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Message: 9600 - Contents - Hide Contents Date: Sat, 31 Jan 2004 06:56:00 Subject: Re: Nonoctave scales in your livingroom From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:> If you don't want to go to the bother of obtaining a musical > instrument based on 3^(1/13), you might try the temperament with TM > basis {3125/3087, 6561/6125}. If you take pure tritaves, you get a > generator of 3^(1/19), or 100.103 cents. If you object that after > twelve generator steps you get to a pretty good octave of 1201.235 > cents, my reply to you is that 41-et has octaves also. Just ignore it. > Pretend it isn't there, and see what happens. exactly.
Message: 9601 - Contents - Hide Contents Date: Sat, 31 Jan 2004 19:36:34 Subject: Re: I guess Pajara's not #2 From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:> So the below was wrong. I forgot that you reverse the order of the > elements to convert a multival wedgie into a multimonzo wedgie! Doing > so would, indeed, give the same rankings as my original L_1 > calculation. But that's gotta be the right norm. The Tenney lattice > is set up to measure complexity, and the norm we always associate > with it is the L_1 norm. Isn't that right? The L_1 norm on the monzo > is what I've been using all along to calculate complexity for the > codimension-1 case, in my graphs and in the "Attn: Gene 2"post . . . To me it seemed there was a reasonable case for either norm, which means you could argue you could use any Lp norm also, since they lie between L1 and L_inf. Do you need me to redo the calculation using the L1 norm? I think it would be a good idea to stick to the normalization by dividing, since for higher limits a linear temperament wedgie is still a bival, so it's easier. ________________________________________________________________________ ________________________________________________________________________ ------------------------------------------------------------------------ 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: 9602 - Contents - Hide Contents Date: Sat, 31 Jan 2004 07:17:27 Subject: Re: 60 for Dave From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:>>>> So that he could understand Gene's badness and my linear badness >>>> in the same form, and propose a compromise. >>>>>> Ah. Is yours the one from the Attn: Gene post? >>>> No, it was the toy "Hermanic" example. >> This, I guess: > > "I thought I'd cull the list of 114 by applying a more stringent > cutoff of 1.355*comp + error < 10.71. This is an arbitrary choice > among the linear functions of complexity and error that could be > chosen" > > You don't say what kind of comp and error you're using.Copied from Gene's "114".>>> That's good to know, but the above is just my value judgement, and >>> as you point out log-flat badness frees us from those, in a sense. >>>> But it results in an infinite number of temperaments, or none at all, >> depending on what level of badness you use as your cutoff. >> ...as I was trying to complain recently, when I said I'd be a lot > more impressed if it didn't need cutoffs.Well you always have at least one cutoff -- namely, the maximum allowed value of the badness function itself -- and of the various badness functions that have been proposed, a few (including log-flat) require one or two additional cutoffs to yield a finite list. A straight line doesn't.
Message: 9603 - Contents - Hide Contents Date: Sat, 31 Jan 2004 07:23:19 Subject: Re: Crunch algorithm From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> > wrote:>> This is similar, if not identical, to Viggo Brun's algorithm that >> Kraig is always referring to . . . >> I'm not so sure--I think possibly Brun's algorithm simply sets out to > find integer relations, not unimodular matricies. I've never seem an > exposition of it, just references to it, and don't know how closely > what people have said about in tuning connections matches what Brun > actually did, which is described as an integer relation algorthm or > something along the lines of Jacbobi-Perron when I read about it > elsewhere.Brun may have done more than one thing. Please make it a habit to check that big tuning biblography. You'd find this: http://www.anaphoria.com/viggo0.PDF - Type Ok * [with cont.] (Wayb.) This language is pretty close to English, but I bet you can just look at the numbers and see what's going on.> However, what I did is identical to the Erv Wilson method-- > he *is* using it to find unimodular matricies, and then inverting > them.Inverting them, huh? Look, he goes all the way out to Orwell. Kraig has insisted this is the best method, but I doubt any algorithm could perform a three-term integer relation search and do as well as brute force for log-flat or whatever badness (except maybe Ferguson- Forcade?) . . . We should ask John Chalmers if he knows the particular Brun reference or if Erv just consulted Mandelbaum.>> See Mandelbaum's book for a full >> exposition . . . >> You'll have to do better than that--is this Joel Mandelbaum, Yes. >and what book? His only:Mandelbaum, M. Joel. Multiple Division of the Octave and the Tonal Resources of the 19-Tone Equal Temperament. PhD Thesis, University of Indiana, 1961, 460 pages. University Microfilms, Ann Arbor MI, 1961.
