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Message: 8225 - Contents - Hide Contents Date: Fri, 14 Nov 2003 01:08:43 Subject: Re: Definition of microtemperament From: Dave Keenan --- In tuning-math@xxxxxxxxxxx.xxxx "George D. Secor" <gdsecor@y...> wrote: I know you've already conceded, but I thought I'd mention this anyway. By sacrificing the ratios of 9 and the ratios of 11 one can find a different optimum generator that treats the 7-limit better (whether you favour minimax or rms). This is at least one thing we mean by 7-limit miracle.

Message: 8226 - Contents - Hide Contents Date: Fri, 14 Nov 2003 12:29:02 Subject: Re: "does not work in the 11-limit" From: Carl Lumma>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.Does that include the diatonic scale in 12-equal?>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.Paul E. has suggested that we only care about collisions if they occur to a consonant interval. That allows the diatonic scale in 12-equal to pass. But incidentally, I'd love a musical example of a hexatonic 11-limit melody where the non-CS "collision" causes a problem with constructing a musical sequence. With all the ink I've spilled on this subject, I'm probably more guilty than anyone of not having come up with musical examples to demonstrate propriety... -Carl

Message: 8227 - Contents - Hide Contents Date: Fri, 14 Nov 2003 22:48:56 Subject: Re: Vals? From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:>>> Yeah 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".well, george tried to explain it, and i had point of agreement and disagreement with him . .>>>> 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.a random scale has essentially a 100% chance of being one.

Message: 8228 - Contents - Hide Contents Date: Fri, 14 Nov 2003 12:35:52 Subject: Re: Vals? From: Carl Lumma>> >'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?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? -Carl

Message: 8229 - Contents - Hide Contents Date: Fri, 14 Nov 2003 22:51:40 Subject: Re: Vals? From: Dave Keenan --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:> --- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" <d.keenan@b...> > wrote: >>> 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. >> just wondering why you keep saying "tET" -- 'If we are told that the > mapping is for a tone equal temperament then . . .' ??I agree it's awkward. Carl objected so vehemently to EDO and I wanted to reserve ET for the most general term (including EDOs ED3s cETs). Perhaps this would be a misuse of ET. Do we have some other term for the most general category of 1D temperaments, i.e. any single generator temperament whether or not it is an integer fraction of any ratio? I guess "1D-temperament" will do.> actually, > and < fit together and create a X (as in times) !Oops. Well we could interpret that as the matrix-product as opposed to the scalar-product (dot-product), but I don't know of any meaning for that in tuning.

Message: 8230 - Contents - Hide Contents Date: Fri, 14 Nov 2003 14:46:26 Subject: Re: Vals? From: Carl Lumma>> >'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. >>no, it's still not a constant structure, since one 3:2 subtends more >notes than the other ('9:6').Oh bloody hell; I had checked this by eye only, and missed that. It's still sitting in my scheme terminal from last night. -Carl

Message: 8231 - Contents - Hide Contents Date: Fri, 14 Nov 2003 21:20:32 Subject: Re: Definition of microtemperament From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" <d.keenan@b...> wrote:> Paul, I assume you were merely arguing that "typically less than 2.8 > cents" is as about as good as any other nearby number as a > just-noticeble-difference, and you wouldn't really mind if the > microtemperament definition was changed to "typically less than 3 cents".of course i wouldn't mind.

Message: 8232 - Contents - Hide Contents Date: Fri, 14 Nov 2003 01:33:47 Subject: Re: Vals? From: Dave Keenan --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:>>> Gene, since you won't say what's desirable about being a >>> standard val... >>>> Purely a matter of being easy to calculate. >> Adding our birthdays together is easy to calculate. There > must be some other reason.No. That's really it. Although I suppose we should say "and it is also the best (i.e. most accurate) mapping when the ET is consistent. So you'd only be interested in mappings other than this easy-to-calculate one, if the ET is inconsistent at the given limit, and we are rarely interested in those anyway, so you can get away with it most of the time. It's a lot more complicated to find the best mapping for an inconsistent ET.> Dave's 'the best approx. to each > element of a chord in n-tET' is better,Err. I don't remember writing that. The "standard" mapping for a tET gives the best approximation to intervals whose ratio contains a single prime number other than 2 (with that odd-prime raised only to the first power). You've certainly gotten hold of some wrong idea about all this, and I don't blame you. I just wish I knew what it is. Standard vals (standard mappings), or vals (mappings) of any kind, have absolutely nothing to do with chords. Except that you can apply vals (mappings) to rational pitches to see where they end up in the temperament that the val (mapping) corresponds to. And chords can contain rational pitches, so vals (mappings) can be applied to chords to see where they end up when so tempered. A val (mapping) for some temperament simply tells you how to map a rational pitch to iterations of a single generator of that temperament (which is the _only_ generator in the case of an ET, and is its step).> but why n should equal > the number of notes in a chord is still a mystery.It's pretty much of a mystery to me too. This is not a necessary property of vals or even of standard vals. It has nothing to do with them. It's just that Gene and George found it interesting to look at how complete chords can be mapped to a single octave of the ET of the same cardinality as the chord. It turns out that the 11-limit otonality can't be. There is no mapping and no voicing of the chord that will do this.

