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Message: 6175 - Contents - Hide Contents Date: Fri, 24 Jan 2003 09:31:29 Subject: Re: A 13-limit comma list From: Graham Breed Gene Ward Smith wrote:> Here is a list of 70 13-limit commas, with epimericity less than 0.25 > and size less than 50 cents. There is a very good chance it is complete, since I took it from a similar list of 499 commas with epimericity less than 0.5 after I could add nothing further to it. > > [36/35, 77/75, 40/39, 128/125, 45/44, 143/140, 49/48, 50/49, 55/54, 56/55, 64/63, 65/64, 66/65, 78/77, 81/80, 245/242, 91/90, 99/98, 100/99, 105/104,121/120, 245/243, 126/125, 275/273, 144/143, 169/168, 176/175, 896/891, 196/195, 1029/1024, 640/637, 225/224, 1188/1183, 243/242, 1573/1568, 325/324, 351/350, 352/351, 364/363, 385/384, 847/845, 441/440, 1375/1372, 540/539, 4000/3993, 625/624, 676/675, 729/728, 2200/2197, 1575/1573, 5632/5625, 1001/1000, 4459/4455, 10985/10976, 1716/1715, 2080/2079, 2401/2400, 3025/3024, 4096/4095, > 4225/4224, 4375/4374, 6656/6655, 140625/140608, 9801/9800, 10648/10647, 196625/196608, 151263/151250, 1990656/1990625, 123201/123200, 5767168/5767125]I left a script running overnight to generate linear temperaments from all combinations (there are nearly a million) of these columns. The results are below. It took 7.8 hours in pure Python (using wedge products, so no Numeric libraries). You could probably knock 2 orders of magnitude off that if you rewrote the whole thing in C, so it'd only be about 5 minutes. Which is still a lot slower than the search by equal temperaments, but that isn't comprehensive yet because I'm not taking all versions of inconsistent temperaments. I might look at it sometime. It's difficult to recognise the temperaments without the ETs as hints, so I don't know if there's anything new or missing. But here they are... 0/1, 15.8 cent generator basis: (0.034482758620689655, 0.013188156675199889) mapping by period and generator: [(29, 0), (46, 0), (67, 1), (81, 1), (100, 1), (107, 1)] mapping by steps: [(29, 0), (46, 0), (-1, 1), (13, 1), (32, 1), (39, 1)] highest interval width: 1 complexity measure: 29 (58 for smallest MOS) highest error: 0.003552 (4.263 cents) unique 1/2, 351.6 cent generator basis: (1.0, 0.29301432925652815) mapping by period and generator: [(1, 0), (1, 2), (-5, 25), (-1, 13), (2, 5), (4, -1)] mapping by steps: [(1, 1), (-1, 1), (-30, -5), (-14, -1), (-3, 2), (5, 4)] highest interval width: 26 complexity measure: 26 (41 for smallest MOS) highest error: 0.006546 (7.855 cents) 1/6, 183.2 cent generator basis: (0.5, 0.15268768272511829) mapping by period and generator: [(2, 0), (5, -6), (8, -11), (5, 2), (6, 3), (8, -2)] mapping by steps: [(10, 2), (1, -1), (-4, -3), (33, 7), (42, 9), (32, 6)] highest interval width: 15 complexity measure: 30 (46 for smallest MOS) highest error: 0.005815 (6.978 cents) 0/1, 103.8 cent generator basis: (0.5, 0.086488824362764394) mapping by period and generator: [(2, 0), (3, 1), (5, -2), (7, -8), (9, -12), (10, -15)] mapping by steps: [(2, 0), (-1, 1), (13, -2), (39, -8), (57, -12), (70, -15)] highest interval width: 17 complexity measure: 34 (46 for smallest MOS) highest error: 0.005094 (6.113 cents) unique 1/7, 186.0 cent generator basis: (0.5, 0.15499541683101503) mapping by period and generator: [(2, 0), (1, 7), (0, 15), (5, 2), (6, 3), (4, 11)] mapping by steps: [(12, 2), (-1, 1), (-15, 0), (28, 5), (33, 6), (13, 4)] highest interval width: 15 complexity measure: 30 (32 for smallest MOS) highest error: 0.005555 (6.666 cents) 0/1, 497.9 cent generator basis: (1.0, 0.41489358489291139) mapping by period and generator: [(1, 0), (2, -1), (-1, 8), (-3, 14), (13, -23), (12, -20)] mapping by steps: [(1, 0), (-1, 1), (23, -8), (39, -14), (-56, 23), (-48, 20)] highest interval width: 37 complexity measure: 37 (41 for smallest MOS) highest error: 0.004468 (5.362 cents) unique 1/10, 310.3 cent generator basis: (1.0, 0.25859402576420476) mapping by period and generator: [(1, 0), (-1, 10), (0, 9), (1, 7), (-3, 25), (5, -5)] mapping by steps: [(9, 1), (1, -1), (9, 0), (16, 1), (-2, -3), (40, 5)] highest interval width: 30 complexity measure: 30 (31 for smallest MOS) highest error: 0.006590 (7.908 cents) 0/1, 475.7 cent generator basis: (1.0, 0.39641218196468747) mapping by period and generator: [(1, 0), (0, 4), (-6, 21), (4, -3), (-12, 39), (-7, 27)] mapping by steps: [(1, 0), (-4, 4), (-27, 21), (7, -3), (-51, 39), (-34, 27)] highest interval width: 42 complexity measure: 42 (43 for smallest MOS) highest error: 0.003409 (4.090 cents) unique 2/5, 339.4 cent generator basis: (1.0, 0.28284398720617654) mapping by period and generator: [(1, 0), (3, -5), (6, -13), (-2, 17), (6, -9), (2, 6)] mapping by steps: [(3, 2), (-1, 1), (-8, -1), (28, 13), (0, 3), (18, 10)] highest interval width: 30 complexity measure: 30 (32 for smallest MOS) highest error: 0.006663 (7.995 cents) 1/3, 166.1 cent generator basis: (0.5, 0.13840115997461999) mapping by period and generator: [(2, 0), (4, -3), (-2, 24), (7, -5), (0, 25), (11, -13)] mapping by steps: [(4, 2), (-1, 1), (68, 22), (-1, 2), (75, 25), (-17, -2)] highest interval width: 38 complexity measure: 76 (94 for smallest MOS) highest error: 0.000971 (1.165 cents) unique 0/1, 83.0 cent generator basis: (0.33333333333333331, 0.06917176981320311) mapping by period and generator: [(3, 0), (6, -6), (8, -5), (8, 2), (11, -3), (14, -14)] mapping by steps: [(3, 0), (-6, 6), (-2, 5), (12, -2), (5, 3), (-14, 14)] highest interval width: 16 complexity measure: 48 (57 for smallest MOS) highest error: 0.002358 (2.830 cents) unique 0/1, 15.8 cent generator basis: (0.027777777777777776, 0.013143428289896716) mapping by period and generator: [(36, 0), (57, 0), (83, 1), (101, 0), (124, 1), (133, 0)] mapping by steps: [(36, 0), (57, 0), (-1, 1), (101, 0), (40, 1), (133, 0)] highest interval width: 1 complexity measure: 36 (72 for smallest MOS) highest error: 0.005995 (7.194 cents) 0/1, 104.9 cent generator basis: (0.5, 0.087412649330450343) mapping by period and generator: [(2, 0), (3, 1), (5, -2), (3, 15), (5, 11), (6, 8)] mapping by steps: [(2, 0), (-1, 1), (13, -2), (-57, 15), (-39, 11), (-26, 8)] highest interval width: 17 complexity measure: 34 (46 for smallest MOS) highest error: 0.006039 (7.247 cents) 1/3, 234.5 cent generator basis: (1.0, 0.19540011144114833) mapping by period and generator: [(1, 0), (1, 3), (-1, 17), (3, -1), (6, -13), (8, -22)] mapping by steps: [(2, 1), (-1, 1), (-19, -1), (7, 3), (25, 6), (38, 8)] highest interval width: 39 complexity measure: 39 (41 for smallest MOS) highest error: 0.005231 (6.277 cents) unique 1/3, 165.8 cent generator basis: (0.5, 0.13817544367976164) mapping by period and generator: [(2, 0), (4, -3), (11, -23), (7, -5), (13, -22), (11, -13)] mapping by steps: [(4, 2), (-1, 1), (-47, -12), (-1, 2), (-40, -9), (-17, -2)] highest interval width: 23 complexity measure: 46 (58 for smallest MOS) highest error: 0.003280 (3.935 cents) unique 4/11, 263.7 cent generator basis: (1.0, 0.21974602053976744) mapping by period and generator: [(1, 0), (4, -11), (1, 6), (5, -10), (5, -7), (7, -15)] mapping by steps: [(8, 3), (-1, 1), (26, 9), (10, 5), (19, 8), (11, 6)] highest interval width: 28 complexity measure: 28 (32 for smallest MOS) highest error: 0.008185 (9.822 cents) 0/1, 565.9 cent generator basis: (1.0, 0.47161343493195518) mapping by period and generator: [(1, 0), (3, -3), (16, -29), (8, -11), (11, -16), (7, -7)] mapping by steps: [(1, 0), (-3, 3), (-42, 29), (-14, 11), (-21, 16), (-7, 7)] highest interval width: 29 complexity measure: 29 (36 for smallest MOS) highest error: 0.010144 (12.173 cents) 1/6, 116.8 cent generator basis: (1.0, 0.097316259752627809) mapping by period and generator: [(1, 0), (1, 6), (3, -7), (3, -2), (2, 15), (0, 38)] mapping by steps: [(5, 1), (-1, 1), (22, 3), (17, 3), (-5, 2), (-38, 0)] highest interval width: 45 complexity measure: 45 (72 for smallest MOS) highest error: 0.003454 (4.145 cents) unique 1/4, 175.9 cent generator basis: (1.0, 0.14656139906015953) mapping by period and generator: [(1, 0), (1, 4), (1, 9), (-1, 26), (2, 10), (4, -2)] mapping by steps: [(3, 1), (-1, 1), (-6, 1), (-29, -1), (-4, 2), (14, 4)] highest interval width: 28 complexity measure: 28 (34 for smallest MOS) highest error: 0.009313 (11.176 cents) 0/1, 476.0 cent generator basis: (1.0, 0.39667339152715836) mapping by period and generator: [(1, 0), (0, 4), (17, -37), (4, -3), (11, -19), (16, -31)] mapping by steps: [(1, 0), (-4, 4), (54, -37), (7, -3), (30, -19), (47, -31)] highest interval width: 45 complexity measure: 45 (48 for smallest MOS) highest error: 0.003774 (4.529 cents) unique Graham
Message: 6176 - Contents - Hide Contents Date: Sat, 25 Jan 2003 13:16:56 Subject: Re: Graham's Top 20 13-limit temperaments From: Graham Breed Gene Ward Smith wrote:> The way I find a kernel basis is to find the invariant commas of the wedgie; that's four commas missing a prime in the 7-limit case, and 10 commas missing two primes in the 11-limit case. Then I LLL reduce that, which gives me a kernel basis, which I can then TM reduce if need be. The trouble is, I don't have a list of all interesting wedgies.I wasn't talking about that. If you have a set of candidate unison vectors, you can check it's complete without doing the full serach.> I don't think the same temperaments will always fall through for you quite this neatly, because the badness of the worst temperaments on the list is no longer so much higher than badness of the best ones.Yes, that's why I always take the best ones. If the filter's strict enough, the badness doesn't matter.> It sounds like I am too slow, and you have bugs which need to be worked out.I've sorted out the bug. I was getting temperaments with the same period and octave-equivalent mapping as mystery, but silly period mappings. But I was rejecting the real mystery because it looked the same. So now I have to reject fewer temperaments, and it's taking 82 minutes, but seems to be working correctly. Graham
Message: 6177 - Contents - Hide Contents Date: Sat, 25 Jan 2003 13:56:07 Subject: Re: Graham's Top 20 13-limit temperaments From: Graham Breed Gene Ward Smith wrote:> What kind of complexity? I didn't find anything nearly as high as 100 for unweighted complexity.I found 103089 such in the 13-limit search, some of them probably duplicates. Graham
Message: 6178 - Contents - Hide Contents Date: Sat, 25 Jan 2003 04:27:21 Subject: Re: A common notation for JI and ETs From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "gdsecor <gdsecor@y...>" <gdsecor@y...> wrote:> ***** HEY IF ANYBODY ELSE OUT THERE IS READING THIS, HERE'S A > QUESTION: What other ETs above 494 besides 612 and 624 would you > want to notate -- ones in which the 5' comma (a.k.a, historical 5- > schisma, 32768:32805) is either a single degree of the ET or vanishes?665, 684, 730, 742 and 836.
