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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. --
{| 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" | 2/1, 3/2
| 3\3<3/1>, 2\3<3/1>, 1\3<3/1>
| 0, 67.970, -67.970
|-
| [[Beta 3/1]]
| [[5edt|5ed3/1]]
| 3.15464876785729
| 5\5<3/1>, 3\5<3/1>, 2\5<3/1>
| 0, -58.827, 58.827
|-
| [[Gamma 3/1]]
| [[8edt|8ed3/1]]
| 5.04743802857166
| 8\8<3/1>, 5\8<3/1>, 3\8<3/1>
| 0, -11.278, 11.278
|-
| [[Alpha 2/1]]
| [[5edo|5ed2/1]]
| 5
| rowspan="3" | 2/1
| rowspan="3" | 3/2, 4/3
| 5\5<2/1>, 3\5<2/1>, 2\5<2/1>
| 0, 18.045, -18.045
|-
| [[Beta 2/1]]
| [[7edo|7ed2/1]]
| 7
| 7\7<2/1>, 4\7<2/1>, 3\7<2/1>
| 0, -16.241, 16.241
|-
| [[Gamma 2/1]]
| [[12edo|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
| rowspan="3" | 5/3
| rowspan="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
|-
| [[Carlos Alpha|Alpha 3/2]]
| [[9edf|9ed3/2]]
| 15.3856016221631
| rowspan="3" | 3/2
| rowspan="3" | 5/4, 6/5
| 9\9<3/2>, 5\9<3/2>, 4\9<3/2>
| 0, 3.661, -3.661
|-
| [[Carlos Beta|Beta 3/2]]
| [[11edf|11ed3/2]]
| 18.8046242048660
| 11\11<3/2>, 6\11<3/2>, 5\11<3/2>
| 0, -3.429, 3.429
|-
| [[Carlos Gamma|Gamma 3/2]]
| [[20edf|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
| rowspan="3" | 7/5
| rowspan="3" | 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
| rowspan="3" | 4/3
| rowspan="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
| rowspan="3" | 9/7
| rowspan="3" | 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
| rowspan="3" | 5/4
| rowspan="3" | 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
| rowspan="3" | 11/9
| rowspan="3" | 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
| rowspan="3" | 6/5
| rowspan="3" | 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
| rowspan="3" | 13/11
| rowspan="3" | 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
| rowspan="3" | 7/6
| rowspan="3" | 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
| rowspan="3" | 15/13
| rowspan="3" | 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
| rowspan="3" | 8/7
| rowspan="3" | 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
|}


{{todo|Table|inline=1|comment=Explain the table. 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.}}
 
== 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"