Message: 8233 - Contents - Hide Contents Date: Fri, 14 Nov 2003 15:06:00 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 >>i 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.You lost me. My 22-val example doesn't reorder the primes! -C.

Message: 8234 - Contents - Hide Contents Date: Fri, 14 Nov 2003 21:29:23 Subject: Re: Vals? From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" <d.keenan@b...> wrote:> 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.just wondering why you keep saying "tET" -- 'If we are told that the mapping is for a tone equal temperament then . . .' ??> Why such an apparently melodic property should be considered important > when applied to a vertical harmony, I don't know.well, my 22 paper does give a bit of indication as to why that might be. by those considerations too, 11-limit hexads would be rather difficult to find a reasonable tonal system for. but my considerations had to do with getting the utonal and otonal complete consonances from the same pattern of scale steps, while george is probably uninterested in the utonalities . . .>> 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.actually, > and < fit together and create a X (as in times) !

Message: 8235 - Contents - Hide Contents Date: Fri, 14 Nov 2003 21:33:52 Subject: Re: Vals? From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" <d.keenan@b...> wrote:> --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:>>>> Gene, since you won't say what's desirable about being a >>>> standard val... >>>>>> Purely a matter of being easy to calculate. >>>> Adding our birthdays together is easy to calculate. There >> must be some other reason. >> No. That's really it. Although I suppose we should say "and it is also > the best (i.e. most accurate) mapping when the ET is consistent. So > you'd only be interested in mappings other than this easy-to- calculate > one, if the ET is inconsistent at the given limit, and we are rarely > interested in those anyway, so you can get away with it most of the > time. It's a lot more complicated to find the best mapping for an > inconsistent ET.right, but gene just did that (two days ago?), and hopefully will do more.> It's just that Gene and George found it interesting to look at how > complete chords can be mapped to a single octave of the ET of the same > cardinality as the chord. It turns out that the 11-limit otonality > can't be. There is no mapping and no voicing of the chord that will do > this.let me just repeat dave and say that this has *nothing* to do with the definition of vals -- it's a separate question that you can safely ignore if you want to understand vals.

Message: 8236 - Contents - Hide Contents Date: Fri, 14 Nov 2003 23:07:16 Subject: Re: Vals? From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" <d.keenan@b...> wrote:> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:>> --- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" <d.keenan@b...> >> wrote: >>>>> 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. >>>> just wondering why you keep saying "tET" -- 'If we are told that the >> mapping is for a tone equal temperament then . . .' ?? >> I agree it's awkward. Carl objected so vehemently to EDO and I wanted > to reserve ET for the most general term (including EDOs ED3s cETs). > Perhaps this would be a misuse of ET. Do we have some other term for > the most general category of 1D temperaments, i.e. any single > generator temperament whether or not it is an integer fraction of any > ratio? I guess "1D-temperament" will do. >>> actually, > and < fit together and create a X (as in times) ! >> Oops. Well we could interpret that as the matrix-product as opposed to > the scalar-product (dot-product), but I don't know of any meaning for > that in tuning.the symbol normally indicates the cross-product, which is extremely useful in tuning: for example, if i take the monzo for the diaschisma [-4 4 -1> and cross it with the (transpose of the?) monzo for the syntonic comma <-11 4 2] i get the val for the et where they both vanish: [12 19 28] not sure how gene would do this notationally, probably i did something terrible, but without it i could not have made those charts . . .

Message: 8237 - Contents - Hide Contents Date: Fri, 14 Nov 2003 01:37:57 Subject: Re: "does not work in the 11-limit" From: Dave Keenan --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:>> 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. You will also >> find stuff about "JI epimorphic", but I don't understand what Manuel >> is up to; it isn't what I expected. >> So there's no val that sends all 11-limit intervals to integers > without collisions?Assuming you mean 11-odd-limit intervals then that's not true, and its not what's being discussed. The "integers" referred to are simply the degrees of some ET, and clearly there are ETs and mappings of primes to them, that give a unique degree for every interval in the diamond, for any odd limit.