Message: 9604 - Contents - Hide Contents Date: Sat, 31 Jan 2004 00:11:40 Subject: Re: 114 7-limit temperaments From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:>>> But not a chain of one and only one generator. >>>> Huh? How not?? >> Because he doesn't temper, the generator varies in size > depending on where you are in the map.That's what I meant by an undistributed commatic unison vector. But there's still one and only one chain, in contradistinction with pajara, augmented, diminished, ennealimmal, etc. . . . which was my point.>> When he does talk about some of these as temperaments (or the related >> just intonation structures with an undistributed commatic unison >> vector), though, they're all single-chain. >> ..of a particular generator in scale steps, not interval size. Yes.
Message: 9605 - Contents - Hide Contents Date: Sat, 31 Jan 2004 08:59:44 Subject: Re: 60 for Dave (was: 41 "Hermanic" 7-limit linear temperaments) From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" <d.keenan@b...> wrote:> Yes. I like that idea too. But by "a wide swath" don't you mean one > that it's easy to put a simple smooth curve thru? Right. > And you must have > some general idea of which way this intergalactic moat must curve.In the 5-limit linear case, it would be really easy to do this if we didn't want to go out to the complexity of schismic and kleismic/hanson (the only argument that would arise would be whether the father and beep couple should be in or out, leading to a grand total of 11 or 9 5-limit LTs). Unfortunately, the low error of schismic has proved tantalizing enough for a few musicians to construct instruments capable of playing the large extended scales that its approximations require. If we consider such complexity justifiable, it seems we should be interested in 15 to 17 5-limit LTs, or 17 to 19 if we include father and beep. The couple residing in "the middle of the road" is 2187;2048 and 3126;2916. With Herman, we could split the difference and select only the better of the pair, 2187;2048 (Dave, have you *heard* Blackwood's 21-equal suite?) . . . I don't think anyone's talked about the 20480;19683 system. But if schismic, and certainly if semisixths, is not too complex to be a useful alternative to strict JI, why shouldn't this system merit some attention from musicians too? I don't think near-JI triads sound enough better than chords in this system (which are purer than those of augmented, porcupine, or diminished) to merit a much higher allowed complexity to generate them linearly. A problem with our plan to have versions of these badness curves for sets of temperaments of different dimensions is that moving to a higher-dimensional tuning system would theoretically increase the badness by an infinite factor. But in practice you never use an infinite swath of the lattice so eventually any complex enough 5- limit linear temperament becomes indistinguishable from a planar system. I'm speaking like you now, Dave! ;)
Message: 9606 - Contents - Hide Contents Date: Sat, 31 Jan 2004 00:19:24 Subject: Re: 114 7-limit temperaments From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:>>> What I remember we gave you a hard time about, was not that linear >>> temperaments are 2-dimensional without octave-equivalence, but that >>> you wanted to call them "planar" (which would have been too >>> confusing a departure from the historical usage). We wanted "Linear >>> temperament" to be the constant name of the musical object which >>> remains essentially the same while its mathematical >>> models vary in dimensionality. >>>> Unfortunately for us, 'linear temperament' has probably never >> referred to a multiple-chains-per-octave system (like pajara, >> diminished, augmented, ennealimmal . . .) before we started using it >> that way, and some of the original users of the term (say, Erv >> Wilson) might be rather upset with this slight generalization. >> I can't remember Erv ever using the term,How about the first line of http://www.anaphoria.com/xen2.PDF - Type Ok * [with cont.] (Wayb.) ?
