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Message: 9750 - Contents - Hide Contents Date: Mon, 02 Feb 2004 23:40:35 Subject: Re: 114 7-limit temperaments From: Graham Breed Me:>> 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. Paul E:> What if you don't assume total octave-equivalence?I don't think it matters. You can't give a fixed number of notes per octave, but some other complexity measure will do. The Farey limit can replace the odd limit. I'd prefer to give less weight to larger intervals, but that's a peripheral issue.> In the Tenney-lattice view of harmony, 'limit' and chord structure is > a more fluid concept.Yes, that's the problem.>>> 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. > > Any examples?Yes, I've got a CGI script that generates examples.>> Oh, yes, I think the 9-limit calculation can be done by giving 3 a >> weight of a half. >> Which calculation are you referring to, exactly?Dave Keenan't original "method for optimally distributing a comma" or whatever it was.>> 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. >> Number of consonances?For minimax error with equal weighting, you share the comma among all the constituent intervals in the given limit. But it's ambiguous how you do the counting. If you take 28:27, that has odd factors 7:3*3*3 So in the 9-limit that gives two intervals, which could be 7:6 and 8:9 or 7:9 and 2:3. If you give 9 half the error, that means 3 has 1/4 of it. For 7:6 to have half, 7 must take the other quarter in the other direction. That means 7:9 has three quarters of the comma, but no 9-limit interval is supposed to have more than half of it. If you give 2:3 half the error, then 9:8 must take the whole error, which is already wrong. So the right thing is to treat 3 as having a weighting of half. Then, the complexity is 1.5 and the minimax should be 2/3 of the original comma. And that worst interval will be 9:8, or anything else with a 9 in it.>> After all, how do you share a comma between 3:2 and 9:8? >> I'm not sure why you're asking this at this point, or what it > means . . . See above.>> I still don't know how the 15-limit would work. > > ?shrug?15 counts as a single consonance, but 25 isn't in the limit so I don't think giving 5 a weight of half will work.>> I'm expecting the limit of this calculation as the odd limit tends > to>> infinity will be the same as this Kees metric. > >> Can you clarify which calculation and which Kees metric you're > talking about?The calculation of sharing the comma to give an odd-limit minimax, and the metric taking the logarithm of the (larger) odd number in the ratio.>> And as the integer limit >> goes to infinity, it'll probably give the Tenney metric. >> I haven't the foggiest idea what you mean. > > All I can say at this point is that n*d seems to be to be a better > criterion to 'limit' than n (integer limit).3 has a weighting of half in the 9-limit because it contributes twice to 9. For a prime p in an n-limit (ignoring other composites like 15) the weighting is 1/floor(log(n)/log(p)) The log(n) is common to all primes, so you can change the weighting to log(n)/floor(log(n)/log(p)) to make it change less drastically with n. As n approaches infinity, this gets more like log(p) -- the Tenney weighting.> I still remain unclear on what you were doing with your octave- > equivalent TOP stuff. Gene ended up interested in the topic later but > you missed each other. I rediscovered your 'worst comma in 12-equal' > when playing around with "orthogonalization" and now figure I must > have misunderstood your code. You weren't searching an infinite > number of commas, but just three, right?I was searching a large number but not infinite. And this was to solve a lower dimensioned temperament, nothing to do with octave equivalence. I don't know how Gene's doing it, but I thought it was some numerical method to calculate the weighted minimax directly. The octave-equivalent TOP is easy -- you use the same weighting formula, but using the log of the larger odd number of n and d, instead of n*d. I don't think TOP really favors simple ratios at all. The weighting is only reflecting the mathematics of composite numbers, which ensure that simple intervals will have a lower error. The result will be close to some average of the odd (or Farey) limits that could apply to that prime limit. So it isn't a fully general case, but is a convenient approximation where it's easier to calculate than the true odd limit optimum. Graham

