View source for Fudging

'''Fudging''', or virtual tempering, is the use of one [[just intonation]] interval to approximate another.

==Fudgers==
Below are listed fudging intervals (or '''fudgers''') which are all ratios between two 15-limit [[tonality diamond]] intervals (listed in the fifth column) which come within a comma of less than eight cents (listed in the fourth column) of a 15-limit tonality diamond interval (listed in the third column). If the [[comma]] is less than 1, the fudger is flat of the interval it approximates; if greater than 1, sharp of it.

===Up-fudge===

{| class="wikitable"
|-
! fudger
! cents
! approximately
! comma(s)
! (example) intervals
! name
|-
| [[242/225]]
| 126.098
| 15/14, 14/13
| 3388/3375, 1573/1575
| [15/11 22/15]
| 
|-
| [[27/25]]
| 133.238
| 14/13, 13/12
| 351/350, 324/325
| [10/9 6/5]
| large limma
|-
| 121/112
| 133.810
| 14/13, 13/12
| 1573/1568, 363/364
| [14/11 11/8]
| 
|-
| 225/208
| 136.010
| 14/13, 13/12
| 225/224, 675/676
| [16/15 15/13]
| 
|-
| [[49/45]]
| 147.428
| 12/11
| 539/540
| [15/14 7/6]
| swetismic neutral second
|-
| [[35/32]]
| 155.140
| 12/11
| 385/384
| [16/15 7/6]
| septimal neutral second
|-
| [[54/49]]
| 168.213
| 11/10
| 540/539
| [7/6 9/7]
| Zalzal's mujannab
|-
| 121/105
| 245.541
| 15/13
| 1573/1575
| [14/11 22/15]
| 
|-
| 140/121
| 252.504
| 15/13
| 364/363
| [11/10 14/11]
| 
|-
| [[81/70]]
| 252.680
| 15/13
| 351/350
| [10/9 9/7]
| 
|-
| [[75/64]]
| 274.582
| 7/6
| 225/224
| [16/15 5/4]
| 
|-
| 33/28
| 284.447
| 13/11
| 363/364
| [12/11 9/7]
| 
|-
| [[32/27]]
| 294.135
| 13/11
| 352/351
| [9/8 4/3]
| 
|-
| 128/105
| 342.905
| 11/9
| 384/385
| [7/4 32/15]
| 
|-
| [[60/49]]
| 350.617
| 11/9
| 540/539
| [7/6 10/7]
| 
|-
| [[49/40]]
| 351.338
| 11/9
| 441/440
| [8/7 7/5]
| 
|-
| [[100/81]]
| 364.807
| 16/13
| 325/324
| [9/5 20/9]
| 
|-
| 121/98
| 364.984
| 16/13
| 1573/1568
| [14/11 11/7]
| 
|-
| [[80/63]]
| 413.578
| 14/11
| 440/441
| [9/8 10/7]
| 
|-
| [[32/25]]
| 427.373
| 9/7
| 224/225
| [5/4 8/5]
| 
|-
| [[35/27]]
| 449.275
| 13/10
| 350/351
| [6/5 14/9]
| 
|-
| 100/77
| 452.484
| 13/10
| 1000/1001
| [11/10 10/7]
| 
|-
| [[49/36]]
| 533.742
| 15/11
| 539/540
| [8/7 14/9]
| 
|-
| 308/225
| 543.606
| 15/11, 11/8
| 3388/3375, 224/225
| [15/14 22/15]
| 
|-
| [[48/35]]
| 546.815
| 11/8
| 384/385
| [7/6 8/5]
| 
|-
| 112/81
| 561.006
| 18/13
| 728/729
| [9/8 14/9]
| 
|-
| 168/121
| 568.145
| 18/13
| 364/363
| [11/7 24/11]
| 
|-
| [[25/18]]
| 568.717
| 18/13
| 325/324
| [6/5 5/3]
| 
|-
| [[36/25]]
| 631.283
| 13/9
| 324/325
| [10/9 8/5]
| 
|-
| 121/84
| 631.855
| 13/9
| 363/364
| [12/11 11/7]
| 
|-
| 81/56
