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Message: 6550

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 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.



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 by Joe Monzo *

are exactly the same pair as the bounding vectors of the Duodene

Definitions of tuning terms: duodene, (c) 1998 by Joe Monzo *

... a *real* periodicity-block, and apparently one whose
12 pitches can "stand in" for the 24 of this torsional-block?
Hmmm...



Very curious,

-monz








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top of page bottom of page up down Message: 6551 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
top of page bottom of page up down Message: 6553 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 *is not* 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.
top of page bottom of page up down Message: 6554 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
top of page bottom of page up down Message: 6555 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?
top of page bottom of page up down Message: 6556 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 by Joe Monzo * What happened to the really nice definition Gene gave?????????????? This should be _at the top_, rather than omitted entirely!!!!!!!!!!!
top of page bottom of page up down Message: 6558 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
top of page bottom of page up down Message: 6559 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
top of page bottom of page up down Message: 6560 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 *
top of page bottom of page up down Message: 6563 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. . . .?
top of page bottom of page up down Message: 6564 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 . . .)?
top of page bottom of page up down Message: 6566 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.
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