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Message: 3550 - Contents - Hide Contents Date: Sun, 27 Jan 2002 22:44:13 Subject: Re: twintone, paultone From: genewardsmith --- In tuning-math@y..., graham@m... wrote:> Using either of the other diaschismic mappings of 34-et, the inconsistent > chords are those of twintone. You certainly do get some of them within > the 10 note MOS.What's an inconsistent chord?>> It shows 34-et as a part of a range of twintone et possibilities. >> Well, that's a radical idea. I'm sure I'd never have worked that out > myself.You seemed to be objecting to it strongly, so I don't know what your point is.> > BTW, what do you make the LLL reduction of > [ 1 1 1] > [-1 0 2] > [ 3 5 6]That might depend on what inner product I use, and I don't know what this is supposed to represent. If I use the standard dot product, I get [ 0 1 0] [ 1 0 1] [-1 0 2]

Message: 3551 - Contents - Hide Contents Date: Sun, 27 Jan 2002 01:00:03 Subject: Re: Proposed dictionary entry: torsion (revised) From: monz Hi Gene,> From: monz <joemonz@xxxxx.xxx> > To: <tuning-math@xxxxxxxxxxx.xxx> > Sent: Saturday, January 26, 2002 10:37 PM > Subject: Re: [tuning-math] Re: Proposed dictionary entry: torsion (revised) > > > Can you integrate that into the "good" definition in the top > part of the page? Then I can delete all the other old junk > in the bottom part. Thanks.Just thought I'd mention ... Even tho I really still don't understand it, because of what I see on the lattice I can intuitively sense how torsion works. And my intuition tells me that torsion is a very important part of getting a better focus on my model of "finity": Definitions of tuning terms: finity, (c) 1998 ... * [with cont.] (Wayb.) I'm thinking that the patterns of unison-vectors that one can see within a torsional block mean something, and this can be modeled mathematically. So I'd really like to keep exploring it until I understand it fully, and to correspondingly expand the Dictionary webpage. I've had the idea to create a book full of 5-limit periodicity- and torsional-blocks, and many of these can go into the webpage. Please, Gene and the others here who do understand torsion, feel free to comment profusely, with lots of 5- and higher-limit examples. I'll try to diagram all of them and include them in the definition, or perhaps I'll make a separate analytical page about torsion. -monz _________________________________________________________ Do You Yahoo!? Get your free @yahoo.com address at Yahoo! Mail - The best web-based email! * [with cont.] (Wayb.)

Message: 3552 - Contents - Hide Contents Date: Sun, 27 Jan 2002 01:27:22 Subject: Re: Proposed dictionary entry: torsion (revised) From: monz> From: monz <joemonz@xxxxx.xxx> > To: <tuning-math@xxxxxxxxxxx.xxx> > Sent: Sunday, January 27, 2002 1:00 AM > Subject: Re: [tuning-math] Re: Proposed dictionary entry: torsion (revised) > > > Even tho I really still don't understand it, because of what > I see on the lattice I can intuitively sense how torsion works. > And my intuition tells me that torsion is a very important > part of getting a better focus on my model of "finity": > > Definitions of tuning terms: finity, (c) 1998 ... * [with cont.] (Wayb.) > > > I'm thinking that the patterns of unison-vectors that one > can see within a torsional block mean something, and this > can be modeled mathematically.Well, OK ... actually I can already see that the unison-vectors inside the torsional-block on my lattice diagram here Definitions of tuning terms: torsion, (c) 2002... * [with cont.] (Wayb.) are exactly the same pair as the bounding vectors of the Duodene Definitions of tuning terms: duodene, (c) 1998... * [with cont.] (Wayb.) ... a *real* periodicity-block, and apparently one whose 12 pitches can "stand in" for the 24 of this torsional-block? Hmmm... Very curious, -monz _________________________________________________________ Do You Yahoo!? Get your free @yahoo.com address at Yahoo! Mail - The best web-based email! * [with cont.] (Wayb.)

