucyTuning? Does it still exist? >> >>>> Well ... the lattice I'm using here is "8ve-equivalent", >> so I can't put any EDOs on them. >> 31-tET or 69-tET or LucyTuning are no more or less "8ve-equivalent" > than fraction-of-a-comma meantones. So I'm not sure what you meant my > this.I simply meant that I don't include 2 as part of the calculation on these lattices, thus I can't graph EDOs (= 2^x).>> But I'm not sure how well it works, because in my lattice >> formula, 2 is the smallest step in ratio-space, and so >> the entire 69-EDO, for example, would be crammed into a >> space smaller than that which separates 1:1 from 3:2, >> making it hard to give any real visual relevance. >> If this whole business really had any acoustical meaning, wouldn't > the 69-tET line simply be somewhere between the 1/3-comma meantone > line and the 2/7-comma meantone line? > >>>> As for LucyTuning, I don't know ... other than the fact >> that it's audibly indistinguishable from 3/10-comma MT, >> which I could easily plot. But as far as actually putting >> pi on my lattice, I don't have a clue. >> I'm hoping this sort of consideration will give you a clue that what > you're plotting isn't actually meaningful.Well ... *I'm* hoping that I can continue to find ways to "warp" my lattices so that these equivalent mathematics *do* correspond visually! Keep helping me! :) -monz _________________________________________________________ Do You Yahoo!? Get your free @yahoo.com address at Yahoo! Mail - The best web-based email! * [with cont.] (Wayb.)

Message: 3579 - Contents - Hide Contents Date: Tue, 29 Jan 2002 08:09:03 Subject: Re: question for Gene From: paulerlich --- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:> --- In tuning-math@y..., "paulerlich" <paul@s...> wrote: >>> Is there any way to directly compare the badnesses of equal >> temperaments and linear temperaments and meaningfully ask the >> question: Which of the linear temperaments that you found (in the 5- >> limit, and whatever other cases you've completed) could be expressed >> by an equal temperament, without pushing the badness over the limit >> you've computed? >> Certainly--just recalculate rms error for the new tuning.Could you do this please? Which of the twenty (?) linear temperaments that you found could thus be expressed?>Complexity will never increase,Complexity of a larger ET must be more than of a smaller ET, otherwise my question makes no sense, as you could always find an infinite number of ETs that pass.

Message: 3580 - Contents - Hide Contents Date: Tue, 29 Jan 2002 14:48:21 Subject: Re: new cylindrical meantone lattice From: monz In my last post, I wrote:> Well ... *I'm* hoping that I can continue to find ways to > "warp" my lattices so that these equivalent mathematics > *do* correspond visually! Keep helping me! :)I should have said "acoustically equivalent mathematics". -monz _________________________________________________________ Do You Yahoo!? Get your free @yahoo.com address at Yahoo! Mail - The best web-based email! * [with cont.] (Wayb.)

Message: 3582 - Contents - Hide Contents Date: Tue, 29 Jan 2002 23:47:28 Subject: Re: twintone, paultone From: clumma>> >he "problem" occurs when modulating from the best approx. of >> one chord to the best approx. of another, and thereby creating >> anomalous (as in, non-existent in JI) commas. >>That won't happen if you confine yourself to a regular >temperamemt, such as the twintone version of 34-et, so >I don't think it is relevant.Let's take a look at what you wrote...>>> You can't hear consistency, so why is this relevant?Seemed relevant from here.>>> A temperament /.../ which maps rational intonation of a given >>> rank in a consistent way to an intonation of smaller rank.So what does it mean to map rational intonation "in a consistent way"? From the Onto page a mathworld, it looks like a regular temperament is just a consistent temperament in the first place. -Carl

