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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"
|+
|+
!
!
!1
!2
!3
!4
!5
!6
!7
!8
!9
|-
| rowspan="9" |3/1
| rowspan="9" |2/1, 3/2
| rowspan="3" |Alpha 3/1
| rowspan="3" |3ed3/1
| rowspan="3" |1.892789
| rowspan="3" |633.985000
| rowspan="3" |1.907395926960071
| rowspan="3" |629.130000247253548
|3/1 mapping: 3\3<3/1>
|equal division error: 0.000
|optimization error: -14.565
|-
|2/1 mapping: 2\3<3/1>
|equal division error: 67.970
|optimization error: 58.260
|-
|3/2 mapping: 1\3<3/1>
|equal division error: -67.970
|optimization error: -72.825
|-
| rowspan="3" |Beta 3/1
| rowspan="3" |5ed3/1
| rowspan="3" |3.154649
| rowspan="3" |380.391000
| rowspan="3" |3.141862316907629
| rowspan="3" |381.939079106781893
|3/1 mapping: 5\5<3/1>
|equal division error: 0.000
|optimization error: 7.740
|-
|2/1 mapping: 3\5<3/1>
|equal division error: -58.827
|optimization error: -54.183
|-
|3/2 mapping: 2\5<3/1>
|equal division error: 58.827
|optimization error: 61.923
|-
| rowspan="3" |Gamma 3/1
| rowspan="3" |8ed3/1
| rowspan="3" |5.047438
| rowspan="3" |237.744375
| rowspan="3" |5.042556213760587
| rowspan="3" |237.974540913461853
|3/1 mapping: 8\8<3/1>
|equal division error: 0.000
|optimization error: 1.841
|-
|2/1 mapping: 5\8<3/1>
|equal division error: -11.278
|optimization error: -10.127
|-
|3/2 mapping: 3\8<3/1>
|equal division error: 11.278
|optimization error: 11.969
|-
| rowspan="9" |2/1
| rowspan="9" |3/2, 4/3
| rowspan="3" |Alpha 2/1
| rowspan="3" |5ed2/1
| rowspan="3" |5.000000
| rowspan="3" |240.000000
| rowspan="3" |5.009912705090773
| rowspan="3" |239.525131601720722
|2/1 mapping: 5\5<2/1>
|equal division error: 0.000
|optimization error: -2.374
|-
|3/2 mapping: 3\5<2/1>
|equal division error: 18.045
|optimization error: 16.620
|-
|4/3 mapping: 2\5<2/1>
|equal division error: -18.045
|optimization error: -18.995
|-
| rowspan="3" |Beta 2/1
| rowspan="3" |7ed2/1
| rowspan="3" |7.000000
| rowspan="3" |171.428571
| rowspan="3" |6.991049802487100
| rowspan="3" |171.648040552234965
|2/1 mapping: 7\7<2/1>
|equal division error: 0.000
|optimization error: 1.536
|-
|3/2 mapping: 4\7<2/1>
|equal division error: -16.241
|optimization error: -15.363
|-
|4/3 mapping: 3\7<2/1>
|equal division error: 16.241
|optimization error: 16.899
|-
| rowspan="3" |Gamma 2/1
| rowspan="3" |12ed2/1
| rowspan="3" |12.000000
| rowspan="3" |100.000000
| rowspan="3" |11.997848091431052
| rowspan="3" |100.017935787755848
|2/1 mapping: 12\12<2/1>
|equal division error: 0.000
|optimization error: 0.215
|-
|3/2 mapping: 7\12<2/1>
|equal division error: -1.955
|optimization error: -1.829
|-
|4/3 mapping: 5\12<2/1>
|equal division error: 1.955
|optimization error: 2.045
|-
|
|
| rowspan="3" |Alpha 5/3
| rowspan="3" |7ed5/3
| rowspan="3" |9.498408
| rowspan="3" |126.336959
| rowspan="3" |9.505833538777849
| rowspan="3" |126.238272015257927
|5/3 mapping: 7\7<5/3>
|equal division error: 0.000
|optimization error: -0.691
