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|equal division error: 11.278
|equal division error: 11.278
|optimization error: 11.969
|optimization error: 11.969
|-
| 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
|-
|10/9 mapping: 21\40<11/9>
|equal division error: -0.015
|optimization error: -0.014
|-
|11/10 mapping: 19\40<11/9>
|equal division error: 0.015
|optimization error: 0.015
|-
| rowspan="3" |Alpha 6/5
| rowspan="3" |21ed6/5
| rowspan="3" |79.837464
| rowspan="3" |15.030537
| rowspan="3" |79.840125772190183
| rowspan="3" |15.030036443379233
|6/5 mapping: 21\21<6/5>
|equal division error: 0.000
|optimization error: -0.011
|-
|11/10 mapping: 11\21<6/5>
|equal division error: 0.332
|optimization error: 0.326
|-
|12/11 mapping: 10\21<6/5>
|equal division error: -0.332
|optimization error: -0.337
|-
| rowspan="3" |Beta 6/5
| rowspan="3" |23ed6/5
| rowspan="3" |87.441032
| rowspan="3" |13.723534
| rowspan="3" |87.438449973495273
| rowspan="3" |13.723939529620542
|6/5 mapping: 23\23<6/5>
|equal division error: 0.000
|optimization error: 0.009
|-
|11/10 mapping: 12\23<6/5>
|equal division error: -0.322
|optimization error: -0.317
|-
|12/11 mapping: 11\23<6/5>
|equal division error: 0.322
|optimization error: 0.326
|-
| rowspan="3" |Gamma 6/5
| rowspan="3" |44ed6/5
| rowspan="3" |167.278497
| rowspan="3" |7.173666
| rowspan="3" |167.278337553931523
| rowspan="3" |7.173672440480304
|6/5 mapping: 44\44<6/5>
|equal division error: 0.000
|optimization error: 0.000
|-
|11/10 mapping: 23\44<6/5>
|equal division error: -0.010
|optimization error: -0.010
|-
|12/11 mapping: 21\44<6/5>
|equal division error: 0.010
|optimization error: 0.010
|-
| rowspan="3" |Alpha 13/11
| rowspan="3" |23ed13/11
| rowspan="3" |95.432477
| rowspan="3" |12.574336
| rowspan="3" |95.434914550823771
| rowspan="3" |12.574014506618971
|13/11 mapping: 23\23<13/11>
|equal division error: 0.000
|optimization error: -0.007
|-
|12/11 mapping: 12\23<13/11>
|equal division error: 0.255
|optimization error: 0.251
|-
|13/12 mapping: 11\23<13/11>
|equal division error: -0.255
|optimization error: -0.259
|-
| rowspan="3" |Beta 13/11
| rowspan="3" |25ed13/11
| rowspan="3" |103.730954
| rowspan="3" |11.568389
| rowspan="3" |103.728582924336770
| rowspan="3" |11.568653173208022
|13/11 mapping: 25\25<13/11>
|equal division error: 0.000
|optimization error: 0.007
|-
|12/11 mapping: 13\25<13/11>
|equal division error: -0.248
|optimization error: -0.245
|-
|13/12 mapping: 12\25<13/11>
|equal division error: 0.248
|optimization error: 0.251
|-
| rowspan="3" |Gamma 13/11
| rowspan="3" |48ed13/11
| rowspan="3" |199.163431
| rowspan="3" |6.025202
| rowspan="3" |199.163297261207502
| rowspan="3" |6.025206534044126
|13/11 mapping: 48\48<13/11>
|equal division error: 0.000
|optimization error: 0.000
|-
|12/11 mapping: 25\48<13/11>
|equal division error: -0.007
|optimization error: -0.007
|-
|13/12 mapping: 23\48<13/11>
|equal division error: 0.007
|optimization error: 0.007
|-
| rowspan="3" |Alpha 7/6
| rowspan="3" |25ed7/6
| rowspan="3" |112.413903
| rowspan="3" |10.674836
| rowspan="3" |112.416150402630623
| rowspan="3" |10.674622780642016
|7/6 mapping: 25\25<7/6>
|equal division error: 0.000
|optimization error: -0.005
|-
|13/12 mapping: 13\25<7/6>
|equal division error: 0.200
|optimization error: 0.197
|-