Message: 6179 - Contents - Hide Contents
Date: Sat, 25 Jan 2003 21:45:47
Subject: Temperament finder update
From: Graham Breed
The optimized script for finding temperaments from unison vectors is at
#!/c/Python22/python.exe * [with cont.] (Wayb.)
You need the latest version of Python, the Numeric extensions and my
temperament library:
# Temperament finding library -- definitions * [with cont.] (Wayb.)
That's been updated to search on all versions of equal temperaments
where we allow inconsistency. So I ran the 13-limit search on both.
The unison vectors
0/1, 16.4 cent generator * [with cont.] (Wayb.)
take about 70 minutes. Searching through pairs of the simplest 100 ETs
with a consistency cutoff of 0.8 scale steps, using
# duplicate selectNumeric.py but do an ET search * [with cont.] (Wayb.)
takes about 50 seconds. Those results
1/2, 16.4 cent generator * [with cont.] (Wayb.)
are the same as for the unison vector search!
I'll look at getting the CGI updated to use the new inconsistency search.
Graham
Message: 6180 - Contents - Hide Contents Date: Sat, 25 Jan 2003 06:09:09 Subject: Graham's Top 20 13-limit temperaments From: Gene Ward Smith Here they are again, in the form I would have given them. Three comments--first, I don't know how this list was filtered. Second, I think 20 is too small a number for the 13-limit. Third, I used unweighted complexity because I don't have geometric complexity in the 13-limit coded as yet. Mystery [0, 29, 29, 29, 29, 46, 46, 46, 46, -14, -33, -40, -19, -26, -7] [[29, 46, 67, 81, 100, 107], [0, 0, 1, 1, 1, 1]] [41.3793103448276, 15.8257880102396] unweighted badness 242.527516 unweighted complexity 22.463303 rms error 2.277984 Hemififth [2, 25, 13, 5, -1, 35, 15, 1, -9, -40, -75, -95, -31, -51, -22] [[1, 1, -5, -1, 2, 4], [0, 2, 25, 13, 5, -1]] [1200., 351.617195107835] unweighted badness 203.445092 unweighted complexity 13.364131 rms error 4.164244 Unidec [12, 22, -4, -6, 4, 7, -40, -51, -38, -71, -90, -72, -3, 26, 36] [[2, 5, 8, 5, 6, 8], [0, -6, -11, 2, 3, -2]] [600., 183.225219270142] unweighted badness 229.903139 unweighted complexity 17.401149 rms error 3.167218 Diaschismic [2, -4, -16, -24, -30, -11, -31, -45, -55, -26, -42, -55, -12, -25, -15] [[2, 3, 5, 7, 9, 10], [0, 1, -2, -8, -12, -15]] [600., 103.786589235317] unweighted badness 251.546815 unweighted complexity 19.682480 rms error 2.880707 Biminortonic [14, 30, 4, 6, 22, 15, -33, -39, -17, -75, -90, -60, 3, 47, 54] [[2, 1, 0, 5, 6, 4], [0, 7, 15, 2, 3, 11]] [600., 185.994500197218] unweighted badness 285.227918 unweighted complexity 17.175564 rms error 4.007057 Schismatic [1, -8, -14, 23, 20, -15, -25, 33, 28, -10, 81, 76, 113, 108, -16] [[1, 2, -1, -3, 13, 12], [0, -1, 8, 14, -23, -20]] [1200., 497.872301871492] unweighted badness 245.881799 unweighted complexity 19.724350 rms error 2.806870 Nonkleismic [10, 9, 7, 25, -5, -9, -17, 5, -45, -9, 27, -45, 46, -40, -110] [[1, -1, 0, 1, -3, 5], [0, 10, 9, 7, 25, -5]] [1200., 310.312830917046] unweighted badness 254.415991 unweighted complexity 15.006665 rms error 4.376411 Acute [4, 21, -3, 39, 27, 24, -16, 48, 28, -66, 18, -15, 120, 87, -51] [[1, 0, -6, 4, -12, -7], [0, 4, 21, -3, 39, 27]] [1200., 475.694618357624] unweighted badness 241.523384 unweighted complexity 22.614155 rms error 2.245891 Amity [5, 13, -17, 9, -6, 9, -41, -3, -28, -76, -24, -62, 84, 46, -54] [[1, 3, 6, -2, 6, 2], [0, -5, -13, 17, -9, 6]] [1200., 339.412784647410] unweighted badness 268.549368 unweighted complexity 15.295424 rms error 4.489333 Subminorsixth [6, -48, 10, -50, 26, -90, -1, -100, 19, 158, 50, 238, -175, 36, 275] [[2, 4, -2, 7, 0, 11], [0, -3, 24, -5, 25, -13]] [600., 166.081391969544] unweighted badness 207.849710 unweighted complexity 43.788126 rms error .717323 Minorsemi [18, 15, -6, 9, 42, -18, -60, -48, 0, -56, -31, 42, 46, 140, 112] [[3, 6, 8, 8, 11, 14], [0, -6, -5, 2, -3, -14]] [400., 83.0061237758442] unweighted badness 231.510127 unweighted complexity 25.260641 rms error 1.823490 Tricontaheximal [0, 36, 0, 36, 0, 57, 0, 57, 0, -101, -41, -133, 101, 0, -133] [[36, 57, 83, 101, 124, 133], [0, 0, 1, 0, 1, 0]] [33.3333333333333, 15.7721139478765] unweighted badness 416.491693 unweighted complexity 25.455844 rms error 3.242837 Spearmint [2, -4, 30, 22, 16, -11, 42, 28, 18, 81, 65, 52, -42, -66, -26] [[2, 3, 5, 3, 5, 6], [0, 1, -2, 15, 11, 8]] [600., 104.895179196541] unweighted badness 288.935571 unweighted complexity 18.477013 rms error 3.637922 Supersupermajor [3, 17, -1, -13, -22, 20, -10, -31, -46, -50, -89, -114, -33, -58, -28] [[1, 1, -1, 3, 6, 8], [0, 3, 17, -1, -13, -22]] [1200., 234.480133729376] unweighted badness 219.395264 unweighted complexity 18.448577 rms error 2.768745 Suprasubminorsixth [6, 46, 10, 44, 26, 59, -1, 49, 19, -106, -57, -110, 89, 36, -73] [[2, 4, 11, 7, 13, 11], [0, -3, -23, -5, -22, -13]] [600., 165.810532415714] unweighted badness 277.161590 unweighted complexity 26.724521 rms error 2.006172 Paraorwell [11, -6, 10, 7, 15, -35, -15, -27, -17, 40, 37, 57, -15, 5, 26] [[1, 4, 1, 5, 5, 7], [0, -11, 6, -10, -7, -15]] [1200., 263.695224647719] unweighted badness 271.488360 unweighted complexity 13.234425 rms error 5.638890 [3, 29, 11, 16, 7, 39, 9, 15, 0, -56, -63, -91, 7, -21, -35] [[1, 3, 16, 8, 11, 7], [0, -3, -29, -11, -16, -7]] [1200., 565.936121918345] unweighted badness 282.926501 unweighted complexity 14.126217 rms error 5.328866 Miracle [6, -7, -2, 15, 38, -25, -20, 3, 38, 15, 59, 114, 49, 114, 76] [[1, 1, 3, 3, 2, 0], [0, 6, -7, -2, 15, 38]] [1200., 116.779511703154] unweighted badness 224.267777 unweighted complexity 21.718656 rms error 2.215733 [4, 9, 26, 10, -2, 5, 30, 2, -18, 35, -8, -38, -62, -102, -44] [[1, 1, 1, -1, 2, 4], [0, 4, 9, 26, 10, -2]] [1200., 175.873678872191] unweighted badness 280.118702 unweighted complexity 13.315405 rms error 5.765149 [4, -37, -3, -19, -31, -68, -16, -44, -64, 97, 84, 65, -43, -76, -37] [[1, 0, 17, 4, 11, 16], [0, 4, -37, -3, -19, -31]] [1200., 476.008069832587] unweighted badness 298.019307 unweighted complexity 25.845696 rms error 2.268099
Message: 6181 - Contents - Hide Contents Date: Sat, 25 Jan 2003 23:46:53 Subject: Re: Graham's Top 20 13-limit temperaments From: wallyesterpaulrus --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith <genewardsmith@j...>" <genewardsmith@j...> wrote:> --- In tuning-math@xxxxxxxxxxx.xxxx Graham Breed <graham@m...> wrote:>> Gene Ward Smith wrote: >>>> Here they are again, in the form I would have given them. >>>> Do you have a script that could have given them, as you do for >> the 11-limit? >> I'm running on Maple, which is more powerful but much slower than >Python, so it's getting to the point where I should really use >something else, or else get you to do it.i have access to a 2.4 GHz machine for running Matlab overnight or for however long it takes. i'd be happy to try whatever algorithms you wish to spell out.
Message: 6182 - Contents - Hide Contents Date: Sat, 25 Jan 2003 06:13:39 Subject: A top 112 list for the 11-limit From: Gene Ward Smith I took my previous comma list, and added the Dirty Dozen commas which were less than 12/11 but greater than 50 cents, and for which the epimericity was less than 0.35. I then fitered the results by requiring that geometric badness be less than 7000, and rms error be less than 75 cents. I ordered by rms error; that way you can find the practical systems by looking in the middle. [102, 210, 216, 222, 96, 56, -1, -88, -211, -124] [[6, 11, 17, 20, 24], [0, -17, -35, -36, -37]] [200.000000000000, 17.5327039660445] bad 3334.010857 comp 954.841112 rms .036009 [64, 172, 102, 146, 124, -18, 10, -246, -256, 57] [[2, -6, -20, -9, -14], [0, 32, 86, 51, 73]] [600.000000000000, 171.934587318404] bad 6258.004071 comp 695.323226 rms .114672 [38, 38, 114, 76, -28, 74, -11, 158, 45, -181] [[38, 60, 88, 106, 131], [0, 1, 1, 3, 2]] [31.5789473684210, 7.15256753882099] bad 2665.433313 comp 386.646417 rms .129891 [82, 28, 120, 155, -146, -40, -38, 200, 263, 20] [[1, -14, -3, -20, -26], [0, 82, 28, 120, 155]] [1200., 228.072122769269] bad 6759.615517 comp 610.722159 rms .153762 [6, -48, -108, -168, -90, -188, -287, -116, -224, -98] [[6, 10, 10, 8, 7], [0, -1, 8, 18, 28]] [200.000000000000, 98.2539292884134] bad 6150.370472 comp 506.518200 rms .191092 Hemiennealimmal [36, 54, 36, 18, 2, -44, -96, -68, -145, -74] [[18, 28, 41, 50, 62], [0, 2, 3, 2, 1]] [66.6666666666667, 17.6128210959325] bad 2055.541657 comp 256.276437 rms .198798 [8, -64, -30, -110, -120, -70, -202, 110, -34, -205] [[2, 2, 14, 10, 23], [0, 4, -32, -15, -55]] [600.000000000000, 175.435046943151] bad 3951.940521 comp 371.595182 rms .205760 [2, -16, 78, 58, -30, 118, 85, 226, 190, -107] [[2, 3, 6, -1, 2], [0, 1, -8, 39, 29]] [600.000000000000, 101.758845066998] bad 3156.846072 comp 322.186595 rms .208485 [44, -10, 6, 79, -118, -114, -27, 42, 218, 201] [[1, 3, 2, 3, 6], [0, -44, 10, -6, -79]] [1200., 38.5948091516174] bad 3815.546956 comp 350.127551 rms .219371 [0, 0, 0, 171, 0, 0, 271, 0, 397, 480] [[171, 271, 397, 480, 592], [0, 0, 0, 0, -1]] [7.01754385964912, 3.33912257213528] bad 6842.169348 comp 496.624465 rms .219692 [6, -48, -108, 3, -90, -188, -16, -116, 173, 