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Message: 3525 - Contents - Hide Contents Date: Fri, 25 Jan 2002 21:00:09 Subject: Linear temperament consistency? From: genewardsmith Does anyone apply the consistency concept to linear temperaments, or even to higher dimensions? It seems like that would make sense also.
Message: 3526 - Contents - Hide Contents Date: Fri, 25 Jan 2002 00:54:59 Subject: Re: twintone, paultone From: paulerlich --- In tuning-math@y..., "clumma" <carl@l...> wrote:>>>> You can't hear consistency, so why is this relevant? >>>>>> You can hear consistency, when neighboring chords involve >>> different approximations to the same interval. >>>> That wouldn't happen in the case Gene is talking about. >> I didn't say it would. Gene's was a general dismissal of > consistency.You can have neighboring chords involve different approximations to the same interval even in a consistent tuning. I see this happening in 76-tET, where one could modulate between twintone, meantone, double-diatonic, as well as other systems.
Message: 3527 - Contents - Hide Contents Date: Fri, 25 Jan 2002 23:23:23 Subject: Re: OUR PAPER From: dkeenanuqnetau --- In tuning-math@y..., "paulerlich" <paul@s...> wrote:> The latest list of 5-limit temperaments is fine by me,Which list is that exactly?> though if > Graham and Dave are into the idea of a stronger penalty for > complexity, sacrificing flatness, I'll side with them against Gene.I'd describe it as giving a weaker reward for tiny errors (sub-cent), but it would have pretty much the same effect.
Message: 3528 - Contents - Hide Contents Date: Fri, 25 Jan 2002 01:25:33 Subject: Re: twintone, paultone From: clumma>You can have neighboring chords involve different approximations >to the same interval even in a consistent tuning. I see this >happening in 76-tET, where one could modulate between twintone, >meantone, double-diatonic, as well as other systems.Example? I don't see how this could happen, unless it involved: () Switching between subsets of the ET, which is cheating. () Invoking higher-order approximations (ie, 10:12:15->16:19:24). -Carl
Message: 3529 - Contents - Hide Contents Date: Fri, 25 Jan 2002 02:51:41 Subject: Re: Blackjack standard... From: paulerlich --- In tuning-math@y..., "jpehrson2" <jpehrson@r...> wrote:> Hi Paul! > > I'm assuming you mean the lattices in the "standard" key C-G-D-A that > Dave Keenan kindly refined for us, yes? > > *That's* the lattice I'm currently using now... > > JosephHi Joseph . . . Well, I think I'll tend to use a different, non-diatonic notation altogether, as the modified Sims notation is a bit too "hairy" for the "pretty" book I'd like to produce. Of course, a separate, more practically oriented Blackjack paper would be great too . . . someday, someday . . .
Message: 3530 - Contents - Hide Contents Date: Fri, 25 Jan 2002 01:31:46 Subject: Re: twintone, paultone From: paulerlich --- In tuning-math@y..., "clumma" <carl@l...> wrote:> I don't see how this could happen, unless it involved: > > () Switching between subsets of the ET, which is cheating.Cheating? Jeez, can't I modulate from diatonic to diminished to whole- tone to augmented in 12-tET?
Message: 3531 - Contents - Hide Contents Date: Fri, 25 Jan 2002 23:36:11 Subject: Re: twintone, paultone From: clumma>>> >ou can't hear consistency, so why is this relevant? >>>> You can hear consistency, when neighboring chords involve >> different approximations to the same interval. > >>I can approximate 1-5/4-3/2-7/4 by 0-18-32-44 or 0-18-32-45 in >55-et, and so I can claim to "hear" inconsistency.Not really. For isolated chords, you just always use the best approximation (say, minimum rms). As long as you're happy with that approximation, and you've based it on the chords you actually want to use, not just the dyads involved (as some early investigators did), then you're golden. The "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. Some people think commas are a "feature not a bug", others prefer to temper them out. Others (apparently both you and I) think both approaches are valid. And, as I said...>> Which is not to say that this is in any way "bad"....even tempering some commas out while inventing news ones can probably be interesting. But for me, as a composer, this is just too confusing. Thus, I restrict myself to consistent ets. Consistency is also useful as a "badness" measure. It may not be ideal for looking at ets up to 10 million, as some optimum "flat" measure may be, but for any kind of goodness per notes you'd actually care about from a pragmatic standpoint, it is more than adequate.>I can also approxmiate it by 0-6-11-15 or 0-6-11-16 in 19-et; >can I also claim to hear the 7-inconsistency of the 19-et? Why >or why not?Depending on the context, the former chord is more likely to approximate 4:5:6:7 or 1/1-5/4-3/2-12/7, and the latter chord 1/1-5/4-3/2-9/5 or 12:15:18:22... in other words, I'd guess these would normally sound like different chords when juxtaposed.>What about both 0-10-18-25 and 0-10-18-26 in the 31-et? Can >you hear inconsistency here?The former chord is clearly 4:5:6:7. The latter chord would attract the same suspects as 0-6-11-16 in 19-tET, and as Paul points out, may be tuned any number of ways in diatonic music since it functions as a dissonance there (1/1-5/4-3/2-16/9 often works well). -Carl
Message: 3532 - Contents - Hide Contents
Date: Fri, 25 Jan 2002 01:54:35
Subject: OUR PAPER
From: paulerlich
Hello?
