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''Fudging'', or virtual tempering, is the use of one [[Just_intonation|just intonation]] interval to approximate another. Below are listed fudging intervals ("fudgers") which are all ratios between two 15-limit [[Tonality_diamond|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.
'''Fudging''', or virtual tempering, is the use of one [[just intonation]] interval to approximate another.


A limit-raising fudger is a p prime limit interval which approximates to a q prime limit consonance, with p<q. An example is 100/77, which is 1001/1000 (1.7 cents) flat of [[13/10|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.
== 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.
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.
 
==Fudging Intervals==
 
===Up-fudge===


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


Line 534: Line 530:
{| class="wikitable"
{| class="wikitable"
|-
|-
| | fudger
! fudger
| | cents
! cents
| | approximately
! approximately
| | comma(s)
! comma(s)
| | (example) intervals
! (example) intervals
| | names
! names
|-
|-
| | 130/121
| 130/121
| | 124.205
| 124.205
| | 15/14, 14/13
| 15/14, 14/13
| | 364/363, 845/847
| 364/363, 845/847
| | [11/10 13/11]
| [11/10 13/11]
| |  
|  
|-
|-
| | [[88/81|88/81]]
| [[88/81]]
| | 143.498
| 143.498
| | 13/12, 12/11
| 13/12, 12/11
| | 352/351, 242/243
| 352/351, 242/243
| | [9/8 11/9]
| [9/8 11/9]
| | undecimal subtone
| undecimal subtone
|-
|-
| | 99/91
| 99/91
| | 145.874
| 145.874
| | 13/12, 12/11
| 13/12, 12/11
| | 1188/1183, 363/364
| 1188/1183, 363/364
| | [13/11 9/7]
| [13/11 9/7]
| |  
|  
|-
|-
| | 256/225
| [[256/225]]
| | 223.463
| 223.463
| | 8/7
| 8/7
| | 224/225
| 224/225
| | [15/8 32/15]
| [15/8 32/15]
| |  
|  
|-
|-
| | 225/196
| 225/196
| | 238.886
| 238.886
| | 8/7
| 8/7
| | 225/224
| 225/224
| | [28/15 15/7]
| [28/15 15/7]
| |  
|  
|-
|-
| | 200/169
| 200/169
| | 291.572
| 291.572
| | 13/11
| 13/11
| | 2200/2197
| 2200/2197
| | [13/10 20/13]
| [13/10 20/13]
| |  
|  
|-
|-
| | 77/65
| 77/65
| | 293.302
| 293.302
| | 13/11
| 13/11
| | 847/845
| 847/845
| | [13/11 7/5]
| [13/11 7/5]
| |  
|  
|-
|-
| | 108/91
| 108/91
| | 296.511
| 296.511
| | 13/11
| 13/11
| | 1188/1183
| 1188/1183
| | [13/12 9/7]
| [13/12 9/7]
| |  
|  
|-
|-
| | 27/22
| [[27/22]]
| | 354.547
| 354.547
| | 11/9, 16/13
| 11/9, 16/13
| | 243/242, 351/352
| 243/242, 351/352
| | [10/9 15/11]
| [10/9 15/11]
| |  
|  
|-
|-
| | 225/182
| 225/182
| | 367.184
| 367.184
| | 16/13
| 16/13
| | 225/224
| 225/224
| | [26/15 15/7]
| [26/15 15/7]
| |  
|  
|-
|-
| | 286/225
| 286/225
| | 415.308
| 415.308
| | 14/11
| 14/11
| | 1573/1575
| 1573/1575
| | [15/13 22/15]
| [15/13 22/15]
| |  
|  
|-
|-
| | 225/176
| 225/176
| | 425.219
| 425.219
| | 14/11
| 14/11
| | 225/224
