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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
|-
|-
| Alpha 3/1
! Name
| 3ed3/1
! Equal division
| 1.892789
! Steps per octave
| 633.985000
! Equave
| 1.907395926960071
! SSC pair
| 629.130000247253548
! Steps (Equave, SSC pair)
| 3/1, 2/1, 3/2
! Errors (cent)
|-
| [[Alpha 3/1]]
| [[3edt|3ed3/1]]
| 1.89278926071437
| rowspan="3" | 3/1
| rowspan="3" | 2/1, 3/2
| 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, 67.9700, -67.9700
| 0, 67.970, -67.970
| -14.5650, 58.2600, -72.8250
|-
|-
| Beta 3/1
| [[Beta 3/1]]
| 5ed3/1
| [[5edt|5ed3/1]]
| 3.154649
| 3.15464876785729
| 380.391000
| 3.141862316907629
| 381.939079106781893
| 3/1, 2/1, 3/2
| 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, -58.8270, 58.8270
| 0, -58.827, 58.827
| 7.7404, -54.1828, 61.9232
|-
|-
| Gamma 3/1
| [[Gamma 3/1]]
| 8ed3/1
| [[8edt|8ed3/1]]
| 5.047438
| 5.04743802857166
| 237.744375
| 5.042556213760587
| 237.974540913461853
| 3/1, 2/1, 3/2
| 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, -11.2781, 11.2781
| 0, -11.278, 11.278
| 1.8413, -10.1273, 11.9686
|-
|-
| Alpha 2/1
| [[Alpha 2/1]]
| 5ed2/1
| [[5edo|5ed2/1]]
| 5.000000
| 5
| 240.000000
| rowspan="3" | 2/1
| 5.009912705090773
| rowspan="3" | 3/2, 4/3
| 239.525131601720722
| 2/1, 3/2, 4/3
| 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, 18.0450, -18.0450
| 0, 18.045, -18.045
| -2.3743, 16.6204, -18.9947
|-
|-
| Beta 2/1
| [[Beta 2/1]]
| 7ed2/1
| [[7edo|7ed2/1]]
| 7.000000
| 7
| 171.428571
| 6.991049802487100
| 171.648040552234965
| 2/1, 3/2, 4/3
| 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, -16.2407, 16.2407
| 0, -16.241, 16.241
| 1.5363, -15.3628, 16.8991
|-
|-
| Gamma 2/1
| [[Gamma 2/1]]
| 12ed2/1
| [[12edo|12ed2/1]]
| 12.000000
| 12
| 100.000000
| 11.997848091431052
| 100.017935787755848
| 2/1, 3/2, 4/3
| 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, -1.9550, 1.9550
| 0, -1.955, 1.955
| 0.2152, -1.8295, 2.0447
|-
|-
| Alpha 5/3
| [[Alpha 5/3]]
| 7ed5/3
| [[7ed5/3]]
| 9.498408
| 9.49840814199707
| 126.336959
| rowspan="3" | 5/3
| 9.505833538777849
| rowspan="3" | 4/3, 5/4
| 126.238272015257927
| 5/3, 4/3, 5/4
| 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, 7.3028, -7.3028
| 0, 7.303, -7.303
| -0.6908, 6.9081, -7.5989
|-
|-
| Beta 5/3
| [[Beta 5/3]]
| 9ed5/3
| [[9ed5/3]]
| 12.212239
| 12.2122390397105
| 98.262079
| 12.205382300878206
| 98.317280886290400
| 5/3, 4/3, 5/4
| 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, -6.7346, 6.7346
| 0, -6.735, 6.735
| 0.4968, -6.4586, 6.9554
|-
|-
| Gamma 5/3
| [[Gamma 5/3]]
| 16ed5/3
| [[16ed5/3]]
| 21.710647
| 21.7106471817076
| 55.272420
| 21.709439921550910
| 55.275493257141231
| 5/3, 4/3, 5/4
| 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, -0.5932, 0.5932
| 0, -0.593, 0.593
| 0.0492, -0.5656, 0.6147
|-
|-
| Alpha 3/2
| [[Carlos Alpha|Alpha 3/2]]
| 9ed3/2
| [[9edf|9ed3/2]]
| 15.385602
| 15.3856016221631
| 77.995000
| rowspan="3" | 3/2
| 15.391523899692793
| rowspan="3" | 5/4, 6/5
| 77.964989550121895
| 3/2, 5/4, 6/5
| 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, 3.6613, -3.6613
| 0, 3.661, -3.661
| -0.2701, 3.5112, -3.7813
|-
|-