Let's push forward, shall we?
Graham, do you agree with the way Gene's doing things?
If so, you guys have a plurality, against Dave and myself, who both
seem to be resisting in different areas.
It's really time to get this stuff published in some form -- who
knows how many university course notes it's appearing in already :)
The latest list of 5-limit temperaments is fine by me, though if
Graham and Dave are into the idea of a stronger penalty for
complexity, sacrificing flatness, I'll side with them against Gene.
Then we'll need similar lists for {2,3,5,7}, {2,3,7}, {2,5,7}, and
{3,5,7} -- always keeping the first prime as the interval of
equivalence, for brevity's sake. Additional useful info would include
a list of proper and improper MOSs (actually, a horagram might be
best) and lattices wherever feasible.
And all this is only part IV of our paper . . .
Message: 3533 - Contents - Hide Contents Date: Fri, 25 Jan 2002 22:41:29 Subject: Re: twintone, paultone From: clumma>>> >ou can't hear consistency, so why is this relevant? >>>> You can hear consistency, when neighboring chords involve >> different approximations to the same interval. > >>I can approximate 1-5/4-3/2-7/4 by 0-18-32-44 or 0-18-32-45 in >55-et, and so I can claim to "hear" inconsistency.Not really. For isolated chords, you just always use the best approximation (say, minimum rms). As long as you're happy with that approximation, and you've based it on the chords you actually want to use, not just the dyads involved (as some early investigators did), you're golden. The "problem" occurs when modulating from the best approx. of one chord to the best approx. of another, which sometimes creates anomalous (as in, non-existent in JI) commas. Some people think commas are a "feature not a bug", others prefer to temper them out. Others (apparently both you and I) think both approaches are valid. And, as I said...>> Which is not to say that this is in any way "bad"....even tempering some commas out while inventing news ones can probably be interesting. But for me, as a composer, this is just too confusing. Thus, I restrict myself to consistent ets. Consistency is also useful as a "badness" measure. It may not be ideal for looking at ets up to 10 million, as some optimum "flat" measure may be, but for any kind of goodness per notes you'd actually care about from a pragmatic standpoint, it is more than adequate.>I can also approxmiate it by 0-6-11-15 or 0-6-11-16 in 19-et; >can I also claim to hear the 7-inconsistency of the 19-et? Why >or why not?Depending on the context, the former chord is more likely to approximate 4:5:6:7 or 1/1-5/4-3/2-12/7, and the latter chord 1/1-5/4-3/2-9/5 or 12:15:18:22... in other words, I'd guess these would normally sound like different chords when juxtaposed.>What about both 0-10-18-25 and 0-10-18-26 in the 31-et? Can >you hear inconsistency here?The former chord is clearly 4:5:6:7. The latter chord would attract the same suspects as 0-6-11-16 in 19-tET, and as Paul points out, may be tuned any number of ways in diatonic music since it functions as a dissonance there (1/1-5/4-3/2-16/9 often works well). -Carl
Message: 3534 - Contents - Hide Contents Date: Fri, 25 Jan 2002 02:06:07 Subject: Re: OUR PAPER From: jpehrson2 --- In tuning-math@y..., "paulerlich" <paul@s...> wrote: Yahoo groups: /tuning-math/message/2958 * [with cont.]> Hello? > > Let's push forward, shall we? > > Graham, do you agree with the way Gene's doing things? > > If so, you guys have a plurality, against Dave and myself, who both > seem to be resisting in different areas. > > It's really time to get this stuff published in some form -- who > knows how many university course notes it's appearing in already :) > > The latest list of 5-limit temperaments is fine by me, though if > Graham and Dave are into the idea of a stronger penalty for > complexity, sacrificing flatness, I'll side with them against Gene. > > Then we'll need similar lists for {2,3,5,7}, {2,3,7}, {2,5,7}, and > {3,5,7} -- always keeping the first prime as the interval of > equivalence, for brevity's sake. Additional useful info would include > a list of proper and improper MOSs (actually, a horagram might be > best) and lattices wherever feasible. > > And all this is only part IV of our paper . . .Well, this is all very exciting, and I saw it posted on the Tuning List. However, it magically disappeared, so I figured Paul meant to post it to Tuning Math instead. I would propose (if I may humbly do that for a micromini second, or a mathmicromini second) that there are actually *two* papers... One the "intense" "real" one, and the other a kind of "synopsis" along the lines of Paul Erlich's *very* fine... in fact *very, very* fine "The Forms of Tonality" which was a very readable and *broadly- based* effort, directed to the larger microtonal community. And it had nice *pictures* too. Whaddya say?? Rather than "diluting" the effort, I think it will just *focus* things on the new developments. Or, similarly, such a "preamble" or "synopsis" could be on the Web similar to Paul's "Gentle Introduction" efforts... Anyway, that's what I'm hoping for... Not to "spoil" the progress over here... but just to share in the excitement! Joseph Pehrson
