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Find a version of the symmetric least-squares error that forms a metric on the set of delta signatures with n terms.
Find a version of the symmetric least-squares error that forms a metric on the set of delta signatures with n terms.


== DR and RTT ==
== DR triads in small edos ==
One may be able to tune a rank-2 regular temperament in such a way that a triad of interest exactly "inherits" its delta signature from a simple JI preimage thereof. This is done by setting up an algebraic equation relating the intervals in the chord to a generator and then solving for the generator that produces proportionally-beating triads. The value to be solved for is the generator's frequency ratio (not its cent value).
Shown below are approximate DR triads in edos < 24 with symmetric least-squared error < 0.05. All delta signatures are in the 3-integer limit.
 
{| class="wikitable sortable"
If we want to optimize a 4:5:6 triad in Meantone, for instance, we want a +1+1 delta signature, or equivalently a 1:1 ratio of frequency deltas between the major third and minor third. Fixing any frequency as the triad's root and letting <math>g</math> be the frequency ratio for the perfect fifth generator for meantone, the minor third in the tempered 4:5:6 triad has a delta of <math>g-g^4/4</math>, and the major third in the same triad has a delta of <math>g^4/4-1</math>. Therefore to ensure that the two deltas form a 1:1 ratio, we must find the appropriate root of the polynomial <math>g^4-2g-2</math> (the difference between the two, simplified to make all coefficients integers). This results in a generator of 1.4945, or about 695.6 cents.
!edo
 
!steps
Note that the equation to solve depends on what chord you want to tune as equal-beating. For example, assuming pure octaves, Meantone admits an equation for tuning the 3:4:5 as equal-beating: <math>g^4+2g-8=0.</math> The latter equation has solution g = 1.4960 = 697.3c. <!--Essentially tempered [[Dyadic chord|dyadic]] triads are also more difficult to tune with simple delta-signatures, since they lack simple JI preimages.-->
!delta signature
 
!naive LS error
Below is a list of temperaments and their various optimizations for proportionally beating chords. They are ordered by highest power in the relevant DR polynomial, with ties broken by leading coefficients, then 2nd term coefficients, 3rd term coefficients, 4th term coefficients, etc. In the case of negative coefficients, only the absolute value is considered.
!symmetric LS error
 
{| class="wikitable"
|-
|-
! colspan="11" | Coefficients of terms
|6
!rowspan=2| Chord
|0, 2, 5
!rowspan=2| Delta signature
| +1+2
!rowspan=2| Temperament
|0.0025
!rowspan=2| Generator (cents)
|0.0026
!rowspan=2| Edos
|-
|-
! g^10
|6
! g^9
|0, 4, 5
! g^8
| +3+1
! g^7
|0.0043
! g^6
|0.0044
! g^5
! g^4
! g^3
! g^2
! g^1
! g^0
|-
|-
|  
|7
|  
|0, 1, 2
|
|
|
|
|
|
| 1
| -1
| -1
| 4:5:6
| +1+1
| +1+1
| {{monzo|1 -2 1}}
|0.0447
| 833.09 (phi)
|0.0449
| 36
|-
|-
|  
|7
|  
|0, 2, 5
|  
| +1+2
|  
|0.0242
|  
|0.0247
|  
|-
|  
|7
|  
|0, 4, 5
| 2
| +3+1
| -1
|0.0273
| -2
|0.0277
| 4:5:6
|-
|7
|0, 4, 6
| +3+2
|0.0046
|0.0047
|-
|8
|0, 1, 2
| +1+1
| +1+1
| Father
|0.0391
| 428.42
|0.0392
| 14
|-
|-
|  
|8
|  
|0, 4, 7
|  
| +1+1
|  
|0.0060
|  
|0.0065
|
|-
|
|8
| 3
|0, 2, 5
| -2
|  
| -2
| 6:7:9
| +1+2
| +1+2
| Beep
|0.0442
| 258.65
|0.0450
| 33, 42, 51
|-
|8
|0, 5, 7
| +2+1
|0.0416
|0.0432
|-
|-
|  
|8
|  
|0, 4, 5
|  
| +3+1
|  
|0.0443
|  
|0.0447
|  
|-
| 1
|8
|  
|0, 4, 6
|  
| +3+2
| -1
|0.0322
| -2
|0.0325
| 4:5:6
|-
|9
|0, 1, 2
| +1+1
| +1+1
| Mavila
|0.0347
| 523.66
|0.0348
| 23, 39
|-
|-
|  
|9
|  
|0, 4, 7
|
|
|
|
| 1
|
|
| -2
| -2
| 4:5:6
| +1+1
| +1+1
| Meantone
|0.0089
| 695.63
|0.0094
| 19, 31, 50
|-
|-
|  
|9
|  
|0, 5, 7
|  
| +2+1
|  
|0.0231
|  
|0.0238
| 1
|-
|
|10
|  
|0, 1, 2
| 2
|
| -4
| 4:5:6
| +1+1
| +1+1
| Porcupine
|0.0312
| 160.89
|0.0313
| 15
|-
|-
|  
|10
|  
|0, 4, 7
|
| +1+1
|
|0.0207
|
|0.0217
| 1
| -4
|
|
|
| 12
| 5:6:9
| +1+3
| Mavila
| 674.90
| 16, 25
|-
|-
|  
|10
|  
|0, 5, 9
|
|
| 1
|
|
|
|
| -4
| -4
| 4:5:6
| +1+1
| +1+1
| Avila
|0.0392
| 660.23
|0.0427
| 20
|-
|-
|  
|10
|  
|0, 2, 7
|  
| +1+3
|  
|0.0463
| 1
|0.0468
| -2
|-
|  
|10
|  
|0, 5, 7
|  
| +2+1
|  
|0.0085
| 2
|0.0087
| 4:5:6
|-
|10
|0, 6, 9
| +3+2
|0.0195
|0.0197
|-
|11
|0, 1, 2
| +1+1
| +1+1
| Hanson
|0.0283
| 317.96
|0.0284
| 19
|-
|-
|  
|11
|  
|0, 4, 7
|
| 1
|
|
|
|
|
| -2
| -4
| 4:5:6
| +1+1
| +1+1
| Uncle
|0.0303
| 467.46
|0.0315
| 18
|-
|-
|  
|11
|  
|0, 5, 9
| 3
| +1+1
|
|0.0265
|
|0.0283
| -4
|
|
|
|
| -16
| 4:5:7
| +1+2
| Mabila
| 527.66
| 25
|-
|-
|  
|11
| 1
|0, 6, 10
|
|
|
|
| -1
|
|
|
| -1
| 4:5:6
| +1+1
| +1+1
| Tetracot
|0.0415
| 176.54
|0.0454
| 34
|-
|11
|0, 3, 8
| +1+2
|0.0454
|0.0465
|-
|-
|  
|11
| 1
|0, 4, 10
|  
| +1+2
| -1
|0.0193
|  
|0.0202
|  
|-
|  
|11
|  
|0, 2, 7
|  
| +1+3
|  
|0.0301
| -4
|0.0304
| 4:5:6
|-
| +1+1
|11
| Sensi
|0, 5, 7
| 442.74
| +2+1
| 19, 65
|0.0034
|0.0034
|-
|-
| 1
|11
|  
|0, 8, 10
|  
| +3+1
|  
|0.0106
|  
|0.0108
|  
|-
|  
|11
| 2
|0, 6, 9
|  
| +3+2
|  
|0.0088
| -8
|0.0089
| 4:5:6
|-
|12
|0, 1, 2
| +1+1
| +1+1
| Orson
|0.0260
| 271.51
|0.0260
| 22, 31, 53
|}
=== Limitations ===
Interpreting simple JI chords as signatures/templates for delta-rational chords is reasonable, as the psychoacoustic effect of DR is more robust to detuning than that of JI for many people, but it is no panacea. If the temperament is too inaccurate, other inversions and voicings of a given JI chord will not have the DR signatures preserved acceptably, and compromising will make all of them inaccurate, though this of course depends on what error tolerance you prefer. This is because DR simplicity is only preserved by inversion and revoicing if the chord in question is low-complexity JI.
 
For example, take 0.0570 (the LSD error of 0-2\11-4\11 as +1+1, approximately equalized 7:8:9) as a somewhat arbitrary but reasonable upper limit of acceptable error. Consider Semaphore temperament (i.e. 2.3.7[14 & 19]). Using a gen of 260.346c, 0-679c-940c is a Semaphore tuning of 4:6:7 that is perfectly +2+1, but inverting the chord yields 0-260c-521c as our 6:7:8, with LSD error 0.0681; the 7:8:12 has an even higher error of 0.123. This is also evident by the fact that we had to use an extreme tuning of Semaphore, which has CWE generator 249.311c and CTE generator 248.126c. If we use the CTE generator, all three inversions are barely acceptable to unacceptable as simple-DR approximations: 4:6:7 has LSD error 0.0574 from +2+1, 6:7:8 has LSD error 0.0652 from +1+1, and 7:8:12 has LSD error 0.0570 from +1+4.
 
== DR triads in small edos ==
Shown below are approximate DR triads in edos < 24 with symmetric least-squared error < 0.05. All delta signatures are in the 3-integer limit.
