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Message: 3025 - Contents - Hide Contents Date: Mon, 07 Jan 2002 05:00:16 Subject: Re: OPTIMAL 5-LIMIT GENERATORS FOR DAVE From: clumma>>> >ight, it's the chinese pentatonic. I threw it in for >>> completeness. >>> >>> Completeness of what? >>>> Of the journey from North to South, and of the survey of >> pentatonic scales in motivic ethnic music of southeast asia. >>You listen to three selections and call that a survey?Uh-huh. I didn't make any statistical claims about it. In fact the only claims I make about it are: () it's better than a 'survey' of the scala scl archive () it's better than a 'survey' without the Thai recording.>>> Give me a strict JI scale to try. >>>> 1/1 5/4 4/3 3/2 15/8 >> >> Sounds like a fine pelog to me. >>Yuck -- reminds me of Charles Lucy. Me too. >Totally inauthentic.But much closer than 5-of-7.>>>> I think the tuning on these discs is closer to JI than 23-tET, >>>> and I don't hear them avoiding a disjoint interval. Do you? >>>>>> Avoiding a disjoint interval? You mean you hear it as 5-of-7? >>> It modulates that much?? >>>> Actually, it doesn't. They seem to stick mostly to I, IV, and >> III (diatonic) with the bass, if you consider those tonics. But >> the melodic stuff does center itself on every degree of the >> scale -- it treats the "bad" 4ths the same as the perfect 4ths. >>What "bad" 4th are you referring to??The one between 15/8 and 4/3 in the scale above.>> To rephrase the question one more time, in what sense are these >> bass notes tonics? Do they change anything about the melody? >> That is, what used to be scale degree 4 is now 1? I say they >> don't. >> Fine -- I don't see how this is related to anything I'm saying.I thought somebody was claiming there was an impetus for 5-limit temperament in Javanese music.>> What I hear is a fixed 1. The melody is a very slow >> series of scale degrees above that 1. On each note of the melody, >> a bunch of ornamentation is hung, which is made of scale arpeggio >> bits. The bass starts and ends on 1, and goes to 3, 2, and >> sometimes 4 (I-IV-III-V diatonic), to provide a sense of >> tension/resolution. >>How do you get I-IV-III-V out of a 5-tone scale? And these chords >sound major or minor to you?I-IV-III-V as if pelog was a 5-out-of diatonic scale. Am I forgetting the case of me numerals? Would be 1-3-2-4 in pelog. 1 and 3 sound major, 2 minor. I can't tell 4. -Carl
Message: 3026 - Contents - Hide Contents Date: Mon, 07 Jan 2002 09:21:35 Subject: Heuristic rules! From: paulerlich --- In tuning-math@y..., "paulerlich" <paul@s...> wrote:> --- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote: >> #1 >>>> 2^-90 3^-15 5^49 >> >> This is not only the the one with lowest badness on the list, it is> the smallest comma, which suggests we are not tapering off, and is > evidence for flatness. >> >> Map: >>>> [ 0 1] >> [49 -6] >> [15 0] >> >> Generators: a = 275.99975/1783 = 113.00046/730; b = 1 >> >> I suggest the "Woolhouse" as a name for this temperament, because> of the 730. Other ets consistent with this are 84, 323, 407, 1053 and > 1460. >> >> badness: 34 >> rms: .000763 >> g: 35.5>> errors: [-.000234, -.001029, -.000796] >> >> #2 32805/32768 Schismic badness=55 >> >> #3 25/24 Neutral thirds badness=82 >> >> #4 15625/15552 Kleismic badness=97 >> >> #5 81/80 Meantone badness=108 >> >> It looks pretty flat so far as this method can show, I think. >> How well do these results back up my now-famous (I hope) heuristic, > which involves only the size of the numbers in, and the difference > between numerator and denominator of, the unison vector? How might we > weight the gens and/or cents measures so that the heuristic will work > perfectly?Let's start with #5 and work our way up to #1: U V W X Y Z unisonvector rms(oct) |n-d|/(d*log(d)) V/U g log(d) X/Y ------------ -------- --------------- ---- ---- ------ --- 80/81 0.003517 0.002809 .799 2.944 4.394 .67 15552/15625 0.0008583 0.0004838 .564 4.546 9.657 .471 24/25 0.024083 0.012427 .516 1.414 3.219 .439 32768/32805 0.00013475 0.00010847 .805 6.976 10.398 .671 [90 15 -49] 6.358e-007 3.44e-007 .541 35.5 78.862 .450 Our current "g" measure is clearly too large when the generator is a fifth, as I've been trying to complain for quite a while now and only Dave Keenan has changes his ways accordingly, and the comparison with the heuristic, though agreeable, suggests that the heuristic is in fact better than our current measure. What if we weighted the intervals unequally in both the g and in the "rms" calculations? Could we get the heuristic to work perfectly? I think that would be very interesting for our paper. Moving on to some relatively "bad" examples . . . Gene, your "Enneadecal" comma should have a power of 2 equal to 14, not 15 as you said, right? U V W X Y Z unisonvector rms(oct) |n-d|/(d*log(d)) V/U g log(d) X/Y ------------ -------- --------------- ---- ---- ------ --- [52 17 -34] 2.864e-005 1.5102e-005 .527 24.042 54.721 .439 128/135 0.01508 0.010571 .701 2.94 4.905 .599 [14 19 -19] 8.733e-005 5.314e-005 .608 15.513 30.579 .507 648/625 0.009217 0.005716 .620 3.27 6.438 .508 [8 14 -13] 0.0002305 0.0001463 .635 11.045 20.923 .528 Here the heuristic seems to work even better (and the V/U and X/Y are well within the range of their values for the top 5).