Message: 3553 - Contents - Hide Contents Date: Sun, 27 Jan 2002 12:36 +0 Subject: Re: twintone, paultone From: graham@xxxxxxxxxx.xx.xx Me:>> No. If you're using a regular temperament, you can't be using 34-et. Gene:> This is just wrong; the whole thing is beginning to seem like another > of those "religion" deal.Close. It's actually a question of terminology. Equal and regular temperaments are different things. Me:>> 34-et is an inconsistent, equal temperament. Gene:> 34-et isn't a regular temperament at all until you define a mapping to > primes according to my proposed definition, which I think would help > clarify all of this confusion.34-et can be used *as* a regular temperament, but 34-et *is not* a regular temperament. Me:> If you're using one of the>> other diaschismic mappings of 34-et, the inconsistent chords will be >> simpler than the regular ones. So what are you going to do? Pretend >> they aren't there? Pretend they're not really 7-limit? Gene:> If you are using a 10-tone subset of 34 et, then they won't be there.Bzzt -- yes they will. Try reading that paragraph a bit more carefully. This is also the first time you've mentioned a 10-note *subset* which would obviously skew towards twintone.> In any case, this is not a new "problem"; it arises in meantone, where > you get augmeted sixth intervals which are much closer to 7/4 than the > 64/63 approximation ones intrinsic to diatonic 7-limit harmony, and so > one has a connitption fit about it.Not new at all. In both cases there's a simplified 7-limit mapping that optimises close to one of the extreme ETs -- 12 in meantone, 22 in diaschismic. The difference with diaschismic is that the "normal" range is covered by two different more-complex mappings, so we can't talk about a "typical" 7-limit diaschismic. But this isn't new either, it's been on my website for a few years. Me:>> If you're not going to make use of the inconsistency, I don't see the >> point in using 34-equal at all. Gene:> The point would be to make use of the superior 5-limit > harmonies--compare the major sixth/minor thirds of 34-et to those of > 22-et, for instance.That's not a sufficient reason. You can get better 5-limit harmonies with a 105.2 cent generator (worst error 3.3 cents compared with 3.9 for 34-equal). So again, why use 34-equal?> If we consider 12-et, with a fifth which is two > cents *flat* to be capable of producing a sort of twintone, we can > certainly accept 34-et. If you look at how the fifth is tempered in > various ets, a whole range of possibilities emerge: > > h12: -1.96 > g34: 3.93 > g56: 5.19 > h22: 7.14 > h54: 9.16 > > There should be something for everyone in there.What does this have to do with the price of eggs? Graham

Message: 3554 - Contents - Hide Contents Date: Sun, 27 Jan 2002 21:37:22 Subject: Re: twintone, paultone From: genewardsmith --- In tuning-math@y..., graham@m... wrote:> 34-et can be used *as* a regular temperament, but 34-et *is not* a regular > temperament.What is your definition of "regular temperament"? I gave mine.>> If you are using a 10-tone subset of 34 et, then they won't be there. >> Bzzt -- yes they will. Try reading that paragraph a bit more carefully. Where? > This is also the first time you've mentioned a 10-note *subset* which > would obviously skew towards twintone.Twintone is the subject of our discussion--how can we skew towards where we already are?>> The point would be to make use of the superior 5-limit >> harmonies--compare the major sixth/minor thirds of 34-et to those of >> 22-et, for instance. >> That's not a sufficient reason. You can get better 5-limit harmonies with > a 105.2 cent generator (worst error 3.3 cents compared with 3.9 for > 34-equal). So again, why use 34-equal?This is simpl an agument that we should never use equal divisions at all. Should I go into reasons why we might want to?>> h12: -1.96 >> g34: 3.93 >> g56: 5.19 >> h22: 7.14 >> h54: 9.16 >> >> There should be something for everyone in there. >> What does this have to do with the price of eggs?It shows 34-et as a part of a range of twintone et possibilities.

