User:Contribution/Successive superparticular complementary pair: Difference between revisions

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{{todo|Finish the article and move it|inline=1|comment=When the article is finished and the table explained, move it to the main root}}
== Context ==


For each pair of superparticular ratios <math>{s1}/{s2}</math>​ and <math>{s2}/{s3}</math>, there exists a ratio <math>{a}/{b}</math> such that <math>{s1}/{s2}</math>​ and <math>{s2}/{s3}</math>​ are <math>{a}/{b}</math> complementary; it is observed that <math>a−b=1</math> or <math>a−b=2</math>.
Read this first: [[Equal-step_tuning#Alpha-beta-gamma_family_of_equal_divisions]]
In other words, for each ratio <math>a/b</math> where <math>a−b=1</math> or <math>a−b=2</math>, there exists a pair of superparticular ratios <math>{s1}/{s2}</math>​ and <math>{s2}/{s3}</math> that are <math>{a}/{b}</math> complementary.


Bellow is a table that show for equal divisions of <math>a/b</math> the cent error in the mapping of superparticular ratios <math>{s1}/{s2}</math>​ and <math>{s2}/{s3}</math> that are <math>a/b</math> complementary.
== The Alpha-Beta-Gamma family ==


We can observe a converging sequence and pattern for low errors: 5, 7, 12; then 7, 9, 16; then 9, 11, 20; then 11, 13, 24; then 13, 15, 28; then 15, 17, 32; then 17, 19, 36; then 19, 21, 40; then 21, 23, 44; etc. --
{{todo|Table|inline=1|comment=Explain the table.}}
{{todo|Pattern|inline=1|comment=Clarify the observed pattern and create a descriptive name for it, such as the "Alpha-Beta-Gamma pattern" or the "Alpha-Beta-Gamma class" when referring to the group of scales. Assign distinct names to each scale within this class. For instance, 5edo might be called "2/1 Alpha", 7edo could be "2/1 Beta", and 12edo could be "2/1 Gamma". Additionally, compute the Dave Benson optimization for each scale as an alternative tuning. Note: 23edo with stretched octave is "7/5 Alpha".
Consider this second version for naming: "Alpha 2/1, Beta 2/1, Gamma 2/1, Alpha 7/5." Consistency and Clarity: The second version ("Alpha 2/1, Beta 2/1, Gamma 2/1, Alpha 7/5") places the descriptive name ("Alpha," "Beta," "Gamma") before the ratio. This makes it clear that "Alpha," "Beta," "Gamma," and so on are categories or types, while "2/1" and "7/5" are specific tunings or ratios within those categories. This ordering helps maintain a logical structure that is easier to follow.}}
{| class="wikitable"
{| class="wikitable"
|+
|-
! colspan="3" | Tuning !! colspan="2" | Intervals !! colspan="2" | Mappings
|-
! Name
! Equal division
! Steps per octave
! Equave
! SSC pair
! Steps (Equave, SSC pair)
! Errors (cent)
|-
|-
| [[Alpha 3/1]]
| [[3edt|3ed3/1]]
| 1.89278926071437
| rowspan="3" | 3/1
| rowspan="3" | 3/1
| rowspan="3" | 2/1, 3/2
| rowspan="3" | 2/1, 3/2
| [[3edt|3ed3/1]]
| [[Alpha 3/1]]
| 1.892789
| 633.985000
| 1.907395926960071
| 629.130000247253548
| 3\3<3/1>, 2\3<3/1>, 1\3<3/1>
| 3\3<3/1>, 2\3<3/1>, 1\3<3/1>
| 0, 67.970001, -67.970001
| 0, 67.970, -67.970
| -14.565000, 58.260000, -72.825001
|-
|-
| [[Beta 3/1]]
| [[5edt|5ed3/1]]
| [[5edt|5ed3/1]]
| [[Beta 3/1]]
| 3.15464876785729
| 3.154649
| 380.391000
| 3.141862316907629
| 381.939079106781893
| 5\5<3/1>, 3\5<3/1>, 2\5<3/1>
| 5\5<3/1>, 3\5<3/1>, 2\5<3/1>
| 0, -58.826999, 58.826999
| 0, -58.827, 58.827
| 7.740395, -54.182763, 61.923157
|-
|-
| [[Gamma 3/1]]
| [[8edt|8ed3/1]]
| [[8edt|8ed3/1]]
| [[Gamma 3/1]]
| 5.04743802857166
| 5.047438
| 237.744375
| 5.042556213760587
| 237.974540913461853
| 8\8<3/1>, 5\8<3/1>, 3\8<3/1>
| 8\8<3/1>, 5\8<3/1>, 3\8<3/1>
| 0, -11.278124, 11.278124
| 0, -11.278, 11.278
| 1.841326, -10.127295, 11.968622
|-
|-
| [[Alpha 2/1]]
| [[5edo|5ed2/1]]
| 5
| rowspan="3" | 2/1
| rowspan="3" | 2/1
| rowspan="3" | 3/2, 4/3
| rowspan="3" | 3/2, 4/3
| [[5edo|5ed2/1]]
| [[Alpha 2/1]]
| 5.000000
| 240.000000
| 5.009912705090773
| 239.525131601720722
| 5\5<2/1>, 3\5<2/1>, 2\5<2/1>
| 5\5<2/1>, 3\5<2/1>, 2\5<2/1>
| 0, 18.044999, -18.044999
| 0, 18.045, -18.045
| -2.374342, 16.620394, -18.994736
|-
|-
| [[Beta 2/1]]
| [[7edo|7ed2/1]]
| [[7edo|7ed2/1]]
| [[Beta 2/1]]
| 7
| 7.000000
| 171.428571
| 6.991049802487100
| 171.648040552234965
| 7\7<2/1>, 4\7<2/1>, 3\7<2/1>
| 7\7<2/1>, 4\7<2/1>, 3\7<2/1>
| 0, -16.240715, 16.240715
| 0, -16.241, 16.241
| 1.536284, -15.362839, 16.899123
|-
|-
| [[Gamma 2/1]]
| [[12edo|12ed2/1]]
| [[12edo|12ed2/1]]
| [[Gamma 2/1]]
| 12
| 12.000000
| 100.000000
| 11.997848091431052
| 100.017935787755848
| 12\12<2/1>, 7\12<2/1>, 5\12<2/1>
| 12\12<2/1>, 7\12<2/1>, 5\12<2/1>
| 0, -1.955001, 1.955001
| 0, -1.955, 1.955
| 0.215229, -1.829450, 2.044680
|-
|-
| [[Alpha 5/3]]
| [[7ed5/3]]
| 9.49840814199707
| rowspan="3" | 5/3
| rowspan="3" | 5/3
| rowspan="3" | 4/3, 5/4
| rowspan="3" | 4/3, 5/4
| [[7ed5/3]]
| [[Alpha 5/3]]
| 9.498408
| 126.336959
| 9.505833538777849
| 126.238272015257927
| 7\7<5/3>, 4\7<5/3>, 3\7<5/3>
| 7\7<5/3>, 4\7<5/3>, 3\7<5/3>
| 0, 7.302837, -7.302837
| 0, 7.303, -7.303
| -0.690809, 6.908089, -7.598898
|-
|-
| [[Beta 5/3]]
| [[9ed5/3]]
| [[9ed5/3]]
| [[Beta 5/3]]
| 12.2122390397105
| 12.212239
| 98.262079
| 12.205382300878206
| 98.317280886290400
| 9\9<5/3>, 5\9<5/3>, 4\9<5/3>
| 9\9<5/3>, 5\9<5/3>, 4\9<5/3>
| 0, -6.734603, 6.734603
| 0, -6.735, 6.735
| 0.496815, -6.458595, 6.955410
|-
|-
| [[Gamma 5/3]]
| [[16ed5/3]]
| [[16ed5/3]]
| [[Gamma 5/3]]
| 21.7106471817076
| 21.710647
| 55.272420
| 21.709439921550910
| 55.275493257141231
| 16\16<5/3>, 9\16<5/3>, 7\16<5/3>
| 16\16<5/3>, 9\16<5/3>, 7\16<5/3>
| 0, -0.593223, 0.593223
| 0, -0.593, 0.593
| 0.049179, -0.565560, 0.614739
|-
|-
| [[Carlos Alpha|Alpha 3/2]]