|-
|
|
|4/3 mapping: 4\7<5/3>
|equal division error: 7.303
|optimization error: 6.908
|-
|
|
|5/4 mapping: 3\7<5/3>
|equal division error: -7.303
|optimization error: -7.599
|-
|
|
| rowspan="3" |Beta 5/3
| rowspan="3" |9ed5/3
| rowspan="3" |12.212239
| rowspan="3" |98.262079
| rowspan="3" |12.205382300878206
| rowspan="3" |98.317280886290400
|5/3 mapping: 9\9<5/3>
|equal division error: 0.000
|optimization error: 0.497
|-
|
|
|4/3 mapping: 5\9<5/3>
|equal division error: -6.735
|optimization error: -6.459
|-
|
|
|5/4 mapping: 4\9<5/3>
|equal division error: 6.735
|optimization error: 6.955
|-
|
|
| rowspan="3" |Gamma 5/3
| rowspan="3" |16ed5/3
| rowspan="3" |21.710647
| rowspan="3" |55.272420
| rowspan="3" |21.709439921550910
| rowspan="3" |55.275493257141231
|5/3 mapping: 16\16<5/3>
|equal division error: 0.000
|optimization error: 0.049
|-
|
|
|4/3 mapping: 9\16<5/3>
|equal division error: -0.593
|optimization error: -0.566
|-
|
|
|5/4 mapping: 7\16<5/3>
|equal division error: 0.593
|optimization error: 0.615
|-
|
|
| rowspan="3" |Alpha 3/2
| rowspan="3" |9ed3/2
| rowspan="3" |15.385602
| rowspan="3" |77.995000
| rowspan="3" |15.391523899692793
| rowspan="3" |77.964989550121895
|3/2 mapping: 9\9<3/2>
|equal division error: 0.000
|optimization error: -0.270
|-
|
|
|5/4 mapping: 5\9<3/2>
|equal division error: 3.661
|optimization error: 3.511
|-
|
|
|6/5 mapping: 4\9<3/2>
|equal division error: -3.661
|optimization error: -3.781
|-
|
|
| rowspan="3" |Beta 3/2
| rowspan="3" |11ed3/2
| rowspan="3" |18.804624
| rowspan="3" |63.814091
| rowspan="3" |18.799073639411081
| rowspan="3" |63.832932569840843
|3/2 mapping: 11\11<3/2>
|equal division error: 0.000
|optimization error: 0.207
|-
|
|
|5/4 mapping: 6\11<3/2>
|equal division error: -3.429
|optimization error: -3.316
|-
|
|
|6/5 mapping: 5\11<3/2>
|equal division error: 3.429
|optimization error: 3.523
|-
|
|
| rowspan="3" |Gamma 3/2
| rowspan="3" |20ed3/2
| rowspan="3" |34.190226
| rowspan="3" |35.097750
| rowspan="3" |34.189454092191388
| rowspan="3" |35.098542280441702
|3/2 mapping: 20\20<3/2>
|equal division error: 0.000
|optimization error: 0.016
|-
|
|
|5/4 mapping: 11\20<3/2>
|equal division error: -0.238
|optimization error: -0.230
|-
|
|
|6/5 mapping: 9\20<3/2>
|equal division error: 0.238
|optimization error: 0.246
|-
|
|
| rowspan="3" |Alpha 7/5
| rowspan="3" |11ed7/5
| rowspan="3" |22.660470
| rowspan="3" |52.955654
| rowspan="3" |22.665391113336561
| rowspan="3" |52.944155871808760
|7/5 mapping: 11\11<7/5>
|equal division error: 0.000
|optimization error: -0.126
|-
|
|
|6/5 mapping: 6\11<7/5>
|equal division error: 2.093
|optimization error: 2.024
|-
|
|
|7/6 mapping: 5\11<7/5>
|equal division error: -2.093
|optimization error: -2.150
|-
|
|
| rowspan="3" |Beta 7/5
| rowspan="3" |13ed7/5
| rowspan="3" |26.780555
| rowspan="3" |44.808630
| rowspan="3" |26.775895108856630
| rowspan="3" |44.816428923157735
|7/5 mapping: 13\13<7/5>
|equal division error: 0.000
|optimization error: 0.101
