Yahoo Tuning Groups Ultimate Backup tuning-math "Ultimate" 5-limit again

I returned to this, and added names, poptimal generators (this time using
everyone's favorite defintion of the minimax generator) and "extensions".
These are defined in a way which is very strict and perhaps a little
arbitrary, but the results seem of some interest. I took the poptimal
generator, found the corresponding val with the lowest badness, extended
it in the way which gave lowest badness, and looked for the lowest
badness 7-limit temperament compatible with this val and the 5-limit
comma. I only go up to Monzimic with this list, which really seems far
enough. To make up for that and make Paul happy, I tacked Miracle on the
end despite the fact that as a 5-limit temperament it's nothing to get
excited about.

Bug 7/31 Extends 15/14 to Bug ([2, 3, 5, 0, 2, 3])
27/25 [[1, 2, 3], [0, -2, -3]] [1200., 268.056438833948093748427143263]
3.739252 35.609240 1861.731473

Pelogic 10/23 Extends 36/35 to Pelogic (aka Hexadecimal)
135/128 [[1, 2, 1], [0, -1, 3]] [1200., 522.862345874111793591855751693]
4.132031 18.077734 1275.365360

Blackwood Universal
256/243 [[5, 8, 12], [0, 0, -1]] [240., 84.6637865678588914278600509674]
5.493061 12.759741 2114.877638

Dicot Universal
25/24 [[1, 1, 2], [0, 2, 1]] [1200., 350.977500432693708872243366367]
3.025593 28.851897 799.108711

Diminished Universal
648/625 [[4, 6, 9], [0, 1, 1]] [300., 94.1343573651111175944350240576]
6.437752 11.060060 2950.938432

Negri 2/19 Extends 49/48 to Negri (aka Tertiathirds)
16875/16384 [[1, 2, 2], [0, -4, 3]] [1200.,
126.238272015257926746682149917]
8.172550 5.942563 3243.743713

Porcupine 11/81 Extends 64/63 to Porcupine
250/243 [[1, 2, 3], [0, -3, -5]] [1200., 162.996026370546548951179738408]
5.948286 7.975801 1678.609846

Augmented Universal
128/125 [[3, 5, 7], [0, -1, 0]] [400., 91.2018560670299909777049249654]
4.828314 9.677666 1089.323984

Magic 19/60 Extends 225/224 to Magic
3125/3072 [[1, 0, 2], [0, 5, 1]] [1200., 379.967949195094816842076920201]
7.741412 4.569472 2119.954990

Quadrafifths 26/177 Extends 245/243 to Octafifths
20000/19683 [[1, 1, 1], [0, 4, 9]] [1200.,
176.282270436412295298990817071]
9.785568 2.504205 2346.540676

Pythagoric Universal
531441/524288 [[12, 19, 28], [0, 0, -1]] [100.,
14.6637865678588914278600509674]
13.183347 1.382394 3167.444999

Meantone 34/81 Extends 126/125 to Meantone
81/80 [[1, 2, 4], [0, -1, -4]] [1200., 503.835154026035812053011163756]
4.132031 4.217731 297.556531

Diaschismic 10/114 Extends 245/243 to Shrutar
2048/2025 [[2, 3, 5], [0, 1, -2]] [600., 105.446531009812541696859310996]
6.271199 2.612822 644.408867

Tertiary 23/285 to Extends 3136/3125 to Tertiary ([3, -12, -30, -26, -56,
-36])
67108864/66430125 [[3, 5, 6], [0, -1, 4]] [400.,
96.7879385616949726317268914802]
15.510107 .905187 3377.402314

Hemisixths 55/149 Various extensions, none much good
78732/78125 [[1, -1, -1], [0, 7, 9]] [1200.,
442.979297439105373735900374126]
12.192182 1.157498 2097.802867

Wuerschmidt 53/164 Various extensions, none much good
393216/390625 [[1, -1, 2], [0, 8, 1]] [1200.,
387.819673068349143521938606127]
12.543123 1.071950 2115.395301

Orwell 43/190 Extends 1029/1024 to Trifokker ([21, -9, -7, -63, -70, 9])
2109375/2097152 [[1, 0, 3], [0, 7, -3]] [1200.,
271.589599585245148575185388331]
12.772341 .800410 1667.723301