| 638.994
| 13/9
| 729/728
| [14/9 9/4]
| 
|-
| [[35/24]]
| 653.185
| 16/11
| 385/384
| [16/15 14/9]
| 
|-
| 143/98
| 654.194
| 16/11
| 1573/1568
| [14/13 11/7]
| 
|-
| 225/154
| 656.394
| 16/11, 22/15
| 225/224, 3375/3388
| [22/15 15/7]
| 
|-
| 77/50
| 747.516
| 20/13
| 1001/1000
| [10/7 11/5]
| 
|-
| [[54/35]]
| 750.725
| 20/13
| 351/350
| [10/9 12/7]
| 
|-
| [[25/16]]
| 772.627
| 14/9
| 225/224
| [16/15 5/3]
| 
|-
| 63/40
| 786.422
| 11/7
| 441/440
| [10/9 7/4]
| 
|-
| 196/121
| 835.016
| 13/8
| 1568/1573
| [11/7 28/11]
| 
|-
| 81/50
| 835.193
| 13/8
| 324/325
| [10/9 9/5]
| 
|-
| 80/49
| 848.662
| 18/11
| 440/441
| [7/5 16/7]
| 
|-
| [[49/30]]
| 849.383
| 18/11
| 539/540
| [15/14 7/4]
| 
|-
| [[105/64]]
| 857.095
| 18/11
| 385/384
| [16/15 7/4] 
| 
|-
| [[27/16]]
| 905.865
| 22/13
| 351/352
| [16/15 9/5]
| 
|-
| 56/33
| 915.553
| 22/13
| 364/363
| [15/14 20/11]
| 
|-
| [[128/75]]
| 925.418
| 12/7
| 224/225
| [5/4 32/15]
| 
|-
| [[140/81]]
| 947.320
| 26/15
| 350/351
| [9/7 20/9]
| 
|-
| 121/70
| 947.496
| 26/15
| 363/364
| [14/11 11/5]
| 
|-
| 210/121
| 954.459
| 26/15
| 1575/1573
| [22/15 28/11]
| 
|-
| 49/27
| 1031.787
| 20/11
| 539/540
| [9/7 7/3]
| 
|-
| [[64/35]]
| 1044.86
| 11/6
| 384/385
| [7/6 32/15]
| 
|-
| 90/49
| 1052.572
| 11/6
| 540/539
| [7/6 15/7]
| 
|-
| 416/225
| 1063.99
| 24/13, 13/7
| 676/675, 224/225
| [15/13 32/15]
| 
|-
| 224/121
| 1066.19
| 24/13, 13/7
| 364/363, 1568/1573
| [11/8 28/11]
| 
|-
| 50/27
| 1066.762
| 24/13, 13/7
| 325/324, 350/351
| [6/5 20/9]
| 
|-
| 117/110
| 106.806
| 16/15
| 351/352
| [10/9 13/11]
| 
|-
| 180/169
| 109.168
| 16/15
| 675/676
| [13/12 15/13]
| 
|-
| [[77/72]]
| 116.234
| 16/15, 15/14
| 385/384, 539/540
| [12/11 7/6]
| undecimal secor
|-
| [[28/25]]
| 196.198
| 9/8
| 224/225
| [15/14 6/5]
| 
|-
| [[224/195]]
| 240.030
| 15/13
| 224/225
| [15/14 16/13]
| 
|-
| 52/45
| 250.304
| 15/13
| 676/675
| [9/8 13/10]
| 
|-
| [[77/60]]
| 431.875
| 9/7
| 539/540
| [15/14 11/8]
| 
|-
| [[110/81]]
| 529.812
| 15/11
| 242/243
| [18/11 20/9]
| 
|-
| 160/117
| 541.876
| 15/11
| 352/351
| [9/8 20/13]
| 
|-
| 117/80
| 658.124
| 22/15
| 351/352
| [10/9 13/8]
| 
|-
| [[81/55]]
| 670.188
| 22/15
| 243/242
| [10/9 18/11]
| 
|-
| 120/77
| 768.125
| 14/9
| 540/539
| [11/10 12/7]
| 
|-
| 45/26
| 949.696
| 26/15
| 675/676
| [16/15 24/13]
| 
|-
| 195/112
| 959.970
| 26/15
| 225/224
| [16/15 13/7]
| 
|-
| [[25/14]]
| 1003.802
| 16/9
| 225/224
| [6/5 15/7]
| 
|-
| 144/77
| 1083.766
| 28/15, 15/8
| 540/539, 384/385
| [7/6 24/11]
| 
|-
| 169/90
| 1090.832
| 15/8
| 676/675
| [15/13 13/6]
| 
|-
| 220/117
| 1093.194
| 15/8
| 352/351
| [13/11 20/9]
| 
|}

