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Message: 4876 - Contents - Hide Contents Date: Fri, 24 May 2002 12:16 +0 Subject: Re: definitions of period, equivalence, etc. (was: Re: graham's line From: graham@xxxxxxxxxx.xx.xx In-Reply-To: <acjitt+tn6m@xxxxxxx.xxx> Gene:>> I'm not sure what the point of it all is. If you leave off octaves >> and just deal with pitch classes, you need to put the octave >> information back into the mix in one way or another. Paul:> this seems to be what graham is questioning.You need to put the octaves in eventually, before you hear actual music. So you can make a credible case for what Gene said there making sense. Existing mathematical theory, however, all seems to be within an octave-equivalent system. We don't have to change that, although it does make some things clearer to put the octaves in. Which would be why Gene and me both independently did that.>> Why not just leave it in? >> graham, that's your cue.What, the bit about nobody saying they care, but the subject keeping on coming up anyway? As this thread was dormant for so long, I thought nobody cared after all. Graham
Message: 4878 - Contents - Hide Contents Date: Fri, 24 May 2002 11:15:55 Subject: Re: definitions of period, equivalence, etc. From: monz> From: <graham@xxxxxxxxxx.xx.xx> > To: <tuning-math@xxxxxxxxxxx.xxx> > Sent: Friday, May 24, 2002 4:16 AM > Subject: [tuning-math] Re: definitions of period, equivalence, etc. > > > What, the bit about nobody saying they care, but the subject keeping on > coming up anyway? As this thread was dormant for so long, I thought > nobody cared after all.it was me who brought it back up again, simply because i'm still not entirely clear on the subtle distinctions. but i haven't been following any of my internet lists carefully for a couple of months now, so i sort of just popped in and brought it up again. (even tho i take a peek everyday, i'm still on semi-hiatus doing "real life" for awhile.) -monz
Message: 4879 - Contents - Hide Contents Date: Fri, 24 May 2002 20:01:12 Subject: Re: latest generalized diatonic review From: Carl Lumma>I should have written [1,25/24,6/5,5/4,36/25,3/2,5/3,9/5], the 46-et >version of which is [0, 3, 12, 15, 24, 27, 34, 39]. However, the >alternative with second degree being approximately 27/25 is very much >worthy of notice also--like star, it is a 126/125-tempered version of >a Fokker block, consisting of two parallel chains of minor thirds, >with a lot of nice harmonic properties. Being a new star, maybe >it's "nova" :)Just for my sanity, fill in the blanks: star = 0 _ _ _ _ _ _ _ 46 nova = 0 _ _ _ _ _ _ _ 46 Thanks! -Carl
Message: 4880 - Contents - Hide Contents Date: Fri, 24 May 2002 22:17:57 Subject: Re: latest generalized diatonic review From: Carl Lumma>> >tar = 0 3 12 15 24 27 34 39 46 >> nova = 0 5 12 15 24 27 34 39 46Thanks, Gene. So what I've been calling star is indeed star. I've added nova to the search. It comes out just above star because its stability is higher while all its other values are the same. It also has one more 3:2 in it (vs. star's extra 8:5). I'll update the search on my site as warranted by the number of new scales. -Carl
Message: 4881 - Contents - Hide Contents Date: Sat, 25 May 2002 15:20:58 Subject: Re: latest generalized diatonic review From: Carl Lumma>> >'ll update the search on my site as warranted by the number of new >> scales. >>Did you see my recent posting to the main list? I'd be interested in >your assessment of Qm(2) and Qm(3).Oh, I guess Qm(2) is possible. I'll have to make it up. It'd be great if you could post Scala files for these. Qm(3) is a mode of this scale:>10-tone Fokker-Lumma, e=27 c=5, in 72-tET >(0 5 12 19 28 35 42 49 58 65) -> ((32 $ 39 % rms) (20 $ 24 % mad))Which brings up an important point: who's keeping track of these discoveries? Perhaps we should begin a database, with keys for both interval and rank-order matrices. For example, Qm(3) interval matrix: ((7 14 21 30 37 44 49 56 63 72) (7 14 23 30 37 42 49 56 65 72) (7 16 23 30 35 42 49 58 65 72) (9 16 23 28 35 42 51 58 65 72) (7 14 19 26 33 42 49 56 63 72) (7 12 19 26 35 42 49 56 65 72) (5 12 19 28 35 42 49 58 65 72) (7 14 23 30 37 44 53 60 67 72) (7 16 23 30 37 46 53 60 65 72) (9 16 23 30 39 46 53 58 65 72)) Qm(3) rank-order matrix: ((2 5 8 12 15 18 20 23 26 29) (2 5 9 12 15 17 20 23 27 29) (2 6 9 12 14 17 20 24 27 29) (3 6 9 11 14 17 21 24 27 29) (2 5 7 10 13 17 20 23 26 29) (2 4 7 10 14 17 20 23 27 29) (1 4 7 11 14 17 20 24 27 29) (2 5 9 12 15 18 22 25 28 29) (2 6 9 12 15 19 22 25 27 29) (3 6 9 12 16 19 22 24 27 29)) Manuel, () What's the best way to get Scala to represent scales as degrees of an et? () As far as inputting scales as et subsets, I do "equal n" and then "select". Is that the Official Way? () Think we could get View -> rank-order matrix? (Yes, I'm using 2.05 now). -Carl