Message: 3583 - Contents - Hide Contents Date: Tue, 29 Jan 2002 21:07:13 Subject: Re: new cylindrical meantone lattice From: monz> From: paulerlich <paul@xxxxxxxxxxxxx.xxx> > To: <tuning-math@xxxxxxxxxxx.xxx> > Sent: Tuesday, January 29, 2002 4:24 PM > Subject: [tuning-math] Re: new cylindrical meantone lattice > > > --- In tuning-math@y..., "monz" <joemonz@y...> wrote: >>> I simply meant that I don't include 2 as part of the calculation >> on these lattices, thus I can't graph EDOs (= 2^x). >> It sounds to me like you're just plugging things into the numbers > without really understanding what they mean. Huh?Most of my lattices don't use 2 as a factor, but it is certainly possible to include 2, and I have done so on occasion. For example, to produce lattices which portray ancient Greek systems, I've occassionaly included 2, because the Greeks really weren't thinking in terms of "8ve"-equivalence, but rather more in terms of tetrachordal equivalence (or similarity, anyway). By including 2 in my formula, and perhaps reversing the prime-lengths so that 2 is the longest (as I wrote in another post), I can show EDOs as well as all the usual JI ratios, and fraction-of-a-comma meantones too.> It's absolutely wonderful that you're going to be producing real > cylindrical lattices. I'm hoping that, at least on some of your > webpages, you won't confuse the reader with your fractional lattice > points -- there is so much valuable information there already without > them -- and I still don't know what they're supposed to mean, > _especially_ once you've wrapped the lattice into a cylinder.Well, now that I finally have a firm understanding of the cylindrical meantone lattice, I can see at least a few of the objections you've been leveling at me: - The lattice of "rational implications" is indeed identical for all meantones. Tempering out the syntonic comma causes the JI lattices to wrap into identical cylinders. - Something I found most interesting: the distance of all meantone pitches as represented by the circumferential lines around the cylinder (the lines which represent the syntonic comma), perpendicular to the cylinder itself, is *also* identical for all meantones. This was a surprise when I first realized it, but upon further reflection, it's an obvious result of the above. *But* ... those agreements noted, I don't understand why you still object to my representation of the various meantone spirals around the cylinder. The different meantone systems are tuned in different ways, and if the difference between any two systems is large enough, it's audible. So what's wrong with showing that visually, by having the meantones slice the cylinder in their own particular way according to the math involved? 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. I would really appreciate much more help with this. I've been trying to understand it for a couple of years, to no avail. And remember ... the whole purpose in making these lattices is to eventually include this capability in my JustMusic software. The point is to be enable the user to create beautiful diagrams which s/he can then manipulate to create beautiful music. (Well, OK ... whatever kind of music s/he wants. Some people really don't like beautiful ...) -monz _________________________________________________________ Do You Yahoo!? Get your free @yahoo.com address at Yahoo! Mail - The best web-based email! * [with cont.] (Wayb.)

Message: 3584 - Contents - Hide Contents Date: Tue, 29 Jan 2002 22:16:25 Subject: Re: new cylindrical meantone lattice From: monz> From: paulerlich <paul@xxxxxxxxxxxxx.xxx> > To: <tuning-math@xxxxxxxxxxx.xxx> > Sent: Tuesday, January 29, 2002 9:24 PM > Subject: [tuning-math] Re: new cylindrical meantone lattice > > > --- In tuning-math@y..., "monz" <joemonz@y...> wrote: >>> By including 2 in my formula, and perhaps reversing the >> prime-lengths so that 2 is the longest (as I wrote in another >> post), I can show EDOs as well as all the usual JI ratios, and >> fraction-of-a-comma meantones too. >> But what happens to the cylinder? It seems that, without necessarily > ignoring all the instances of the number 2, you would want to make > these diagrams octave-invariant, wouldn't you?I'll have to try to answer this question and the previous (which I snipped) more fully another time. But I have one here: what the heck is the difference between "octave equivalent" and "octave invariant"? Is there a difference?>> *But* ... those agreements noted, I don't understand why >> you still object to my representation of the various meantone >> spirals around the cylinder. >> What do those spirals represent?The actual mathematical tuning of the fraction-of-a-comma meantones.> And what does it say about LucyTuning and ETs that you can't > construct such spirals from _these_ meantones?I says that while LucyTuning and meantone-like ETs are audibly indistinguishable from certain fraction-of-a-comma meantones, they are mathematically entirely different. Again, I refer you to my (very vague but seemingly always getting clearer) ideas on finity. Xenharmonic Bridges in effect here.>> The different meantone systems are tuned in different ways, >> and if the difference between any two systems is large enough, >> it's audible. So what's wrong with showing that visually, >> by having the meantones slice the cylinder in their own >> particular way according to the math involved? >> (I mentioned another way of showing that visually above. A slight > acoustical difference merits at most a slight visual difference, IMO).OK, Paul, I can buy that! As I've said before many times, I'd love to enlist your help and for the two of us to work together to create some really killer lattice formulae. You know that I'm very fond of my particular formula, but I'm open-minded and willing to revise it, or better, to create new kinds of lattices from scratch.>> 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. . . . > > If you insist on having the spirals, then you _need_ to find a way to > get them to work for LucyTuning and Golden meantone and ET meantones, > etc. Otherwise I can't imagine how they could _possibly_ be > meaningful. (P.S. Congratulations on discovering them -- they may be > an original Monzo contribution!)Hmmm ... yes, I'm thinking that the meantone-spiral thing really might be a useful new contribution to tuning theory. Thanks for the acknowledgement! I'd definitely like to see more about what you say here. Hope this sparks an interest in several of you, so that a real discussion might ensue. I'm still pretty confused about the "non-uniqueness" business myself.>> And remember ... the whole purpose in making these lattices >> is to eventually include this capability in my JustMusic software. >> Well . . . meantones aren't "Just" by any of the definitions > proposed. So maybe a new name is in order?Yeah, I've been sweating over that name ever since having been on the tuning list for about a year. But then again ... one of the things I seek to show, which I interpret as being along the same lines as your Hypothesis, is that all of our various tuning systems (in other words, the way "finity" works) have some ultimate basis in the manipulation of the prime series, which is at the heart of my whole JustMusic theory and software design. Maybe "PrimeMusic" would be better? But it doesn't have the same ring, the logo wouldn't look as cool, and ... JustMusic happens to have the same initials as JoeMonzo, which makes abbreviations in emails very convenient for me! :) -monz _________________________________________________________ Do You Yahoo!? Get your free @yahoo.com address at Yahoo! Mail - The best web-based email! * [with cont.] (Wayb.)