Message: 3535 - Contents - Hide Contents Date: Fri, 25 Jan 2002 02:09:42 Subject: Re: OUR PAPER From: paulerlich --- In tuning-math@y..., "jpehrson2" <jpehrson@r...> wrote:> One the "intense" "real" one, and the other a kind of "synopsis" > along the lines of Paul Erlich's *very* fine... in fact *very, very* > fine "The Forms of Tonality" which was a very readable and *broadly- > based* effort, directed to the larger microtonal community. And it > had nice *pictures* too. > > Whaddya say??I'll see to that -- but of course that'll be a paper (or book, encompassing "The Forms of Tonality" too) I produce *alone*. As you can imagine, the lattices I created for Blackjack and that you are already using will appear in it . . .> > Rather than "diluting" the effort, I think it will just *focus* > things on the new developments. Or, similarly, such a "preamble" > or "synopsis" could be on the Web similar to Paul's "Gentle > Introduction" efforts... Yup!
Message: 3536 - Contents - Hide Contents Date: Fri, 25 Jan 2002 02:17:11 Subject: Re: OUR PAPER From: jpehrson2 --- In tuning-math@y..., "paulerlich" <paul@s...> wrote: Yahoo groups: /tuning-math/message/2960 * [with cont.]> --- In tuning-math@y..., "jpehrson2" <jpehrson@r...> wrote: >>> One the "intense" "real" one, and the other a kind of "synopsis" >> along the lines of Paul Erlich's *very* fine... in fact *very, > very*>> fine "The Forms of Tonality" which was a very readable and *broadly- >> based* effort, directed to the larger microtonal community. And it >> had nice *pictures* too. >> >> Whaddya say?? >> I'll see to that -- but of course that'll be a paper (or book, > encompassing "The Forms of Tonality" too) I produce *alone*. As you > can imagine, the lattices I created for Blackjack and that you are > already using will appear in it . . . >>>> Rather than "diluting" the effort, I think it will just *focus* >> things on the new developments. Or, similarly, such a "preamble" >> or "synopsis" could be on the Web similar to Paul's "Gentle >> Introduction" efforts... > > Yup! Great, Paul!I'll be anxious to see all this! Go team! JP
Message: 3537 - Contents - Hide Contents Date: Fri, 25 Jan 2002 02:43:51 Subject: Blackjack standard... From: jpehrson2 --- In tuning-math@y..., "paulerlich" <paul@s...> wrote: Yahoo groups: /tuning-math/message/2960 * [with cont.]> --- In tuning-math@y..., "jpehrson2" <jpehrson@r...> wrote: >>> One the "intense" "real" one, and the other a kind of "synopsis" >> along the lines of Paul Erlich's *very* fine... in fact *very, > very*>> fine "The Forms of Tonality" which was a very readable and *broadly- >> based* effort, directed to the larger microtonal community. And it >> had nice *pictures* too. >> >> Whaddya say?? >> I'll see to that -- but of course that'll be a paper (or book, > encompassing "The Forms of Tonality" too) I produce *alone*. As you > can imagine, the lattices I created for Blackjack and that you are > already using will appear in it . . . >> Hi Paul!I'm assuming you mean the lattices in the "standard" key C-G-D-A that Dave Keenan kindly refined for us, yes? *That's* the lattice I'm currently using now... Joseph
Message: 3538 - Contents - Hide Contents Date: Fri, 25 Jan 2002 04:17:20 Subject: Re: twintone, paultone From: genewardsmith --- In tuning-math@y..., "clumma" <carl@l...> wrote:> I wrote... >>>> You can't hear consistency, so why is this relevant? >>>> You can hear consistency, when neighboring chords involve >> different approximations to the same interval. >> Which is not to say that this is in any way "bad".I can approximate 1-5/4-3/2-7/4 by 0-18-32-44 or 0-18-32-45 in 55-et, and so I can claim to "hear" inconsistency. I can also approxmiate it by 0-6-11-15 or 0-6-11-16 in 19-et; can I also claim to hear the 7-inconsistency of the 19-et? Why or why not? What about both 0-10-18-25 and 0-10-18-26 in the 31-et? Can you hear inconsistency here?
