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Message: 11250 - Contents - Hide Contents Date: Sun, 04 Jul 2004 08:53:30 Subject: Re: NOT tuning From: Graham Breed Carl Lumma wrote:>>>> except where TOP already had pure octaves, in >>>> which case it would actually change! >>>>>> That's impossible given the criterion of NOT. >>> >>> Maybe I don't comprehend you. >>>> Some examples of this method of tuning would be nice, and >> a definition even better. > >> Which method? Graham's? I think he gave examples. > > Graham, what's a good word to search for? I know I have that > post. I think I replied to it.I searched my local folder for "Kees metric" and found a post on 2nd Feb that you can work back from. I didn't originally know I was using a Kees metric, so you won't find that post. I think it must be different to NOT, partly because Gene mentioned some problems that I'm sure I'd already solved. Graham

Message: 11251 - Contents - Hide Contents Date: Sun, 04 Jul 2004 14:22:18 Subject: A chart of syntonic comma temperaments From: Herman Miller http://www.io.com/~hmiller/png/syntonic.png - Type Ok * [with cont.] (Wayb.) This is a chart of 7-limit temperaments that temper out the syntonic comma 81;80. The horizontal axis is deviation from 3:1 and the vertical axis is the deviation from 7:1. This time I limited the list to 7-limit consistent ET's.

Message: 11252 - Contents - Hide Contents Date: Sun, 04 Jul 2004 09:07:47 Subject: Re: NOT tuning From: Graham Breed Paul Erlich wrote:> --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote: > > >>> Meantone >>>>>> 5-limit: 698.0187 (43, 55, 98, 153, 251, 404) >>> >>> 7-limit: 697.6458 (31, 43, 934, 977; 0.0286 cents flatter than 43) >>>> Hmm... I dunno, this seems a bit far from the old-style rms >> optimum. >> >> -Carl > >> Carl, when Graham investigated this same question here a few months > ago, he concluded that pure-octaves TOP would be a uniform stretching > or compression of TOP, except where TOP already had pure octaves, in > which case it would actually change!You can always define the method to give the same answer for pure-odd ratios. But yes, for the 5-limit it should give quarter comma meantone, because the 81:80 is shared between the four factors of 3 in the numerator. It's clearly doing something different. I haven't defined the 7-limit result because I don't generally know how to do 7-limit linear TOP. What I do have is: Minimax 696.58 RMS (7) 696.65 RMS (9) 696.44 PORMSWE 697.22 The last one you may recall is my alternative to TOP. Here, the octave is stretched by 1.24 cents. I can't generalize it to the octave-equivalent case (which is why I switched to odd limits in the first place). But you can always unstretch the octave, which here gives a fifth of 696.49 cents. Either there's a systematic error in all my calculations, or Gene's result is perverse. Graham

Message: 11253 - Contents - Hide Contents Date: Sun, 04 Jul 2004 19:49:10 Subject: Re: A chart of syntonic comma temperaments From: monz hi Herman, --- In tuning-math@xxxxxxxxxxx.xxxx Herman Miller <hmiller@I...> wrote:> http://www.io.com/~hmiller/png/syntonic.png - Type Ok * [with cont.] (Wayb.) > > This is a chart of 7-limit temperaments that temper out > the syntonic comma 81;80. The horizontal axis is deviation > from 3:1 and the vertical axis is the deviation from 7:1. > This time I limited the list to 7-limit consistent ET's.so according to the criteria in this chart, the temperament with the lowest error for both 3 and 7 is 36-ET gawel? i've been missing a lot on the tuning lists until lately, so i don't even know about the gawel family. -monz

