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Message: 9775 - Contents - Hide Contents Date: Tue, 03 Feb 2004 01:48:27 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:>>> --- 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.What range of exponents are acceptable to you? Isn't 1 near the (geometric) middle of them?> >> I suggest >> 50/49 >> 49/48 >> 64/63 >> 81/80 >> 126/125 >> 225/224 >> 245/243 >> 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? 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) and go all the way up to include 21/20 (which seems lidicrous to me).
Message: 9776 - Contents - Hide Contents Date: Tue, 03 Feb 2004 01:52:00 Subject: Re: Back to the 5-limit cutoff From: Dave Keenan --- In tuning-math@xxxxxxxxxxx.xxxx "David Bowen" <dmb0317@f...> wrote:> Sorry for my delay in entering this discussion, but I'm a Digest subscriber. ...Hi David. It's good to hear from someone other than the usual suspects. Thanks.
Message: 9777 - Contents - Hide Contents Date: Tue, 03 Feb 2004 01:58:33 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:>> 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 . . . Seriously, I think all kinds of novel effects could be obtained if 21/20 vanished, 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 . . .
Message: 9778 - Contents - Hide Contents Date: Tue, 03 Feb 2004 03:31:10 Subject: TOP Equal Temperament graphs! (was: Re: Cross-check for TOP 5-limit 12-equal) From: Dave Keenan --- 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:Thanks for doing these Paul. 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. 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. 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. That's another dimension of usefulness that we're not considering -- 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. 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). I'm guessing Gene would be happy with that too, since it looks more like a log-flat badness cutoff with additional cutoffs on error and complexity.
Message: 9779 - Contents - Hide Contents Date: Tue, 03 Feb 2004 04:44:10 Subject: Re: Back to the 5-limit cutoff From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:> There's a history in the literature of using ratios to notate pitches.I'm aware of that. Pitches are specified by numbers. Ratios between pitches are specified by numbers. This sort of thing typical of numbers in any application, and we are also used to seeing logariths specified by numbers, and in fact a whole hell of a lot of things specified by numbers. Why not simply treat numbers as if they were numbers?> 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.It makes just as much sense that way as the other way.
Message: 9780 - Contents - Hide Contents Date: Tue, 03 Feb 2004 04:45:01 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: >>>> 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. 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. Otherwise we have a reductio ad absurdum. By the way, the TOP tuning of the 21/20 planar temperament has the following errors in the primes (to the nearest cent). 2 +10 c 3 -15 c 5 +23 c 7 -27 c So we have the following large errors in certain intervals 2:3 -25 c 7:10 +60 c 3:4 +35 c 5:7 -50 c 3:5 +38 c 4:7 -47 c The approximations of 3:4 and 5:7 are the same interval, so are the approximations of 3:5 and 4:7, and 2:3 is the same as 7:10.
Message: 9781 - Contents - Hide Contents Date: Tue, 03 Feb 2004 04:46:33 Subject: Re: Back to the 5-limit cutoff From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:> --- 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?4/5/6 = (4/5)/6 = 4/30 = 2/15, so no. I write it 1--5/4--3/2 mostly, but I have no objection colons for chords, where it serves a purpose.
Message: 9782 - Contents - Hide Contents Date: Tue, 03 Feb 2004 04:57:56 Subject: Re: Back to the 5-limit cutoff From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" <d.keenan@b...> wrote:>> The only thing which might qualify as microtempering is schismic, >> which I presume is the idea. > > And kleismic.Not in my book, but whatever.> 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. (c)is a reason why colons are inferior. (d) is hilarious. There is no canonical order, which is one of the difficulties this this business.> > 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.