Septathirds 31/673 No good extensions
4294967296/4271484375 [[1, 2, 2], [0, -9, 7]] [1200.,
55.2754932571412314963954609732]
18.573955 .483108 3095.692488

Kleismic 65/246 Extends 5120/5103 to Countercatakleismic ([6, 5, -31, -6,
-66, -86])
15625/15552 [[1, 0, 1], [0, 6, 5]] [1200.,
317.079675185758890225628070818]
9.338935 1.029625 838.631548

Amity 58/205 Extends 5120/5103 to Amity
1600000/1594323 [[1, 3, 6], [0, -5, -13]] [1200.,
339.508825625715624367834924710]
13.794200 .383104 1005.555381

Parakleismic 31/118 Extends 3136/3125 to Parakleismic ([13, 14, 35, -8,
19, 42])
1224440064/1220703125
[[1, 5, 6], [0, -13, -14]] [1200., 315.250913337821936408197840098]
21.322672 .276603 2681.521263

Vulture 128/323 Extends 4375/4374 to Vulture ([4, 21, -56, 24, -100,
-189])
10485760000/10460353203
[[1, 0, -6], [0, 4, 21]] [1200., 475.542233398945960632986914825]
21.733049 .153767 1578.433204

Semisuper 30/506 Nothing much good
6115295232/6103515625
[[2, 4, 5], [0, -7, -3]] [600., 71.1460635722374759764193142621]
21.207625 .194018 1850.624306

Enneadecal Universal
19073486328125/19042491875328
[[19, 30, 44], [0, 1, 1]] [63.1578947368421052631578947368,
7.29225210195322285759291880280]
30.579320 .104784 2996.244873

Semitonic 95/1019 No good extensions
295578376007080078125/295147905179352825856
[[1, 0, 4], [0, 17, -18]] [1200., 111.875426120872633513689333181]
38.845486 .058853 3449.774562

Tricot 233/494 Extends 4375/4374 to Tricot ([3, 29, -95, 39, -159, -302])
68719476736000/68630377364883
[[1, 3, 16], [0, -3, -29]] [1200., 565.988014913065527948022354197]
30.550812 .057500 1639.596150

Schismic 120/289 Extends 4375/4374 to Infraschismic ([1, -8, 39, -15, 59,
113])
32805/32768
[[1, 2, -1], [0, -1, 8]] [1200., 498.272487171563819993901705714]
9.459948 .161693 136.885775

Counterschismic 237/571 No good extensions
2954312706550833698643/2951479051793528258560
[[1, 2, 21], [0, -1, -45]] [1200., 498.082318148218414995068857757]
48.911647 .026391 3088.065497

Hemithird 232/1441 Extends 4375/4374 to Infrahemithird ([15, -2, 113,
-38, 137, 268])
274877906944/274658203125
[[1, 4, 2], [0, -15, 2]] [1200., 193.199614933859969427837273253]
24.977022 .060822 947.732642

Minortone 196/1289 Extends 2460375/2458624 to Hemiminortone
([30, 70, 129, 32, 109, 103]. Minortone is [17, 35, -21, 16, -81, -147])
50031545098999707/50000000000000000
[[1, -1, -3], [0, 17, 35]] [1200., 182.466089137089694182158775289]
38.845486 .025466 1492.763207

Ennnealimmal 68/1665 Extends 2401/2400 to Ennealimmal
7629394531250/7625597484987
[[9, 15, 22], [0, -2, -3]] [133.333333333333333333333333333,
49.0088197863290461293795242156]
33.653272 .025593 975.428947

Glum 103/935 Nothing much good
2475880078570760549798248448/2474715001881122589111328125
[[1, 5, 1], [0, -31, 12]] [1200., 132.194510561451335831533197063]
55.785793 .014993 2602.883149

Kwasy 182/1342 Extends 4375/4374 to Hemikwasy ([16, -10, 152, -53, 196,
381])
9010162353515625/9007199254740992
[[2, 1, 6], [0, 8, -5]] [600., 162.741892126380267669129153916]
31.255737 .017725 541.228379

Mum 341/730 Extends 4375/4374 to Mum ([33, 25, 131, -37, 115, 234])
116450459770592056836096/116415321826934814453125
[[1, 17, 14], [0, -33, -25]] [1200., 560.546969532517954992081849041]
50.788153 .012388 1622.898233

Bum 237/901 Extends 4375/4374 to Bum ([51, 52, 149, -36, 93, 200])
444089209850062616169452667236328125/444002166576103304796646509039845376
[[1, 15, 16], [0, -51, -52]] [1200., 315.647874693157629813083932838]
82.462759 .004660 2613.109284

Monzimic 116/559 Extends 4375/4374 to Monzimic ([2, 37, -134, 54, -218,
-415])
450359962737049600/450283905890997363
[[1, 2, 10], [0, -2, -37]] [1200., 249.018447894645757478665305415]
39.665603 .005738 358.125500

...