| [[9edf|9ed3/2]]
| 15.3856016221631
| rowspan="3" | 3/2
| rowspan="3" | 3/2
| rowspan="3" | 5/4, 6/5
| rowspan="3" | 5/4, 6/5
| [[9edf|9ed3/2]]
| [[Alpha 3/2]]
| 15.385602
| 77.995000
| 15.391523899692793
| 77.964989550121895
| 9\9<3/2>, 5\9<3/2>, 4\9<3/2>
| 9\9<3/2>, 5\9<3/2>, 4\9<3/2>
| 0, 3.661287, -3.661287
| 0, 3.661, -3.661
| -0.270095, 3.511234, -3.781329
|-
|-
| [[Carlos Beta|Beta 3/2]]
| [[11edf|11ed3/2]]
| [[11edf|11ed3/2]]
| [[Beta 3/2]]
| 18.8046242048660
| 18.804624
| 63.814091
| 18.799073639411081
| 63.832932569840843
| 11\11<3/2>, 6\11<3/2>, 5\11<3/2>
| 11\11<3/2>, 6\11<3/2>, 5\11<3/2>
| 0, -3.429168, 3.429168
| 0, -3.429, 3.429
| 0.207257, -3.316118, 3.523376
|-
|-
| [[Carlos Gamma|Gamma 3/2]]
| [[20edf|20ed3/2]]
| [[20edf|20ed3/2]]
| [[Gamma 3/2]]
| 34.1902258270291
| 34.190226
| 35.097750
| 34.189454092191388
| 35.098542280441702
| 20\20<3/2>, 11\20<3/2>, 9\20<3/2>
| 20\20<3/2>, 11\20<3/2>, 9\20<3/2>
| 0, -0.238463, 0.238463
| 0, -0.238, 0.238
| 0.015845, -0.229749, 0.245594
|-
|-
| [[Alpha 7/5]]
| [[11ed7/5]]
| 22.6604698881676
| rowspan="3" | 7/5
| rowspan="3" | 7/5
| rowspan="3" | 6/5, 7/6
| rowspan="3" | 6/5, 7/6
| [[11ed7/5]]
| [[Alpha 7/5]]
| 22.660470
| 52.955654
| 22.665391113336561
| 52.944155871808760
| 11\11<7/5>, 6\11<7/5>, 5\11<7/5>
| 11\11<7/5>, 6\11<7/5>, 5\11<7/5>
| 0, 2.092636, -2.092636
| 0, 2.093, -2.093
| -0.126478, 2.023648, -2.150126
|-
|-
| [[Beta 7/5]]
| [[13ed7/5]]
| [[13ed7/5]]
| [[Beta 7/5]]
| 26.7805553223799
| 26.780555
| 44.808630
| 26.775895108856630
| 44.816428923157735
| 13\13<7/5>, 7\13<7/5>, 6\13<7/5>
| 13\13<7/5>, 7\13<7/5>, 6\13<7/5>
| 0, -1.980876, 1.980876
| 0, -1.981, 1.981
| 0.101383, -1.926285, 2.027668
|-
|-
| [[Gamma 7/5]]
| [[24ed7/5]]
| [[24ed7/5]]
| [[Gamma 7/5]]
| 49.4410252105475
| 49.441025
| 24.271341
| 49.440489621601243
| 24.271604290013001
| 24\24<7/5>, 13\24<7/5>, 11\24<7/5>
| 24\24<7/5>, 13\24<7/5>, 11\24<7/5>
| 0, -0.113849, 0.113849
| 0, -0.114, 0.114
| 0.006310, -0.110431, 0.116742
|-
|-
| [[Alpha 4/3]]
| [[13ed4/3]]
| 31.3224709154917
| rowspan="3" | 4/3
| rowspan="3" | 4/3
| rowspan="3" | 7/6, 8/7
| rowspan="3" | 7/6, 8/7
| [[13ed4/3]]
| [[Alpha 4/3]]
| 31.322471
| 38.311154
| 31.326679032092577
| 38.306007437643215
| 13\13<4/3>, 7\13<4/3>, 6\13<4/3>
| 13\13<4/3>, 7\13<4/3>, 6\13<4/3>
| 0, 1.307171, -1.307171
| 0, 1.307, -1.307
| -0.066902, 1.271146, -1.338049
|-
|-
| [[Beta 4/3]]
| [[15ed4/3]]
| [[15ed4/3]]
| [[Beta 4/3]]
| 36.1413125947981
| 36.141313
| 33.203000
| 36.137297503882719
| 33.206689013506551
| 15\15<4/3>, 8\15<4/3>, 7\15<4/3>
| 15\15<4/3>, 8\15<4/3>, 7\15<4/3>
| 0, -1.246906, 1.246906
| 0, -1.247, 1.247
| 0.055336, -1.217393, 1.272730
|-
|-
| [[Gamma 4/3]]
| [[28ed4/3]]
| [[28ed4/3]]
| [[Gamma 4/3]]
| 67.4637835102899
| 67.463784
| 17.787321
| 67.463390164664623
| 17.787425106728855
| 28\28<4/3>, 15\28<4/3>, 13\28<4/3>
| 28\28<4/3>, 15\28<4/3>, 13\28<4/3>
| 0, -0.061085, 0.061085
| 0, -0.061, 0.061
| 0.002904, -0.059529, 0.062433
|-
|-
| [[Alpha 9/7]]
| [[15ed9/7]]
| 41.3713123417559
| rowspan="3" | 9/7
| rowspan="3" | 9/7
| rowspan="3" | 8/7, 9/8
| rowspan="3" | 8/7, 9/8
| [[15ed9/7]]
| [[Alpha 9/7]]
| 41.371312
| 29.005606
| 41.374987163985893
| 29.003030145820039
| 15\15<9/7>, 8\15<9/7>, 7\15<9/7>
| 15\15<9/7>, 8\15<9/7>, 7\15<9/7>
| 0, 0.870757, -0.870757
| 0, 0.871, -0.871
| -0.038643, 0.850148, -0.888791
|-
|-
| [[Beta 9/7]]
| [[17ed9/7]]
| [[17ed9/7]]
| [[Beta 9/7]]
| 46.8874873206567
| 46.887487
| 25.593182
| 46.883960906871343
| 25.595107085419638
| 17\17<9/7>, 9\17<9/7>, 8\17<9/7>
| 17\17<9/7>, 9\17<9/7>, 8\17<9/7>
| 0, -0.835455, 0.835455
| 0, -0.835, 0.835
| 0.032725, -0.818130, 0.850855
|-
|-
| [[Gamma 9/7]]
| [[32ed9/7]]
| [[32ed9/7]]
| [[Gamma 9/7]]
| 88.2587996624126
| 88.258800
| 13.596378
| 88.258498580415662
| 13.596424359141285
| 32\32<9/7>, 17\32<9/7>, 15\32<9/7>
| 32\32<9/7>, 17\32<9/7>, 15\32<9/7>
| 0, -0.035668, 0.035668
| 0, -0.036, 0.036
| 0.001484, -0.034879, 0.036364
|-
|-
| [[Alpha 5/4]]
| [[17ed5/4]]
| 52.8068232315916
| rowspan="3" | 5/4
| rowspan="3" | 5/4
| rowspan="3" | 9/8, 10/9
| rowspan="3" | 9/8, 10/9
| [[17ed5/4]]
| [[Alpha 5/4]]
| 52.806823
| 22.724336
| 52.810084374305705
| 22.722932830303330
| 17\17<5/4>, 9\17<5/4>, 8\17<5/4>
| 17\17<5/4>, 9\17<5/4>, 8\17<5/4>
| 0, 0.609023, -0.609023
| 0, 0.609, -0.609
| -0.023856, 0.596394, -0.620249
|-
|-
| [[Beta 5/4]]
| [[19ed5/4]]
| [[19ed5/4]]
| [[Beta 5/4]]
| 59.0193906706024
| 59.019391
| 20.332301
| 59.016247125030467
| 20.333383745288099
| 19\19<5/4>, 10\19<5/4>, 9\19<5/4>
| 19\19<5/4>, 10\19<5/4>, 9\19<5/4>
| 0, -0.586994, 0.586994
| 0, -0.587, 0.587
| 0.020577, -0.576164, 0.596742
|-
|-
| [[Gamma 5/4]]
| [[36ed5/4]]
| [[36ed5/4]]
| [[Gamma 5/4]]
| 111.826213902194
| 111.826214
| 10.730936
| 111.825976049765954
| 10.730959320810789
| 36\36<5/4>, 19\36<5/4>, 17\36<5/4>
| 36\36<5/4>, 19\36<5/4>, 17\36<5/4>
| 0, -0.022208, 0.022208
| 0, -0.022, 0.022
| 0.000822, -0.021775, 0.022596
|-
|-
| [[Alpha 11/9]]
| [[19ed11/9]]
| 65.6288971357202
| rowspan="3" | 11/9
| rowspan="3" | 11/9
| rowspan="3" | 10/9, 11/10
| rowspan="3" | 10/9, 11/10
| [[19ed11/9]]
| [[Alpha 11/9]]
| 65.628897
| 18.284628
| 65.631828119476568
| 18.283811900157846
| 19\19<11/9>, 10\19<11/9>, 9\19<11/9>
| 19\19<11/9>, 10\19<11/9>, 9\19<11/9>
| 0, 0.442572, -0.442572
| 0, 0.443, -0.443
| -0.015515, 0.434407, -0.449921
|-
|-
| [[Beta 11/9]]
| [[21ed11/9]]
| [[21ed11/9]]
| [[Beta 11/9]]
| 72.5372020973750
| 72.537202
| 16.543235
| 72.534366561494206
| 16.543881981552112
| 21\21<11/9>, 11\21<11/9>, 10\21<11/9>
| 21\21<11/9>, 11\21<11/9>, 10\21<11/9>
| 0, -0.428124, 0.428124