Message: 3027 - Contents - Hide Contents Date: Mon, 07 Jan 2002 02:18:22 Subject: Re: tetrachordality From: paulerlich --- In tuning-math@y..., "clumma" <carl@l...> wrote:>>> The interval pattern stuff (the L-L-s of conventional theory) is >>> a relative pitch thing... ? >>>> Not sure what you mean. >> I'm trying to understand the psychoacoustical basis for the > version in your paper, and recent posts about ethnic scales > on the main list (x+x+y, etc.).You mean when I talk about tetrachordal (now homotetrachordal) octave species of ethnic scales?> And I'm trying to understand > the lack, if any, of a psychoacoustical basis for my stuff > (absolute pitches being transposed by a 3:2).Psychoacoustic basis? Well, when you hear the step pattern repeated at a 4:3 or a 3:2, it spells a more coherent tonal organization than when you hear it repeated at some other interval. The patterns sound more similar to one another in the first case, so the information can be "compressed" more efficiently, if that makes any sense . . .
Message: 3028 - Contents - Hide Contents Date: Mon, 07 Jan 2002 05:51:58 Subject: A 72-et decatonic From: genewardsmith There were 66 scales of this type, but one was something of a standout, so I'll give just it: [0, 5, 14, 19, 28, 33, 42, 49, 58, 63] [5, 9, 5, 9, 5, 9, 7, 9, 5, 9] edges 11 24 34 connectivity 0 4 6
Message: 3029 - Contents - Hide Contents Date: Mon, 07 Jan 2002 09:23:16 Subject: Re: OPTIMAL 5-LIMIT GENERATORS FOR DAVE From: paulerlich --- In tuning-math@y..., "clumma" <carl@l...> wrote:>>> I've never heard a voice >> >> Voice? >> As in, part or parts in the music sharing the same rhythm.So what does the sentence, "I've never heard a voice in the music that was triads, Paul." mean? You haven't heard parallel triads? Me either!