Message: 9607 - Contents - Hide Contents Date: Sat, 31 Jan 2004 09:17:41 Subject: Re: Crunch algorithm From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:> This language is pretty close to English, but I bet you can just look > at the numbers and see what's going on.This language you claim is a lot like English is called Norwegian. My mother once translated a book from Norwegian, despite the fact that she only knew Swedish, but I don't know Swedish either. However! It is clear, as you say, that Brun is doing what Wilson was doing and what I was calling crunch, sans the inversion part--that is, complete with unimodular matricies. I think I can do better than Brun for this project anyway, by using the square==>triangle triangle==>square scheme. I'll test that.> Mandelbaum, M. Joel. Multiple Division of the Octave and the Tonal > Resources of the 19-Tone Equal Temperament. PhD Thesis, University of > Indiana, 1961, 460 pages. University Microfilms, Ann Arbor MI, 1961.Ouch! Did you pay for 460 pages of University Microfilms? Anyway, it's easy to see why Brun got noticed by music theorists--he wrote about it, referencing Barbour and going on from there.
Message: 9608 - Contents - Hide Contents Date: Sat, 31 Jan 2004 01:31:55 Subject: kleismic v. hanson (was Re: 60 for Dave) From: Carl Lumma>In the 5-limit linear case, it would be really easy to do this if we >didn't want to go out to the complexity of schismic and >kleismic/hansonWhile I think it would be nice to name this after Larry Hanson (and I certainly agreeable to the idea), my preference is to keep kleismic, since it tells the name of the comma involved, and has a fairly-well established body of use. What say everybody? -Carl
Message: 9609 - Contents - Hide Contents Date: Sat, 31 Jan 2004 09:42:58 Subject: the choice of wedgie-norm greatly impacts miracle's ranking From: Paul Erlich This time I'll try L_1 (multimonzo interpretation?) instead of L_infinity (multival interpretation?) to get complexity from the wedgie. Let's see how it affects the rankings -- we don't need to worry about scaling because Gene's badness measure is multiplicative . . . The top 10 get re-ordered as follows, though this is probably not the new top 10 overall . . . 1.> Number 1 Ennealimmal > > [18, 27, 18, 1, -22, -34] [[9, 15, 22, 26], [0, -2, -3, -2]] > TOP tuning [1200.036377, 1902.012656, 2786.350297, 3368.723784] > TOP generators [133.3373752, 49.02398564] > bad: 4.918774 comp: 11.628267 err: .03637739.8287 -> bad = 57.7058 2.> Number 2 Meantone > > [1, 4, 10, 4, 13, 12] [[1, 2, 4, 7], [0, -1, -4, -10]] > TOP tuning [1201.698521, 1899.262909, 2790.257556, 3370.548328] > TOP generators [1201.698520, 504.1341314] > bad: 21.551439 comp: 3.562072 err: 1.69852111.7652 -> bad = 235.1092 3.> Number 9 Miracle > > [6, -7, -2, -25, -20, 15] [[1, 1, 3, 3], [0, 6, -7, -2]] > TOP tuning [1200.631014, 1900.954868, 2784.848544, 3368.451756] > TOP generators [1200.631014, 116.7206423] > bad: 29.119472 comp: 6.793166 err: .63101421.1019 --> bad = 280.9843 4.> Number 7 Dominant Seventh > > [1, 4, -2, 4, -6, -16] [[1, 2, 4, 2], [0, -1, -4, 2]] > TOP tuning [1195.228951, 1894.576888, 2797.391744, 3382.219933] > TOP generators [1195.228951, 495.8810151] > bad: 28.744957 comp: 2.454561 err: 4.7710497.9560 -> bad = 301.9952 5.> Number 3 Magic > > [5, 1, 12, -10, 5, 25] [[1, 0, 2, -1], [0, 5, 1, 12]] > TOP tuning [1201.276744, 1903.978592, 2783.349206, 3368.271877] > TOP generators [1201.276744, 380.7957184] > bad: 23.327687 comp: 4.274486 err: 1.27674415.5360 -> bad = 308.1642 6.> Number 4 Beep > > [2, 3, 1, 0, -4, -6] [[1, 2, 3, 3], [0, -2, -3, -1]] > TOP tuning [1194.642673, 1879.486406, 2819.229610, 3329.028548] > TOP generators [1194.642673, 254.8994697] > bad: 23.664749 comp: 1.292030 err: 14.1761054.7295 -> bad = 317.0935 7.> Number 6 Pajara > > [2, -4, -4, -11, -12, 2] [[2, 3, 5, 6], [0, 1, -2, -2]] > TOP tuning [1196.893422, 1901.906680, 2779.100462, 3377.547174] > TOP generators [598.4467109, 106.5665459] > bad: 27.754421 comp: 2.988993 err: 3.10657810.4021 -> bad = 336.1437 8.