Message: 11254 - Contents - Hide Contents Date: Sun, 04 Jul 2004 00:02:20 Subject: Re: A map of starling space From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx Herman Miller <hmiller@I...> wrote:> I made a chart of ET's with the comma 126;125, and drew lines on it to > illustrate some of the landmark starling temperaments.Great! This is something I've suggested Paul do from time to time; I didn't know you were up to this level of graphic design.> Any major starling-related temperaments that I'm missing?Here's a list of 21 starling temperaments, ordered in terms of increasing Graham-TOP badness: [4, 4, 4, -3, -5, -2] [[4, 6, 9, 11], [0, 1, 1, 1]] 4 5.871540 93.944647 [6, 5, 3, -6, -12, -7] [[1, 0, 1, 2], [0, 6, 5, 3]] 6 3.187309 114.743119 [10, 9, 7, -9, -17, -9] [[1, -1, 0, 1], [0, 10, 9, 7]] 10 1.171542 117.154200 [9, 5, -3, -13, -30, -21] [[1, 1, 2, 3], [0, 9, 5, -3]] 12 1.049791 151.169891 [1, 4, 10, 4, 13, 12] [[1, 2, 4, 7], [0, -1, -4, -10]] 10 1.698521 169.852100 [3, 0, -6, -7, -18, -14] [[3, 5, 7, 8], [0, -1, 0, 2]] 9 2.939961 238.136875 [7, 9, 13, -2, 1, 5] [[1, -1, -1, -2], [0, 7, 9, 13]] 13 1.610469 272.169318 [7, 4, -2, -10, -23, -16] [[1, 1, 2, 3], [0, 7, 4, -2]] 9 4.771049 386.454969 [2, -4, -16, -11, -31, -26] [[2, 3, 5, 7], [0, 1, -2, -8]] 18 1.267597 410.701556 [11, 13, 17, -5, -4, 3] [[1, 3, 4, 5], [0, -11, -13, -17]] 17 1.485250 429.237250 [17, 18, 20, -11, -16, -4] [[1, 6, 7, 8], [0, -17, -18, -20]] 20 1.127925 451.169872 [5, 1, -7, -10, -25, -19] [[1, 0, 2, 5], [0, 5, 1, -7]] 12 3.148011 453.313549 [8, 1, -13, -17, -43, -33] [[1, -1, 2, 7], [0, 8, 1, -13]] 21 1.116045 492.175800 [1, -1, -5, -4, -11, -9] [[1, 2, 2, 1], [0, -1, 1, 5]] 6 14.789095 532.407425 [3, 5, 9, 1, 6, 7] [[1, 2, 3, 4], [0, -3, -5, -9]] 9 6.584324 533.330263 [8, 13, 23, 2, 14, 17] [[1, 2, 3, 4], [0, -8, -13, -23]] 23 1.024522 541.972271 [1, -8, -26, -15, -44, -38] [[1, 2, -1, -8], [0, -1, 8, 26]] 27 1.021376 744.583261 [5, 8, 14, 1, 8, 10] [[1, 2, 3, 4], [0, -5, -8, -14]] 14 4.143252 812.077399 [4, 9, 19, 5, 19, 19] [[1, 1, 1, 0], [0, 4, 9, 19]] 19 2.289528 826.519647 [1, -3, -11, -7, -20, -17] [[1, 2, 1, -2], [0, -1, 3, 11]] 12 7.214635 1038.907446 [0, 5, 15, 8, 24, 21] [[5, 8, 12, 15], [0, 0, -1, -3]] 15 5.665687 1274.779472> The original starling map, like most of the others we've seen, is based > on graphing the deviations from 3/1 and 5/1. Possibly a more useful map > for a 7-limit temperament family is one that shows the deviations from > 7/1. So I created a second map. > > http://www.io.com/~hmiller/png/starling-map2.png - Type Ok * [with cont.] (Wayb.)You could also graph the deviation from the 3 and 5 of planar starling in some tuning; if you use the rms values, the 3 comes to 1899.984 cents, the 5 to 2789.270 cents, and the 7 follows from 7=125/18. Using this as the origin makes more sense, really.