| 0, -0.428, 0.428
| 0.013581, -0.421010, 0.434591
|-
|-
| [[Gamma 11/9]]
| [[40ed11/9]]
| [[40ed11/9]]
| [[Gamma 11/9]]
| 138.166099233095
| 138.166099
| 8.685199
| 138.165906595462172
| 8.685210625176124
| 40\40<11/9>, 21\40<11/9>, 19\40<11/9>
| 40\40<11/9>, 21\40<11/9>, 19\40<11/9>
| 0, -0.014543, 0.014543
| 0, -0.015, 0.015
| 0.000484, -0.014289, 0.014773
|-
|-
| [[Alpha 6/5]]
| [[21ed6/5]]
| 79.8374643554025
| rowspan="3" | 6/5
| rowspan="3" | 6/5
| rowspan="3" | 11/10, 12/11
| rowspan="3" | 11/10, 12/11
| [[21ed6/5]]
| [[Alpha 6/5]]
| 79.837464
| 15.030537
| 79.840125772190183
| 15.030036443379233
| 21\21<6/5>, 11\21<6/5>, 10\21<6/5>
| 21\21<6/5>, 11\21<6/5>, 10\21<6/5>
| 0, 0.331684, -0.331684
| 0, 0.332, -0.332
| -0.010522, 0.326172, -0.336694
|-
|-
| [[Beta 6/5]]
| [[23ed6/5]]
| [[23ed6/5]]
| [[Beta 6/5]]
| 87.4410323892504
| 87.441032
| 13.723534
| 87.438449973495273
| 13.723939529620542
| 23\23<6/5>, 12\23<6/5>, 11\23<6/5>
| 23\23<6/5>, 12\23<6/5>, 11\23<6/5>
| 0, -0.321818, 0.321818
| 0, -0.322, 0.322
| 0.009322, -0.316954, 0.326276
|-
|-
| [[Gamma 6/5]]
| [[44ed6/5]]
| [[44ed6/5]]
| [[Gamma 6/5]]
| 167.278496744653
| 167.278497
| 7.173666
| 167.278337553931523
| 7.173672440480304
| 44\44<6/5>, 23\44<6/5>, 21\44<6/5>
| 44\44<6/5>, 23\44<6/5>, 21\44<6/5>
| 0, -0.009919, 0.009919
| 0, -0.010, 0.010
| 0.000300, -0.009762, 0.010063
|-
|-
| [[Alpha 13/11]]
| [[23ed13/11]]
| 95.4324773621886
| rowspan="3" | 13/11
| rowspan="3" | 13/11
| rowspan="3" | 12/11, 13/12
| rowspan="3" | 12/11, 13/12
| [[23ed13/11]]
| [[Alpha 13/11]]
| 95.432477
| 12.574336
| 95.434914550823771
| 12.574014506618971
| 23\23<13/11>, 12\23<13/11>, 11\23<13/11>
| 23\23<13/11>, 12\23<13/11>, 11\23<13/11>
| 0, 0.254969, -0.254969
| 0, 0.255, -0.255
| -0.007386, 0.251116, -0.258501
|-
|-
| [[Beta 13/11]]
| [[25ed13/11]]
| [[25ed13/11]]
| [[Beta 13/11]]
| 103.730953654553
| 103.730954
| 11.568389
| 103.728582924336770
| 11.568653173208022
| 25\25<13/11>, 13\25<13/11>, 12\25<13/11>
| 25\25<13/11>, 13\25<13/11>, 12\25<13/11>
| 0, -0.248004, 0.248004
| 0, -0.248, 0.248
| 0.006610, -0.244567, 0.251177
|-
|-
| [[Gamma 13/11]]
| [[48ed13/11]]
| [[48ed13/11]]
| [[Gamma 13/11]]
| 199.163431016741
| 199.163431
| 6.025202
| 199.163297261207502
| 6.025206534044126
| 48\48<13/11>, 25\48<13/11>, 23\48<13/11>
| 48\48<13/11>, 25\48<13/11>, 23\48<13/11>
| 0, -0.006996, 0.006996
| 0, -0.007, 0.007
| 0.000194, -0.006895, 0.007089
|-
|-
| [[Alpha 7/6]]
| [[25ed7/6]]
| 112.413902640048
| rowspan="3" | 7/6
| rowspan="3" | 7/6
| rowspan="3" | 13/12, 14/13
| rowspan="3" | 13/12, 14/13
| [[25ed7/6]]
| [[Alpha 7/6]]
| 112.413903
| 10.674836
| 112.416150402630623
| 10.674622780642016
| 25\25<7/6>, 13\25<7/6>, 12\25<7/6>
| 25\25<7/6>, 13\25<7/6>, 12\25<7/6>
| 0, 0.200210, -0.200210
| 0, 0.200, -0.200
| -0.005336, 0.197435, -0.202771
|-
|-
| [[Beta 7/6]]
| [[27ed7/6]]
| [[27ed7/6]]
| [[Beta 7/6]]
| 121.407014851252
| 121.407015
| 9.884108
| 121.404823766036118
| 9.884286000962910
| 27\27<7/6>, 14\27<7/6>, 13\27<7/6>
| 27\27<7/6>, 14\27<7/6>, 13\27<7/6>
| 0, -0.195154, 0.195154
| 0, -0.195, 0.195
| 0.004816, -0.192657, 0.197473
|-
|-
| [[Gamma 7/6]]
| [[52ed7/6]]
| [[52ed7/6]]
| [[Gamma 7/6]]
| 233.820917491300
| 233.820917
| 5.132133
| 233.820803527976982
| 5.132135301452842
| 52\52<7/6>, 27\52<7/6>, 25\52<7/6>
| 52\52<7/6>, 27\52<7/6>, 25\52<7/6>
| 0, -0.005075, 0.005075
| 0, -0.005, 0.005
| 0.000130, -0.005008, 0.005138
|-
|-
| [[Alpha 15/13]]
| [[27ed15/13]]
| 130.781715879411
| rowspan="3" | 15/13
| rowspan="3" | 15/13
| rowspan="3" | 14/13, 15/14
| rowspan="3" | 14/13, 15/14
| [[27ed15/13]]
| [[Alpha 15/13]]
| 130.781716
| 9.175595
| 130.783801507844919
| 9.175448229557843
| 27\27<15/13>, 14\27<15/13>, 13\27<15/13>
| 27\27<15/13>, 14\27<15/13>, 13\27<15/13>
| 0, 0.160079, -0.160079
| 0, 0.160, -0.160
| -0.003951, 0.158031, -0.161981
|-
|-
| [[Beta 15/13]]
| [[29ed15/13]]
| [[29ed15/13]]
| [[Beta 15/13]]
| 140.469250388997
| 140.469250
| 8.542795
| 140.467213664559518
| 8.542918797162452
| 29\29<15/13>, 15\29<15/13>, 14\29<15/13>
| 29\29<15/13>, 15\29<15/13>, 14\29<15/13>
| 0, -0.156321, 0.156321
| 0, -0.156, 0.156
| 0.003592, -0.154463, 0.158055
|-
|-
| [[Gamma 15/13]]
| [[56ed15/13]]
| [[56ed15/13]]
| [[Gamma 15/13]]
| 271.250966268408
| 271.250966
| 4.423947
| 271.250868008139347
| 4.423948976871078
| 56\56<15/13>, 29\56<15/13>, 27\56<15/13>
| 56\56<15/13>, 29\56<15/13>, 27\56<15/13>
| 0, -0.003771, 0.003771
| 0, -0.004, 0.004
| 0.000090, -0.003724, 0.003814
|-
|-
| [[Alpha 8/7]]
| [[29ed8/7]]
| 150.535899020849
| rowspan="3" | 8/7
| rowspan="3" | 8/7
| rowspan="3" | 15/14, 16/15
| rowspan="3" | 15/14, 16/15
| [[29ed8/7]]
| [[Alpha 8/7]]
| 150.535899
| 7.971520
| 150.537844310638475
| 7.971417456488689
| 29\29<8/7>, 15\29<8/7>, 14\29<8/7>
| 29\29<8/7>, 15\29<8/7>, 14\29<8/7>
| 0, 0.129999, -0.129999
| 0, 0.130, -0.130
| -0.002987, 0.128454, -0.131441
|-
|-
| [[Beta 8/7]]
| [[31ed8/7]]
| [[31ed8/7]]
| [[Beta 8/7]]
| 160.917685160217
| 160.917685
| 7.457229
| 160.915782495277457
| 7.457316997698579
| 31\31<8/7>, 16\31<8/7>, 15\31<8/7>
| 31\31<8/7>, 16\31<8/7>, 15\31<8/7>
| 0, -0.127147, 0.127147
| 0, -0.127, 0.127
| 0.002733, -0.125736, 0.128470
|-
|-
| [[Gamma 8/7]]
| [[60ed8/7]]
| [[60ed8/7]]
| [[Gamma 8/7]]
| 311.453584181066
| 311.453584
| 3.852902
| 311.453498588281532
| 3.852902617691610
| 60\60<8/7>, 31\60<8/7>, 29\60<8/7>
| 60\60<8/7>, 31\60<8/7>, 29\60<8/7>
| 0, -0.002860, 0.002860
| 0, -0.003, 0.003
| 0.000064, -0.002827, 0.002891
|}
|}
{{todo|Temperaments|inline=1|comment=Compute the temperaments associated to each Alpha-Beta-Gamma scales.}}


