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Message: 8200 - Contents - Hide Contents Date: Fri, 14 Nov 2003 16:20:11 Subject: Re: "does not work in the 11-limit" (was:: Vals?) From: Manuel Op de Coul Gene wrote:>You will also find stuff about "JI epimorphic", but I don't understand >what Manuel is up to; it isn't what I expected.Can you give an example of what you expected to be different? I thought I implemented the epimorphism you discussed on this list. Manuel

Message: 8201 - Contents - Hide Contents Date: Fri, 14 Nov 2003 22:17:59 Subject: Re: Vals? From: Dave Keenan --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:> I am not picking nits. This is very important to the math. I don't > understand what your objection is anyway--we have a lot of defintions > of tuning and scale terms tossed around which are neither clear nor > elegant sounding. "Val" is short, and I gave it a precise definition, > thereby doing what I wish other people would more often do, judging > by the definitions in Joe's dictionary.I thought we just agreed that you wouldn't worry too much if my explanations didn't capture the precise pure-math meaning, and in return I wouldn't worry that you give definitions that are incomprehensible to most tuning-math readers But if I've made a serious mistake I really need to know: What do the integers [the val's coefficients] represent in tuning terms? What are they counting?

Message: 8202 - Contents - Hide Contents Date: Fri, 14 Nov 2003 23:59:44 Subject: Re: Vals? From: Dave Keenan --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> > wrote: >>>> That's Gene's proposal (that we should write them that way, not > that>>> we should call them bras and kets) and it seems like a reasonable >> one >>> to me. >>>> it'll help me, since i'm used to them. >> Sounds like we are achieving consensus on something! >>>> Except I wonder how we should write a complete mapping matrix for > a >> more-than-1D temperament. >>>> a matrix is a matrix, not a bra or a ket. i never understood >> covariant vs. contravariant, though . . . >> Let's leave matricies alone.But mappings for more-than-1D temperaments are naturally expressed as matrices that map prime-exponent vectors to generator-count vectors. In general a temperament has a prime-mapping matrix "M" that has a column for each prime and a row for each generator. And a ratio has a prime-exponent-vector (monzo) "a" of the same width. If we want to know how many of each generator correspond to that ratio in that temperament we simply calculate the matrix product transpose(M*transpose(a)), or simply M*a if the monzo is already a column vector and the result can be a column vector.> As for covariant vs contravariant, if > you change the basis for monzos to something other than primes, you > have to make a complimentary change in basis for the val basis. The > standard basis is that monzos have a basis e_2, e_3 etc. > corresponding to primes, and vals v_2, v_3 corresponding to (whether > we want to call them that or not) padic valuations.Sure. BWDIMAATT? :-) Never mind. I don't think we need to worry about it. We just need to remember that mappings and monzos are different kinds of things. They have different "units" as it were.

Message: 8203 - Contents - Hide Contents Date: Fri, 14 Nov 2003 02:17:37 Subject: Re: Vals? From: Carl Lumma>> >oicing shouldn't matter, since the voicing of the thing you're >> mapping to (an ET) doesn't matter. If I set... >> >> 1= 9/8 >> 2= 5/4 >> 3= 11/8 >> 4= 3/2 >> 5= 7/4 >> 6= 2/1 >> >> ...can you show me the problem? >>3/2 is 4 steps, so 9/4 is 8 steps, so 9/8 is 2 steps--except it is >also 1 step, contradiction.You may be wondering why I ask a question and then answer it in the same message. If I'm wrong it saves a volley because the reader knows what I was trying to do. -Carl

Message: 8204 - Contents - Hide Contents Date: Fri, 14 Nov 2003 17:28:34 Subject: Re: Vals? From: monz hey paul, --- In tuning-math@xxxxxxxxxxx.xxxx "monz" <monz@a...> wrote:> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> > wrote:>> p.s. would it be OK for me to attempt a modification of your page >> >> Definitions of tuning terms: EDO prime error, ... * [with cont.] (Wayb.) >> >> ? i realized that you *do* have the signed errors of the >> primes in the text, despite your use of absolute values in >> the graph. so if i just added the signed errors for the >> odds that you omitted, i could then quickly locate any >> inconsistency, since inconsistency occurs if and only if >> the signed relative error of one odd differs by over 50% >> from the signed relative error of another odd. for example, >> in 43-equal, the error on 7 is, as you show, +28%; the error >> on 9 is double that on 3, so about -30%; the difference >> between these two signed percentages (and thus the implied >> error on 9:7) is 58%; so 43-equal is inconsistent in the >> 9-limit. what do you think? i think the page would be >> sorely misleading, and much less useful, without this >> information. > > >> paul, you already know that i think that the information > given on my "EDO prime error" page is useful as it is. > > but if you envision a better version of the page, sure, > send me the modification. you know i trust your judgment > on tuning matters! :)take a look at this: Definitions of tuning terms: EDO 11-odd-limit ... * [with cont.] (Wayb.) -monz

