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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"
Retrieved from "https://en.xen.wiki/w/User:Contribution/Successive_superparticular_complementary_pair"