Message: 9783 - Contents - Hide Contents Date: Wed, 04 Feb 2004 09:08:14 Subject: TOP Equal Temperament graphs! (was: Re: Cross-check for TOP 5-limit 12-equal) From: Dave Keenan --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:> --- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" <d.keenan@b...> > wrote: > So 12-equal makes it in for you at 7 but not at 11?12-equal doesn't really make it in as 7-limit for me personally, but I was trying to keep you happy too.>> Looking at these has disposed me more towards linear moats and less >> towards quadratic ones, >> So it worked!Yes. And tomorrow someone might come up with something to convince me of something different. :-)
Message: 9784 - Contents - Hide Contents Date: Wed, 04 Feb 2004 21:38:36 Subject: Re: Duals to ems optimization From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote: >>> It's not surprising given how Gene set it up: with the same weighting >> that gives equilateral triangles and tetrahedra in the 5-limit and 7- >> limit lattices . . . >> Actually, isosceles triangles.Huh? You said this was the *unweighted* optimization scheme! Also, you never followed up on the loose ends of this thread, including the post "Attn: Gene" (not 2).> The fifth gets a length of log(3) (or > cents(3) or whatever log you are using) and the major and minor thirds > have the same length, log(5).You're seem to be contradicting yourself now -- in the post you're replying to, it said (your writing): ||5/4|| = ||7/4|| = ||11/8|| = sqrt(11).
Message: 9785 - Contents - Hide Contents Date: Wed, 04 Feb 2004 09:16:27 Subject: Paul's 32 again From: Gene Ward Smith Here it is, sorted by a hyperbolic cutoff function in the log(err)-log(complexity) plane. This was fudged so as to put ennealimmal at the bottom of the list. The cuttoff would be some value of (15 - ln(TOP error)-15)(40 - ln(L1 TOP complexity)). I could try to fudge father instead, or eqifudge them. [1, 4, 10, 4, 13, 12] [7, 9, 13, -2, 1, 5] [2, -4, -4, -11, -12, 2] [5, 1, 12, -10, 5, 25] [3, 0, -6, -7, -18, -14] [4, -3, 2, -14, -8, 13] [1, 9, -2, 12, -6, -30] [6, 5, 3, -6, -12, -7] [2, 8, 1, 8, -4, -20] [2, 8, 8, 8, 7, -4] [1, 4, -2, 4, -6, -16] [7, -3, 8, -21, -7, 27] [1, -8, -14, -15, -25, -10] [6, -7, -2, -25, -20, 15] [10, 9, 7, -9, -17, -9] [4, 4, 4, -3, -5, -2] [3, 0, 6, -7, 1, 14] [9, 5, -3, -13, -30, -21] [6, 5, 22, -6, 18, 37] [0, 5, 0, 8, 0, -14] [4, 2, 2, -6, -8, -1] [2, 1, 6, -3, 4, 11] [0, 0, 7, 0, 11, 16] [16, 2, 5, -34, -37, 6] [1, -3, -4, -7, -9, -1] [2, 25, 13, 35, 15, -40] [5, 13, -17, 9, -41, -76] [1, -1, 3, -4, 2, 10] [2, 3, 1, 0, -4, -6] [1, 4, 3, 4, 2, -4] [13, 14, 35, -8, 19, 42] [18, 27, 18, 1, -22, -34]
Message: 9786 - Contents - Hide Contents Date: Wed, 04 Feb 2004 21:40:27 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:>>> 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 . . . =( >> It was just a suggestion. I wrote "perhaps we should" and "e.g.". > > What does "=(" mean? > > I'm guessing you think it's a bad idea.It's a picture of me succumbing to your authority.
Message: 9787 - Contents - Hide Contents Date: Wed, 04 Feb 2004 01:26:22 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.Looks like a 'constellation' -- with 12 stars :( :)> 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.I also would have liked to see 43 and 50, but I suppose these are just 'footnoats' . . .> 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.Hmm . . . I see a rivulet, not a moat . . . 58 has gotten a lot more attention than 39 . . . but I was actually referring indirectly to how, in these graphs, the density of temperaments is not constant along each equicomplexity line, as in the comma graphs . . .> 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.So 12-equal makes it in for you at 7 but not at 11?> Looking at these has disposed me more towards linear moats and less > towards quadratic ones,So it worked!
Message: 9788 - Contents - Hide Contents
Date: Wed, 04 Feb 2004 09:42:44
Subject: Convex hull of Paul's 32 in log(err)log(comp) plane
From: Gene Ward Smith
Here's the convex hull; one way to do a cutoff is simply to use it for
that purpose; but we could find a smooth function which worked like it.