== Coincidence? ==


As a coincidence (?), all Alpha scales are (s1 + s2)ED(a / b), all Beta scales are (s2 + s3)ED(a / b), and all Gamma scales are (s1 + s2 + s2 + s3)ED(a / b).
== The converging Alpha-Beta-Gamma sequence ==
 
As a fact, for each <math>n\ge 2</math>, equal divisions of <math>R_n=\dfrac{n+1}{n-1}</math> where low errors appear for <math>S_n=\dfrac{n+1}{n}</math> and <math>B_n=\dfrac{n}{n-1}</math> forms a converging sequence and pattern, with the happy equal divisions of <math>R_n</math> being:
* '''Alpha:''' <math>k_\alpha=2n-1</math>
* '''Beta:''' <math>k_\beta=2n+1</math>
* '''Gamma:''' <math>k_\gamma=4n=k_\alpha+k_\beta</math>
 
In this sequence, the errors are lower and lower.
 
{{todo|Why this pattern|inline=1|comment=Explain why divisions of ratios where low errors appear for successive superparticular complementary pair make this pattern appears.}}


{| class="wikitable sortable right-1 left-2 right-3 left-4 right-5 left-6 right-7 left-8 right-9 left-10 right-11 left-12 right-13 left-14 right-15 left-16 right-17 left-18 right-19 left-20"
{| class="wikitable sortable right-1 left-2 right-3 left-4 right-5 left-6 right-7 left-8 right-9 left-10 right-11 left-12 right-13 left-14 right-15 left-16 right-17 left-18 right-19 left-20"