{| class="wikitable sortable"
!edo
!steps
!delta signature
!naive LS error
!symmetric LS error
|-
|-
|6
|12
|0, 2, 5
|0, 4, 7
| +1+2
| +1+1
|0.0025
|0.0383
|0.0026
|0.0395
|-
|-
|6
|12
|0, 4, 5
|0, 5, 9
| +3+1
|0.0043
|0.0044
|-
|7
|0, 1, 2
| +1+1
| +1+1
|0.0447
|0.0159
|0.0449
|0.0168
|-
|-
|7
|12
|0, 2, 5
|0, 3, 8
| +1+2
| +1+2
|0.0242
|0.0321
|0.0247
|0.0327
|-
|-
|7
|12
|0, 4, 5
|0, 4, 10
| +3+1
| +1+2
|0.0273
|0.0025
|0.0277
|0.0026
|-
|-
|7
|12
|0, 4, 6
|0, 2, 7
| +3+2
| +1+3
|0.0046
|0.0165
|0.0047
|0.0166
|-
|-
|8
|12
|0, 1, 2
|0, 3, 10
| +1+1
| +1+3
|0.0391
|0.0310
|0.0392
|0.0316
|-
|-
|8
|12
|0, 4, 7
|0, 2, 3
| +1+1
| +2+1
|0.0060
|0.0489
|0.0065
|0.0490
|-
|-
|8
|12
|0, 2, 5
|0, 5, 7
| +1+2
| +2+1
|0.0442
|0.0132
|0.0450
|0.0133
|-
|-
|8
|12
|0, 5, 7
|0, 8, 11
| +2+1
| +2+1
|0.0416
|0.0125
|0.0432
|0.0130
|-
|-
|8
|12
|0, 4, 5
|0, 8, 10
| +3+1
| +3+1
|0.0443
|0.0043
|0.0447
|0.0044
|-
|-
|8
|12
|0, 4, 6
|0, 6, 9
| +3+2
| +3+2
|0.0322
|0.0322
|0.0325
|0.0325
|-
|-
|9
|13
|0, 1, 2
|0, 1, 2
| +1+1
| +1+1
|0.0347
|0.0240
|0.0348
|0.0240
|-
|-
|9
|13
|0, 4, 7
|0, 2, 4
| +1+1
| +1+1
|0.0089
|0.0481
|0.0094
|0.0485
|-
|-
|9
|13
|0, 5, 7
|0, 4, 7
| +2+1
|0.0231
|0.0238
|-
|10
|0, 1, 2
| +1+1
| +1+1
|0.0312
|0.0451
|0.0313
|0.0462
|-
|-
|10
|13
|0, 4, 7
|0, 5, 9
| +1+1
| +1+1
|0.0207
|0.0070
|0.0217
|0.0074
|-
|-
|10
|13
|0, 5, 9
|0, 7, 12
| +1+1
| +1+1
|0.0392
|0.0087
|0.0427
|0.0095
|-
|-
|10
|13
|0, 2, 7
|0, 3, 8
| +1+3
| +1+2
|0.0463
|0.0207
|0.0468
|0.0210
|-
|13
|0, 4, 10
| +1+2
|0.0119
|0.0122
|-
|-
|10
|13
|0, 5, 7
|0, 5, 12
| +2+1
| +1+2
|0.0085
|0.0215
|0.0087
|0.0225
|-
|-
|10
|13
|0, 6, 9
|0, 2, 7
| +3+2
| +1+3
|0.0195
|0.0049
|0.0197
|0.0050
|-
|-
|11
|13
|0, 1, 2
|0, 3, 10
| +1+1
| +1+3
|0.0283
|0.0145
|0.0284
|0.0147
|-
|-
|11
|13
|0, 4, 7
|0, 2, 3
| +1+1
| +2+1
|0.0303
|0.0451
|0.0315
|0.0452
|-
|-
|11
|13
|0, 5, 9
|0, 5, 7
| +1+1
| +2+1
|0.0265
|0.0214
|0.0283
|0.0217
|-
|-
|11
|13
|0, 6, 10
|0, 8, 11
| +1+1
| +2+1
|0.0415
|0.0005
|0.0454
|0.0005
|-
|-
|11
|13
|0, 3, 8
| +1+2
|0.0454
|0.0465
|-
|11
|0, 4, 10
| +1+2
|0.0193
|0.0202
|-
|11
|0, 2, 7
| +1+3
|0.0301
|0.0304
|-
|11
|0, 5, 7
| +2+1
|0.0034
|0.0034
|-
|11
|0, 8, 10
|0, 8, 10
| +3+1
| +3+1
|0.0106
|0.0167
|0.0108
|0.0170
|-
|-
|11
|13
|0, 6, 9
|0, 8, 12
| +3+2
| +3+2
|0.0088
|0.0275
|0.0089
|0.0279
|-
|-
|12
|14
|0, 1, 2
|0, 1, 2
| +1+1
| +1+1
|0.0260
|0.0222
|0.0260
|0.0223
|-
|-
|12
|14
|0, 4, 7
|0, 2, 4
| +1+1
| +1+1
|0.0383
|0.0447
|0.0395
|0.0449
|-
|-
|12
|14
|0, 5, 9
|0, 5, 9
| +1+1
| +1+1
|0.0159
|0.0006
|0.0168
|0.0006
|-
|-
|12
|14
|0, 3, 8
|0, 6, 11
| +1+2
| +1+1
|0.0321
|0.0401
|0.0327
|0.0426
|-
|-
|12
|14
|0, 7, 12
| +1+1
|0.0186
|0.0201
|-
|14
|0, 1, 3
| +1+2
|0.0468
|0.0469
|-
|14
|0, 3, 8
| +1+2
|0.0110
|0.0111
|-
|14
|0, 4, 10
|0, 4, 10
| +1+2
| +1+2
|0.0025
|0.0242
|0.0026
|0.0247
|-
|14
|0, 5, 12
| +1+2
|0.0363
|0.0377
|-
|-
|12
|14
|0, 2, 7
|0, 2, 7
| +1+3
| +1+3
|0.0165
|0.0050
|0.0166
|0.0050
|-
|-
|12
|14
|0, 3, 10
|0, 3, 10
| +1+3
| +1+3
|0.0310
|0.0002
|0.0316
|0.0002
|-
|14
|0, 4, 13
| +1+3
|0.0294
|0.0301
|-
|-
|12
|14
|0, 2, 3
|0, 2, 3
| +2+1
| +2+1
|0.0489
|0.0418
|0.0490
|0.0419
|-
|-
|12
|14
|0, 5, 7
|0, 5, 7
| +2+1
| +2+1
|0.0132
|0.0284
|0.0133
|0.0287
|-
|-
|12
|14
|0, 8, 11
|0, 8, 11
| +2+1
| +2+1
|0.0125
|0.0116
|0.0130
|0.0119
|-
|-
|12
|14
|0, 8, 10
|0, 4, 9
| +3+1
| +2+3
|0.0043
|0.0461
|0.0044
|0.0464
|-
|14
|0, 7, 9
| +3+1
|0.0393
|0.0397
|-
|14
|0, 8, 10
| +3+1
|0.0273
|0.0277
|-
|-
|12
|14
|0, 6, 9
|0, 8, 12
| +3+2
| +3+2
|0.0322
|0.0046
|0.0325
|0.0047
|-
|-
|13
|15
|0, 1, 2
|0, 1, 2
| +1+1
| +1+1
|0.0240
|0.0208
|0.0240
|0.0208
|-
|-
|13
|15
|0, 2, 4
|0, 2, 4
| +1+1
| +1+1
|0.0481
|0.0417
|0.0485
|0.0419
|-
|-
|13
|15
|0, 4, 7
|0, 5, 9
| +1+1
| +1+1
|0.0451
|0.0071
|0.0462
|0.0074
|-
|-
|13
|15
|0, 5, 9
|0, 6, 11
| +1+1
| +1+1
|0.0070
|0.0319
|0.0074
|0.0336
|-
|-
|13
|15
|0, 7, 12
|0, 7, 12
| +1+1
| +1+1
|0.0087
|0.0273
|0.0095
|0.0291
|-
|15
|0, 8, 14
| +1+1
|0.0149
|0.0164
|-
|-
|13
|15
|0, 3, 8
|0, 1, 3
| +1+2
| +1+2
|0.0207
|0.0437
|0.0210
|0.0437
|-
|-
|13
|15
|0, 4, 10
|0, 3, 8
| +1+2
| +1+2
|0.0119
|0.0025
|0.0122
|0.0025
|-
|-
|13
|15
|0, 5, 12
|0, 4, 10
| +1+2
| +1+2
|0.0215
|0.0349
|0.0225
|0.0355
|-
|-
|13
|15
|0, 2, 7
|0, 2, 7
| +1+3
| +1+3
|0.0049
|0.0137
|0.0050
|0.0137
|-
|-
|13
|15
|0, 3, 10
|0, 3, 10
| +1+3
| +1+3
|0.0145
|0.0122
|0.0147
|0.0123
|-
|15
|0, 4, 13
| +1+3
|0.0134
|0.0137
|-
|-
|13
|15
|0, 2, 3
|0, 2, 3
| +2+1
| +2+1
|0.0451
|0.0390
|0.0452
|0.0391
|-
|-
|13
|15
|0, 5, 7
|0, 5, 7
| +2+1
| +2+1
|0.0214
|0.0345
|0.0217
|0.0348
|-
|15
|0, 7, 10
| +2+1
|0.0415
|0.0423
|-
|-
|13
|15
|0, 8, 11
|0, 8, 11
| +2+1
| +2+1
|0.0005
|0.0211
|0.0005
|0.0216
|-
|-
|13
|15
|0, 8, 10
|0, 3, 7
| +3+1
| +2+3
|0.0167
|0.0496
|0.0170
|0.0498
|-
|-
|13
|15
|0, 8, 12
|0, 4, 9
| +3+2
| +2+3
|0.0275
|0.0294
|0.0279
|0.0296
|-
|15
|0, 5, 11
| +2+3
|0.0356
|0.0359
|-
|15
|0, 7, 9
| +3+1
|0.0304
|0.0306
|-
|15
|0, 8, 10
| +3+1
|0.0364
|0.0368
|-
|15
|0, 8, 12
| +3+2
|0.0151
|0.0152
|-
|-
|14
|16
|0, 1, 2
|0, 1, 2
| +1+1
| +1+1
|0.0222
|0.0195
|0.0223
|0.0195
|-
|-
|14
|16
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| +1+1
|0.0447
|0.0391
|0.0449
|0.0392
|-
|-
|14
|16
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| +1+1
|0.0006
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|0.0006
|0.0132
|-
|-
|14
|16
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| +1+1
|0.0401
|0.0248
|0.0426
|0.0259
|-
|-
|14
|16
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| +1+1
|0.0186
|0.0348
|0.0201
|0.0368
|-
|16
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|0.0060
|0.0065
|-
|16
|0, 9, 15
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|0.0410
|-
|-
|14
|16
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| +1+2
|0.0468
|0.0410