Beep
[2, 3, 1, 0, -4, -6] [4.729523732, 14.17610523]
{21/20, 28/27}
[1, 4, 3, 4, 2, -4] [5.572287934, 14.25364200]
Blackwood
[0, 5, 0, 8, 0, -14] [6.474937224, 7.239629324]
Dominant seventh
[1, 4, -2, 4, -6, -16] [7.955968622, 4.771049000]
Meantone
[1, 4, 10, 4, 13, 12] [11.76517777, 1.698521000]
Miracle
[6, -7, -2, -25, -20, 15] [21.10188070, .6310140000]
Hemiwuerschmidt
[16, 2, 5, -34, -37, 6] [31.21187526, .3079970486]
Parakleismic
[13, 14, 35, -8, 19, 42] [39.58593953, .2619341054]
Ennealimmal
[18, 27, 18, 1, -22, -34] [39.82871923, .3637700000e-1]
Message: 9789 - Contents - Hide Contents Date: Wed, 04 Feb 2004 21:50:11 Subject: Re: Paul's 32 again From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:> Here it is,I put this list together in a fairly absurd manner from the information you gave, because it was the best I could do. I appreciate your great efforts to help out, but the most valuable thing you could do in this case would be to come up with a new list. The reason Dave and I wanted it in single-line format was so that we could graph it and make decisions based on it. Starting with the same 32 defeats the whole purpose, I'm afraid.
Message: 9790 - Contents - Hide Contents Date: Wed, 04 Feb 2004 05:13:50 Subject: Acceptance regions From: Gene Ward Smith For a linear temperament, let e be the absolute value of the error and c the complexity. Then the log-flat badness is e c^(n/(n-2)), where n is the number of primes (with 2 being the number of generators; in general it becomes e c^(n/(n-g))). If we set x = log(e), y = log(c) then bounds on error, complexity and badness become x <= a, y <= b, y+(n/(n-2))x <= c This defines a triangular region in the xy plane. We could define a region with smooth boundry instead, in particular an ellipse. If we took a set of temperaments we wanted on our list, and analyzed them statistically, we might have an idea of what region we are looking for. One way might be to do principle component analysis, and convert the data set into something we can draw a nice circle around. All of which leads to the quesiton, which temperaments do we start out with?
Message: 9791 - Contents - Hide Contents Date: Wed, 04 Feb 2004 09:59:16 Subject: Re: Convex hull of Paul's 32 in log(err)log(comp) plane From: Gene Ward Smith Sorry, that was was in the err-comp plane, not the logs. That's interesting also, but a whole other story. Hemiforths [2, 8, 1, 8, -4, -20] [2.416311999, 1.299876081] Injera [2, 8, 8, 8, 7, -4] [2.478014205, 1.276118547] Superpythagorean [1, 9, -2, 12, -6, -30] [2.669372146, .8770838286] Magic [5, 1, 12, -10, 5, 25] [2.743162410, .2443130871] Catakleismic [6, 5, 22, -6, 18, 37] [3.223959018, -.6229571593] Amity [5, 13, -17, 9, -41, -76] [3.640951092, -1.286970429]
Message: 9792 - Contents - Hide Contents Date: Wed, 04 Feb 2004 22:04:07 Subject: Re: Paul32 ordered by a beep-ennealimmal measure From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:> Here are the same temperaments, ordered by error * complexity^(2.8). > If the exponent was 2.7996... then ennealimma and beep would be the > same, but why get fancy? I'm thinking an error cutoff of 15 and a > badness cutoff of 4200 might work, looking at this; that would include > schismic. More ruthlessly, we might try 3500. Really savage would be > 3000; bye-bye miracle.Strange; miracle was #3 according to log-flat, wasn't it?