Message: 8238 - Contents - Hide Contents Date: Fri, 14 Nov 2003 15:07:20 Subject: Re: Vals? From: Carl Lumma>> >ight, got that, just don't see why it's a "problem". >>well, george tried to explain it, and i had point of agreement and >disagreement with him . .Without an musical example (referenced or constructed), I'm skeptical. -Carl

Message: 8239 - Contents - Hide Contents Date: Fri, 14 Nov 2003 21:37:36 Subject: Re: Vals? From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" <d.keenan@b...> wrote:> --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:>> Why the same card.? >> Dunno. But it's too easy otherwise. >>>> 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. >>>> It's proper and for that matter seems to fit to 6-tET reasonably >> well. >> OK. Well maybe I got something wrong about what's being claimed. I'd > better let the experts sort it out.it's not strictly proper.>> So are monzos are now kets written [ ... > ? >> and vals are bras written < ... ] ? >> 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.> 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 . . .

Message: 8240 - Contents - Hide Contents Date: Fri, 14 Nov 2003 01:53:48 Subject: Re: Vals? From: Dave Keenan --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:> Why the same card.?Dunno. But it's too easy otherwise.>> 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. >> It's proper and for that matter seems to fit to 6-tET reasonably > well.OK. Well maybe I got something wrong about what's being claimed. I'd better let the experts sort it out.> So are monzos are now kets written [ ... > ? > and vals are bras written < ... ] ?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. Except I wonder how we should write a complete mapping matrix for a more-than-1D temperament. I suppose < ...... ...... ...... ] would be best, although to put them on one line we could use < ...... ; ...... ; ...... ]

Message: 8241 - Contents - Hide Contents Date: Fri, 14 Nov 2003 14:48:24 Subject: Re: Vals? From: Carl Lumma>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.What happens if we change the bases to odd numbers. There's no longer a unique monzo for any given interval, which seems bad. What other bases did you have in mind? -Carl

Message: 8242 - Contents - Hide Contents Date: Fri, 14 Nov 2003 21:43:02 Subject: Re: Definition of microtemperament From: Paul Erlich --- 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.> 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.> 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: 8243 - Contents - Hide Contents Date: Fri, 14 Nov 2003 21:46:00 Subject: Re: Integrating the Riemann-Siegel Zeta function and ets From: Paul Erlich this is really hot, and i wish i understood it . . . maybe if manfred schroeder wrote a book on it . . . --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:> The Riemann-Siegel Zeta function Z(t) is defined here: > > Riemann-Siegel Functions -- from MathWorld * [with cont.] > > For our purposes we want to change scales, setting t = 2 pi x / ln (2), > and use Z(x) instead. I integrated |Z(x)| between successive zeros, for > zeros up to 100.409754. Below I list every inteval between zeros where > the integral is greater than one. > > 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. > > 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, > though it starts out looking as if it is planning on being one. I > think I'll write to some people more expert than I am and inquire. > > If we take the values over one by decades, we get: > > 1-10: 2, 3, 5, 7, 10 > 11-20: 12, 15, 17, 19 > 21-30: 22, 24, 26, 27, 29 > 31-40: 31, 34, 36 > 41-50: 41, 43, 46, 50 > 51-50: 53, 58, 60 > 61-70: 63, 65, 68 > 71-80: 72, 77, 80 > 81-90: 84, 87, 89 > 91-100: 94, 96, 99 > > It seems the density may be falling off slowly. > > > [1.559311781, 2.319105165] 1.103823 > [2.759142784, 3.356405400] 1.044063 > [4.779747405, 5.295822634] 1.131648 > [6.710827976, 7.183072612] 1.162332 > [9.797225769, 10.20350285] 1.082282 > [11.82260542, 12.24853409] 1.269599 > [14.86604170, 15.23665791] 1.104057 > [16.88134757, 17.22203271] 1.032175 > [18.74431544, 19.13037920] 1.313799 > [21.84461333, 22.20308465] 1.258178 > [23.84734791, 24.16705528] 1.092055 > [25.78054223, 26.09283267] 1.031155 > [26.92536457, 27.26360905] 1.185939 > [28.77144315, 29.07689211] 1.000619 > [30.80395665, 31.16093004] 1.403777 > [33.89177893, 34.21059373] 1.241437 > [35.83815669, 36.12289081] 1.028887 > [40.82320329, 41.15537120] 1.423937 > [42.89664942, 43.18457394] 1.035628 > [45.83210532, 46.15561125] 1.356067 > [49.79781990, 50.08281814] 1.111229 > [52.83584779, 53.15446302] 1.486620 > [57.92538202, 58.23716835] 1.358357 > [59.77541720, 60.04861404] 1.131000 > [62.88678487, 63.14811332] 1.049023 > [64.88227375, 65.16035560] 1.269821 > [67.90486013, 68.18884771] 1.254592 > [71.78033774, 72.10918271] 1.625363 > [76.85025545, 77.12671468] 1.311364 > [79.93215353, 80.20726288] 1.247325 > [83.87267811, 84.13938972] 1.241945 > [86.87178850, 87.15758094] 1.439474 > [88.90088275, 89.15353029] 1.124501 > [93.84133446, 94.11907762] 1.394050 > [95.82981785, 96.06991440] 1.045052 > [98.91449741, 99.20014010] 1.510412