Message: 8205 - Contents - Hide Contents Date: Fri, 14 Nov 2003 22:20:09 Subject: Re: Vals? From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> > wrote: >>>> That's Gene's proposal (that we should write them that way, not > that>>> we should call them bras and kets) and it seems like a reasonable >> one >>> to me. >>>> it'll help me, since i'm used to them. >> Sounds like we are achieving consensus on something! >>>> Except I wonder how we should write a complete mapping matrix for > a >> more-than-1D temperament. >>>> a matrix is a matrix, not a bra or a ket. i never understood >> covariant vs. contravariant, though . . . >> Let's leave matricies alone. As for covariant vs contravariant, if > you change the basis for monzos to something other than primes, you > have to make a complimentary change in basis for the val basis. The > standard basis is that monzos have a basis e_2, e_3 etc. > corresponding to primes, and vals v_2, v_3 corresponding to (whether > we want to call them that or not) padic valuations.so how can i tell which one is covariant and which one is contravariant?

Message: 8206 - Contents - Hide Contents Date: Fri, 14 Nov 2003 10:34:39 Subject: Re: Integrating the Riemann-Siegel Zeta function and ets From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:> If we look at successively larger values, we get 2, 5, 7, 12, 19, 31, > 41, 53, 72 ..., and this makes a lot of sense to me. The so-called > "Omega theorems", about the rate of growth of the high values of > |Z(x)|, do not seem strong enough to show this is an infinte list...Never mind. They are strong enough...