|-
|
|
|6/5 mapping: 7\13<7/5>
|equal division error: -1.981
|optimization error: -1.926
|-
|
|
|7/6 mapping: 6\13<7/5>
|equal division error: 1.981
|optimization error: 2.028
|-
|
|
| rowspan="3" |Gamma 7/5
| rowspan="3" |24ed7/5
| rowspan="3" |49.441025
| rowspan="3" |24.271341
| rowspan="3" |49.440489621601243
| rowspan="3" |24.271604290013001
|7/5 mapping: 24\24<7/5>
|equal division error: 0.000
|optimization error: 0.006
|-
|
|
|6/5 mapping: 13\24<7/5>
|equal division error: -0.114
|optimization error: -0.110
|-
|
|
|7/6 mapping: 11\24<7/5>
|equal division error: 0.114
|optimization error: 0.117
|-
|
|
| rowspan="3" |Alpha 4/3
| rowspan="3" |13ed4/3
| rowspan="3" |31.322471
| rowspan="3" |38.311154
| rowspan="3" |31.326679032092577
| rowspan="3" |38.306007437643215
|4/3 mapping: 13\13<4/3>
|equal division error: 0.000
|optimization error: -0.067
|-
|
|
|7/6 mapping: 7\13<4/3>
|equal division error: 1.307
|optimization error: 1.271
|-
|
|
|8/7 mapping: 6\13<4/3>
|equal division error: -1.307
|optimization error: -1.338
|-
|
|
| rowspan="3" |Beta 4/3
| rowspan="3" |15ed4/3
| rowspan="3" |36.141313
| rowspan="3" |33.203000
| rowspan="3" |36.137297503882719
| rowspan="3" |33.206689013506551
|4/3 mapping: 15\15<4/3>
|equal division error: 0.000
|optimization error: 0.055
|-
|
|
|7/6 mapping: 8\15<4/3>
|equal division error: -1.247
|optimization error: -1.217
|-
|
|
|8/7 mapping: 7\15<4/3>
|equal division error: 1.247
|optimization error: 1.273
|-
|
|
| rowspan="3" |Gamma 4/3
| rowspan="3" |28ed4/3
| rowspan="3" |67.463784
| rowspan="3" |17.787321
| rowspan="3" |67.463390164664623
| rowspan="3" |17.787425106728855
|4/3 mapping: 28\28<4/3>
|equal division error: 0.000
|optimization error: 0.003
|-
|
|
|7/6 mapping: 15\28<4/3>
|equal division error: -0.061
|optimization error: -0.060
|-
|
|
|8/7 mapping: 13\28<4/3>
|equal division error: 0.061
|optimization error: 0.062
|-
|
|
| rowspan="3" |Alpha 9/7
| rowspan="3" |15ed9/7
| rowspan="3" |41.371312
| rowspan="3" |29.005606
| rowspan="3" |41.374987163985893
| rowspan="3" |29.003030145820039
|9/7 mapping: 15\15<9/7>
|equal division error: 0.000
|optimization error: -0.039
|-
|
|
|8/7 mapping: 8\15<9/7>
|equal division error: 0.871
|optimization error: 0.850
|-
|
|
|9/8 mapping: 7\15<9/7>
|equal division error: -0.871
|optimization error: -0.889
|-
|
|
| rowspan="3" |Beta 9/7
| rowspan="3" |17ed9/7
| rowspan="3" |46.887487
| rowspan="3" |25.593182
| rowspan="3" |46.883960906871343
| rowspan="3" |25.595107085419638
|9/7 mapping: 17\17<9/7>
|equal division error: 0.000
|optimization error: 0.033
|-
|
|
|8/7 mapping: 9\17<9/7>
|equal division error: -0.835
|optimization error: -0.818
|-
|
|
|9/8 mapping: 8\17<9/7>
|equal division error: 0.835
|optimization error: 0.851
|-
|
|
| rowspan="3" |Gamma 9/7
| rowspan="3" |32ed9/7
| rowspan="3" |88.258800
| rowspan="3" |13.596378
| rowspan="3" |88.258498580415662
| rowspan="3" |13.596424359141285
|9/7 mapping: 32\32<9/7>
|equal division error: 0.000
