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Message: 10600 - Contents - Hide Contents Date: Wed, 10 Mar 2004 19:33:50 Subject: Re: Graham on contorsion From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx Graham Breed <graham@m...> wrote:> The question is perfectly meaningful, and if you don't understand it you > can't have been paying attention. Go back and re-read the message in > which I asked it.My point was that it shouldn't give either family. If you read the stuff I've been posting off and on about white and black keys, that treats the question as one about scales which removes consideration of temperament altogether. If you want to introduce temperament, then starting from 5/24 and asking what temperament that seems to be can be done, and at that point you might also enforce compatibilty with a pair of vals if you wanted to. I really dislike the idea that a 5-val and a 19-val in the 5-limit ought to produce a contorted "temperament", but if you want to go that route you can do it using wedgies if you like, since the information you deem so crucial is found simply from the fact that 5 is about 1/4 of 19, and not (like 7 and 17) between 1/2 and 1/3.> If you don't want contorsion, then you shouldn't be following this > thread, which is about contorsion (and clearly says so in the subject line).I wrote the subject line, so presumably I know what it is about--which were your remarks suggesting contorted "temperaments" were dubious at best.

Message: 10601 - Contents - Hide Contents Date: Wed, 10 Mar 2004 19:53:14 Subject: Re: Octave equivalent calculations (Was: Hanzos From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx Graham Breed <graham@m...> wrote:> Well, nice to see that you now think my program is "a good thing". But > how does that suddenly stop it from being a temperament program? It > produces, as far as I can tell, a homomorphic mapping from what may be > rational numbers to a tone group with a smaller rank, in line with your > definition here: > > Regular Temperaments * [with cont.] (Wayb.)No, in fact it doesn't. Half of the notes are not being mapped to. If there was something to correspond to 2/sqrt(3), half of a fourth, you would be OK but the whole point of contorsion is that there isn't.> That's somewhat vague as to whether contorsion is allowed. First it > says that an "icon" has to be epimorphic.Nothing about it is vague that I can see. The epimorphic mapping, as stated, is from the p-limit for some prime p to the icon. But then "given an icon" we> have to "find a homomorphic mapping".And the homomorphic mapping is from the icon to cents or something of that sort. This is the tuning stage and it would not make sense to ask that this be epimorphic. Well, you can certainly find the> mapping for Vicentino's 7&31 system using the icon for meantone, so I > suppose it must be a temperament.How do you propose to map the 5-limit to Vicentino's system? You get a map to only half of it.> Still if you've now decided that temperaments don't have contorsion, > that's fine, we'll find another name for the things like temperaments > that may have contorsion. Contemperaments? > Yes, producing contorted quasi-temperaments from wedge products may be > pointless. That could be why nobody, except you, has ever advocated > doing so.And why is producing contemperaments from wedge products, which can easily be done, any more pointless than producing them any other way?> Anyway, those who want contorsion-free temperaments will be pleased to > learn that the module at > > # Temperament finding library -- definitions * [with cont.] (Wayb.) > > has been updated with a method to remove any contorsion from a > "LinearTemperament" object. Good!

Message: 10603 - Contents - Hide Contents Date: Wed, 10 Mar 2004 20:52:21 Subject: Re: Graham on contorsion From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx Graham Breed <graham@m...> wrote:> If you asked for 7&24 or 24&31 than you should expect to get Vicentino's > system.How do you figure that? 24+31=55, so you should expect to get a 55-equal system. The penultimate convergent for 31/24 is 22/17, so I would expect to get (17+22)/(24+31) = 39/55. In the 5-limit that means contorsion, and in the 7-limit, "Number 101", the temperament with commas generated by 81/80 and 6144/6125, and which maps 7/64 to -11 half-fifth generator steps. 7&24 is similar, but not identical; the penultimate convergent to 24/7 is 7/2, and (2+7)/(7+24) = 9/31; this time instead of 2/"half-fifth" we get the half-fifth itself, and instead of 55 we get 31; the comments on temperaments being the same. If the program automatically removed contorsion, you would be> rightly surprised to get a result that wasn't consistent with 24-equal.Since 24 equal is contorted in the 5-limit and is a contorted meantone system, why would you be surprised to get meantone? Anyway, any linear combination of 12 and 19 should do; if I stick in 17<12 19 28| - 3<19 30 44| = <109 173 253| I would expect to learn it is a meantone system. If I put it together with <31 49 72| I am happy to learn it is meantone; I am not so interested to find it has 22-contorsion and 22 generators of 9/140 give me an 8/3, but I can live with the information. As for scales, I think I am better ignoring the vals looking at 9/140 directly. That would suggest a microtemperament with 2401/2400 and 65625/65536 as commas in the 7-limit.