Message: 8207 - Contents - Hide Contents Date: Fri, 14 Nov 2003 18:41:40 Subject: Re: "does not work in the 11-limit" (was:: Vals?) From: George D. Secor This is in reply to two messages: Yahoo groups: /tuning-math/message/7634 * [with cont.] --- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" <d.keenan@b...> wrote:> --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:>> --- In tuning-math@xxxxxxxxxxx.xxxx "monz" <monz@a...> wrote: >>>>> can you *please* give a very detailed explanation of what >>> you're saying? ... with lots and lots of 11-limit examples >>> that don't work and 3-, 5-, 7-, 9-, 13-limit examples that do? >>> >>> thanks. >>>> Here are the 5, 7, 9, 11 and 13 limit complete otonal chords as Scala >> scale files. If you run "data" on them, you will find that 5, 7, 9 >> and 13 give Constant Structure scales, and 11 does not. > ... >> Bravo! Gene, this is an excellent minimal-math explanation! > > It certainly is a curious fact. And I'm looking forward to hearing > from George, why he thinks it matters so much musically that he > wouldn't consider using an 11-limit tuning.I'm not sure that I said that I wouldn't consider it, only that I have never been (and am still not) interested in it. Yahoo groups: /tuning-math/message/7630 * [with cont.] --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:> maybe george would be better equipped to explain why this is > musically so significant. > ...Gene's observation about constant structure is not only relevant, but it goes right to the heart of the matter. Having said that, I guess that the question, then, is: Is it important that a musical scale be a constant structure, and if so, why? I've never really thought very much about this, because for me this was something that seemed to be fairly obvious: that a musical scale that is not a constant structure will tend to result in confusion or disorientation by an inherent contradiction between the acoustical properties of certain intervals and their identity (or ability to function) as members (i.e., degrees or steps) of that scale. (By "scale", I am referring to a set of tones that may be used to write a simple melody. A scale may or may not be a subset of a larger set of tones, which I would call a "tuning". Thus a major or minor scale could be considered a subset of the pythagorean tuning, meantone temperament, or 12-ET, for example.) If I'm using a pentatonic scale made from a 9-limit otonal chord: 8 : 9 : 10 : 12 : 14 : 16 then I have two intervals each of 2:3 (both pentatonic "4ths") and 3:4 (both pentatonic "3rds"). Likewise, if I'm using a heptatonic scale made from a 13-limit otonal chord: 8 : 9 : 10 : 11 : 12 : 13 : 14 : 16 then I have two intervals each of 2:3 (both heptatonic 5ths) and 3:4 (both heptatonic 4ths). So far, so good. But if I try to use hexatonic scale made from an 11-limit otonal chord: 8 : 9 : 10 : 11 : 12 : 14 : 16 then one of my 2:3s is a hexatonic "5th" and the other is a hexatonic "4th", and likewise one of my 3:4s is a hexatonic "4th" and the other is a hexatonic "3rd". Most attempts to transfer a melodic figure beginning on a certain scale degree to another scale degree (such as is required in the musical device called a "sequence") will tend to produce undesirable consequences (such as listener disorientation) due to the fact that the 2:3 and/or 3:4 must switch degree-roles in the process. Now we could go on to ask why this scale-member identity or functionality is so important, and this is the point at which I really had to dig deep for an answer. I believe that, at least with the examples given above, it has something to do with the role that the simplest ratios of 3 play in establishing the roots of chords. If a chord contains a *single* 2:3 or 3:4 (whether just or tempered), then I can almost guarantee that the tone represented by the 3 will *never* be heard as the root of the chord. (There are instances, e.g., 8:10:15, that a tone not in the 2:3 interval will be perceived as the root, but that's not critical to the point that I'm making.) It is this property of the simple ratios of 3 that makes it possible to *invert* many conventional triads and seventh chords *without* changing our *perception* of which note of the chord functions as the *root*. So it would not have been possible for the methods of conventional (5- limit) harmony to have reached such sophistication if the major and minor scales were not constant structures, because our whole method of building chords (by 3rds) has depended on the fact that the simple ratios of 3 would always be heptatonic 4ths and 5ths and that the simple ratios of 5 would always be 3rds and 6ths. (Don't make the mistake of calling the augmented 2nd of a harmonic minor scale a minor 3rd, because it isn't; the two intervals just happen to be the the same size in 12-ET, but the meantone temperament will reveal that they are different.) One can similarly demonstrate that pentatonic melodies are perceived as coherent, because pentatonic scales that contain multiple 3:4s and 2:3s are also constant structures. But take away the property of constant structure while retaining multiple 2:3s, and you invite confusion and disorientation. If you want 11-limit otonal harmony in a conherent scale, then I think it will have to be at least heptatonic and that you're going to have to fill that extra position with something or other, such as: 8:9:10:11:12:27/2:14:16. Hmmm, that's really not a bad choice, if you'll notice that 22:27:32 is an isoharmonic triad. I remember that this scale works very nicely in 31-ET, since the 27/2:16:20 ends up as an ordinary minor triad. Likewise, you can have constant-structure 17 and 19-limit otonal scales: 16:17:18:20:21:22:24:26:28:30:32 (with chords built in decatonic "4ths") and 16:17:18:19:20:21:22:24:25:26:28:30:32 --George

Message: 8208 - Contents - Hide Contents Date: Fri, 14 Nov 2003 22:39:33 Subject: Re: Definition of microtemperament From: monz hi paul, --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:> --- In tuning-math@xxxxxxxxxxx.xxxx "monz" <monz@a...> wrote: >> hi paul, >>>> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> >> wrote: >>>>> You mean 11-odd limit? Well, meantone contained >>> excellent approximations to ratios of 7, but practically >>> no one considered them consonant historically. >> >>>> that's not true, and you know it: meantone gave good >> approximations to a 4:5:7 triad in its "augmented-6th" >> chord, which was used a *lot* in the "common-practice" >> era. >> but not as a consonance -- so what i was saying is true.OK, you're right.>> true, no-one at the time analyzed these chords as >> consonant 4:5:7 chords, >> but huygens *did* find these ratios in augmented sixth chord.oops ... i *knew* you'd catch me on that one!>> but in meantone, that's what >> they were, and they were perfectly acceptable in >> both theory and practice. >> i didn't say they were unacceptable -- plenty of > not-so-easy-to-ratio-analyse sonorities were acceptable > as dissonances as well -- just not considered consonant, > that is, it was not used as a chord to resolve a dissonant > chord to, but rather it was used as a chord that would > resolve *to* a consonant chord.OK, now i understand perfectly what you were saying. the augmented-6th chord was always used as a "dissonant" chord which had to resolve, as you say. -monz