Message: 11255 - Contents - Hide Contents Date: Sun, 04 Jul 2004 14:31:51 Subject: Re: dual, and inner product space (was: Gene's mail server) From: monz hi Gene, --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote: >>> What do you mean? In quantum mechanics, the bracket >> or "inner product" is simply the application of a >> linear functional acting on a vector. >> In QM state vectors are unit vectors in a complex > Hilbert space, meaning there is a Hermitian inner product > on the space. An eigenvector for a Hermitian linear operator > (ie, an "observable") with discrete spectrum will *also* > be a state vector if normalized to a unit vector. They > live in the same space, so the eigenvector will be a > wave function like the state vector, if those are wave > functions. If you take the absolute value of the inner > product and square it, you get the probability of a > measurement coming up with with corresponding eigenvalue > as a result of the measurement of the observable > (= Hermitian operator.) If you don't have a discrete spectrum > you need to resort to spectral theory, but it's basically > similar. The upshot is that the eigenvectors are bounded > linear functionals on the states, but since we are in an > inner product space we can identify these with states. > It's like identifying a row vector with a column vector. > > Applications to music? I dunno; but the discrete spectrum > business is intriging. Someone should tune up a hydrogen atom > when taken down enough octaves, I guess.i understand very little of what you wrote here, but ever since i came up with the idea of finity and bridging in 1998 ... Definitions of tuning terms: finity, (c) 1998 ... * [with cont.] (Wayb.) Tonalsoft Encyclopaedia of Tuning - bridging, ... * [with cont.] (Wayb.) ... what i *do* understand about QM has had me believing that it might have some application to music. i.e., there's a sort of "uncertainty principle" with regard to our perception and comprehension of pitch / tuning / harmony. -monz ________________________________________________________________________ ________________________________________________________________________ ------------------------------------------------------------------------ Yahoo! Groups Links <*> To visit your group on the web, go to: Yahoo groups: /tuning-math/ * [with cont.] <*> To unsubscribe from this group, send an email to: tuning-math-unsubscribe@xxxxxxxxxxx.xxx <*> Your use of Yahoo! Groups is subject to: Yahoo! Terms of Service * [with cont.] (Wayb.)

Message: 11256 - Contents - Hide Contents Date: Sun, 04 Jul 2004 19:51:29 Subject: Re: A chart of syntonic comma temperaments From: monz oops ... of course, i see that 36-ET is also catler and mothra, as well as gawel. i don't know about either of those two, either. -monz --- In tuning-math@xxxxxxxxxxx.xxxx "monz" <monz@a...> wrote:> hi Herman, > > --- In tuning-math@xxxxxxxxxxx.xxxx Herman Miller <hmiller@I...> > wrote: > >> http://www.io.com/~hmiller/png/syntonic.png - Type Ok * [with cont.] (Wayb.) >>>> This is a chart of 7-limit temperaments that temper out >> the syntonic comma 81;80. The horizontal axis is deviation >> from 3:1 and the vertical axis is the deviation from 7:1. >> This time I limited the list to 7-limit consistent ET's. > > >

Message: 11257 - Contents - Hide Contents Date: Sun, 04 Jul 2004 00:12:47 Subject: Re: Gene's mail server From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote: >>> It says that the bra is a covariant 1-vector, and the ket is a >> contravariant one-form. It also says that the combination of the two, >> as in <v|w>, is an inner product. >> I know people talk that way. I also know it is very confusing to > people trying to learn this stuff. But probably not worth worrying > about in connection with your paper. I'd also flush all of that stuff > about 1-forms from your brain immediately.Well, I remember trying to understand differential forms when cramming for a big freshman math final, but really there's nothing to flush. I just looked at Robert Griffiths' book "Consistent Quantum Mechanics", and it introduces kets explitictly as linear functionals, and then the bracket product as the "inner product". Luckily (or perhaps unluckily?), the music theory doesn't have to deal with taking complex conjugates.

Message: 11258 - Contents - Hide Contents Date: Sun, 04 Jul 2004 15:02:14 Subject: Re: A map of starling space From: Herman Miller Gene Ward Smith wrote:> --- In tuning-math@xxxxxxxxxxx.xxxx Herman Miller <hmiller@I...> wrote: >>> I made a chart of ET's with the comma 126;125, and drew lines on it to >> illustrate some of the landmark starling temperaments. > >> Great! This is something I've suggested Paul do from time to time; I > didn't know you were up to this level of graphic design. > >>> Any major starling-related temperaments that I'm missing? > >> Here's a list of 21 starling temperaments, ordered in terms of > increasing Graham-TOP badness: > [7, 9, 13, -2, 1, 5] > [[1, -1, -1, -2], [0, 7, 9, 13]] > 13 1.610469 272.169318Semisixths. I should add that one at least. Some of the others are ones that I've seen before, but don't know much about.