Message: 4882 - Contents - Hide Contents Date: Sat, 25 May 2002 23:53:58 Subject: Re: 7-limit temperament constraints From: dkeenanuqnetau --- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:> Does anyone want to propose any guidelines for 7-limit temperamentsearches? What must be included, and what should not be? Yes. If you provide me with a wide-open list (using only your badness cutoff) of maybe 40 or 50 temperaments, I'll try to find some points where my gradual cutoffs agree with some sharp ones. It would be easiest for me if your list was in a form easily imported into Excel, i.e. everything relating to a single temperament being on a single line, with no brackets. Regards,
Message: 4883 - Contents - Hide Contents Date: Sat, 25 May 2002 04:27:30 Subject: definitions of period, equivalence, etc. (was: Re: graham's line From: genewardsmith --- In tuning-math@y..., graham@m... wrote:> You need to put the octaves in eventually, before you hear actual music. > So you can make a credible case for what Gene said there making sense.By "putting the octaves back in" I meant you must do this to define the temperament. If you define a temperament by the octave-equivalent mapping [0,1,4], how do you know it isn't the 160/81 temperament, and not the 81/80? You need explicitly or implicitly to do a further calculation, using real arithmetic and not just algebra, to make this system workable. It seems obviously better to me to simply define the temperament unambiguously in the first place.> Existing mathematical theory, however, all seems to be within an > octave-equivalent system.This makes no sense at all to me.
Message: 4884 - Contents - Hide Contents Date: Sat, 25 May 2002 04:30:32 Subject: Re: latest generalized diatonic review From: genewardsmith --- In tuning-math@y..., Carl Lumma <carl@l...> wrote:> Just for my sanity, fill in the blanks: > > star = 0 3 12 15 24 27 34 39 46 > nova = 0 5 12 15 24 27 34 39 46
Message: 4885 - Contents - Hide Contents Date: Sat, 25 May 2002 07:18:52 Subject: Re: latest generalized diatonic review From: genewardsmith --- In tuning-math@y..., Carl Lumma <carl@l...> wrote:> I'll update the search on my site as warranted by the number of new > scales.Did you see my recent posting to the main list? I'd be interested in your assessment of Qm(2) and Qm(3).
Message: 4886 - Contents - Hide Contents Date: Sat, 25 May 2002 02:25:38 Subject: Re: latest generalized diatonic review From: Carl Lumma>> >'ll update the search on my site as warranted by the number of new >> scales. >>Did you see my recent posting to the main list? I'd be interested in your >assessment of Qm(2) and Qm(3).I get the main list and Columbia in digest format, so it may be another day until I see your post. -Carl
Message: 4887 - Contents - Hide Contents Date: Sat, 25 May 2002 10:44 +0 Subject: Re: definitions of period, equivalence, etc. (was: Re: graham's line From: graham@xxxxxxxxxx.xx.xx genewardsmith wrote:> By "putting the octaves back in" I meant you must do this to define the > temperament. If you define a temperament by the octave-equivalent > mapping [0,1,4], how do you know it isn't the 160/81 temperament, and > not the 81/80? You need explicitly or implicitly to do a further > calculation, using real arithmetic and not just algebra, to make this > system workable. It seems obviously better to me to simply define the > temperament unambiguously in the first place.The octave equivalent mapping is [1, 4] not [0, 1, 4]. Both 160/81 and 81/80 are tempered out. What difference does it make? How do you know if it's 81/80 or 80/81 tempered out in an octave-specific meantone?>> Existing mathematical theory, however, all seems to be within an >> octave-equivalent system. >> This makes no sense at all to me.I forgot that Karp left the octaves in. But Fokker and Rothenberg are all octave-equivalent. Graham
Message: 4888 - Contents - Hide Contents Date: Sat, 25 May 2002 11:56:29 Subject: 7-limit temperament constraints From: genewardsmith Does anyone want to propose any guidelines for 7-limit temperament searches? What must be included, and what should not be?