Message: 3585 - Contents - Hide Contents Date: Wed, 30 Jan 2002 20:55:30 Subject: Re: question for Gene From: paulerlich Awaiting a response . . . --- In tuning-math@y..., "paulerlich" <paul@s...> wrote:> --- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:>> --- In tuning-math@y..., "paulerlich" <paul@s...> wrote: >>>>> Is there any way to directly compare the badnesses of equal >>> temperaments and linear temperaments and meaningfully ask the >>> question: Which of the linear temperaments that you found (in the > 5->>> limit, and whatever other cases you've completed) could be > expressed>>> by an equal temperament, without pushing the badness over the > limit >>> you've computed? >>>> Certainly--just recalculate rms error for the new tuning. >> Could you do this please? Which of the twenty (?) linear temperaments > that you found could thus be expressed? >>> Complexity will never increase, >

Message: 3586 - Contents - Hide Contents Date: Wed, 30 Jan 2002 21:21:48 Subject: Some Minkowsi reduced bases From: genewardsmith --- In tuning-math@y..., "paulerlich" <paul@s...> wrote: Perhaps Gene can> provide the 5-limit Minkowski reduced basis, which would have only > two commas. I bet the diaschisma is one of them . . .Here you go. Enjoy! 7: [25/24, 81/80] 12: [81/80, 128/125] 19: [81/80, 3125/3072] 22: [250/243, 2048/2025] 31: [81/80, 393216/390625] 34: [2048/2025, 15625/15552] 41: [3125/3072, 20000/19683] 46: [2028/2025, 78732/78125] 53: [15625/15552, 32805/32768] 65: [32805/32768, 78732/78125] 72: [15625/15552, 531441/524288] 84: [78732/78125, 531441/524288] 87: [15625/15552, 67108864/66430125] 99: [393216/390625, 1600000/1594323] 118: [32805/32768, 1224440064/1220703125] 20000/19683, 531441/524288 and 67108864/66430125 weren't on my temperament list, so I'd better check if they should have been.

Message: 3587 - Contents - Hide Contents Date: Wed, 30 Jan 2002 22:11:54 Subject: Re: Some Minkowsi reduced bases From: paulerlich --- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:> --- In tuning-math@y..., "paulerlich" <paul@s...> wrote: > > Perhaps Gene can>> provide the 5-limit Minkowski reduced basis, which would have only >> two commas. I bet the diaschisma is one of them . . . >> Here you go. Enjoy! > > 7: [25/24, 81/80] > 12: [81/80, 128/125] > 19: [81/80, 3125/3072] > 22: [250/243, 2048/2025]How about 15? Herman Miller wrote some wonderful music in 15-tET exploiting the 250:243.>20000/19683, 531441/524288 and 67108864/66430125 weren't on my >temperament list,The first two are called the "minimal diesis" and "Pythagorean comma".>so I'd better check if they should have been.I had hoped your 5-limit search was "airtight" already . . . oh well.