Message: 11259 - Contents - Hide Contents Date: Sun, 04 Jul 2004 00:15:53 Subject: Re: from linear to equal From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "Kalle Aho" <kalleaho@m...> wrote:> Hi, > > Linear temperaments (or 2-dimensional tunings) are infinitely > extendable. Once you extend a linear temperament eonugh you'll start > getting different pitches that nevertheless are more or less > indistinguishable from each other. Even before that you'll get > approximations that are better than those the linear temperament is > supposed to give. > > So what would be a good place to close the circle and go from linear > to equal? > > For TOP tempered linear temperaments I suggest closing the circle > when you start getting better approximations to the primes for which > the tuning is optimized.Not a bad idea. I don't think any of my horagrams go further than this, although 5:4 is slightly better in TOP Catler, and maybe there's another similar example somewhere. You'd have to make your criterion a little more precise -- are you assuming that the scales grow in one direction, or in both directions, as you apply the generator more and more times?

Message: 11260 - Contents - Hide Contents Date: Sun, 04 Jul 2004 20:28:07 Subject: Copop hanson From: Gene Ward Smith 7, 9 and 11 limit hanson/catakleismic have a copop generator of 123/466. 5-limit hanson has a range a bit above, with poptimals of 246, 299, 352. The minimal 5-limit poptimal is 3^(1/6), and the maximum poptimal for 7 and 9 is 56^(1/22); in between these two are 53, 125, 178... 197 is copoptimal for 7 and 9, 269 for 7 and 11, and 466 for 9 and 11. 3^(1/6) is both the maximum kleismic popimal and the minimal hanson poptimal, so 53 is good for both. 123/466 gives a minor third generator 1.0969 cents sharp, fifth 1.5358 cents flat, major third 2.6227 cents flat, 7/4 0.5856 cents flat, and 11/8 2.82001 cents flat. You do slightly better than the 72-et version of hanson with 466, and therefore hanson in the 7 or 11 limit beats miracle slightly in tuning accuracy, but of course it is much more complex.

Message: 11261 - Contents - Hide Contents Date: Sun, 04 Jul 2004 00:18:29 Subject: Re: dual, and inner product space (was: Gene's mail server) From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:> --- In tuning-math@xxxxxxxxxxx.xxxx "monz" <monz@a...> wrote: >>> what is the dual? if you could explain it along >> the lines of my "prime-space" definition, that >> would help me. >> Paul and I have been tossing this back and forth at each other: > > Dual Vector Space -- from MathWorld * [with cont.] > > Linear Function -- from MathWorld * [with cont.] > >> Definitions of tuning terms: prime-space, (c) ... * [with cont.] (Wayb.) >> >>>>> That is how you ended up with a bracket product >>> despite the fact that no inner product space is >>> being discussed, or would make any sense in the context. >> >>>> what's an "inner product space"? > > Vector Space -- from MathWorld * [with cont.] > Vector space - Wikipedia, the free encyclopedia * [with cont.] (Wayb.) > > Inner Product Space -- from MathWorld * [with cont.] > Inner product space - Wikipedia, the free ency... * [with cont.] (Wayb.) >> However, since we aren't using inner product spaces it is only vector > spaces which need concern us. Other relevant encyclopedia pages are > for abelian group > > Abelian group - Wikipedia, the free encyclopedia * [with cont.] (Wayb.) > > You could also look up bra-ket notation, but because it assumes we are > in an inner product space and we are not, it may not be that great. > This is my problem with Paul wanting to use "inner product" for the > bracket--we don't actually have an inner product, whereas in quantum > mechanics we do. This isn't QM, thank heavens.What do you mean? In quantum mechanics, the bracket or "inner product" is simply the application of a linear functional acting on a vector.

