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Message: 6525

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.


top of page bottom of page up down Message: 6526 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
top of page bottom of page up down Message: 6527 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... > > Joseph Hi 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 . . .
top of page bottom of page up down Message: 6528 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?
top of page bottom of page up down Message: 6529 Date: Fri, 25 Jan 2002 23:36:11 Subject: Re: twintone, paultone From: clumma >>>You 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
top of page bottom of page up down Message: 6530 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 . . .
top of page bottom of page up down Message: 6531 Date: Fri, 25 Jan 2002 22:41:29 Subject: Re: twintone, paultone From: clumma >>>You 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
top of page bottom of page up down Message: 6532 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 * > 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
top of page bottom of page up down Message: 6533 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!
top of page bottom of page up down Message: 6534 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 * > --- 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
top of page bottom of page up down Message: 6535 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 * > --- 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
top of page bottom of page up down Message: 6539 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
top of page bottom of page up down Message: 6540 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
top of page bottom of page up down Message: 6541 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
top of page bottom of page up down Message: 6542 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 by Joe Monzo * 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 Setup *
top of page bottom of page up down Message: 6543 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 by Joe Monzo * -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: more on the duodene * 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 Setup *
top of page bottom of page up down Message: 6544 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 by Joe Monzo * > > > > -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 Setup *
top of page bottom of page up down Message: 6549 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 by Joe Monzo * 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 Setup *
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