|0.0469
|0.0410
|-
|-
|14
|16
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|0, 3, 8
| +1+2
| +1+2
|0.0110
|0.0049
|0.0111
|0.0050
|-
|-
|14
|16
|0, 4, 10
|0, 4, 10
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| +1+2
|0.0242
|0.0442
|0.0247
|0.0450
|-
|16
|0, 5, 13
| +1+2
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|0.0398
|-
|-
|14
|16
|0, 5, 12
|0, 6, 15
| +1+2
| +1+2
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|0.0257
|0.0377
|0.0268
|-
|-
|14
|16
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| +1+3
|0.0050
|0.0213
|0.0050
|0.0213
|-
|-
|14
|16
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|0, 3, 10
| +1+3
| +1+3
|0.0002
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|0.0002
|0.0233
|-
|-
|14
|16
|0, 4, 13
|0, 4, 13
| +1+3
| +1+3
|0.0294
|0.0007
|0.0301
|0.0008
|-
|-
|14
|16
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| +2+1
|0.0418
|0.0365
|0.0419
|0.0366
|-
|-
|14
|16
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| +2+1
|0.0284
|0.0398
|0.0287
|0.0401
|-
|16
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|0.0340
|-
|-
|14
|16
|0, 8, 11
|0, 8, 11
| +2+1
| +2+1
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|0.0294
|0.0119
|0.0300
|-
|-
|14
|16
|0, 10, 14
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|0.0416
|0.0432
|-
|16
|0, 11, 15
| +2+1
|0.0010
|0.0010
|-
|16
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|-
|16
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|0, 4, 9
| +2+3
| +2+3
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|0.0148
|0.0464
|0.0149
|-
|-
|14
|16
|0, 7, 9
|0, 5, 11
| +3+1
| +2+3
|0.0393
|0.0178
|0.0397
|0.0179
|-
|-
|14
|16
|0, 6, 13
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|0.0348
|0.0352
|-
|16
|0, 7, 9
| +3+1
|0.0227
|0.0228
|-
|16
|0, 8, 10
|0, 8, 10
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| +3+1
|0.0273
|0.0443
|0.0277
|0.0447
|-
|16
|0, 12, 15
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|0.0162
|0.0166
|-
|-
|14
|16
|0, 8, 12
|0, 8, 12
| +3+2
| +3+2
|0.0046
|0.0322
|0.0047
|0.0325
|-
|16
|0, 10, 15
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|0.0325
|0.0330
|-
|-
|15
|17
|0, 1, 2
|0, 1, 2
| +1+1
| +1+1
|0.0208
|0.0183
|0.0208
|0.0183
|-
|-
|15
|17
|0, 2, 4
|0, 2, 4
| +1+1
| +1+1
|0.0417
|0.0367
|0.0419
|0.0369
|-
|-
|15
|17
|0, 5, 9
|0, 5, 9
| +1+1
| +1+1
|0.0071
|0.0179
|0.0074
|0.0184
|-
|-
|15
|17
|0, 6, 11
|0, 6, 11
| +1+1
| +1+1
|0.0319
|0.0185
|0.0336
|0.0192
|-
|-
|15
|17
|0, 7, 12
|0, 7, 12
| +1+1
| +1+1
|0.0273
|0.0414
|0.0291
|0.0434
|-
|-
|15
|17
|0, 8, 14
|0, 8, 14
| +1+1
| +1+1
|0.0149
|0.0019
|0.0164
|0.0020
|-
|-
|15
|17
|0, 9, 15
| +1+1
|0.0454
|0.0494
|-
|17
|0, 9, 16
| +1+1
|0.0327
|0.0360
|-
|17
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|0, 1, 3
| +1+2
| +1+2
|0.0437
|0.0386
|0.0437
|0.0386
|-
|-
|15
|17
|0, 3, 8
|0, 3, 8
| +1+2
| +1+2
|0.0025
|0.0115
|0.0025
|0.0116
|-
|-
|15
|17
|0, 4, 10
|0, 5, 13
| +1+2
| +1+2
|0.0349
|0.0281
|0.0355
|0.0288
|-
|-
|15
|17
|0, 2, 7
|0, 6, 15
| +1+3
| +1+2
|0.0137
|0.0134
|0.0137
|0.0139
|-
|17
|0, 2, 7
| +1+3
|0.0280
|0.0281
|-
|-
|15
|17
|0, 3, 10
|0, 3, 10
| +1+3
| +1+3
|0.0122
|0.0328
|0.0123
|0.0331
|-
|-
|15
|17
|0, 4, 13
|0, 4, 13
| +1+3
| +1+3
|0.0134
|0.0133
|0.0137
|0.0135
|-
|17
|0, 5, 16
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|0.0164
|0.0168
|-
|-
|15
|17
|0, 2, 3
|0, 2, 3
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| +2+1
|0.0390
|0.0344
|0.0391
|0.0344
|-
|-
|15
|17
|0, 5, 7
|0, 5, 7
| +2+1
| +2+1
|0.0345
|0.0444
|0.0348
|0.0447
|-
|-
|15
|17
|0, 7, 10
|0, 7, 10
| +2+1
| +2+1
|0.0415
|0.0264
|0.0423
|0.0267
|-
|-
|15
|17
|0, 8, 11
|0, 8, 11
| +2+1
| +2+1
|0.0211
|0.0366
|0.0216
|0.0373
|-
|17
|0, 10, 14
| +2+1
|0.0318
|0.0328
|-
|17
|0, 11, 15
| +2+1
|0.0110
|0.0114
|-
|17
|0, 12, 16
| +2+1
|0.0477
|0.0499
|-
|-
|15
|17
|0, 3, 7
|0, 3, 7
| +2+3
| +2+3
|0.0496
|0.0283
|0.0498
|0.0284
|-
|-
|15
|17
|0, 4, 9
|0, 4, 9
| +2+3
| +2+3
|0.0294
|0.0020
|0.0296
|0.0020
|-
|-
|15
|17
|0, 5, 11
|0, 5, 11
| +2+3
| +2+3
|0.0356
|0.0021
|0.0359
|0.0021
|-
|17
|0, 6, 13
| +2+3
|0.0163
|0.0164
|-
|17
|0, 7, 15
| +2+3
|0.0389
|0.0394
|-
|-
|15
|17
|0, 7, 9
|0, 3, 4
| +3+1
| +3+1
|0.0304
|0.0499
|0.0306
|0.0500
|-
|-
|15
|17
|0, 8, 10
|0, 7, 9
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|0.0159
|0.0160
|-
|17
|0, 11, 14
| +3+1
|0.0474
|0.0483
|-
|17
|0, 12, 15
| +3+1
|0.0053
|0.0054
|-
|17
|0, 13, 16
| +3+1
| +3+1
|0.0364
|0.0310
|0.0368
|0.0318
|-
|-
|15
|17
|0, 8, 12
|0, 8, 12
| +3+2
| +3+2
|0.0151
|0.0472
|0.0152
|0.0476
|-
|-
|16
|17
|0, 1, 2
|0, 10, 15
| +3+2
|0.0133
|0.0135
|-
|18
|0, 1, 2
| +1+1
| +1+1
|0.0195
|0.0173
|0.0195
|0.0173
|-
|-
|16
|18
|0, 2, 4
|0, 2, 4
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| +1+1
|0.0391
|0.0347
|0.0392
|0.0348
|-
|-
|16
|18
|0, 5, 9
|0, 5, 9
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| +1+1
|0.0129
|0.0224
|0.0132
|0.0229
|-
|-
|16
|18
|0, 6, 11
|0, 6, 11
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| +1+1
|0.0248
|0.0129
|0.0259
|0.0134
|-
|-
|16
|18
|0, 7, 12
|0, 7, 12
| +1+1
| +1+1
|0.0348
|0.0472
|0.0368
|0.0493
|-
|-
|16
|18
|0, 7, 13
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|0.0430
|0.0452
|-
|18
|0, 8, 14
|0, 8, 14
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| +1+1
|0.0060
|0.0089
|0.0065
|0.0094
|-
|18
|0, 9, 16
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|0.0246
|0.0267
|-
|-
|16
|18
|0, 9, 15
|0, 10, 17
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| +1+1
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|0.0145
|0.0410
|0.0160
|-
|-
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|18
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| +1+2
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|0.0364