Message: 11262 - Contents - Hide Contents Date: Sun, 04 Jul 2004 20:34:41 Subject: Re: A chart of syntonic comma temperaments From: monz hi Herman, --- In tuning-math@xxxxxxxxxxx.xxxx Herman Miller <hmiller@I...> wrote:> http://www.io.com/~hmiller/png/syntonic.png - Type Ok * [with cont.] (Wayb.) > > This is a chart of 7-limit temperaments that temper out > the syntonic comma 81;80. The horizontal axis is deviation > from 3:1 and the vertical axis is the deviation from 7:1. > This time I limited the list to 7-limit consistent ET's.are the numbers associated with each temperament family on this chart wedgies? if not, then what are they? on the meantone one, <<1, 4, 10, 4, 13, 12|| , the "1, 4, 10" part at least looks familiar as the generator mapping for primes 3, 5, and 7. am i on the right track? please explain one, using the meantone one as an example. if your triangular arrangement is appropriate, please show that as well. thanks. -monz

Message: 11263 - Contents - Hide Contents Date: Sun, 04 Jul 2004 00:21:49 Subject: Re: from linear to equal From: Paul Erlich 9-limit should also be considered when you're going "poptimal". --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:> --- In tuning-math@xxxxxxxxxxx.xxxx "Kalle Aho" <kalleaho@m...> wrote: >>> So what would be a good place to close the circle and go from linear >> to equal? >> >> For TOP tempered linear temperaments I suggest closing the circle >> when you start getting better approximations to the primes for which >> the tuning is optimized. >> >> What are your thoughts about this? >> For pure octave tunings, a system I sometimes use is to close at a > "poptimal" generator. A generator is "poptimal" for a certain set of > octave-eqivalent consonances if there is some exponent p, 2 <= p <= > infinity, such that the sum of the pth powers of the absolute value of > the errors over the set of consonances is minimal. This is convenient > for Scala score files, since the notes are now represented by > (reasonably small) integers. I also sometimes use it when cooking up a > Scala scl file (just did, in fact, over on the tuning list) though in > that case it makes little difference. > > If you follow this system, 5-limit meantone closes for 81, 7-limit > meantone for 31, and 11-limit meantone for 31. 5 and 7 taken together > are 1/4-comma exactly, which doesn't close; 5 and 11 taken together > closes at 112, and 7 and 11 of course also at 31. One rarely > encounters problems; even a microtemperament like ennealimmal closes > at 1053, which is perfectly reasonable for Scala applications; one > does, however, need to ensure the division is divisible by 9. > > A different naming convention than using TOP tuning would be to give > the same name iff the poptimal ranges intersect. This isn't very > convenient in practice, due to the difficulty of computing the > poptimal range, but clearly it leads to quite different results. > Miracle, for instance, has the same TOP tuning in the 5, 7 and 11 > limits, but while the 5 and 7 limit poptimal ranges intersect, the 5 > and 11 or 7 and 11 ranges apparently do not, though as I say computing > these is a pain, so I may have the range too small. In any case, > miracle closes at 175 in the 5 and 7 limits, and at 401 in the 11- limit.

Message: 11264 - Contents - Hide Contents Date: Sun, 04 Jul 2004 21:46:17 Subject: Copop hemiwuerschmidt From: Gene Ward Smith If you take the poptimal range for wuerschmidt and divide it by 2, and compare to the poptimal range for 7 or 9 limit hemiwuerschmidt, you find they are almost the same. 229 works as a poptimal et for hemiwuerschmidt in the 7 or 9 limit, and wuerschmidt in the 5 limit. In the 11-limit, we get a temperament with the same TOP tuning, 11-limit hemiwuerschmidt, from <<16 2 5 -90 -34 -37 -198 6 -216 -270|| This has a common poptimal generator with 5, 7, and 9 limit (hemi)wuerschmidt of 6^(1/16) for hemiwuerschmidt, 6^(1/8) for wuerschmidt, so 229 works well here also, and 359 is almost exact. However, it has a high complexity and so a high badness, <<16 2 5 40 -34 -37 8 6 86 95|| is much better in terms of badness; 421 is poptimal for this. This temperament has a TM basis of {243/242, 441/440, 3136/3125} and is reasonable in terms of badness; it can be described as 31&130. The 421 val for it, derivable from the 68/421 generator, is <421 667 978 1182 1457| I've listed this before as hemiwuerschmidt, but the tuning is too different. Worseschmidt?

