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Message: 10604 Date: Wed, 10 Mar 2004 20:57:12 Subject: Re: Graham on contorsion From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx Graham Breed <graham@m...> wrote: > Paul Erlich wrote: > > My rambling thoughts: > > > > If the wedge product of the monzos (i.e., bimonzo) of two commas lead > > to torsion, they lead to torsion; one should make a big deal about > > this fact but not sweep it under the rug. Similarly, if the wedge > > product of the breeds (i.e., cross-breed) of two ETs lead to > > contorsion, they lead to contorsion; one should make a big deal about > > this fact but not sweep it under the rug. It's fine to then define > > the "wedgie" (why don't we call this the smith) as either of these > > wedge products with the (con)torsion removed by dividing through by > > gcd, and then insist that true temperaments correspond to a > > wedgie/smith. > > But the two cases are different. There is at least one historically > important instance of contorsion: Vicentino's enharmonic of 1555. It's about equally meaningful to say that there are at least two historically important instances of torsion: Helmholtz's schismic-24, and Groven's schismic-36. > Torsion of unison vectors is a different matter. I don't know of any > cases, theoretical or otherwise, in which torsion is desired in a > tempered MOS. I'm not even sure what it would mean. A periodicity > block with torsion certainly doesn't correspond to an MOS with > contorsion. As we've discussed before, tempering a torsional block results in an nMOS wherever tempering its non-torsional equivalent would result in an MOS -- such as the two schismic cases just mentioned. > Where you don't supply a chromatic unison vector, it's even > more likely that you simply wanted a torsion free scale that tempers out > all the commatic unison vectors, and that's what you get. Maybe if you > supplied the chromatic unison vector, you would want to be warned of > torsion. But you aren't. So there. Sounds like you're assuming linear.
Message: 10605 Date: Wed, 10 Mar 2004 21:00:28 Subject: Re: Please remind me From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "Paul G Hjelmstad" <paul.hjelmstad@u...> wrote: > I need a quick refresher. Would someone please remind me how > period-and-generators are found, from two temperaments (say 12&19) > without using wedgies (Not that I have anything against them...) One way: The period is 1 octave divided by the gcd of the 2 numbers. The generator corresponds to the interval in one ET that is the closest to an interval in the other ET, without being exactly identical.
Message: 10611 Date: Wed, 10 Mar 2004 22:55:51 Subject: Re: "Vicentino" temperament From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote: > --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> > wrote: > > I'm proposing "Vicentino" as a name for the 7 and 11 limit > > temperaments (with identical TOP generators, so they should get the > > same name) one obtains from the 9/31 generator. The obvious objection > > to this is that 7-limit, and certainly 11-limit, is not what he had in > > mind. > > Actually, I'm not so sure this is true--didn't someone say Vicentino > considered 11/9 to be a consonance? No -- at least not if you mean he made any reference to any ratios of 11. He did find neutral thirds somewhat more consonant than most of the other novel intervals of his 31-tone system, though.
Message: 10613 Date: Wed, 10 Mar 2004 23:52:56 Subject: Re: Dual L1 norm deep hole scales From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote: > --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> > wrote: > > --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> > wrote: > > > > > The raison d'etre for these scales would seem to be if someone's > > > looking for a good JI scale to use, without regard to its melodic > > > structure. What does he or she care how we measure error? > > > > It seems to me the duality provides another and convincing raison > d'etre. > > Can you outline this raison d'etre, please? I have no idea what it > could be. > > > > > The > > > > Hahn norm corresponds to the minimax error, > > > > > > I thought the "Hahn scales" we were coming up with were minimax > in > > > terms of note-classes. So wouldn't the dual of that be L1 in > terms of > > > error? > > > > The situation is confusing, because we've got three different (at > > minimum) norms to contend with. We have a norm which is Linf > (minimax) > > or Euclidean (rms.) This leads to *another* norm by applying it to > the > > consonances. > > > > Starting with the Euclidean norm (norm #1) (x^2+y^2+z^2)^(1/2), > > How is this used below? > > > we > > apply it to 3,5,7,5/3,7/3,7/5 and get the following: > > > > x^2+y^2+z^2+(y-x)^2+(z-x)^2+(z-y)^2 = 3(x^2+y^2+z^2)-2(xy+xz+yz) > > > > Taking the square root of this is norm #2. > > This is the rms error criterion (or rms 'loss function'), right? > > > Now form the symmetric > > matrix for the above, and take the inverse: > > > > [[3,-1,-1],[-1,3,-1],[-1,-1,3]]^(-1) = > > [[1/2,1/4,1/4],[1/4,1/2,1/4],[1/4,1/4,1/2]] > > > > The quadratic form for this last is > > > > (a^2+b^2+c^2+ab+ac+bc)/2, which gives us norm #2; of course we can > > rescale by multiplying by 2. > > So you're saying the dual of rms error is euclidean norm in the > symmetric oct-tet lattice, yes? > > > The same situation, three *different* norms, we find if we start > with > > norm #1 being the L_inf norm. Then norm #2 has a unit ball which is > > the convex hull of the twelve consonances--ie, a cuboctahedron. From > > the 14 faces of this we get norm #3. > > I guess I must have been wrong above. What's the difference between > this latter #2 and #3? And what about duality and the fact that on > your Tenney page, you say that the dual of L1 is L_inf, but today you > seem to be saying something different (is it the triangularity of the > lattice that alters the situation)? Still wondering about all of this, Paul