382] [[3, 4, 13, 22, 10], [0, 2, -16, -36, 1]] [400.000000000000, 150.873707762761] bad 5244.564640 comp 413.748659 rms .228287 [10, -80, 48, -52, -150, 48, -117, 336, 156, -312] [[2, 2, 14, 0, 13], [0, 5, -40, 24, -26]] [600.000000000000, 140.349907913464] bad 6968.405461 comp 476.669358 rms .239573 [8, -64, -30, 61, -120, -70, 69, 110, 363, 275] [[1, 1, 7, 5, -1], [0, 8, -64, -30, 61]] [1200., 87.7196164895498] bad 4849.969367 comp 382.009103 rms .241148 [20, -30, -10, -80, -94, -72, -196, 61, -82, -190] [[10, 16, 23, 28, 34], [0, -2, 3, 1, 8]] [120.000000000000, 8.93614137390901] bad 3203.569881 comp 288.351390 rms .254545 [6, -36, -84, -132, -71, -150, -230, -94, -182, -80] [[6, 10, 11, 10, 10], [0, -1, 6, 14, 22]] [200.000000000000, 97.7979183891198] bad 5608.137305 comp 399.243757 rms .259073 [58, -33, -2, -16, -187, -166, -226, 88, 77, -38] [[1, 25, -11, 2, -3], [0, -58, 33, 2, 16]] [1200., 484.447475312528] bad 6976.828740 comp 418.239598 rms .298275 Ennealimmal [18, 27, 18, 144, 1, -22, 166, -34, 241, 342] [[9, 15, 22, 26, 37], [0, -2, -3, -2, -16]] [133.333333333333, 48.8643746439101] bad 6729.608278 comp 388.997739 rms .324647 [2, -57, -28, 46, -95, -50, 66, 95, 304, 226] [[1, 1, 19, 11, -10], [0, 2, -57, -28, 46]] [1200., 351.114995249826] bad 4873.582264 comp 316.738528 rms .331141 [34, -24, 64, -28, -117, 6, -162, 216, 18, -300] [[2, -3, 9, -6, 12], [0, 17, -12, 32, -14]] [600.000000000000, 217.771906630368] bad 6646.001898 comp 375.555942 rms .339966 [14, 6, 74, 52, -23, 78, 34, 155, 100, -110] [[2, 4, 5, 10, 10], [0, -7, -3, -37, -26]] [600.000000000000, 71.1037242406681] bad 3709.724191 comp 258.983392 rms .352551 [24, -9, -66, 12, -70, -172, -64, -128, 59, 262] [[3, 2, 8, 16, 9], [0, 8, -3, -22, 4]] [400.000000000000, 137.769752421577] bad 5047.036505 comp 310.384815 rms .354706 [16, 84, 46, 98, 96, 28, 100, -129, -63, 116] [[2, 4, 9, 8, 12], [0, -8, -42, -23, -49]] [600.000000000000, 62.2185174715750] bad 6841.803268 comp 362.672040 rms .370949 [38, -3, 8, 64, -93, -94, -30, 27, 159, 152] [[1, -7, 3, 1, -11], [0, 38, -3, 8, 64]] [1200., 271.110310111484] bad 4668.828025 comp 282.799004 rms .383188 [42, 47, 34, 33, -23, -64, -93, -53, -86, -25] [[1, -13, -14, -9, -8], [0, 42, 47, 34, 33]] [1200., 416.714284033583] bad 6156.791010 comp 260.478466 rms .579519 Octoid [24, 32, 40, 24, -5, -4, -45, 3, -55, -71] [[8, 13, 19, 23, 28], [0, -3, -4, -5, -3]] [150.000000000000, 16.0721625542327] bad 4139.349022 comp 173.261857 rms .768706 Hemiamity [10, 26, -34, -28, 18, -82, -79, -152, -155, 39] [[2, 1, -1, 13, 13], [0, 5, 13, -17, -14]] [600.000000000000, 260.565078207829] bad 6616.465084 comp 211.396985 rms .881985 [2, -16, -40, -60, -30, -69, -102, -48, -84, -30] [[2, 3, 6, 9, 12], [0, 1, -8, -20, -30]] [600.000000000000, 101.618164889910] bad 5760.166901 comp 182.650266 rms .979643 [18, -14, 30, -20, -64, -3, -94, 109, 2, -160] [[2, 4, 4, 7, 6], [0, -9, 7, -15, 10]] [600.000000000000, 55.2942867561727] bad 6760.326288 comp 198.527159 rms 1.000617 [23, -1, 13, 42, -55, -44, -13, 33, 101, 73] [[1, 9, 2, 7, 17], [0, -23, 1, -13, -42]] [1200., 386.859261176094] bad 5380.665067 comp 171.779435 rms 1.013641 [18, 39, 42, 9, 20, 16, -48, -12, -114, -120] [[3, 2, 1, 2, 9], [0, 6, 13, 14, 3]] [400.000000000000, 183.522617686842] bad 6297.037328 comp 184.847442 rms 1.049818 [24, 20, 16, -12, -24, -42, -102, -19, -97, -89] [[4, 6, 9, 11, 14], [0, 6, 5, 4, -3]] [300.000000000000, 16.8775244689307] bad 5185.657720 comp 161.127182 rms 1.086899 [1, 33, 27, -18, 50, 40, -32, -30, -156, -144] [[1, 2, 16, 14, -4], [0, -1, -33, -27, 18]] [1200., 497.374746314971] bad 6259.261050 comp 177.573572 rms 1.115730 [30, 13, 14, 3, -49, -62, -99, -4, -38, -40] [[1, -13, -4, -4, 2], [0, 30, 13, 14, 3]] [1200., 583.380644844016] bad 6326.911305 comp 172.871604 rms 1.179376 Unidec [12, 22, -4, -6, 7, -40, -51, -71, -90, -3] [[2, 5, 8, 5, 6], [0, -6, -11, 2, 3]] [600.000000000000, 183.182783130202] bad 3535.629452 comp 117.775665 rms 1.249417 Minorsemi [18, 15, -6, 9, -18, -60, -48, -56, -31, 46] [[3, 6, 8, 8, 11], [0, -6, -5, 2, -3]] [400.000000000000, 83.1441780396820] bad 4104.955170 comp 122.582132 rms 1.357052 [12, 34, 20, 30, 26, -2, 6, -49, -48, 15] [[2, 4, 7, 7, 9], [0, -6, -17, -10, -15]] [600.000000000000, 83.1977089999346] bad 5359.187207 comp 137.542589 rms 1.462302 [3, -24, -1, 28, -45, -10, 34, 65, 148, 82] [[1, 1, 7, 3, -2], [0, 3, -24, -1, 28]] [1200., 233.937160252768] bad 6259.999409 comp 148.705092 rms 1.499793 [6, 29, -2, -21, 32, -20, -54, -86, -149, -52] [[1, 4, 14, 2, -5], [0, -6, -29, 2, 21]] [1200., 483.287995701320] bad 6718.809689 comp 151.808898 rms 1.555238 Wizard [12, -2, 20, -6, -31, -2, -51, 52, -7, -86] [[2, 1, 5, 2, 8], [0, 6, -1, 10, -3]] [600.000000000000, 216.784022806644] bad 3830.786398 comp 107.160572 rms 1.584515 [18, 27, 18, 45, 1, -22, 9, -34, 11, 64] [[9, 15, 22, 26, 33], [0, -2, -3, -2, -5]] [133.333333333333, 49.5985795043903] bad 6795.385375 comp 149.438324 rms 1.614770 [18, -9, 18, 9, -56, -22, -48, 67, 52, -37] [[9, 14, 21, 25, 31], [0, 2, -1, 2, 1]] [133.333333333333, 17.0160504901062] bad 6325.840257 comp 139.340563 rms 1.689101 Catakleismic [6, 5, 22, -21, -6, 18, -54, 37, -66, -135] [[1, 0, 1, -3, 9], [0, 6, 5, 22, -21]] [1200., 316.707652223641] bad 4805.477932 comp 117.818038 rms 1.697137 Hemiwuerschmidt [16, 2, 5, 40, -34, -37, 8, 6, 86, 95] [[1, -1, 2, 2, -3], [0, 16, 2, 5, 40]] [1200., 193.827642802708] bad 6485.787550 comp 137.781843 rms 1.764586 Hemithird [15, -2, -5, 22, -38, -50, -17, -6, 58, 79] [[1, 4, 2, 2, 7], [0, -15, 2, 5, -22]] [1200., 193.222638998759] bad 4937.466578 comp 116.683657 rms 1.772097 [6, -19, -26, -21, -44, -58, -54, -7, 17, 31] [[1, 2, 1, 1, 2], [0, -6, 19, 26, 21]] [1200., 83.3216367301528] bad 5534.523125 comp 120.483135 rms 1.883084 Duodecimal [0, 12, 24, 36, 19, 38, 57, 22, 42, 18] [[12, 19, 28, 34, 42], [0, 0, -1, -2, -3]] [100.000000000000, 16.8004014246913] bad 4602.747842 comp 107.748365 rms 1.886540 [6, 5, 22, 51, -6, 18, 60, 37, 101, 67] [[1, 0, 1, -3, -10], [0, 6, 5, 22, 51]] [1200., 316.660461986628] bad 6980.736242 comp 137.961373 rms 1.895129 Miracle [6, -7, -2, 15, -25, -20, 3, 15, 59, 49] [[1, 1, 3, 3, 2], [0, 6, -7, -2, 15]] [1200., 116.672264296056] bad 2362.204791 comp 71.868304 rms 1.901466 [17, 6, 15, 27, -30, -24, -16, 18, 42, 24] [[1, -5, 0, -3, -7], [0, 17, 6, 15, 27]] [1200., 464.880312700637] bad 6229.034819 comp 111.479674 rms 2.412281 Slender [13, -10, 6, 17, -46, -27, -18, 42, 74, 27] [[1, 2, 2, 3, 4], [0, -13, 10, -6, -17]] [1200., 38.3548342420224] bad 6321.956488 comp 111.749905 rms 2.438407 Schismatic [1, -8, -14, 23, -15, -25, 33, -10, 81, 113] [[1, 2, -1, -3, 13], [0, -1, 8, 14, -23]] [1200., 497.816548051118] bad 5478.851625 comp 102.323143 rms 2.447559 [11, 1, -19, -17, -24, -61, -65, -47, -43, 18] [[1, -2, 2, 9, 9], [0, 11, 1, -19, -17]] [1200., 391.092374744452] bad 6894.100227 comp 108.843088 rms 2.778494 [6, -12, 10, -14, -33, -1, -43, 57, 9, -74] [[2, 4, 3, 7, 5], [0, -3, 6, -5, 7]] [600.000000000000, 165.152290841724] bad 6063.419881 comp 96.498734 rms 2.986631 Supersupermajor [3, 17, -1, -13, 20, -10, -31, -50, -89, -33] [[1, 1, -1, 3, 6], [0, 3, 17, -1, -13]] [1200., 234.451570241274] bad 5412.588259 comp 89.805421 rms 3.005389 Diaschismic [2, -4, -16, -24, -11, -31, -45, -26, -42, -12] [[2, 3, 5, 7, 9], [0, 1, -2, -8, -12]] [600.000000000000, 103.783535998448] bad 4368.478167 comp 76.308394 rms 3.182069 Nonkleismic [10, 9, 7, 25, -9, -17, 5, -9, 27, 46] [[1, -1, 0, 1, -3], [0, 10, 9, 7, 25]] [1200., 310.147077475992] bad 4830.505212 comp 79.065160 rms 3.316530 [12, 5, -9, 1, -20, -48, -40, -35, -15, 34] [[1, -4, 0, 7, 3], [0, 12, 5, -9, 1]] [1200., 558.663023804566] bad 6881.869686 comp 85.507274 rms 4.146688 Quartaminorthirds [9, 5, -3, 7, -13, -30, -20, -21, -1, 30] [[1, 1, 2, 3, 3], [0, 9, 5, -3, 7]] [1200., 77.9320061599426] bad 4041.237407 comp 59.805237 rms 4.418576 Magic [5, 1, 12, -8, -10, 5, -30, 25, -22, -64] [[1, 0, 2, -1, 6], [0, 5, 1, 12, -8]] [1200., 380.713812625105] bad 4474.854563 comp 61.027896 rms 4.730404 Tritonic [5, -11, -12, -3, -29, -33, -22, 3, 31, 33] [[1, 4, -3, -3, 2], [0, -5, 11, 12, 3]] [1200., 580.274408362853] bad 6158.168741 comp 70.204409 rms 5.154394 Schismic [1, -8, -14, -18, -15, -25, -32, -10, -14, -2] [[1, 2, -1, -3, -4], [0, -1, 8, 14, 18]] [1200., 497.529640593025] bad 4970.055835 comp 60.776340 rms 5.290179 Superkleismic [9, 10, -3, 2, -5, -30, -28, -35, -30, 16] [[1, 4, 5, 2, 4], [0, -9, -10, 3, -2]] [1200., 321.939679549928] bad 5706.061898 comp 65.931245 rms 5.302952 Orwell [7, -3, 8, 2, -21, -7, -21, 27, 15, -22] [[1, 0, 3, 1, 3], [0, 7, -3, 8, 2]] [1200., 271.444627221787] bad 4352.766535 comp 54.544189 rms 5.548615 Meanpop [1, 4, 10, -13, 4, 13, -24, 12, -44, -71] [[1, 2, 4, 7, -2], [0, -1, -4, -10, 13]] [1200., 503.595073255734] bad 5420.225629 comp 61.580856 rms 5.644271 Supermajor seconds [3, 12, -1, -8, 12, -10, -23, -36, -60, -19] [[1, 1, 0, 3, 5], [0, 3, 12, -1, -8]] [1200., 231.991673973517] bad 6304.096121 comp 62.196165 rms 6.456792 Meantone [1, 4, 10, 18, 4, 13, 25, 12, 28, 16] [[1, 2, 4, 7, 11], [0, -1, -4, -10, -18]] [1200., 