Miracle 41/422 Extends 225/224 to Miracle
34171875/33554432
[[1, 1, 3], [0, 6, -7]] [1200., 116.578231256479]
14.2507126003310 1.98070788903097 5732.31669654049

--- In tuning-math@yahoogroups.com, "Gene Ward Smith
<genewardsmith@j...>" <genewardsmith@j...> wrote:
> --- In tuning-math@yahoogroups.com, "wallyesterpaulrus
> <wallyesterpaulrus@y...>" <wallyesterpaulrus@y...> wrote:
>
> > probably i goofed. can you help fill in all the data for the row
> that
> > i would need to add to this table:
> >
> > /tuning/database?
> > method=reportRows&tbl=10&sortBy=5&sortDir=up
>
> The comma is 2^(-69) 3^45 5^(-1). Is that all you need?

here's the contents of the database (in very non-database form --
hopefully your modem will heal soon). note that this is intended to
go along with the charts on monz's equal temperament dictionary entry:

"comma name(s)", "as 2^x*3^y*5^z, [x y z]
is . . .", "numerator", "denominator", "cents", "temperament name
(s)", "sample ETs/MOSs (notes per octave)", "[period, optimal
generator] (cents)", "[2 3 5] map in terms of [per., gen.]", "optimum
RMS error (cents)"
" large limma", "0 3 -2", "27", "25", "133.237575", "beep", "4, 5,
9", "[1200, 268.056439]", "[[1,0] [2,-2] [3,-3]]", "35.609240"
" small diesis", "-10 -1
5", "3125", "3072", "29.613568", "magic", "3, 16, 19, 22, 25, 35, 41,
60, 63, 79", "[1200, 379.967949]", "[[1,0] [0,5] [2,1]]", "4.569472"
" minimal diesis", "5 -9
4", "20000", "19683", "27.659848", "tetracot", "7, 27, 34, 41, 48,
61, 75", "[1200, 176.282270]", "[[1,0] [1,4] [1,9]]", "2.504205"
" diaschisma", "11 -4 -2", "2048", "2025", "19.552569", "diaschismic,
5-limit pajara", "12, 22, 34, 46, 56, 58, 70, 78, 80, 90", "[600,
105.446531]", "[[2,0] [3,1] [5,-2]]", "2.612822"
" pythagorean comma", "-19 12
0", "531441", "524288", "23.460010", "aristoxenean", "12, 48, 60, 72,
84, 96", "[100, 14.663787]", "[[12,0] [19,0] [28,-1]]", "1.382394"
"-", "2 9 -7", "78732", "78125", "13.399010", "semisixths", "19, 27,
46, 65, 73, 84", "[1200, 442.979297]", "[[1,0] [-1,7] [-
1,9]]", "1.157498"
"-", "26 -12 -3", "67108864", "66430125", "17.598848", "misty", "12,
63, 75, 87, 99, 205", "[400, 96.787939]", "[[3,0] [5,-1]
[6,4]]", "0.905187"
" würschmidt's comma", "17 1 -
8", "393216", "390625", "11.445290", "würschmidt", "28, 31, 34, 37,
65, 71, 96, 99", "[1200, 387.819673]", "[[1,0] [-1,8]
[2,1]]", "1.071950"
" kleisma", "-6 -5 6", "15625", "15552", "8.107279", "kleismic,
hanson", "15, 19, 23, 34, 53, 72, 83, 87, 91, 125", "[1200,
317.079675]", "[[1,0] [0,6] [1,5]]", "1.029625"
" semicomma", "-21 3
7", "2109375", "2097152", "10.061000", "orwell", "9, 22, 31, 53, 75,
84, 97, 128, 243", "[1200, 271.589600]", "[[1,0] [0,7] [3,-
3]]", "0.800410"
"-", "9 -13 5", "1600000", "1594323", "6.1535581", "amity", "39, 46,
53, 60, 99, 152, 205, 311", "[1200, 339.508826]", "[[1,0] [3,-5] [6,-
13]]", "0.383104"
" minor chroma, classic chromatic semitone", "-3 -1
2", "25", "24", "70.672427", "dicot", "3, 4, 7, 10", "[1200.