Message: 10604 - Contents - Hide Contents 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 - Contents - Hide Contents 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: 10606 - Contents - Hide Contents Date: Wed, 10 Mar 2004 21:43:25 Subject: "Vicentino" temperament From: Gene Ward Smith 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. Paul will probably have a strong opinion one way or another; if that opinion is "no" I hope he offers an alternative. Vicentino 7-limit Wedgie: <2 8 -11 8 -23 -48| TM commas: {81/80, 6144/6125} Mapping: [<1 1 9 6|, <0 2 8 -11|] Vicentino 11-limit Wedgie: <2 8 -11 5 8 -23 1 -48 -16 52| TM commas: {81/80, 121/120, 176/175} Mapping: [<1 1 0 6 2|, <0 2 8 -11 5|] TOP generators: [1201.698520 348.7821945] TOP tuning: [1201.698521 1899.262909 2790.257556 3373.586984 4147.308013]

Message: 10607 - Contents - Hide Contents Date: Wed, 10 Mar 2004 21:50:03 Subject: Re: Please remind me From: Gene Ward Smith --- 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...)Do you mean from two vals for 12 and 19, or simply from 12 and 19? If the latter, the method I've been talking about for two relatively prime numbers such as 12 and 19--call them a and b--is to take the penultimate continued fraction convergent to a/b, call it u/v. Then (u+v)/(a+b) is a generator of (a+b)-equal corresponding to a and b. In the case of 12 and 19, 7/11 is the penultimate convergent to 12/19, and so we get (7+11)/(12+19) = 18/31 as a generator.

Message: 10608 - Contents - Hide Contents Date: Wed, 10 Mar 2004 22:00:30 Subject: Re: Please remind me From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:> --- 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.Using continued fractions, this would go, for a and b not necessarily relatively prime, period = 1/gcd(a,b), and the generator is found as before. For instance, the penultimate convergent to 10/12 is 4/5, and so the generator is (4+5)/(10+12), or 9/22 (pajara with a fourth as generator.)

Message: 10609 - Contents - Hide Contents Date: Wed, 10 Mar 2004 22:23:09 Subject: Re: "Vicentino" temperament From: Gene Ward Smith --- 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? Given that it happens to be the generator of 11-limit "Vicentino" this seems like a key question.

Message: 10611 - Contents - Hide Contents 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: 10612 - Contents - Hide Contents Date: Wed, 10 Mar 2004 23:31:31 Subject: Re: "Vicentino" temperament From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:> --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> > wrote:>> 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.Half of a 1/4-comma meantone fifth is 5^(1/8), which is less than a cent away from 11/9, so this seems like a pretty good reason to give the 11-limit temperament with this generator and 81/80 as a comma his name. If we halve a 1/3-comma meantone fifth instead we get a truly tiny difference--0.0148 cents--related to the comma (11/9)^6/(10/3) = 1171561/1171470, but this seems more relevant to microtemperaments than anything using 81/80. Someone wanting to try out methods for getting periods and generators might try 342&494, with this comma and a smaller than ordinary period. Of course, we could just use the no-sevens temperament from 121/120 and 81/80 I suppose.