Message: 4889 - Contents - Hide Contents Date: Sat, 25 May 2002 12:10:23 Subject: definitions of period, equivalence, etc. (was: Re: graham's line From: genewardsmith --- In tuning-math@y..., graham@m... wrote:> genewardsmith wrote: > The octave equivalent mapping is [1, 4] not [0, 1, 4].It's [0, 1, 4] after you put the octave info in. Both 160/81 and> 81/80 are tempered out. What difference does it make? How do you know if > it's 81/80 or 80/81 tempered out in an octave-specific meantone?You don't--they are the same. However, 160/81 defines a completely different, and very bad, temperament; you propose to ignore it, which certainly makes sense, but you still need octave information even to see it is 81/80, and not 160/81, we are looking at. You can do some of the math on pitch classes, but then you can't tell the difference between a small interval and a large one, which is important for commas, obviously.>>> Existing mathematical theory, however, all seems to be within an >>> octave-equivalent system. >>>> This makes no sense at all to me. >> I forgot that Karp left the octaves in. But Fokker and Rothenberg are all > octave-equivalent.Does it matter?
Message: 4890 - Contents - Hide Contents Date: Sat, 25 May 2002 13:48 +0 Subject: Re: definitions of period, equivalence, etc. (was: Re: graham's line From: graham@xxxxxxxxxx.xx.xx Me:>> The octave equivalent mapping is [1, 4] not [0, 1, 4]. Gene:> It's [0, 1, 4] after you put the octave info in.In which case it isn't octave equivalent, is it? Me:> Both 160/81 and>> 81/80 are tempered out. What difference does it make? How do you >> know if it's 81/80 or 80/81 tempered out in an octave-specific >> meantone? Gene:> You don't--they are the same. However, 160/81 defines a completely > different, and very bad, temperament; you propose to ignore it, which > certainly makes sense, but you still need octave information even to > see it is 81/80, and not 160/81, we are looking at. You can do some of > the math on pitch classes, but then you can't tell the difference > between a small interval and a large one, which is important for > commas, obviously.[1, 4] uniquely defines an octave-equivalent temperament. Give an example of this "completely different" temperament you say exists. You can tell the difference between a large and small interval if you take all intervals modulo the octave, as Dave suggested at the start of this thread.> Does it matter?No. Nobody cares. But we keep arguing about it anyway. Graham
Message: 4891 - Contents - Hide Contents Date: Sun, 26 May 2002 00:58:15 Subject: Re: latest generalized diatonic review From: genewardsmith --- In tuning-math@y..., Carl Lumma <carl@l...> wrote:>>> I'll update the search on my site as warranted by the number of new >>> scales. >>>> Did you see my recent posting to the main list? I'd be interested in >> your assessment of Qm(2) and Qm(3). >> Oh, I guess Qm(2) is possible. I'll have to make it up. It'd be > great if you could post Scala files for these.Here are my own personal Scala files for them: ! qm2.scl ! [0, 7, 23, 30, 42, 49, 65] Qm(2) 7-note quasi-miracle scale 7 ! 116.6666667 383.3333333 500. 700. 816.6666667 1083.333333 2/1 ! qm3a.scl ! [0, 7, 16, 23, 30, 35, 42, 49, 58, 65] Qm(3) 10-note quasi-miracle scale, mode A 10 ! 116.6666667 266.6666667 383.3333333 500. 583.3333333 700. 816.6666667 966.6666667 1083.333333 2/1 ! qm3b.scl ! [0, 7, 14, 23, 30, 37, 42, 49, 56, 65] Qm(3) 10-note quasi-miracle scale, mode B 10 ! 116.6666667 233.3333333 383.3333333 500. 616.6666667 700. 816.6666667 933.3333333 1083.333333 2/1> Qm(3) is a mode of this scale: >>> 10-tone Fokker-Lumma, e=27 c=5, in 72-tET >> (0 5 12 19 28 35 42 49 58 65) -> ((32 $ 39 % rms) (20 $ 24 % mad)) >> Which brings up an important point: who's keeping track of these > discoveries?Nobody, I think. Where is the Fokker-Lumma 2-parameter family described?