Message: 10616 Date: Thu, 11 Mar 2004 10:04:17 Subject: Re: "Vicentino" temperament From: Graham Breed Paul Erlich wrote: > No -- at least not if you mean he made any reference to any ratios of > 11. He did find neutral thirds somewhat more consonant than most of > the other novel intervals of his 31-tone system, though. There is one reference to a ratio of 11, in Book II, Chapter 4 (p.124) where he gives the major tenth as 11:8, that is an octave above 6:5. Possibly he's adding 6:5 to 5:3. Otherwise, along with the usual extended 5-limit ratios, he gives the following for intervals in his enharmonic scale: 14:13 for a diatonic semitone (Book V, Chapter 60, p.433) 21:20 for a chromatic semitone (Book V, Chapter 60, p.433) 13:12 for a neutral tone (Book V, Chapter 61, p.434 and again on p.435) 8:7 for a supermajor second (Book V, Chapter 61, p.436) 5.5:4.5 for a neutral third (Book V, Chapter 62, p.437) 4.5:3.5 for a supermajor third (Book V, Chapter 62, p.439) I'd have to check through to see if he ever gives 16:15 and 25:24 for the semitones. Certainly in this section of Book V he doesn't. 14:13 and 13:12 probably come from dividing the interval 7:6 arithmetically. 5.5:4.5 is probably the average of 5:4 and 6:5. If he mulitiplied through by 2, he'd get 11:9. But he doesn't, and says it's irrational. 4.4:3.5 is probably the average of 5:4 and 4:3, and is the same as 9:7. Neutral thirds aren't used as vertical intervals in any of his example compositions. However, because he uses implicit accidentals (and there are misprints) it'd be difficult to know if he did intend one. Still, the implication of such a method is that 5-limit harmony is assumed. The examples in Book III, Chapter 50 (p.208) are probably intended to show neutral thirds (accidentals are implied). In each case, a step of a chromatic semitone is broken into two dieses, so the progression is minor -> neutral -> major. There are two passages concerning neutral thirds in music. Book I, Chapter 28 compares the neutral third to the major third, but only in melody. Book V, Chapter 8 (pp.336-7) has the famous paragraph> "If a player fails to pay attention to the proximate and most proximate consonances, he will be deceived by them, for they are so proximate to imperfect consonances that they seem identical to them. Thus, when playing the archicembalo, you may use the third larger than the minor third, that is, the proximate third that is one minor diesis larger than the minor third. This step resembles the major third without being a major third, and the minor third without being a minor third. The minor third we use below the low A la mi re [1A] is the second G sol re ut [2F#]. Its proximate is on the third F fa ut on the fourth rank [4F^], and it seems better than the minor third because it is not as weak as the minor third in comparison to the major third. Still, the proximate is somewhat weaker than the major third because it is smaller by one enharmonic diesis. Thus, the proximate or most proximate to the minor third sounds acceptable and can be played. I believe that some people sing proximate and most proximate thirds as they sharpen these minor and major consonances when performing compositions, and they do not create discords despite the fact that the former are not the same size as the latter." There's a translator or editor's note to the effect that he might be talking about the second tuning of the archicembalo here. So the (most) proximate minor third will be an exact 6:5 (is that right?) rather than a neutral third. That would make him a tad more obtuse than usual, but I wouldn't rule it out. Graham ________________________________________________________________________ ________________________________________________________________________ ------------------------------------------------------------------------ Yahoo! Groups Links <*> To visit your group on the web, go to: Yahoo groups: /tuning-math/ * <*> To unsubscribe from this group, send an email to: tuning-math-unsubscribe@xxxxxxxxxxx.xxx <*> Your use of Yahoo! Groups is subject to: Yahoo! Terms of Service *
Message: 10617 Date: Fri, 12 Mar 2004 18:05:07 Subject: Re: Dual L1 norm deep hole scales From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote: > --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote: > > > Still wondering about all of this, > > Paul > > That's not a very specific question. Everything with a question mark after it was a question.
Message: 10618 Date: Fri, 12 Mar 2004 18:09:56 Subject: Re: Breeding From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote: > My 2401/2400-planar temperament piece is coming along, but it seems to > me that if someone is actually going to present a piece of music in > it, it should have a name. Somehow Graham acquired the rights to this > important comma, By drawing lattices projected along its direction, similar to what you and Paul Hj. were just discussing.
Message: 10620 Date: Fri, 12 Mar 2004 18:46:49 Subject: Re: Breeding From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote: > --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote: > > --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> > > wrote: > > > > My 2401/2400-planar temperament piece is coming along, but it seems > > to > > > me that if someone is actually going to present a piece of music in > > > it, it should have a name. Somehow Graham acquired the rights to > > this > > > important comma, > > > > By drawing lattices projected along its direction, similar to what > > you and Paul Hj. were just discussing. > > When was that? During the Miracle rediscovery (2001).
Message: 10621 Date: Fri, 12 Mar 2004 20:11:49 Subject: Re: Breeding From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote: > My 2401/2400-planar temperament piece is coming along, but it seems to > me that if someone is actually going to present a piece of music in > it, it should have a name. Somehow Graham acquired the rights to this > important comma, which means some name with a "breed" in it is a > possibility. Breeding? Why "-ing"? This seems to suggest a process; in fact I used this term to describe a process here: Yahoo groups: /tuning/message/28558 * not to mention much more recently on this list.
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