Message: 8209 - Contents - Hide Contents Date: Fri, 14 Nov 2003 10:36:59 Subject: Re: Vals? From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:>>> So are monzos are now kets written [ ... > ? >>> and vals are bras written < ... ] ? >>>> I think that's a good suggestion. It is a standard (especially in >> physics), clever notation due to Dirac. We just don't worry about >> complex numbers and certainly not about quantum mechanics. >> But Dave is right that you weren't suggesting using the names > "bra" and "ket", right?No--that would really lead to confusion anyway.

Message: 8210 - Contents - Hide Contents Date: Fri, 14 Nov 2003 20:00:05 Subject: Re: Vals? From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:>> But Dave is right that you weren't suggesting using the names >> "bra" and "ket", right? >> No--that would really lead to confusion anyway.Speaking of which, I should probably have used < ... | for the val and | ... > for the monzo, as being the actual bra-ket notation. The idea is that then you just stick them together, without putting a dot in the middle, for the product: < ... | ... > notating the product.

Message: 8211 - Contents - Hide Contents Date: Fri, 14 Nov 2003 22:42:19 Subject: Re: Definition of microtemperament From: Paul Erlich but the appearance of this chord so easily in the meantone series certainly makes one wonder what direction western music might have progressed in had the movement for closure at 12 notes not won out. --- In tuning-math@xxxxxxxxxxx.xxxx "monz" <monz@a...> wrote:> hi paul, > > > --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> > wrote:>> --- In tuning-math@xxxxxxxxxxx.xxxx "monz" <monz@a...> wrote: >>> hi paul, >>>>>> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> >>> wrote: >>>>>>> You mean 11-odd limit? Well, meantone contained >>>> excellent approximations to ratios of 7, but practically >>>> no one considered them consonant historically. >>> >>>>>> that's not true, and you know it: meantone gave good >>> approximations to a 4:5:7 triad in its "augmented-6th" >>> chord, which was used a *lot* in the "common-practice" >>> era. >>>> but not as a consonance -- so what i was saying is true. > >> OK, you're right. > >>>> true, no-one at the time analyzed these chords as >>> consonant 4:5:7 chords, >>>> but huygens *did* find these ratios in augmented sixth chord. > > >> oops ... i *knew* you'd catch me on that one! > >>>> but in meantone, that's what >>> they were, and they were perfectly acceptable in >>> both theory and practice. >>>> i didn't say they were unacceptable -- plenty of >> not-so-easy-to-ratio-analyse sonorities were acceptable >> as dissonances as well -- just not considered consonant, >> that is, it was not used as a chord to resolve a dissonant >> chord to, but rather it was used as a chord that would >> resolve *to* a consonant chord. > >

Message: 8212 - Contents - Hide Contents Date: Fri, 14 Nov 2003 00:11:15 Subject: Re: Definition of microtemperament From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:> You mean 11-odd limit? Well, meantone contained excellent > approximations to ratios of 7, but practically no one considered them > consonant historically. So i see no problem with considering miracle > a 7-limit temperament if someone uses it in a style where ratios of > 11 or their approximation are used as dissonances.Which, in fact, I have already done, which was why I was using 175 and not 72.

Message: 8213 - Contents - Hide Contents Date: Fri, 14 Nov 2003 10:43:06 Subject: Re: Vals? From: Dave Keenan --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:> --- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" <d.keenan@b...> > wrote:>> A prime-mapping (or val with log-prime basis) simply maps each prime >> number (or strictly-speaking the logarithm of each prime number) to > an>> integer multiple of some interval (log of frequency ratio) that we >> call a generator. >> This is absolutely not what I mean by a val, which maps to integers.I think we're picking nits here. What do the integers represent in tuning terms? What are they counting?

Message: 8214 - Contents - Hide Contents Date: Fri, 14 Nov 2003 14:40:16 Subject: Re: Integrating the Riemann-Siegel Zeta From: Carl Lumma Roger that! I was going to post something to this effect, but I still haven't studied the mathworld entry. -Carl>this is really hot, and i wish i understood it . . . maybe if manfred >schroeder wrote a book on it . . . > // >>>> The point of this business is to give what you might call a generic >> goodness measure for ets; meaning one not attached to any particular >> prime limit. The result seems better than what we get for maximal >> values of |Z(x)|, and much better than what we can glean from gaps >> between the zeros.