Message: 11265 - Contents - Hide Contents Date: Sun, 04 Jul 2004 00:34:55 Subject: Re: A map of starling space From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:> --- In tuning-math@xxxxxxxxxxx.xxxx Herman Miller <hmiller@I...> wrote:>> I made a chart of ET's with the comma 126;125, and drew lines on it to >> illustrate some of the landmark starling temperaments. >> Great! This is something I've suggested Paul do from time to time;I must have missed those; I remember a request to plot according to the generator mapping.> Here's a list of 21 starling temperaments, ordered in terms of > increasing Graham-TOP badness: > > [4, 4, 4, -3, -5, -2] > [[4, 6, 9, 11], [0, 1, 1, 1]] > 4 5.871540 93.944647Dimisept in my paper> [6, 5, 3, -6, -12, -7] > [[1, 0, 1, 2], [0, 6, 5, 3]] > 6 3.187309 114.743119Keenan in my paper> [10, 9, 7, -9, -17, -9] > [[1, -1, 0, 1], [0, 10, 9, 7]] > 10 1.171542 117.154200Myna in my paper> [9, 5, -3, -13, -30, -21] > [[1, 1, 2, 3], [0, 9, 5, -3]] > 12 1.049791 151.169891 > [1, 4, 10, 4, 13, 12] > [[1, 2, 4, 7], [0, -1, -4, -10]] > 10 1.698521 169.852100Meantone in my paper> [3, 0, -6, -7, -18, -14] > [[3, 5, 7, 8], [0, -1, 0, 2]] > 9 2.939961 238.136875Augene in my paper> [7, 9, 13, -2, 1, 5] > [[1, -1, -1, -2], [0, 7, 9, 13]] > 13 1.610469 272.169318Sensisept in my paper

Message: 11266 - Contents - Hide Contents Date: Sun, 04 Jul 2004 22:28:08 Subject: Re: A chart of syntonic comma temperaments From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "monz" <monz@a...> wrote:> are the numbers associated with each temperament family > on this chart wedgies? if not, then what are they? Wedgies, yes. > on the meantone one, <<1, 4, 10, 4, 13, 12|| , > the "1, 4, 10" part at least looks familiar as the > generator mapping for primes 3, 5, and 7. am i on > the right track? > > please explain one, using the meantone one as an example.1 4 10 This is how many generator steps, assuming octave periods, it takes to get to 3, 5, and 7 4 13 This is how many generator steps, assuming tritave (3) periods, it takes to get to 5 and 7 12 3 is the number of generator steps, using a period of 5^(1/4) (1/4 meantone fifth) to get to 7; 12 is 3*4 Mostly we are just interested in 1, 4 and 10, but leaving off the rest makes things hard to compute. It is a lot like leaving the 2 off a comma, and just giving the octave equivalent comma; and in fact the two proceedures are related. Anything of the form <<1 4 10 a b 4b - 10a|| is a valid wedgie, but not much good unless it is meantone. For instance, <<1 4 10 4 14 16|| has TM basis {81/80, 252/125} and <<1 4 10 5 15 10|| has TM basis {160/81, 125/63}. Both of these are octave equivalent to {81/80, 126/125}. If we take <<1 4 10 0 0 0|| we get a TM basis consisting of odd ratios, {81/5, 225/7}. This may be regarded as the octave equivalent form of meantone, and its easy to get meantone from it by reducing the commas to the half-octave. You can take any generator mapping you like at will, fill it out with zeros to get an octave-equivalent wedgie, reduce the commas to the half-octave, take the wedgie from this, and you have a temperament for that generator mapping. Mostly, the results aren't interesting.