|optimization error: 0.001
|-
|
|
|8/7 mapping: 17\32<9/7>
|equal division error: -0.036
|optimization error: -0.035
|-
|
|
|9/8 mapping: 15\32<9/7>
|equal division error: 0.036
|optimization error: 0.036
|-
|
|
| rowspan="3" |Alpha 5/4
| rowspan="3" |17ed5/4
| rowspan="3" |52.806823
| rowspan="3" |22.724336
| rowspan="3" |52.810084374305705
| rowspan="3" |22.722932830303330
|5/4 mapping: 17\17<5/4>
|equal division error: 0.000
|optimization error: -0.024
|-
|
|
|9/8 mapping: 9\17<5/4>
|equal division error: 0.609
|optimization error: 0.596
|-
|
|
|10/9 mapping: 8\17<5/4>
|equal division error: -0.609
|optimization error: -0.620
|-
|
|
| rowspan="3" |Beta 5/4
| rowspan="3" |19ed5/4
| rowspan="3" |59.019391
| rowspan="3" |20.332301
| rowspan="3" |59.016247125030467
| rowspan="3" |20.333383745288099
|5/4 mapping: 19\19<5/4>
|equal division error: 0.000
|optimization error: 0.021
|-
|
|
|9/8 mapping: 10\19<5/4>
|equal division error: -0.587
|optimization error: -0.576
|-
|
|
|10/9 mapping: 9\19<5/4>
|equal division error: 0.587
|optimization error: 0.597
|-
|
|
| rowspan="3" |Gamma 5/4
| rowspan="3" |36ed5/4
| rowspan="3" |111.826214
| rowspan="3" |10.730936
| rowspan="3" |111.825976049765954
| rowspan="3" |10.730959320810789
|5/4 mapping: 36\36<5/4>
|equal division error: 0.000
|optimization error: 0.001
|-
|
|
|9/8 mapping: 19\36<5/4>
|equal division error: -0.022
|optimization error: -0.022
|-
|
|
|10/9 mapping: 17\36<5/4>
|equal division error: 0.022
|optimization error: 0.023
|-
|
|
| rowspan="3" |Alpha 11/9
| rowspan="3" |19ed11/9
| rowspan="3" |65.628897
| rowspan="3" |18.284628
| rowspan="3" |65.631828119476568
| rowspan="3" |18.283811900157846
|11/9 mapping: 19\19<11/9>
|equal division error: 0.000
|optimization error: -0.016
|-
|
|
|10/9 mapping: 10\19<11/9>
|equal division error: 0.443
|optimization error: 0.434
|-
|
|
|11/10 mapping: 9\19<11/9>
|equal division error: -0.443
|optimization error: -0.450
|-
|
|
| rowspan="3" |Beta 11/9
| rowspan="3" |21ed11/9
| rowspan="3" |72.537202
| rowspan="3" |16.543235
| rowspan="3" |72.534366561494206
| rowspan="3" |16.543881981552112
|11/9 mapping: 21\21<11/9>
|equal division error: 0.000
|optimization error: 0.014
|-
|
|
|10/9 mapping: 11\21<11/9>
|equal division error: -0.428
|optimization error: -0.421
|-
|
|
|11/10 mapping: 10\21<11/9>
|equal division error: 0.428
|optimization error: 0.435
|-
|
|
| rowspan="3" |Gamma 11/9
| rowspan="3" |40ed11/9
| rowspan="3" |138.166099
| rowspan="3" |8.685199
| rowspan="3" |138.165906595462172
| rowspan="3" |8.685210625176124
|11/9 mapping: 40\40<11/9>
|equal division error: 0.000
|optimization error: 0.000
|-
|-
|
! colspan="3" | Tuning !! colspan="2" | Intervals !! colspan="2" | Mappings
|
|10/9 mapping: 21\40<11/9>
|equal division error: -0.015
|optimization error: -0.014
|-
|-
|
! Name
|
! Equal division
|11/10 mapping: 19\40<11/9>
! Steps per octave
|equal division error: 0.015
! Equave
|optimization error: 0.015
! SSC pair