Message: 9751 - Contents - Hide Contents Date: Mon, 02 Feb 2004 03:55:55 Subject: Re: 7-limit horagrams From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:>>> Beautiful! I take it the green lines are proper scales? >>> >>> -C. >>>> Guess again (it's easy)! >> Obviously not easy enough if we've had to exchange three > messages about it. > > -CarlThen you can't actually be looking at the horagrams ;) ________________________________________________________________________ ________________________________________________________________________ ------------------------------------------------------------------------ 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: 9752 - Contents - Hide Contents Date: Mon, 02 Feb 2004 08:30:11 Subject: Re: Back to the 5-limit cutoff 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:>>> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> >> wrote: >>>>>>> Sure, why not let *both* exponents and >>> >>> and what? >>>> and both constants vary when optimizing a moat . . . >> OK. But I'd like to limit the exponent to between 1 and 2 inclusive.I'd go lower too. Meanwhile, looks like Gene might be calling us crazy for considering any exponent above 1 . . . ________________________________________________________________________ ________________________________________________________________________ ------------------------------------------------------------------------ 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: 9753 - Contents - Hide Contents Date: Mon, 02 Feb 2004 22:57:41 Subject: Re: finding a moat in 7-limit commas a bit tougher . . . From: Herman Miller On Tue, 03 Feb 2004 01:58:33 -0000, "Paul Erlich" <perlich@xxx.xxxx.xxx> wrote:>> Why not. I have enough trouble wondering why anyone would use a >> 5-limit _linear_ temperament that was non-unique, 7-limit planar >> stretches my credibility even further. Can you propose a scale or >> finite tuning in these that you think might be useful as an >> approximation of 7-limit JI? >>Not right now, must jet soon . . . This is Herman's department, or >maybe Gene's . . .Tempering 36/35 gives you a perfectly symmetrical 4:5:6:7:9 chord, which could be of some use; the 6/5 is the same as the 7/6. Looks similar to 12-ET; a 12-note subset might be appropriate. TOP tuning: [1195.264647, 1894.449645, 2797.308862, 3382.119722] The difference between this tuning and top dominant seventh is VERY minor. TOP 36/35 TOP Dominant Seventh 9/8 203.105 203.47 5/4 406.78 406.94 3/2 699.185 699.35 5/3 902.859 902.82 7/4 991.59 991.76 2/1 1195.265 1195.23 Tempering out 28/27 gives you something close to 5-ET, but unevenly spaced; a series of three fifths gives you a 7/2. Adding a just 5/1 gives you sharp thirds like those of 15-ET. TOP tuning: [1193.415676, 1912.390908, 2786.313714, 3350.341372] TOP 28/27 TOP Blackwood 3/2 718.975 717.54 7/4 963.51 956.72 2/1 1193.416 1195.9 -- see my music page ---> ---<The Music Page * [with cont.] (Wayb.)>-- hmiller (Herman Miller) "If all Printers were determin'd not to print any @io.com email password: thing till they were sure it would offend no body, \ "Subject: teamouse" / there would be very little printed." -Ben Franklin ________________________________________________________________________ ________________________________________________________________________ ------------------------------------------------------------------------ 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: 9754 - Contents - Hide Contents Date: Mon, 02 Feb 2004 23:23:12 Subject: Re: finding a moat in 7-limit commas a bit tougher . . . From: Dave Keenan --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote: And could you please multiply the vertical axis numbers by 1200. I'm getting tired of doing this mentally all the time, to make them mean something. ________________________________________________________________________ ________________________________________________________________________ ------------------------------------------------------------------------ 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: 9755 - Contents - Hide Contents Date: Tue, 03 Feb 2004 06:14:19 Subject: Re: Back to the 5-limit cutoff From: Dave Keenan --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:> --- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" <d.keenan@b...> > wrote:>> Have you read this? >> A note on mathematical notation for musical in... * [with cont.] (Wayb.) >> If so, are there particular parts of it that make no sense to you? >> (a) makes no sense, because the colon has even more potential for > confusing the hell out of the uninitianted.This doesn't seem to have been borne out by the last decade on the tuning lists. Why should using different notations for different musical objects _add_ to confusion? I understand that the use of the same notation for points and vectors continues to cause confusion for math and physics students to this day. Which is one reason the Geometric Algebra is such a good idea.> (c)is a reason why colons are inferior.Well you can say that, but unless you can explain it I'll just assume you're having a bad day. It seems to me that it is common in mathematics (although far from universal) to use symbols which are laterally symmetrical to represent commutative operators (*, +) and laterally asymmetrical to represent noncommutative operators (/, superscripting for exponentiation). Seems like a good idea to me. Seems like it has mnemonic value. But in any case, a better reason to use colon for intervals is that people have long been using colons to express the ratios of the frequencies in chords of 3 or more notes, so why not for 2 notes. Colons are in common use in everyday life to separate the numbers giving the relative proportions of 2 or more quantities. You'll find them on DVD covers and ads for TVs (screen aspect ratios) and on bags of cement (cement:sand:gravel).> (d) is hilarious. There is no canonical order, which is one of the > difficulties this this business.Where have you found someone writing an extended ratio for a chord of 3 or more notes with the big numbers on the left, in the past 50 years? How many examples of this can you find on the tuning lists?>> In software, the safe way to turn these colonic thingies into real >> numbers is always to divide the big one by the small one. >> >> real(a:b) = max(a,b)/min(a,b) >> That's nice. The way to turn a/b into a real number is to divide the > top one by the bottom one--except, hey, it's already a real number.No one's disputing that. If we could agree on the order for intervals (dyads), as we have for triads and larger, I could say the same about a:b.

