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Message: 3650 - Contents - Hide Contents Date: Fri, 1 Feb 2002 12:58 +00 Subject: Re: interval of equivalence, unison-vector, period From: graham@xxxxxxxxxx.xx.xx In-Reply-To: <a3d5gb+ldpj@xxxxxxx.xxx> genewardsmith wrote:> If you take a set of unison vectors defining an equal temperament, as > for instance {50/49, 64/63, 245/243} and now add 2 to the set, then {2, > 50/49, 64/63, 245/243} generates a kernel K such that N7/K = C22--we > have a map of the 7-limit to a cyclic group of order 22--which is a > torsion group, since everything has finite order. All elements are > torsion elements, and we have a finite group, so this is rather > different than a block with torsion elements.But it still involves torsion?> If we take 50/49^64/63, the wedgie for twintone, and wedge it with > 245/245 we get the 7-limit val h22 of the 22-et, which of course > defines a temperament.Okay, I assume you mean 245/243.>>> h22 = (wedgie^temper.WedgableRatio(245,243)).equalTemperament() >>> h22.basis(22, 35, 51, 62)> If we wedge the twintone wedgie with 2 instead, > we also get a val--the mapping to generators of the non-octave > generator of twintone. Right.>>> g0 = (wedgie^{(0,):1}).equalTemperament() >>> g0.basis(0, 1, -2, -2)> This is *not* a temperament, or at least not one > I'm interested in hearing, so 2 is not acting as a unison, which is > hardly a surprise.Of course it's a temperament. It's twintone/paultone/pajara. The octave is acting as a unison, but it's more complicated than that. As it has torsion, it's actually half an octave that's acting as a commatic unison vector. This isn't a problem for normal unison vectors because half a unison is still a unison. All you find is that you get twice as many notes as you expected. But half an octave is very different from an octave. So our commatic unison vector is, in a sense, the tritone. (BTW, in an octave-equivalent system, half a unison is a half-octave as well as a unison. This is obvious if you think of octave-equivalent frequency space as a Hilbert space, and remember that half the pitch is the same as the square root of the frequency.) The first entry being zero tells us that the octave is clearly acting as a unison. In fact, two tritone-unisons. The second entry tells us that the 3:1 is a generator. That's a semitone larger than an octave plus a tritone. As octaves and tritones are unisons, that means it's a semitone larger than a unison. And two notes differing by a unison must be the same, so the generator is a semitone. The third entry tells us that 5:1 is -2 generators. That means it's a unison less two semitones. In this case, octaves and tritones are unisons. 5:4 and 5:1 only differ by octaves, which means they only differ by unisons, so 5:4 is also a unison less two semitones. A tritone is a unison. So 5:4 is a tritone less two semitones. The fourth entry tells us that 7:1 is also -2 generators. As the generator is still a semitone and octaves are still unisons, that means 7:4 is an octave less two semitones. This is all an accurate description of paultone/twintone/pajara in a tritone-equivalent system. So what's the problem? In general, making an octave a unison vector is like imposing octave equivalence. That's actually quite similar to something Fokker said. But hey, octave equivalence is a strange concept to pull out of the hat, so perhaps you're still not convinced. Well, remember the unison vector Gene gave describes the equal temperament h22, and the octave as unison vector describes something I called g0. Well, I will now show that g0 is indeed an equal temperament. It's a consistent mapping to a temperament with no equal steps to the octave. Another unison vector that isn't a unison could be 3:1. Tempering this out gives two tone equal temperament:>>> g2 = (wedgie^{(1,):1}).equalTemperament() >>> g2.basis(2, 0, 11, 12) g0 and g2 together define paultone/twintone/pajara>>> (g0^g2).complement() == wedgie 1It's also possible to combine g0 and g2 to get h22.>>> g22 = g2 + g0 >>> for each in range(10):... g22 = g22 + g2 ...>>> for each in range(34):... g22 = g22 + g0 ...>>> h22.basis(22, 35, 51, 62)>>> g22.basis[22, 35, 51, 62] Unfortunately, I didn't set my library up to allow equal temperaments to be multiplied by integers. Also, addition causes the basis to be a list rather than a tuple. But if it weren't for these things, that could have been written>>> (g0*35 + g2*11) == h22 1I'm assuming we all accept that 12-equal is a temperament. It's also consistent with paultone/twintone/pajara.>>> h12 = temper.PrimeET(12, temper.primes[:3]) >>> h12.basis[12, 19, 28, 34]>>> wedgie.complement()^h12 {}If h12 and h22 are pajara-consistent ETs, so must h10 be>>> h10 = h22-h12 >>> h10.basis[10, 16, 23, 28] and h2 (not the prime mapping, but still distinct from g2)>>> h2 = h12-h10 >>> h2.basis[2, 3, 5, 6] We can also get another 2 note equal temperament from h12 and h22, which I'll have to call i2 because I'm running out of letters.>>> h8 = h10-h2 >>> h6 = h8-h2 >>> h4 = h6-h2 >>> i2 = h4-h2 >>> i2.basis[2, 4, 3, 4] As we have two 2 note equal temperaments, the difference between them must be an equal temperament with no notes.>>> i0 = i2-h2 >>> i0.basis[0, 1, -2, -2] Hey, that's the same as g0 above!>>> g0.basis(0, 1, -2, -2) So the pajara equal temperament you get with the octave as a commatic unison vector certainly looks like an equal temperament as I understand the concept. Perhaps, specifically, it's a regular equal temperament as not any no notes are allowed. Unless somebody wants to stipulate that an equal temperament has to have more than no notes, but I don't remember that ever being mentioned before. Graham