Message: 3555 - Contents - Hide Contents Date: Sun, 27 Jan 2002 21:43:01 Subject: Re: twintone, paultone From: paulerlich --- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:> --- In tuning-math@y..., graham@m... wrote: >>> 34-et can be used *as* a regular temperament, but 34-et *isnot* a regular>> temperament. >> What is your definition of "regular temperament"? I gave mine.Yours seems to fit perfectly with the existing literature. I have no idea what definition Graham might be going by. You may note that many of the decatonic "keys" in the 22-tone well-temperament in my paper are quite similar to 34-tET in intonation.

Message: 3556 - Contents - Hide Contents Date: Sun, 27 Jan 2002 22:02 +0 Subject: Re: twintone, paultone From: graham@xxxxxxxxxx.xx.xx genewardsmith wrote:> What is your definition of "regular temperament"? I gave mine.In fact, I've noticed that I have no idea what a "regular temperament" is. So, I'd better beat a hasty retreat. I can't find your definition either, though ...>>> If you are using a 10-tone subset of 34 et, then they won't be >>> there.>> Bzzt -- yes they will. Try reading that paragraph a bit more >> carefully. > > Where?The one you cut out:> If you're using one of the>> other diaschismic mappings of 34-et, the inconsistent chords will be >> simpler than the regular ones. So what are you going to do? Pretend >> they aren't there? Pretend they're not really 7-limit?Using either of the other diaschismic mappings of 34-et, the inconsistent chords are those of twintone. You certainly do get some of them within the 10 note MOS.>> This is also the first time you've mentioned a 10-note *subset* which >> would obviously skew towards twintone. >> Twintone is the subject of our discussion--how can we skew towards > where we already are?What? At this time of night I can't even understand why that objection's bogus. You're saying because we're discussing something it must be right?>>> The point would be to make use of the superior 5-limit >>> harmonies--compare the major sixth/minor thirds of 34-et to those >>> of 22-et, for instance. >>>> That's not a sufficient reason. You can get better 5-limit harmonies >> with a 105.2 cent generator (worst error 3.3 cents compared with 3.9 >> for 34-equal). So again, why use 34-equal? >> This is simpl an agument that we should never use equal divisions at > all. Should I go into reasons why we might want to?You could do.> It shows 34-et as a part of a range of twintone et possibilities.Well, that's a radical idea. I'm sure I'd never have worked that out myself. BTW, what do you make the LLL reduction of [ 1 1 1] [-1 0 2] [ 3 5 6] ? Graham

Message: 3557 - Contents - Hide Contents Date: Sun, 27 Jan 2002 22:06:35 Subject: Re: twintone, paultone From: paulerlich --- In tuning-math@y..., graham@m... wrote:> BTW, what do you make the LLL reduction of > > [ 1 1 1] > [-1 0 2] > [ 3 5 6] > > ?Is the first column 2 or 3? How about the Minkowski reduction?

Message: 3558 - Contents - Hide Contents Date: Mon, 28 Jan 2002 06:01:40 Subject: Re: Proposed dictionary entry: torsion (revised) From: paulerlich --- In tuning-math@y..., "monz" <joemonz@y...> wrote:> > OK, fair enough. I decided to go ahead and make the lattice > diagram of your example after all. Here's the latest definition: > > Definitions of tuning terms: torsion, (c) 2002... * [with cont.] (Wayb.)What happened to the really nice definition Gene gave?????????????? This should be _at the top_, rather than omitted entirely!!!!!!!!!!!