! Steps (Equave, SSC pair)
! Errors (cent)
|-
|-
|
| [[Alpha 3/1]]
|
| [[3edt|3ed3/1]]
| rowspan="3" |Alpha 6/5
| 1.89278926071437
| rowspan="3" |21ed6/5
| rowspan="3" | 3/1
| rowspan="3" |79.837464
| rowspan="3" | 2/1, 3/2
| rowspan="3" |15.030537
| 3\3<3/1>, 2\3<3/1>, 1\3<3/1>
| rowspan="3" |79.840125772190183
| 0, 67.970, -67.970
| rowspan="3" |15.030036443379233
|6/5 mapping: 21\21<6/5>
|equal division error: 0.000
|optimization error: -0.011
|-
|-
|
| [[Beta 3/1]]
|
| [[5edt|5ed3/1]]
|11/10 mapping: 11\21<6/5>
| 3.15464876785729
|equal division error: 0.332
| 5\5<3/1>, 3\5<3/1>, 2\5<3/1>
|optimization error: 0.326
| 0, -58.827, 58.827
|-
|-
|
| [[Gamma 3/1]]
|
| [[8edt|8ed3/1]]
|12/11 mapping: 10\21<6/5>
| 5.04743802857166
|equal division error: -0.332
| 8\8<3/1>, 5\8<3/1>, 3\8<3/1>
|optimization error: -0.337
| 0, -11.278, 11.278
|-
|-
|
| [[Alpha 2/1]]
|
| [[5edo|5ed2/1]]
| rowspan="3" |Beta 6/5
| 5
| rowspan="3" |23ed6/5
| rowspan="3" | 2/1
| rowspan="3" |87.441032
| rowspan="3" | 3/2, 4/3
| rowspan="3" |13.723534
| 5\5<2/1>, 3\5<2/1>, 2\5<2/1>
| rowspan="3" |87.438449973495273
| 0, 18.045, -18.045
| rowspan="3" |13.723939529620542
|6/5 mapping: 23\23<6/5>
|equal division error: 0.000
|optimization error: 0.009
|-
|-
|
| [[Beta 2/1]]
|
| [[7edo|7ed2/1]]
|11/10 mapping: 12\23<6/5>
| 7
|equal division error: -0.322
| 7\7<2/1>, 4\7<2/1>, 3\7<2/1>
|optimization error: -0.317
| 0, -16.241, 16.241
|-
|-
|
| [[Gamma 2/1]]
|
| [[12edo|12ed2/1]]
|12/11 mapping: 11\23<6/5>
| 12
|equal division error: 0.322
| 12\12<2/1>, 7\12<2/1>, 5\12<2/1>
|optimization error: 0.326
| 0, -1.955, 1.955
|-
|-
|
| [[Alpha 5/3]]
|
| [[7ed5/3]]
| rowspan="3" |Gamma 6/5
| 9.49840814199707
| rowspan="3" |44ed6/5
| rowspan="3" | 5/3
| rowspan="3" |167.278497
| rowspan="3" | 4/3, 5/4
| rowspan="3" |7.173666
| 7\7<5/3>, 4\7<5/3>, 3\7<5/3>
| rowspan="3" |167.278337553931523
| 0, 7.303, -7.303
| rowspan="3" |7.173672440480304
|6/5 mapping: 44\44<6/5>
|equal division error: 0.000
|optimization error: 0.000
|-
|-
|
| [[Beta 5/3]]
|
| [[9ed5/3]]
|11/10 mapping: 23\44<6/5>
| 12.2122390397105
|equal division error: -0.010
| 9\9<5/3>, 5\9<5/3>, 4\9<5/3>
|optimization error: -0.010
| 0, -6.735, 6.735
|-
|-
|
| [[Gamma 5/3]]
|
| [[16ed5/3]]
|12/11 mapping: 21\44<6/5>
| 21.7106471817076
|equal division error: 0.010
| 16\16<5/3>, 9\16<5/3>, 7\16<5/3>
|optimization error: 0.010
| 0, -0.593, 0.593
|-
|-
|
| [[Carlos Alpha|Alpha 3/2]]
|
| [[9edf|9ed3/2]]
| rowspan="3" |Alpha 13/11
| 15.3856016221631
| rowspan="3" |23ed13/11
| rowspan="3" | 3/2
| rowspan="3" |95.432477
| rowspan="3" | 5/4, 6/5
| rowspan="3" |12.574336
| 9\9<3/2>, 5\9<3/2>, 4\9<3/2>
| rowspan="3" |95.434914550823771
| 0, 3.661, -3.661
| rowspan="3" |12.574014506618971
|13/11 mapping: 23\23<13/11>
|equal division error: 0.000
|optimization error: -0.007
|-
|-
|
| [[Carlos Beta|Beta 3/2]]
|
| [[11edf|11ed3/2]]
|12/11 mapping: 12\23<13/11>
| 18.8046242048660
|equal division error: 0.255
| 11\11<3/2>, 6\11<3/2>, 5\11<3/2>