Message: 3539 - Contents - Hide Contents Date: Fri, 25 Jan 2002 04:21:48 Subject: Re: OUR PAPER From: genewardsmith --- In tuning-math@y..., "paulerlich" <paul@s...> wrote:> Hello? > > Let's push forward, shall we? > > Graham, do you agree with the way Gene's doing things? > > If so, you guys have a plurality, against Dave and myself, who both > seem to be resisting in different areas.One possibility would be for me to write up my own approach to the theory part, and have that as a separate paper. Would they publish it?
Message: 3540 - Contents - Hide Contents Date: Fri, 25 Jan 2002 09:15:43 Subject: Two more Japanese papers From: genewardsmith Here is a brief account of the other two papers John sent me. The first is another by Kazuo Kondo. Here he decides to construct the periodicity block for the kleisma and the schisma, and seems to want to detemper it into 53 equal or something of the sort. However, he somehow manages to convince himself the PB has 54 tones, even though it should have 53, so he completely wigs out and removes a corner from the block, and then treats it as if it was a problem in chemistry concerning imperfect crystals. He produces a lot of nonsense about non-Riemannian geometry and screw dislocations, which may be describing his mental processes, and he ends by an unattributed quotation saying that tempering out commas is a good idea. The second paper is non-Kondo, and makes more sense than anything Kondo seems capable of. The author, Nobuyuki Otsu, takes sixteen temperaments, such as meantone, Kirnberger 1-3, Werckmeister 1, C and 3, and so forth, and subjects them to various kinds of statistical analysis designed to show relationships. A cluster analysis gives a dendrogram, (sometimes called cladogram) of the relationships in the form of a tree. Factor analysis sorts of methods, involving eigenvalue decomposition, are also applied, and he shows diagrams of the projections onto the first few eigenspaces. He also applies the same analysis to the tones themselves, so that you caan find out if C would rather hang with G or with F#. The results, unsurprisingly, tell us that meantones are like other meantones, and well-temperaments like other well-temperaments, but the details might interest someone.