Message: 3651 - Contents - Hide Contents Date: Fri, 1 Feb 2002 01:06:18 Subject: Re: Fwd: Re: interval of equivalence, unison-vector, periodd From: monz> From: paulerlich <paul@xxxxxxxxxxxxx.xxx> > To: <tuning-math@xxxxxxxxxxx.xxx> > Sent: Friday, February 01, 2002 1:01 AM > Subject: [tuning-math] Fwd: Re: interval of equivalence, unison-vector, period > > > Recall that a particular unison vector (or product of unison vectors, > etc.) is candidate for torsion if it's a power (square, cube, etc.) > of some other interval. Let's say you don't keep track of the factors > of 2 making up the unison vectors. ... > <etc.> > > But a tritone is no kind of unison vector! Instead, it (as 1/2 > octave) becomes the _period_ for any system involving the 50:49 > unison vector. > > Catchin' on?YUP! Thanks, Paul! This is the kind of "come on Joey, give me your hand" explanation that I really need sometimes when I'm buried in this math stuff. -monz _________________________________________________________ Do You Yahoo!? Get your free @yahoo.com address at Yahoo! Mail - The best web-based email! * [with cont.] (Wayb.)

Message: 3652 - Contents - Hide Contents Date: Fri, 01 Feb 2002 04:27:55 Subject: Re: interval of equivalence, unison-vector, period From: genewardsmith --- In tuning-math@y..., "paulerlich" <paul@s...> wrote:>> If the octave is a chromatic unison vector in twintone, >> How odd to call an octave a chromatic unison vector. When a note is > altered by a chromatic unison vector, it is supposed to undergo a > small but nonzero change in pitch. In neither the octave-invariant > nor the octave-specific case is this true for the octave! >>> it also gets >> divided into two equal parts because of torsion. >> Warning -- this does not agree with the definition of torsion that > Gene was talking about.If you take a set of unison vectors defining an equal temperament, as for instance {50/49, 64/63, 245/243} and now add 2 to the set, then {2, 50/49, 64/63, 245/243} generates a kernel K such that N7/K = C22--we have a map of the 7-limit to a cyclic group of order 22--which is a torsion group, since everything has finite order. All elements are torsion elements, and we have a finite group, so this is rather different than a block with torsion elements. If we take 50/49^64/63, the wedgie for twintone, and wedge it with 245/245 we get the 7-limit val h22 of the 22-et, which of course defines a temperament. If we wedge the twintone wedgie with 2 instead, we also get a val--the mapping to generators of the non-octave generator of twintone. This is *not* a temperament, or at least not one I'm interested in hearing, so2 is not acting as a unison, which is hardly a surprise.

Message: 3653 - Contents - Hide Contents Date: Fri, 1 Feb 2002 12:58 +00 Subject: Re: interval of equivalence, unison-vector, period From: graham@xxxxxxxxxx.xx.xx In-Reply-To: <a3cadl+f19b@xxxxxxx.xxx> Me:>> If the octave is a chromatic unison vector in twintone, Paul:> How odd to call an octave a chromatic unison vector. When a note is > altered by a chromatic unison vector, it is supposed to undergo a > small but nonzero change in pitch. In neither the octave-invariant > nor the octave-specific case is this true for the octave!We've been through this before. Although an octave isn't a unison vector because it isn't small, we still don't have a word for something that's like a unison vector but small. So I have to keep calling them "things like unison vectors". Do you have a mathematical definition for "like a unison"? Me:>> it also gets >> divided into two equal parts because of torsion. Paul:> Warning -- this does not agree with the definition of torsion that > Gene was talking about. How so?Is 25:24 a unison vector? Does it have torsion when used with the Miracle wedgie? Does it define a decimal scale? Is it always divided into two equal parts in other Miracle tunings? Where am I going wrong? Graham

Message: 3654 - Contents - Hide Contents Date: Fri, 1 Feb 2002 22:42:02 Subject: Re: interval of equivalence, unison-vector, period From: monz> From: Graham Breed <graham@xxxxxxxxxx.xx.xx> > To: <tuning-math@xxxxxxxxxxx.xxx> > Sent: Friday, February 01, 2002 2:23 PM > Subject: [tuning-math] Re: interval of equivalence, unison-vector, period > > > "period". The thread actually started because Monz was > confused about the difference between "unison vector", > "equivalence interval" and "period". He was right to be > because they are very similar. Hi Graham,No need to use the past tense here ... halfway thru today's spate of messages on this subject, I'm more confused now than ever. :) and :( at the same time. -monz _________________________________________________________ Do You Yahoo!? Get your free @yahoo.com address at Yahoo! Mail - The best web-based email! * [with cont.] (Wayb.)