Latest revision as of 00:21, 28 October 2025

Context

Read this first: Equal-step_tuning#Alpha-beta-gamma_family_of_equal_divisions

The Alpha-Beta-Gamma family

Tuning Intervals Mappings
Name Equal division Steps per octave Equave SSC pair Steps (Equave, SSC pair) Errors (cent)
Alpha 3/1 3ed3/1 1.89278926071437 3/1 2/1, 3/2 3\3<3/1>, 2\3<3/1>, 1\3<3/1> 0, 67.970, -67.970
Beta 3/1 5ed3/1 3.15464876785729 5\5<3/1>, 3\5<3/1>, 2\5<3/1> 0, -58.827, 58.827
Gamma 3/1 8ed3/1 5.04743802857166 8\8<3/1>, 5\8<3/1>, 3\8<3/1> 0, -11.278, 11.278
Alpha 2/1 5ed2/1 5 2/1 3/2, 4/3 5\5<2/1>, 3\5<2/1>, 2\5<2/1> 0, 18.045, -18.045
Beta 2/1 7ed2/1 7 7\7<2/1>, 4\7<2/1>, 3\7<2/1> 0, -16.241, 16.241
Gamma 2/1 12ed2/1 12 12\12<2/1>, 7\12<2/1>, 5\12<2/1> 0, -1.955, 1.955
Alpha 5/3 7ed5/3 9.49840814199707 5/3 4/3, 5/4 7\7<5/3>, 4\7<5/3>, 3\7<5/3> 0, 7.303, -7.303
Beta 5/3 9ed5/3 12.2122390397105 9\9<5/3>, 5\9<5/3>, 4\9<5/3> 0, -6.735, 6.735
Gamma 5/3 16ed5/3 21.7106471817076 16\16<5/3>, 9\16<5/3>, 7\16<5/3> 0, -0.593, 0.593
Alpha 3/2 9ed3/2 15.3856016221631 3/2 5/4, 6/5 9\9<3/2>, 5\9<3/2>, 4\9<3/2> 0, 3.661, -3.661
Beta 3/2 11ed3/2 18.8046242048660 11\11<3/2>, 6\11<3/2>, 5\11<3/2> 0, -3.429, 3.429
Gamma 3/2 20ed3/2 34.1902258270291 20\20<3/2>, 11\20<3/2>, 9\20<3/2> 0, -0.238, 0.238
Alpha 7/5 11ed7/5 22.6604698881676 7/5 6/5, 7/6 11\11<7/5>, 6\11<7/5>, 5\11<7/5> 0, 2.093, -2.093
Beta 7/5 13ed7/5 26.7805553223799 13\13<7/5>, 7\13<7/5>, 6\13<7/5> 0, -1.981, 1.981
Gamma 7/5 24ed7/5 49.4410252105475 24\24<7/5>, 13\24<7/5>, 11\24<7/5> 0, -0.114, 0.114
Alpha 4/3 13ed4/3 31.3224709154917 4/3 7/6, 8/7 13\13<4/3>, 7\13<4/3>, 6\13<4/3> 0, 1.307, -1.307
Beta 4/3 15ed4/3 36.1413125947981 15\15<4/3>, 8\15<4/3>, 7\15<4/3> 0, -1.247, 1.247
Gamma 4/3 28ed4/3 67.4637835102899 28\28<4/3>, 15\28<4/3>, 13\28<4/3> 0, -0.061, 0.061
Alpha 9/7 15ed9/7 41.3713123417559 9/7 8/7, 9/8 15\15<9/7>, 8\15<9/7>, 7\15<9/7> 0, 0.871, -0.871
Beta 9/7 17ed9/7 46.8874873206567 17\17<9/7>, 9\17<9/7>, 8\17<9/7> 0, -0.835, 0.835
Gamma 9/7 32ed9/7 88.2587996624126 32\32<9/7>, 17\32<9/7>, 15\32<9/7> 0, -0.036, 0.036
Alpha 5/4 17ed5/4 52.8068232315916 5/4 9/8, 10/9 17\17<5/4>, 9\17<5/4>, 8\17<5/4> 0, 0.609, -0.609
Beta 5/4 19ed5/4 59.0193906706024 19\19<5/4>, 10\19<5/4>, 9\19<5/4> 0, -0.587, 0.587
Gamma 5/4 36ed5/4 111.826213902194 36\36<5/4>, 19\36<5/4>, 17\36<5/4> 0, -0.022, 0.022
Alpha 11/9 19ed11/9 65.6288971357202 11/9 10/9, 11/10 19\19<11/9>, 10\19<11/9>, 9\19<11/9> 0, 0.443, -0.443
Beta 11/9 21ed11/9 72.5372020973750 21\21<11/9>, 11\21<11/9>, 10\21<11/9> 0, -0.428, 0.428
Gamma 11/9 40ed11/9 138.166099233095 40\40<11/9>, 21\40<11/9>, 19\40<11/9> 0, -0.015, 0.015
Alpha 6/5 21ed6/5 79.8374643554025 6/5 11/10, 12/11 21\21<6/5>, 11\21<6/5>, 10\21<6/5> 0, 0.332, -0.332
Beta 6/5 23ed6/5 87.4410323892504 23\23<6/5>, 12\23<6/5>, 11\23<6/5> 0, -0.322, 0.322
Gamma 6/5 44ed6/5 167.278496744653 44\44<6/5>, 23\44<6/5>, 21\44<6/5> 0, -0.010, 0.010
Alpha 13/11 23ed13/11 95.4324773621886 13/11 12/11, 13/12 23\23<13/11>, 12\23<13/11>, 11\23<13/11> 0, 0.255, -0.255
Beta 13/11 25ed13/11 103.730953654553 25\25<13/11>, 13\25<13/11>, 12\25<13/11> 0, -0.248, 0.248
Gamma 13/11 48ed13/11 199.163431016741 48\48<13/11>, 25\48<13/11>, 23\48<13/11> 0, -0.007, 0.007
Alpha 7/6 25ed7/6 112.413902640048 7/6 13/12, 14/13 25\25<7/6>, 13\25<7/6>, 12\25<7/6> 0, 0.200, -0.200
Beta 7/6 27ed7/6 121.407014851252 27\27<7/6>, 14\27<7/6>, 13\27<7/6> 0, -0.195, 0.195
Gamma 7/6 52ed7/6 233.820917491300 52\52<7/6>, 27\52<7/6>, 25\52<7/6> 0, -0.005, 0.005
Alpha 15/13 27ed15/13 130.781715879411 15/13 14/13, 15/14 27\27<15/13>, 14\27<15/13>, 13\27<15/13> 0, 0.160, -0.160
Beta 15/13 29ed15/13 140.469250388997 29\29<15/13>, 15\29<15/13>, 14\29<15/13> 0, -0.156, 0.156
Gamma 15/13 56ed15/13 271.250966268408 56\56<15/13>, 29\56<15/13>, 27\56<15/13> 0, -0.004, 0.004
Alpha 8/7 29ed8/7 150.535899020849 8/7 15/14, 16/15 29\29<8/7>, 15\29<8/7>, 14\29<8/7> 0, 0.130, -0.130
Beta 8/7 31ed8/7 160.917685160217 31\31<8/7>, 16\31<8/7>, 15\31<8/7> 0, -0.127, 0.127
Gamma 8/7 60ed8/7 311.453584181066 60\60<8/7>, 31\60<8/7>, 29\60<8/7> 0, -0.003, 0.003


The converging Alpha-Beta-Gamma sequence

As a fact, for each [math]\displaystyle{ n\ge 2 }[/math], equal divisions of [math]\displaystyle{ R_n=\dfrac{n+1}{n-1} }[/math] where low errors appear for [math]\displaystyle{ S_n=\dfrac{n+1}{n} }[/math] and [math]\displaystyle{ B_n=\dfrac{n}{n-1} }[/math] forms a converging sequence and pattern, with the happy equal divisions of [math]\displaystyle{ R_n }[/math] being:

  • Alpha: [math]\displaystyle{ k_\alpha=2n-1 }[/math]
  • Beta: [math]\displaystyle{ k_\beta=2n+1 }[/math]
  • Gamma: [math]\displaystyle{ k_\gamma=4n=k_\alpha+k_\beta }[/math]

In this sequence, the errors are lower and lower.

Todo: Why this pattern

Explain why divisions of ratios where low errors appear for successive superparticular complementary pair make this pattern appears.