Message: 9756 - Contents - Hide Contents Date: Tue, 03 Feb 2004 23:53:27 Subject: Re: finding a moat in 7-limit commas a bit tougher . . . 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 the obtaining of relative consonance by using timbres of poorly >>> defined pitch in massive otonalities is a sufficient criterion for >>> temperament-hood (JI approximation) then please give me a non- >> trivial>>> planar tuning that _doesn't_ work like that. >>>> Since the consensus seems to be that father (16:15) doesn't work that >> way, then the 16:15 and 15:14 planar temperaments probably won't work >> that way either . . . >> There is a consensus that "father" isn't an approximation of 5-limit > JI, but I don't think that is because anyone tried it with large > otonalities using inharmonic timbres. It may well work in that way. > > Perhaps we should limit such tests to otonalities having at most one > note per prime (or odd) in the limit. e.g. If you can't make a > convincing major triad then it aint 5-limit. And you can't use > scale-spectrum timbres although you can use inharmonics that have no > relation to the scale.yes, mastuuuhhhhh . . . =(

Message: 9757 - Contents - Hide Contents Date: Tue, 03 Feb 2004 19:53:41 Subject: TOP Equal Temperament graphs! (was: Re: Cross-check for TOP 5-limit 12-equal) 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 "Paul Erlich" <perlich@a...> >> wrote: > >> I used this latter complexity measure to create these graphs: >> I'm not familiar enough with 3-limit harmony (or rather ignoring > 5-limit harmony) to comment on this, but I think I could be happy with > a straight line cutoff here. > >> Yahoo groups: /tuning_files/files/et5.gif * [with cont.] >> For this I'd go for a cutoff that just includes 15, 29, 46, 53, which > has a good enough straight-line moat, but admittedly it would be > widened slightly by using an exponent slightly less than 1. > >> Yahoo groups: /tuning_files/files/et7.gif * [with cont.] >> Here I assume you are referring to the difficulty of finding a moat > that includes both 12 and 72 and keeps out things like 58 and 39. > > To me, this is just evidence that 72-ET would not be of much interest > as a 7-limit temperament (due to its complexity) if it wasn't for the > fact that it is a subdivision of 12-ET. So we could justify its > inclusion an an historical special case whether it was inside any moat > or not.Same goes for Waage in the 5- (where it's been called Aristoxenean) and 7-limit lineat temperament cases, as I've advocated mentioning before. We can thus mention 72-equal and the Waages as being extra- moat examples with "12-ness" . . .> Here we can include 22, 31, 41, 46, and 72 with a straight line, but > admittedly it would be a somewhat wider moat if the exponent was made > slightly less than one. > > Looking at these has disposed me more towards linear moats and less > towards quadratic ones, but only slightly toward powers slightly less > than one. > > If I revisit the 5-limit linear temperament plot and look for good > straight (or near-straight) moats, I find there are none that would > include 2187/2048 that I could accept, because they would either mean > including too much dross at the high complexity end of things, or > would make 25/24 and 135/128 look far better than the marginal things > that they are.It definitely looks like 2187/2048 should be de-moated :). We can mention Blackwood's use of it in a footnoat . . .> But I could accept a straight line (or one with exponent slightly less > than 1) that excluded not only 2187/2048 and 3125/2916 but also > 6561/6250 and 20480/19683, and included semisixths (78732/78125).Maybe an even wider such moat would include wuerschmidt, aristoxenean/waage, amity, and orwell.