Message: 4892 - Contents - Hide Contents Date: Sun, 26 May 2002 01:09:12 Subject: Re: 7-limit temperament constraints From: genewardsmith --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:> It would be easiest for me if your list was in a form easily imported > into Excel, i.e. everything relating to a single temperament being on > a single line, with no brackets.I use brackets for matricies and wedgies--are you saying use parens instead? What should be on this single line?
Message: 4893 - Contents - Hide Contents Date: Sun, 26 May 2002 00:47:32 Subject: Re: latest generalized diatonic review From: Carl Lumma>Here are my own personal Scala files for them: Thanks.>> Qm(3) is a mode of this scale: >>>>> 10-tone Fokker-Lumma, e=27 c=5, in 72-tET >>> (0 5 12 19 28 35 42 49 58 65) -> ((32 $ 39 % rms) (20 $ 24 % mad)) >>>> Which brings up an important point: who's keeping track of these >> discoveries? >>Nobody, I think. Where is the Fokker-Lumma 2-parameter family described?Don't know what you mean by 2-parameter. AFAIK Fokker-Lumma refers to any scale with a 225:224 in the map. -Carl
Message: 4894 - Contents - Hide Contents Date: Sun, 26 May 2002 11:53:04 Subject: definitions of period, equivalence, etc. (was: Re: graham's line From: genewardsmith --- In tuning-math@y..., graham@m... wrote:> [1, 4] uniquely defines an octave-equivalent temperament.What's your definition of an octave-equivalent temperament? Give an example> of this "completely different" temperament you say exists.You simply stick 160/81 into the usual machine and turn the crank: Mapping [ 1 0] [ 0 1] [-5 4] rms = 231 g = 2.94 badness = 5897 generator = 2219 cents You can tell> the difference between a large and small interval if you take all > intervals modulo the octave, as Dave suggested at the start of this > thread.That fails to distinguish 160/81 (large) from 80/81 (small.) You need a standard reduction; but then you are back to my point--you are using real arithmetic to get back the information you first left out, hardly a sensible proceedure.
Message: 4895 - Contents - Hide Contents Date: Sun, 26 May 2002 21:24 +0 Subject: Re: definitions of period, equivalence, etc. (was: Re: graham's line From: graham@xxxxxxxxxx.xx.xx genewardsmith wrote:> What's your definition of an octave-equivalent temperament?A temperament where notes separated by an octave are considered equivalent.> You simply stick 160/81 into the usual machine and turn the crank: > > Mapping > > [ 1 0] > [ 0 1] > [-5 4] > > rms = 231 g = 2.94 badness = 5897 generator = 2219 centsBut there you're giving an octave-specific definition of a temperament you say won't work in octave-equivalent space. Obviously an octave equivalent system can't do octave-specific things, but it works fine on its own terms.> You can tell>> the difference between a large and small interval if you take all >> intervals modulo the octave, as Dave suggested at the start of this >> thread. >> That fails to distinguish 160/81 (large) from 80/81 (small.) You need a > standard reduction; but then you are back to my point--you are using > real arithmetic to get back the information you first left out, hardly > a sensible proceedure.Yes, it fails to distinguish them. That's because they're the same. We told it we wanted octave equivalence, so we can hardly be surprised if that's what we get. So what's the problem? Graham
Message: 4896 - Contents - Hide Contents Date: Mon, 27 May 2002 03:57:04 Subject: definitions of period, equivalence, etc. (was: Re: graham's line From: genewardsmith --- In tuning-math@y..., graham@m... wrote:> genewardsmith wrote:>> What's your definition of an octave-equivalent temperament?> A temperament where notes separated by an octave are considered > equivalent.That seems to be saying the tone group is a circle group, R/1200R if we use cents. This means that the group is not ordered and its image under the log map does not embed into a field, both of which don't help you. On the plus side, it is a topological group with an invariant metric, which gives us a notion of closeness. I'd say from a mathematician's point of view, having made things harder in this way, we would want a payoff of some kind.> But there you're giving an octave-specific definition of a temperament you > say won't work in octave-equivalent space.I give octave-specific definitions of everything; you are the one saying it might be better not to. Obviously an octave equivalent> system can't do octave-specific things, but it works fine on its own > terms.It works when it works? It seems to me it works because you can lift it to octave-specific in cases of practical interest. Why bother to do the heavy lifting? What's the payoff?