> Number 10 Orwell > > [7, -3, 8, -21, -7, 27] [[1, 0, 3, 1], [0, 7, -3, 8]] > TOP tuning [1199.532657, 1900.455530, 2784.117029, 3371.481834] > TOP generators [1199.532657, 271.4936472] > bad: 30.805067 comp: 5.706260 err: .94606119.9797 -> bad = 377.6573 9.> Number 8 Schismic > > [1, -8, -14, -15, -25, -10] [[1, 2, -1, -3], [0, -1, 8, 14]] > TOP tuning [1200.760625, 1903.401919, 2784.194017, 3371.388750] > TOP generators [1200.760624, 498.1193303] > bad: 28.818558 comp: 5.618543 err: .91290420.2918 --> bad = 375.8947 10.> Number 5 Augmented > > [3, 0, 6, -7, 1, 14] [[3, 5, 7, 9], [0, -1, 0, -2]] > TOP tuning [1199.976630, 1892.649878, 2799.945472, 3385.307546] > TOP generators [399.9922103, 107.3111730] > bad: 27.081145 comp: 2.147741 err: 5.8708798.3046 -> bad = 404.8933
Message: 9610 - Contents - Hide Contents Date: Sat, 31 Jan 2004 00:37:28 Subject: pelogic and kleismic/hanson From: Paul Erlich See http://www.anaphoria.com/keygrid.PDF - Type Ok * [with cont.] (Wayb.) page 7 seems to be using some pelog terminology; anyone familiar with it? page 10 provides further support for my proposal of "hanson" as a name for kleismic (i.e., dave keenan's chain-of-minor-thirds scale)
Message: 9611 - Contents - Hide Contents Date: Sat, 31 Jan 2004 09:46:10 Subject: kleismic v. hanson (was Re: 60 for Dave) From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:>> In the 5-limit linear case, it would be really easy to do this if we >> didn't want to go out to the complexity of schismic and >> kleismic/hanson >> While I think it would be nice to name this after Larry Hanson (and > I certainly agreeable to the idea), my preference is to keep kleismic, > since it tells the name of the comma involved,A rare feature, and Pythagorean doesn't eat the Pythagorean comma, for example . . .> and has a fairly-well > established body of use. What say everybody? > > -CarlFairly-well established body of use?
Message: 9612 - Contents - Hide Contents Date: Sat, 31 Jan 2004 02:09:07 Subject: Re: kleismic v. hanson From: Carl Lumma>> >nd has a fairly-well >> established body of use. What say everybody? >>Fairly-well established body of use? Yes. -Carl ________________________________________________________________________ ________________________________________________________________________ ------------------------------------------------------------------------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: 9613 - Contents - Hide Contents Date: Sun, 01 Feb 2004 05:30:04 Subject: Re: The true top 32 in log-flat? From: Carl Lumma>There's something VERY CREEPY about my complexity values.Wow dude. What sort of DES are these? Not the smallest apparently. -Carl
Message: 9614 - Contents - Hide Contents Date: Sun, 01 Feb 2004 14:13:25 Subject: Re: 114 7-limit temperaments From: Carl Lumma>> >ou're using temperaments to construct scales, aren't you? >>Not me, for the most part. I think the non-keyboard composer is >simply being ignored in these discussions, and I think I'll stand >up for him.How *are* you constructing scales, and what does it have to do with keyboards? -Carl
Message: 9615 - Contents - Hide Contents Date: Sun, 01 Feb 2004 21:28:52 Subject: Re: 114 7-limit temperaments From: Carl Lumma At 09:16 PM 2/1/2004, you wrote:>--- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:>>>>>>> Such distinctions may be important for *scales*, but for >>>>>>> temperaments, I'm perfectly happy not to have to worry about >>>>>>> them. Any reasons I shouldn't be? >>>>>>>>>>>> You're using temperaments to construct scales, aren't you? >>>>>>>>>> Not necessarily -- they can be used directly to construct >music,>>>> ??? Doing so creates a scale. >>>>>> A 108-tone scale? >>>> "Scale" is a term with a definition. I was simply using it. You >> meant (and thought I meant?) "diatonic", or "diatonic scale", >> maybe. >>Did you mean a 108-tone scale? Yes. -C.