===Neutral/ambiguous===

{| class="wikitable"
|-
! fudger
! cents
! approximately
! comma(s)
! (example) intervals
! names
|-
| 130/121
| 124.205
| 15/14, 14/13
| 364/363, 845/847
| [11/10 13/11]
| 
|-
| [[88/81]]
| 143.498
| 13/12, 12/11
| 352/351, 242/243
| [9/8 11/9]
| undecimal subtone
|-
| 99/91
| 145.874
| 13/12, 12/11
| 1188/1183, 363/364
| [13/11 9/7]
| 
|-
| [[256/225]]
| 223.463
| 8/7
| 224/225
| [15/8 32/15]
| 
|-
| 225/196
| 238.886
| 8/7
| 225/224
| [28/15 15/7]
| 
|-
| 200/169
| 291.572
| 13/11
| 2200/2197
| [13/10 20/13]
| 
|-
| 77/65
| 293.302
| 13/11
| 847/845
| [13/11 7/5]
| 
|-
| 108/91
| 296.511
| 13/11
| 1188/1183
| [13/12 9/7]
| 
|-
| [[27/22]]
| 354.547
| 11/9, 16/13
| 243/242, 351/352
| [10/9 15/11]
| 
|-
| 225/182
| 367.184
| 16/13
| 225/224
| [26/15 15/7]
| 
|-
| 286/225
| 415.308
| 14/11
| 1573/1575
| [15/13 22/15]
| 
|-
| 225/176
| 425.219
| 14/11
| 225/224
| [16/15 15/11]
| 
|-
| 135/104
| 451.651
| 13/10
| 675/676
| [16/15 18/13]
| 
|-
| 220/169
| 456.576
| 13/10
| 2200/2197
| [13/11 20/13]
| 
|-
| 176/135
| 459.139
| 13/10
| 352/351
| [9/8 22/15]
| 
|-
| 224/165
| 529.239
| 15/11
| 224/225
| [15/14 16/11]
| 
|-
| 143/105
| 534.751
| 15/11
| 1573/1575
| [14/13 22/15]
| 
|-
| 91/66
| 556.081
| 11/8, 18/13
| 364/363, 1183/1188
| [11/7 13/6]
| 
|-
| 135/98
| 554.527
| 11/8
| 540/539
| [14/9 15/7]
| 
|-
| 104/75
| 565.945
| 18/13
| 676/675
| [15/13 8/5]
| 
|-
| [[45/32]]
| 590.224
| 7/5
| 225/224
| [16/15 3/2]
| 
|-
| [[64/45]]
| 609.776
| 10/7
| 224/225
| [9/8 8/5]
| 
|-
| 75/52
| 634.055
| 13/9
| 675/676
| [16/15 20/13]
| 
|-
| 132/91
| 643.919
| 13/9, 16/11
| 1188/1183, 363/364
| [13/12 11/7]
| 
|-
| 196/135
| 645.473
| 16/11
| 539/540
| [15/14 14/9]
| 
|-
| 210/143
| 665.249
| 22/15
| 1575/1573
| [22/15 28/13]
| 
|-
| 165/112
| 670.761
| 22/15
| 225/224
| [16/15 11/7]
| 
|-
| 135/88
| 740.861
| 20/13
| 351/352
| [16/15 18/11]
| 
|-
| 169/110
| 743.424
| 20/13
| 2197/2200
| [20/13 26/11]
| 
|-
| 208/135
| 748.349
| 20/13
| 676/675
| [9/8 26/15]
| 
|-
| 352/225
| 774.781
| 11/7
| 224/225
| [15/11 32/15]
| 
|-
| 225/143
| 784.692
| 11/7
| 1575/1573
| [22/15 30/13]
| 
|-
| 364/225
| 832.816
| 13/8
| 224/225
| [15/14 26/15]
| 
|-
| 44/27
| 845.453
| 13/8, 18/11
| 352/351, 242/243
| [12/11 16/9]
| 
|-
| 91/54
| 903.489
| 22/13
| 1183/1188
| [9/7 13/6]
| 
|-
| 130/77
| 906.698
| 22/13
| 845/847
| [14/13 20/11]
| 
|-
| 169/100
| 908.428
| 22/13
| 2197/2200
| [20/13 13/5]
| 
|-
| 392/225
| 961.114
| 7/4
| 224/225
| [15/14 28/15]
| 
|-
| 225/128
| 976.537
| 7/4
| 225/224
| [16/15 15/8]
| 
|-
| 81/44
| 1056.502
| 11/6, 24/13
| 243/242, 351/352
| [11/9 9/4]
| 
|-
| 225/121
| 1073.902
| 13/7, 28/15
| 1575/1573, 3375/3388
| [22/15 30/11]
| 
|-
| 121/65
| 1075.795
| 13/7, 28/15
| 847/845, 363/364
| [13/11 11/5]
| 
|}