Message: 3588 - Contents - Hide Contents Date: Wed, 30 Jan 2002 22:24:30 Subject: Re: Some Minkowsi reduced bases From: genewardsmith --- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:> 20000/19683, 531441/524288 and 67108864/66430125 weren't on my temperament list, so I'd better check if they should have been. 20000/19683Map: [[0, 4, 9], [1, 1, 1]] Ets: 7, 27, 34, 41, 75 Generators: a = 6.023/41 = 176.2823 cents; b = 1 badness 649 rms 2.504 g 6.38 531441/524288 Map: [[0, 0, 1], [12, 19, 24]] Ets: 12, 24, 36, 48, 60, 72, 84 ... Generators: a = 385.3362 cents, b = 100 cents badness 1300 rms 1.382 g 9.80 67108864/66430125 Map: [[0, -1, 4], [3, 6, 2]] Ets: 12, 75, 87, 99, 111 Generators: a = 496.7879 cents; b = 400 cents badness 1280 rms .9052 g 11.225

Message: 3589 - Contents - Hide Contents Date: Wed, 30 Jan 2002 22:46:07 Subject: Re: Some Minkowsi reduced bases From: genewardsmith --- In tuning-math@y..., "paulerlich" <paul@s...> wrote:> How about 15? Herman Miller wrote some wonderful music in 15-tET > exploiting the 250:243.5: [16/15, 27/25] 15: [128/125, 250/243] 171: [32805/32768, 2 5^18 3^(-27)]> I had hoped your 5-limit search was "airtight" already . . . oh well.It should be, but it never hurts to check. None of the new temperaments were under my 500 limit.

Message: 3590 - Contents - Hide Contents Date: Wed, 30 Jan 2002 22:56:33 Subject: Re: Some Minkowsi reduced bases From: paulerlich --- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:> --- In tuning-math@y..., "paulerlich" <paul@s...> wrote: >>> How about 15? Herman Miller wrote some wonderful music in 15-tET >> exploiting the 250:243.> 15: [128/125, 250/243]There it is!> None of the new >temperaments were under my 500 limit.I noticed that. BTW, have you had a chance to think about my "question for Gene"?

Message: 3591 - Contents - Hide Contents Date: Wed, 30 Jan 2002 17:33:01 Subject: ET that does adaptive-JI? From: monz In a discussion from August, in which Bob Wendell was looking for a nice consistent ET to supplement the use of cents and the consensus was the 111-tET fit his criteria, Paul wrote:> tuning-math message 837 > From: "Paul Erlich" <paul@s...> > Date: Thu Aug 23, 2001 5:18 pm > Subject: Re: EDO consistency and accuracy tables (was: A little research...) > > > Right . . . but I think the whole idea of 72 or 111 or 121 as a least- > common-denominator way of describing ideal musical practice kind of > falters when _adaptive JI_ comes into the picture . . . doesn't it?I see that there are follow-ups which I haven't yet read, but in case they don't quite address this question, I'm very curious -- what kind of criteria would have to be set to find an ET that *does* work for adaptive-JI? -monz _________________________________________________________ Do You Yahoo!? Get your free @yahoo.com address at Yahoo! Mail - The best web-based email! * [with cont.] (Wayb.)

Message: 3592 - Contents - Hide Contents Date: Wed, 30 Jan 2002 17:29:05 Subject: interval of equivalence, period, unison-vector From: monz Hi guys, I've been diligently studying the tuning-math archives, and am really confused about one thing. (OK, many things ... but let's start here...)> tuning-math message 823 > From: graham@m... > Date: Thu Aug 23, 2001 7:22 am > Subject: Re: Interpreting Graham's matrix Yahoo groups: /tuning-math/message/823?expand=1 * [with cont.] > > The things that make this system different to the one > before is that it isn't unitary, and only one column of > the inverse depends on the first generator. It's the second > criterion that allows us to draw the non-arbitrary > distinction between "interval of equivalence" and > "unison vector", and so throw away the former.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? -monz _________________________________________________________ Do You Yahoo!? Get your free @yahoo.com address at Yahoo! Mail - The best web-based email! * [with cont.] (Wayb.)