|0.0410
|0.0365
|-
|-
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|18
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| +1+2
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|0.0173
|0.0050
|0.0174
|-
|-
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|18
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| +1+2
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|0.0186
|0.0450
|0.0191
|-
|-
|16
|18
|0, 5, 13
|0, 6, 15
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| +1+2
|0.0386
|0.0025
|0.0398
|0.0026
|-
|-
|16
|18
|0, 6, 15
|0, 7, 17
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| +1+2
|0.0257
|0.0038
|0.0268
|0.0040
|-
|-
|16
|18
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| +1+3
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|0.0340
|0.0213
|0.0341
|-
|-
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|18
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| +1+3
| +1+3
|0.0231
|0.0414
|0.0233
|0.0417
|-
|-
|16
|18
|0, 4, 13
|0, 4, 13
| +1+3
| +1+3
|0.0007
|0.0244
|0.0008
|0.0248
|-
|18
|0, 5, 16
| +1+3
|0.0028
|0.0028
|-
|-
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|18
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| +2+1
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|0.0324
|0.0366
|0.0325
|-
|-
|16
|18
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| +2+1
|0.0398
|0.0485
|0.0401
|0.0488
|-
|-
|16
|18
|0, 7, 10
|0, 7, 10
| +2+1
| +2+1
|0.0334
|0.0201
|0.0340
|0.0204
|-
|-
|16
|18
|0, 8, 11
|0, 8, 11
| +2+1
| +2+1
|0.0294
|0.0430
|0.0300
|0.0437
|-
|-
|16
|18
|0, 10, 14
|0, 10, 14
| +2+1
| +2+1
|0.0416
|0.0231
|0.0432
|0.0238
|-
|-
|16
|18
|0, 11, 15
|0, 11, 15
| +2+1
| +2+1
|0.0010
|0.0198
|0.0010
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|-
|-
|16
|18
|0, 3, 7
|0, 3, 7
| +2+3
| +2+3
|0.0383
|0.0194
|0.0384
|0.0194
|-
|-
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|18
|0, 4, 9
|0, 4, 9
| +2+3
| +2+3
|0.0148
|0.0094
|0.0149
|0.0095
|-
|-
|16
|18
|0, 5, 11
|0, 5, 11
| +2+3
| +2+3
|0.0178
|0.0119
|0.0179
|0.0120
|-
|-
|16
|18
|0, 6, 13
|0, 6, 13
| +2+3
| +2+3
|0.0348
|0.0002
|0.0352
|0.0002
|-
|-
|16
|18
|0, 7, 15
| +2+3
|0.0198
|0.0201
|-
|18
|0, 8, 17
| +2+3
|0.0453
|0.0461
|-
|18
|0, 3, 4
| +3+1
|0.0471
|0.0472
|-
|18
|0, 7, 9
|0, 7, 9
| +3+1
| +3+1
|0.0227
|0.0099
|0.0228
|0.0099
|-
|-
|16
|18
|0, 8, 10
|0, 11, 14
| +3+1
| +3+1
|0.0443
|0.0380
|0.0447
|0.0386
|-
|-
|16
|18
|0, 12, 15
|0, 12, 15
| +3+1
| +3+1
|0.0162
|0.0043
|0.0166
|0.0044
|-
|18
|0, 13, 16
| +3+1
|0.0407
|0.0417
|-
|-
|16
|18
|0, 8, 12
|0, 5, 8
| +3+2
| +3+2
|0.0322
|0.0495
|0.0325
|0.0497
|-
|-
|16
|18
|0, 10, 15
|0, 10, 15
| +3+2
| +3+2
|0.0325
|0.0036
|0.0330
|0.0036
|-
|-
|17
|19
|0, 1, 2
|0, 1, 2
| +1+1
| +1+1
|0.0183
|0.0164
|0.0183
|0.0164
|-
|-
|17
|19
|0, 2, 4
|0, 2, 4
| +1+1
| +1+1
|0.0367
|0.0329
|0.0369
|0.0330
|-
|-
|17
|19
|0, 3, 6
| +1+1
|0.0494
|0.0498
|-
|19
|0, 5, 9
|0, 5, 9
| +1+1
| +1+1
|0.0179
|0.0264
|0.0184
|0.0269
|-
|-
|17
|19
|0, 6, 11
|0, 6, 11
| +1+1
| +1+1
|0.0185
|0.0079
|0.0192
|0.0082
|-
|-
|17
|19
|0, 7, 12
|0, 7, 13
| +1+1
| +1+1
|0.0414
|0.0371
|0.0434
|0.0387
|-
|-
|17
|19
|0, 8, 14
|0, 8, 14
| +1+1
| +1+1
|0.0019
|0.0151
|0.0020
|0.0159
|-
|-
|17
|19
|0, 9, 15
|0, 9, 16
| +1+1
| +1+1
|0.0454
|0.0174
|0.0494
|0.0187
|-
|-
|17
|19
|0, 9, 16
|0, 10, 17
| +1+1
| +1+1
|0.0327
|0.0220
|0.0360
|0.0239
|-
|-
|17
|19
|0, 1, 3
|0, 1, 3
| +1+2
| +1+2
|0.0386
|0.0345
|0.0386
|0.0345
|-
|-
|17
|19
|0, 3, 8
|0, 3, 8
| +1+2
| +1+2
|0.0115
|0.0225
|0.0116
|0.0227
|-
|19
|0, 4, 11
| +1+2
|0.0433
|0.0439
|-
|-
|17
|19
|0, 5, 13
|0, 5, 13
| +1+2
| +1+2
|0.0281
|0.0102
|0.0288
|0.0104
|-
|-
|17
|19
|0, 6, 15
|0, 6, 15
| +1+2
| +1+2
|0.0134
|0.0073
|0.0139
|0.0075
|-
|19
|0, 7, 17
| +1+2
|0.0149
|0.0155
|-
|-
|17
|19
|0, 2, 7
|0, 2, 7
| +1+3
| +1+3
|0.0280
|0.0393
|0.0281
|0.0394
|-
|-
|17
|19
|0, 3, 10
|0, 3, 10
| +1+3
| +1+3
|0.0328
|0.0492
|0.0331
|0.0495
|-
|-
|17
|19
|0, 4, 13
|0, 4, 13
| +1+3
| +1+3
|0.0133
|0.0345
|0.0135
|0.0349
|-
|-
|17
|19
|0, 5, 16
|0, 5, 16
| +1+3
| +1+3
|0.0164
|0.0095
|0.0168
|0.0097
|-
|-
|17
|19
|0, 2, 3
|0, 2, 3
| +2+1
| +2+1
|0.0344
|0.0307
|0.0344
|0.0307
|-
|-
|17
|19
|0, 5, 7
| +2+1
|0.0444
|0.0447
|-
|17
|0, 7, 10
|0, 7, 10
| +2+1
| +2+1
|0.0264
|0.0145
|0.0267
|0.0147
|-
|-
|17
|19
|0, 8, 11
|0, 8, 11
| +2+1
| +2+1
|0.0366
|0.0487
|0.0373
|0.0494
|-
|-
|17
|19
|0, 10, 14
|0, 10, 14
| +2+1
| +2+1
|0.0318
|0.0154
|0.0328
|0.0158
|-
|-
|17
|19
|0, 11, 15
|0, 11, 15
| +2+1
| +2+1
|0.0110
|0.0277
|0.0114
|0.0285
|-
|-
|17
|19
|0, 12, 16
|0, 13, 18
| +2+1
| +2+1
|0.0477
|0.0327
|0.0499
|0.0342
|-
|-
|17
|19
|0, 3, 7
|0, 3, 7
| +2+3
| +2+3
|0.0283
|0.0115
|0.0284
|0.0115
|-
|-
|17
|19
|0, 4, 9
|0, 4, 9
| +2+3
| +2+3
|0.0020
|0.0197
|0.0020
|0.0197
|-
|-
|17
|19
|0, 5, 11
|0, 5, 11
| +2+3
| +2+3
|0.0021
|0.0244
|0.0021
|0.0245
|-
|-
|17
|19
|0, 6, 13
|0, 6, 13
| +2+3
| +2+3
|0.0163
|0.0150
|0.0164
|0.0151
|-
|-
|17
|19
|0, 7, 15
|0, 7, 15
| +2+3
| +2+3
|0.0389
|0.0028
|0.0394
|0.0028
|-
|-
|17
|19
|0, 8, 17
| +2+3
|0.0260
|0.0264
|-
|19
|0, 3, 4
|0, 3, 4
| +3+1
| +3+1
|0.0499
|0.0446
|0.0500
|0.0446
|-
|-
|17
|19
|0, 7, 9
|0, 7, 9
| +3+1
| +3+1
|0.0159
|0.0045
|0.0160
|0.0045
|-
|-
|17
|19
|0, 11, 14
|0, 11, 14
| +3+1
| +3+1
|0.0474
|0.0296
|0.0483
|0.0300
|-
|-
|17
|19
|0, 12, 15
|0, 12, 15
| +3+1
| +3+1
|0.0053
|0.0128
|0.0054
|0.0130
|-
|-
|17
|19
|0, 13, 16
|0, 5, 8
| +3+1
| +3+2
|0.0310
|0.0412
|0.0318
|0.0413
|-
|-
|17
|19
|0, 8, 12
|0, 7, 11
| +3+2
| +3+2
|0.0472
|0.0464
|0.0476
|0.0466
|-
|-
|17
|19
|0, 10, 15
|0, 10, 15
| +3+2
| +3+2
|0.0133
|0.0187
|0.0135
|0.0189
|-
|19
|0, 12, 18
| +3+2
|0.0359
|0.0365
|-
|-
|18
|20
|0, 1, 2
|0, 1, 2
| +1+1
| +1+1
|0.0173
|0.0156
|0.0173
|0.0156
|-
|-
|18
|20
|0, 2, 4
|0, 2, 4
| +1+1
| +1+1
|0.0347