|optimization error: 0.251
| 0, -3.429, 3.429
|-
|-
|
| [[Carlos Gamma|Gamma 3/2]]
|
| [[20edf|20ed3/2]]
|13/12 mapping: 11\23<13/11>
| 34.1902258270291
|equal division error: -0.255
| 20\20<3/2>, 11\20<3/2>, 9\20<3/2>
|optimization error: -0.259
| 0, -0.238, 0.238
|-
|-
|
| [[Alpha 7/5]]
|
| [[11ed7/5]]
| rowspan="3" |Beta 13/11
| 22.6604698881676
| rowspan="3" |25ed13/11
| rowspan="3" | 7/5
| rowspan="3" |103.730954
| rowspan="3" | 6/5, 7/6
| rowspan="3" |11.568389
| 11\11<7/5>, 6\11<7/5>, 5\11<7/5>
| rowspan="3" |103.728582924336770
| 0, 2.093, -2.093
| rowspan="3" |11.568653173208022
|13/11 mapping: 25\25<13/11>
|equal division error: 0.000
|optimization error: 0.007
|-
|-
|
| [[Beta 7/5]]
|
| [[13ed7/5]]
|12/11 mapping: 13\25<13/11>
| 26.7805553223799
|equal division error: -0.248
| 13\13<7/5>, 7\13<7/5>, 6\13<7/5>
|optimization error: -0.245
| 0, -1.981, 1.981
|-
|-
|
| [[Gamma 7/5]]
|
| [[24ed7/5]]
|13/12 mapping: 12\25<13/11>
| 49.4410252105475
|equal division error: 0.248
| 24\24<7/5>, 13\24<7/5>, 11\24<7/5>
|optimization error: 0.251
| 0, -0.114, 0.114
|-
|-
|
| [[Alpha 4/3]]
|
| [[13ed4/3]]
| rowspan="3" |Gamma 13/11
| 31.3224709154917
| rowspan="3" |48ed13/11
| rowspan="3" | 4/3
| rowspan="3" |199.163431
| rowspan="3" | 7/6, 8/7
| rowspan="3" |6.025202
| 13\13<4/3>, 7\13<4/3>, 6\13<4/3>
| rowspan="3" |199.163297261207502
| 0, 1.307, -1.307
| rowspan="3" |6.025206534044126
|13/11 mapping: 48\48<13/11>
|equal division error: 0.000
|optimization error: 0.000
|-
|-
|
| [[Beta 4/3]]
|
| [[15ed4/3]]
|12/11 mapping: 25\48<13/11>
| 36.1413125947981
|equal division error: -0.007
| 15\15<4/3>, 8\15<4/3>, 7\15<4/3>
|optimization error: -0.007
| 0, -1.247, 1.247
|-
|-
|
| [[Gamma 4/3]]
|
| [[28ed4/3]]
|13/12 mapping: 23\48<13/11>
| 67.4637835102899
|equal division error: 0.007
| 28\28<4/3>, 15\28<4/3>, 13\28<4/3>
|optimization error: 0.007
| 0, -0.061, 0.061
|-
|-
|
| [[Alpha 9/7]]
|
| [[15ed9/7]]
| rowspan="3" |Alpha 7/6
| 41.3713123417559
| rowspan="3" |25ed7/6
| rowspan="3" | 9/7
| rowspan="3" |112.413903
| rowspan="3" | 8/7, 9/8
| rowspan="3" |10.674836
| 15\15<9/7>, 8\15<9/7>, 7\15<9/7>
| rowspan="3" |112.416150402630623
| 0, 0.871, -0.871
| rowspan="3" |10.674622780642016
|7/6 mapping: 25\25<7/6>
|equal division error: 0.000
|optimization error: -0.005
|-
|-
|
| [[Beta 9/7]]
|
| [[17ed9/7]]
|13/12 mapping: 13\25<7/6>
| 46.8874873206567
|equal division error: 0.200
| 17\17<9/7>, 9\17<9/7>, 8\17<9/7>
|optimization error: 0.197
| 0, -0.835, 0.835
|-
|-
|
| [[Gamma 9/7]]
|
| [[32ed9/7]]
|14/13 mapping: 12\25<7/6>
| 88.2587996624126
|equal division error: -0.200
| 32\32<9/7>, 17\32<9/7>, 15\32<9/7>
|optimization error: -0.203
| 0, -0.036, 0.036
|-
|-
|
| [[Alpha 5/4]]
|
| [[17ed5/4]]
| rowspan="3" |Beta 7/6
| 52.8068232315916
| rowspan="3" |27ed7/6
| rowspan="3" | 5/4
| rowspan="3" |121.407015
| rowspan="3" | 9/8, 10/9
| rowspan="3" |9.884108
| 17\17<5/4>, 9\17<5/4>, 8\17<5/4>
| rowspan="3" |121.404823766036118
| 0, 0.609, -0.609