Message: 3541 - Contents - Hide Contents Date: Fri, 25 Jan 2002 11:11 +0 Subject: Re: Our Paper From: graham@xxxxxxxxxx.xx.xx In-Reply-To: <a2qdsr+8o3t@xxxxxxx.xxx> paulerlich wrote:> Graham, do you agree with the way Gene's doing things? > > If so, you guys have a plurality, against Dave and myself, who both > seem to be resisting in different areas.What differences are you seeing? I thought we were in broad agreement. On suggestion I would like to make, though. If the intention of this paper is to concentrate on unison vectors, I'd like it to avoid mentioning the method of constructing linear temperaments from equal temperaments. So long as acknowledgements are in place, you can leave me off that one. Then I can write the ET method up, along with whoever's interested, for a future issue of Xenharmonikon. This would concentrate more on the practicality than the maths (most of which you'll already have covered) and so has to be left until I have more practical experience of the scales.> The latest list of 5-limit temperaments is fine by me, though if > Graham and Dave are into the idea of a stronger penalty for > complexity, sacrificing flatness, I'll side with them against Gene.That's all froth as far as I'm concerned.> Then we'll need similar lists for {2,3,5,7}, {2,3,7}, {2,5,7}, and > {3,5,7} -- always keeping the first prime as the interval of > equivalence, for brevity's sake. Additional useful info would include > a list of proper and improper MOSs (actually, a horagram might be > best) and lattices wherever feasible.I can calculate these, with whatever metrics you like, if you can't work out the Python code. I'm thinking of adding CGIs to do this kind of thing, but it would mean restructuring the code. I still have to do the automatic search on unison vectors as well. Is that a priority?> And all this is only part IV of our paper . . .It is getting bloated. I suggest you decide what really needs to be published now, and get cracking. Or perhaps an introduction to some of the new temperaments for the imminent Xenharmonikon, and leave the mathematical details for a formal journal (but get your foot in the door as soon as possible). Graham
Message: 3542 - Contents - Hide Contents Date: Sat, 26 Jan 2002 13:32 +0 Subject: Re: twintone, paultone From: graham@xxxxxxxxxx.xx.xx Carl:>> The "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. Gene:> 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.No. If you're using a regular temperament, you can't be using 34-et. 34-et is an inconsistent, equal temperament. 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? If you're not going to make use of the inconsistency, I don't see the point in using 34-equal at all. Graham
Message: 3543 - Contents - Hide Contents Date: Sat, 26 Jan 2002 13:32 +0 Subject: Re: Linear temperament consistency? From: graham@xxxxxxxxxx.xx.xx genewardsmith wrote:> Does anyone apply the consistency concept to linear temperaments, or > even to higher dimensions? It seems like that would make sense also.I've thought about it. You'd have to confine it to a particular scale, because any non-just interval can be approximated better with enough steps of an irrational generator. One detail is that the 31 note MOS of 1/4 comma meantone would be inconsistent. The unofficial fifth is tuned better than the official one. Linear mappings of equal temperaments will tend to be ambiguous, because you can get to the same intervals by going round the circle in the other direction. Graham
Message: 3544 - Contents - Hide Contents Date: Sat, 26 Jan 2002 22:37:38 Subject: Re: Proposed dictionary entry: torsion (revised) From: monz Hey Gene,> From: genewardsmith <genewardsmith@xxxx.xxx> > To: <tuning-math@xxxxxxxxxxx.xxx> > Sent: Saturday, January 26, 2002 8:04 PM > Subject: [tuning-math] Re: Proposed dictionary entry: torsion (revised) > > > --- In tuning-math@y..., "monz" <joemonz@y...> wrote: >>> Is there some special reason to use the ... >> >> UVs = >> <648/625, 2048/2025> = [3 4 -4], [11 -4 -2] >> >> adj = >> [-24 0 0] >> [-38 -2 4] >> [-56 4 4] >> >> ... PB as an example, instead of the one I already put into >> the definition? >> I wanted an example, and I cooked this one up, that's all. > The only advantage of it I can see is that it uses simpler commas.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.) I'd like to leave in the bit which explains how to calculate the torsion factor from the gcd of the determinants of the minors. 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. -monz _________________________________________________________ Do You Yahoo!? Get your free @yahoo.com address at Yahoo! Mail - The best web-based email! * [with cont.] (Wayb.)
Message: 3545 - Contents - Hide Contents Date: Sat, 26 Jan 2002 13:09:45 Subject: Re: Proposed dictionary entry: torsion (revised) From: monz Hi Gene,> From: genewardsmith <genewardsmith@xxxx.xxx> > To: <tuning-math@xxxxxxxxxxx.xxx> > Sent: Friday, January 25, 2002 11:46 AM > Subject: [tuning-math] Proposed dictionary entry: torsion (revised) > > > torsion > > Torsion describes a condition wherein an > independent set of n unison vectors {u1, u2, ..., un} > (<uvector.htm>) defines a non-epimophic (epimorphic.htm>) > periodicity block, because of the existence a torsion > element, meaning an interval which is not the product > > u1^e1 u2^e2 ... un^en > > of the set of unison vectors raised to (positive, > negative or zero) integral powers, but some integer > power of which is. An example would be a block > defined by 648/625 and 2048/2025; here 81/80 is > not a product of these commas, but > (81/80)^2 = (648/625) (2048/2025)^(-1).Thanks for the revised definition! Definitions of tuning terms: torsion, (c) 2002... * [with cont.] (Wayb.) -24 0 0 -38 -2 4 -56 4 4 Is there some special reason to use the ... UVs = <648/625, 2048/2025> = [3 4 -4], [11 -4 -2] adj = [-24 0 0] [-38 -2 4] [-56 4 4] ... PB as an example, instead of the one I already put into the definition? -- that one is also the same one which Paul used as an illustration when the torsion discussion first began on this list: UVs = <128/125, 32805/32768> = [7 0 -3], [-15 8 1] adj = [24 0 0] [38 1 3] [56 -8 0] My website already has several webpages and lattice diagrams of this PB, and I'd like to link to them. For example, see the first graphic at: Internet Express - Quality, Affordable Dial Up... * [with cont.] (Wayb.) If I use your PB in the definition, I'll have to create new diagrams for it. ... Not that I don't want to do that anyway ... but since I already have diagrams of a torsional PB, I'd like to employ them right away as illustration. Also, please tell me if I should keep anything that appears below the row of asterisks in the definition. Otherwise it's trash, but I don't fully understand torsion yet, so I'm being careful and only deleting what you tell me to delete. -monz _________________________________________________________ Do You Yahoo!? Get your free @yahoo.com address at Yahoo! Mail - The best web-based email! * [with cont.] (Wayb.)