|0.0312
|0.0348
|0.0313
|-
|-
|18
|20
|0, 3, 6
| +1+1
|0.0469
|0.0472
|-
|20
|0, 5, 9
|0, 5, 9
| +1+1
| +1+1
|0.0224
|0.0300
|0.0229
|0.0305
|-
|-
|18
|20
|0, 6, 11
|0, 6, 11
| +1+1
| +1+1
|0.0129
|0.0035
|0.0134
|0.0036
|-
|-
|18
|20
|0, 7, 12
| +1+1
|0.0472
|0.0493
|-
|18
|0, 7, 13
|0, 7, 13
| +1+1
| +1+1
|0.0430
|0.0317
|0.0452
|0.0330
|-
|-
|18
|20
|0, 8, 14
|0, 8, 14
| +1+1
| +1+1
|0.0089
|0.0207
|0.0094
|0.0217
|-
|-
|18
|20
|0, 9, 16
|0, 9, 16
| +1+1
| +1+1
|0.0246
|0.0109
|0.0267
|0.0116
|-
|-
|18
|20
|0, 10, 17
|0, 10, 17
| +1+1
| +1+1
|0.0145
|0.0287
|0.0160
|0.0310
|-
|20
|0, 10, 18
| +1+1
|0.0392
|0.0427
|-
|20
|0, 11, 19
| +1+1
|0.0036
|0.0039
|-
|-
|18
|20
|0, 1, 3
|0, 1, 3
| +1+2
| +1+2
|0.0364
|0.0328
|0.0365
|0.0328
|-
|-
|18
|20
|0, 3, 8
|0, 3, 8
| +1+2
| +1+2
|0.0173
|0.0272
|0.0174
|0.0274
|-
|-
|18
|20
|0, 4, 11
| +1+2
|0.0369
|0.0374
|-
|20
|0, 5, 13
|0, 5, 13
| +1+2
| +1+2
|0.0186
|0.0026
|0.0191
|0.0026
|-
|-
|18
|20
|0, 6, 15
|0, 6, 15
| +1+2
| +1+2
|0.0025
|0.0162
|0.0026
|0.0166
|-
|-
|18
|20
|0, 7, 17
|0, 7, 17
| +1+2
| +1+2
|0.0038
|0.0250
|0.0040
|0.0259
|-
|-
|18
|20
|0, 2, 7
|0, 7, 18
| +1+3
| +1+2
|0.0340
|0.0467
|0.0341
|0.0485
|-
|-
|18
|20
|0, 3, 10
|0, 8, 19
| +1+2
|0.0271
|0.0283
|-
|20
|0, 2, 7
| +1+3
| +1+3
|0.0414
|0.0442
|0.0417
|0.0443
|-
|-
|18
|20
|0, 4, 13
|0, 4, 13
| +1+3
| +1+3
|0.0244
|0.0436
|0.0248
|0.0440
|-
|20
|0, 4, 14
| +1+3
|0.0463
|0.0468
|-
|-
|18
|20
|0, 5, 16
|0, 5, 16
| +1+3
| +1+3
|0.0028
|0.0206
|0.0028
|0.0209
|-
|-
|18
|20
|0, 2, 3
|0, 6, 19
| +2+1
| +1+3
|0.0324
|0.0076
|0.0325
|0.0078
|-
|-
|18
|20
|0, 5, 7
|0, 2, 3
| +2+1
| +2+1
|0.0485
|0.0292
|0.0488
|0.0292
|-
|-
|18
|20
|0, 7, 10
|0, 7, 10
| +2+1
| +2+1
|0.0201
|0.0095
|0.0204
|0.0096
|-
|-
|18
|20
|0, 8, 11
| +2+1
|0.0430
|0.0437
|-
|18
|0, 10, 14
|0, 10, 14
| +2+1
| +2+1
|0.0231
|0.0085
|0.0238
|0.0087
|-
|-
|18
|20
|0, 11, 15
|0, 11, 15
| +2+1
| +2+1
|0.0198
|0.0348
|0.0205
|0.0357
|-
|-
|18
|20
|0, 13, 18
| +2+1
|0.0238
|0.0247
|-
|20
|0, 14, 19
| +2+1
|0.0087
|0.0091
|-
|20
|0, 3, 7
|0, 3, 7
| +2+3
| +2+3
|0.0194
|0.0043
|0.0194
|0.0043
|-
|-
|18
|20
|0, 4, 9
|0, 4, 9
| +2+3
| +2+3
|0.0094
|0.0289
|0.0095
|0.0290
|-
|-
|18
|20
|0, 5, 11
|0, 5, 11
| +2+3
| +2+3
|0.0119
|0.0356
|0.0120
|0.0358
|-
|-
|18
|20
|0, 6, 13
|0, 6, 13
| +2+3
| +2+3
|0.0002
|0.0283
|0.0002
|0.0285
|-
|-
|18
|20
|0, 7, 15
|0, 7, 15
| +2+3
| +2+3
|0.0198
|0.0125
|0.0201
|0.0127
|-
|-
|18
|20
|0, 8, 17
|0, 8, 17
| +2+3
| +2+3
|0.0453
|0.0086
|0.0461
|0.0087
|-
|20
|0, 9, 19
| +2+3
|0.0335
|0.0341
|-
|-
|18
|20
|0, 3, 4
|0, 3, 4
| +3+1
| +3+1
|0.0471
|0.0423
|0.0472
|0.0424
|-
|-
|18
|20
|0, 7, 9
|0, 7, 9
| +3+1
| +3+1
|0.0099
|0.0003
|0.0099
|0.0003
|-
|-
|18
|20
|0, 11, 14
|0, 11, 14
| +3+1
| +3+1
|0.0380
|0.0221
|0.0386
|0.0223
|-
|-
|18
|20
|0, 12, 15
|0, 12, 15
| +3+1
| +3+1
|0.0043
|0.0205
|0.0044
|0.0208
|-
|-
|18
|20
|0, 13, 16
|0, 5, 8
| +3+1
| +3+2
|0.0407
|0.0336
|0.0417
|0.0337
|-
|-
|18
|20
|0, 5, 8
|0, 7, 11
| +3+2
| +3+2
|0.0495
|0.0361
|0.0497
|0.0362
|-
|-
|18
|20
|0, 10, 15
|0, 10, 15
| +3+2
| +3+2
|0.0036
|0.0322
|0.0036
|0.0325
|-
|-
|19
|20
|0, 1, 2
|0, 12, 18
| +1+1
| +3+2
|0.0164
|0.0195
|0.0164
|0.0197
|-
|20
|0, 13, 19
| +3+2
|0.0460
|0.0468
|-
|21
|0, 1, 2
| +1+1
|0.0148
|0.0148
|-
|-
|19
|21
|0, 2, 4
|0, 2, 4
| +1+1
| +1+1
|0.0329
|0.0297
|0.0330
|0.0298
|-
|-
|19
|21
|0, 3, 6
|0, 3, 6
| +1+1
| +1+1
|0.0494
|0.0447
|0.0498
|0.0449
|-
|-
|19
|21
|0, 5, 9
|0, 5, 9
| +1+1
| +1+1
|0.0264
|0.0333
|0.0269
|0.0338
|-
|-
|19
|21
|0, 6, 11
|0, 6, 11
| +1+1
| +1+1
|0.0079
|0.0006
|0.0082
|0.0006
|-
|-
|19
|21
|0, 7, 13
|0, 7, 13
| +1+1
| +1+1
|0.0371
|0.0269
|0.0387
|0.0278
|-
|-
|19
|21
|0, 8, 14
|0, 8, 14
| +1+1
| +1+1
|0.0151
|0.0257
|0.0159
|0.0268
|-
|-
|19
|21
|0, 9, 16
|0, 9, 16
| +1+1
| +1+1
|0.0174
|0.0051
|0.0187
|0.0054
|-
|-
|19
|21
|0, 10, 17
|0, 10, 17
| +1+1
| +1+1
|0.0220
|0.0348
|0.0239
|0.0372
|-
|-
|19
|21
|0, 1, 3
|0, 10, 18
| +1+2
| +1+1
|0.0345
|0.0325
|0.0345
|0.0351
|-
|21
|0, 11, 19
| +1+1
|0.0033
|0.0036
|-
|21
|0, 12, 20
| +1+1
|0.0350
|0.0387
|-
|21
|0, 1, 3
| +1+2
|0.0312
|0.0313
|-
|-
|19
|21
|0, 3, 8
|0, 3, 8
| +1+2
| +1+2
|0.0225
|0.0315
|0.0227
|0.0317
|-
|-
|19
|21
|0, 4, 11
|0, 4, 11
| +1+2
| +1+2
|0.0433
|0.0311
|0.0439
|0.0315
|-
|-
|19
|21
|0, 5, 13
|0, 5, 13
| +1+2
| +1+2
|0.0102
|0.0043
|0.0104
|0.0044
|-
|-
|19
|21
|0, 6, 15
|0, 6, 15
| +1+2
| +1+2
|0.0073
|0.0242
|0.0075
|0.0247
|-
|-
|19
|21
|0, 7, 17
|0, 7, 17
| +1+2
| +1+2
|0.0149
|0.0341
|0.0155
|0.0351
|-
|-
|19
|21
|0, 2, 7
|0, 7, 18
| +1+3
| +1+2
|0.0393
|0.0372
|0.0394
|0.0385
|-
|-
|19
|21
|0, 3, 10
|0, 8, 19
| +1+3
| +1+2
|0.0492
|0.0372
|0.0495
|0.0388
|-
|-
|19
|21
|0, 4, 13
|0, 8, 20
| +1+2
|0.0290
|0.0303
|-
|21
|0, 1, 4
| +1+3
| +1+3
|0.0345
|0.0476
|0.0349
|0.0477
|-
|-
|19
|21
|0, 5, 16
|0, 2, 7
| +1+3
| +1+3
|0.0095
|0.0485
|0.0097
|0.0486
|-
|-
|19
|21
|0, 2, 3
|0, 3, 11
| +2+1
| +1+3
|0.0307
|0.0475
|0.0307
|0.0478
|-
|-
|19
|21
|0, 7, 10
|0, 4, 14
| +1+3
|0.0378
|0.0382
|-
|21
|0, 5, 16
| +1+3
|0.0306
|0.0311
|-
|21
|0, 5, 17
| +1+3
|0.0451
|0.0459
|-
|21
|0, 6, 19
| +1+3
|0.0042
|0.0043
|-
|21
|0, 2, 3
| +2+1
| +2+1
|0.0145
|0.0278
|0.0147
|0.0278
|-
|-
|19
|21
|0, 8, 11
|0, 7, 10
| +2+1
| +2+1
|0.0487
|0.0050
|0.0494
|0.0051
|-
|-
|19
|21
|0, 10, 14
|0, 10, 14
| +2+1
| +2+1
|0.0154
|0.0023
|0.0158
|0.0023
|-
|-
|19
|21
|0, 11, 15
|0, 11, 15
| +2+1
| +2+1
|0.0277
|0.0411
|0.0285
|0.0421
|-
|-
|19
|21
|0, 13, 18
|0, 13, 18
| +2+1
| +2+1
|0.0327
|0.0157
|0.0342
|0.0163
|-
|21
|0, 14, 19
| +2+1
|0.0169
|0.0176
|-
|21
|0, 15, 20
| +2+1
|0.0458
|0.0480
|-
|-
|19
|21
|0, 3, 7
|0, 3, 7
| +2+3
| +2+3
|0.0115
|0.0022
|0.0115
|0.0022
|-
|-
|19
|21
|0, 4, 9
|0, 4, 9
| +2+3
| +2+3
|0.0197
|0.0372
|0.0197
|0.0373
|-
|-
|19
|21
|0, 5, 11
|0, 5, 11
| +2+3
| +2+3
|0.0244
|0.0458