350.977500]", "[[1,0] [1,2] [2,1]]", "28.851897"
"-", "32 -7 -
9", "4294967296", "4271484375", "9.491569", "escapade", "22, 43, 65,
87, 152, 217", "[1200, 55.275493]", "[[1,0] [2,-9] [2,7]]", "0.483108"
" parakleisma", "8 14 -
13", "1224440064", "1220703125", "5.291732", "parakleismic", "19, 42,
61, 80, 99, 118, 217", "[1200, 315.250913]", "[[1,0] [5,13] [6,-
14]]", "0.276603"
" schisma", "-15 8 1", "32805", "32768", "1.953721", "schismic,
helmholtz/groven", "29, 41, 53, 65, 77, 89, 118, 171, 200", "[1200,
498.272487]", "[[1,0] [2,-1] [-1,8]]", "0.161693"
"-", "24 -21
4", "10485760000", "10460353203", "4.199837", "vulture", "48, 53, 58,
217, 270, 323", "[1200, 475.542233]", "[[1,0] [0,4] [-
6,21]]", "0.153767"
"-", "23 6 -
14", "6115295232", "6103515625", "3.338011", "semisuper", "16, 18,
34, 50, 84, 118, 152, 270, 388", "[600, 71.146064]", "[[2,0] [4,-7]
[5,-3]]", "0.194018"
"'19-tone comma'", "-14 -19
19", "19073486328125", "19042491875328", "2.815547", "enneadecal", "19
, 152, 171, 323, 494, 665", "[63.157894, 7.292252]", "[[19,0] [30,1]
[44,1]]", "0.104784"
"-", "39 -29
3", "68719476736000", "68630377364883", "2.246116", "tricot", "53,
388, 441, 494, 547, 600", "[1200, 565.988015]", "[[1,0] [3,-3] [16,-
29]]", "0.057500"
"-", "-68 18
17", "295578376007080078125", "295147905179352825856", "2.523151", "va
voom", "118, 311, 547, 665", "[1200, 111.875426]", "[[1,0] [0,17] [4,-
18]]", "0.058853"
"-", "38 -2 -
15", "274877906944", "274658203125", "1.384290", "semithirds", "118,
205, 323, 441, 559, 1000", "[1200, 193.199615]", "[[1,0] [4,-15] [2,
2]]", "0.060822"
" ennealimma", "1 -27
18", "7629394531250", "7625597484987", "0.861826", "ennealimmal", "72,
99, 171, 243, 270, 441, 612", "[133.333333, 49,008820]", "[[9,0]
[15,-2] [22,-3]]", "0.025593"
" major chroma, major limma, limma ascendant", "-7 3
1", "135", "128", "92.178716", "pelogic", "7, 9, 16, 23", "[1200,
522.862346]", "[[1,0] [2,-1] [1,3]]", "18.077734"
"-", "-16 35 -
17", "50031545098999707", "50000000000000000", "1.091895", "minortone"
, "125, 171, 217, 388, 559, 730, 901", "[1200, 182.466089]", "[[1,0]
[-1,17] [-3,35]]", "0.025466"
"-", "-53 10
16", "9010162353515625", "9007199254740992", "0.569430", "kwazy", "118
, 494, 612, 730, 1342", "[600, 162.741892]", "[[2,0] [1,8] [6,-
5]]", "0.017725"
"-", "91 -12 -
31", "2475880078570760549798248448", "2474715001881122589111328125", "
0.814860", "astro", "118, 1171, 2224", "[1200, 132.194511]", "[[1,0]
[5,-31] [1,12]]", "0.014993"
"-", "37 25 -
33", "116450459770592056836096", "116415321826934814453125", "0.522464
", "whoosh", "441, 730, 1171, 3072", "[1200, 560.546970]", "[[1,0]
[17,-33] [14,-25]]", "0.012388"
" monzisma", "54 -37
2", "450359962737049600", "450283905890997363", "0.292396", "monzismic
", "53, 559, 612, 665, 1171, 1783", "[1200, 249.018448]", "[[1,0] [2,-
2] [10,-37]]", "0.005738"
"-", "-36 -52
51", "444089209850062616169452667236328125", "444002166576103304796646
509039845376", "0.339362", "egads", "441, 1342, 1783, 3125", "[1200,