Message: 3655 - Contents - Hide Contents Date: Fri, 01 Feb 2002 05:42:53 Subject: Re: any ideas? From: paulerlich --- In tuning-math@y..., "D.Stearns" <STEARNS@C...> wrote:> Paul, > > Well of course you're absolutely right about context, but I'm more > interested in a broad generalization here.I like avoiding generalizations in this context. But in general, I'd get the 1/1-3/2 happening, and then if there were more than one possibility, I'd look for rare intervals to resplve to, and rare step- sizes to act as signposts or leading-tone figurations toward, notes in the tonic triad (or its equivalent).

Message: 3656 - Contents - Hide Contents Date: Fri, 01 Feb 2002 09:06:27 Subject: Re: new cylindrical meantone lattice From: paulerlich --- In tuning-math@y..., "monz" <joemonz@y...> wrote:>> OK, I'm in favor of thinking in this direction, as it addresses at >> least one of the three objections I raised in a post to you a few >> minutes ago. > >> And that objection would be this one, correct? : >>> (c) the fact that you can't yet plot >> non-rational-fraction-of-a-comma meantones this way, >> though the _angle_ part should be just as meaningful >> for those. > >> So, how about a formula that plots 19-EDO as, literally, > a close cousin to 1/3-comma meantone spiral? How does > take something that's roots of 2, and change it into > "8ve"-equivalent fractional powers of 3 and 5?Well, maybe there's another way to get the right spiral.> Now about your other two objections: > >>> (a) the density of points along the line, which doesn't >> appear to be meaningful; > >> I'm hoping that the post I just sent before this one, > about composer choosing particular flavors of meantone, > addresses this one.Not at all -- I was referring to the fact that, for example, in 5/18- comma meantone, the points on the spiral are rather far apart from one another -- that doesn't seem particularly meaningful.>> (b) the fact that you have to pin the spiral to a particular >> "1/1" origin, which ruins the rotational symmetry of the >> cylindrical meantone lattice > >> I've already said elsewhere that the spiral doesn't have > to be pinned to anything. It can float anywhere the user > wants it. What's important is the angle of the spiral, > as you've noted.So maybe a set of arrows (say from every _true_ lattice point) pointing at that angle would be preferable to a spiral.

Message: 3657 - Contents - Hide Contents Date: Fri, 01 Feb 2002 20:52:14 Subject: Re: interval of equivalence, unison-vector, period From: genewardsmith --- In tuning-math@y..., "paulerlich" <paul@s...> wrote:> Oops, I meant a non-octave linear temperament, approximately 11 > periods per octave?As long as 2 is represented, it seems to me any temperament is an octave temperament. The basis I gave was for a fifth and a tritone below a fifth, and I could if I wanted make the fifth a pure fifth, but I could do that, and temper octaves, in the octave basis also. There are three considerations: interval of equivalence of a scale using a given temperament, a basis of generators for the temperament, and the tuning of the temperament. This are independent.

Message: 3658 - Contents - Hide Contents Date: Fri, 1 Feb 2002 22:50:48 Subject: Re: interval of equivalence, unison-vector, period From: monz> From: paulerlich <paul@xxxxxxxxxxxxx.xxx> > To: <tuning-math@xxxxxxxxxxx.xxx> > Sent: Friday, February 01, 2002 3:56 PM > Subject: [tuning-math] Re: interval of equivalence, unison-vector, period > >>> I would say it's acting as a generator, but if you make 2 a unison >> it becomes a torsion element, since its square is an octave. >> This, along with my message to Monzo this morning, seems to show the > very real problems with considering 2 a unison!How is 2^2 an octave? By definition, it's simply 2. Now you guys have really lost me. (I'm glad I understand at least most of what was posted here up to last September!) -monz _________________________________________________________ Do You Yahoo!? Get your free @yahoo.com address at Yahoo! Mail - The best web-based email! * [with cont.] (Wayb.)