|0.0245
|0.0460
|-
|-
|19
|21
|0, 6, 13
|0, 6, 13
| +2+3
| +2+3
|0.0150
|0.0403
|0.0151
|0.0406
|-
|-
|19
|21
|0, 7, 15
|0, 7, 15
| +2+3
| +2+3
|0.0028
|0.0264
|0.0028
|0.0266
|-
|-
|19
|21
|0, 8, 17
|0, 8, 17
| +2+3
| +2+3
|0.0260
|0.0071
|0.0264
|0.0072
|-
|21
|0, 9, 19
| +2+3
|0.0160
|0.0162
|-
|-
|19
|21
|0, 3, 4
|0, 3, 4
| +3+1
| +3+1
|0.0446
|0.0403
|0.0446
|0.0403
|-
|-
|19
|21
|0, 7, 9
|0, 7, 9
| +3+1
| +3+1
|0.0045
|0.0046
|0.0045
|0.0046
|-
|-
|19
|21
|0, 11, 14
|0, 11, 14
| +3+1
| +3+1
|0.0296
|0.0153
|0.0300
|0.0155
|-
|-
|19
|21
|0, 12, 15
|0, 12, 15
| +3+1
| +3+1
|0.0128
|0.0273
|0.0130
|0.0277
|-
|21
|0, 16, 20
| +3+1
|0.0191
|0.0196
|-
|-
|19
|21
|0, 5, 8
|0, 5, 8
| +3+2
| +3+2
|0.0412
|0.0268
|0.0413
|0.0269
|-
|-
|19
|21
|0, 7, 11
|0, 7, 11
| +3+2
| +3+2
|0.0464
|0.0268
|0.0466
|0.0269
|-
|21
|0, 9, 14
| +3+2
|0.0480
|0.0483
|-
|-
|19
|21
|0, 10, 15
|0, 10, 15
| +3+2
| +3+2
|0.0187
|0.0444
|0.0189
|0.0448
|-
|-
|19
|21
|0, 12, 18
|0, 12, 18
| +3+2
| +3+2
|0.0359
|0.0046
|0.0365
|0.0047
|-
|-
|20
|22
|0, 1, 2
|0, 1, 2
| +1+1
| +1+1
|0.0156
|0.0141
|0.0156
|0.0141
|-
|-
|20
|22
|0, 2, 4
|0, 2, 4
| +1+1
| +1+1
|0.0312
|0.0283
|0.0313
|0.0284
|-
|-
|20
|22
|0, 3, 6
|0, 3, 6
| +1+1
| +1+1
|0.0469
|0.0426
|0.0472
|0.0429
|-
|-
|20
|22
|0, 5, 9
|0, 5, 9
| +1+1
| +1+1
|0.0300
|0.0362
|0.0305
|0.0367
|-
|-
|20
|22
|0, 6, 11
|0, 6, 11
| +1+1
| +1+1
|0.0035
|0.0042
|0.0036
|0.0043
|-
|-
|20
|22
|0, 7, 13
|0, 7, 13
| +1+1
| +1+1
|0.0317
|0.0225
|0.0330
|0.0232
|-
|-
|20
|22
|0, 8, 14
|0, 8, 14
| +1+1
| +1+1
|0.0207
|0.0303
|0.0217
|0.0315
|-
|22
|0, 8, 15
| +1+1
|0.0461
|0.0481
|-
|-
|20
|22
|0, 9, 16
|0, 9, 16
| +1+1
| +1+1
|0.0109
|0.0003
|0.0116
|0.0003
|-
|-
|20
|22
|0, 10, 17
|0, 10, 17
| +1+1
| +1+1
|0.0287
|0.0403
|0.0310
|0.0428
|-
|-
|20
|22
|0, 10, 18
|0, 10, 18
| +1+1
| +1+1
|0.0392
|0.0265
|0.0427
|0.0283
|-
|-
|20
|22
|0, 11, 19
|0, 11, 19
| +1+1
| +1+1
|0.0036
|0.0096
|0.0039
|0.0104
|-
|22
|0, 12, 20
| +1+1
|0.0415
|0.0454
|-
|22
|0, 12, 21
| +1+1
|0.0182
|0.0201
|-
|-
|20
|22
|0, 1, 3
|0, 1, 3
| +1+2
| +1+2
|0.0328
|0.0298
|0.0328
|0.0298
|-
|-
|20
|22
|0, 3, 8
|0, 3, 8
| +1+2
| +1+2
|0.0272
|0.0354
|0.0274
|0.0355
|-
|-
|20
|22
|0, 4, 11
|0, 4, 11
| +1+2
| +1+2
|0.0369
|0.0259
|0.0374
|0.0261
|-
|-
|20
|22
|0, 5, 13
|0, 5, 13
| +1+2
| +1+2
|0.0026
|0.0106
|0.0026
|0.0108
|-
|-
|20
|22
|0, 6, 15
|0, 6, 15
| +1+2
| +1+2
|0.0162
|0.0315
|0.0166
|0.0321
|-
|22
|0, 6, 16
| +1+2
|0.0454
|0.0465
|-
|-
|20
|22
|0, 7, 17
|0, 7, 17
| +1+2
| +1+2
|0.0250
|0.0424
|0.0259
|0.0435
|-
|-
|20
|22
|0, 7, 18
|0, 7, 18
| +1+2
| +1+2
|0.0467
|0.0286
|0.0485
|0.0295
|-
|-
|20
|22
|0, 8, 19
|0, 8, 19
| +1+2
| +1+2
|0.0271
|0.0465
|0.0283
|0.0482
|-
|22
|0, 8, 20
| +1+2
|0.0193
|0.0202
|-
|-
|20
|22
|0, 2, 7
|0, 9, 21
| +1+3
| +1+2
|0.0442
|0.0459
|0.0443
|0.0481
|-
|-
|20
|22
|0, 4, 13
|0, 1, 4
| +1+3
| +1+3
|0.0436
|0.0455
|0.0440
|0.0455
|-
|22
|0, 3, 11
| +1+3
|0.0415
|0.0418
|-
|-
|20
|22
|0, 4, 14
|0, 4, 14
| +1+3
| +1+3
|0.0463
|0.0301
|0.0468
|0.0304
|-
|-
|20
|22
|0, 5, 16
|0, 5, 16
| +1+3
| +1+3
|0.0206
|0.0398
|0.0209
|0.0404
|-
|22
|0, 5, 17
| +1+3
|0.0357
|0.0363
|-
|-
|20
|22
|0, 6, 19
|0, 6, 19
| +1+3
| +1+3
|0.0076
|0.0151
|0.0078
|0.0154
|-
|-
|20
|22
|0, 2, 3
|0, 2, 3
| +2+1
| +2+1
|0.0292
|0.0265
|0.0292
|0.0265
|-
|-
|20
|22
|0, 7, 10
|0, 7, 10
| +2+1
| +2+1
|0.0095
|0.0010
|0.0096
|0.0010
|-
|22
|0, 9, 13
| +2+1
|0.0478
|0.0485
|-
|-
|20
|22
|0, 10, 14
|0, 10, 14
| +2+1
| +2+1
|0.0085
|0.0034
|0.0087
|0.0034
|-
|-
|20
|22
|0, 11, 15
|0, 11, 15
| +2+1
| +2+1
|0.0348
|0.0469
|0.0357
|0.0478
|-
|22
|0, 12, 17
| +2+1
|0.0457
|0.0469
|-
|-
|20
|22
|0, 13, 18
|0, 13, 18
| +2+1
| +2+1
|0.0238
|0.0085
|0.0247
|0.0088
|-
|-
|20
|22
|0, 14, 19
|0, 14, 19
| +2+1
| +2+1
|0.0087
|0.0242
|0.0091
|0.0251
|-
|-
|20
|22
|0, 3, 7
|0, 3, 7
| +2+3
| +2+3
|0.0043
|0.0080
|0.0043
|0.0081
|-
|-
|20
|22
|0, 4, 9
|0, 4, 9
| +2+3
| +2+3
|0.0289
|0.0448
|0.0290
|0.0449
|-
|-
|20
|22
|0, 5, 11
|0, 7, 15
| +2+3
| +2+3
|0.0356
|0.0390
|0.0358
|0.0393
|-
|-
|20
|22
|0, 6, 13
|0, 8, 17
| +2+3
| +2+3
|0.0283
|0.0214
|0.0285
|0.0216
|-
|-
|20
|22
|0, 7, 15
|0, 9, 19
| +2+3
| +2+3
|0.0125
|0.0000
|0.0127
|0.0000
|-
|-
|20
|22
|0, 8, 17
|0, 10, 21
| +2+3
| +2+3
|0.0086
|0.0240
|0.0087
|0.0244
|-
|-
|20
|22
|0, 9, 19
| +2+3
|0.0335
|0.0341
|-
|20
|0, 3, 4
|0, 3, 4
| +3+1
| +3+1
|0.0423
|0.0384
|0.0424
|0.0384
|-
|-
|20
|22
|0, 7, 9
|0, 7, 9
| +3+1
| +3+1
|0.0003
|0.0085
|0.0003
|0.0085
|-
|-
|20
|22
|0, 11, 14
|0, 11, 14
| +3+1
| +3+1
|0.0221
|0.0092
|0.0223
|0.0093
|-
|-
|20
|22
|0, 12, 15
|0, 12, 15
| +3+1
| +3+1
|0.0205
|0.0335
|0.0208
|0.0339
|-
|22
|0, 15, 19
| +3+1
|0.0415
|0.0423
|-
|22
|0, 16, 20
| +3+1
|0.0106
|0.0108
|-
|22
|0, 17, 21
| +3+1
|0.0171
|0.0175
|-
|-
|20
|22
|0, 5, 8
|0, 5, 8
| +3+2
| +3+2
|0.0336
|0.0207
|0.0337
|0.0207
|-
|-
|20
|22
|0, 7, 11
|0, 7, 11
| +3+2
| +3+2
|0.0361
|0.0184
|0.0362
|0.0184
|-
|-
|20
|22
|0, 10, 15
|0, 9, 14
| +3+2
| +3+2
|0.0322
|0.0372
|0.0325
|0.0374
|-
|-
|20
|22
|0, 12, 18
|0, 12, 18
| +3+2
| +3+2
|0.0195
|0.0088
|0.0197
|0.0089
|-
|-
|20
|22
|0, 13, 19
|0, 14, 21
| +3+2
| +3+2
|0.0460
|0.0384
|0.0468
|0.0391
|-
|-
|21
|23
|0, 1, 2
|0, 1, 2
| +1+1
| +1+1
|0.0148
|0.0135
|0.0148
|0.0135
|-
|-
|21
|23
|0, 2, 4
|0, 2, 4
| +1+1
| +1+1
|0.0297
|0.0271
|0.0298
|0.0272
|-
|-
|21
|23
|0, 3, 6
|0, 3, 6
| +1+1
| +1+1
|0.0447
|0.0408
|0.0449
|0.0410
|-
|-
|21
|23
|0, 5, 9
|0, 5, 9
| +1+1
| +1+1
|0.0333
|0.0389
|0.0338
|0.0394
|-
|-
|21
|23
|0, 6, 11
|0, 6, 11
| +1+1
| +1+1
|0.0006
|0.0076
|0.0006
|0.0077
|-
|-
|21
|23
|0, 7, 13
|0, 7, 13
| +1+1
| +1+1
|0.0269
|0.0185
|0.0278
|0.0190
|-
|-
|21
|23
|0, 8, 14
|0, 8, 14
| +1+1
| +1+1
|0.0257
|0.0345
|0.0268
|0.0357
|-
|-
|21
|23
|0, 8, 15
| +1+1
|0.0415
|0.0431
|-
|23
|0, 9, 16
|0, 9, 16
| +1+1
| +1+1
|0.0051