Message: 11267 - Contents - Hide Contents Date: Sun, 04 Jul 2004 01:08:33 Subject: Re: dual, and inner product space (was: Gene's mail server) From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:> What do you mean? In quantum mechanics, the bracket or "inner > product" is simply the application of a linear functional acting on a > vector.In QM state vectors are unit vectors in a complex Hilbert space, meaning there is a Hermitian inner product on the space. An eigenvector for a Hermitian linear operator (ie, an "observable") with discrete spectrum will *also* be a state vector if normalized to a unit vector. They live in the same space, so the eigenvector will be a wave function like the state vector, if those are wave functions. If you take the absolute value of the inner product and square it, you get the probability of a measurement coming up with with corresponding eigenvalue as a result of the measurement of the observable (= Hermitian operator.) If you don't have a discrete spectrum you need to resort to spectral theory, but it's basically similar. The upshot is that the eigenvectors are bounded linear functionals on the states, but since we are in an inner product space we can identify these with states. It's like identifying a row vector with a column vector. Applications to music? I dunno; but the discrete spectrum business is intriging. Someone should tune up a hydrogen atom when taken down enough octaves, I guess.

Message: 11268 - Contents - Hide Contents Date: Sun, 04 Jul 2004 17:48:11 Subject: Re: A chart of syntonic comma temperaments From: Herman Miller monz wrote:> are the numbers associated with each temperament family > on this chart wedgies? if not, then what are they? > > on the meantone one, <<1, 4, 10, 4, 13, 12|| , > the "1, 4, 10" part at least looks familiar as the > generator mapping for primes 3, 5, and 7. am i on > the right track? > > please explain one, using the meantone one as an example. > if your triangular arrangement is appropriate, please > show that as well. thanks.Yes, you could use the triangular arrangement: . 1 4 10 (note that <<1, 4, 4|| is the wedgie for 81;80) . 4 13 . 12 Gene explained a couple of days ago about how the first row of the wedgie in this form is related to the mapping (depending on how the generators are defined, you might need to negate them, as in the case of meantone with a fourth as the generator, and divide by the number of periods in an octave). Looking at an 11-limit version of meantone, <<1, 4, 10, -13, 4, 13, -24, 12, -44, -71||, you can see the 7-limit meantone wedgie in the triangular arrangement: . 1 4 10 -13 . 4 13 -24 . 12 -44 . -71 Since meantone on the chart is shown as a line between 12 and 19, you can get any other information you need from Graham Breed's temperament finder (Temperament Finder * [with cont.] (Wayb.)): put in "12", "19", and "7" into the boxes, and this is what you get: 13/31, 503.4 cent generator basis: (1.0, 0.419517976278) mapping by period and generator: [(1, 0), (2, -1), (4, -4), (7, -10)] mapping by steps: [(19, 12), (30, 19), (44, 28), (53, 34)] highest interval width: 10 complexity measure: 10 (12 for smallest MOS) highest error: 0.004480 (5.377 cents) unique Similarly, "mothra" <<3, 12, -1, 12, -10, -36|| is shown as a line from 5 to 26. Follow this link for the stats: Temperament result * [with cont.] (Wayb.)&et2=26&limit=7

Message: 11269 - Contents - Hide Contents Date: Sun, 04 Jul 2004 01:16:39 Subject: Re: Temperaments with a 7/5 generator From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx Gene W Smith <genewardsmith@j...> wrote:> Linear temperaments with a generator which is itself a consonant interval > seem to me of particular interest, so I thought I would explore what is > out there for generators of about 7/5. While nothing dramatic turned up, > these systems might be of interest. > > > [3, -5, -6, -1, -15, -18, -12, 0, 15, 18] > <56/55, 64/63, 77/75> > badness = 326 rms = 13.78 > > [3, -5, -6, 0, 18, -15] > <64/63, 392/375> > badness = 532 rms = 14.78 > > 7/15 < 15/32 < 8/17 > > 15/32 is a nearly exact 11-limit generator; 11/8 is closer than 7/5 to > this generator, which is convenient. > > > > [3, 12, 11, -1, 12, 9, -12, -8, -44, -41] > <56/55, 81/80, 540/539> > badness = 404 rms = 12.62 > > [3, 12, 11, -8, -9, 12] > <81/80, 686/675> > badness = 634 rms = 9.05 > > 8/17 < 17/36 < 9/19 > > 17/36 is nearly exact 11-generator; again, 11/8 is closer. These two are > the same in the 17-et.17/36 or 19/36. Gawel.