Error (abs, ¢) on successive superparticular complementary pair in equal divisions
#ed3/1 2/1, 3/2 error #ed2/1 3/2, 4/3 error #ed5/3 4/3, 5/4 error #ed3/2 5/4, 6/5 error #ed7/5 6/5, 7/6 error #ed4/3 7/6, 8/7 error #ed9/7 8/7, 9/8 error #ed5/4 9/8, 10/9 error #ed11/9 10/9, 11/10 error #ed6/5 11/10, 12/11 error
1ed3/1 701.96 1ed2/1 498.04 1ed5/3 386.31 1ed3/2 315.64 1ed7/5 266.87 1ed4/3 231.17 1ed9/7 203.91 1ed5/4 182.40 1ed11/9 165.00 1ed6/5 150.64
2ed3/1 249.02 2ed2/1 101.96 2ed5/3 55.87 2ed3/2 35.34 2ed7/5 24.39 2ed4/3 17.85 2ed9/7 13.63 2ed5/4 10.75 2ed11/9 8.70 2ed6/5 7.18
3ed3/1 67.97 3ed2/1 98.04 3ed5/3 91.53 3ed3/2 81.66 3ed7/5 72.70 3ed4/3 65.16 3ed9/7 58.88 3ed5/4 53.63 3ed11/9 49.20 3ed6/5 45.42
4ed3/1 226.47 4ed2/1 101.96 4ed5/3 55.87 4ed3/2 35.34 4ed7/5 24.39 4ed4/3 17.85 4ed9/7 13.63 4ed5/4 10.75 4ed11/9 8.70 4ed6/5 7.18
5ed3/1 58.83 5ed2/1 18.04 5ed5/3 32.57 5ed3/2 34.86 5ed7/5 33.87 5ed4/3 31.96 5ed9/7 29.88 5ed5/4 27.88 5ed11/9 26.04 5ed6/5 24.38
6ed3/1 67.97 6ed2/1 98.04 6ed5/3 55.87 6ed3/2 35.34 6ed7/5 24.39 6ed4/3 17.85 6ed9/7 13.63 6ed5/4 10.75 6ed11/9 8.70 6ed6/5 7.18
7ed3/1 113.17 7ed2/1 16.24 7ed5/3 7.30 7ed3/2 14.80 7ed7/5 17.22 7ed4/3 17.73 7ed9/7 17.45 7ed5/4 16.84 7ed11/9 16.12 7ed6/5 15.36
8ed3/1 11.28 8ed2/1 48.04 8ed5/3 54.68 8ed3/2 35.34 8ed7/5 24.39 8ed4/3 17.85 8ed9/7 13.63 8ed5/4 10.75 8ed11/9 8.70 8ed6/5 7.18
9ed3/1 67.97 9ed2/1 35.29 9ed5/3 6.73 9ed3/2 3.66 9ed7/5 7.98 9ed4/3 9.82 9ed9/7 10.54 9ed5/4 10.71 9ed11/9 10.60 9ed6/5 10.35
10ed3/1 58.83 10ed2/1 18.04 10ed5/3 32.57 10ed3/2 34.86 10ed7/5 24.39 10ed4/3 17.85 10ed9/7 13.63 10ed5/4 10.75 10ed11/9 8.70 10ed6/5 7.18
11ed3/1 10.34 11ed2/1 47.41 11ed5/3 15.67 11ed3/2 3.43 11ed7/5 2.09 11ed4/3 4.79 11ed9/7 6.14 11ed5/4 6.81 11ed11/9 7.09 11ed6/5 7.16
12ed3/1 67.97 12ed2/1 1.96 12ed5/3 17.83 12ed3/2 23.16 12ed7/5 24.16 12ed4/3 17.85 12ed9/7 13.63 12ed5/4 10.75 12ed11/9 8.70 12ed6/5 7.18
13ed3/1 29.57 13ed2/1 36.51 13ed5/3 21.85 13ed3/2 8.34 13ed7/5 1.98 13ed4/3 1.31 13ed9/7 3.10 13ed5/4 4.11 13ed11/9 4.66 13ed6/5 4.96
14ed3/1 22.69 14ed2/1 16.24 14ed5/3 7.30 14ed3/2 14.80 14ed7/5 17.22 14ed4/3 17.73 14ed9/7 13.63 14ed5/4 10.75 14ed11/9 8.70 14ed6/5 7.18
15ed3/1 58.83 15ed2/1 18.04 15ed5/3 26.39 15ed3/2 11.94 15ed7/5 4.97 15ed4/3 1.25 15ed9/7 0.87 15ed5/4 2.12 15ed11/9 2.88 15ed6/5 3.34
16ed3/1 11.28 16ed2/1 26.96 16ed5/3 0.59 16ed3/2 8.54 16ed7/5 12.02 16ed4/3 13.28 16ed9/7 13.56 16ed5/4 10.75 16ed11/9 8.70 16ed6/5 7.18
17ed3/1 30.68 17ed2/1 3.93 17ed5/3 22.17 17ed3/2 14.69 17ed7/5 7.25 17ed4/3 3.20 17ed9/7 0.84 17ed5/4 0.61 17ed11/9 1.52 17ed6/5 2.10
18ed3/1 37.69 18ed2/1 31.38 18ed5/3 6.73 18ed3/2 3.66 18ed7/5 7.98 18ed4/3 9.82 18ed9/7 10.54 18ed5/4 10.71 18ed11/9 8.70 18ed6/5 7.18
19ed3/1 1.23 19ed2/1 7.22 19ed5/3 13.95 19ed3/2 16.86 19ed7/5 9.06 19ed4/3 4.74 19ed9/7 2.18 19ed5/4 0.59 19ed11/9 0.44 19ed6/5 1.12
20ed3/1 36.27 20ed2/1 18.04 20ed5/3 11.65 20ed3/2 0.24 20ed7/5 4.74 20ed4/3 7.05 20ed9/7 8.12 20ed5/4 8.56 20ed11/9 8.67 20ed6/5 7.18
21ed3/1 22.60 21ed2/1 16.24 21ed5/3 7.30 21ed3/2 14.80 21ed7/5 10.52 21ed4/3 5.99 21ed9/7 3.27 21ed5/4 1.56 21ed11/9 0.43 21ed6/5 0.33
22ed3/1 10.34 22ed2/1 7.14 22ed5/3 15.67 22ed3/2 3.43 22ed7/5 2.09 22ed4/3 4.79 22ed9/7 6.14 22ed5/4 6.81 22ed11/9 7.09 22ed6/5 7.16
23ed3/1 40.41 23ed2/1 23.69 23ed5/3 1.81 23ed3/2 10.44 23ed7/5 11.72 23ed4/3 7.02 23ed9/7 4.17 23ed5/4 2.36 23ed11/9 1.15 23ed6/5 0.32
24ed3/1 11.28 24ed2/1 1.96 24ed5/3 17.83 24ed3/2 6.09 24ed7/5 0.11 24ed4/3 2.90 24ed9/7 4.50 24ed5/4 5.34 24ed11/9 5.78 24ed6/5 5.97
25ed3/1 17.25 25ed2/1 18.04 25ed5/3 2.80 25ed3/2 6.78 25ed7/5 10.57 25ed4/3 7.89 25ed9/7 4.93 25ed5/4 3.03 25ed11/9 1.75 25ed6/5 0.87
26ed3/1 29.57 26ed2/1 9.65 26ed5/3 12.16 26ed3/2 8.34 26ed7/5 1.98 26ed4/3 1.31 26ed9/7 3.10 26ed5/4 4.11 26ed11/9 4.66 26ed6/5 4.96
27ed3/1 2.47 27ed2/1 9.16 27ed5/3 6.73 27ed3/2 3.66 27ed7/5 7.98 27ed4/3 8.63 27ed9/7 5.57 27ed5/4 3.60 27ed11/9 2.27 27ed6/5 1.34
28ed3/1 22.69 28ed2/1 16.24 28ed5/3 7.30 28ed3/2 10.27 28ed7/5 3.58 28ed4/3 0.06 28ed9/7 1.91 28ed5/4 3.04 28ed11/9 3.71 28ed6/5 4.09
29ed3/1 19.48 29ed2/1 1.49 29ed5/3 10.12 29ed3/2 0.97 29ed7/5 5.74 29ed4/3 7.91 29ed9/7 6.13 29ed5/4 4.09 29ed11/9 2.71 29ed6/5 1.74
30ed3/1 4.57 30ed2/1 18.04 30ed5/3 3.09 30ed3/2 11.46 30ed7/5 4.97 30ed4/3 1.25 30ed9/7 0.87 30ed5/4 2.12 30ed11/9 2.88 30ed6/5 3.34
31ed3/1 27.07 31ed2/1 5.18 31ed5/3 13.07 31ed3/2 1.37 31ed7/5 3.80 31ed4/3 6.25 31ed9/7 6.61 31ed5/4 4.52 31ed11/9 3.10 31ed6/5 2.09
32ed3/1 11.28 32ed2/1 10.54 32ed5/3 0.59 32ed3/2 8.54 32ed7/5 6.18 32ed4/3 2.28 32ed9/7 0.04 32ed5/4 1.32 32ed11/9 2.16 32ed6/5 2.68