Message: 8215 - Contents - Hide Contents Date: Fri, 14 Nov 2003 20:09:05 Subject: Re: Vals? From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" <d.keenan@b...> wrote:> I think we're picking nits here. What do the integers represent in > tuning terms? What are they counting?I am not picking nits. This is very important to the math. I don't understand what your objection is anyway--we have a lot of defintions of tuning and scale terms tossed around which are neither clear nor elegant sounding. "Val" is short, and I gave it a precise definition, thereby doing what I wish other people would more often do, judging by the definitions in Joe's dictionary.

Message: 8216 - Contents - Hide Contents Date: Fri, 14 Nov 2003 00:14:10 Subject: Re: "does not work in the 11-limit" (was:: Vals?) From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:> maybe george would be better equipped to explain why this is > musically so significant.Could be, though I've made very extensive use of it as a compositional tool. That goes all the way back to my 1980s paper.

Message: 8217 - Contents - Hide Contents Date: Fri, 14 Nov 2003 03:08:25 Subject: Re: Vals? From: Carl Lumma>I think we're picking nits here. What do the integers represent in >tuning terms? What are they counting?Not necessarily the most musically-obvious generator, if you're coming at vals in the context of linear temperaments. I don't think integers is what confuses me. I'm still wondering about 6. In 22, the 11-prime-limit val consistently maps the 9/8, and the resulting hexad taken as a scale is a Constant Structure. 22 is even generally 11-limit consistent. Why not use 22? -Carl

Message: 8218 - Contents - Hide Contents Date: Fri, 14 Nov 2003 12:11:13 Subject: Re: Vals? From: Carl Lumma>Speaking of which, I should probably have used < ... | for the val >and | ... > for the monzo, as being the actual bra-ket notation.The only reason I put it the other way was so that bra-ket would tell you the position of the square brackets. But you're using a pipe instead of a square bracket, so this way the word still tells you the location of the most bracket-like things, the angle bracket. -Carl

Message: 8219 - Contents - Hide Contents Date: Fri, 14 Nov 2003 14:43:21 Subject: Re: Vals? From: Carl Lumma>> >k, now we're on the right track, but I'm still not grokking >> you. I started with six rationals and ended up with 6 integers. >> What's the problem? >>Are your integers consecutive?No, and that's part of the def. of standard val, but what motivates it? -Carl

Message: 8220 - Contents - Hide Contents Date: Fri, 14 Nov 2003 01:00:34 Subject: Re: Vals? From: Dave Keenan --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:> [Dave Keenan:]>> So did it become a lot clearer what a val was, when you figured >> out that it was a prime mapping? >> Everything I've ever figured out about vals made them clearer, > obviously. The words "prime mapping" wouldn't have helped a bit.This seems like a logical contradiction. You say that figuring out that vals were prime mappings made them clearer. Wouldn't calling them "prime mappings" have helped you figure out that they were prime mappings? And wouldn't this have happened sooner if they had been called prime mappings instead of vals? I guess what you're saying is that this wasn't a very important factor in your understanding of the concept.>> Did you understand what they were mapping the primes _to_ at that >> time? >> I still don't think I do.A prime-mapping (or val with log-prime basis) simply maps each prime number (or strictly-speaking the logarithm of each prime number) to an integer multiple of some interval (log of frequency ratio) that we call a generator. If we are told that the mapping is for a tET then _which_ tET it is for can be read straight out of the mapping, as the coefficient for the prime 2 (the first coefficient). And the generator is simply one step of that tET. If we were told it was for an ET3 then we could read off what ET3 it was for as the second coefficient, and the generator is that fraction of the tritave. If however we are told that the mapping is for an arbitrary equal temperament, a cET, then we would have to solve for the generator that minimises some error function such as max-absolute (minimax) or rms (sum of squares). It's fairly simple to do this numerically in Excel using the Solver add-in. Let me know if you want more details on that. In an ET, the generator is the step. But not necessarily so in higher D temperaments If we were told the mapping was one row (Gene says we can forget that "column" stuff) of a linear temperament mapping, then to solve for the generator that this row maps to, we would either need to know what the other generator was, or what its mapping was, e.g. maybe the other generator is the period and we are told it is an exact octave.> But Gene's talking about finding vals for limits!!!He's just abbreviating excessively and assuming the meaning will be clear from your readings of his previous postings in the same thread. He's really talking about finding vals-with-log-prime-basis (prime-mappings) that map the complete chord of each limit to a tET with the same cardinality. It's all about how evenly-spaced the chords are. Try the 6 possible possible voicings of the 11-limit otonality, that fit within an octave, and you'll see that none of them are very even. Why such an apparently melodic property should be considered important when applied to a vertical harmony, I don't know.> Note that I have no idea what the bra ket notation stuff is about.It's just a way of distinguishing prime-mappings (vals) from prime-exponent-vectors (monzos) without having to say it in words every time. It only makes sense to multiply mappings by exponent-vectors, not any other combination and these brackets try to make that clear because ] and [ fit together, but > and <, > and [, ] and < do not.> Then I don't know why the standard 11-limit mapping wouldn't be > identical to the standard 11-prime-limit mapping. It is. > Anyway, saying mapping instead of val is already confusing me here.Sorry. I've given both in several places above. As far as tuning is concerned the only important difference I can see is that in the case of temperaments with more than one generator, "the mapping" (unqualified) refers to the whole matrix (all the rows, one per generator). There's no such thing as "the val" for such a temperament. In this case a val is apparently only one row. But even there, "the val for generator x" is the same as "the mapping for generator x". So the term val isn't actually needed.