Message: 9758 - Contents - Hide Contents Date: Tue, 03 Feb 2004 23:57:07 Subject: Re: Comma reduction? From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "Paul G Hjelmstad" <paul.hjelmstad@u...> wrote:> Well, this might seem like a dumb question, but I need to ask it. > When generating commas from a linear temperament, such as 12&19, > you obtain 1 comma in the 5-limit, 2 commas in the 7-limit, 3 commas > in the 11-limit and so on. Gene shew me how to generate commas from > wedgies, to obtain the subgroup commas, but mentioned that these > results are not necc. linearly independent, and something about using > Hermite reduction to simplify all of this.. > > Are the 2 commas in the 7-limit always linearly independent?Yes, they are never 'collinear'.> How > are they generated, (from wedgies OR matrices)?You can pick them off the tree. We've been looking at some of the 'fruits' here.> Also, was told > that the complement of a wedge product in the 5-limit is the same > as the cross-product, how does this work in the 7-limit?There's a 4-d cross product, but all I know is that when switching from val (bra vector) basis to monzo (ket vector) basis, the wedge product reverses order, and some :( of the signs change . . .

Message: 9759 - Contents - Hide Contents Date: Tue, 03 Feb 2004 20:01:03 Subject: Re: finding a moat in 7-limit commas a bit tougher . . . 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:>>> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> >> wrote: >>>>>> I definitely wouldn't want to throw out 28/27, 36/35 . . . >>>>>> Gene, I hope you're happy I'm using slashes here. I agree there >> isn't>>> likely to be any confusion in this discussion since we're not >> talking>>> about individual pitches at all. >>> >>> Why not. I have enough trouble wondering why anyone would use a >>> 5-limit _linear_ temperament that was non-unique, 7-limit planar >>> stretches my credibility even further. Can you propose a scale or >>> finite tuning in these that you think might be useful as an >>> approximation of 7-limit JI? >>>> Not right now, must jet soon . . . This is Herman's department, or >> maybe Gene's . . . >>>>> Moat-wise, I can see my way to adding 36/35 and 128/125. That >> probably>>> gives the biggest moat possible (percentage-wise) particularly if >> you>>> use an exponent greater than 1. Unless you were to have one with an >>> exponent less than 1 (which I don't like) >>>> Maybe you'll reconsider when you look at the ET graphs I just posted. >>>>> and go all the way up to >>> include 21/20 (which seems lidicrous to me). >>>> It doesn't seem that lidicrous :) to me . . . >> Definition of "lidicrous": so ludicrous that you can't type correctly. ;-) >>> Seriously, I think all >> kinds of novel effects could be obtained if 21/20 vanished, >> "All kinds of novel effects" is one thing and "approximating 7-limit > JI" is another. > >> and if>> you used full 1:2:3:4:5:6:7:8:9:10 chords, there would certainly be >> no confusion over what the chords were 'representing' -- you might >> simply have to use the kinds of timbres that George and I were >> talking about . . . Maybe Herman would like to entertain us with some >> sort of example . . . >> It's the lack of counterexamples I'm more worried about. > > I understand you claim that 12-ET is an approximation of JI for all > limits.Well, I don't know about 13-limit, but in 11-limit for example, there appear to be instances where it works, and 19-equal can work with two different mappings (and thus two different "stretch" factors) in the 11-limit . . .> If the obtaining of relative consonance by using timbres of poorly > defined pitch in massive otonalities is a sufficient criterion for > temperament-hood (JI approximation) then please give me a non- trivial > planar tuning that _doesn't_ work like that.Since the consensus seems to be that father (16:15) doesn't work that way, then the 16:15 and 15:14 planar temperaments probably won't work that way either . . .