Message: 3560 - Contents - Hide Contents Date: Mon, 28 Jan 2002 11:14 +0 Subject: consistent mappings, LLL, Re: twintone, paultone From: graham@xxxxxxxxxx.xx.xx In-Reply-To: <a31vrt+4sau@xxxxxxx.xxx> genewardsmith wrote:> What's an inconsistent chord?I meant chords inconsistent with the "official" mapping but which approximate some interval better. So in 34-as-twintone, anything from h34 is inconsistent. Using the other diaschismic mappings (h34&h46 or h34&h22 which could be written as h46&h58 and h56&h22 respectively using only consistent ETs) it's chords from g34 that are inconsistent.>>> It shows 34-et as a part of a range of twintone et possibilities. >>>> Well, that's a radical idea. I'm sure I'd never have worked that out >> myself. >> You seemed to be objecting to it strongly, so I don't know what your > point is.What am I objecting to now? I thought it was what I said before but you described as "ridiculous". Still, it was something I hadn't said that was ridiculous then as well. I can't find any more 7-limit diaschismics covering 34 with a complexity of less than 34. The closest is g34&h46 (prime mapping of 46 with alternative mapping of 34) which has a complexity of exactly 34 (so 36 notes for two complete otonalities) and a minimax error of 4 cents. Period/generator mapping [(2, 0), (3, 1), (5, -2), (3, 15)] and my wedge invariant (2, -4, 30, -11, 42, 81) (this is different to Gene's wedge invariants which I still don't know how to get). Oh, time for a new conjecture: The linear temperament formed by combining two consistent equal temperaments will never have a higher complexity than the number of notes in the more complex ET. This is using the same definitions of complexity and consistency as my program. Anybody care to prove/refute it?> That might depend on what inner product I use, and I don't know what > this is supposed to represent. If I use the standard dot product, I get ...Yes, that's what I meant. It's one of the examples from the book, and my function doesn't get it right. So thanks for confirming that I'm wrong and not the book. I'm not clear what to do with the function when it is working. It takes square matrices, but the matrices formed by commatic unison vectors aren't square. Do you have an inner product that works for octave-equivalent harmony? At least then we could get a reduced basis for a periodicity block. Also, the book covers simultaneous Diophantine approximations (of which more sometime) but not simultaneous linear Diophantine equations. So I still don't know how to get an original basis without torsion. Any clues? Graham

Message: 3561 - Contents - Hide Contents Date: Mon, 28 Jan 2002 11:40 +0 Subject: Re: twintone, paultone, 34 From: graham@xxxxxxxxxx.xx.xx In-Reply-To: <200201280951.LAA65774@xxxxxx.xxx.xxxxx.xxx> Robert C Valentine wrote:> Or, (and this is probably the RIGHT answer) > > 2 * best( 5/4 ) = octave_reduced( 4 * best( 4/3 ) )That looks right, but it's usually done as best(5:4) = half_octave_reduced( 2 * best(4:3) ) because all diaschismics are divisible by two (in terms of notes to the octave).> So, I have tried in the past to express it as a system in 3, 5, > and 13 (since 13 is notably absent from other EDOs I intend using) > > 3 5 13 > [ 1 0 2 ] 2 * best( 13/8 ) = normalized( best( 4/3 ) ) > [ -4 2 0 ] from above... > [ ] got another ? does it come out to 34 ?By combining 58 (which is fully 13-limit consistent) and 34, I get best (16:13) = tritone_reduced(2*best(3:2)) and best (16:13) = tritone_reduced(4*best(8:5)) but that gets quite hairy. Graham

Message: 3562 - Contents - Hide Contents Date: Mon, 28 Jan 2002 18:30:38 Subject: Re: twintone, paultone, 34 From: paulerlich --- In tuning-math@y..., Robert C Valentine <BVAL@I...> wrote:> > Due to my math skills this would probably be more appropriate > on the main list, but due to the topic here and what I think > my questions may elicit, I'll put it here. > > First, some "Bob Valentine oriented definitions for EDOs". > > Meantone := best(5/4) = octave_reduced( 4 * best(3/2) ) > Schismic := best(5/4) = octave_reduced( 8 * best(4/3) ) > > diaschismic := [SNIP] > Or, (and this is probably the RIGHT answer) > > 2 * best( 5/4 ) = octave_reduced( 4 * best( 4/3 ) )That's right! If you ever forget these things, just look up the ratios: Schisma = 32805:32768 Diaschisma = 2048:2025 Then you can always work out the equivalencies. These apply not only to EDOs, but also to equal temperaments. Now, can you figure out how "kleismic" is defined? Hint: the kleisma = 15625:15552 You can always look up commas here: Stichting Huygens-Fokker: List of intervals * [with cont.] (Wayb.)