|14/13 mapping: 12\25<7/6>
|equal division error: -0.200
|optimization error: -0.203
|-
| rowspan="3" |Beta 7/6
| rowspan="3" |27ed7/6
| rowspan="3" |121.407015
| rowspan="3" |9.884108
| rowspan="3" |121.404823766036118
| rowspan="3" |9.884286000962910
|7/6 mapping: 27\27<7/6>
|equal division error: 0.000
|optimization error: 0.005
|-
|13/12 mapping: 14\27<7/6>
|equal division error: -0.195
|optimization error: -0.193
|-
|14/13 mapping: 13\27<7/6>
|equal division error: 0.195
|optimization error: 0.197
|-
| rowspan="3" |Gamma 7/6
| rowspan="3" |52ed7/6
| rowspan="3" |233.820917
| rowspan="3" |5.132133
| rowspan="3" |233.820803527976982
| rowspan="3" |5.132135301452842
|7/6 mapping: 52\52<7/6>
|equal division error: 0.000
|optimization error: 0.000
|-
|13/12 mapping: 27\52<7/6>
|equal division error: -0.005
|optimization error: -0.005
|-
|14/13 mapping: 25\52<7/6>
|equal division error: 0.005
|optimization error: 0.005
|-
| rowspan="3" |Alpha 15/13
| rowspan="3" |27ed15/13
| rowspan="3" |130.781716
| rowspan="3" |9.175595
| rowspan="3" |130.783801507844919
| rowspan="3" |9.175448229557843
|15/13 mapping: 27\27<15/13>
|equal division error: 0.000
|optimization error: -0.004
|-
|14/13 mapping: 14\27<15/13>
|equal division error: 0.160
|optimization error: 0.158
|-
|15/14 mapping: 13\27<15/13>
|equal division error: -0.160
|optimization error: -0.162
|-
| rowspan="3" |Beta 15/13
| rowspan="3" |29ed15/13
| rowspan="3" |140.469250
| rowspan="3" |8.542795
| rowspan="3" |140.467213664559518
| rowspan="3" |8.542918797162452
|15/13 mapping: 29\29<15/13>
|equal division error: 0.000
|optimization error: 0.004
|-
|14/13 mapping: 15\29<15/13>
|equal division error: -0.156
|optimization error: -0.154
|-
|15/14 mapping: 14\29<15/13>
|equal division error: 0.156
|optimization error: 0.158
|-
| rowspan="3" |Gamma 15/13
| rowspan="3" |56ed15/13
| rowspan="3" |271.250966
| rowspan="3" |4.423947
| rowspan="3" |271.250868008139347
| rowspan="3" |4.423948976871078
|15/13 mapping: 56\56<15/13>
|equal division error: 0.000
|optimization error: 0.000
|-
|14/13 mapping: 29\56<15/13>
|equal division error: -0.004
|optimization error: -0.004
|-
|15/14 mapping: 27\56<15/13>
|equal division error: 0.004
|optimization error: 0.004
|-
| rowspan="3" |Alpha 8/7
| rowspan="3" |29ed8/7
| rowspan="3" |150.535899
| rowspan="3" |7.971520
| rowspan="3" |150.537844310638475
| rowspan="3" |7.971417456488689
|8/7 mapping: 29\29<8/7>
|equal division error: 0.000
|optimization error: -0.003
|-
|15/14 mapping: 15\29<8/7>
|equal division error: 0.130
|optimization error: 0.128
|-
|16/15 mapping: 14\29<8/7>
|equal division error: -0.130
|optimization error: -0.131
|-
| rowspan="3" |Beta 8/7
| rowspan="3" |31ed8/7
| rowspan="3" |160.917685
| rowspan="3" |7.457229
| rowspan="3" |160.915782495277457
| rowspan="3" |7.457316997698579
|8/7 mapping: 31\31<8/7>
|equal division error: 0.000
|optimization error: 0.003
|-
|15/14 mapping: 16\31<8/7>
|equal division error: -0.127
|optimization error: -0.126
|-
|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
|-
|}
|}
{{todo|Temperaments|inline=1|comment=Compute the temperaments associated to each Alpha-Beta-Gamma scales.}}
{{todo|Temperaments|inline=1|comment=Compute the temperaments associated to each Alpha-Beta-Gamma scales.}}
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