Message: 3593 - Contents - Hide Contents Date: Wed, 30 Jan 2002 18:35:04 Subject: Re: new cylindrical meantone lattice From: monz ----- Original Message ----- From: paulerlich <paul@xxxxxxxxxxxxx.xxx> To: <tuning-math@xxxxxxxxxxx.xxx> Sent: Wednesday, January 30, 2002 11:10 AM Subject: [tuning-math] Re: new cylindrical meantone lattice> --- In tuning-math@y..., "monz" <joemonz@y...> wrote: >>>>> From: paulerlich <paul@s...> >>> To: <tuning-math@y...> >>> Sent: Tuesday, January 29, 2002 9:24 PM >>> Subject: [tuning-math] Re: new cylindrical meantone lattice >>> >>> >>> --- In tuning-math@y..., "monz" <joemonz@y...> wrote: >>>>> what the heck is the difference between "octave equivalent" and >> "octave invariant"? Is there a difference? >> Not that I can think of right now.I just posted something else about this a few minutes ago. I'm totally confused, in studying this list's archives, about the differences between "octave-invariant" and "octave-equivalent", and don't understand what differences there are between these and unison-vectors and periods.>> >>>>>> *But* ... those agreements noted, I don't understand why >>>> you still object to my representation of the various meantone >>>> spirals around the cylinder. >>>>>> What do those spirals represent? >>>> The actual mathematical tuning of the fraction-of-a-comma meantones. >> Hmm . . . >>>> And what does it say about LucyTuning and ETs that you can't >>> construct such spirals from _these_ meantones? >> >>>> I says that while LucyTuning and meantone-like ETs are audibly >> indistinguishable from certain fraction-of-a-comma meantones, >> they are mathematically entirely different. > > Not really.Huh? Take a look at the "ratios" I post below. 2^(2/10) * 3^(-2/10) * 5^(3/10) is *mathematically* totally different from 2^(2pi+1)/4pi, even tho they're acoustically identical.>> Again, I refer you to my (very vague but seemingly always >> getting clearer) ideas on finity. Xenharmonic Bridges in >> effect here. >> Can you elaborate, please? Sure.(Well, I'll be cornswoggled -- that doesn't mean anything ... I just realized I had explained this on an updated definition I made for "LucyTuning", but I never uploaded it until now!) It's all in the Dictionary now at Definitions of tuning terms: LucyTuning, (c) 2... * [with cont.] (Wayb.) But I'll post it anyway:>> This [LucyTuning] generator or "5th" is composed of three >> Large (3L) plus one small note (s), i.e. (3L+s) >> = (~190.986*3) + (~122.535) = ~695.493 cents or ratio of >> >> 2^(3/2pi) * ( 2/(2^(5/2pi)) )^(1/2) >> >> = 2^( (2pi + 1) / 4pi ) >> >> = 2^( 1/2 + 1/4pi ) >> >> = ~1.494412. >> >> >> This generator is audibly indistinguishable from that >> of 3/10-comma quasi-meantone: >> >> 2^(2/10) * 3^(-2/10) * 5^(3/10) 3/10-comma quasi-meantone "5th" >> - 2^(2pi+1)/4pi Lucytuning "5th" >> ------------------------------------------ >> 2^(-12pi-10)/40pi * 3^(-2/10) * 5^(3/10) = ~0.010148131 cent = ~1/99 centThat last number is the "ratio" of the tiny xenharmonic bridge I'm talking about. 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.> 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? Please elaborate.>>>> 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. 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.)>> 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. 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. 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. That's just a brief outline. Sure do hope to see you over there! (and anyone else on this list who is intrigued but hasn't signed up yet) -monz _________________________________________________________ Do You Yahoo!? Get your free @yahoo.com address at Yahoo! Mail - The best web-based email! * [with cont.] (Wayb.)

Message: 3594 - Contents - Hide Contents Date: Wed, 30 Jan 2002 00:24:37 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: Tuesday, January 29, 2002 2:30 PM >> Subject: [tuning-math] Re: new cylindrical meantone lattice >> >> >> --- In tuning-math@y..., "monz" <joemonz@y...> wrote: >>>>>> And what is it for meantones like 31-tET >>>> or 69-tET or LucyTuning? Does it still exist? >>> >>>>>> Well ... the lattice I'm using here is "8ve-equivalent", >>> so I can't put any EDOs on them. >>>> 31-tET or 69-tET or LucyTuning are no more or less "8ve- equivalent" >> than fraction-of-a-comma meantones. So I'm not sure what you meant my >> this. > >> I simply meant that I don't include 2 as part of the calculation > on these lattices, thus I can't graph EDOs (= 2^x).It sounds to me like you're just plugging things into the numbers without really understanding what they mean.> Well ... *I'm* hoping that I can continue to find ways to > "warp" my lattices so that these equivalent mathematics > *do* correspond visually! Keep helping me! :)It's absolutely wonderful that you're going to be producing real cylindrical lattices. I'm hoping that, at least on some of your webpages, you won't confuse the reader with your fractional lattice points -- there is so much valuable information there already without them -- and I still don't know what they're supposed to mean, _especially_ once you've wrapped the lattice into a cylinder.