Message: 3030 - Contents - Hide Contents Date: Mon, 07 Jan 2002 02:31:26 Subject: Re: OPTIMAL 5-LIMIT GENERATORS FOR DAVE From: paulerlich --- In tuning-math@y..., "clumma" <carl@l...> wrote:>>> >>> s represents the blur of the spectral components coming in. >>> How could an inharmonic timbre change that? >>>> When we're dealing with a dyad consisting of complex tones, and >> trying to apply harmonic entropy to that dyad, s is decreased >> below the value that sine waves in place of the complex tones >> would imply. The more inharmonic the timbre, the less s is >> decreased below the sine-wave case. >> Still doesn't explain how. You need a way for data from the > combination-sensitive stuff to improve the spectral stuff coming > off the cochlea.There's less ambiguity as to the possible ratio-intepretations.> I don't think it works that way.It does. Devise a baby mathematical model for it and you'll see.> The "accuracy" > of the "fundamental" is improved as the spectral components get > closer to just, as the harmonic entropy calc. itself correctly > models. But to change s in this way is a fudge, in my opinion.It would be nice to derive the change in s mathematically, and any baby model will do so. However, until the model is more fully developed, I'm hesitant to put forward any exact formulas about how s changes.> With harmonic timbres, h.e. on the fundamentals is a good > approximation of things, but with inharmonic timbres, all spectral > components need to be put in to the h.e. calculation. Jacking up > s may approximate this, but it would be a fudge.Again, if the inharmonicities are only about 50 cents, I thing it's a pretty darn good one.> Anyway, there is now psychoacoustic evidence for harmonic entropy. > In fact, it looks like it perfectly models what happens in > populations of "combination-sensitive" neurons in the inferior > colliculus. At least in bats. I plan on posting to > harmonic_entropy on this as soon as I can get the citations > together.You might want to mention this to J on the main tuning list, who seems to think that Terhardt and I are the only two people in the world who believe there is such a thing as "virtual pitch".>>>>> Yes, to me, pelog sounds like a I and a III with a 4th in >>>>> the middle. But the music seems to use a fixed tonic, with >>>>> not much in the way of triadic structure. >>>>>>>> How about 5-limit intervals? >>>>>> Not sure what you're asking. >>>> Not much in the way of 5-limit intervals? >> I think the large 2nd approximates a 5:4, and the perfect > 4th a 3:2, with some tempering to reduce the roughness of > these intervals on the instrumentation used (as opposed to > tempering to improve the consonance of these interval in > different modes, to distribute any commas, etc.).Why don't you experiment with this with a number of timbres. I know you can't, but I can, and believe me, reducing roughness does absolutely nothing to capture an authentic Gamelan sound.>>> Right, it's the chinese pentatonic. I threw it in for >>> completeness. >>>> Completeness of what? >> Of the journey from North to South, and of the survey of > pentatonic scales in motivic ethnic music of southeast asia.You listen to three selections and call that a survey?> And it was informative; nothing about the music changed as > we went from Pelog, to the hybrid, to the chinese pentatonic > _except_ the scale. You could transcribe the notes and wind > up with the same stuff, more or less.Probably to people in that part of the world Beethoven and the blues sound the same. Go live in Southeast Asia for twelve years and then get back to me.>>> Western music uses progressions of four fifths and expects >>> to wind up on a major third. >>>> These don't have to be triadic, harmonic progression. >> I guess not. But there's a big difference in how this stuff > is used. The Indonesia music is motivic, not modal. At least, > I follow the pitches and their positions in the scale, not the > intervals of the scale and there relation to one another.Right, those, you might say, are "incidental". Still, they give the sound a certain texture.> The > harmonic motion is used to render some consonance, and some > tension/release action, but that's it. It's a backdrop to the > motivic material.The same is true in Western music, in a certain sense.>>> I didn't notice anything like this >>> for the [1 -3] map (right?) on the cited discs. >>>> [3 1]. It's not something you should expect to hear as a triadic >> harmonic progression. It's simply the way the 5-limit intervals >> fit together in the scale. If they didn't, the scale, and the >> music that depends on it, wouldn't work. >> [3 1]? I thought these maps expressed each odd identity, from > three to the limit, increasing from left to right, in numbers > of generators. Thus up one 3:2 for the 3:2, and down three 3:2s > for the 5:4.I though you meant the unison vector. OK ,[1 -3] map looks right.>>> But I think a lot of the other stuff that goes along with >>> harmonic music is missing from this music. Western music >>> requires meantone. The pelog 5-limit map is far more extreme, >>> but what suffers in this music as we change the tuning from >>> 5-of- 7, to 23, to 16, all the way to strict JI? >>>> 23 and 16 give you the Pelog sound. 7 doesn't. >> By gods, you're right! 7-of-5 doesn't sound like pelog at all. 5-of-7? >>> Give me a strict JI scale to try. >> 1/1 5/4 4/3 3/2 15/8 > > Sounds like a fine pelog to me.Yuck -- reminds me of Charles Lucy. Totally inauthentic.>>> I think the tuning on these discs is closer to JI than 23-tET, >>> and I don't hear them avoiding a disjoint interval. Do you? >>>> Avoiding a disjoint interval? You mean you hear it as 5-of-7? >> It modulates that much?? >> Actually, it doesn't. They seem to stick mostly to I, IV, and > III (diatonic) with the bass, if you consider those tonics. But > the melodic stuff does center itself on every degree of the > scale -- it treats the "bad" 4ths the same as the perfect 4ths.What "bad" 4th are you referring to??> To rephrase the question one more time, in what sense are these > bass notes tonics? Do they change anything about the melody? > That is, what used to be scale degree 4 is now 1? I say they > don't.Fine -- I don't see how this is related to anything I'm saying.> What I hear is a fixed 1. The melody is a very slow > series of scale degrees above that 1. On each note of the melody, > a bunch of ornamentation is hung, which is made of scale arpeggio > bits. The bass starts and ends on 1, and goes to 3, 2, and > sometimes 4 (I-IV-III-V diatonic), to provide a sense of > tension/resolution.How do you get I-IV-III-V out of a 5-tone scale? And these chords sound major or minor to you?