Message: 3659 - Contents - Hide Contents Date: Fri, 01 Feb 2002 05:48:14 Subject: Re: interval of equivalence, unison-vector, period From: paulerlich --- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:> --- In tuning-math@y..., "paulerlich" <paul@s...> wrote: >>>> If the octave is a chromatic unison vector in twintone, >>>> How odd to call an octave a chromatic unison vector. When a note is >> altered by a chromatic unison vector, it is supposed to undergo a >> small but nonzero change in pitch. In neither the octave- invariant >> nor the octave-specific case is this true for the octave! >>>>> it also gets >>> divided into two equal parts because of torsion. >>>> Warning -- this does not agree with the definition of torsion that >> Gene was talking about. >> If you take a set of unison vectors defining an equal temperament, >as for instance {50/49, 64/63, 245/243} and now add 2 to the set, >then {2, 50/49, 64/63, 245/243} generates a kernel K such that N7/K >= C22--we have a map of the 7-limit to a cyclic group of order 22--This, I think, corresponds to how Graham thinks of things, and how I _used_ to think of things, before I understood torsion in the period- is-1/2-or-1/9-or-1/N-octave sense.>which is a torsion group, since everything has finite order. All >elements are torsion elements, and we have a finite group,And this could happen just as well for a group with a prime number of elements, such as {2, 25/24, 81/80} -> C7.>so this is rather different than a block with torsion elements.Yes it is. Now we really need to revise the definition of torsion :(, and think of different names for these two things.> If we take 50/49^64/63, the wedgie for twintone, and wedge it with > 245/245 we get the 7-limit val h22 of the 22-et, which of course >defines a temperament. If we wedge the twintone wedgie with 2 >instead, we also get a val--the mapping to generators of the non- >octave generator of twintone.Can you go into this in more detail, pretty please with sugar on top?

Message: 3660 - Contents - Hide Contents Date: Fri, 01 Feb 2002 09:11:55 Subject: Re: interval of equivalence, unison-vector, period From: paulerlich --- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:> --- In tuning-math@y..., "paulerlich" <paul@s...> wrote: >>>> 50/49^245/243 = [6,10,10,-5,1,2] with generators >>> a = 3.0143/22 = 164.4176 cents and b = 1/2 >> >> Glassic >> Good name--where does it come from?Sorry -- that's the wrong name. Glassic has b = 1 . . . my piece "Glassic" uses it.>>> 64/63^245/243 = [1,9,-2,-30,6,12] with generators a = 8.9763/22 = >>> 489.6152 cents and b = 1 >>>> "Big fifth" -- a unique facet of 22 >> Not really; I would say it is even more characteristic of 27 or 49.OK, you're right.>>> A triple wedge product of three intervals will be a val, but it >>> doesn't have to be an equal temperament val. >>> What other kinds are there? >> Vals are the dual concept to intervals. We have prime numberintervals, such as octave or twelvth, and the dual to those are the> p-adic valuations vp, which count the number of powers of p(positive or negative) in the prime factorization of a rational number. An interval is a finite Z-linear combination of primes; that is, it is p1^e1 * p2^e2 ... pk^ek for certain primes pn and certain integers en. A val is a finite Z-linear combination of p-adic valuations:> e1 v1 + e2 v2 + ... + ek vk. Ah! >Dual to the comma, or small interval is, more or less, an et val. >Another type of val of interest are the maps of generators to primes.What's the dual to that kind of val?>>> Mapping 2 to 1, and both 5 and 7 to 1/9 does not strike me as much>> of a temperament. >> >> Well . . . I'm lost . . . does this have anything to do with what you >> were once showing about your process, where for a "linear" or 2D >> temperament, you started off with two generators, but then found a >> different generator basis pair where you forced one member to be an >> octave? > > RightGRAHAM, TAKE NOTE!>--this val would be the starting point for that process, not a >temperament.But it's a kind of non-octave-repeating scale, isn't it?? It's something that would be of interest if we didn't assume that "octaves sound the same", right?> I can do the same sort of thing starting from > [-2,-2,-7,-8], where I end up with > > [-2 2] > [-2 3] > [-7 5] > [-8 6] > > as a mapping from generators to primes; here "b" is a wide fifth >and "a" is a tritone below that.Wouldn't that just be a non-octave ET, approximately 11 tones per octave?

Message: 3661 - Contents - Hide Contents Date: Fri, 01 Feb 2002 21:29:06 Subject: Re: interval of equivalence, unison-vector, period From: paulerlich --- In tuning-math@y..., graham@m... wrote:> The octave is acting as a unison, but it's more complicated than that. As > it has torsion, it's actually half an octave that's acting as a commatic > unison vector.No offense, Graham, but could you at least invent some terminology that makes sense for what you're talking about, instead of misappropriating terminology that makes no sense the way you're using it? Half an octave does not act a commatic unison vector here -- this is very frustrating because I thought I had spent dozens of posts explaining to you what a commatic unison vector is, and convincing you that an octave isn't one and a fifth isn't one . . . did all that arguing make no impression on you? A chromatic unison vector is a generalized "augmented unison". Nothing else.> (BTW, in an octave-equivalent system, half a unison is a half- octave as > well as a unison. This is obvious if you think of octave- equivalent > frequency space as a Hilbert space, and remember that half the pitch is > the same as the square root of the frequency.)Huh? So if the frequency is 6400 Hz, the square root of that is 80, and that's half the pitch??> A tritone is a > unison.Right . . .> This is all an accurate description of paultone/twintone/pajara in a > tritone-equivalent system. So what's the problem? In general, making an > octave a unison vector is like imposing octave equivalence. That's > actually quite similar to something Fokker said.Please fill us in!