502.999427606597] bad 4405.132983 comp 49.575532 rms 6.584357 [2, 8, -11, 5, 8, -23, 1, -48, -16, 52] [[1, 1, 0, 6, 2], [0, 2, 8, -11, 5]] [1200., 348.593670912599] bad 6970.647424 comp 64.720989 rms 6.681355 [4, 16, 9, 10, 16, 3, 2, -24, -32, -3] [[1, 3, 8, 6, 7], [0, -4, -16, -9, -10]] [1200., 425.850157863434] bad 6772.320977 comp 62.039252 rms 6.965623 [1, 9, -2, 16, 12, -6, 22, -30, 6, 52] [[1, 2, 6, 2, 10], [0, -1, -9, 2, -16]] [1200., 489.922901740158] bad 6990.389258 comp 58.536187 rms 7.921255 Double wide [8, 6, 6, -4, -9, -13, -34, -3, -30, -32] [[2, 5, 6, 7, 6], [0, -4, -3, -3, 2]] [600.000000000000, 274.687303196174] bad 6195.802215 comp 53.474677 rms 8.163015 Pajara [2, -4, -4, -12, -11, -12, -26, 2, -14, -20] [[2, 3, 5, 6, 8], [0, 1, -2, -2, -6]] [600.000000000000, 107.105867266486] bad 4125.050690 comp 38.122013 rms 9.552922 Porcupine [3, 5, -6, 4, 1, -18, -4, -28, -8, 32] [[1, 2, 3, 2, 4], [0, -3, -5, 6, -4]] [1200., 162.926556663970] bad 5765.207417 comp 41.067070 rms 11.793935 Pajarous [2, -4, -4, 10, -11, -12, 9, 2, 37, 42] [[2, 3, 5, 6, 6], [0, 1, -2, -2, 5]] [600.000000000000, 109.882784790765] bad 6667.906223 comp 43.767076 rms 12.267148 Tripletone [3, 0, -6, -6, -7, -18, -20, -14, -14, 4] [[3, 5, 7, 8, 10], [0, -1, 0, 2, 2]] [400.000000000000, 87.7987973107273] bad 4275.784995 comp 32.722273 rms 12.772525 Injera [2, 8, 8, 12, 8, 7, 12, -4, 0, 6] [[2, 3, 4, 5, 6], [0, 1, 4, 4, 6]] [600.000000000000, 91.3378934339818] bad 5930.390623 comp 38.784565 rms 13.344995 [3, -5, -6, -1, -15, -18, -12, 0, 15, 18] [[1, 3, 0, 0, 3], [0, -3, 5, 6, 1]] [1200., 562.608972647194] bad 5472.363478 comp 36.251932 rms 13.781284 [6, 0, 3, 3, -14, -12, -16, 7, 7, -2] [[3, 4, 7, 8, 10], [0, 2, 0, 1, 1]] [400.000000000000, 152.119884700576] bad 5804.051786 comp 36.468676 rms 14.472091 Kleismic [6, 5, 3, -2, -6, -12, -24, -7, -22, -16] [[1, 0, 1, 2, 4], [0, 6, 5, 3, -2]] [1200., 317.610475585507] bad 6369.686860 comp 38.560870 rms 14.472383 Dominant Seventh [1, 4, -2, -6, 4, -6, -13, -16, -28, -10] [[1, 2, 4, 2, 1], [0, -1, -4, 2, 6]] [1200., 495.145082636305] bad 4962.157739 comp 28.253374 rms 18.933026 Meanenneadecal [1, 4, 10, 6, 4, 13, 6, 12, 0, -18] [[1, 2, 4, 7, 6], [0, -1, -4, -10, -6]] [1200., 504.558724588958] bad 6252.411315 comp 32.422449 rms 18.965801 [4, -3, 2, 5, -14, -8, -6, 13, 22, 7] [[1, 2, 2, 3, 4], [0, -4, 3, -2, -5]] [1200., 126.510501480109] bad 6873.457917 comp 33.942918 rms 19.316421 [4, 2, 2, 10, -6, -8, 2, -1, 16, 21] [[2, 4, 5, 6, 9], [0, -2, -1, -1, -5]] [600.000000000000, 252.994745923802] bad 6414.557575 comp 32.426498 rms 19.453599 [5, 3, 7, 4, -7, -3, -11, 8, -1, -13] [[1, 1, 2, 2, 3], [0, 5, 3, 7, 4]] [1200., 141.164897164495] bad 6400.766040 comp 32.195552 rms 19.644403 [2, -6, 1, -2, -14, -4, -10, 19, 16, -9] [[1, 2, 1, 3, 3], [0, -2, 6, -1, 2]] [1200., 259.236678540745] bad 6745.990413 comp 32.886454 rms 19.984071 [0, 5, 0, -5, 8, 0, -8, -14, -29, -14] [[5, 8, 12, 14, 17], [0, 0, -1, 0, 1]] [240.000000000000, 82.5021142499568] bad 5925.447275 comp 28.649813 rms 22.089446 Opossum [3, 5, 9, 4, 1, 6, -4, 7, -8, -20] [[1, 2, 3, 4, 4], [0, -3, -5, -9, -4]] [1200., 159.564330324400] bad 6767.545993 comp 30.993567 rms 22.129858 [4, 2, 2, -4, -6, -8, -20, -1, -16, -18] [[2, 4, 5, 6, 6], [0, -2, -1, -1, 2]] [600.000000000000, 257.288990758189] bad 6037.202663 comp 28.553621 rms 22.632565 Septimal [0, 0, 7, 0, 0, 11, 0, 16, 0, -24] [[7, 11, 16, 20, 24], [0, 0, 0, -1, 0]] [171.428571428572, 85.5877110103187] bad 5184.550217 comp 26.058106 rms 22.636347 Pajaric [2, -4, -4, 0, -11, -12, -7, 2, 14, 14] [[2, 3, 5, 6, 7], [0, 1, -2, -2, 0]] [600.000000000000, 106.675554557680] bad 6293.616955 comp 26.330282 rms 27.006882 [2, 1, -4, 5, -3, -12, 1, -12, 8, 28] [[1, 1, 2, 4, 2], [0, 2, 1, -4, 5]] [1200., 353.356606337850] bad 6651.131431 comp 27.157168 rms 27.107413 Diminished [4, 4, 4, 0, -3, -5, -14, -2, -14, -14] [[4, 6, 9, 11, 14], [0, 1, 1, 1, 0]] [300.000000000000, 114.119995039099] bad 6306.500152 comp 26.212063 rms 27.265894 [3, 0, 6, 6, -7, 1, -1, 14, 14, -4] [[3, 5, 7, 9, 11], [0, -1, 0, -2, -2]] [400.000000000000, 110.279883784335] bad 6837.646281 comp 27.334513 rms 27.566887 [1, 4, -2, 6, 4, -6, 6, -16, 0, 24] [[1, 2, 4, 2, 6], [0, -1, -4, 2, -6]] [1200., 502.810610708562] bad 6771.748415 comp 26.432118 rms 28.872265 [3, 0, -3, -3, -7, -13, -15, -7, -7, 2] [[3, 5, 7, 8, 10], [0, -1, 0, 1, 1]] [400.000000000000, 119.412265830049] bad 6959.564859 comp 23.686181 rms 35.625175 [1, -3, -4, -1, -7, -9, -5, -1, 8, 11] [[1, 2, 1, 1, 3], [0, -1, 3, 4, 1]] [1200., 526.844177546525] bad 4967.542223 comp 19.028176 rms 36.627981 [2, 1, 6, 5, -3, 4, 1, 11, 8, -7] [[1, 1, 2, 1, 2], [0, 2, 1, 6, 5]] [1200., 355.041079645760] bad 6442.538585 comp 21.934212 rms 37.484665 Arnold [1, 4, -2, -1, 4, -6, -5, -16, -16, 4] [[1, 2, 4, 2, 3], [0, -1, -4, 2, 1]] [1200., 501.833702411116] bad 6618.437371 comp 21.483576 rms 39.863722 Pentoid [2, 3, 1, -2, 0, -4, -10, -6, -15, -9] [[1, 2, 3, 3, 3], [0, -2, -3, -1, 2]] [1200., 262.017672736483] bad 4792.041040 comp 17.620981 rms 40.160927 [2, 3, 1, 7, 0, -4, 4, -6, 6, 16] [[1, 2, 3, 3, 5], [0, -2, -3, -1, -7]] [1200., 265.050484486844] bad 6958.659717 comp 21.366135 rms 42.297592 [2, 1, -4, -2, -3, -12, -10, -12, -8, 8] [[1, 1, 2, 4, 4], [0, 2, 1, -4, -2]] [1200., 351.665197943778] bad 6935.910001 comp 21.310039 rms 42.344437 [1, -1, 3, -4, -4, 2, -10, 10, -6, -22] [[1, 2, 2, 4, 2], [0, -1, 1, -3, 4]] [1200., 455.251489802704] bad 6476.838112 comp 20.253838 rms 43.037876 [1, -1, 3, 4, -4, 2, 3, 10, 13, 1] [[1, 2, 2, 4, 5], [0, -1, 1, -3, -4]] [1200., 455.784552043733] bad 4694.936247 comp 16.401858 rms 44.341253 [2, -2, -2, 0, -8, -9, -7, 1, 7, 7] [[2, 3, 5, 6, 7], [0, 1, -1, -1, 0]] [600.000000000000, 145.338368444940] bad 5731.615600 comp 17.991842 rms 46.396405 [1, -1, -5, -4, -4, -11, -10, -9, -6, 6] [[1, 2, 2, 1, 2], [0, -1, 1, 5, 4]] [1200., 449.014384102829] bad 6924.969904 comp 20.026539 rms 46.889405 Meanertone [1, 4, 3, -1, 4, 2, -5, -4, -16, -13] [[1, 2, 4, 4, 3], [0, -1, -4, -3, 1]] [1200., 503.381925649482] bad 6235.745071 comp 18.648791 rms 47.548543 [1, 4, 3, 6, 4, 2, 6, -4, 0, 6] [[1, 2, 4, 4, 6], [0, -1, -4, -3, -6]] [1200., 507.069242052444] bad 6926.660782 comp 18.987849 rms 51.254378 [2, 1, 3, -2, -3, -1, -10, 4, -8, -16] [[1, 1, 2, 2, 4], [0, 2, 1, 3, -2]] [1200., 336.439285434486] bad 6018.139480 comp 17.254513 rms 52.234534 [2, 1, -1, -2, -3, -7, -10, -5, -8, -2] [[1, 1, 2, 3, 4], [0, 2, 1, -1, -2]] [1200., 336.420671507429] bad 4702.538268 comp 14.859260 rms 52.360492 [0, 0, 0, 5, 0, 0, 8, 0, 12, 14] [[5, 8, 12, 14, 17], [0, 0, 0, 0, 1]] [240.000000000000, 99.1170177787140] bad 4584.543716 comp 14.521183 rms 53.042760 [1, -3, -2, -1, -7, -6, -5, 4, 8, 4] [[1, 2, 1, 2, 3], [0, -1, 3, 2, 1]] [1200., 510.517333273116] bad 5513.617839 comp 15.875495 rms 54.982509 [2, 1, 3, 5, -3, -1, 1, 4, 8, 4] [[1, 1, 2, 2, 2], [0, 2, 1, 3, 5]] [1200., 344.800380607378] bad 5859.936960 comp 16.340735 rms 55.689539 [2, -1, 1, -2, -6, -4, -10, 5, -1, -9] [[1, 2, 2, 3, 3], [0, -2, 1, -1, 2]] [1200., 265.122370321608] bad 6271.548770 comp 16.000568 rms 61.728037
Message: 6183 - Contents - Hide Contents Date: Sat, 25 Jan 2003 08:12:59 Subject: Re: Graham's Top 20 13-limit temperaments From: Graham Breed Gene Ward Smith wrote:> Here they are again, in the form I would have given them.Do you have a script that could have given them, as you do for the 11-limit?> Three comments--first, I don't know how this list was filtered. Second, I think 20 is too small a number for the 13-limit. Third, I used unweighted complexity because I don't have geometric complexity in the 13-limit coded as yet.Complexity<100, RMS error < 6 or so cents. Why so? How many have you tuned up? Graham
Message: 6184 - Contents - Hide Contents Date: Sat, 25 Jan 2003 08:40:10 Subject: Re: Graham's Top 20 13-limit temperaments From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx Graham Breed <graham@m...> wrote:> Gene Ward Smith wrote:>> Here they are again, in the form I would have given them. >> Do you have a script that could have given them, as you do for the 11-limit?I'm running on Maple, which is more powerful but much slower than Python, so it's getting to the point where I should really use something else, or else get you to do it.>> Three comments--first, I don't know how this list was filtered. Second, I think 20 is too small a number for the 13-limit. Third, I used unweighted complexity because I don't have geometric complexity in the 13-limit coded as yet.> Complexity<100, RMS error < 6 or so cents.What kind of complexity? I didn't find anything nearly as high as 100 for unweighted complexity.> Why so? How many have you tuned up?As I say, I think you might be better for the job at this point. If I calculated the coefficients for determining geometric complexity, would you try that? As for why 20 is too small, as we go up in prime limit, the number of reasonable systems increases; it more and more becomes the case that the lists will be completely different if we use slightly different complexity or error measures. I think we should take in a bigger haul at least to start out with.
