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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|Tables|inline=1|comment=Explain the tables.}}
 
{{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.}}
 
{{todo|Add data|inline=1|comment=Include columns for the following optimizations: Pure Equave, Dave Benson, TE, POTE, and CTE. To prevent the table from becoming overcrowded, choose between using cents or EDO. Cents are preferable since these are EDONOI. Retain the EDO column for pure equaves to provide a sense of scale, but place it outside the optimization subgroup column.}}


{| class="wikitable"
{| class="wikitable"
|+
|+
|-
|-
! colspan="3" | Tuning !! colspan="2" | Intervals !! colspan="2" | Mapping !! colspan="2" | Various optimizations (cents)
! colspan="3" | Tuning !! colspan="2" | Intervals !! colspan="2" | Mappings
|-
|-
! Name
! Name
Line 25: Line 15:
! Equave
! Equave
! SSC pair
! SSC pair
! Mappings (Equave, SSC pair)
! Steps (Equave, SSC pair)
! Errors (cent)
! Errors (cent)
! Pure Equave
! Dave Benson
|-
|-
| [[Alpha 3/1]]
| [[Alpha 3/1]]
Line 37: Line 25:
| 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, 67.970, -67.970
| 0, 67.970, -67.970
| 633.985000288462
| 629.130000247254
|-
|-
| [[Beta 3/1]]
| [[Beta 3/1]]
Line 45: Line 31:
| 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, -58.827, 58.827
| 0, -58.827, 58.827
| 380.391000173077
| 381.939079106782
|-
|-
| [[Gamma 3/1]]
| [[Gamma 3/1]]
Line 53: Line 37:
| 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, -11.278, 11.278
| 0, -11.278, 11.278
| 237.744375108173
| 237.974540913462
|-
|-
| [[Alpha 2/1]]
| [[Alpha 2/1]]
| [[5edo|5ed2/1]]
| [[5edo|5ed2/1]]
| 5.
| 5
| rowspan="3" | 2/1
| rowspan="3" | 2/1
| rowspan="3" | 3/2, 4/3
| rowspan="3" | 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, 18.045, -18.045
| 0, 18.045, -18.045
| 240.
| 239.525131601721
|-
|-
| [[Beta 2/1]]
| [[Beta 2/1]]
| [[7edo|7ed2/1]]
| [[7edo|7ed2/1]]
| 7.
| 7
| 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, -16.241, 16.241
| 0, -16.241, 16.241
| 171.428571428571
| 171.648040552235
|-
|-
| [[Gamma 2/1]]
| [[Gamma 2/1]]
| [[12edo|12ed2/1]]
| [[12edo|12ed2/1]]
| 12.
| 12
| 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, -1.955, 1.955
| 0, -1.955, 1.955
| 100.
| 100.017935787756
|-
|-
| [[Alpha 5/3]]
| [[Alpha 5/3]]
Line 89: Line 65:
| 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, 7.303, -7.303
| 0, 7.303, -7.303
| 126.336958999921
| 126.238272015258
|-
|-
| [[Beta 5/3]]
| [[Beta 5/3]]
Line 97: Line 71:
| 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, -6.735, 6.735
| 0, -6.735, 6.735
| 98.2620792221608
| 98.3172808862904
|-
|-
| [[Gamma 5/3]]
| [[Gamma 5/3]]
Line 105: Line 77:
| 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, -0.593, 0.593
| 0, -0.593, 0.593
| 55.2724195624655
| 55.2754932571412
|-
|-
| [[Carlos Alpha|Alpha 3/2]]
| [[Carlos Alpha|Alpha 3/2]]
Line 115: Line 85:
| 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, 3.661, -3.661
| 0, 3.661, -3.661
| 77.9950000961542
| 77.9649895501219
|-
|-
| [[Carlos Beta|Beta 3/2]]
| [[Carlos Beta|Beta 3/2]]
Line 123: Line 91:
| 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, -3.429, 3.429
| 0, -3.429, 3.429
| 63.8140909877625
| 63.8329325698408
|-
|-
| [[Carlos Gamma|Gamma 3/2]]
| [[Carlos Gamma|Gamma 3/2]]
Line 131: Line 97:
| 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, -0.238, 0.238
| 0, -0.238, 0.238
| 35.0977500432694
| 35.0985422804417
|-
|-
| [[Alpha 7/5]]
| [[Alpha 7/5]]
Line 141: Line 105:
| 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, 2.093, -2.093
| 0, 2.093, -2.093
| 52.9556538731173
| 52.9441558718088
|-
|-
| [[Beta 7/5]]
| [[Beta 7/5]]
Line 149: Line 111:
| 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, -1.981, 1.981
| 0, -1.981, 1.981
| 44.8086302003300
| 44.8164289231577
|-
|-
| [[Gamma 7/5]]
| [[Gamma 7/5]]
Line 157: Line 117:
| 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, -0.114, 0.114
| 0, -0.114, 0.114
| 24.2713413585121
| 24.2716042900130
|-
|-
| [[Alpha 4/3]]
| [[Alpha 4/3]]
Line 167: Line 125:
| 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, 1.307, -1.307
| 0, 1.307, -1.307
| 38.3111537795856
| 38.3060074376432
|-
|-
| [[Beta 4/3]]
| [[Beta 4/3]]
Line 175: Line 131:
| 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, -1.247, 1.247
| 0, -1.247, 1.247
| 33.2029999423075
| 33.2066890135066
|-
|-
| [[Gamma 4/3]]
| [[Gamma 4/3]]
Line 183: Line 137:
| 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, -0.061, 0.061
| 0, -0.061, 0.061