Message: 9760 - Contents - Hide Contents Date: Tue, 03 Feb 2004 00:00:39 Subject: Re: finding a moat in 7-limit commas a bit tougher . . . From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" <d.keenan@b...> wrote:> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote: >> Yahoo groups: /tuning_files/files/Erlich/plana... * [with cont.] >> And you can safely cut it off above 25/24. Since this was marginal as > a 5-limit linear temperament it isn't going to fare any better as a > 7-limit planar merely by adding some just ratio of 7 as a second > generator.Sometimes a perfect fifth that sounds awful by itself can yield a lovely major triad when a note is added.

Message: 9761 - Contents - Hide Contents Date: Tue, 03 Feb 2004 20:34:13 Subject: Re: Back to the 5-limit cutoff From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" <d.keenan@b...> wrote:> --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> > wrote:>> --- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" <d.keenan@b...> >> wrote:>>> Have you read this? >>> A note on mathematical notation for musical in... * [with cont.] (Wayb.) >>> If so, are there particular parts of it that make no sense to you? >>>> (a) makes no sense, because the colon has even more potential for >> confusing the hell out of the uninitianted. >> This doesn't seem to have been borne out by the last decade on the > tuning lists. Why should using different notations for different > musical objects _add_ to confusion?This reminds me of Gene recently saying distinguishing "odd-limit" vs. "prime-limit" adds to confusion. I note that he's now making this distinction himself here: Gene Ward Smith * [with cont.] (Wayb.)>> (d) is hilarious. There is no canonical order, which is one of the >> difficulties this this business. >> Where have you found someone writing an extended ratio for a chord of > 3 or more notes with the big numbers on the left, in the past 50 years? > > How many examples of this can you find on the tuning lists?Perhaps more to the point, when musicians break down a chord by listing its notes, they go low-to-high. What's C half-diminished seventh? C-Eb-Gb-Bb, the answer always comes back in that order.

Message: 9762 - Contents - Hide Contents Date: Tue, 03 Feb 2004 00:08:54 Subject: Re: finding a moat in 7-limit commas a bit tougher . . . From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" <d.keenan@b...> wrote:> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote: >> Yahoo groups: /tuning_files/files/Erlich/plana... * [with cont.] > > Paul, > > Please do another one of these without the labels, so we have a chance > of eyeballing the moats.My eyeballs are telling me the same thing as when the labels were there: Yahoo groups: /tuning_files/files/Erlich/myemo... * [with cont.]

Message: 9763 - Contents - Hide Contents Date: Tue, 03 Feb 2004 21:42:55 Subject: Re: finding a moat in 7-limit commas a bit tougher . . . 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 the obtaining of relative consonance by using timbres of poorly >> defined pitch in massive otonalities is a sufficient criterion for >> temperament-hood (JI approximation) then please give me a non- > trivial>> planar tuning that _doesn't_ work like that. >> Since the consensus seems to be that father (16:15) doesn't work that > way, then the 16:15 and 15:14 planar temperaments probably won't work > that way either . . .There is a consensus that "father" isn't an approximation of 5-limit JI, but I don't think that is because anyone tried it with large otonalities using inharmonic timbres. It may well work in that way. Perhaps we should limit such tests to otonalities having at most one note per prime (or odd) in the limit. e.g. If you can't make a convincing major triad then it aint 5-limit. And you can't use scale-spectrum timbres although you can use inharmonics that have no relation to the scale.

Message: 9765 - Contents - Hide Contents Date: Tue, 03 Feb 2004 00:18:15 Subject: Re: Back to the 5-limit cutoff From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:>> By the way, if you use 81/80 instead of 80:81, you are not going to >> be inconsistent with that other fellow who uses 81:80 for the exact >> same ratio. You will aslo be specifying an actual number. Numbers are >> nice. This whole obsession with colons makes me want to give the >> topic a colostomy. I have read no justification for it which makes >> any sense to me. >> There's a history in the literature of using ratios to notate pitches. > Normally around here we use them to notate intervals, but confusion > between the two has caused tragic misunderstandings and more than a > few flame wars. So we adopted colon notation for intervals. I have > no idea what the idea behind putting the smaller number first is, > and I don't approve of it. > > -CarlBut it's almost always done that way for chords of more than 2 notes, e.g., 4:5:6 . . .