Message: 6185 - Contents - Hide Contents Date: Sat, 25 Jan 2003 10:47:50 Subject: Re: Graham's Top 20 13-limit temperaments From: Graham Breed Gene Ward Smith wrote:> I'm running on Maple, which is more powerful but much slower than Python, so it's getting to the point where I should really use something else, or else get you to do it.I'd have thought Maple would be better optimized for numerical work. Anyway, I do have a version now using Numerical Python, which links to C and Fortran libraries for matrix operations. I can now do the 13-limit search within an hour, but there are bugs -- I'm not getting a correct period mapping for mystery. It wouldn't be that difficult to convert the script to a more efficient language like C++ or Java, with a suitable numerical library. I can invert matrices in C++, but that isn't optimized code. Currently I still use my high-level Python library to do the RMS optimization, but I think that can be re-written.>> Complexity<100, RMS error < 6 or so cents. > >> What kind of complexity? I didn't find anything nearly as high as 100 for unweighted complexity.It's the usual max-min complexity. And not calculated correctly because I'm only using primes. Maybe nothing's that high. I'm not recording what I throw away. A lot of inaccurate temperaments were getting in, which is why I tightened up the error cutoff.> As I say, I think you might be better for the job at this point. If I calculated the coefficients for determining geometric complexity, would you try that?Yes, if you give me an algorithm I can copy. Ideally Python code or pseudocode. If it's only a question of checking that the unison vectors can produce all the temperaments you're interested, there are more efficient ways of doing it. Start with the wedgie for the temperament, and you can filter for unison vectors that are consistent with the temperament. Then you only need to take subsets of those until you get one that's linearly independent.> As for why 20 is too small, as we go up in prime limit, the number of reasonable systems increases; it more and more becomes the case that the lists will be completely different if we use slightly different complexity or error measures. I think we should take in a bigger haul at least to start out with.Good algorithms will be more likely to agree on the best temperaments than the mediocre ones. The best thing's to restrict the range of complexities and errors so the same temperaments will fall through. I can also run the ET and unison vector searches with the same parameters. Oh, and if two sets of results are held in RAM they can be compared automatically, so larger sets can be used. Graham
Message: 6186 - Contents - Hide Contents Date: Sat, 25 Jan 2003 11:30:29 Subject: Re: Graham's Top 20 13-limit temperaments From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx Graham Breed <graham@m...> wrote:> If it's only a question of checking that the unison vectors can produce > all the temperaments you're interested, there are more efficient ways of > doing it. Start with the wedgie for the temperament, and you can filter > for unison vectors that are consistent with the temperament. Then you > only need to take subsets of those until you get one that's linearly > independent.The way I find a kernel basis is to find the invariant commas of the wedgie; that's four commas missing a prime in the 7-limit case, and 10 commas missing two primes in the 11-limit case. Then I LLL reduce that, which gives me a kernel basis, which I can then TM reduce if need be. The trouble is, I don't have a list of all interesting wedgies.>> As for why 20 is too small, as we go up in prime limit, the number of reasonable systems increases; it more and more becomes the case that the lists will be completely different if we use slightly different complexity or error measures. I think we should take in a bigger haul at least to start out with.> Good algorithms will be more likely to agree on the best temperaments > than the mediocre ones. The best thing's to restrict the range of > complexities and errors so the same temperaments will fall through.I don't think the same temperaments will always fall through for you quite this neatly, because the badness of the worst temperaments on the list is no longer so much higher than badness of the best ones. It sounds like I am too slow, and you have bugs which need to be worked out.
Message: 6187 - Contents - Hide Contents Date: Sun, 26 Jan 2003 11:44:56 Subject: Graham's top 20, with standard vals From: Gene Ward Smith This is the same list as before, only this time I included all of the standard vals consistent with the temperament. The good news is this: (1) 18 out of the 20 systems can be defined by wedging two standard vals (2) In no case is it neccessary to use vals where the number of divisions of the octave of one is more than twice that of another (3) The vals could have been filtered. Here are the ets from 30 to 130, with badness less than 1.6: [31, 36, 37, 41, 43, 44, 46, 50, 53, 56, 58, 62, 63, 68, 70, 72, 77, 79, 80, 87, 94, 103, 111, 113, 118, 121, 130] We see that this would have more than sufficed to get our 18. The bad news is we missed two; one had the biggest error and second lowest complexity on the list, and the other had a very small period of 1/36 octave. A separate system for dealing with small periods may be required. The other bad news is that we are skating on thin ice, as many of the temperaments had only the minimum two standard vals. Mystery [0, 29, 29, 29, 29, 46, 46, 46, 46, -14, -33, -40, -19, -26, -7] [[29, 46, 67, 81, 100, 107], [0, 0, 1, 1, 1, 1]] [41.3793103448276, 15.8257880102396] [29, 58, 87, 145, 232] unweighted badness 242.527516 unweighted complexity 22.463303 rms error 2.277984 Hemififth [2, 25, 13, 5, -1, 35, 15, 1, -9, -40, -75, -95, -31, -51, -22] [[1, 1, -5, -1, 2, 4], [0, 2, 25, 13, 5, -1]] [1200., 351.617195107835] [41, 58] unweighted badness 203.445092 unweighted complexity 13.364131 rms error 4.164244 Unidec [12, 22, -4, -6, 4, 7, -40, -51, -38, -71, -90, -72, -3, 26, 36] [[2, 5, 8, 5, 6, 8], [0, -6, -11, 2, 3, -2]] [600., 183.225219270142] [26, 46, 72] unweighted badness 229.903139 unweighted complexity 17.401149 rms error 3.167218 Diaschismic [2, -4, -16, -24, -30, -11, -31, -45, -55, -26, -42, -55, -12, -25, -15] [[2, 3, 5, 7, 9, 10], [0, 1, -2, -8, -12, -15]] [600., 103.786589235317] [46, 58] unweighted badness 251.546815 unweighted complexity 19.682480 rms error 2.880707 Biminortonic [14, 30, 4, 6, 22, 15, -33, -39, -17, -75, -90, -60, 3, 47, 54] [[2, 1, 0, 5, 6, 4], [0, 7, 15, 2, 3, 11]] [600., 185.994500197218] [26, 58, 84, 110] unweighted badness 285.227918 unweighted complexity 17.175564 rms error 4.007057 Schismatic [1, -8, -14, 23, 20, -15, -25, 33, 28, -10, 81, 76, 113, 108, -16] [[1, 2, -1, -3, 13, 12], [0, -1, 8, 14, -23, -20]] [1200., 497.872301871492] [41, 53, 94, 135] unweighted badness 245.881799 unweighted complexity 19.724350 rms error 2.806870 Nonkleismic [10, 9, 7, 25, -5, -9, -17, 5, -45, -9, 27, -45, 46, -40, -110] [[1, -1, 0, 1, -3, 5], [0, 10, 9, 7, 25, -5]] [1200., 310.312830917046] [31, 58] unweighted badness 254.415991 unweighted complexity 15.006665 rms error 4.376411 Acute [4, 21, -3, 39, 27, 24, -16, 48, 28, -66, 18, -15, 120, 87, -51] [[1, 0, -6, 4, -12, -7], [0, 4, 21, -3, 39, 27]] [1200., 475.694618357624] [53, 58, 111] unweighted badness 241.523384 unweighted complexity 22.614155 rms error 2.245891 Amity [5, 13, -17, 9, -6, 9, -41, -3, -28, -76, -24, -62, 84, 46, -54] [[1, 3, 6, -2, 6, 2], [0, -5, -13, 17, -9, 6]] [1200., 339.412784647410] [7, 39, 46, 53, 99] unweighted badness 268.549368 unweighted complexity 15.295424 rms error 4.489333 Subminorsixth [6, -48, 10, -50, 26, -90, -1, -100, 19, 158, 50, 238, -175, 36, 275] [[2, 4, -2, 7, 0, 11], [0, -3, 24, -5, 25, -13]] [600., 166.081391969544] [36, 94, 130, 224, 318, 354, 542, 578] unweighted badness 207.849710 unweighted complexity 43.788126 rms error .717323 Minorsemi [18, 15, -6, 9, 42, -18, -60, -48, 0, -56, -31, 42, 46, 140, 112] [[3, 6, 8, 8, 11, 14], [0, -6, -5, 2, -3, -14]] [400., 83.0061237758442] [15, 72, 87, 159] unweighted badness 231.510127 unweighted complexity 25.260641 rms error 1.823490 Tricontaheximal [0, 36, 0, 36, 0, 57, 0, 57, 0, -101, -41, -133, 101, 0, -133] [[36, 57, 83, 101, 124, 133], [0, 0, 1, 0, 1, 0]] [33.3333333333333, 15.7721139478765] [72] unweighted badness 416.491693 unweighted complexity 25.455844 rms error 3.242837 Spearmint [2, -4, 30, 22, 16, -11, 42, 28, 18, 81, 65, 52, -42, -66, -26] [[2, 3, 5, 3, 5, 6], [0, 1, -2, 15, 11, 8]] [600., 104.895179196541] [46, 80, 126] unweighted badness 288.935571 unweighted complexity 18.477013 rms error 3.637922 Supersupermajor [3, 17, -1, -13, -22, 20, -10, -31, -46, -50, -89, -114, -33, -58, -28] [[1, 1, -1, 3, 6, 8], [0, 3, 17, -1, -13, -22]] [1200., 234.480133729376] [41, 46, 87, 128, 133] unweighted badness 219.395264 unweighted complexity 18.448577 rms error 2.768745 Suprasubminorsixth [6, 46, 10, 44, 26, 59, -1, 49, 19, -106, -57, -110, 89, 36, -73] [[2, 4, 11, 7, 13, 11], [0, -3, -23, -5, -22, -13]] [600., 165.810532415714] [58, 94] unweighted badness 277.161590 unweighted complexity 26.724521 rms error 2.006172 Paraorwell [11, -6, 10, 7, 15, -35, -15, -27, -17, 40, 37, 57, -15, 5, 26] [[1, 4, 1, 5, 5, 7], [0, -11, 6, -10, -7, -15]] [1200., 263.695224647719] [9, 41, 50, 91] unweighted badness 271.488360 unweighted complexity 13.234425 rms error 5.638890 [3, 29, 11, 16, 7, 39, 9, 15, 0, -56, -63, -91, 7, -21, -35] [[1, 3, 16, 8, 11, 7], [0, -3, -29, -11, -16, -7]] [1200., 565.936121918345] [53, 70] unweighted badness 282.926501 unweighted complexity 14.126217 rms error 5.328866 Miracle [6, -7, -2, 15, 38, -25, -20, 3, 38, 15, 59, 114, 49, 114, 76] [[1, 1, 3, 3, 2, 0], [0, 6, -7, -2, 15, 38]] [1200., 116.779511703154] [41, 72, 113] unweighted badness 224.267777 unweighted complexity 21.718656 rms error 2.215733 [4, 9, 26, 10, -2, 5, 30, 2, -18, 35, -8, -38, -62, -102, -44] [[1, 1, 1, -1, 2, 4], [0, 4, 9, 26, 10, -2]] [1200., 175.873678872191] [41] unweighted badness 280.118702 unweighted complexity 13.315405 rms error 5.765149 [4, -37, -3, -19, -31, -68, -16, -44, -64, 97, 84, 65, -43, -76, -37] [[1, 0, 17, 4, 11, 16], [0, 4, -37, -3, -19, -31]] [1200., 476.008069832587] [58, 63, 121, 184] unweighted badness 298.019307 unweighted complexity 25.845696 rms error 2.268099
Message: 6188 - Contents - Hide Contents Date: Sun, 26 Jan 2003 11:52:24 Subject: Re: hi everybodyyyyyyyyyyyyyyyyyyyyyyyyy From: Dave Keenan Hi Paul, I think I understand the point you're making, and it's a good one, but I don't think you have described the situation correctly. --- In tuning-math@xxxxxxxxxxx.xxxx "wallyesterpaulrus <wallyesterpaulrus@y...>" <wallyesterpaulrus@y...> wrote:> on this list, > > one group of people is talking about equal temperaments which belong > to important families of tunings (each family being where a > particular set of unison vectors vanishes),You mean the folks searching for and cataloging good temperaments, in particular linear temperaments (LTs) at various odd limits?> another group is concerned with notating equal temperaments according > to where the potential unison vectors lie relative to their chains of > fifths,You mean the "Common notation .." thread. I don't know what you mean by "potential" unison vectors. Only commas that _don't_ vanish are of any use in notating a temperament.> and the two groups are not talking to one another. > > am i perceiving the situation correctly?I was actively involved in the linear temperament effort up until 7-limit. Gene has contributed to the Common notation thread regarding notating linear temperaments. But I agree he seems to have been following the idea that a temperament can be notated adequately using the notation for its most representative ET, and apparently assuming that George and I have already found the best notation for that ET (within the constraints we have imposed upon ourselves). neither of which may e true. Although chains of fifths are the backbone of the sagittal notation, this does not prevent it from notating ETs in LT-specific ways, so the same ET can be notated differently depending on which LT you are considering it as. I recently gave some examples in a "Notating Linear Temperaments" thread (or some such), but no one responded or carried it forward.