===Down-fudge===

{| class="wikitable"
|-
! fudger
! cents
! approximately
! commas
! (example) intervals
|-
| 128/117
| 155.562
| 12/11
| 352/351
| [9/8 16/13]
|-
| 169/154
| 160.911
| 11/10
| 845/847
| [14/13 13/11]
|-
| 100/91
| 163.274
| 11/10
| 1000/1001
| [13/10 10/7]
|-
| 182/165
| 169.767
| 11/10
| 364/363
| [15/14 13/11]
|-
| 72/65
| 177.069
| 10/9
| 324/325
| [13/12 6/5]
|-
| 195/176
| 177.478
| 10/9
| 351/352
| [16/15 13/11]
|-
| 49/44
| 186.334
| 10/9
| 441/440
| [8/7 14/11]
|-
| 39/35
| 187.343
| 10/9
| 351/350
| [14/13 6/5]
|-
| 135/121
| 189.543
| 10/9
| 243/242
| [11/9 15/11]
|-
| 121/108
| 196.771
| 9/8
| 242/243
| [12/11 11/9]
|-
| 55/49
| 199.980
| 9/8
| 440/441
| [14/11 10/7]
|-
| 91/81
| 201.534
| 9/8
| 728/729
| [9/7 13/9]
|-
| 169/150
| 206.473
| 9/8
| 676/675
| [15/13 13/10]
|-
| 44/39
| 208.835
| 9/8
| 352/351
| [13/12 11/9]
|-
| 154/135
| 227.965
| 8/7
| 539/540
| [15/14 11/9]
|-
| 63/55
| 235.104
| 8/7
| 441/440
| [10/9 14/11]
|-
| 55/48
| 235.677
| 8/7
| 385/384
| [16/15 11/9]
|-
| 121/104
| 262.108
| 7/6
| 363/364
| [13/11 11/8]
|-
| 64/55
| 262.368
| 7/6
| 384/385
| [5/4 16/11]
|-
| 90/77
| 270.080
| 7/6
| 540/539
| [11/10 9/7]
|-
| 117/100
| 271.810
| 7/6
| 351/350
| [10/9 13/10]
|-
| 198/169
| 274.173
| 7/6
| 1188/1183
| [13/11 18/13]
|-
| 140/117
| 310.702
| 6/5
| 350/351
| [9/7 20/13]
|-
| 77/64
| 320.144
| 6/5
| 385/384
| [8/7 11/8]
|-
| 65/54
| 320.976
| 6/5
| 325/324
| [6/5 13/9]
|-
| 135/112
| 323.353
| 6/5
| 225/224
| [16/15 9/7]
|-
| 39/32
| 342.483
| 11/9
| 351/352
| [16/15 13/10]
|-
| 56/45
| 378.602
| 5/4
| 224/225
| [15/14 4/3]
|-
| 81/65
| 380.979
| 5/4
| 324/325
| [10/9 18/13]
|-
| 96/77
| 381.811
| 5/4
| 384/385
| [7/6 16/11]
|-
| 169/135
| 388.877
| 5/4
| 676/675
| [15/13 13/9]
|-
| 33/26
| 412.745
| 14/11
| 363/364
| [13/12 11/8]
|-
| 143/112
| 423.020
| 14/11
| 1573/1568
| [14/13 11/8]
|-
| 216/169
| 424.810
| 14/11
| 1188/1183
| [13/12 18/13]
|-
| 169/132
| 427.782
| 9/7
| 1183/1188
| [22/13 13/6]
|-
| 50/39
| 430.145
| 9/7
| 350/351
| [6/5 20/13]
|-
| 104/81
| 432.708
| 9/7
| 728/729
| [9/8 13/9]
|-
| 165/128
| 439.587
| 9/7
| 385/384
| [16/15 11/8]
|-
| 156/121
| 439.847
| 9/7
| 364/363
| [22/13 24/11]
|-
| 117/88
| 493.120
| 4/3
| 351/352
| [11/9 13/8]
|-
| 121/91
| 493.282
| 4/3
| 363/364
| [13/11 11/7]
|-
| 225/169
| 495.482
| 4/3
| 675/676
| [26/15 30/13]
|-
| 162/121
| 505.184
| 4/3
| 243/242
| [11/9 18/11]
|-
| 75/56
| 505.757
| 4/3
| 225/224
| [16/15 10/7]
|-
| 196/143
| 545.806
| 11/8
| 1568/1573
| [11/7 28/13]
|-
| 169/121
| 578.419
| 7/5
| 845/847
| [22/13 26/11]
|-
| 88/63
| 578.582
| 7/5
| 440/441
| [9/8 11/7]
|-
| 200/143
| 580.782
| 7/5
| 1000/1001
| [11/10 20/13]
|-
| 108/77
| 585.721
| 7/5
| 540/539
| [7/6 18/11]