Message: 4897 - Contents - Hide Contents Date: Mon, 27 May 2002 07:26:03 Subject: Re: 7-limit temperament constraints From: dkeenanuqnetau --- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:> --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote: >>> It would be easiest for me if your list was in a form easily imported >> into Excel, i.e. everything relating to a single temperament being on >> a single line, with no brackets. >> I use brackets for matricies and wedgies--are you saying use parens instead?No. I meant use nothing of that kind, although it's not a big deal if you do. Doesn't matter much if you want to have commas too. The main thing is to have one temperament per line.> What should be on this single line? At least:Generators in 1:2 Generators in 1:3 Generators in 1:5 Generators in 1:7 Periods in 1:2 Periods in 1:3 Periods in 1:5 Periods in 1:7 RMS-optimum generator (cents) RMS error (cents) Thanks.
Message: 4898 - Contents - Hide Contents Date: Mon, 27 May 2002 14:30:55 Subject: Re: another keyboard and notation system (for manuel) From: manuel.op.de.coul@xxxxxxxxxxx.xxx Paul wrote:>this one for 15-equal: >The Pentadecaphonic system - 5. Simulation sof... * [with cont.] (Wayb.) >unbeknowest to the author of the page (who tried to shoehorn >traditional diatonic and pentatonic scales into 15-equal), this is a >great keyboard for porcupine music in 15-equal.Nice, I'll add it with the name P15. I already have another layout for Porcupine, used with notation systems M37 and M59, which can be tried in the latest version. Manuel
Message: 4899 - Contents - Hide Contents Date: Mon, 27 May 2002 12:53:25 Subject: A 7-limit best list From: genewardsmith This is a first pass at a 7-limit best list. The first entry is the mapping matrix, the second period and generator, the third (unweighted) rms generator steps to consonances, the fourth (unweighted) rms error. Badness is not listed to save space, but is less than 300; generator steps are less than 40, and rms error less than 50 cents. The ordering is by badness, lowest to highest. [[9, 1, 1, 12], [0, 2, 3, 2]] [133, 884] 16.0156 .13045 [[1, 3, 2, 3], [0, 22, -5, 3]] [1200, -77] 16.6933 .25334 [[1, 15, 4, 7], [0, 16, 2, 5]] [1200, -1006] 10.09125 .87536 [[1, 1, 3, 3], [0, 6, -7, -2]] [1200, 117] 7.60482 1.6374 [[1, 1, 7, 5], [0, 4, -32, -15]] [1200, 175] 23.13367 .18381 [[1, 0, 0, 2], [0, 2, 3, 1]] [1200, 929] 1.825742 34.5661 [[1, 21, 13, 13], [0, 40, 22, 21]] [1200, -583] 23.13367 .22219 [[2, 0, 3, 4], [0, 2, 1, 1]] [600, 951] 2.3094 23.945 [[1, 9, 9, 8], [0, 10, 9, 7]] [1200, -890] 6.377 3.32016 [[1, 0, -4, -13], [0, 1, 4, 10]] [1200, 1896] 6.36396 