Message: 9616 - Contents - Hide Contents Date: Sun, 01 Feb 2004 05:43:05 Subject: Re: The true top 32 in log-flat? From: Carl Lumma>>> >OP generators [1201.698520, 504.1341314]So how are these generators being chosen? Hermite? I confess I don't know how to 'refactor' a generator basis. -Carl
Message: 9617 - Contents - Hide Contents Date: Sun, 01 Feb 2004 14:51:02 Subject: Re: 114 7-limit temperaments From: Carl Lumma>> > don't think that's quite what Partch says. Manuel, at least, has >> always insisted that simpler ratios need to be tuned more accurately, >> and harmonic entropy and all the other discordance functions I've >> seen show that the increase in discordance for a given amount of >> mistuning is greatest for the simplest intervals. >>Did you ever track down what Partch said?Observation One: The extent and intensity of the influence of a magnet is in inverse proportion to its ratio to 1. "To be taken in conjunction with the following" Observation Two: The intensity of the urge for resolution is in direct proportion to the proximity of the temporarily magnetized tone to the magnet.>It also shows that, if all intervals are equally mistuned, the more >complex ones will have the highest entropy.? The more complex ones already have the highest entropy. You mean they gain the most entropy from the mistuning? I think Paul's saying the entropy gain is about constant per mistuning of either complex or simple putative ratios.>Once you've established that the 9-limit intervals are playable and >audible, it may make sense to weight the simple ones higher because >you expect to use them more. You could even generate the weights >statistically from the score.I was thinking about this last night before I passed out. If you tally the number of each dyad at every beat in a piece of music and average, I think you'd find the most common dyads are octaves, to be followed by fifths and so on. Thus if consonance really *does* deteriorate at the same rate for all ratios as Paul claims, one would place less mistuning on the simple ratios because they occur more often. This is, I believe, what TOP does. -Carl
Message: 9618 - Contents - Hide Contents Date: Sun, 01 Feb 2004 13:52:34 Subject: Re: 7-limit horagrams From: Carl Lumma Beautiful! I take it the green lines are proper scales? -C.
Message: 9619 - Contents - Hide Contents Date: Sun, 01 Feb 2004 00:16:14 Subject: kleismic v. hanson (was Re: 60 for Dave) From: Dave Keenan --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:>> In the 5-limit linear case, it would be really easy to do this if we >> didn't want to go out to the complexity of schismic and >> kleismic/hanson >> While I think it would be nice to name this after Larry Hanson (and > I certainly agreeable to the idea), my preference is to keep kleismic, > since it tells the name of the comma involved, and has a fairly-well > established body of use. What say everybody? > > -CarlI have to agree with Carl. While Hanson may well be a perfectly appropriate name for it, when a name has been as extensively used as "kleismic" (even if only in these archives), you'd not only need to show what's right about the new name, but also what's wrong about the old name.
Message: 9620 - Contents - Hide Contents Date: Sun, 01 Feb 2004 17:15:20 Subject: Re: 60 for Dave (was: 41 "Hermanic" 7-limit linear temperaments) From: Graham Breed Herman:>> Schismic and kleismic/hanson start being useful (barely) around 12 > > notes, >>> but the tiny size of schismic steps beyond 12 notes is a drawback > > until you >>> get to around 41 notes when the steps are a bit more evenly spaced. Paul E:> Others may feel differently. Schismic-17 is a favorite of Wilson and > others and closely resembles the medieval Arabic system; Helmholtz > and Groven used 24 and 36 notes, respectively. Justin White seemed to > be most interested in the 29-note version.I played a lot with the 29 note scale, and it worked fine. It's nice to have some small intervals to throw in when you want them. It's a serious contender because it's still based on fifths, so it's familiar and can work with simple adaptations of regular notation. But that also makes it easy to find and so overrated. It should be included in a 9-limit, or weighted 7-limit, list anyway. Graham
Message: 9621 - Contents - Hide Contents Date: Sun, 01 Feb 2004 02:25:19 Subject: Re: I guess Pajara's not #2 From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:> That would be excellent, and then I could make a graph forDave . . . Do I need to push any farther towards either less accurate or more complex temperaments to make everyone happy?