Message: 6189 - Contents - Hide Contents
Date: Sun, 26 Jan 2003 00:26:12
Subject: Re: A common notation for JI and ETs
From: David C Keenan
Hi George,
I agree with most of your suggestions re single ASCII characters for
sagittal. Previously I figured there were so few approximate up-down pairs
that the left-right pairs <> [] {} had to get used somewhere. Sure they're
laterally confusable, but so are / and \. However I think you've shown that
we can get by without using them.
How's this? (in order of size relative to strict Pythagorean)
'| ' 5'-comma sharp 32768:32805
.! . 5'-comma flat
|( ` 5:7-comma sharp 5103:5120
!( , 5:7-comma flat
~| ~ 17-comma sharp 2176:2187
~! $ or z 17-comma flat
~|( h 17'-comma sharp 4096:4131
~!( y 17'-comma flat
/| / 5-comma sharp 80:81
\! \ 5-comma flat
|) f 7-comma sharp 63:64
!) t 7-comma flat
|\ & 55-comma sharp 54:55
!/ % 55-comma flat
(| ? 7:11-comma sharp 45056:45927
(! j 7:11-comma flat
(|( d 5:11-comma sharp 44:45
(!( q 5:11-comma flat
//| // 25-diesis sharp 6400:6561
\\! \\ 25-diesis flat
/|) n 13-diesis sharp 1024:1053
\!) u 13-diesis flat
/|\ ^ 11-diesis sharp 32:33
\!/ v 11-diesis flat
(|) @ 11'-diesis sharp 704:729
(!) U or o 11'-diesis flat
(|\ m 13'-diesis sharp 26:27
(!/ w 13'-diesis flat
/||\ # apotome sharp 2048:2187
\!!/ b apotome flat
Which do you prefer out of $ or z and U or o?
I note that some folk have in the past used t for the tartini half-sharp
and d for the backwards-flat (meaning half-flat), but I don't think that
should stop us using them in other ways here.
>For 217-ET that's 12 pairs of characters. I don't think I would want
>to see ) paired with (, for example.
OK. Let's not use ( or ) at all in the single ASCII character version.
> For the 5:7 comma why not ( for
>up and { for down, and the 19 comma would be ) and }.
No. I find absolutely nothing to suggest which is up and which is down in
each pair, or even that they _are_ a pair.
> Maybe a 17 comma
>up could be S and down s, or would that be better for the 23 comma?
I don't like using uppercase-lowercase pairs. Too confusable. I know we're
using x and X in the multi-ASCII, but these should be very rare and will
often have additional direction cues provided by straight flags, e.g. \x/ /X\
The 23-comma is so rare it doesn't need a single ASCII character.
>After that it gets more difficult.
Yes. No need to go beyond 17.
>Do we really need shorthand ascii notation for anything more than
>straight and convex-flag symbols?
It would be nice to have the full 217-ET set provided they're reasonably
memorable, or figure-out-able.
> (Or are you intending to combine
>those single-character ascii symbols in any way?)
No.
> I think that it
>would get pretty complicated (hence difficult to remember), and the
>result would have very resemblance to what sagittal symbols look
>like.
Well how do you think we're doing above?
> Wouldn't it be more productive just to use the sagittal ascii
>system that we're already using for these things?
I'm not sure what you mean by "productive". The thing to do is to provide a
key or legend like those above, whenever you use these ASCII symbols.
-- Dave Keenan
Brisbane, Australia
Dave Keenan's Home Page * [with cont.] (Wayb.)
Message: 6190 - Contents - Hide Contents Date: Sun, 26 Jan 2003 13:00:42 Subject: Re: Graham's top 20, with standard vals From: Graham Breed Gene Ward Smith wrote:>This is the same list as before, only this time I included all of the standard vals consistent with the temperament. The good news is this: > >(1) 18 out of the 20 systems can be defined by wedging two standard val >So these have to be consistent?>(2) In no case is it neccessary to use vals where the number of divisions of the octave of one is more than twice that of another >Now that is interesting..>(3) The vals could have been filtered. Here are the ets from 30 to 130, with badness less than 1.6: > >[31, 36, 37, 41, 43, 44, 46, 50, 53, 56, 58, 62, 63, 68, 70, 72, 77, 79, 80, 87, 94, 103, 111, 113, 118, 121, 130] > >We see that this would have more than sufficed to get our 18 >I misread that, and generated 27 ETs with a cutoff of 0.6 scale step errors. [17, 26, 27, 29, 31, 34, 36, 41, 43, 46, 50, 53, 58, 60, 62, 63, 65, 72, 77, 80, 87, 94, 103, 104, 111, 113, 121] They give me all 20 temperaments (it took 5 seconds). And also one more, h26&h104, with a generator of 9.8 and a period of 1/26 octaves. Generator mapping [1 2 0 0 1]. It breaks your rule because 26*2<104. I'll have missed it before because I wasn't going high enough to get 104-equal. If I'd taken 104 ETs instead of 100 I would have got it. It takes a whole minute to search through 200 ETs.>The bad news is we missed two; one had the biggest error and second lowest complexity on the list, and the other had a very small period of 1/36 octave. A separate system for dealing with small periods may be required. The other bad news is that we are skating on thin ice, as many of the temperaments had only the minimum two standard vals. > >Was that one of them? The error's 2.5 cents (minimax) and the complexity's 52. So it doesn't sound like either. I can't find another one that was missing. Graham
Message: 6191 - Contents - Hide Contents Date: Sun, 26 Jan 2003 14:25:03 Subject: Re: hi everybodyyyyyyyyyyyyyyyyyyyyyyyyy From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan <d.keenan@u...>" <d.keenan@u...> wrote:> Gene has contributed to the Common notation thread regarding notating > linear temperaments. But I agree he seems to have been following the > idea that a temperament can be notated adequately using the notation > for its most representative ET, and apparently assuming that George > and I have already found the best notation for that ET (within the > constraints we have imposed upon ourselves). neither of which may e true.Oh. Well, darn.> Although chains of fifths are the backbone of the sagittal notation, > this does not prevent it from notating ETs in LT-specific ways, so the > same ET can be notated differently depending on which LT you are > considering it as. I recently gave some examples in a "Notating Linear > Temperaments" thread (or some such), but no one responded or carried > it forward.I can't follow these things when they involve the symbols, not at least unless I have a key handy (which I don't.)
Message: 6192 - Contents - Hide Contents Date: Sun, 26 Jan 2003 14:31:04 Subject: Re: Graham's top 20, with standard vals From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx Graham Breed <graham@m...> wrote:> Gene Ward Smith wrote: >>> This is the same list as before, only this time I included all of the standard vals consistent with the temperament. The good news is this: >> >> (1) 18 out of the 20 systems can be defined by wedging two standard val >>> So these have to be consistent?Vals are consistent by definition. What I meant was, wedging the val with the wedgie will give a zero vector.>> The bad news is we missed two; one had the biggest error and second lowest complexity on the list, and the other had a very small period of 1/36 octave. A separate system for dealing with small periods may be required. The other bad news is that we are skating on thin ice, as many of the temperaments had only the minimum two standard vals. >> >>> Was that one of them? The error's 2.5 cents (minimax) and the > complexity's 52. So it doesn't sound like either. I can't find another > one that was missing.The missing ones are the two on the list with only one standard val; you may not be using only standard vals. I wish you would explain this. By "standard" I mean the mapping is determined by rounding log2(p) for prime p to the nearest integer.
Message: 6193 - Contents - Hide Contents Date: Sun, 26 Jan 2003 01:54:43 Subject: Re: Temperament finder update From: Carl Lumma>You need the latest version of Python, the Numeric extensionsnumpy or Numarray? -C.> 0/1, 16.4 cent generator * [with cont.] (Wayb.) > 1/2, 16.4 cent generator * [with cont.] (Wayb.) Awesome!
Message: 6194 - Contents - Hide Contents Date: Sun, 26 Jan 2003 15:25:21 Subject: Re: Graham's top 20, with standard vals From: Graham Breed Gene Ward Smith wrote:> Vals are consistent by definition. What I meant was, wedging the val with the wedgie will give a zero vector.Say what? I thought a val was the complement of a vector.> The missing ones are the two on the list with only one standard val; you may not be using only standard vals. I wish you would explain this. By "standard" I mean the mapping is determined by rounding > log2(p) for prime p to the nearest integer.I did explain this -- I'm not only taking nearest prime approximations any more. Tricontaheximal is h72&h36. I don't see why you missed that. The other one, where you only give 41, requires an alternative mapping of 34-equal. This is actually more accurate than the nearest-prime mapping, and so the only one I use in the search. [34, 54, 79, 96, 118, 126] I didn't realize alternative mappings were getting in with such a low cutoff. g104 is another one, so here's the temperament that the unison vectors don't seem to cover: 1/5, 9.8 cent generator basis: (0.038461538461538464, 0.0081713150481090846) mapping by period and generator: [(26, 0), (41, 1), (60, 2), (73, 0), (90, 0), (96, 1)] mapping by steps: [(104, 26), (165, 41), (242, 60), (292, 73), (360, 90), (385, 96)] highest interval width: 2 complexity measure: 52 (78 for smallest MOS) highest error: 0.002107 (2.528 cents) unique I've updated the scripts at Linear Temperament Finding Home * [with cont.] (Wayb.) to use alternative mappings now. Graham
Message: 6195 - Contents - Hide Contents Date: Sun, 26 Jan 2003 02:38:59 Subject: Re: Temperament finder update From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx Graham Breed <graham@m...> wrote:> The optimized script for finding temperaments from unison vectors is at > > #!/c/Python22/python.exe * [with cont.] (Wayb.) > > You need the latest version of Python, the Numeric extensions and my > temperament library: > > # Temperament finding library -- definitions * [with cont.] (Wayb.)Last time I tried this I couldn't get it to work. I see about it again.