|-
| 77/54
| 614.279
| 10/7
| 539/540
| [12/11 14/9]
|-
| 143/100
| 619.218
| 10/7
| 1001/1000
| [20/13 11/5]
|-
| 63/44
| 621.418
| 10/7
| 441/440
| [8/7 18/11]
|-
| 242/169
| 621.581
| 10/7
| 847/845
| [13/11 22/13]
|-
| 72/49
| 666.258
| 22/15
| 540/539
| [7/6 12/7]
|-
| 112/75
| 694.243
| 3/2
| 224/225
| [15/14 8/5]
|-
| 121/81
| 694.816
| 3/2
| 242/243
| [18/11 22/9]
|-
| 338/225
| 704.518
| 3/2
| 676/675
| [15/13 26/15]
|-
| 182/121
| 706.718
| 3/2
| 364/363
| [11/7 26/11]
|-
| 176/117
| 706.880
| 3/2
| 352/351
| [9/8 22/13]
|-
| 121/78
| 760.153
| 14/9
| 363/364
| [12/11 22/13]
|-
| 256/165
| 760.413
| 14/9
| 384/385
| [11/8 32/15]
|-
| 81/52
| 767.292
| 14/9
| 729/728
| [13/9 9/4]
|-
| 39/25
| 769.855
| 14/9
| 351/350
| [10/9 26/15]
|-
| 264/169
| 772.218
| 14/9
| 1188/1183
| [13/12 22/13]
|-
| 169/108
| 775.190
| 11/7
| 1183/1188
| [18/13 13/6]
|-
| 224/143
| 776.980
| 11/7
| 1568/1573
| [11/8 28/13]
|-
| 52/33
| 787.255
| 11/7
| 364/363
| [11/10 26/15]
|-
| 270/169
| 811.123
| 8/5
| 675/676
| [13/9 30/13]
|-
| 77/48
| 818.189
| 8/5
| 385/384
| [12/11 7/4]
|-
| 130/81
| 819.021
| 8/5
| 325/324
| [18/13 20/9]
|-
| 45/28
| 821.398
| 8/5
| 225/224
| [16/15 12/7]
|-
| 64/39
| 857.517
| 18/11
| 352/351
| [13/12 16/9]
|-
| 224/135
| 876.647
| 5/3
| 224/225
| [15/14 16/9]
|-
| 108/65
| 879.024
| 5/3
| 324/325
| [13/12 9/5]
|-
| 128/77
| 879.856
| 5/3
| 384/385
| [11/8 16/7]
|-
| 117/70
| 889.298
| 5/3
| 351/350
| [14/13 9/5]
|-
| 169/99
| 925.827
| 12/7
| 1183/1188
| [18/13 26/11]
|-
| 200/117
| 928.190
| 12/7
| 350/351
| [13/10 20/9]
|-
| 77/45
| 929.920
| 12/7
| 539/540
| [15/14 11/6]
|-
| 55/32
| 937.632
| 12/7
| 385/384
| [16/15 11/6]
|-
| 208/121
| 937.892
| 12/7
| 364/363
| [11/8 26/11]
|-
| 96/55
| 964.323
| 7/4
| 384/385
| [11/9 32/15]
|-
| 110/63
| 964.896
| 7/4
| 440/441
| [14/11 20/9]
|-
| 135/77
| 972.035
| 7/4
| 540/539
| [11/9 15/7]
|-
| 39/22
| 991.165
| 16/9
| 351/352
| [11/9 13/6]
|-
| 300/169
| 993.527
| 16/9
| 675/676
| [13/10 30/13]
|-
| 162/91
| 998.466
| 16/9
| 729/728
| [13/9 18/7]
|-
| 98/55
| 1000.02
| 16/9
| 441/440
| [10/7 28/11]
|-
| 216/121
| 1003.229
| 16/9
| 243/242
| [11/9 24/11]
|-
| 242/135
| 1010.457
| 9/5
| 242/243
| [15/11 22/9]
|-
| 70/39
| 1012.657
| 9/5
| 350/351
| [6/5 28/13]
|-
| 88/49
| 1013.666
| 9/5
| 440/441
| [14/11 16/7]
|-
| 352/195
| 1022.522
| 9/5
| 352/351
| [13/11 32/15]
|-
| 65/36
| 1022.931
| 9/5
| 325/324
| [6/5 13/6]
|-
| 165/91
| 1030.233
| 20/11
| 363/364
| [13/11 15/7]
|-
| 91/50
| 1036.726
| 20/11
| 1001/1000
| [10/7 13/5]
|-
| 308/169
| 1039.089
| 20/11
| 847/845
| [13/11 28/13]
|-
| 117/64
| 1044.438
| 11/6
| 351/352
| [16/13 9/4]
|-
| 182/99
| 1054.126
| 11/6, 24/13
| 364/363, 1183/1188
| [9/7 26/11]
|}