Message: 3563 - Contents - Hide Contents Date: Mon, 28 Jan 2002 19:46:16 Subject: consistent mappings, LLL, Re: twintone, paultone From: genewardsmith --- In tuning-math@y..., graham@m... wrote:> [(2, 0), (3, 1), (5, -2), (3, 15)] > > and my wedge invariant > > (2, -4, 30, -11, 42, 81) > > (this is different to Gene's wedge invariants which I still don't know how > to get).What about a compromise? You change the order of the basis elements, so that this reads (2, -4, 30, 81, 42, -11). The point is to make the wedgie calclulated from the period matrix or from two ets the same as the wedgie calculated from two commas. In return, I could normalize by your method. Mine was chosen to correspond with the 5-limit, where a wedgie is just the comma of the temperament, and where the obvious way to normalize is to make the comma greater than one. However this isn't going to work in the 11-limit, and we should decide on a single system which may as well be the one you are using now.> > Oh, time for a new conjecture: > > The linear temperament formed by combining two consistent equal > temperaments will never have a higher complexity than the number of notes > in the more complex ET. > > This is using the same definitions of complexity and consistency as my > program. Anybody care to prove/refute it?Have you done enough calculations to check its plausibility? It sounds plausible, though.> I'm not clear what to do with the function when it is working. It takes > square matrices, but the matrices formed by commatic unison vectors aren't > square.You should be able to change the code to nonsquare easily enough. If that doesn't work, try filling out the matrix with rows of zeros.

Message: 3564 - Contents - Hide Contents Date: Mon, 28 Jan 2002 20:01:25 Subject: consistent mappings, LLL, Re: twintone, paultone From: genewardsmith --- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:>> This is using the same definitions of complexity and consistency as my >> program. Anybody care to prove/refute it? >> Have you done enough calculations to check its plausibility? It sounds plausible, though.It occurs to me that the theorems I've already posted here should suffice for this. I'll get back to it.

Message: 3565 - Contents - Hide Contents Date: Tue, 29 Jan 2002 12:15 +0 Subject: Re: consistent mappings, LLL, Re: twintone, paultone From: graham@xxxxxxxxxx.xx.xx In-Reply-To: <a349q8+dd3a@xxxxxxx.xxx> Me:>> (2, -4, 30, -11, 42, 81) >> >> (this is different to Gene's wedge invariants which I still don't >> know how to get). Gene:> What about a compromise? You change the order of the basis elements, so > that this reads (2, -4, 30, 81, 42, -11). The point is to make the > wedgie calclulated from the period matrix or from two ets the same as > the wedgie calculated from two commas.Ah! Well, in my case this isn't true. This is the first time I've seen you acknowledge the problem. The wedgie above is {(2, 3): 81, (0, 1): 2, (1, 3): 42, (0, 3): 30, (0, 2): -4, (1, 2): -11} which has the invariant (2, -4, 30, -11, 42, 81) its complement is {(2, 3): 2, (1, 2): 30, (1, 3): 4, (0, 3): -11, (0, 2): -42, (0, 1): 81} with invariant (81, -42, -11, 30, 4, 2) How am I supposed to know which of these to take? They're dimensionally identical. It looks like your invariant function magically removes the distinction, but like I said before, I don't know how to do that. I did make a mistake above. It's {(2, 3): 2, (1, 2): 30, (1, 3): 4, (0, 3): -11, (0, 2): -42, (0, 1): 81} that gives the period/generator mapping [(2, 0), (0, 1), (11, -2), (-42, 15)] Whereas {(2, 3): 81, (0, 1): 2, (1, 3): 42, (0, 3): 30, (0, 2): -4, (1, 2): -11} gives [(1, 0), (-74, 81), (38, -42), (10, -11)] which has a stonking 3256 cent 7-limit minimax, but is still legally defined. How do you tell which is "correct"? I really don't know. Gene:> In return, I could normalize by > your method. Mine was chosen to correspond with the 5-limit, where a > wedgie is just the comma of the temperament, and where the obvious way > to normalize is to make the comma greater than one. However this isn't > going to work in the 11-limit, and we should decide on a single system > which may as well be the one you are using now.All I do is take the simpler of a "pair" of wedgies, which is ambiguous only in the 7 prime limit. Then, I sort each basis into ascending order and sort the whole thing in ascending order of bases. Me:>> The linear temperament formed by combining two consistent equal >> temperaments will never have a higher complexity than the number of >> notes in the more complex ET. >> >> This is using the same definitions of complexity and consistency as >> my program. Anybody care to prove/refute it? Gene:> Have you done enough calculations to check its plausibility? It sounds > plausible, though.I haven't found any exceptions yet, but haven't done a systematic search. It's easy to refute if you don't enforce consistency. Pretty much anything can be written as g0&g1 with the right g0, g1 and period. Me:>> I'm not clear what to do with the function when it is working. It >> takes square matrices, but the matrices formed by commatic unison >> vectors aren't square. Gene:> You should be able to change the code to nonsquare easily enough. If > that doesn't work, try filling out the matrix with rows of zeros.The former might work. I'll have to see if I can get my brain around it. The latter certainly won't. The Gram-Schmidt orthogonalization involves dividing by inner products, which will be zero if a row is zero. Graham