Message: 6196 - Contents - Hide Contents Date: Sun, 26 Jan 2003 15:37:21 Subject: Standard vals for my top 112 11-limit list From: Gene Ward Smith I found the standard vals for these, and listed them together with a filtered list with a badness cutoff of 1.5 and a size cutoff (which I shouldn't have done) of 20. Clearly 1.5 is too much filtering for this list. [102, 210, 216, 222, 96, 56, -1, -88, -211, -124] [[6, 11, 17, 20, 24], [0, -17, -35, -36, -37]] [342, 480, 822, 1164, 1506, 1848, 2190, 2532, 2670, 2874, 3354, 4038, 4722, 4860, 5202, 5886, 6228, 7050, 7734, 8418, 8898, 9582, 9924, 11430, 13278, 13620] [342, 1506, 1848, 2190] [64, 172, 102, 146, 124, -18, 10, -246, -256, 57] [[2, -6, -20, -9, -14], [0, 32, 86, 51, 73]] [328, 342, 670, 1012, 1354, 1696, 2038, 2366, 3050, 3720] [342, 1012] [38, 38, 114, 76, -28, 74, -11, 158, 45, -181] [[38, 60, 88, 106, 131], [0, 1, 1, 3, 2]] [152, 190, 342, 494, 646, 836, 1178, 1330, 1520, 1824, 1862, 2014, 2166, 2850, 3002, 3496] [152, 190, 342, 494, 836, 1178] [82, 28, 120, 155, -146, -40, -38, 200, 263, 20] [[1, -14, -3, -20, -26], [0, 82, 28, 120, 155]] [121, 342, 463, 805, 1147, 1268, 1489, 1831, 1952] [121, 342] [6, -48, -108, -168, -90, -188, -287, -116, -224, -98] [[6, 10, 10, 8, 7], [0, -1, 8, 18, 28]] [12, 342, 354, 696, 1038, 1380, 1722, 1734, 2064, 2418] [342] Hemiennealimmal [36, 54, 36, 18, 2, -44, -96, -68, -145, -74] [[18, 28, 41, 50, 62], [0, 2, 3, 2, 1]] [72, 126, 198, 270, 342, 414, 468, 486, 612, 738, 756, 882, 954, 1008, 1098, 1152, 1296, 1422, 1494, 1566, 1692, 1962, 2034, 2232] [72, 198, 270, 342, 612] [8, -64, -30, -110, -120, -70, -202, 110, -34, -205] [[2, 2, 14, 10, 23], [0, 4, -32, -15, -55]] [130, 212, 342, 472, 602, 814, 1156, 1286, 1498, 1758, 1970, 2100, 2230] [130, 342] [2, -16, 78, 58, -30, 118, 85, 226, 190, -107] [[2, 3, 6, -1, 2], [0, 1, -8, 39, 29]] [106, 118, 224, 342, 460, 566, 790, 802, 908, 1014, 1250] [118, 224, 342, 566] [44, -10, 6, 79, -118, -114, -27, 42, 218, 201] [[1, 3, 2, 3, 6], [0, -44, 10, -6, -79]] [31, 280, 311, 342, 373, 591, 653, 715, 964, 995, 1275, 1337] [31, 311, 342, 653] [0, 0, 0, 171, 0, 0, 271, 0, 397, 480] [[171, 271, 397, 480, 592], [0, 0, 0, 0, -1]] [171, 342, 513, 855, 1197, 1368] [342] [6, -48, -108, 3, -90, -188, -16, -116, 173, 382] [[3, 4, 13, 22, 10], [0, 2, -16, -36, 1]] [159, 183, 342, 501, 525, 708, 867, 1209] [159, 183, 342] [10, -80, 48, -52, -150, 48, -117, 336, 156, -312] [[2, 2, 14, 0, 13], [0, 5, -40, 24, -26]] [94, 154, 248, 342, 436, 590, 778, 932] [94, 342] [8, -64, -30, 61, -120, -70, 69, 110, 363, 275] [[1, 1, 7, 5, -1], [0, 8, -64, -30, 61]] [41, 260, 301, 342, 383, 424, 643, 725, 1067] [41, 342, 383] [20, -30, -10, -80, -94, -72, -196, 61, -82, -190] [[10, 16, 23, 28, 34], [0, -2, 3, 1, 8]] [130, 140, 270, 400, 410, 670, 680, 940, 950, 1210] [130, 270, 400] [6, -36, -84, -132, -71, -150, -230, -94, -182, -80] [[6, 10, 11, 10, 10], [0, -1, 6, 14, 22]] [12, 270, 282, 552, 822, 1092] [270] [58, -33, -2, -16, -187, -166, -226, 88, 77, -38] [[1, 25, -11, 2, -3], [0, -58, 33, 2, 16]] [52, 109, 161, 270, 379, 431, 592, 649, 701, 919, 971, 1189, 1241] [270] Ennealimmal [18, 27, 18, 144, 1, -22, 166, -34, 241, 342] [[9, 15, 22, 26, 37], [0, -2, -3, -2, -16]] [270, 369, 639, 909, 1179] [270] [2, -57, -28, 46, -95, -50, 66, 95, 304, 226] [[1, 1, 19, 11, -10], [0, 2, -57, -28, 46]] [41, 188, 229, 270, 311, 352, 499, 581, 663, 769, 851, 892, 1121, 1203, 1391, 1432, 1473, 1514, 1661, 1931, 1972, 2013, 2054, 2201] [41, 270, 311] [34, -24, 64, -28, -117, 6, -162, 216, 18, -300] [[2, -3, 9, -6, 12], [0, 17, -12, 32, -14]] [22, 226, 248, 270, 292, 314, 518, 562, 766, 788, 832, 1058, 1102, 1328, 1372, 1642] [22, 270] [14, 6, 74, 52, -23, 78, 34, 155, 100, -110] [[2, 4, 5, 10, 10], [0, -7, -3, -37, -26]] [118, 152, 270, 388, 422, 658, 692, 928, 962, 1198, 1232, 1468, 1502, 1738] [118, 152, 270, 422] [24, -9, -66, 12, -70, -172, -64, -128, 59, 262] [[3, 2, 8, 16, 9], [0, 8, -3, -22, 4]] [87, 96, 183, 270, 357, 444, 453, 627, 723, 897, 984, 993, 1167, 1263, 1437, 1533, 1707] [87, 183, 270] [16, 84, 46, 98, 96, 28, 100, -129, -63, 116] [[2, 4, 9, 8, 12], [0, -8, -42, -23, -49]] [58, 212, 270, 328, 482, 598, 752] [58, 270] [38, -3, 8, 64, -93, -94, -30, 27, 159, 152] [[1, -7, 3, 1, -11], [0, 38, -3, 8, 64]] [31, 208, 239, 270, 301, 509, 571, 748, 779, 841, 1049, 1111, 1319, 1381, 1589] [31, 270] [42, 47, 34, 33, -23, -64, -93, -53, -86, -25] [[1, -13, -14, -9, -8], [0, 42, 47, 34, 33]] [72, 95, 167, 239, 311, 383, 455, 550, 694, 861] [72, 311, 383] Octoid [24, 32, 40, 24, -5, -4, -45, 3, -55, -71] [[8, 13, 19, 23, 28], [0, -3, -4, -5, -3]] [72, 80, 152, 224, 296, 368, 376, 520] [72, 80, 152, 224, 296] Hemiamity [10, 26, -34, -28, 18, -82, -79, -152, -155, 39] [[2, 1, -1, 13, 13], [0, 5, 13, -17, -14]] [46, 106, 152, 198, 244, 350] [46, 152, 198] [2, -16, -40, -60, -30, -69, -102, -48, -84, -30] [[2, 3, 6, 9, 12], [0, 1, -8, -20, -30]] [12, 118, 130, 248, 366, 378, 626] [118, 130] [18, -14, 30, -20, -64, -3, -94, 109, 2, -160] [[2, 4, 4, 7, 6], [0, -9, 7, -15, 10]] [22, 86, 108, 130, 152, 282] [22, 130, 152] [23, -1, 13, 42, -55, -44, -13, 33, 101, 73] [[1, 9, 2, 7, 17], [0, -23, 1, -13, -42]] [31, 121, 152, 183, 214, 397] [31, 121, 152, 183] [18, 39, 42, 9, 20, 16, -48, -12, -114, -120] [[3, 2, 1, 2, 9], [0, 6, 13, 14, 3]] [72, 111, 183, 255, 327, 399, 438, 582] [72, 111, 183] [24, 20, 16, -12, -24, -42, -102, -19, -97, -89] [[4, 6, 9, 11, 14], [0, 6, 5, 4, -3]] [4, 68, 72, 76, 140, 212, 284] [72] [1, 33, 27, -18, 50, 40, -32, -30, -156, -144] [[1, 2, 16, 14, -4], [0, -1, -33, -27, 18]] [41, 70, 111, 152, 193, 234, 345] [41, 111, 152] [30, 13, 14, 3, -49, -62, -99, -4, -38, -40] [[1, -13, -4, -4, 2], [0, 30, 13, 14, 3]] [35, 37, 72, 109, 181, 253, 290] [72] Unidec [12, 22, -4, -6, 7, -40, -51, -71, -90, -3] [[2, 5, 8, 5, 6], [0, -6, -11, 2, 3]] [26, 46, 72, 118, 164, 190] [26, 46, 72, 118, 190] Minorsemi [18, 15, -6, 9, -18, -60, -48, -56, -31, 46] [[3, 6, 8, 8, 11], [0, -6, -5, 2, -3]] [15, 57, 72, 87, 102, 159, 231] [72, 87, 159] [12, 34, 20, 30, 26, -2, 6, -49, -48, 15] [[2, 4, 7, 7, 9], [0, -6, -17, -10, -15]] [58, 72, 130, 202, 274, 332, 346, 476] [58, 72, 130] [3, -24, -1, 28, -45, -10, 34, 65, 148, 82] [[1, 1, 7, 3, -2], [0, 3, -24, -1, 28]] [41, 77, 118, 159, 200] [41, 118, 159] [6, 29, -2, -21, 32, -20, -54, -86, -149, -52] [[1, 4, 14, 2, -5], [0, -6, -29, 2, 21]] [5, 72, 77, 149] [72] Wizard [12, -2, 20, -6, -31, -2, -51, 52, -7, -86] [[2, 1, 5, 2, 8], [0, 6, -1, 10, -3]] [22, 50, 72, 94, 116, 122, 166] [22, 50, 72, 94] [18, 27, 18, 45, 1, -22, 9, -34, 11, 64] [[9, 15, 22, 26, 33], [0, -2, -3, -2, -5]] [72, 171, 243, 315] [72] [18, -9, 18, 9, -56, -22, -48, 67, 52, -37] [[9, 14, 21, 25, 31], [0, 2, -1, 2, 1]] [9, 54, 63, 72, 81, 135] [72] Catakleismic [6, 5, 22, -21, -6, 18, -54, 37, -66, -135] [[1, 0, 1, -3, 9], [0, 6, 5, 22, -21]] [19, 53, 72, 91, 125] [72] Hemiwuerschmidt [16, 2, 5, 40, -34, -37, 8, 6, 86, 95] [[1, -1, 2, 2, -3], [0, 16, 2, 5, 40]] [31, 130, 161, 192, 223, 291, 353, 452] [31, 130] Hemithird [15, -2, -5, 22, -38, -50, -17, -6, 58, 79] [[1, 4, 2, 2, 7], [0, -15, 2, 5, -22]] [31, 56, 87, 118, 143, 149] [31, 87, 118] [6, -19, -26, -21, -44, -58, -54, -7, 17, 31] [[1, 2, 1, 1, 2], [0, -6, 19, 26, 21]] [29, 43, 72, 115, 158, 187] [29, 72] Duodecimal [0, 12, 24, 36, 19, 38, 57, 22, 42, 18] [[12, 19, 28, 34, 42], [0, 0, -1, -2, -3]] [12, 72, 84, 156, 228] [72] [6, 5, 22, 51, -6, 18, 60, 37, 101, 67] [[1, 0, 1, -3, -10], [0, 6, 5, 22, 51]] [72, 197] [72] Miracle [6, -7, -2, 15, -25, -20, 3, 15, 59, 49] [[1, 1, 3, 3, 2], [0, 6, -7, -2, 15]] [10, 31, 41, 72, 103, 113, 175] [31, 41, 72] [17, 6, 15, 27, -30, -24, -16, 18, 42, 24] [[1, -5, 0, -3, -7], [0, 17, 6, 15, 27]] [31, 49, 80, 111, 142, 173] [31, 80, 111] Slender [13, -10, 6, 17, -46, -27, -18, 42, 74, 27] [[1, 2, 2, 3, 4], [0, -13, 10, -6, -17]] [31, 32, 63, 94, 125] [31, 94] Schismatic [1, -8, -14, 23, -15, -25, 33, -10, 81, 113] [[1, 2, -1, -3, 13], [0, -1, 8, 14, -23]] [41, 53, 94, 135] [41, 94] [11, 1, -19, -17, -24, -61, -65, -47, -43, 18] [[1, -2, 2, 9, 9], [0, 11, 1, -19, -17]] [3, 43, 46, 89] [46, 89] [6, -12, 10, -14, -33, -1, -43, 57, 9, -74] [[2, 4, 3, 7, 5], [0, -3, 6, -5, 7]] [22, 36, 58, 80] [22, 36, 58, 80] Supersupermajor [3, 17, -1, -13, 20, -10, -31, -50, -89, -33] [[1, 1, -1, 3, 6], [0, 3, 17, -1, -13]] [5, 41, 46, 87, 128, 133, 169] [41, 46, 87] Diaschismic [2, -4, -16, -24, -11, -31, -45, -26, -42, -12] [[2, 3, 5, 7, 9], [0, 1, -2, -8, -12]] [12, 46, 58] [46, 58] Nonkleismic [10, 9, 7, 25, -9, -17, 5, -9, 27, 46] [[1, -1, 0, 1, -3], [0, 10, 9, 7, 25]] [31, 58, 89, 120, 151] [31, 58, 89] [12, 5, -9, 1, -20, -48, -40, -35, -15, 34] [[1, -4, 0, 7, 3], [0, 12, 5, -9, 1]] [15, 