Message: 10613 - Contents - Hide Contents 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: 10614 - Contents - Hide Contents Date: Thu, 11 Mar 2004 02:25:15 Subject: Vicentino #2 From: Gene Ward Smith It occurs to me there is at least one other temperament family one might want to tack Vicentino's name onto. In the 5-limit, it would be the temperament deriving from the comma 2^72/5^31. As a temperamnet this isn't much, but out of it you get something which looks like Vicentino's adaptive tuning scheme, with two meantones separated by 1/4 comma--the mapping being [<31 49 72|, <0 1 0|]. The extensions to the 7 and 11 limits seem better; the 11-limit mapping would be [<31 49 72 87 107|, <0 1 0 0 2|], with commas 441/440, 3136/3125, and 41503/41472. In the 7-limit one has 3136/3125 and 823543/819200. Either gives something which is supported by 186, 217 or 248 and which is the temperament related to using 217-et for Vicentino-style adaptive tuning. The TOP generators are not exactly the same for 5, 7, or 11 limits so you'd take your pick. Does anyone care to suggest a name before I go and call it "adaptive"?

Message: 10615 - Contents - Hide Contents Date: Thu, 11 Mar 2004 02:26:22 Subject: Re: Dual L1 norm deep hole scales From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:> Still wondering about all of this, > PaulThat's not a very specific question.

Message: 10616 - Contents - Hide Contents 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/ * [with cont.] <*> 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 * [with cont.] (Wayb.)

Message: 10617 - Contents - Hide Contents 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 - Contents - Hide Contents 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: 10619 - Contents - Hide Contents Date: Fri, 12 Mar 2004 18:44:14 Subject: Re: Breeding From: Gene Ward Smith --- 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?

Message: 10620 - Contents - Hide Contents 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 - Contents - Hide Contents 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 * [with cont.] not to mention much more recently on this list.

Message: 10622 - Contents - Hide Contents Date: Fri, 12 Mar 2004 21:04:15 Subject: Re: Breeding From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:> Why "-ing"? This seems to suggest a process; in fact I used this term > to describe a process here: > > Yahoo groups: /tuning/message/28558 * [with cont.] > > not to mention much more recently on this list.It's lousy, I agree. Are you suggesting just "Breed", or what would you suggest? ________________________________________________________________________ ________________________________________________________________________ ------------------------------------------------------------------------ Yahoo! Groups Links <*> To visit your group on the web, go to: Yahoo groups: /tuning-math/ * [with cont.] <*> 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 * [with cont.] (Wayb.)

Message: 10623 - Contents - Hide Contents Date: Sat, 13 Mar 2004 21:41:04 Subject: A 9-limit cube scale From: Gene Ward Smith If you take the 27 tetrads of the chord cube of side 2 and extend them to complete 9-limit o- or utonalities, you get a scale of 47 notes. Once again we have one step of size 2401/2400, and so now tempering this out gives us 46 notes, not 31 as before. If we look at commas of less than 10 cents which lead to approximate consonances, we get in order 2401/2400, 5120/5103, 225/224 and 1029/1024. Tempering out both 2401/2400 and 5120/5103 leads to hemififths; like hemiwuerschmidt this is a temperament 99 does a good job for, though 140 or 239 do even better. Another interesting tuning is the minimax, which has pure major thirds--the hemififths answer to 1/4 comma meantone. The 46 scale degrees in terms of hemififths range from -35 to 50 generator steps, which spans most of 99.

Message: 10624 - Contents - Hide Contents Date: Sat, 13 Mar 2004 22:03:36 Subject: 1605632/1594323 tempering From: Gene Ward Smith This the no-fives comma of hemififths; 2^15 3^(-13) 7^2. The difference between the TOP tuning for this and for hemififths is a negligable < 0.05 cents for 2, 3, and 7, and as a {2,3,7}-linear temperament it's pretty well identifiable as hemififths. Since Michael Harrison loves 3, likes 7, and seems to be willing to live without 5, tempering 24 notes to hemififths seems like a plausible tuning method for Harrison style music if one can live with fifths a cent sharp.

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