Message: 3031 - Contents - Hide Contents Date: Mon, 07 Jan 2002 06:20:48 Subject: Re: Optimal 5-Limit Generators For Dave From: paulerlich --- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:> --- In tuning-math@y..., "paulerlich" <paul@s...> wrote: >>>> Wouldn't it be nice to say whether or not Fokker's methods would>> have worked if he>>> had run into torsion? >>> I'm pretty sure the answer is no. Gene? >> I don't know they are. What would he have done in the case of the >24-note business which was our first example? I'd guess he would just leave it as a 24-tone JI scale, but we'll never know, as he only ever published 12-, 19-, 22-, 31-, 41-, and 53- tone PBs (and left them all as JI scales).
Message: 3032 - Contents - Hide Contents Date: Mon, 07 Jan 2002 09:33:54 Subject: Distinct p-limit intervals and ets From: genewardsmith Here's an interesting computation--the first ets which distinguish all the p-limit intervals for p = 5,7,11,13 and 17. Anyone who cares to may join in the fun by filling in the gaps, or extending the list of p-limits. 5: 9,11,12,14,15,16,18,19 ... 7: 27,31,35,36,37,40,41,42 ... 11: 58,65,72,73,80,84,87,89 ... 13: 87,94,95,103,111,113,118,120 ... 17: 149,151,159,161,163,169,170,171 ...
Message: 3033 - Contents - Hide Contents Date: Mon, 07 Jan 2002 06:24:47 Subject: Re: Some 12-tone, 2-step 46-et scales From: paulerlich --- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:> --- In tuning-math@y..., "paulerlich" <paul@s...> wrote: >>> Sure. Now are there some 46-tET commas you did not take into account? >> You didn't answer my "hyper-torus" point on the tuning listyet . . .> > I wasn't clear what you meant, but there are topological >considerations which come into graph theory. A graph can be a planar >graph, for instance, or a graph on a quotient (cylinder or torus), >so it can have a genus--it might be a graph on something with >negative curvature. I plan on reading some graph theory and seeing >if anything I run across suggests some application.It shouldn't be so complicated. You can draw a big giant graph for all of 72-tET, can't you?
Message: 3034 - Contents - Hide Contents Date: Mon, 07 Jan 2002 09:38:40 Subject: Re: Distinct p-limit intervals and ets From: paulerlich --- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:> Here's an interesting computation--the first ets which distinguishall the p-limit intervals for p = 5,7,11,13 and 17. There are an infinite number of 5-prime limit intervals. You must mean odd limit, in which case you should include 9 and 15.>Anyone who cares to may join in the fun by filling in the gaps, or >extending the list of p-limits. > > 5: 9,11,12,14,15,16,18,19 ... > > 7: 27,31,35,36,37,40,41,42 ... > > 11: 58,65,72,73,80,84,87,89 ... > > 13: 87,94,95,103,111,113,118,120 ... > > 17: 149,151,159,161,163,169,170,171 ...This is what we call "uniqueness", and Manuel has compiled vast tables of both this and "consistency" for both odd and integer limits, and fractional ETs. Please see Definitions of tuning terms: unique, (c) 1998 ... * [with cont.] (Wayb.)
Message: 3035 - Contents - Hide Contents Date: Mon, 07 Jan 2002 06:28:36 Subject: Re: Some 12-tone, 2-step 46-et scales From: genewardsmith --- In tuning-math@y..., "paulerlich" <paul@s...> wrote:> It shouldn't be so complicated. You can draw a big giant graph for > all of 72-tET, can't you?Certainly. What are you getting at, though?