315.647874]", "[[1,0] [15,-51] [16,-52]]", "0.004660"
"-", "-107 47
14", "162285243890121480027996826171875", "162259276829213363391578010
288128", "0.277035", "fortune", "612, 1901, 2513, 3125", "[1200,
221.567865]", "[[1,0] [-1,14] [11,-47]]", "0.003542"
"-", "-17 62 -
35", "381520424476945831628649898809", "381469726562500000000000000000
", "0.230068", "senior", "171, 1000, 1171, 1342, 2513, 3684", "[1200,
322.801387]", "[[1,0] [11,-35] [19,-62]]", "0.003022"
"-", "144 -22 -
47", "22300745198530623141535718272648361505980416", "2229758394562963
9856633730232715606689453125", "0.245429", "gross", "118, 1783, 1901,
3684", "[1200, 91.531021]", "[[1,0] [-2,47] [4,-22]]", "0.002842"
"-", "-90 -15
49", "17763568394002504646778106689453125", "1776308649528226802416196
7871623168", "0.046966", "pirate", "730, 1783, 2513, 4296", "[1200,
185.754179]", "[[1,0] [-6,49] [0,15]]", "0.000761"
" limma, pythagorean minor second", "8 -5
0", "256", "243", "90.224996", "blackwood", "5, 10, 15, 25", "[240,
84.663787]", "[[5,0] [8,0] [12.-1]]", "12.759741"
"-", "71 -99
37", "171798691840000000000000000000000000000000000000", "171792506910
670443678820376588540424234035840667", "0.062327", "raider", "1171,
3125, 4296", "[1200, 343.296099]", "[[1,0] [-9,37] [-
26,99]]", "0.000511"
" atom of kirnberger", "161 -84 -
12", "2923003274661805836407369665432566039311865085952", "29229773394
92680612451840826835216578535400390625", "0.015361", "atomic", "(12,)
600, 612, 3072, 3684, 4296", "[100, 1.955169]", "[[12,0] [19,1] [28,-
7]]", "0.000120"
" diatonic semitone", "4 -1 -
1", "16", "15", "111.731285", "father", "3, 5, 8", "[1200,
442.179356]", "[[1,0] [2,-1] [2,1]]", "45.614107"
" major diesis", "3 4 -
4", "648", "625", "62.565148", "diminished, 'octatonic'", "4, 8, 12,
16, 28", "[300, 94.134357]", "[[4,0] [6,1] [9,1]]", "11.060060"
" maximal diesis", "1 -5
3", "250", "243", "49.166137", "porcupine", "7, 8, 15, 22, 29, 37,
59", "[1200, 162.996026]", "[[1,0] [2,-3], [3,-5]]", "7.975801"
" diesis, great diesis, minor diesis", "7 0 -
3", "128", "125", "41.058858", "augmented, diesic", "3, 9, 12, 15,
18, 27, 39, 42", "[400, 91.201856]", "[[3,0] [5,-1]
[7,0]]", "9.677666"
"-", "-14 3 4", "16875", "16384", "51.119858", "negri", "9, 10, 19,
28, 29", "[1200, 126.238272]", "[[1,0] [2,-4] [2,3]]", "5.942563"
" comma, syntonic comma, comma of didymus", "-4 4 -
1", "81", "80", "21.506290", "meantone, 'diatonic'", "5, 7, 12, 19,
26, 31, 43, 45, 50, 55, 69, 74, 81, 88", "[1200,
503.835154]", "[[1,0] [2,-1] [4,-4]]", "4.217731"

"Ultimate" 5-limit again

🔗Gene W Smith <genewardsmith@juno.com>

🔗wallyesterpaulrus <wallyesterpaulrus@yahoo.com> <wallyesterpaulrus@yahoo.com>

🔗wallyesterpaulrus <wallyesterpaulrus@yahoo.com> <wallyesterpaulrus@yahoo.com>

🔗Gene W Smith <genewardsmith@juno.com>

🔗Gene Ward Smith <genewardsmith@juno.com> <genewardsmith@juno.com>

🔗wallyesterpaulrus <wallyesterpaulrus@yahoo.com> <wallyesterpaulrus@yahoo.com>