| rowspan="3" |9.884286000962910
|7/6 mapping: 27\27<7/6>
|equal division error: 0.000
|optimization error: 0.005
|-
|-
|
| [[Beta 5/4]]
|
| [[19ed5/4]]
|13/12 mapping: 14\27<7/6>
| 59.0193906706024
|equal division error: -0.195
| 19\19<5/4>, 10\19<5/4>, 9\19<5/4>
|optimization error: -0.193
| 0, -0.587, 0.587
|-
|-
|
| [[Gamma 5/4]]
|
| [[36ed5/4]]
|14/13 mapping: 13\27<7/6>
| 111.826213902194
|equal division error: 0.195
| 36\36<5/4>, 19\36<5/4>, 17\36<5/4>
|optimization error: 0.197
| 0, -0.022, 0.022
|-
|-
|
| [[Alpha 11/9]]
|
| [[19ed11/9]]
| rowspan="3" |Gamma 7/6
| 65.6288971357202
| rowspan="3" |52ed7/6
| rowspan="3" | 11/9
| rowspan="3" |233.820917
| rowspan="3" | 10/9, 11/10
| rowspan="3" |5.132133
| 19\19<11/9>, 10\19<11/9>, 9\19<11/9>
| rowspan="3" |233.820803527976982
| 0, 0.443, -0.443
| rowspan="3" |5.132135301452842
|7/6 mapping: 52\52<7/6>
|equal division error: 0.000
|optimization error: 0.000
|-
|-
|
| [[Beta 11/9]]
|
| [[21ed11/9]]
|13/12 mapping: 27\52<7/6>
| 72.5372020973750
|equal division error: -0.005
| 21\21<11/9>, 11\21<11/9>, 10\21<11/9>
|optimization error: -0.005
| 0, -0.428, 0.428
|-
|-
|
| [[Gamma 11/9]]
|
| [[40ed11/9]]
|14/13 mapping: 25\52<7/6>
| 138.166099233095
|equal division error: 0.005
| 40\40<11/9>, 21\40<11/9>, 19\40<11/9>
|optimization error: 0.005
| 0, -0.015, 0.015
|-
|-
|
| [[Alpha 6/5]]
|
| [[21ed6/5]]
| rowspan="3" |Alpha 15/13
| 79.8374643554025
| rowspan="3" |27ed15/13
| rowspan="3" | 6/5
| rowspan="3" |130.781716
| rowspan="3" | 11/10, 12/11
| rowspan="3" |9.175595
| 21\21<6/5>, 11\21<6/5>, 10\21<6/5>
| rowspan="3" |130.783801507844919
| 0, 0.332, -0.332
| rowspan="3" |9.175448229557843
|15/13 mapping: 27\27<15/13>
|equal division error: 0.000
|optimization error: -0.004
|-
|-
|
| [[Beta 6/5]]
|
| [[23ed6/5]]
|14/13 mapping: 14\27<15/13>
| 87.4410323892504
|equal division error: 0.160
| 23\23<6/5>, 12\23<6/5>, 11\23<6/5>
|optimization error: 0.158
| 0, -0.322, 0.322
|-
|-
|
| [[Gamma 6/5]]
|
| [[44ed6/5]]
|15/14 mapping: 13\27<15/13>
| 167.278496744653
|equal division error: -0.160
| 44\44<6/5>, 23\44<6/5>, 21\44<6/5>
|optimization error: -0.162
| 0, -0.010, 0.010
|-
|-
|
| [[Alpha 13/11]]
|
| [[23ed13/11]]
| rowspan="3" |Beta 15/13
| 95.4324773621886
| rowspan="3" |29ed15/13
| rowspan="3" | 13/11
| rowspan="3" |140.469250
| rowspan="3" | 12/11, 13/12
| rowspan="3" |8.542795
| 23\23<13/11>, 12\23<13/11>, 11\23<13/11>
| rowspan="3" |140.467213664559518
| 0, 0.255, -0.255
| rowspan="3" |8.542918797162452
|15/13 mapping: 29\29<15/13>
|equal division error: 0.000
|optimization error: 0.004
|-
|-
|
| [[Beta 13/11]]
|
| [[25ed13/11]]
|14/13 mapping: 15\29<15/13>
| 103.730953654553
|equal division error: -0.156
| 25\25<13/11>, 13\25<13/11>, 12\25<13/11>
|optimization error: -0.154
| 0, -0.248, 0.248
|-
|-
|
| [[Gamma 13/11]]
|
| [[48ed13/11]]
|15/14 mapping: 14\29<15/13>
| 199.163431016741
|equal division error: 0.156
| 48\48<13/11>, 25\48<13/11>, 23\48<13/11>
|optimization error: 0.158