28, 43, 58, 73] [58] Quartaminorthirds [9, 5, -3, 7, -13, -30, -20, -21, -1, 30] [[1, 1, 2, 3, 3], [0, 9, 5, -3, 7]] [15, 16, 31, 46, 61, 77] [31, 46] Magic [5, 1, 12, -8, -10, 5, -30, 25, -22, -64] [[1, 0, 2, -1, 6], [0, 5, 1, 12, -8]] [19, 22, 41, 60, 63, 85, 104] [22, 41] Tritonic [5, -11, -12, -3, -29, -33, -22, 3, 31, 33] [[1, 4, -3, -3, 2], [0, -5, 11, 12, 3]] [2, 29, 31, 33, 64] [29, 31] Schismic [1, -8, -14, -18, -15, -25, -32, -10, -14, -2] [[1, 2, -1, -3, -4], [0, -1, 8, 14, 18]] [12, 29, 41] [29, 41] Superkleismic [9, 10, -3, 2, -5, -30, -28, -35, -30, 16] [[1, 4, 5, 2, 4], [0, -9, -10, 3, -2]] [15, 26, 41, 56, 71, 97] [26, 41] Orwell [7, -3, 8, 2, -21, -7, -21, 27, 15, -22] [[1, 0, 3, 1, 3], [0, 7, -3, 8, 2]] [9, 22, 31, 40, 53, 75] [22, 31] Meanpop [1, 4, 10, -13, 4, 13, -24, 12, -44, -71] [[1, 2, 4, 7, -2], [0, -1, -4, -10, 13]] [19, 31, 50, 81] [31, 50] Supermajor seconds [3, 12, -1, -8, 12, -10, -23, -36, -60, -19] [[1, 1, 0, 3, 5], [0, 3, 12, -1, -8]] [5, 26, 31, 57, 88] [26, 31] Meantone [1, 4, 10, 18, 4, 13, 25, 12, 28, 16] [[1, 2, 4, 7, 11], [0, -1, -4, -10, -18]] [12, 31, 43, 74, 105] [31] [2, 8, -11, 5, 8, -23, 1, -48, -16, 52] [[1, 1, 0, 6, 2], [0, 2, 8, -11, 5]] [7, 24, 31, 38, 55] [24, 31] [4, 16, 9, 10, 16, 3, 2, -24, -32, -3] [[1, 3, 8, 6, 7], [0, -4, -16, -9, -10]] [31] [31] [1, 9, -2, 16, 12, -6, 22, -30, 6, 52] [[1, 2, 6, 2, 10], [0, -1, -9, 2, -16]] [22, 49] [22] Double wide [8, 6, 6, -4, -9, -13, -34, -3, -30, -32] [[2, 5, 6, 7, 6], [0, -4, -3, -3, 2]] [4, 18, 22, 26, 48] [22, 26] Pajara [2, -4, -4, -12, -11, -12, -26, 2, -14, -20] [[2, 3, 5, 6, 8], [0, 1, -2, -2, -6]] [12, 22] [22] Porcupine [3, 5, -6, 4, 1, -18, -4, -28, -8, 32] [[1, 2, 3, 2, 4], [0, -3, -5, 6, -4]] [7, 8, 15, 22, 37, 59] [22] Pajarous [2, -4, -4, 10, -11, -12, 9, 2, 37, 42] [[2, 3, 5, 6, 6], [0, 1, -2, -2, 5]] [10, 22, 32, 54] [22] Tripletone [3, 0, -6, -6, -7, -18, -20, -14, -14, 4] [[3, 5, 7, 8, 10], [0, -1, 0, 2, 2]] [3, 12, 15] [] Injera [2, 8, 8, 12, 8, 7, 12, -4, 0, 6] [[2, 3, 4, 5, 6], [0, 1, 4, 4, 6]] [12, 26] [26] [3, -5, -6, -1, -15, -18, -12, 0, 15, 18] [[1, 3, 0, 0, 3], [0, -3, 5, 6, 1]] [2, 13, 15] [] [6, 0, 3, 3, -14, -12, -16, 7, 7, -2] [[3, 4, 7, 8, 10], [0, 2, 0, 1, 1]] [6, 9, 15, 24, 39] [24] Kleismic [6, 5, 3, -2, -6, -12, -24, -7, -22, -16] [[1, 0, 1, 2, 4], [0, 6, 5, 3, -2]] [4, 15, 19, 34] [] Dominant Seventh [1, 4, -2, -6, 4, -6, -13, -16, -28, -10] [[1, 2, 4, 2, 1], [0, -1, -4, 2, 6]] [5, 12] [] Meanenneadecal [1, 4, 10, 6, 4, 13, 6, 12, 0, -18] [[1, 2, 4, 7, 6], [0, -1, -4, -10, -6]] [12, 19] [] [4, -3, 2, 5, -14, -8, -6, 13, 22, 7] [[1, 2, 2, 3, 4], [0, -4, 3, -2, -5]] [9, 10, 19] [] [4, 2, 2, 10, -6, -8, 2, -1, 16, 21] [[2, 4, 5, 6, 9], [0, -2, -1, -1, -5]] [10] [] [5, 3, 7, 4, -7, -3, -11, 8, -1, -13] [[1, 1, 2, 2, 3], [0, 5, 3, 7, 4]] [9] [] [2, -6, 1, -2, -14, -4, -10, 19, 16, -9] [[1, 2, 1, 3, 3], [0, -2, 6, -1, 2]] [9] [] [0, 5, 0, -5, 8, 0, -8, -14, -29, -14] [[5, 8, 12, 14, 17], [0, 0, -1, 0, 1]] [5, 10, 15] [] Opossum [3, 5, 9, 4, 1, 6, -4, 7, -8, -20] [[1, 2, 3, 4, 4], [0, -3, -5, -9, -4]] [15] [] [4, 2, 2, -4, -6, -8, -20, -1, -16, -18] [[2, 4, 5, 6, 6], [0, -2, -1, -1, 2]] [4] [] Septimal [0, 0, 7, 0, 0, 11, 0, 16, 0, -24] [[7, 11, 16, 20, 24], [0, 0, 0, -1, 0]] [7] [] Pajaric [2, -4, -4, 0, -11, -12, -7, 2, 14, 14] [[2, 3, 5, 6, 7], [0, 1, -2, -2, 0]] [2, 10, 12] [] [2, 1, -4, 5, -3, -12, 1, -12, 8, 28] [[1, 1, 2, 4, 2], [0, 2, 1, -4, 5]] [7, 10, 17] [] Diminished [4, 4, 4, 0, -3, -5, -14, -2, -14, -14] [[4, 6, 9, 11, 14], [0, 1, 1, 1, 0]] [4, 12] [] [3, 0, 6, 6, -7, 1, -1, 14, 14, -4] [[3, 5, 7, 9, 11], [0, -1, 0, -2, -2]] [9, 12, 21] [] [1, 4, -2, 6, 4, -6, 6, -16, 0, 24] [[1, 2, 4, 2, 6], [0, -1, -4, 2, -6]] [7, 12] [] [3, 0, -3, -3, -7, -13, -15, -7, -7, 2] [[3, 5, 7, 8, 10], [0, -1, 0, 1, 1]] [3, 9] [] [1, -3, -4, -1, -7, -9, -5, -1, 8, 11] [[1, 2, 1, 1, 3], [0, -1, 3, 4, 1]] [2, 9, 11] [] [2, 1, 6, 5, -3, 4, 1, 11, 8, -7] [[1, 1, 2, 1, 2], [0, 2, 1, 6, 5]] [10] [] Arnold [1, 4, -2, -1, 4, -6, -5, -16, -16, 4] [[1, 2, 4, 2, 3], [0, -1, -4, 2, 1]] [5, 7] [] Pentoid [2, 3, 1, -2, 0, -4, -10, -6, -15, -9] [[1, 2, 3, 3, 3], [0, -2, -3, -1, 2]] [4, 5, 9, 14] [] [2, 3, 1, 7, 0, -4, 4, -6, 6, 16] [[1, 2, 3, 3, 5], [0, -2, -3, -1, -7]] [9] [] [2, 1, -4, -2, -3, -12, -10, -12, -8, 8] [[1, 1, 2, 4, 4], [0, 2, 1, -4, -2]] [3, 7] [] [1, -1, 3, -4, -4, 2, -10, 10, -6, -22] [[1, 2, 2, 4, 2], [0, -1, 1, -3, 4]] [3, 7] [] [1, -1, 3, 4, -4, 2, 3, 10, 13, 1] [[1, 2, 2, 4, 5], [0, -1, 1, -3, -4]] [5] [] [2, -2, -2, 0, -8, -9, -7, 1, 7, 7] [[2, 3, 5, 6, 7], [0, 1, -1, -1, 0]] [2, 6] [] [1, -1, -5, -4, -4, -11, -10, -9, -6, 6] [[1, 2, 2, 1, 2], [0, -1, 1, 5, 4]] [3] [] Meanertone [1, 4, 3, -1, 4, 2, -5, -4, -16, -13] [[1, 2, 4, 4, 3], [0, -1, -4, -3, 1]] [5] [] [1, 4, 3, 6, 4, 2, 6, -4, 0, 6] [[1, 2, 4, 4, 6], [0, -1, -4, -3, -6]] [5] [] [2, 1, 3, -2, -3, -1, -10, 4, -8, -16] [[1, 1, 2, 2, 4], [0, 2, 1, 3, -2]] [4, 7] [] [2, 1, -1, -2, -3, -7, -10, -5, -8, -2] [[1, 1, 2, 3, 4], [0, 2, 1, -1, -2]] [3, 4] [] [0, 0, 0, 5, 0, 0, 8, 0, 12, 14] [[5, 8, 12, 14, 17], [0, 0, 0, 0, 1]] [5] [] [1, -3, -2, -1, -7, -6, -5, 4, 8, 4] [[1, 2, 1, 2, 3], [0, -1, 3, 2, 1]] [2, 7] [] [2, 1, 3, 5, -3, -1, 1, 4, 8, 4] [[1, 1, 2, 2, 2], [0, 2, 1, 3, 5]] [7] [] [2, -1, 1, -2, -6, -4, -10, 5, -1, -9] [[1, 2, 2, 3, 3], [0, -2, 1, -1, 2]] [1, 4] []
Message: 6197 - Contents - Hide Contents Date: Sun, 26 Jan 2003 02:49:16 Subject: Re: Temperament finder update From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx Graham Breed <graham@m...> wrote:> takes about 50 seconds. Those results > > 1/2, 16.4 cent generator * [with cont.] (Wayb.) > > are the same as for the unison vector search!Not that surprising, but the comma search seems more likely to catch oddball systems. I suppose I could try to identify these.
Message: 6198 - Contents - Hide Contents Date: Sun, 26 Jan 2003 15:46:50 Subject: Re: Vals vs commas From: Graham Breed Gene Ward Smith wrote:> We run into a problem with the strictly val approach if fewer than two standard vals cover the temperament in question; this can happen because it is of relatively low complexity, or because it has a small period. One way out of this is to include non-standard vals--that is, look at various second best choices for mappings to primes; this however, increases the computational burden also.It doesn't add much to the computational burden. You can still search with the same number of ETs, because the new, simpler ones are as likely to give a hit as the old, accurate ones. Even using twice as many ETs only makes the search 4 times as hard. The vector search is orders of magnitude harder, and gets worse the more primes you use. It may be possible to tame it by pruning branches, but that'll also make it much more complex.> Graham reports that he didn't find he needed the comma list approach. > I can't quite see how this happened, since the Tricontaheximal temperament, with a period of 33 1/3 cents, has only one standard val, namely 72. Other temperaments are skating close to the wire by having only two standard vals--Hemififths with 41 and 58, and Diaschismic with 46 and 58 among the top four, which is all I have checked.Because I'm using non-standard vals. And this Tricontaheximal works anyway. Inconsistent ETs are more important the higher the odd limit, as simple temperaments can be missed. Probably non-standard vals will be important at the same time. I wasn't confident the ET search would compete with the vectors until I implemented them. There are other ways of doing the search that should complete with the pairs of ETs. One is to go through the list of ETs and choose each number of steps as the generator for a linear temperament. The number of LTs you look at is roughly proportional to the number of ETs, so it's equivalent to the pairs search in complexity. The list of ETs should be smaller. You also have to look at some alternative mappings with the same ET and generator. The other way is to start with a huge list of 5-limit linear temperaments (which is easy) and keep adding primes. Reject any LTs that are too complex or inaccurate as you go along. Graham
Message: 6199 - Contents - Hide Contents Date: Sun, 26 Jan 2003 02:50:04 Subject: Re: Graham's Top 20 13-limit temperaments From: Carl Lumma>i have access to a 2.4 GHz machine for running Matlab overnight or >for however long it takes. i'd be happy to try whatever algorithms >you wish to spell out.One page claims Matlab is implemented in C. I seem to think Maple is implemented in Maple, but I can't find that in the manual now. I'd be surprised if either of them were faster than python, but I could very well be wrong. -Carl
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