33ed3/1 10.34 33ed2/1 11.05 33ed5/3 11.13 33ed3/2 3.43 33ed7/5 2.09 33ed4/3 4.79 33ed9/7 6.14 33ed5/4 4.90 33ed11/9 3.44 33ed6/5 2.40
34ed3/1 25.26 34ed2/1 3.93 34ed5/3 3.84 34ed3/2 5.96 34ed7/5 7.25 34ed4/3 3.20 34ed9/7 0.84 34ed5/4 0.61 34ed11/9 1.52 34ed6/5 2.10
35ed3/1 4.49 35ed2/1 16.24 35ed5/3 7.30 35ed3/2 5.25 35ed7/5 0.58 35ed4/3 3.50 35ed9/7 5.01 35ed5/4 5.23 35ed11/9 3.74 35ed6/5 2.67
36ed3/1 15.14 36ed2/1 1.96 36ed5/3 6.73 36ed3/2 3.66 36ed7/5 7.98 36ed4/3 4.01 36ed9/7 1.55 36ed5/4 0.02 36ed11/9 0.95 36ed6/5 1.58
37ed3/1 17.70 37ed2/1 11.56 37ed5/3 3.89 37ed3/2 6.88 37ed7/5 0.77 37ed4/3 2.34 37ed9/7 4.01 37ed5/4 4.91 37ed11/9 4.01 37ed6/5 2.92
38ed3/1 1.23 38ed2/1 7.22 38ed5/3 9.32 38ed3/2 1.61 38ed7/5 6.27 38ed4/3 4.74 38ed9/7 2.18 38ed5/4 0.59 38ed11/9 0.44 38ed6/5 1.12
39ed3/1 19.20 39ed2/1 5.74 39ed5/3 0.82 39ed3/2 8.34 39ed7/5 1.98 39ed4/3 1.31 39ed9/7 3.10 39ed5/4 4.11 39ed11/9 4.25 39ed6/5 3.14
40ed3/1 11.28 40ed2/1 11.96 40ed5/3 10.46 40ed3/2 0.24 40ed7/5 4.74 40ed4/3 5.40 40ed9/7 2.75 40ed5/4 1.10 40ed11/9 0.01 40ed6/5 0.71
41ed3/1 6.12 41ed2/1 0.48 41ed5/3 1.94 41ed3/2 7.47 41ed7/5 3.07 41ed4/3 0.37 41ed9/7 2.29 41ed5/4 3.38 41ed11/9 4.01 41ed6/5 3.33
42ed3/1 22.60 42ed2/1 12.33 42ed5/3 7.30 42ed3/2 1.91 42ed7/5 3.35 42ed4/3 5.87 42ed9/7 3.27 42ed5/4 1.56 42ed11/9 0.43 42ed6/5 0.33
43ed3/1 5.75 43ed2/1 4.28 43ed5/3 4.45 43ed3/2 5.48 43ed7/5 4.06 43ed4/3 0.47 43ed9/7 1.55 43ed5/4 2.72 43ed11/9 3.42 43ed6/5 3.51
44ed3/1 10.34 44ed2/1 7.14 44ed5/3 4.43 44ed3/2 3.43 44ed7/5 2.09 44ed4/3 4.79 44ed9/7 3.74 44ed5/4 1.97 44ed11/9 0.80 44ed6/5 0.01
45ed3/1 16.56 45ed2/1 8.62 45ed5/3 6.73 45ed3/2 3.66 45ed7/5 4.97 45ed4/3 1.25 45ed9/7 0.87 45ed5/4 2.12 45ed11/9 2.88 45ed6/5 3.34
46ed3/1 0.94 46ed2/1 2.39 46ed5/3 1.81 46ed3/2 4.82 46ed7/5 0.94 46ed4/3 3.81 46ed9/7 4.17 46ed5/4 2.36 46ed11/9 1.15 46ed6/5 0.32
47ed3/1 14.01 47ed2/1 12.59 47ed5/3 8.83 47ed3/2 2.00 47ed7/5 5.79 47ed4/3 1.95 47ed9/7 0.25 47ed5/4 1.58 47ed11/9 2.39 47ed6/5 2.89
48ed3/1 11.28 48ed2/1 1.96 48ed5/3 0.59 48ed3/2 6.09 48ed7/5 0.11 48ed4/3 2.90 48ed9/7 4.50 48ed5/4 2.70 48ed11/9 1.46 48ed6/5 0.61
49ed3/1 3.28 49ed2/1 8.25 49ed5/3 7.30 49ed3/2 0.48 49ed7/5 5.33 49ed4/3 2.60 49ed9/7 0.31 49ed5/4 1.07 49ed11/9 1.94 49ed6/5 2.48
50ed3/1 17.25 50ed2/1 5.96 50ed5/3 2.80 50ed3/2 6.78 50ed7/5 1.08 50ed4/3 2.07 50ed9/7 3.77 50ed5/4 3.03 50ed11/9 1.75 50ed6/5 0.87
51ed3/1 6.62 51ed2/1 3.93 51ed5/3 4.83 51ed3/2 0.93 51ed7/5 4.17 51ed4/3 3.20 51ed9/7 0.84 51ed5/4 0.61 51ed11/9 1.52 51ed6/5 2.10
52ed3/1 7.01 52ed2/1 9.65 52ed5/3 4.84 52ed3/2 5.16 52ed7/5 1.98 52ed4/3 1.31 52ed9/7 3.10 52ed5/4 3.32 52ed11/9 2.02 52ed6/5 1.11
53ed3/1 15.76 53ed2/1 0.07 53ed5/3 2.54 53ed3/2 2.23 53ed7/5 3.09 53ed4/3 3.75 53ed9/7 1.32 53ed5/4 0.18 53ed11/9 1.13 53ed6/5 1.75
54ed3/1 2.47 54ed2/1 9.16 54ed5/3 6.73 54ed3/2 3.66 54ed7/5 2.81 54ed4/3 0.60 54ed9/7 2.48 54ed5/4 3.55 54ed11/9 2.27 54ed6/5 1.34
55ed3/1 10.34 55ed2/1 3.77 55ed5/3 0.41 55ed3/2 3.43 55ed7/5 2.09 55ed4/3 4.27 55ed9/7 1.77 55ed5/4 0.22 55ed11/9 0.78 55ed6/5 1.42
56ed3/1 11.28 56ed2/1 5.19 56ed5/3 7.30 56ed3/2 2.27 56ed7/5 3.58 56ed4/3 0.06 56ed9/7 1.91 56ed5/4 3.04 56ed11/9 2.50 56ed6/5 1.55
57ed3/1 1.23 57ed2/1 7.22 57ed5/3 1.56 57ed3/2 4.55 57ed7/5 1.16 57ed4/3 4.00 57ed9/7 2.18 57ed5/4 0.59 57ed11/9 0.44 57ed6/5 1.12
58ed3/1 13.32 58ed2/1 1.49 58ed5/3 5.12 58ed3/2 0.97 58ed7/5 4.30 58ed4/3 0.67 58ed9/7 1.37 58ed5/4 2.57 58ed11/9 2.71 58ed6/5 1.74
59ed3/1 7.25 59ed2/1 9.91 59ed5/3 3.40 59ed3/2 5.59 59ed7/5 0.30 59ed4/3 3.26 59ed9/7 2.57 59ed5/4 0.93 59ed11/9 0.13 59ed6/5 0.84
60ed3/1 4.57 60ed2/1 1.96 60ed5/3 3.09 60ed3/2 0.24 60ed7/5 4.74 60ed4/3 1.25 60ed9/7 0.87 60ed5/4 2.12 60ed11/9 2.88 60ed6/5 1.92
61ed3/1 15.18 61ed2/1 6.24 61ed5/3 5.12 61ed3/2 4.94 61ed7/5 0.51 61ed4/3 2.56 61ed9/7 2.93 61ed5/4 1.25 61ed11/9 0.16 61ed6/5 0.58
62ed3/1 3.61 62ed2/1 5.18 62ed5/3 1.19 62ed3/2 1.37 62ed7/5 3.80 62ed4/3 1.78 62ed9/7 0.40 62ed5/4 1.71 62ed11/9 2.51 62ed6/5 2.09
63ed3/1 7.59 63ed2/1 2.81 63ed5/3 6.73 63ed3/2 3.66 63ed7/5 1.27 63ed4/3 1.92 63ed9/7 3.27 63ed5/4 1.56 63ed11/9 0.43 63ed6/5 0.33
64ed3/1 11.28 64ed2/1 8.21 64ed5/3 0.59 64ed3/2 2.43 64ed7/5 2.92 64ed4/3 2.28 64ed9/7 0.04 64ed5/4 1.32 64ed11/9 2.16 64ed6/5 2.25
65ed3/1 0.31 65ed2/1 0.42 65ed5/3 5.36 65ed3/2 2.46 65ed7/5 1.98 65ed4/3 1.31 65ed9/7 3.10 65ed5/4 1.84 65ed11/9 0.68 65ed6/5 0.10
66ed3/1 10.34 66ed2/1 7.14 66ed5/3 2.27 66ed3/2 3.43 66ed7/5 2.09 66ed4/3 2.76 66ed9/7 0.45 66ed5/4 0.95 66ed11/9 1.83 66ed6/5 2.38