|0.0051
|0.0054
|0.0053
|-
|-
|21
|23
|0, 10, 17
|0, 10, 17
| +1+1
| +1+1
|0.0348
|0.0453
|0.0372
|0.0478
|-
|-
|21
|23
|0, 10, 18
|0, 10, 18
| +1+1
| +1+1
|0.0325
|0.0210
|0.0351
|0.0223
|-
|-
|21
|23
|0, 11, 19
|0, 11, 19
| +1+1
| +1+1
|0.0033
|0.0153
|0.0036
|0.0164
|-
|-
|21
|23
|0, 12, 20
|0, 11, 20
| +1+1
| +1+1
|0.0350
|0.0448
|0.0387
|0.0484
|-
|-
|21
|23
|0, 12, 21
| +1+1
|0.0118
|0.0129
|-
|23
|0, 13, 22
| +1+1
|0.0176
|0.0195
|-
|23
|0, 1, 3
|0, 1, 3
| +1+2
| +1+2
|0.0312
|0.0285
|0.0313
|0.0285
|-
|-
|21
|23
|0, 3, 8
|0, 3, 8
| +1+2
| +1+2
|0.0315
|0.0389
|0.0317
|0.0391
|-
|-
|21
|23
|0, 4, 11
|0, 4, 11
| +1+2
| +1+2
|0.0311
|0.0211
|0.0315
|0.0213
|-
|-
|21
|23
|0, 5, 13
|0, 5, 13
| +1+2
| +1+2
|0.0043
|0.0163
|0.0044
|0.0166
|-
|-
|21
|23
|0, 6, 15
|0, 6, 15
| +1+2
| +1+2
|0.0242
|0.0381
|0.0247
|0.0388
|-
|-
|21
|23
|0, 7, 17
|0, 6, 16
| +1+2
| +1+2
|0.0341
|0.0385
|0.0351
|0.0393
|-
|-
|21
|23
|0, 7, 18
|0, 7, 18
| +1+2
| +1+2
|0.0372
|0.0207
|0.0385
|0.0213
|-
|-
|21
|23
|0, 8, 19
|0, 8, 20
| +1+2
| +1+2
|0.0372
|0.0105
|0.0388
|0.0109
|-
|-
|21
|23
|0, 8, 20
|0, 9, 22
| +1+2
| +1+2
|0.0290
|0.0059
|0.0303
|0.0062
|-
|-
|21
|23
|0, 1, 4
|0, 1, 4
| +1+3
| +1+3
|0.0476
|0.0435
|0.0477
|0.0435
|-
|-
|21
|23
|0, 2, 7
| +1+3
|0.0485
|0.0486
|-
|21
|0, 3, 11
|0, 3, 11
| +1+3
| +1+3
|0.0475
|0.0360
|0.0478
|0.0362
|-
|-
|21
|23
|0, 4, 14
|0, 4, 14
| +1+3
| +1+3
|0.0378
|0.0230
|0.0382
|0.0232
|-
|-
|21
|23
|0, 5, 16
|0, 5, 16
| +1+3
| +1+3
|0.0306
|0.0482
|0.0311
|0.0488
|-
|-
|21
|23
|0, 5, 17
|0, 5, 17
| +1+3
| +1+3
|0.0451
|0.0271
|0.0459
|0.0275
|-
|-
|21
|23
|0, 6, 19
|0, 6, 19
| +1+3
| +1+3
|0.0042
|0.0250
|0.0043
|0.0254
|-
|23
|0, 6, 20
| +1+3
|0.0403
|0.0412
|-
|23
|0, 7, 22
| +1+3
|0.0012
|0.0013
|-
|-
|21
|23
|0, 2, 3
|0, 2, 3
| +2+1
| +2+1
|0.0278
|0.0253
|0.0278
|0.0253
|-
|-
|21
|23
|0, 7, 10
|0, 7, 10
| +2+1
| +2+1
|0.0050
|0.0028
|0.0051
|0.0028
|-
|23
|0, 9, 13
| +2+1
|0.0428
|0.0433
|-
|-
|21
|23
|0, 10, 14
|0, 10, 14
| +2+1
| +2+1
|0.0023
|0.0085
|0.0023
|0.0086
|-
|-
|21
|23
|0, 11, 15
|0, 12, 17
| +2+1
| +2+1
|0.0411
|0.0392
|0.0421
|0.0401
|-
|-
|21
|23
|0, 13, 18
|0, 13, 18
| +2+1
| +2+1
|0.0157
|0.0019
|0.0163
|0.0020
|-
|-
|21
|23
|0, 14, 19
|0, 14, 19
| +2+1
| +2+1
|0.0169
|0.0309
|0.0176
|0.0320
|-
|-
|21
|23
|0, 15, 20
|0, 16, 22
| +2+1
| +2+1
|0.0458
|0.0199
|0.0480
|0.0208
|-
|-
|21
|23
|0, 3, 7
|0, 3, 7
| +2+3
| +2+3
|0.0022
|0.0134
|0.0022
|0.0134
|-
|-
|21
|23
|0, 4, 9
|0, 8, 17
| +2+3
| +2+3
|0.0372
|0.0344
|0.0373
|0.0347
|-
|-
|21
|23
|0, 5, 11
|0, 9, 19
| +2+3
| +2+3
|0.0458
|0.0146
|0.0460
|0.0148
|-
|-
|21
|23
|0, 6, 13
|0, 10, 21
| +2+3
| +2+3
|0.0403
|0.0079
|0.0406
|0.0080
|-
|-
|21
|23
|0, 7, 15
|0, 3, 4
| +2+3
| +3+1
|0.0264
|0.0367
|0.0266
|0.0367
|-
|-
|21
|23
|0, 8, 17
| +2+3
|0.0071
|0.0072
|-
|21
|0, 9, 19
| +2+3
|0.0160
|0.0162
|-
|21
|0, 3, 4
| +3+1
|0.0403
|0.0403
|-
|21
|0, 7, 9
|0, 7, 9
| +3+1
| +3+1
|0.0046
|0.0121
|0.0046
|0.0121
|-
|-
|21
|23
|0, 11, 14
|0, 11, 14
| +3+1
| +3+1
|0.0153
|0.0036
|0.0155
|0.0037
|-
|-
|21
|23
|0, 12, 15
|0, 12, 15
| +3+1
| +3+1
|0.0273
|0.0392
|0.0277
|0.0396
|-
|-
|21
|23
|0, 15, 19
| +3+1
|0.0338
|0.0344
|-
|23
|0, 16, 20
|0, 16, 20
| +3+1
| +3+1
|0.0191
|0.0028
|0.0196
|0.0028
|-
|23
|0, 17, 21
| +3+1
|0.0249
|0.0255
|-
|-
|21
|23
|0, 5, 8
|0, 5, 8
| +3+2
| +3+2
|0.0268
|0.0150
|0.0269
|0.0151
|-
|-
|21
|23
|0, 7, 11
|0, 7, 11
| +3+2
| +3+2
|0.0268
|0.0107
|0.0269
|0.0107
|-
|-
|21
|23
|0, 9, 14
|0, 9, 14
| +3+2
| +3+2
|0.0480
|0.0274
|0.0483
|0.0276
|-
|-
|21
|23
|0, 10, 15
|0, 12, 18
| +3+2
| +3+2
|0.0444
|0.0210
|0.0448
|0.0213
|-
|-
|21
|23
|0, 12, 18
|0, 14, 21
| +3+2
| +3+2
|0.0046
|0.0240
|0.0047
|0.0244
|-
|-
|22
|23
|0, 1, 2
|0, 15, 22
| +1+1
| +3+2
|0.0141
|0.0325
|0.0141
|0.0331
|-
|}
|22
 
|0, 2, 4
== DR and RTT ==
| +1+1
One may be able to tune a rank-2 regular temperament in such a way that a triad of interest exactly "inherits" its delta signature from a simple JI preimage thereof. This is done by setting up an algebraic equation relating the intervals in the chord to a generator and then solving for the generator that produces proportionally-beating triads. The value to be solved for is the generator's frequency ratio (not its cent value).
|0.0283
 
|0.0284
If we want to optimize a 4:5:6 triad in Meantone, for instance, we want a +1+1 delta signature, or equivalently a 1:1 ratio of frequency deltas between the major third and minor third. Fixing any frequency as the triad's root and letting <math>g</math> be the frequency ratio for the perfect fifth generator for meantone, the minor third in the tempered 4:5:6 triad has a delta of <math>g-g^4/4</math>, and the major third in the same triad has a delta of <math>g^4/4-1</math>. Therefore to ensure that the two deltas form a 1:1 ratio, we must find the appropriate root of the polynomial <math>g^4-2g-2</math> (the difference between the two, simplified to make all coefficients integers). This results in a generator of 1.4945, or about 695.6 cents.
 
Note that the equation to solve depends on what chord you want to tune as equal-beating. For example, assuming pure octaves, Meantone admits an equation for tuning the 3:4:5 as equal-beating: <math>g^4+2g-8=0.</math> The latter equation has solution g = 1.4960 = 697.3c. <!--Essentially tempered [[Dyadic chord|dyadic]] triads are also more difficult to tune with simple delta-signatures, since they lack simple JI preimages.-->
 
Below is a list of temperaments and their various optimizations for proportionally beating chords. They are ordered by highest power in the relevant DR polynomial, with ties broken by leading coefficients, then 2nd term coefficients, 3rd term coefficients, 4th term coefficients, etc. In the case of negative coefficients, only the absolute value is considered.