Message: 3662 - Contents - Hide Contents Date: Fri, 1 Feb 2002 22:58:08 Subject: 171-EDO, Vogel (was: 7-limit MT reduced bases for ets) From: monz> From: genewardsmith <genewardsmith@xxxx.xxx> > To: <tuning-math@xxxxxxxxxxx.xxx> > Sent: Friday, February 01, 2002 7:34 PM > Subject: [tuning-math] Re: 7-limit MT reduced bases for ets > > > --- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote: >>> 140: [2401/2400, 5120/5103, 15625/15552] >> I accidentally left off > > 171: [2401/2400, 4375/4374, 32805/32768] > > Wouldn't want to do that--look at those three high-powered commas!Right! 171-EDO has been strongly advocated by prolific German tuning theorist Martin Vogel, who has even had a keyboard built for it for use in his classes. The only detailed version of Vogel's work in English is: Vogel, Martin. _Die Lehre von den Tonbeziehungen_. Orpheus, Band 16, Verlag für systematische Musikwissenschaft, Bonn-Bad Godesberg, 1975, 480 pages. English translation _On the Relations of Tone_ by Vincent Jean Kisselbach and edited by Carl A. Poldy, Verlag für systematische Musikwissenschaft, Bonn-Bad Godesberg, 1993. (note that the title is an homage to Helmholtz) -monz _________________________________________________________ Do You Yahoo!? Get your free @yahoo.com address at Yahoo! Mail - The best web-based email! * [with cont.] (Wayb.)

Message: 3663 - Contents - Hide Contents Date: Fri, 01 Feb 2002 05:53:28 Subject: Re: Approximate consonances of Parch's 43 tone scale From: genewardsmith --- In tuning-math@y..., "paulerlich" <paul@s...> wrote:>> What's the generator and period of catakleismic? From Yahoo groups: /tuning-math/message/2791 * [with cont.] 20. Catakleismic[6, 5, 22, -21, -6, 18, -54, 37, -66, -135] [225/224, 385/384, 540/539, 4375/4374] ets 19, 72 [[0, 6, 5, 22, -21], [1, 0, 1, -3, 9]] [.2639230436, 1] a = 19.0025/72 = 316.7076522 cents badness 271.0589693 rms 1.697136764 g 20.98979344 Because of the recent discussion of the 152 et, I thought I would also repost this for your consideration: 3. Octoid [24, 32, 40, 24, -5, -4, -45, 3, -55, -71] [540/539, 3025/3024, 4375/4374, 9801/9800] ets 72, 80, 152, 224, 296 [[0, -3, -4, -5, -3], [8, 16, 23, 28, 31]] [.1383934690, 1/8] a = 9.9643/72 = 31.0001/224 = 166.0721626 badness 147.3854996 rms .7687062948 g 23.42160176>> How many notes in contiguous, (equal-length?) chains of generators >> does each of these need to encompass Partch's 'Genesis' scale?As in Genesis of a Music?> What's the answer for MIRACLE? Wasn't it an non-'Genesis' 43-tone > scale that MIRACLE comprised in 45 consecutive notes in a chain of > generators?I don't know. I got the scale I analyzed from a web search; I didn't know there was more than one 43 tone Partch scale.> This is really interesting, as it makes one wonder, to what extend > was Secor's original proposal "unique" or "best".Miracle does appear here.

Message: 3664 - Contents - Hide Contents Date: Fri, 01 Feb 2002 09:13:11 Subject: Re: interval of equivalence, unison-vector, period From: paulerlich --- In tuning-math@y..., "paulerlich" <paul@s...> wrote:>> I can do the same sort of thing starting from >> [-2,-2,-7,-8], where I end up with >> >> [-2 2] >> [-2 3] >> [-7 5] >> [-8 6] >> >> as a mapping from generators to primes; here "b" is a wide fifth >> and "a" is a tritone below that. >> Wouldn't that just be a non-octave ET, approximately 11 tones per > octave?Oops, I meant a non-octave linear temperament, approximately 11 periods per octave?

Message: 3665 - Contents - Hide Contents Date: Fri, 01 Feb 2002 21:31:19 Subject: Re: interval of equivalence, unison-vector, period From: paulerlich --- In tuning-math@y..., graham@m... wrote:> Where am I going wrong?I'm not saying you're wrong, only that your methods are different from Gene's -- most recently exemplified with the case that he considered "not a temperament" and you considered "22-tET".