3.665 [[1, 0, 3, 1], [0, 7, -3, 8]] [1200, 271] 7.572 2.589 [[4, 0, 3, 5], [0, 1, 1, 1]] [300, 1885] 2.828 19.137 [[1, 1, -5, -1], [0, 2, 25, 13]] [1200, 352] 16.289 .585 [[19, 0, 14, -37], [0, 1, 1, 3]] [63, 1901] 33.811 .1402 [[1, 31, 0, 9], [0, 38, -3, 8]] [1200, -929] 26.517 .2287 [[1, 1, 2, 2], [0, 2, 1, 3]] [1200, 322] 1.8257 48.926 [[1, 0, 1, 2], [0, 6, 5, 3]] [1200, 317] 3.7416 12.2738 [[1, 1, 2, 3], [0, 9, 5, -3]] [1200, 78] 7.5167 3.0659 [[1, 27, 24, 20], [0, 34, 29, 23]] [1200, -897] 21.2446 .4048 [[1, 11, 42, 25], [0, 14, 59, 33]] [1200, -807] 36.1201 .1439 [[4, 0, 4, 7], [0, 6, 5, 4]] [300, 317] 14.8773 .8816 [[2, 0, 11, 12], [0, 1, -2, -2]] [600, 1908] 4.2427 10.903 [[2, 4, 5, 6], [0, 28, 12, 13]] [600, -18] 32.445 .1871 [[5, 8, 0, 14], [0, 0, 1, 0]] [240, 2789] 3.5356 15.8153 [[1, 0, 15, -59], [0, 1, -8, 39]] [1200, 1901] 29.7769 .22341 [[10, 0, 47, 36], [0, 2, -3, -1]] [120, 951] 29.439 .2289 [[3, 0, 45, 94], [0, 1, -8, -18]] [400, 1901] 37.370 .1469 [[1, 2, 2, 3], [0, 4, -3, 2]] [1200, -126] 4.223 12.1886 [[1, 0, 4, 6], [0, 1, -1, -2]] [1200, 1921] 1.826 65.953 [[1, 1, 1, 2], [0, 8, 18, 11]] [1200, 88] 10.5435 2.0643 [[3, 0, 7, 18], [0, 1, 0, -2]] [400, 1911] 5.3385 8.1007 [[1, 45, 39, 32], [0, 58, 49, 39]] [1200, -898] 36.1202 .18298 [[1, 0, 4, 2], [0, 2, -2, 1]] [1200, 981] 2.4152 41.5247 [[2, 1, 3, 4], [0, 4, 3, 3]] [600, 326] 4.899 10.1323 [[2, 4, 7, 7], [0, 6, 17, 10]] [600, -83] 20.1825 .60032 [[1, 0, 2, -1], [0, 5, 1, 12]] [1200, 381] 7.7028 4.139 [[1, 1, 0, 3], [0, 3, 12, -1]] [1200, 232] 8.3667 3.579 [[1, 0, -4, 6], [0, 1, 4, -2]] [1200, 1902] 3.5356 20.163 [[3, 5, 7, 8], [0, 7, 1, -12]] [400, -14] 33.764 .2218 [[1, 0, 4, -2], [0, 1, -1, 3]] [1200, 1940] 2.415 43.6595 [[1, 27, 11, 40], [0, 41, 14, 60]] [1200, -744] 38.0416 .1869 [[1, 3, 6, -2], [0, 5, 13, -17]] [1200, -339] 17.9397 .84588 [[3, 0, 7, -1], [0, 1, 0, 2]] [400, 1889] 4.062 16.597 [[1, 4, 2, 2], [0, 15, -2, -5]] [1200, -193] 12.596 1.7312 [[3, 5, 7, 0], [0, 0, 0, 1]] [400, 3406] 2.1213 61.3125 [[3, 0, 7, 6], [0, 2, 0, 1]] [400, 956] 4.062 16.787 [[1, 0, 4, 1], [0, 1, -1, 1]] [1200, 2025] 1.354 154.263 [[1, 11, -3, 20], [0, 23, -13, 42]] [1200, -491] 34.506 .2393 [[1, 0, 15, 25], [0, 1, -8, -14]] [1200, 1902] 10.02497 2.859 [[1, 5, 5, 5], [0, 14, 11, 9]] [1200, -293] 8.5245 4.007 [[1, 16, 32, -15], [0, 17, 35, -21]] [1200, -1017] 33.811 .2558 [[1, 0, 1, -3], [0, 6, 5, 22]] [1200, 317] 13.4846 1.61056 [[1, 1, 5, 4], [0, 2, -9, -4]] [1200, 356] 6.8678 6.2453 [[1, 6, 8, 11], [0, 7, 9, 13]] [1200, -756] 7.692 5.053 [[1, 0, -12, 6], [0, 1, 9, -2]] [1200, 1910] 6.831 6.410
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