Message: 8221 - Contents - Hide Contents Date: Fri, 14 Nov 2003 14:42:38 Subject: Re: Vals? From: Carl Lumma>> >eah well, the choice of 6 here still hasn't been accounted for. >>i explained why it was 3 in the case of 5-limit, and here it's the >same -- 6 is the number of notes in the 11-limit complete otonality! >this has nothing to do with the definition of vals, it's just one >particular problem that gene and george happen to be interested in.Right, got that, just don't see why it's a "problem".>>> So it's proper, but not a constant structure. I was under the >>> misapprehension that proper always implied constant structure, i.e >>> that propriety was a stronger condition. Hmm. // >>> However, the Enharmonic of Archytas is. Translate the scale 28/27 x >>> 36/35 x 5/4 x 9/8 x 28/27 x 36/35 x 5/4 into cents and generate the >>> D-matrix. >>> >>> 63 49 386 204 63 49 386 >>> 112 435 590 267 112 435 449 >>> 498 639 653 316 498 498 498 >>> 702 702 702 702 561 547 884 >>> 765 751 1088 765 610 933 1088 >>> 814 1137 1151 814 996 1137 1151 >>> 1200 ....... >> >> -Carl >>this is an example of . . . ?A (wildly) improper constant structure. -C.

Message: 8222 - Contents - Hide Contents Date: Fri, 14 Nov 2003 20:11:36 Subject: Re: Vals? From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:> I'm still wondering about 6. In 22, the 11-prime-limit val > consistently maps the 9/8, and the resulting hexad taken as a > scale is a Constant Structure. 22 is even generally 11-limit > consistent. Why not use 22?You started with 6 and ended up with 22. Where is your 22 note scale/chord?

Message: 8223 - Contents - Hide Contents Date: Fri, 14 Nov 2003 12:25:40 Subject: Re: Vals? From: Carl Lumma I wrote...>> So it's proper, but not a constant structure. I was under the >> misapprehension that proper always implied constant structure, i.e >> that propriety was a stronger condition. Hmm. >>Nope. The set of all non-CS scales is equivalennt to the set of >all non-strictly-proper scales.Sorry, it was too late. There improper non-CS scales. There's no convenient way to express CS with Rothenberg's language that I know of. -Carl

Message: 8224 - Contents - Hide Contents Date: Fri, 14 Nov 2003 22:47:39 Subject: Re: Vals? From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:>>> Ok, now we're on the right track, but I'm still not grokking >>> you. I started with six rationals and ended up with 6 integers. >>> What's the problem? >>>> Are your integers consecutive? >> No, and that's part of the def. of standard val, but what > motivates it? > > -Carli can't make heads or tails of this question. the standard val puts the primes in order because it's easy to remember them that way. you could put them in a different order but you would have to remember which entry refers to which prime. so i don't see what there is to motivate.

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