Message: 11270 - Contents - Hide Contents Date: Sun, 04 Jul 2004 22:56:43 Subject: Re: NOT tuning From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:>>>> Meantone >>>> >>>> 5-limit: 698.0187 (43, 55, 98, 153, 251, 404) >>>> >>>> 7-limit: 697.6458 (31, 43, 934, 977; 0.0286 cents flatter than 43) >>>>>> Hmm... I dunno, this seems a bit far from the old-style rms >>> optimum. >>> >>> -Carl >>>> Carl, when Graham investigated this same question here a few months >> ago, he concluded that pure-octaves TOP would be a uniform stretching >> or compression of TOP, >> That seems obvious for ETs....But it's not what Gene's definition gives you.>>> except where TOP already had pure octaves, in >> which case it would actually change! >> That's impossible given the criterion of NOT. > > Maybe I don't comprehend you.I didn't say NOT, I said "Graham" and "pure-octaves TOP".

Message: 11271 - Contents - Hide Contents Date: Sun, 04 Jul 2004 01:19:09 Subject: Re: NOT tuning From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:>> Meantone >> >> 5-limit: 698.0187 (43, 55, 98, 153, 251, 404) >> >> 7-limit: 697.6458 (31, 43, 934, 977; 0.0286 cents flatter than 43) >> Hmm... I dunno, this seems a bit far from the old-style rms > optimum. > > -CarlCarl, when Graham investigated this same question here a few months ago, he concluded that pure-octaves TOP would be a uniform stretching or compression of TOP, except where TOP already had pure octaves, in which case it would actually change!

Message: 11272 - Contents - Hide Contents Date: Sun, 04 Jul 2004 22:57:57 Subject: Re: NOT tuning From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:> --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote: >>>> except where TOP already had pure octaves, in >>> which case it would actually change! >>>> That's impossible given the criterion of NOT. >> >> Maybe I don't comprehend you. >> Some examples of this method of tuning would be nice, and a definition > even better.I looked again and Graham said Feb 2nd.

Message: 11273 - Contents - Hide Contents Date: Sun, 04 Jul 2004 23:01:05 Subject: Re: from linear to equal From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:> --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:>>>> 9-limit should also be considered when you're going "poptimal". >>>>>> True enough. Alas, even though we have the same wedgie, commas and >>> tuning map, the poptimal range need not even overlap. Orwell is a >>> typical example--there seems to be no overlap from 7 to 9, and none >>> between 11 and 9, but the others are OK. So, 5 and 9 overlap, and >>> have 43/190 as a common generator, but 7 and 9, no. >>>> This is AWESOME. Seriously, if you had come to me in a past >> life and asked me to imagine the most heinously interesting >> thing ever, for torturing curious folks in purgatory or >> something, I wouldn't have come up with the half of this >> temperaments thing. >> Har. You think that is bad, try this: two different 11-limit linear > temperaments are the meantone variants meantone or meanpop (sharing > the same TOP tuning with the 7-limit temperament) and huygens (sharing > the same NOT tuning with the 7-limit temperament.)Isn't that a ridiculous name for an 11-limit temperament?

Message: 11274 - Contents - Hide Contents Date: Sun, 04 Jul 2004 23:06:04 Subject: Re: from linear to equal From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "Kalle Aho" <kalleaho@m...> wrote:> --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> > wrote: >>> For pure octave tunings, a system I sometimes use is to close at a >> "poptimal" generator. A generator is "poptimal" for a certain set of >> octave-eqivalent consonances if there is some exponent p, 2 <= p <= >> infinity, such that the sum of the pth powers of the absolute value > of>> the errors over the set of consonances is minimal. >> This is quite an interesting approach. What makes poptimal >generators > good?Not much, IMHO -- the "true" value of p in any situation will be some number, not an infinite range of numbers.> And why can't p be 1?My graphs show p going even slightly below 1, and I think this is more than appropriate when you look at the kinds of discordance curves Bill Sethares predicts and George Secor prefers. Very sharp spikes at the simple ratios.> These results are interesting. Do these poptimal generators make > these linear temperaments close exactly at these ETs?!"Poptimal" doesn't imply uniqueness the way "optimal" does. Any generator within a certain finite range will be poptimal for a given situation. So you have to "feed in" ET generators at the beginning if you want the circle(s) to close.

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