| Beta 3/2
| [[Carlos Beta|Beta 3/2]]
| 11ed3/2
| [[11edf|11ed3/2]]
| 18.804624
| 18.8046242048660
| 63.814091
| 18.799073639411081
| 63.832932569840843
| 3/2, 5/4, 6/5
| 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, -3.4292, 3.4292
| 0, -3.429, 3.429
| 0.2073, -3.3161, 3.5234
|-
|-
| Gamma 3/2
| [[Carlos Gamma|Gamma 3/2]]
| 20ed3/2
| [[20edf|20ed3/2]]
| 34.190226
| 34.1902258270291
| 35.097750
| 34.189454092191388
| 35.098542280441702
| 3/2, 5/4, 6/5
| 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, -0.2385, 0.2385
| 0, -0.238, 0.238
| 0.0158, -0.2297, 0.2456
|-
|-
| Alpha 7/5
| [[Alpha 7/5]]
| 11ed7/5
| [[11ed7/5]]
| 22.660470
| 22.6604698881676
| 52.955654
| rowspan="3" | 7/5
| 22.665391113336561
| rowspan="3" | 6/5, 7/6
| 52.944155871808760
| 7/5, 6/5, 7/6
| 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, 2.0926, -2.0926
| 0, 2.093, -2.093
| -0.1265, 2.0236, -2.1501
|-
|-
| Beta 7/5
| [[Beta 7/5]]
| 13ed7/5
| [[13ed7/5]]
| 26.780555
| 26.7805553223799
| 44.808630
| 26.775895108856630
| 44.816428923157735
| 7/5, 6/5, 7/6
| 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, -1.9809, 1.9809
| 0, -1.981, 1.981
| 0.1014, -1.9263, 2.0277
|-
|-
| Gamma 7/5
| [[Gamma 7/5]]
| 24ed7/5
| [[24ed7/5]]
| 49.441025
| 49.4410252105475
| 24.271341
| 49.440489621601243
| 24.271604290013001
| 7/5, 6/5, 7/6
| 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, -0.1138, 0.1138
| 0, -0.114, 0.114
| 0.0063, -0.1104, 0.1167
|-
|-
| Alpha 4/3
| [[Alpha 4/3]]
| 13ed4/3
| [[13ed4/3]]
| 31.322471
| 31.3224709154917
| 38.311154
| rowspan="3" | 4/3
| 31.326679032092577
| rowspan="3" | 7/6, 8/7
| 38.306007437643215
| 4/3, 7/6, 8/7
| 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, 1.3072, -1.3072
| 0, 1.307, -1.307
| -0.0669, 1.2711, -1.3380
|-
|-
| Beta 4/3
| [[Beta 4/3]]
| 15ed4/3
| [[15ed4/3]]
| 36.141313
| 36.1413125947981
| 33.203000
| 36.137297503882719
| 33.206689013506551
| 4/3, 7/6, 8/7
| 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, -1.2469, 1.2469
| 0, -1.247, 1.247
| 0.0553, -1.2174, 1.2727
|-
|-
| Gamma 4/3
| [[Gamma 4/3]]
| 28ed4/3
| [[28ed4/3]]
| 67.463784
| 67.4637835102899
| 17.787321
| 67.463390164664623
| 17.787425106728855
| 4/3, 7/6, 8/7
| 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, -0.0611, 0.0611
| 0, -0.061, 0.061
| 0.0029, -0.0595, 0.0624
|-
|-
| Alpha 9/7
| [[Alpha 9/7]]
| 15ed9/7
| [[15ed9/7]]
| 41.371312
| 41.3713123417559
| 29.005606
| rowspan="3" | 9/7
| 41.374987163985893
| rowspan="3" | 8/7, 9/8
| 29.003030145820039
| 9/7, 8/7, 9/8
| 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, 0.8708, -0.8708
| 0, 0.871, -0.871
| -0.0386, 0.8501, -0.8888
|-
|-
| Beta 9/7
| [[Beta 9/7]]
| 17ed9/7
| [[17ed9/7]]
| 46.887487
| 46.8874873206567
| 25.593182
| 46.883960906871343
| 25.595107085419638
| 9/7, 8/7, 9/8
| 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, -0.8355, 0.8355
| 0, -0.835, 0.835
| 0.0327, -0.8181, 0.8509
|-
|-
| Gamma 9/7
| [[Gamma 9/7]]
| 32ed9/7
| [[32ed9/7]]
| 88.258800
| 88.2587996624126
| 13.596378
| 88.258498580415662
| 13.596424359141285
| 9/7, 8/7, 9/8
| 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, -0.0357, 0.0357
| 0, -0.036, 0.036