Message: 3595 - Contents - Hide Contents Date: Wed, 30 Jan 2002 20:16:43 Subject: Re: ET that does adaptive-JI? From: monz> From: paulerlich <paul@xxxxxxxxxxxxx.xxx> > To: <tuning-math@xxxxxxxxxxx.xxx> > 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? Or is there some special interval between chains? Does it also work equivalently as chains of 19-EDO, since that's so close to 1/3-comma MT? I find this really interesting. -monz _________________________________________________________ Do You Yahoo!? Get your free @yahoo.com address at Yahoo! Mail - The best web-based email! * [with cont.] (Wayb.)

Message: 3596 - Contents - Hide Contents Date: Wed, 30 Jan 2002 00:26:49 Subject: Re: twintone, paultone From: paulerlich --- In tuning-math@y..., "clumma" <carl@l...> wrote:> So what does it mean to map rational intonation "in a consistent > way"? From the Onto page a mathworld, it looks like a regular > temperament is just a consistent temperament in the first place.I'm sure Gene would be happy with this statement. Only thing, it's not consistent in the 'TTTTTT, footnote 8' sense of necessarily using the best available approximation to every 'consonant' interval.

Message: 3597 - Contents - Hide Contents Date: Wed, 30 Jan 2002 20:12:08 Subject: Re: ET that does adaptive-JI? From: monz> From: paulerlich <paul@xxxxxxxxxxxxx.xxx> > To: <tuning-math@xxxxxxxxxxx.xxx> > Sent: Wednesday, January 30, 2002 8:08 PM > Subject: [tuning-math] Re: ET that does adaptive-JI? > > > --- In tuning-math@y..., "monz" <joemonz@y...> wrote: > >>> I see that there are follow-ups which I haven't yet read, >> but in case they don't quite address this question, I'm >> very curious -- what kind of criteria would have to be set >> to find an ET that *does* work for adaptive-JI? > >> Thanks for asking this now, Monz. > > 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. > > In addition, it acts as a strict 11-limit JI system, with the maximum > error in the consonant intervals and in the Tonality Diamond pitches > is 2.23 cents -- 0.68 cents through the 5-limit. > > Finally, and most importantly, it contains 76-tET, which contains all > the linear temperaments that interest me most, as they support > omnitetrachordal scales (and are 7-limit): > 1. Meantone within 19-tET subsets > 2. Pajara > 3. Double-Diatonic within 38-tET subsets > as well as the non-tetrachordal > 4. Kleismic within 19-tET subsets (see > 11 note chain-of-minor-thirds scale * [with cont.] (Wayb.)). > The full 152-tET could probably support a large number of other > interesting linear temperaments. > > Anyway, this is all just an example of how ETs can be used > consistently by exploiting their inconsistency ;) though 152-tET _is_ > consistent through the 11-limit (see above).Hmmm ... can you elaborate more on that bit about "exploiting their inconsistency"? I don't quite get that. Anyway, 152-EDO seems like a fantastic tuning to play around with! Why have I read nothing about it before? -monz _________________________________________________________ Do You Yahoo!? Get your free @yahoo.com address at Yahoo! Mail - The best web-based email! * [with cont.] (Wayb.)

Message: 3598 - Contents - Hide Contents Date: Wed, 30 Jan 2002 04:29:12 Subject: Re: twintone, paultone From: clumma>> >o what does it mean to map rational intonation "in a consistent >> way"? From the Onto page a mathworld, it looks like a regular >> temperament is just a consistent temperament in the first place. >>I'm sure Gene would be happy with this statement.So then what's the point of saying that inconsistency won't cause any problems in a regular temperament? -Carl

Message: 3599 - Contents - Hide Contents Date: Wed, 30 Jan 2002 04:37:46 Subject: Re: twintone, paultone From: paulerlich --- In tuning-math@y..., "clumma" <carl@l...> wrote:>>> So what does it mean to map rational intonation "in a consistent >>> way"? From the Onto page a mathworld, it looks like a regular >>> temperament is just a consistent temperament in the first place. >>>> I'm sure Gene would be happy with this statement. >> So then what's the point of saying that inconsistency won't cause > any problems in a regular temperament?There's a point if 'inconsistency' is defined in the TTTTTT, footnote 8 sense.

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