Message: 9766 - Contents - Hide Contents Date: Tue, 03 Feb 2004 00:19:42 Subject: Re: Back to the 5-limit cutoff From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:> This whole obsession with colons makes me want to give the > topic a colostomy. I have read no justification for it which makes > any sense to me.Would you write a major triad as 4/5/6 or 6/5/4? I hope not?

Message: 9767 - Contents - Hide Contents Date: Tue, 03 Feb 2004 00:33:45 Subject: Re: finding a moat in 7-limit commas a bit tougher . . . From: Dave Keenan --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:> --- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" <d.keenan@b...> > wrote:>> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> > wrote: >>> Yahoo groups: /tuning_files/files/Erlich/plana... * [with cont.] >> >> Paul, >> >> Please do another one of these without the labels, so we have a > chanceThanks Paul. Fascinating to look at, isn't it. So organic. Some order, some randomness. I think that planar temperaments are inherently less useful than linear (which are less useful than equal). This is mostly due to the melodic dimension, which Herman mentions all the time, but we are completely ignoring (except in so far as harmonic complexity implies melodic complexity). We are not measuring things like evenness and transposability when deciding what is in and what is out. And that's OK. We have to learn to crawl before we can walk. But because planar are inherently less even and less transposable than linear I think there are only a very few interesting or useful 7-limit planars. Since you favour linear moats, I suggest 50/49 49/48 64/63 81/80 126/125 225/224 245/243

Message: 9768 - Contents - Hide Contents Date: Tue, 03 Feb 2004 00:58:18 Subject: Re: finding a moat in 7-limit commas a bit tougher . . . 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:>>> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> >> wrote: >>>> Yahoo groups: /tuning_files/files/Erlich/plana... * [with cont.] >>> >>> Paul, >>> >>> Please do another one of these without the labels, so we have a >> chance>>> of eyeballing the moats. >> >> Yahoo groups: /tuning_files/files/Erlich/plana... * [with cont.] >> Thanks Paul. Fascinating to look at, isn't it. So organic. Some order, > some randomness. > > I think that planar temperaments are inherently less useful than > linear (which are less useful than equal).I completely agree if you replace "less useful" with "more complex".> This is mostly due to the > melodic dimension, which Herman mentions all the time, but we are > completely ignoring (except in so far as harmonic complexity implies > melodic complexity).I disagree that it's about an ignored melodic dimension. Instead, it's as I said before, these complexity values are not directly comparable, because what's the length of an area? What's the area of a volume.> We are not measuring things like evenness and > transposability when deciding what is in and what is out. And that's > OK. We have to learn to crawl before we can walk.Well, we're definitely agreed that a 7-limit planar temperament based on a particular comma is quite a bit more complex than a 5-limit linear temperament based on that same comma.> But because planar are inherently less even and less transposable than > linear I think there are only a very few interesting or useful 7- limit > planars.Sure. I kind of figured the ragismic planar deserved to be in there, but I wouldn't insist on it.> Since you favour linear moats,Where did you get that idea? Curved is fine too.> I suggest > 50/49 > 49/48 > 64/63 > 81/80 > 126/125 > 225/224 > 245/243I definitely wouldn't want to throw out 28/27, 36/35 . . .