| 0.0015, -0.0349, 0.0364
|-
|-
| Alpha 5/4
| [[Alpha 5/4]]
| 17ed5/4
| [[17ed5/4]]
| 52.806823
| 52.8068232315916
| 22.724336
| rowspan="3" | 5/4
| 52.810084374305705
| rowspan="3" | 9/8, 10/9
| 22.722932830303330
| 5/4, 9/8, 10/9
| 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, 0.6090, -0.6090
| 0, 0.609, -0.609
| -0.0239, 0.5964, -0.6202
|-
|-
| Beta 5/4
| [[Beta 5/4]]
| 19ed5/4
| [[19ed5/4]]
| 59.019391
| 59.0193906706024
| 20.332301
| 59.016247125030467
| 20.333383745288099
| 5/4, 9/8, 10/9
| 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, -0.5870, 0.5870
| 0, -0.587, 0.587
| 0.0206, -0.5762, 0.5967
|-
|-
| Gamma 5/4
| [[Gamma 5/4]]
| 36ed5/4
| [[36ed5/4]]
| 111.826214
| 111.826213902194
| 10.730936
| 111.825976049765954
| 10.730959320810789
| 5/4, 9/8, 10/9
| 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, -0.0222, 0.0222
| 0, -0.022, 0.022
| 0.0008, -0.0218, 0.0226
|-
|-
| Alpha 11/9
| [[Alpha 11/9]]
| 19ed11/9
| [[19ed11/9]]
| 65.628897
| 65.6288971357202
| 18.284628
| rowspan="3" | 11/9
| 65.631828119476568
| rowspan="3" | 10/9, 11/10
| 18.283811900157846
| 11/9, 10/9, 11/10
| 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, 0.4426, -0.4426
| 0, 0.443, -0.443
| -0.0155, 0.4344, -0.4499
|-
|-
| Beta 11/9
| [[Beta 11/9]]
| 21ed11/9
| [[21ed11/9]]
| 72.537202
| 72.5372020973750
| 16.543235
| 72.534366561494206
| 16.543881981552112
| 11/9, 10/9, 11/10
| 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, -0.4281, 0.4281
| 0, -0.428, 0.428
| 0.0136, -0.4210, 0.4346
|-
|-
| Gamma 11/9
| [[Gamma 11/9]]
| 40ed11/9
| [[40ed11/9]]
| 138.166099
| 138.166099233095
| 8.685199
| 138.165906595462172
| 8.685210625176124
| 11/9, 10/9, 11/10
| 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, -0.0145, 0.0145
| 0, -0.015, 0.015
| 0.0005, -0.0143, 0.0148
|-
|-
| Alpha 6/5
| [[Alpha 6/5]]
| 21ed6/5
| [[21ed6/5]]
| 79.837464
| 79.8374643554025
| 15.030537
| rowspan="3" | 6/5
| 79.840125772190183
| rowspan="3" | 11/10, 12/11
| 15.030036443379233
| 6/5, 11/10, 12/11
| 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, 0.3317, -0.3317
| 0, 0.332, -0.332
| -0.0105, 0.3262, -0.3367
|-
|-
| Beta 6/5
| [[Beta 6/5]]
| 23ed6/5
| [[23ed6/5]]
| 87.441032
| 87.4410323892504
| 13.723534
| 87.438449973495273
| 13.723939529620542
| 6/5, 11/10, 12/11
| 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, -0.3218, 0.3218
| 0, -0.322, 0.322
| 0.0093, -0.3170, 0.3263
|-
|-
| Gamma 6/5
| [[Gamma 6/5]]
| 44ed6/5
| [[44ed6/5]]
| 167.278497
| 167.278496744653
| 7.173666
| 167.278337553931523
| 7.173672440480304
| 6/5, 11/10, 12/11
| 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, -0.0099, 0.0099
| 0, -0.010, 0.010
| 0.0003, -0.0098, 0.0101
|-
|-
| Alpha 13/11
| [[Alpha 13/11]]
| 23ed13/11
| [[23ed13/11]]
| 95.432477
| 95.4324773621886
| 12.574336
| rowspan="3" | 13/11
| 95.434914550823771
| rowspan="3" | 12/11, 13/12
| 12.574014506618971
| 13/11, 12/11, 13/12
| 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, 0.2550, -0.2550
| 0, 0.255, -0.255
| -0.0074, 0.2511, -0.2585
|-
|-
| Beta 13/11
| [[Beta 13/11]]
| 25ed13/11
| [[25ed13/11]]
| 103.730954
| 103.730953654553
| 11.568389
| 103.728582924336770
| 11.568653173208022