| 17.7873213976647
| 17.7874251067289
|-
|-
| [[Alpha 9/7]]
| [[Alpha 9/7]]
Line 193: Line 145:
| 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, 0.871, -0.871
| 0, 0.871, -0.871
| 29.0056063507767
| 29.0030301458200
|-
|-
| [[Beta 9/7]]
| [[Beta 9/7]]
Line 201: Line 151:
| 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, -0.835, 0.835
| 0, -0.835, 0.835
| 25.5931820742147
| 25.5951070854196
|-
|-
| [[Gamma 9/7]]
| [[Gamma 9/7]]
Line 209: Line 157:
| 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, -0.036, 0.036
| 0, -0.036, 0.036
| 13.5963779769266
| 13.5964243591413
|-
|-
| [[Alpha 5/4]]
| [[Alpha 5/4]]
Line 219: Line 165:
| 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, 0.609, -0.609
| 0, 0.609, -0.609
| 22.7243361096962
| 22.7229328303033
|-
|-
| [[Beta 5/4]]
| [[Beta 5/4]]
Line 227: Line 171:
| 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, -0.587, 0.587
| 0, -0.587, 0.587
| 20.3323007297281
| 20.3333837452881
|-
|-
| [[Gamma 5/4]]
| [[Gamma 5/4]]
Line 235: Line 177:
| 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, -0.022, 0.022
| 0, -0.022, 0.022
| 10.7309364962454
| 10.7309593208108
|-
|-
| [[Alpha 11/9]]
| [[Alpha 11/9]]
Line 245: Line 185:
| 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, 0.443, -0.443
| 0, 0.443, -0.443
| 18.2846284544201
| 18.2838119001578
|-
|-
| [[Beta 11/9]]
| [[Beta 11/9]]
Line 253: Line 191:
| 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, -0.428, 0.428
| 0, -0.428, 0.428
| 16.5432352682849
| 16.5438819815521
|-
|-
| [[Gamma 11/9]]
| [[Gamma 11/9]]
Line 261: Line 197:
| 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, -0.015, 0.015
| 0, -0.015, 0.015
| 8.68519851584955
| 8.68521062517612
|-
|-
| [[Alpha 6/5]]
| [[Alpha 6/5]]
Line 271: Line 205:
| 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, 0.332, -0.332
| 0, 0.332, -0.332
| 15.0305374762168
| 15.0300364433792
|-
|-
| [[Beta 6/5]]
| [[Beta 6/5]]
Line 279: Line 211:
| 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, -0.322, 0.322
| 0, -0.322, 0.322
| 13.7235342174153
| 13.7239395296205
|-
|-
| [[Gamma 6/5]]
| [[Gamma 6/5]]
Line 287: Line 217:
| 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, -0.010, 0.010
| 0, -0.010, 0.010
| 7.17366561364892
| 7.17367244048030
|-
|-
| [[Alpha 13/11]]
| [[Alpha 13/11]]
Line 297: Line 225:
| 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, 0.255, -0.255
| 0, 0.255, -0.255
| 12.5743356262850
| 12.5740145066190
|-
|-
| [[Beta 13/11]]
| [[Beta 13/11]]
Line 305: Line 231:
| 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, -0.248, 0.248
| 0, -0.248, 0.248
| 11.5683887761822
| 11.5686531732080
|-
|-
| [[Gamma 13/11]]
| [[Gamma 13/11]]
Line 313: Line 237:
| 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, -0.007, 0.007
| 0, -0.007, 0.007
| 6.02520248759487
| 6.02520653404413
|-
|-
| [[Alpha 7/6]]
| [[Alpha 7/6]]
Line 323: Line 245:
| 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, 0.200, -0.200
| 0, 0.200, -0.200
| 10.6748362241495
| 10.6746227806420
|-
|-
| [[Beta 7/6]]
| [[Beta 7/6]]
Line 331: Line 251:
| 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, -0.195, 0.195
| 0, -0.195, 0.195
| 9.88410761495324
| 9.88428600096291
|-
|-
| [[Gamma 7/6]]
| [[Gamma 7/6]]
Line 339: Line 257:
| 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, -0.005, 0.005
| 0, -0.005, 0.005
| 5.13213280007188
| 5.13213530145284
|-
|-
| [[Alpha 15/13]]
| [[Alpha 15/13]]
Line 349: Line 265:
| 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, 0.160, -0.160
| 0, 0.160, -0.160
| 9.17559455410784
| 9.17544822955784
|-
|-
| [[Beta 15/13]]
| [[Beta 15/13]]
Line 357: Line 271:
| 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, -0.156, 0.156
| 0, -0.156, 0.156
| 8.54279492968661
| 8.54291879716245
|-
|-
| [[Gamma 15/13]]
| [[Gamma 15/13]]
Line 365: Line 277:
| 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, -0.004, 0.004
| 0, -0.004, 0.004
| 4.42394737430199
| 4.42394897687108
|-
|-
| [[Alpha 8/7]]
| [[Alpha 8/7]]
Line 375: Line 285:
| 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, 0.130, -0.130
| 0, 0.130, -0.130
| 7.97152046658190
| 7.97141745648869
|-
|-
| [[Beta 8/7]]
| [[Beta 8/7]]
Line 383: Line 291:
| 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, -0.127, 0.127
| 0, -0.127, 0.127
| 7.45722882357662
| 7.45731699769858
|-
|-
| [[Gamma 8/7]]
| [[Gamma 8/7]]
Line 391: Line 297:
| 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, -0.003, 0.003
| 0, -0.003, 0.003
| 3.85290155884792
| 3.85290261769161
|}
|}
{{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"