Message: 9793 - Contents - Hide Contents Date: Wed, 04 Feb 2004 05:22:31 Subject: Re: Comma reduction? From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:>> Are the 2 commas in the 7-limit always linearly independent? >> Yes, they are never 'collinear'.By definition of a 7-limit linear temperament.>> 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.Trees don't work for me. You can get them from direct comma searches or extract them out of temperaments, etc.>> 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?The complement of a 2-val is a 2-monzo, and vice-versa, which just involves reordering. ~<<||l1 l2 l3 l4 l5 l6|| = <<l6 -l5 l4 l3 -l2 l1||
Message: 9794 - Contents - Hide Contents
Date: Wed, 04 Feb 2004 11:08:11
Subject: Convex hull yet again
From: Gene Ward Smith
For some reason it didn't come out right; this seems to be correct. It
gives us a compact poygonal region, we can use the top border as a
badness cutoff and the bottom one (in case anyone thinks goodness
should be punished) as a goodness cutoff. The goodness line runs from
Beep to Ennealimmal, and an alternative approach is simply to take a
line with the same slope and use it as a badness cutoff. No specific
error and complexity cutoffs would be used. I think I'll see if that
works. This would probably end up making Beep and Ennealimmal come out
at the top of the list, if people can stand that idea, but we shall
see. The bad line runs up from Ennealimmal (in error) to Parakleismic,
tnen to Valentine, Pelogic, the {21/20, 28/27} temperament and then
around to Beep. This suggests drawing a line between what you think
are the best two temperaments in the log(error) log(complexity) plane
and using a line with that slope as a badness cutoff might work.
Beep
[2, 3, 1, 0, -4, -6] [1.55382450640435, 2.65155778136775]
Ennealimmal
[18, 27, 18, 1, -22, -34] [3.68458824092559, -3.31382430859740]
Parakleismic
[13, 14, 35, -8, 19, 42] [3.67847399261317, -1.33966326804887]
Valentine
[9, 5, -3, -13, -30, -21] [3.10891281481728, .485909326707073e-1]
Pelogic
[1, -3, -4, -7, -9, -1] [2.00498200415831, 2.27563066979103]
{21/20, 28/27}
[1, 4, 3, 4, 2, -4] [1.71780572949088, 2.65701243004446]
Message: 9795 - Contents - Hide Contents
Date: Wed, 04 Feb 2004 22:02:51
Subject: Some convex hull badness measures
From: Gene Ward Smith
I took all pairs n,m<=100 of integers and calculated the corresponding
wedgie; excluding [0,0,0,0,0,0] gave me 2280 wedgies. The convex hull
of this in the log(comp)-log(error) plane has 16 verticies. If I
exclude the slopes greater than -2 (the logflat slope) and those going
between the "bad" verticies, I am left with two possibilities:
Line from the {9/8, 15/14} temperament to the {10/9, 16/15}
temperament, with slope -5.99, and the line from {10/9, 16/15} to
ennealimmal, with slope -2.69; the first something which would make
sense only if you were interested in high-error, low-complexity.
If I take only temperaments with error less than that of {9/8, 15/14}
and redo the proceedure, I get a line from {15/14, 25/24} to
ennealimmal with a slope of -2.79. If I now boot {15/14, 25/24}, I get
the beep-ennealimmal line, with slope -2.88. Booting beep gives me a
blackwood-ennealimmal line with slope -2.91. If I boot blackwood, I
get a more complicated situation again; I have the unusuably large
slope of
-42.23 from augmented to dominant seventh, and then a line of slope
-3.03 running from dominant seventh to ennealimmal.
I think this suggests it is better to treat ennealimmal as a friend
than an enemy; rather than trying to cook up a way to exclude it, it
seems to work to make its very strong goodness a basis for seeing what
kinds of badness functions make sense. One question is do we want to
keep beep, or exclude it? What about father?
Message: 9796 - Contents - Hide Contents Date: Wed, 04 Feb 2004 05:31:09 Subject: Re: Back to the 5-limit cutoff From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:> 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.)What I said was that odd limit always should mean a means consonance, so a "9-odd-limit interval" should always mean, and does always mean when I use it, a 9-odd-limit consonance; and saying "9-odd-limit" means intervals, chords, etc. are based on such consonances. Do you have a problem with that?