Message: 9769 - Contents - Hide Contents Date: Tue, 03 Feb 2004 01:04:58 Subject: Re: Back to the 5-limit cutoff From: Dave Keenan --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:> --- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" <d.keenan@b...> > wrote: >>> My favourite cutoff for 5-limit temperaments is now. >> >> (error/8.13)^2 + (complexity/30.01)^2 < 1 >> Where do these numbers come floating in from--why 30.01, and not just > 30, for instance?So schismic just squeaks in. The point of this cutoff is that it is the inner edge of a moat, i.e. there is no other temperament outside of it for quite a way. i.e. you can increase either or both of those divisors quite a bit (and also change the exponents) without including any more temperaments. That's what makes it a good cutoff. However, you may prefer to see the cutoff given as a curve that runs thru the _middle_ of the moat. That would be fine with me too.>> meantone 80:81 >> augmented 125:128 >> porcupine 243:250 >> diaschismic 2025:2048 >> diminished 625:648 >> magic 3072:3125 >> blackwood 243:256 >> kleismic 15552:15625 >> pelogic 128:135 >> 6561/6250 6250:6561 >> quartafifths (tetracot) 19683:20000 >> negri 16384:16875 >> 2187/2048 2048:2187 >> neutral thirds (dicot) 24:25 >> superpythag 19683:20480 >> schismic 32768:32805 >> 3125/2916 2916:3125 >> The only thing which might qualify as microtempering is schismic, > which I presume is the idea.And kleismic. The curve was designed to just barely include the last four on the list. I wouldn't mind if semisixths was included too, as Paul would like.> It looks OK at first glance, and could > even be shorted on the high-error side without upsetting me any.You mean like leaving out dicot and pelogic. This wouldn't upset me either, but I know it would upset Paul so they'd better stay.> By the way, if you use 81/80 instead of 80:81, you are not going to > be inconsistent with that other fellow who uses 81:80 for the exact > same ratio. You will aslo be specifying an actual number. Numbers are > nice. This whole obsession with colons makes me want to give the > topic a colostomy. Hee hee.I guess that makes you a slasher. :-)> I have read no justification for it which makes > any sense to me.Have you read this? A note on mathematical notation for musical in... * [with cont.] (Wayb.) If so, are there particular parts of it that make no sense to you? In software, the safe way to turn these colonic thingies into real numbers is always to divide the big one by the small one. real(a:b) = max(a,b)/min(a,b)

Message: 9770 - Contents - Hide Contents Date: Tue, 03 Feb 2004 01:23:09 Subject: Re: finding a moat in 7-limit commas a bit tougher . . . From: Dave Keenan --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:> --- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" <d.keenan@b...> > wrote:>> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> > wrote: >>> Yahoo groups: /tuning_files/files/Erlich/plana... * [with cont.] >> >> Paul, >> >> Please do another one of these without the labels, so we have a > chance>> of eyeballing the moats. >> My eyeballs are telling me the same thing as when the labels were > there:Maybe so, but it's also conceivable that one could be prejudiced by the slope on the labels, so it's better not to have them when moat-spotting. Although we still need the labelled one to refer to. Woah! 42 7-limit planars, when we only have 17 or 18 5-limit linears. No way. I can see several linear moats less inclusive than that one that are much wider (especially when the width is counted as the percentage of the distance to the origin). It would be good to draw two parallel lines showing the two sides of the moat (and even fill with colour between them) so we can easily see the moat width.

Message: 9771 - Contents - Hide Contents Date: Tue, 03 Feb 2004 23:01:08 Subject: Re: Duals to ems optimization From: Carl Lumma>For any set of consonances C we want to do an rms optimization for, we >can find a corresponding Euclidean norm on the val space (or >octave-excluding subspace if we are interested in the odd limit) by >taking the sum of terms > >(c2 x2 + c3 x3 + ... + cp xp)^2 > >for each monzo |c2 c3 ... cp> in C. If we want something corresponded >to weighted optimization we would add weights, and if we wanted the >odd limit, the consonances in C can be restricted to quotients of odd >integers, in which case c2 will always be zero. // >In the 11-limit and beyond, of course, things become more complicated >because we will want to introduce ratios of odd numbers which are not >necessarily primes. If we take ratios of odd numbers up to 11 for our >set of consonances, we get > >sqrt(20x3^2+5x5^2-2x7x11-6x3x5+5x7^2+5x11^2-6x3x7-2x5x7-2x5x11-6x3x11) > >as our norm on vals, and correspondingly, > >sqrt(18e3^2+36e3e5+36e3e7+36e3e11+62e5^2+58e5e7+58e5e11// > >as our norm on octave classes. This norm is not altogether >satisfactory; for instance it gives a length of sqrt(44) to 5/3 and >6/5, and a length of sqrt(62) to 5/4. This suggests to me that there >is something a little dubious in theory about using unweighted rms >optimization, at least in the 11 limit and beyond. An alternative rms >optimization scheme would be to use dual of the norm I've been using >on octave classes as the norm for a weighted rms optimization. > >In the 5-limit, this norm on octave classes is > >sqrt(p3e3^2 + p3e3e5 + p5e5^2) > >where p3 = log2(3), p5 = log2(5). The dual norm on vals is > >sqrt(p5x3^2 - p3x3x5 + p3x5^2) > >These norms will weigh lower prime errors a little higher than higher >prime errors, which of course is also what TOP does. Now I need a >catchy name for them.Have you checked that this weighted version is ok at the 11-limit (ie doesn't make 5/4 shorter than 5/3)? Also: If you leave in the c2 term does this optimize over "all the intervals" like TOP? It seems to me a comparison of... (1) TOP (2) rms-TOP (3) odd-limit TOP (4) rms odd-limit TOP ...has not been done. I'm happy to give up on unweighted TOPs. -Carl