Message: 3666 - Contents - Hide Contents Date: Fri, 1 Feb 2002 23:04:02 Subject: Re: 171-EDO, Vogel (was: 7-limit MT reduced bases for ets) From: monz> From: monz <joemonz@xxxxx.xxx> > To: <tuning-math@xxxxxxxxxxx.xxx> > Sent: Friday, February 01, 2002 10:58 PM > Subject: [tuning-math] 171-EDO, Vogel (was: 7-limit MT reduced bases for ets) > > >>> From: genewardsmith <genewardsmith@xxxx.xxx> >> To: <tuning-math@xxxxxxxxxxx.xxx> >> Sent: Friday, February 01, 2002 7:34 PM >> Subject: [tuning-math] Re: 7-limit MT reduced bases for ets >> >> >> --- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote: >>>>> 140: [2401/2400, 5120/5103, 15625/15552] >>>> I accidentally left off >> >> 171: [2401/2400, 4375/4374, 32805/32768] >> >> Wouldn't want to do that--look at those three high-powered commas! > >> Right! 171-EDO has been strongly advocated by prolific > German tuning theorist Martin Vogel, who has even had > a keyboard built for it for use in his classes.I should have added, in keeping with the original subject line, that Vogel is an enthusiastic advocate of prime-factor 7 in tuning. His analyses sometimes include higher primes, but from what I've read he doesn't seem too interested in 11 or higher. But early on in his career he wrote this: Vogel, Martin. _Zahl 7 in der Spekulativen Musiktheorie_. unpublished PhD thesis, 1955 That titles translates as "The Number 7 in Speculative Music-Theory". -monz _________________________________________________________ Do You Yahoo!? Get your free @yahoo.com address at Yahoo! Mail - The best web-based email! * [with cont.] (Wayb.)

Message: 3667 - Contents - Hide Contents Date: Fri, 1 Feb 2002 01:13:04 Subject: Re: new cylindrical meantone lattice From: monz> From: paulerlich <paul@xxxxxxxxxxxxx.xxx> > To: <tuning-math@xxxxxxxxxxx.xxx> > Sent: Friday, February 01, 2002 1:03 AM > Subject: [tuning-math] Re: new cylindrical meantone lattice > > > > So you're deriving 2/9-comma meantone as an optimal meantone?Optimal for the *Duodene*, specifically. And I'm not really sure if it's an optimal, just guessing.> What weights are you using to do so? I showed the meantones > that I could easily derive from various weightings of the basic > consonant intervals (M3/m6, m3/M6, p4/p5), and 2/9-comma wasn't > one of them . . .Didn't do it that way at all. Simply looked at the lattice of the shifted-boundary Duodene PB and saw that 2/9-comma slashed right across the middle of it. Since the angle of the meantone line on the flat lattice (and of the spiral on the cylindrical) graphically shows the tempering of the meantone in relation to the nearest JI pitches, I moved it around until it was centered perfectly within the shifted PB, and it seemed to distribute the error the most evenly. So I suppose it's only optimal for the intervals within the Duodene PB. If that's the case, then it says to me that 2/9-comma meantone will tend to imply JI tonal structures which derive from the Duodene. -monz _________________________________________________________ Do You Yahoo!? Get your free @yahoo.com address at Yahoo! Mail - The best web-based email! * [with cont.] (Wayb.)

Message: 3668 - Contents - Hide Contents Date: Fri, 01 Feb 2002 06:07:24 Subject: Re: Approximate consonances of Parch's 43 tone scale From: paulerlich --- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:> 3. Octoid > > [24, 32, 40, 24, -5, -4, -45, 3, -55, -71] > > [540/539, 3025/3024, 4375/4374, 9801/9800] > > ets 72, 80, 152, 224, 296 > > [[0, -3, -4, -5, -3], [8, 16, 23, 28, 31]] > > [.1383934690, 1/8] > > a = 9.9643/72Equivalently, about 1/72 oct. -- right?> = 31.0001/224 = 166.0721626 > > badness 147.3854996This 11-limit badness is not directly comparable to 5-limit badness for 5-limit temperaments, is it?> rms .7687062948 > g 23.42160176Certainly looks like an efficient way of getting 11-limit harmony in 152-tET! Sort of the 152-tET version of MIRACLE, but more accurate. This implies sort of an adaptive 8-tET scheme, where more and more 16- cent-apart 8-tET chains are needed the more expansive the JI chords you want to play. The only JI interval in a single 8-tET chain is the 11/6 (and equivalents). Certainly a major aspect of 152-tET that I overlooked.>> This is really interesting, as it makes one wonder, to what extend >> was Secor's original proposal "unique" or "best". >> Miracle does appear here.I know, but how do the others compare, in terms of number of "restored" consonant intervals and chords (hexads on down), and in terms of minimax 11-limit error?