Message: 9797 - Contents - Hide Contents Date: Wed, 04 Feb 2004 11:35:29 Subject: Paul32 ordered by a beep-ennealimmal measure From: Gene Ward Smith Here are the same temperaments, ordered by error * complexity^(2.8). If the exponent was 2.7996... then ennealimma and beep would be the same, but why get fancy? I'm thinking an error cutoff of 15 and a badness cutoff of 4200 might work, looking at this; that would include schismic. More ruthlessly, we might try 3500. Really savage would be 3000; bye-bye miracle. Could anyone stand having ennealimmal but not septimal miracle? [2, 3, 1, 0, -4, -6] 1099.12138564414 [18, 27, 18, 1, -22, -34] 1099.95303893104 [0, 5, 0, 8, 0, -14] 1352.62028295398 [1, -1, 3, -4, 2, 10] 1429.37604841752 [1, 4, -2, 4, -6, -16] 1586.91771200691 [1, 4, 10, 4, 13, 12] 1689.45478545707 [1, 4, 3, 4, 2, -4] 1749.12068096674 [4, 4, 4, -3, -5, -2] 1926.26540601954 [2, -4, -4, -11, -12, 2] 2188.88089121908 [3, 0, 6, -7, 1, 14] 2201.89086653167 [4, 2, 2, -6, -8, -1] 2306.67858004561 [2, 1, 6, -3, 4, 11] 2392.13957455646 [0, 0, 7, 0, 11, 16] 2580.68840117061 [1, -3, -4, -7, -9, -1] 2669.32332096936 [5, 1, 12, -10, 5, 25] 2766.02839153075 [7, 9, 13, -2, 1, 5] 2852.99148620301 [3, 0, -6, -7, -18, -14] 3181.79058955503 [2, 8, 1, 8, -4, -20] 3182.90486667288 [6, -7, -2, -25, -20, 15] 3222.09329766576 [4, -3, 2, -14, -8, 13] 3448.99841914489 [6, 5, 3, -6, -12, -7] 3680.09543007275 [2, 8, 8, 8, 7, -4] 3694.34379334901 [7, -3, 8, -21, -7, 27] 4145.42670717715 [1, -8, -14, -15, -25, -10] 4177.54976222602 [1, 9, -2, 12, -6, -30] 4235.79245013082 [6, 5, 22, -6, 18, 37] 4465.45695002232 [16, 2, 5, -34, -37, 6] 4705.89445909217 [10, 9, 7, -9, -17, -9] 5386.21535996825 [9, 5, -3, -13, -30, -21] 6333.11068687830 [2, 25, 13, 35, 15, -40] 6657.51406862505 [5, 13, -17, 9, -41, -76] 7388.58309989111 [13, 14, 35, -8, 19, 42] 7785.85505014731
Message: 9798 - Contents - Hide Contents Date: Wed, 04 Feb 2004 22:05:38 Subject: Re: Comma reduction? From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Paul G Hjelmstad" <paul.hjelmstad@u...> wrote:> Thanks. Are they called 2-val and 2-monzo because they are "linear" > or is there some other reason?2-vals are two vals wedged, 2-monzos are two monzos wedged. The former is linear unless it reduces to the zero wedgie, the latter is linear only in the 7-limit.
Message: 9799 - Contents - Hide Contents Date: Wed, 04 Feb 2004 05:26:43 Subject: Re: Comma reduction? From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote: >>>> Are the 2 commas in the 7-limit always linearly independent? >>>> Yes, they are never 'collinear'. >> By definition of a 7-limit linear temperament. > >>> 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. >> Trees don't work for me. You can get them from direct comma searches > or extract them out of temperaments, etc. >>>> 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? >> The complement of a 2-val is a 2-monzo, and vice-versa, which just > involves reordering. > > ~<<||l1 l2 l3 l4 l5 l6|| = <<l6 -l5 l4 l3 -l2 l1||Shouldn't that be ~||l1 l2 l3 l4 l5 l6>> = <<l6 -l5 l4 l3 -l2 l1|| or something?
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