Message: 3566 - Contents - Hide Contents Date: Tue, 29 Jan 2002 10:19:43 Subject: Re: Proposed dictionary entry: torsion (revised) From: monz> From: paulerlich <paul@xxxxxxxxxxxxx.xxx> > To: <tuning-math@xxxxxxxxxxx.xxx> > Sent: Sunday, January 27, 2002 10:01 PM > Subject: [tuning-math] Re: Proposed dictionary entry: torsion (revised) > >>> ... Here's the latest definition: >> >> Definitions of tuning terms: torsion, (c) 2002... * [with cont.] (Wayb.) >> What happened to the really nice definition Gene gave?????????????? > > This should be _at the top_, rather than omitted entirely!!!!!!!!!!!The definition I now have at the top (apparently the same one you commented on here) is the latest one Gene sent to the tuning-math list: Message 2973 From: "genewardsmith" <genewardsmith@j...> Date: Fri Jan 25, 2002 2:46 pm Subject: Proposed dictionary entry: torsion (revised) Yahoo groups: /tuning-math/message/2973 * [with cont.] (... did you click "refresh/reload"?) -monz _________________________________________________________ Do You Yahoo!? Get your free @yahoo.com address at Yahoo! Mail - The best web-based email! * [with cont.] (Wayb.)

Message: 3567 - Contents - Hide Contents Date: Tue, 29 Jan 2002 20:27:15 Subject: Re: Proposed dictionary entry: torsion (revised) From: paulerlich --- In tuning-math@y..., "monz" <joemonz@y...> wrote:>>> From: paulerlich <paul@s...> >> To: <tuning-math@y...> >> Sent: Sunday, January 27, 2002 10:01 PM >> Subject: [tuning-math] Re: Proposed dictionary entry: torsion (revised) >> >>>>> ... Here's the latest definition: >>> >>> Definitions of tuning terms: torsion, (c) 2002... * [with cont.] (Wayb.) >>>> What happened to the really nice definition Gene gave?????????????? >> >> This should be _at the top_, rather than omitted entirely!!!!!!!!!!! > > >> The definition I now have at the top (apparently the same one > you commented on here) is the latest one Gene sent to the > tuning-math list: > > > Message 2973 > From: "genewardsmith" <genewardsmith@j...> > Date: Fri Jan 25, 2002 2:46 pm > Subject: Proposed dictionary entry: torsion (revised) > Yahoo groups: /tuning-math/message/2973 * [with cont.] > > > (... did you click "refresh/reload"?) > > > > -monzWho is going to understand that definition? You, Monz? I meant the definition in Yahoo groups: /tuning-math/message/2937 * [with cont.] but possibly changing "products of the proposed set of unison vectors" to "members of the group generated by the unison vectors".