67ed3/1 7.73 67ed2/1 3.45 67ed5/3 3.53 67ed3/2 1.33 67ed7/5 2.65 67ed4/3 0.74 67ed9/7 2.60 67ed5/4 2.10 67ed11/9 0.92 67ed6/5 0.12
68ed3/1 2.71 68ed2/1 3.93 68ed5/3 3.84 68ed3/2 4.37 68ed7/5 1.31 68ed4/3 3.20 68ed9/7 0.84 68ed5/4 0.61 68ed11/9 1.52 68ed6/5 2.10
69ed3/1 12.84 69ed2/1 6.30 69ed5/3 1.81 69ed3/2 0.27 69ed7/5 3.28 69ed4/3 0.20 69ed9/7 2.13 69ed5/4 2.36 69ed11/9 1.15 69ed6/5 0.32
70ed3/1 4.49 70ed2/1 0.90 70ed5/3 5.33 70ed3/2 4.78 70ed7/5 0.58 70ed4/3 3.50 70ed9/7 1.20 70ed5/4 0.28 70ed11/9 1.23 70ed6/5 1.83
71ed3/1 5.46 71ed2/1 7.90 71ed5/3 0.19 71ed3/2 0.73 71ed7/5 3.87 71ed4/3 0.31 71ed9/7 1.69 71ed5/4 2.59 71ed11/9 1.36 71ed6/5 0.52
72ed3/1 11.28 72ed2/1 1.96 72ed5/3 5.55 72ed3/2 3.66 72ed7/5 0.11 72ed4/3 2.90 72ed9/7 1.55 72ed5/4 0.02 72ed11/9 0.95 72ed6/5 1.58
73ed3/1 1.51 73ed2/1 4.89 73ed5/3 1.35 73ed3/2 1.68 73ed7/5 3.54 73ed4/3 0.79 73ed9/7 1.27 73ed5/4 2.48 73ed11/9 1.56 73ed6/5 0.70
74ed3/1 8.00 74ed2/1 4.66 74ed5/3 3.89 74ed3/2 2.61 74ed7/5 0.77 74ed4/3 2.34 74ed9/7 1.87 74ed5/4 0.31 74ed11/9 0.69 74ed6/5 1.35
75ed3/1 8.11 75ed2/1 2.04 75ed5/3 2.80 75ed3/2 2.58 75ed7/5 2.80 75ed4/3 1.25 75ed9/7 0.87 75ed5/4 2.12 75ed11/9 1.75 75ed6/5 0.87
76ed3/1 1.23 76ed2/1 7.22 76ed5/3 2.32 76ed3/2 1.61 76ed7/5 1.39 76ed4/3 1.81 76ed9/7 2.18 76ed5/4 0.59 76ed11/9 0.44 76ed6/5 1.12
77ed3/1 10.34 77ed2/1 0.66 77ed5/3 4.18 77ed3/2 3.43 77ed7/5 2.09 77ed4/3 1.68 77ed9/7 0.49 77ed5/4 1.79 77ed11/9 1.93 77ed6/5 1.03
78ed3/1 5.18 78ed2/1 5.74 78ed5/3 0.82 78ed3/2 0.66 78ed7/5 1.98 78ed4/3 1.31 78ed9/7 2.48 78ed5/4 0.85 78ed11/9 0.21 78ed6/5 0.91
79ed3/1 3.77 79ed2/1 3.22 79ed5/3 5.49 79ed3/2 4.24 79ed7/5 1.42 79ed4/3 2.09 79ed9/7 0.14 79ed5/4 1.47 79ed11/9 2.10 79ed6/5 1.19
80ed3/1 11.28 80ed2/1 3.04 80ed5/3 0.59 80ed3/2 0.24 80ed7/5 2.54 80ed4/3 0.83 80ed9/7 2.68 80ed5/4 1.10 80ed11/9 0.01 80ed6/5 0.71
81ed3/1 2.47 81ed2/1 5.66 81ed5/3 4.18 81ed3/2 3.66 81ed7/5 0.79 81ed4/3 2.48 81ed9/7 0.20 81ed5/4 1.17 81ed11/9 2.02 81ed6/5 1.34
82ed3/1 6.12 82ed2/1 0.48 82ed5/3 1.94 82ed3/2 1.09 82ed7/5 3.07 82ed4/3 0.37 82ed9/7 2.29 82ed5/4 1.33 82ed11/9 0.23 82ed6/5 0.51
83ed3/1 8.41 83ed2/1 6.48 83ed5/3 2.74 83ed3/2 2.72 83ed7/5 0.18 83ed4/3 2.85 83ed9/7 0.53 83ed5/4 0.88 83ed11/9 1.76 83ed6/5 1.48
84ed3/1 0.04 84ed2/1 1.96 84ed5/3 3.23 84ed3/2 1.91 84ed7/5 3.35 84ed4/3 0.06 84ed9/7 1.91 84ed5/4 1.56 84ed11/9 0.43 84ed6/5 0.33
85ed3/1 8.30 85ed2/1 3.93 85ed5/3 1.36 85ed3/2 1.83 85ed7/5 0.40 85ed4/3 2.66 85ed9/7 0.84 85ed5/4 0.61 85ed11/9 1.52 85ed6/5 1.61
86ed3/1 5.75 86ed2/1 4.28 86ed5/3 4.45 86ed3/2 2.69 86ed7/5 2.71 86ed4/3 0.47 86ed9/7 1.55 86ed5/4 1.77 86ed11/9 0.62 86ed6/5 0.16
87ed3/1 2.39 87ed2/1 1.49 87ed5/3 0.04 87ed3/2 0.97 87ed7/5 0.95 87ed4/3 2.19 87ed9/7 1.13 87ed5/4 0.35 87ed11/9 1.28 87ed6/5 1.74
88ed3/1 10.34 88ed2/1 6.50 88ed5/3 4.43 88ed3/2 3.43 88ed7/5 2.09 88ed4/3 0.87 88ed9/7 1.20 88ed5/4 1.97 88ed11/9 0.80 88ed6/5 0.01
89ed3/1 3.26 89ed2/1 0.83 89ed5/3 1.21 89ed3/2 0.16 89ed7/5 1.48 89ed4/3 1.74 89ed9/7 1.41 89ed5/4 0.10 89ed11/9 1.06 89ed6/5 1.68
90ed3/1 4.57 90ed2/1 4.71 90ed5/3 3.09 90ed3/2 3.66 90ed7/5 1.50 90ed4/3 1.25 90ed9/7 0.87 90ed5/4 2.12 90ed11/9 0.98 90ed6/5 0.17
91ed3/1 8.67 91ed2/1 3.05 91ed5/3 2.42 91ed3/2 0.62 91ed7/5 1.98 91ed4/3 1.31 91ed9/7 1.68 91ed5/4 0.14 91ed11/9 0.84 91ed6/5 1.49
92ed3/1 0.94 92ed2/1 2.39 92ed5/3 1.81 92ed3/2 2.81 92ed7/5 0.94 92ed4/3 1.61 92ed9/7 0.56 92ed5/4 1.84 92ed11/9 1.15 92ed6/5 0.32
93ed3/1 6.62 93ed2/1 5.18 93ed5/3 3.56 93ed3/2 1.37 93ed7/5 2.46 93ed4/3 0.90 93ed9/7 1.94 93ed5/4 0.37 93ed11/9 0.64 93ed6/5 1.30
94ed3/1 6.22 94ed2/1 0.17 94ed5/3 0.58 94ed3/2 2.00 94ed7/5 0.40 94ed4/3 1.95 94ed9/7 0.25 94ed5/4 1.58 94ed11/9 1.31 94ed6/5 0.47
95ed3/1 1.23 95ed2/1 5.41 95ed5/3 4.64 95ed3/2 2.09 95ed7/5 2.92 95ed4/3 0.50 95ed9/7 2.18 95ed5/4 0.59 95ed11/9 0.44 95ed6/5 1.12
96ed3/1 8.53 96ed2/1 1.96 96ed5/3 0.59 96ed3/2 1.22 96ed7/5 0.11 96ed4/3 2.28 96ed9/7 0.04 96ed5/4 1.32 96ed11/9 1.46 96ed6/5 0.61
97ed3/1 3.93 97ed2/1 3.20 97ed5/3 3.40 97ed3/2 2.77 97ed7/5 2.64 97ed4/3 0.12 97ed9/7 2.07 97ed5/4 0.80 97ed11/9 0.25 97ed6/5 0.95
98ed3/1 3.28 98ed2/1 4.00 98ed5/3 1.72 98ed3/2 0.48 98ed7/5 0.61 98ed4/3 2.48 98ed9/7 0.31 98ed5/4 1.07 98ed11/9 1.61 98ed6/5 0.74
99ed3/1 8.88 99ed2/1 1.08 99ed5/3 2.20 99ed3/2 3.43 99ed7/5 2.09 99ed4/3 0.24 99ed9/7 1.75 99ed5/4 1.00 99ed11/9 0.07 99ed6/5 0.79
100ed3/1 1.77 100ed2/1 5.96 100ed5/3 2.80 100ed3/2 0.24 100ed7/5 1.08 100ed4/3 2.07 100ed9/7 0.58 100ed5/4 0.84 100ed11/9 1.72 100ed6/5 0.87
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