Message: 8244 - Contents - Hide Contents Date: Fri, 14 Nov 2003 23:12:02 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? >>> >>> -Carl >>>> i 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. >> You lost me. My 22-val example doesn't reorder the primes! > > -C.what does that have to do with the definition of standard val? the definition, as dave keenan explained it, seems perfectly well motivated, if not always optimal. in a special case it is turned toward the particular problem of an odd-limit complete otonality, but that should be a separate concern . . .

Message: 8245 - Contents - Hide Contents Date: Fri, 14 Nov 2003 02:01:57 Subject: Re: Vals? From: Dave Keenan --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:>> 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. >> Yes, I know this. But why integers?Because that's what a temperament is. A mapping from rational pitches to integer numbers of some (usually irrational) generator (or generators). If they didn't have to be integers then there would be nothing to it.> And why can't there be collisions?I don't know what you're referring to here. I don't think I said anything about collisions. I suspect that's just part of the particular investigation. It's certainly nothing to do with vals or mappings per se. They can certainly result in collisions.> And in what sense could the order in which the identities of a chord > are considered have any bearing on things?If you bring everything back to the first octave then probably no bearing at all.

Message: 8246 - Contents - Hide Contents Date: Fri, 14 Nov 2003 15:12:23 Subject: Re: Vals? From: Carl Lumma>the symbol normally indicates the cross-product, which is extremely >useful in tuning: for example, if i take the monzo for the diaschisma > >[-4 4 -1> > >and cross it with the (transpose of the?) monzo for the syntonic comma > ><-11 4 2]Huh; I thought all monzos were supposed to be written | ... > from here on out.>i get the val for the et where they both vanish: > >[12 19 28] > >not sure how gene would do this notationally, probably i did >something terrible, but without it i could not have made those >charts . . .And I thought all vals were supposed to be written < ... |. Did I miss something? -Carl

Message: 8247 - Contents - Hide Contents Date: Fri, 14 Nov 2003 21:49:05 Subject: Re: Vals? From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:>> Yes! Congratulations! >> Yeah 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.>> 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. > > I remember surprise when John Chalmers first pointed this out > to me, by way of this example: > > [private communication]>> 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 ....... > > -Carlthis is an example of . . . ?

Message: 8248 - Contents - Hide Contents Date: Fri, 14 Nov 2003 02:32:00 Subject: Re: Vals? From: Dave Keenan --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:>> cardinality as the chord. It turns out that the 11-limit otonality >> can't be. There is no mapping and no voicing of the chord that will >> do this. >> Voicing shouldn't matter, since the voicing of the thing you're > mapping to (an ET) doesn't matter.You're probably right, forget about voicing.> 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? Lessee, would the val would be > (the parens pending clarification on the ketbra situation)... > > (val 0 10 14 17 21)Except for the typo where you have a 0 instead of a 6. Yes <6 10, 14 17 21] is the standard prime-mapping for 6-tET.> and reversing the process to get the above I only need to worry > about 9/8, which is (monzo -3 2 0 0 0). It looks like this gives > me a 20-18 = 2,Yes, we can write <6 10, 14 17 21].[-3 2, 0 0 0> = 2 (step generators).> which is supposed to be 5/4. Is *this* the > problem? Yes! Congratulations!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.

Message: 8249 - Contents - Hide Contents Date: Fri, 14 Nov 2003 19:59:56 Subject: Re: Vals? From: Carl Lumma>> >ithout an musical example (referenced or constructed), I'm >skeptical. >>What do you mean by an example?A short melody which doesn't sounds a certain way when tuned in a CS scale, and then, by changing as little else as possible, change things for the worse by warping the scale until its non-CS. Would be ideal. But even an example of music where the failure of "sequence" (even from a theoretical point of view, even without hearing it) would help to communicate what "sequence" is. -Carl

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8900
8950

8200 -
**8225 -**