Message: 3568 - Contents - Hide Contents Date: Tue, 29 Jan 2002 13:15:27 Subject: new cylindrical meantone lattice From: monz I've added an important new lattice to my "meantone" Dictionary entry, at the bottom: Internet Express - Quality, Affordable Dial Up... * [with cont.] (Wayb.) This attempts to show visually how a meantone "wraps" the lattice into a cylinder, thus closing one of the theoretically infinite dimensions of the JI lattice. Does anyone know the math that will apply sines and cosines, to warp the lattice-lines and points so that they actually *look* like they're sitting on the curved face of a cylinder? -monz _________________________________________________________ Do You Yahoo!? Get your free @yahoo.com address at Yahoo! Mail - The best web-based email! * [with cont.] (Wayb.)

Message: 3569 - Contents - Hide Contents Date: Tue, 29 Jan 2002 21:35:37 Subject: Re: new cylindrical meantone lattice From: paulerlich --- In tuning-math@y..., "monz" <joemonz@y...> wrote:> > I've added an important new lattice to my "meantone" > Dictionary entry, at the bottom: > > Internet Express - Quality, Affordable Dial Up... * [with cont.] (Wayb.) > > > This attempts to show visually how a meantone "wraps" > the lattice into a cylinder, thus closing one of the > theoretically infinite dimensions of the JI lattice. > > > Does anyone know the math that will apply sines and > cosines, to warp the lattice-lines and points so that > they actually *look* like they're sitting on the > curved face of a cylinder?Well, assuming you're looking at the cylinder in a direction perpendicular to its axis, it's pretty easy -- first of all, only use a slice of the lattice running perpendicular to, and with width of one, syntonic comma . . . then divide that width into 360 degrees . . . then, for each point, you can use either the sine or the cosine of that angle to determine the final horizontal position, while the vertical position remains the same! Of course, that will have no "perspective" effect, nor will it impart any sense of "translucency" or "opacity" to the surface of the cylinder . . .

Message: 3570 - Contents - Hide Contents Date: Tue, 29 Jan 2002 21:39:18 Subject: Re: new cylindrical meantone lattice From: paulerlich --- In tuning-math@y..., "paulerlich" <paul@s...> wrote:> --- In tuning-math@y..., "monz" <joemonz@y...> wrote: >>>> I've added an important new lattice to my "meantone" >> Dictionary entry, at the bottom: >> >> Internet Express - Quality, Affordable Dial Up... * [with cont.] (Wayb.)I still have a problem with the way you're plotting the "meantone chain", i.e., "each meantone chain itself . . ." What is the meaning of it? And what is it for meantones like 31-tET or 69-tET or LucyTuning? Does it still exist?