| 0, -0.007, 0.007
|-
|-
|
| [[Alpha 7/6]]
|
| [[25ed7/6]]
| rowspan="3" |Gamma 15/13
| 112.413902640048
| rowspan="3" |56ed15/13
| rowspan="3" | 7/6
| rowspan="3" |271.250966
| rowspan="3" | 13/12, 14/13
| rowspan="3" |4.423947
| 25\25<7/6>, 13\25<7/6>, 12\25<7/6>
| rowspan="3" |271.250868008139347
| 0, 0.200, -0.200
| rowspan="3" |4.423948976871078
|15/13 mapping: 56\56<15/13>
|equal division error: 0.000
|optimization error: 0.000
|-
|-
|
| [[Beta 7/6]]
|
| [[27ed7/6]]
|14/13 mapping: 29\56<15/13>
| 121.407014851252
|equal division error: -0.004
| 27\27<7/6>, 14\27<7/6>, 13\27<7/6>
|optimization error: -0.004
| 0, -0.195, 0.195
|-
|-
|
| [[Gamma 7/6]]
|
| [[52ed7/6]]
|15/14 mapping: 27\56<15/13>
| 233.820917491300
|equal division error: 0.004
| 52\52<7/6>, 27\52<7/6>, 25\52<7/6>
|optimization error: 0.004
| 0, -0.005, 0.005
|-
|-
|
| [[Alpha 15/13]]
|
| [[27ed15/13]]
| rowspan="3" |Alpha 8/7
| 130.781715879411
| rowspan="3" |29ed8/7
| rowspan="3" | 15/13
| rowspan="3" |150.535899
| rowspan="3" | 14/13, 15/14
| rowspan="3" |7.971520
| 27\27<15/13>, 14\27<15/13>, 13\27<15/13>
| rowspan="3" |150.537844310638475
| 0, 0.160, -0.160
| rowspan="3" |7.971417456488689
|8/7 mapping: 29\29<8/7>
|equal division error: 0.000
|optimization error: -0.003
|-
|-
|
| [[Beta 15/13]]
|
| [[29ed15/13]]
|15/14 mapping: 15\29<8/7>
| 140.469250388997
|equal division error: 0.130
| 29\29<15/13>, 15\29<15/13>, 14\29<15/13>
|optimization error: 0.128
| 0, -0.156, 0.156
|-
|-
|
| [[Gamma 15/13]]
|
| [[56ed15/13]]
|16/15 mapping: 14\29<8/7>
| 271.250966268408
|equal division error: -0.130
| 56\56<15/13>, 29\56<15/13>, 27\56<15/13>
|optimization error: -0.131
| 0, -0.004, 0.004
|-
|-
|
| [[Alpha 8/7]]
|
| [[29ed8/7]]
| rowspan="3" |Beta 8/7
| 150.535899020849
| rowspan="3" |31ed8/7
| rowspan="3" | 8/7
| rowspan="3" |160.917685
| rowspan="3" | 15/14, 16/15
| rowspan="3" |7.457229
| 29\29<8/7>, 15\29<8/7>, 14\29<8/7>
| rowspan="3" |160.915782495277457
| 0, 0.130, -0.130
| rowspan="3" |7.457316997698579
|8/7 mapping: 31\31<8/7>
|equal division error: 0.000
|optimization error: 0.003
|-
|-
|
| [[Beta 8/7]]
|
| [[31ed8/7]]
|15/14 mapping: 16\31<8/7>
| 160.917685160217
|equal division error: -0.127
| 31\31<8/7>, 16\31<8/7>, 15\31<8/7>
|optimization error: -0.126
| 0, -0.127, 0.127
|-
|
|
|16/15 mapping: 15\31<8/7>
|equal division error: 0.127
|optimization error: 0.128
|-
|
|
| rowspan="3" |Gamma 8/7
| rowspan="3" |60ed8/7
| rowspan="3" |311.453584
| rowspan="3" |3.852902
| rowspan="3" |311.453498588281532
| rowspan="3" |3.852902617691610
|8/7 mapping: 60\60<8/7>
|equal division error: 0.000
|optimization error: 0.000
|-
|
|
|15/14 mapping: 31\60<8/7>
|equal division error: -0.003
|optimization error: -0.003
|-
|
|
|16/15 mapping: 29\60<8/7>
|equal division error: 0.003
|optimization error: 0.003
|-
|-
| [[Gamma 8/7]]
| [[60ed8/7]]
| 311.453584181066
| 60\60<8/7>, 31\60<8/7>, 29\60<8/7>
| 0, -0.003, 0.003
|}
|}
{{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"