Message: 9622 - Contents - Hide Contents Date: Sun, 01 Feb 2004 18:16:40 Subject: Re: 114 7-limit temperaments From: Graham Breed Paul Erlich wrote:> I don't think that's quite what Partch says. Manuel, at least, has > always insisted that simpler ratios need to be tuned more accurately, > and harmonic entropy and all the other discordance functions I've > seen show that the increase in discordance for a given amount of > mistuning is greatest for the simplest intervals.Did you ever track down what Partch said? Harmonic entropy can obviously be used to prove whatever you like. It also shows that the troughs get narrower the more complex the limit, so it takes a smaller mistuning before the putative ratio becomes irrelevant. It also shows that, if all intervals are equally mistuned, the more complex ones will have the highest entropy. So they're the ones for which the mistuning is most problematic, and where you should start for optimization.> Such distinctions may be important for *scales*, but for > temperaments, I'm perfectly happy not to have to worry about them. > Any reasons I shouldn't be?You're using temperaments to construct scales, aren't you? If you don't want more than 18 notes in your scale, miracle is a contender in the 7-limit but not the 9-limit. And if you don't want errors more than 6 cents, you can use meantone in the 7-limit but not the 9-limit. There's no point in using intervals that are uselessly complex or inaccurate so you need to know whether you want the wider 9-limit when choosing the temperament.> Tenney weighting can be conceived of in other ways than you're > conceiving of it. For example, if you're looking at 13-limit, it > suffices to minimize the maximum weighted error of {13:8, 13:9, > 13:10, 13:11, 13:12, 14:13} or any such lattice-spanning set of > intervals. Here the weights are all very close (13:8 gets 1.12 times > the weight of 14:13), *all* the ratios are ratios of 13 so simpler > intervals are not directly weighted *at all*, and yet the TOP result > will still be the same as if you just used the primes. I think TOP is > far more robust than you're giving it credit for.It's really an average over all odd-limit minimaxes. And the higher you get probably the less difference it makes -- but then the harder the consonances will be to hear anyway. For the special case of 7 vs 9 limit, which is the most important, it seems to make quite a difference. Once you've established that the 9-limit intervals are playable and audible, it may make sense to weight the simple ones higher because you expect to use them more. You could even generate the weights statistically from the score. But there are usually so few usable temperaments in any situation, you may as well consider each one individually and subjectively. Oh, yes, I think the 9-limit calculation can be done by giving 3 a weight of a half. That places 9 on an equal footing with 5 and 7, and I think it works better than vaguely talking about the number of consonances. After all, how do you share a comma between 3:2 and 9:8? I still don't know how the 15-limit would work. I'm expecting the limit of this calculation as the odd limit tends to infinity will be the same as this Kees metric. And as the integer limit goes to infinity, it'll probably give the Tenney metric. But as the integers don't get much beyond 10, infinity isn't really an important consideration. Not that it does much harm either, because the minimax always depends on the most complex intervals, which will have roughly equal weighting. The same as octave specific metrics give roughly the same results as odd-limit style octave equivalent ones if you allow for octave stretching. Graham
Message: 9623 - Contents - Hide Contents Date: Sun, 01 Feb 2004 02:47:52 Subject: Re: 60 for Dave (was: 41 "Hermanic" 7-limit linear temperaments) From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx Herman Miller <hmiller@I...> wrote:> So the complexity of kleismic in a musically useful sense isn't really > comparable to schismic; this is one thing that the horagrams are useful > for.The horagrams assume distributionally even, octave-repeating scales; our complexity measures don't.
Message: 9624 - Contents - Hide Contents Date: Sun, 01 Feb 2004 18:56:18 Subject: Re: The true top 32 in log-flat? From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:> There's something VERY CREEPY about my complexity values. I'm going > to have to accept this as *the* correct scaling for complexity (I'm > already convinced this is the correct formulation too, i.e. L_1 norm, > for the time being) . . .That's great, Paul. So what's the scaling?
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