Message: 3571 - Contents - Hide Contents Date: Tue, 29 Jan 2002 13:48:34 Subject: Re: Proposed dictionary entry: torsion (revised) From: monz> From: paulerlich <paul@xxxxxxxxxxxxx.xxx> > To: <tuning-math@xxxxxxxxxxx.xxx> > Sent: Tuesday, January 29, 2002 12:27 PM > Subject: [tuning-math] Re: Proposed dictionary entry: torsion (revised) > >>>>> ... Here's the latest definition: >>>> >>>> Definitions of tuning terms: torsion, (c) 2002... * [with cont.] (Wayb.) >>> >>>> The definition I now have at the top (apparently the same one >> you commented on here) is the latest one Gene sent to the >> tuning-math list: >> >> >> Message 2973 >> From: "genewardsmith" <genewardsmith@j...> >> Date: Fri Jan 25, 2002 2:46 pm >> Subject: Proposed dictionary entry: torsion (revised) >> Yahoo groups: /tuning-math/message/2973 * [with cont.] >> >> >> (... did you click "refresh/reload"?) >> >> >> >> -monz >> Who is going to understand that definition? You, Monz? I meant the > definition in > > Yahoo groups: /tuning-math/message/2937 * [with cont.] > > but possibly changing "products of the proposed set of unison > vectors" to "members of the group generated by the unison vectors".OK, well, the definition I previously put at the top is still the latest revision from Gene. I haven't heard from him on this for a while now, so the Dictionary entry for "torsion" continues to grow and get messier. I've now included Paul's favorite definition at the top, and labeled it as such, with everything else still intact below it. Any comments on the lattice diagram and description of it which I added to illustrate Gene's example? I found it interesting that the torsion element appears as a line which divides the PB in half. Is this typical? Omnipresent? -monz _________________________________________________________ Do You Yahoo!? Get your free @yahoo.com address at Yahoo! Mail - The best web-based email! * [with cont.] (Wayb.)

Message: 3572 - Contents - Hide Contents Date: Tue, 29 Jan 2002 14:02:05 Subject: Re: new cylindrical meantone lattice From: monz> From: paulerlich <paul@xxxxxxxxxxxxx.xxx> > To: <tuning-math@xxxxxxxxxxx.xxx> > Sent: Tuesday, January 29, 2002 1:39 PM > Subject: [tuning-math] Re: new cylindrical meantone lattice > > > --- In tuning-math@y..., "paulerlich" <paul@s...> wrote:>> --- In tuning-math@y..., "monz" <joemonz@y...> wrote: >>>>>> I've added an important new lattice to my "meantone" >>> Dictionary entry, at the bottom: >>> >>> Internet Express - Quality, Affordable Dial Up... * [with cont.] (Wayb.) >> I still have a problem with the way you're plotting the "meantone > chain", i.e., > > "each meantone chain itself . . ." > > What is the meaning of it?I'm trying to show that, even acknowledging that the meantone cylinder is *always the same* (which I mention on the webpage), the way each fraction-of-a-comma meantone "cuts" across the cylinder is different. I don't know if there's any special meaning to that, and I know that you've argued that there isn't. But the actual amount of tempering in a meantone is an acoustical reality which can be modeled as a mathematical property that's easy to see on this lattice as the angle of spiral which the meantone chain produces across the face of the cylinder. Would you like me to create a 1/6-comma example for contrast? Using the view that's on my webpage, the 1/4-comma chain has an angle something like this: -._ '-._ '-._ Whereas 1/6-comma is nearly vertical.> 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. But my formula can easily include 2 as a prime-factor, in which case the whole lattice is stretched vertically, and EDOs simply form vertical chains. 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. 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. -monz _________________________________________________________ Do You Yahoo!? Get your free @yahoo.com address at Yahoo! Mail - The best web-based email! * [with cont.] (Wayb.)

Message: 3573 - Contents - Hide Contents Date: Tue, 29 Jan 2002 06:46:57 Subject: question for Gene From: paulerlich 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? 'Cents/error' will always increase, and 'gens/complexity' will often increase as well, but may conceivably decrease . . . or maybe this isn't meaningful at all. . . .?

Message: 3574 - Contents - Hide Contents Date: Tue, 29 Jan 2002 22:06:01 Subject: Re: Proposed dictionary entry: torsion (revised) From: paulerlich --- In tuning-math@y..., "monz" <joemonz@y...> wrote:> Is this typical?Yes -- try the 24-tone {diesis, schisma} case.> Omnipresent?No -- it depends which UVs you use to construct the parellelepiped. For example, you could restate your 24-tone {diesis, schisma} one with a {6561:6400, 128:125} basis, and then the syntonic-comma- squared will _not_ go from one diagonal of the block to the other. Try it!

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