| 13/11, 12/11, 13/12
| 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, -0.2480, 0.2480
| 0, -0.248, 0.248
| 0.0066, -0.2446, 0.2512
|-
|-
| Gamma 13/11
| [[Gamma 13/11]]
| 48ed13/11
| [[48ed13/11]]
| 199.163431
| 199.163431016741
| 6.025202
| 199.163297261207502
| 6.025206534044126
| 13/11, 12/11, 13/12
| 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, -0.0070, 0.0070
| 0, -0.007, 0.007
| 0.0002, -0.0069, 0.0071
|-
|-
| Alpha 7/6
| [[Alpha 7/6]]
| 25ed7/6
| [[25ed7/6]]
| 112.413903
| 112.413902640048
| 10.674836
| rowspan="3" | 7/6
| 112.416150402630623
| rowspan="3" | 13/12, 14/13
| 10.674622780642016
| 7/6, 13/12, 14/13
| 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, 0.2002, -0.2002
| 0, 0.200, -0.200
| -0.0053, 0.1974, -0.2028
|-
|-
| Beta 7/6
| [[Beta 7/6]]
| 27ed7/6
| [[27ed7/6]]
| 121.407015
| 121.407014851252
| 9.884108
| 121.404823766036118
| 9.884286000962910
| 7/6, 13/12, 14/13
| 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, -0.1952, 0.1952
| 0, -0.195, 0.195
| 0.0048, -0.1927, 0.1975
|-
|-
| Gamma 7/6
| [[Gamma 7/6]]
| 52ed7/6
| [[52ed7/6]]
| 233.820917
| 233.820917491300
| 5.132133
| 233.820803527976982
| 5.132135301452842
| 7/6, 13/12, 14/13
| 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, -0.0051, 0.0051
| 0, -0.005, 0.005
| 0.0001, -0.0050, 0.0051
|-
|-
| Alpha 15/13
| [[Alpha 15/13]]
| 27ed15/13
| [[27ed15/13]]
| 130.781716
| 130.781715879411
| 9.175595
| rowspan="3" | 15/13
| 130.783801507844919
| rowspan="3" | 14/13, 15/14
| 9.175448229557843
| 15/13, 14/13, 15/14
| 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, 0.1601, -0.1601
| 0, 0.160, -0.160
| -0.0040, 0.1580, -0.1620
|-
|-
| Beta 15/13
| [[Beta 15/13]]
| 29ed15/13
| [[29ed15/13]]
| 140.469250
| 140.469250388997
| 8.542795
| 140.467213664559518
| 8.542918797162452
| 15/13, 14/13, 15/14
| 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, -0.1563, 0.1563
| 0, -0.156, 0.156
| 0.0036, -0.1545, 0.1581
|-
|-
| Gamma 15/13
| [[Gamma 15/13]]
| 56ed15/13
| [[56ed15/13]]
| 271.250966
| 271.250966268408
| 4.423947
| 271.250868008139347
| 4.423948976871078
| 15/13, 14/13, 15/14
| 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, -0.0038, 0.0038
| 0, -0.004, 0.004
| 0.0001, -0.0037, 0.0038
|-
|-
| Alpha 8/7
| [[Alpha 8/7]]
| 29ed8/7
| [[29ed8/7]]
| 150.535899
| 150.535899020849
| 7.971520
| rowspan="3" | 8/7
| 150.537844310638475
| rowspan="3" | 15/14, 16/15
| 7.971417456488689
| 8/7, 15/14, 16/15
| 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, 0.1300, -0.1300
| 0, 0.130, -0.130
| -0.0030, 0.1285, -0.1314
|-
|-
| Beta 8/7
| [[Beta 8/7]]
| 31ed8/7
| [[31ed8/7]]
| 160.917685
| 160.917685160217
| 7.457229
| 160.915782495277457
| 7.457316997698579
| 8/7, 15/14, 16/15
| 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, -0.1271, 0.1271
| 0, -0.127, 0.127
| 0.0027, -0.1257, 0.1285
|-
|-
| Gamma 8/7
| [[Gamma 8/7]]
| 60ed8/7
| [[60ed8/7]]
| 311.453584
| 311.453584181066
| 3.852902
| 311.453498588281532
| 3.852902617691610
| 8/7, 15/14, 16/15
| 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, -0.0029, 0.0029
| 0, -0.003, 0.003
| 0.0001, -0.0028, 0.0029
|}
|}
{{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"