[[Category:15-odd-limit]]
[[Category:Method]]
[[Category:Terms]]
[[Category:Lists of intervals]]
[[Category:Just intonation]]

== Limit-raising and limit-lowering fudgers ==
A limit-raising fudger is a p prime limit interval which approximates to a q prime limit consonance, with p&lt;q. An example is 100/77, which is 1001/1000 (1.7 cents) flat of [[13/10]], and which arises in 11-limit scales as the interval between 11/10 and 10/7, giving 13-limit harmony "for free", so to speak. 

A limit-lowering fudger is an interval such as the marvelous fourth, 75/56, which is 225/224 (7.7 cents) sharp of 4/3, and which arises very often in 7-limit JI scales as the interval between 16/15 and 10/7, 7/5 and 15/8, 8/5 and 15/7, and 28/15 and 5/2, giving a 3-limit interval approximated in the 7-limit.

== Tempering fudging commas ==
By [[tempering out]] the fudging commas, the error in the fudging may be distributed evenly. Since 225/224, 385/384 and 540/539 occur so commonly as fudging commas, marvel tempering in particular is often an excellent means to introduce smooth 7- and 11-limit harmonies into 5- or 7- limit scales. Adding 441/440 to that list results in miracle tempering, another excellent smoothing option. 

However, indiscriminate tempering can lead to problems. For example, [[112/75]], 121/81, 338/225, 182/121 and 176/117 are all fudged fifths, with commas [[676/675]], [[364/363]], [[352/351]], [[243/242]] and [[225/224]]. Tempering them all out leads to 34d tempering, a rather crude system by comparison.