 
{| class="wikitable"
|-
|-
|22
! colspan="11" | Coefficients of terms
|0, 3, 6
!rowspan=2| Chord
| +1+1
!rowspan=2| Delta signature
|0.0426
!rowspan=2| Temperament
|0.0429
!rowspan=2| Generator (cents)
!rowspan=2| Edos
|-
|-
|22
! g^10
|0, 5, 9
! g^9
| +1+1
! g^8
|0.0362
! g^7
|0.0367
! g^6
! g^5
! g^4
! g^3
! g^2
! g^1
! g^0
|-
|-
|22
|  
|0, 6, 11
|  
|
|
|
|
|
|
| 1
| -1
| -1
| 4:5:6
| +1+1
| +1+1
|0.0042
| {{monzo|1 -2 1}}
|0.0043
| 833.09 (phi)
| 36
|-
|-
|22
|  
|0, 7, 13
|  
|
|
|
|
|
|
| 2
| -1
| -2
| 4:5:6
| +1+1
| +1+1
|0.0225
| Father
|0.0232
| 428.42
| 14
|-
|-
|22
|  
|0, 8, 14
|  
| +1+1
|
|0.0303
|
|0.0315
|
|
|
| 3
| -2
|
| -2
| 6:7:9
| +1+2
| Beep
| 258.65
| 33, 42, 51
|-
|-
|22
|  
|0, 8, 15
|  
|
|
|
|
| 1
|
|
| -1
| -2
| 4:5:6
| +1+1
| +1+1
|0.0461
| Mavila
|0.0481
| 523.66
| 23, 39
|-
|-
|22
|  
|0, 9, 16
|  
|
|
|
|
| 1
|
|
| -2
| -2
| 4:5:6
| +1+1
| +1+1
|0.0003
| Meantone
|0.0003
| 695.63
| 19, 31, 50
|-
|-
|22
|  
|0, 10, 17
|  
| +1+1
|  
|0.0403
|
|0.0428
|
|-
| 1
|22
|  
|0, 10, 18
|  
| 2
|
| -4
| 4:5:6
| +1+1
| +1+1
|0.0265
| Porcupine
|0.0283
| 160.89
| 15
|-
|-
|22
|  
|0, 11, 19
|  
| +1+1
|
|0.0096
|
|0.0104
|
| 1
| -4
|
|
|
| 12
| 5:6:9
| +1+3
| Mavila
| 674.90
| 16, 25
|-
|-
|22
|  
|0, 12, 20
|  
|
|
| 1
|
|
|
|
| -4
| -4
| 4:5:6
| +1+1
| +1+1
|0.0415
| Avila
|0.0454
| 660.23
| 20
|-
|-
|22
|  
|0, 12, 21
|  
|
|
| 1
| -2
|
|
|
|
| 2
| 4:5:6
| +1+1
| +1+1
|0.0182
| Hanson
|0.0201
| 317.96
| 19
|-
|-
|22
|  
|0, 1, 3
|  
| +1+2
|
|0.0298
| 1
|0.0298
|
|
|
|
|
| -2
| -4
| 4:5:6
| +1+1
| Uncle
| 467.46
| 18
|-
|-
|22
|  
|0, 3, 8
|  
| 3
|
|
| -4
|
|
|
|
| -16
| 4:5:7
| +1+2
| +1+2
|0.0354
| Mabila
|0.0355
| 527.66
| 25
|-
|-
|22
|  
|0, 4, 11
| 1
| +1+2
|
|0.0259
|
|0.0261
|
|
| -1
|
|
|
| -1
| 4:5:6
| +1+1
| Tetracot
| 176.54
| 34
|-
|-
|22
|  
|0, 5, 13
| 1
| +1+2
|
|0.0106
| -1
|0.0108
|  
|-
|  
|22
|  
|0, 6, 15
|  
| +1+2
|  
|0.0315
|
|0.0321
| -4
| 4:5:6
| +1+1
| Sensi
| 442.74
| 19, 65
|-
|-
|22
| 1
|0, 6, 16
|  
| +1+2
|  
|0.0454
|  
|0.0465
|  
|-
|  
|22
|  
|0, 7, 17
| 2
| +1+2
|  
|0.0424
|  
|0.0435
| -8
|-
| 4:5:6
|22
| +1+1
|0, 7, 18
| Orson
| +1+2
| 271.51
|0.0286
| 22, 31, 53
|0.0295
|-
|22
|0, 8, 19
| +1+2
|0.0465
|0.0482
|-
|22
|0, 8, 20
| +1+2
|0.0193
|0.0202
|-
|22
|0, 9, 21
| +1+2
|0.0459
|0.0481
|-
|22
|0, 1, 4
| +1+3
|0.0455
|0.0455
|-
|22
|0, 3, 11
| +1+3
|0.0415
|0.0418
|-
|22
|0, 4, 14
| +1+3
|0.0301
|0.0304
|-
|22
|0, 5, 16
| +1+3
|0.0398
|0.0404
|-
|22
|0, 5, 17
| +1+3
|0.0357
|0.0363
|-
|22
|0, 6, 19
| +1+3
|0.0151
|0.0154
|-
|22
|0, 2, 3
| +2+1
|0.0265
|0.0265
|-
|22
|0, 7, 10
| +2+1
|0.0010
|0.0010
|-
|22
|0, 9, 13
| +2+1
|0.0478
|0.0485
|-
|22
|0, 10, 14
| +2+1
|0.0034
|0.0034
|-
|22
|0, 11, 15
| +2+1
|0.0469
|0.0478
|-
|22
|0, 12, 17
| +2+1
|0.0457
|0.0469
|-
|22
|0, 13, 18
| +2+1
|0.0085
|0.0088
|-
|22
|0, 14, 19
| +2+1
|0.0242
|0.0251
|-
|22
|0, 3, 7
| +2+3
|0.0080
|0.0081
|-
|22
|0, 4, 9
| +2+3
|0.0448
|0.0449
|-
|22
|0, 7, 15
| +2+3
|0.0390
|0.0393
|-
|22
|0, 8, 17
| +2+3
|0.0214
|0.0216
|-
|22
|0, 9, 19
| +2+3
|0.0000
|0.0000
|-
|22
|0, 10, 21
| +2+3
|0.0240
|0.0244
|-
|22
|0, 3, 4
| +3+1
|0.0384
|0.0384
|-
|22
|0, 7, 9
| +3+1
|0.0085
|0.0085
|-
|22
|0, 11, 14
| +3+1
|0.0092
|0.0093
|-
|22
|0, 12, 15
| +3+1
|0.0335
|0.0339
|-
|22
|0, 15, 19
| +3+1
|0.0415
|0.0423
|-
|22
|0, 16, 20
| +3+1
|0.0106
|0.0108
|-
|22
|0, 17, 21
| +3+1
|0.0171
|0.0175
|-
|22
|0, 5, 8
| +3+2
|0.0207
|0.0207
|-
|22
|0, 7, 11
| +3+2
|0.0184
|0.0184
|-
|22
|0, 9, 14
| +3+2
|0.0372
|0.0374
|-
|22
|0, 12, 18
| +3+2
|0.0088
|0.0089
|-
|22
|0, 14, 21
| +3+2
|0.0384
|0.0391
|-
|23
|0, 1, 2
| +1+1
|0.0135
|0.0135
|-
|23
|0, 2, 4
| +1+1
|0.0271
|0.0272
|-
|23
|0, 3, 6
| +1+1
|0.0408
|0.0410
|-
|23
|0, 5, 9
| +1+1
|0.0389
|0.0394
|-
|23
|0, 6, 11
| +1+1
|0.0076
|0.0077
|-
|23
|0, 7, 13
| +1+1
|0.0185
|0.0190
|-
|23
|0, 8, 14
| +1+1
|0.0345
|0.0357
|-
|23
|0, 8, 15
| +1+1
|0.0415
|0.0431
|-
|23
|0, 9, 16
| +1+1
|0.0051
|0.0053
|-
|23
|0, 10, 17
| +1+1
|0.0453
|0.0478
|-
|23
|0, 10, 18
| +1+1
|0.0210
|0.0223
|-
|23
|0, 11, 19
| +1+1
|0.0153
|0.0164
|-
|23
|0, 11, 20
| +1+1
|0.0448
|0.0484
|-
|23
|0, 12, 21
| +1+1
|0.0118
|0.0129
|-
|23
|0, 13, 22
| +1+1
|0.0176
|0.0195
|-
|23
|0, 1, 3
| +1+2
|0.0285
|0.0285
|-
|23
|0, 3, 8
| +1+2
|0.0389
|0.0391
|-
|23
|0, 4, 11
| +1+2
|0.0211
|0.0213
|-
|23
|0, 5, 13
| +1+2
|0.0163
|0.0166
|-
|23
|0, 6, 15
| +1+2
|0.0381
|0.0388
|-
|23
|0, 6, 16
| +1+2
|0.0385
|0.0393
|-
|23
|0, 7, 18
| +1+2
|0.0207
|0.0213
|-
|23
|0, 8, 20
| +1+2
|0.0105
|0.0109
|-
|23
|0, 9, 22
| +1+2
|0.0059
|0.0062
|-
|23
|0, 1, 4
| +1+3
|0.0435
|0.0435
|-
|23
|0, 3, 11
| +1+3
|0.0360
|0.0362
|-
|23
|0, 4, 14
| +1+3
|0.0230
|0.0232
|-
|23
|0, 5, 16
| +1+3
|0.0482
|0.0488
|-
|23
|0, 5, 17
| +1+3
|0.0271
|0.0275
|-
|23
|0, 6, 19
| +1+3
|0.0250
|0.0254
|-
|23
|0, 6, 20
| +1+3
|0.0403
|0.0412
|-
|23
|0, 7, 22
| +1+3
|0.0012
|0.0013
|-
|23
|0, 2, 3
| +2+1
|0.0253
|0.0253
|-
|23
|0, 7, 10
| +2+1
|0.0028
|0.0028
|-
|23
|0, 9, 13
| +2+1
|0.0428
|0.0433
|-
|23
|0, 10, 14
| +2+1
|0.0085
|0.0086
|-
|23
|0, 12, 17
| +2+1
|0.0392
|0.0401
|-
|23
|0, 13, 18
| +2+1
|0.0019
|0.0020
|-
|23
|0, 14, 19
| +2+1
|0.0309
|0.0320
|-
|23
|0, 16, 22
| +2+1
|0.0199
|0.0208
|-
|23
|0, 3, 7
| +2+3
|0.0134
|0.0134
|-
|23
|0, 8, 17
| +2+3
|0.0344
|0.0347
|-
|23
|0, 9, 19
| +2+3
|0.0146
|0.0148
|-
|23
|0, 10, 21
| +2+3
|0.0079
|0.0080
|-
|23
|0, 3, 4
| +3+1
|0.0367
|0.0367
|-
|23
|0, 7, 9
| +3+1
|0.0121
|0.0121
|-
|23
|0, 11, 14
| +3+1
|0.0036
|0.0037
|-
|23
|0, 12, 15
| +3+1
|0.0392
|0.0396
|-
|23
|0, 15, 19
| +3+1
|0.0338
|0.0344
|-
|23
|0, 16, 20
| +3+1
|0.0028
|0.0028
|-
|23
|0, 17, 21
| +3+1
|0.0249
|0.0255
|-
|23
|0, 5, 8
| +3+2
|0.0150
|0.0151
|-
|23
|0, 7, 11
| +3+2
|0.0107
|0.0107
|-
|23
|0, 9, 14
| +3+2
|0.0274
|0.0276
|-
|23
|0, 12, 18
| +3+2
|0.0210
|0.0213
|-
|23
|0, 14, 21
| +3+2
|0.0240
|0.0244
|-
|23
|0, 15, 22
| +3+2
|0.0325
|0.0331
|}
|}
=== Limitations ===
Interpreting simple JI chords as signatures/templates for delta-rational chords is reasonable, as the psychoacoustic effect of DR is more robust to detuning than that of JI for many people, but it is no panacea. If the temperament is too inaccurate, other inversions and voicings of a given JI chord will not have the DR signatures preserved acceptably, and compromising will make all of them inaccurate, though this of course depends on what error tolerance you prefer. This is because DR simplicity is only preserved by inversion and revoicing if the chord in question is low-complexity JI.
For example, take 0.0570 (the LSD error of 0-2\11-4\11 as +1+1, approximately equalized 7:8:9) as a somewhat arbitrary but reasonable upper limit of acceptable error. Consider Semaphore temperament (i.e. 2.3.7[14 & 19]). Using a gen of 260.346c, 0-679c-940c is a Semaphore tuning of 4:6:7 that is perfectly +2+1, but inverting the chord yields 0-260c-521c as our 6:7:8, with LSD error 0.0681; the 7:8:12 has an even higher error of 0.123. This is also evident by the fact that we had to use an extreme tuning of Semaphore, which has CWE generator 249.311c and CTE generator 248.126c. If we use the CTE generator, all three inversions are barely acceptable to unacceptable as simple-DR approximations: 4:6:7 has LSD error 0.0574 from +2+1, 6:7:8 has LSD error 0.0652 from +1+1, and 7:8:12 has LSD error 0.0570 from +1+4.


== Higher-order differences of frequency ==
== Higher-order differences of frequency ==
Retrieved from "https://en.xen.wiki/w/Delta-rational_chord"