Message: 9772 - Contents - Hide Contents Date: Tue, 03 Feb 2004 01:31:55 Subject: Re: finding a moat in 7-limit commas a bit tougher . . . 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:>>> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> >> wrote: >>>> Yahoo groups: /tuning_files/files/Erlich/plana... * [with cont.] >>> >>> Paul, >>> >>> Please do another one of these without the labels, so we have a >> chance>>> of eyeballing the moats. >>>> My eyeballs are telling me the same thing as when the labels were >> there: >> Maybe so, but it's also conceivable that one could be prejudiced by > the slope on the labels, so it's better not to have them when > moat-spotting. Although we still need the labelled one to refer to. > >> Yahoo groups: /tuning_files/files/Erlich/myemo... * [with cont.] > > Woah! 42 7-limit planars, when we only have 17 or 18 5-limit linears. > No way.I agree it's probably too large a number for the purposes of a paper, but I'm not sure there have to be fewer than the number of 5-limit linears. There is simply a greater variety in the 7-limit planar world, which works against the fact that we'd use a considerably lower bound on the most complex comma in 7-limit planar vs. 5-limit linear.

Message: 9773 - Contents - Hide Contents Date: Tue, 03 Feb 2004 01:39:14 Subject: TOP Equal Temperament graphs! (was: Re: Cross-check for TOP 5-limit 12-equal) From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:> Wedgie norm for 12-equal: > > Take the two unison vectors > > |7 0 -3> > |-4 4 -1> > > Now find the determinant, and the "area" it represents, in each of > the basis planes: > > |7 0| = 28*(e23) -> 28/lg2(5) = 12.059 > |-4 4| > > |7 -3| = -19*(e25) -> 19/lg2(3) = 11.988 > |-4 -1| > > |0 -3| = 12*(e35) -> 12 = 12 > |4 -1| > > sum = 36.047 > > If I just use the maximum (L_inf = 12.059) as a measure of notes per > acoustical octave, then I "predict" tempered octaves of 1194.1 cents. > If I use the sum (L_1), dividing by the "mystery constant" 3, > I "predict" tempered octaves of 1198.4 cents. Neither one is the TOP > value . . . :( . . . but what sorts of error criteria, if any, *do* > they optimize? > > So the cross-checking I found for the 3-limit case in "Attn: Gene 2" > Yahoo groups: /tuning-math/message/8799 * [with cont.] > doesn't seem to work in the 5-limit ET case for either the L_1 or > L_inf norms. > > However, if I just add the largest and smallest values above: > > 28/lg2(5)+19/lg2(3) > > I do predict the correct tempered octave (aside from a factor of 2), > > 1197.67406985219 cents. > > So what sort of norm, if any, did I use to calculate complexity this > time? It's related to how we temper for TOP . . .I used this latter complexity measure to create these graphs: Yahoo groups: /tuning_files/files/et3.gif * [with cont.] Yahoo groups: /tuning_files/files/et5.gif * [with cont.] Yahoo groups: /tuning_files/files/et7.gif * [with cont.] Yahoo groups: /tuning_files/files/et11.gif * [with cont.]

Message: 9774 - Contents - Hide Contents Date: Tue, 03 Feb 2004 23:11:52 Subject: Re: Duals to ems optimization From: Carl Lumma>Also: If you leave in the c2 term does this optimize over "all the >intervals" like TOP?Or do you just get integer limit? -Carl

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