Message: 3036 - Contents - Hide Contents Date: Mon, 07 Jan 2002 09:43:02 Subject: Re: Distinct p-limit intervals and ets From: genewardsmith --- In tuning-math@y..., "paulerlich" <paul@s...> wrote:> --- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:>> Here's an interesting computation--the first ets which distinguish> all the p-limit intervals for p = 5,7,11,13 and 17. > > There are an infinite number of 5-prime limit intervals. You must > mean odd limit, in which case you should include 9 and 15.Thus my mention of gaps.
Message: 3037 - Contents - Hide Contents Date: Mon, 07 Jan 2002 06:40:56 Subject: Re: OPTIMAL 5-LIMIT GENERATORS FOR DAVE From: paulerlich --- In tuning-math@y..., "clumma" <carl@l...> wrote:>>>> Right, it's the chinese pentatonic. I threw it in for >>>> completeness. >>>> >>>> Completeness of what? >>>>>> Of the journey from North to South, and of the survey of >>> pentatonic scales in motivic ethnic music of southeast asia. >>>> You listen to three selections and call that a survey? >> Uh-huh. I didn't make any statistical claims about it. In > fact the only claims I make about it are: > > () it's better than a 'survey' of the scala scl archiveMaybe. We still haven't heard which scales Dave used, and where they came from. There's lots of other research -- on the internet alone.>>>> Give me a strict JI scale to try. >>>>>> 1/1 5/4 4/3 3/2 15/8 >>> >>> Sounds like a fine pelog to me. >>>> Yuck -- reminds me of Charles Lucy. > > Me too. > >> Totally inauthentic. >> But much closer than 5-of-7.5-of-7 gives you neutral thirds.>>>>> I think the tuning on these discs is closer to JI than 23-tET, >>>>> and I don't hear them avoiding a disjoint interval. Do you? >>>>>>>> Avoiding a disjoint interval? You mean you hear it as 5-of-7? >>>> It modulates that much?? >>>>>> Actually, it doesn't. They seem to stick mostly to I, IV, and >>> III (diatonic) with the bass, if you consider those tonics. But >>> the melodic stuff does center itself on every degree of the >>> scale -- it treats the "bad" 4ths the same as the perfect 4ths. >>>> What "bad" 4th are you referring to?? >> The one between 15/8 and 4/3 in the scale above.So the fact that they're not avoiding it helps prove I'm right! It's a GOOD fourth in Pelog, since 135:128 vanishes! It's a bad fourth in JI, so _if_ the tuning were JI, _then_ there might be a tendency to avoid that interval.>>> To rephrase the question one more time, in what sense are these >>> bass notes tonics? Do they change anything about the melody? >>> That is, what used to be scale degree 4 is now 1? I say they >>> don't. >>>> Fine -- I don't see how this is related to anything I'm saying. >> I thought somebody was claiming there was an impetus for 5-limit > temperament in Javanese music.Still don't see any relationship.>>> What I hear is a fixed 1. The melody is a very slow >>> series of scale degrees above that 1. On each note of the melody, >>> a bunch of ornamentation is hung, which is made of scale arpeggio >>> bits. The bass starts and ends on 1, and goes to 3, 2, and >>> sometimes 4 (I-IV-III-V diatonic), to provide a sense of >>> tension/resolution. >>>> How do you get I-IV-III-V out of a 5-tone scale? And these chords >> sound major or minor to you? >> I-IV-III-V as if pelog was a 5-out-of diatonic scale. Am I > forgetting the case of me numerals?I think so.> Would be 1-3-2-4 in pelog. 1 and 3 sound major,So C major and E major ???> 2 minor. I > can't tell 4.
Message: 3038 - Contents - Hide Contents Date: Mon, 07 Jan 2002 09:43:03 Subject: Re: Distinct p-limit intervals and ets From: paulerlich Monz, the links to the tables are outdated. Manuel, could you provide the updated links?
Message: 3039 - Contents - Hide Contents Date: Mon, 07 Jan 2002 06:42:11 Subject: Re: A 72-et decatonic From: paulerlich --- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:> There were 66 scales of this type, but one was something of astandout, so I'll give just it:> > [0, 5, 14, 19, 28, 33, 42, 49, 58, 63] > [5, 9, 5, 9, 5, 9, 7, 9, 5, 9] > edges 11 24 34 connectivity 0 4 6Are you now taking into account _all_ the consonances of 72-tET?