Message: 3669 - Contents - Hide Contents Date: Fri, 01 Feb 2002 21:33:09 Subject: Re: interval of equivalence, unison-vector, period From: paulerlich --- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:> --- In tuning-math@y..., "paulerlich" <paul@s...> wrote: >>> Oops, I meant a non-octave linear temperament, approximately 11 >> periods per octave? >> As long as 2 is represented, it seems to me any temperament is an >octave temperament. The basis I gave was for a fifth and a tritone >below a fifth, and I could if I wanted make the fifth a pure fifth,>> but I could do that, and temper octaves, in the octave basis also.>There are three considerations: interval of equivalence of a scale >using a given temperament, a basis of generators for the >temperament, and the tuning of the temperament. This are independent.So why did you say "this was not a temperament"? And isn't it true that, if you took it out to, say, 10 notes per approximate octave, and tuned the octaves pure, it would _not_ be an octave-repeating scale? This seems to be the point Graham is missing.

Message: 3670 - Contents - Hide Contents Date: Fri, 01 Feb 2002 06:35:45 Subject: Re: interval of equivalence, unison-vector, period From: genewardsmith --- In tuning-math@y..., "paulerlich" <paul@s...> wrote:> And this could happen just as well for a group with a prime number of > elements, such as {2, 25/24, 81/80} -> C7. Yes, indeedy.>> so this is rather different than a block with torsion elements. >> Yes it is. Now we really need to revise the definition of torsion :(, > and think of different names for these two things.Why do we need to worry about it?> Can you go into this in more detail, pretty please with sugar on top?I'm not sure what you are asking for, so let's see if this does it: The MT reduced basis for 22 et in the 7-limit is {50/49, 64/63, 245/243}. If I take these in pairs and wedge them, I get three temperaments instead, which can also be thought of as a defining basis for 22-et: 50/49^64/63 = [-2,4,4,-2,-12,11] -- twintone 50/49^245/243 = [6,10,10,-5,1,2] with generators a = 3.0143/22 = 164.4176 cents and b = 1/2 64/63^245/243 = [1,9,-2,-30,6,12] with generators a = 8.9763/22 = 489.6152 cents and b = 1 I can now wedge these with 2, and get triple wedge products. A triple wedgeproduct of three intervals will be a val, but it doesn't have to be an equal temperament val. 50/49^64/63^2 = [0,2,-4,-4] 50/49^245/243^2 = [0,-6,-10,-10] 64/63^245/243^2 = [0,-1,-9,2] This is giving us the non-octave part of the generator map. We could also wedge with other intervals of equivalence besides 2, and get what the corresponding temperament would be then; for instance 50/49^64/63^3/2 = [-2,-2,-7,-8] We can then use this mapping to primes (or [2,2,7,8], which seems nicer andwhich a different order of the triple product would have given us) to define a version of this temperament based on the fifth as an interval of equivalence. Note that both 2 and 3 are mapped to the same number, in this case 2, which is a consequence of 3 = 2 (3/2), and corresponds to the way that 2 maps to 0 in the previous triple wedge products.>> This is *not* a temperament, or at least not one I'm interested in >> hearing, so 2 is not acting as a unison, which is hardly a surprise.Mapping 2 to 1, and both 5 and 7 to 1/9 does not strike me as much of a temperament.

Message: 3671 - Contents - Hide Contents Date: Fri, 01 Feb 2002 09:15:33 Subject: Re: new cylindrical meantone lattice From: paulerlich --- In tuning-math@y..., "monz" <joemonz@y...> wrote:>>> From: paulerlich <paul@s...> >> To: <tuning-math@y...> >> Sent: Friday, February 01, 2002 1:03 AM >> Subject: [tuning-math] Re: new cylindrical meantone lattice >> >> >> >> So you're deriving 2/9-comma meantone as an optimal meantone? > >> Optimal for the *Duodene*, specifically. > And I'm not really sure if it's an optimal, just guessing. > >>> What weights are you using to do so? I showed the meantones >> that I could easily derive from various weightings of the basic >> consonant intervals (M3/m6, m3/M6, p4/p5), and 2/9-comma wasn't >> one of them . . . > >> Didn't do it that way at all. Simply looked at the lattice > of the shifted-boundary Duodene PB and saw that 2/9-comma > slashed right across the middle of it.Hmm . . . so you're not using only the consonant intervals, as you said you were. Anyway, can you show me how it slashes right down the middle, which some other meantone doesn't?> Since the angle > of the meantone line on the flat lattice (and of the spiral > on the cylindrical) graphically shows the tempering of the > meantone in relation to the nearest JI pitches, I moved it > around until it was centered perfectly within the shifted PB, > and it seemed to distribute the error the most evenly.Couldn't any other meantone do exactly the same thing?