Message: 3546 - Contents - Hide Contents Date: Sat, 26 Jan 2002 13:30:58 Subject: Re: Proposed dictionary entry: torsion (revised) From: monz ----- Original Message ----- From: monz <joemonz@xxxxx.xxx> To: <tuning-math@xxxxxxxxxxx.xxx> Sent: Saturday, January 26, 2002 1:09 PM Subject: Re: [tuning-math] Proposed dictionary entry: torsion (revised)> Thanks for the revised definition! > Definitions of tuning terms: torsion, (c) 2002... * [with cont.] (Wayb.) > > > > -24 0 0 > -38 -2 4 > -56 4 4 > > Is there some special reason to use the ...Oops ... my bad. That matrix didn't belong there, it means nothing where it is, and should have been deleted. Sorry. -monz _________________________________________________________ Do You Yahoo!? Get your free @yahoo.com address at Yahoo! Mail - The best web-based email! * [with cont.] (Wayb.)
Message: 3547 - Contents - Hide Contents Date: Sat, 26 Jan 2002 22:19:51 Subject: Re: twintone, paultone From: genewardsmith --- In tuning-math@y..., graham@m... wrote:> No. If you're using a regular temperament, you can't be using 34-et.This is just wrong; the whole thing is beginning to seem like another of those "religion" deal.> 34-et is an inconsistent, equal temperament.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. 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?If you are using a 10-tone subset of 34 et, then they won't be there. 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. In the 34-et twintone, 96 is mapped to 96/54 mod 17 = -2, the familiar 64/63 approximation, whereas 95 maps to 95/54 mod 17 = -8, which isn't even a part of the 10 or 12 note twintone scales. This is quite analogous to the situation with the diatonic and standard septimal versions of 7-limit meantone; if g31 is the map [31,49,72,88] instead of the usual h31 of [31,49,72,87], then h12^g31 gives the temperament [-1,-4,2,16,-6,-4] rather than h12^h31 = [-1,-4,-10,-12,13,-4]. The first is the diatonic version of 7-limit meantone, and may be regarded as the standard Western temperament of the last few centuries; the second uses the much better version of 7/4 which the 31-et allows, but it makes no appearance on the diatonic scale, as a glance at the wedgie shows. 31 equal can deal with either.> If you're not going to make use of the inconsistency, I don't see the > point in using 34-equal at all.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. 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.
Message: 3548 - Contents - Hide Contents Date: Sat, 26 Jan 2002 02:18:05 Subject: Re: twintone, paultone From: genewardsmith --- In tuning-math@y..., "clumma" <carl@l...> wrote:> The "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.
Message: 3549 - Contents - Hide Contents Date: Sun, 27 Jan 2002 04:04:43 Subject: Re: Proposed dictionary entry: torsion (revised) From: genewardsmith --- In tuning-math@y..., "monz" <joemonz@y...> wrote:> Is there some special reason to use the ... > > UVs = > <648/625, 2048/2025> = [3 4 -4], [11 -4 -2] > > adj = > [-24 0 0] > [-38 -2 4] > [-56 4 4] > > ... PB as an example, instead of the one I already put into > the definition?I wanted an example, and I cooked this one up, that's all. The only advantage of it I can see is that it uses simpler commas.
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