Message: 3040 - Contents - Hide Contents Date: Mon, 07 Jan 2002 09:54:07 Subject: Re: Heuristic rules! From: genewardsmith --- In tuning-math@y..., "paulerlich" <paul@s...> wrote:> U V W X Y Z > unisonvector rms(oct) |n-d|/(d*log(d)) V/U g log(d) X/Y > ------------ -------- --------------- ---- ---- ------ --- > 80/81 0.003517 0.002809 .799 2.944 4.394 .67 > 15552/15625 0.0008583 0.0004838 .564 4.546 9.657 .471 > 24/25 0.024083 0.012427 .516 1.414 3.219 .439 > 32768/32805 0.00013475 0.00010847 .805 6.976 10.398 .671 > [90 15 -49] 6.358e-007 3.44e-007 .541 35.5 78.862 .450I was going to ask what this was, but I see when I follow-up the table makes more sense.> Our current "g" measure is clearly too large when the generator is a > fifth, Why?as I've been trying to complain for quite a while now and only> Dave Keenan has changes his ways accordingly, and the comparison with > the heuristic, though agreeable, suggests that the heuristic is in > fact better than our current measure.Better for what? If you mean as a badness measure, g isn't one. What if we weighted the> intervals unequally in both the g and in the "rms" calculations?We could, but I was never clear how exactly you wanted this done.
Message: 3041 - Contents - Hide Contents Date: Mon, 07 Jan 2002 06:43:49 Subject: Re: Some 12-tone, 2-step 46-et scales From: paulerlich --- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:> --- In tuning-math@y..., "paulerlich" <paul@s...> wrote: >>> It shouldn't be so complicated. You can draw a big giant graph for >> all of 72-tET, can't you? >> Certainly. What are you getting at, though?If you do that, you take _all_ the commas of 72-tET into account.
Message: 3042 - Contents - Hide Contents Date: Mon, 07 Jan 2002 10:08:01 Subject: Re: Heuristic rules! From: paulerlich --- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:> --- In tuning-math@y..., "paulerlich" <paul@s...> wrote:>> U V W X Y Z >> unisonvector rms(oct) |n-d|/(d*log(d)) V/U g log(d) X/Y >> ------------ -------- --------------- ---- ---- ------ --- >> 80/81 0.003517 0.002809 .799 2.944 4.394 .67 >> 15552/15625 0.0008583 0.0004838 .564 4.546 9.657 .471 >> 24/25 0.024083 0.012427 .516 1.414 3.219 .439 >> 32768/32805 0.00013475 0.00010847 .805 6.976 10.398 .671 >> [90 15 -49] 6.358e-007 3.44e-007 .541 35.5 78.862 .450 >> I was going to ask what this was, but I see when I follow-up thetable makes more sense.>>> Our current "g" measure is clearly too large when the generator is a >> fifth, > > Why?Because of two systems with the same "g" measure, the one which has a generator of a 3:2 is musically simpler. Movements by 3:2 are easier to hear than movements by 5:3 or 5:4.> as I've been trying to complain for quite a while now and only>> Dave Keenan has changes his ways accordingly, and the comparison with >> the heuristic, though agreeable, suggests that the heuristic is in >> fact better than our current measure. >> Better for what? If you mean as a badness measure, g isn't one.I mean as a measure of the musical complexity of the system.>> What if we weighted the >> intervals unequally in both the g and in the "rms" calculations? >> We could, but I was never clear how exactly you wanted this done.The suggestion I gave you in the past was, in the g measure, to weight the 3:2 by 1/log(3), 5:4 by 1/log(5), and 5:3 by 1/log(5). This is what Dave Keenan is now using. It's unclear to me that an root-mean-square calculation for g is exactly what we want, though it's probably a good approximation. But my nagging suspicion is that some perfectly musically reasonable formulations of g, and of 'rms' or the cents-error measure, should conform to my heuristics _exactly_, or at least way better than the current measures are. You're the mathematician, can't you figure this out? Read my original justification for the heuristics again if you need to (message #1437), and consider the van Prooijen lattice: Searching Small Intervals * [with cont.] (Wayb.) and Searching Small Intervals * [with cont.] (Wayb.) except for the last graph. This should be doable!