Message: 3672 - Contents - Hide Contents Date: Fri, 1 Feb 2002 22:23:20 Subject: Re: interval of equivalence, unison-vector, period From: Graham Breed Me:>> The octave is acting as a unison, but it's more complicated that that. As >> it has torsion, it's actually half an octave that's acting as a commatic >> unison vector. Paul:> No offense, Graham, but could you at least invent some terminology > that makes sense for what you're talking about, instead of > misappropriating terminology that makes no sense the way you're using > it? Half an octave does not act a commatic unison vector here -- this > is very frustrating because I thought I had spent dozens of posts > explaining to you what a commatic unison vector is, and convincing > you that an octave isn't one and a fifth isn't one . . . did all that > arguing make no impression on you?I'm fully aware that an octave is not a unison vector. I've said so before and I didn't say otherwise in that quote. All I said is that it (or the tritone) acts as a unison vector. Which it does. As far as the algebra's concerned, it's exactly like a unison vector. I could invent terminology, but don't need to here because the tritone is already what we're calling the "period". The thread actually started because Monz was confused about the difference between "unison vector", "equivalence interval" and "period". He was right to be because they are very similar. Paul:> A chromatic unison vector is a generalized "augmented unison". > Nothing else.Well, can you think of a word for something that acts like a unison vector but isn't? To cover the meanings of "unison vector", "generator", "period" and "equivalence interval"? Me:>> (BTW, in an octave-equivalent system, half a unison is a half- octave as >> well as a unison. This is obvious if you think of octave-equivalent >> frequency space as a Hilbert space, and remember that half the pitch is >> the same as the square root of the frequency.) Paul:> Huh? So if the frequency is 6400 Hz, the square root of that is 80, > and that's half the pitch??Hmm, something wrong there. I meant the square root of a frequency *ratio*. Yes? That seems to make sense. Me:>> A tritone is a unison. Paul:> Right . . .That's what the algebra says if you assume an octave is a unison vector (which it isn't, but I was assuming it in that passage). Me:>> octave a unison vector is like imposing octave equivalence. That's >> actually quite similar to something Fokker said. Paul:> Please fill us in!In <A.D. Fokker: Unison Vectors and Periodicity Bl... * [with cont.] (Wayb.)>, "By common general agreement all notes differing by an arbitrary number of octaves only, are considered as unison, and as one and the same note." I mentioned this last time round as well. Graham

Message: 3673 - Contents - Hide Contents Date: Fri, 1 Feb 2002 01:19:47 Subject: Re: new cylindrical meantone lattice From: monz> From: paulerlich <paul@xxxxxxxxxxxxx.xxx> > To: <tuning-math@xxxxxxxxxxx.xxx> > Sent: Friday, February 01, 2002 1:06 AM > Subject: [tuning-math] Re: new cylindrical meantone lattice > >>> So, how about a formula that plots 19-EDO as, literally, >> a close cousin to 1/3-comma meantone spiral? How does >> take something that's roots of 2, and change it into >> "8ve"-equivalent fractional powers of 3 and 5? >> Well, maybe there's another way to get the right spiral.I'm all ears.>>> Now about your other two objections: >> >>>>> (a) the density of points along the line, which doesn't >>> appear to be meaningful; >> >>>> I'm hoping that the post I just sent before this one, >> about composer choosing particular flavors of meantone, >> addresses this one. >> Not at all -- I was referring to the fact that, for example, > in 5/18-comma meantone, the points on the spiral are rather > far apart from one another -- that doesn't seem particularly > meaningful.OK, the only way I can respond to this properly is to go ahead and create a 5/18-comma lattice and examine it. That's not going to happen until tomorrow. But try to remember this if I don't respond soon, because I do actually want to make that lattice and have a look, and try to answer you.>>>> (b) the fact that you have to pin the spiral to a particular >>> "1/1" origin, which ruins the rotational symmetry of the >>> cylindrical meantone lattice >> >>>> I've already said elsewhere that the spiral doesn't have >> to be pinned to anything. It can float anywhere the user >> wants it. What's important is the angle of the spiral, >> as you've noted. >> So maybe a set of arrows (say from every _true_ lattice point) > pointing at that angle would be preferable to a spiral.Well, I think arrows are a good idea, sure. But again, I'd leave the choice of spirals or arrows up to the user. The whole idea behind JustMusic is to give people a visual way to create music, by manipulating graphs of the sonic math. In effect, the computer screen becomes a lattice instrument. So I want it to be as flexible as possible, and want to leave as many decisions as possible to the user. -monz _________________________________________________________ Do You Yahoo!? Get your free @yahoo.com address at Yahoo! Mail - The best web-based email! * [with cont.] (Wayb.)

Message: 3674 - Contents - Hide Contents Date: Fri, 01 Feb 2002 06:50:31 Subject: Re: Approximate consonances of Parch's 43 tone scale From: genewardsmith --- In tuning-math@y..., "paulerlich" <paul@s...> wrote:>> [.1383934690, 1/8] >> >> a = 9.9643/72 >> Equivalently, about 1/72 oct. -- right?Right. It suggests a temperament of 72 in terms of the 224-et, with a generator of 3/224 and another of 1/8.>> = 31.0001/224 = 166.0721626 >> >> badness 147.3854996 >> This 11-limit badness is not directly comparable to 5-limit badness > for 5-limit temperaments, is it?Nope; it's similar in a way, because of the flatness condition.

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