Message: 3043 - Contents - Hide Contents Date: Mon, 07 Jan 2002 06:55:56 Subject: Re: OPTIMAL 5-LIMIT GENERATORS FOR DAVE From: clumma>> > thought somebody was claiming there was an impetus for 5-limit >> temperament in Javanese music. >>Still don't see any relationship.I'm trying to show that the things in Western music that led to temperament are absent in Indonesian music.>>> How do you get I-IV-III-V out of a 5-tone scale? And these chords >>> sound major or minor to you? >>>> I-IV-III-V as if pelog was a 5-out-of diatonic scale. Am I >> forgetting the case of me numerals? >>I think so. >>> Would be 1-3-2-4 in pelog. 1 and 3 sound major, >> So C major and E major ???C and F major.>> 2 minor. E minor. >> I can't tell 4. G something. -Carl
Message: 3044 - Contents - Hide Contents Date: Mon, 07 Jan 2002 10:09:58 Subject: Re: Heuristic rules! From: paulerlich I wrote,> Searching Small Intervals * [with cont.] (Wayb.) and > Searching Small Intervals * [with cont.] (Wayb.)Oops -- the second link should be lattice orientation * [with cont.] (Wayb.)
Message: 3045 - Contents - Hide Contents Date: Mon, 07 Jan 2002 07:01:14 Subject: Re: OPTIMAL 5-LIMIT GENERATORS FOR DAVE From: paulerlich --- In tuning-math@y..., "clumma" <carl@l...> wrote:>>> I thought somebody was claiming there was an impetus for 5-limit >>> temperament in Javanese music. >>>> Still don't see any relationship. >> I'm trying to show that the things in Western music that led > to temperament are absent in Indonesian music.So what? They may have had their own reasons, and inharmonicity makes the situations rather different. Gamelan intervals are "pastelized", as Margo Schulter says.>>>>> How do you get I-IV-III-V out of a 5-tone scale? And these chords >>>> sound major or minor to you? >>>>>> I-IV-III-V as if pelog was a 5-out-of diatonic scale. Am I >>> forgetting the case of me numerals? >>>> I think so. >>>>> Would be 1-3-2-4 in pelog. 1 and 3 sound major, >>>> So C major and E major ??? >> C and F major.How did the note A get in there?>>> 2 minor. > > E minor. >>>> I can't tell 4. > > G something.You mean you can't hear which notes make up the chord?
Message: 3046 - Contents - Hide Contents Date: Mon, 07 Jan 2002 10:15:45 Subject: Enneadecal? From: paulerlich Gene, you didn't reply: 'Gene, your "Enneadecal" comma should have a power of 2 equal to 14, not 15 as you said, right?' I'm referring to your results in message #2121.
Message: 3047 - Contents - Hide Contents Date: Mon, 07 Jan 2002 07:02:26 Subject: Re: OPTIMAL 5-LIMIT GENERATORS FOR DAVE From: paulerlich Carl, it seemed you yourself just gave evidence for the 135:128 vanishing, didn't you?
Message: 3048 - Contents - Hide Contents Date: Mon, 07 Jan 2002 10:52:25 Subject: Re: Enneadecal? From: genewardsmith --- In tuning-math@y..., "paulerlich" <paul@s...> wrote:> Gene, you didn't reply: > > 'Gene, your "Enneadecal" comma should have a power of 2 equal to 14, > not 15 as you said, right?'If that's what makes it small, of course. If not, obviously not.
Message: 3049 - Contents - Hide Contents Date: Mon, 07 Jan 2002 07:03:41 Subject: Re: Some 12-tone, 2-step 46-et scales From: genewardsmith --- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:> I was facinated to discover that the 7,5 system did a little better than the completely symmetrical 6,6 system.Here are the graphs. Looking at these, 12 might be a good place to center.> [0, 4, 8, 12, 16, 20, 23, 27, 31, 35, 39, 43] > [4, 4, 4, 4, 4, 3, 4, 4, 4, 4, 4, 3] > edges 24 24 40 connectivity 3 3 6 Yahoo groups: /tuning-math/files/Gene/graph/g5... * [with cont.] Yahoo groups: /tuning-math/files/Gene/graph/g7... * [with cont.] Yahoo groups: /tuning-math/files/Gene/graph/g1... * [with cont.] > [0, 4, 8, 12, 16, 20, 24, 27, 31, 35, 39, 43] > [4, 4, 4, 4, 4, 4, 3, 4, 4, 4, 4, 3] > edges 24 25 41 connectivity 3 3 6 Yahoo groups: /tuning-math/files/Gene/graph/g5... * [with cont.] Yahoo groups: /tuning-math/files/Gene/graph/g7... * [with cont.] Yahoo groups: /tuning-math/files/Gene/graph/g1... * [with cont.]
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