| 225/224
| | [16/15 15/11]
| [16/15 15/11]
| |  
|  
|-
|-
| | 135/104
| 135/104
| | 451.651
| 451.651
| | 13/10
| 13/10
| | 675/676
| 675/676
| | [16/15 18/13]
| [16/15 18/13]
| |  
|  
|-
|-
| | 220/169
| 220/169
| | 456.576
| 456.576
| | 13/10
| 13/10
| | 2200/2197
| 2200/2197
| | [13/11 20/13]
| [13/11 20/13]
| |  
|  
|-
|-
| | 176/135
| 176/135
| | 459.139
| 459.139
| | 13/10
| 13/10
| | 352/351
| 352/351
| | [9/8 22/15]
| [9/8 22/15]
| |  
|  
|-
|-
| | 224/165
| 224/165
| | 529.239
| 529.239
| | 15/11
| 15/11
| | 224/225
| 224/225
| | [15/14 16/11]
| [15/14 16/11]
| |  
|  
|-
|-
| | 143/105
| 143/105
| | 534.751
| 534.751
| | 15/11
| 15/11
| | 1573/1575
| 1573/1575
| | [14/13 22/15]
| [14/13 22/15]
| |  
|  
|-
|-
| | 91/66
| 91/66
| | 556.081
| 556.081
| | 11/8, 18/13
| 11/8, 18/13
| | 364/363, 1183/1188
| 364/363, 1183/1188
| | [11/7 13/6]
| [11/7 13/6]
| |  
|  
|-
|-
| | 135/98
| 135/98
| | 554.527
| 554.527
| | 11/8
| 11/8
| | 540/539
| 540/539
| | [14/9 15/7]
| [14/9 15/7]
| |  
|  
|-
|-
| | 104/75
| 104/75
| | 565.945
| 565.945
| | 18/13
| 18/13
| | 676/675
| 676/675
| | [15/13 8/5]
| [15/13 8/5]
| |  
|  
|-
|-
| | 45/32
| [[45/32]]
| | 590.224
| 590.224
| | 7/5
| 7/5
| | 225/224
| 225/224
| | [16/15 3/2]
| [16/15 3/2]
| |  
|  
|-
|-
| | 64/45
| [[64/45]]
| | 609.776
| 609.776
| | 10/7
| 10/7
| | 224/225
| 224/225
| | [9/8 8/5]
| [9/8 8/5]
| |  
|  
|-
|-
| | 75/52
| 75/52
| | 634.055
| 634.055
| | 675/676
| 13/9
| | 13/9
| 675/676
| | [16/15 20/13]
| [16/15 20/13]
| |  
|  
|-
|-
| | 132/91
| 132/91
| | 643.919
| 643.919
| | 13/9, 16/11
| 13/9, 16/11
| | 1188/1183, 363/364
| 1188/1183, 363/364
| | [13/12 11/7]
| [13/12 11/7]
| |  
|  
|-
|-
| | 196/135
| 196/135
| | 645.473
| 645.473
| | 16/11
| 16/11
| | 539/540
| 539/540
| | [15/14 14/9]
| [15/14 14/9]
| |  
|  
|-
|-
| | 210/143
| 210/143
| | 665.249
| 665.249
| | 22/15
| 22/15
| | 1575/1573
| 1575/1573
| | [22/15 28/13]
| [22/15 28/13]
| |  
|  
|-
|-
| | 165/112
| 165/112
| | 670.761
| 670.761
| | 22/15
| 22/15
| | 225/224
| 225/224
| | [16/15 11/7]
| [16/15 11/7]
| |  
|  
|-
|-
| | 135/88
| 135/88
| | 740.861
| 740.861
| | 20/13
| 20/13
| | 351/352
| 351/352
| | [16/15 18/11]
| [16/15 18/11]
| |  
|  
|-
|-
| | 169/110
| 169/110
| | 743.424
| 743.424
| | 20/13
| 20/13
| | 2197/2200
| 2197/2200
| | [20/13 26/11]
| [20/13 26/11]
| |  
|  
|-
|-
| | 208/135
| 208/135
| | 748.349
| 748.349
| | 20/13
| 20/13
| | 676/675
| 676/675
| | [9/8 26/15]
| [9/8 26/15]
| |  
|  
|-
|-
| | 352/225
| 352/225
| | 774.781
| 774.781
| | 11/7
| 11/7
| | 224/225
| 224/225
| | [15/11 32/15]
| [15/11 32/15]
| |  
|  
|-
|-
| | 225/143
| 225/143
| | 784.692
| 784.692
| | 11/7
| 11/7
| | 1575/1573
| 1575/1573
| | [22/15 30/13]
| [22/15 30/13]
| |  
|  
|-
|-
| | 364/225
| 364/225
| | 832.816
| 832.816
| | 13/8
| 13/8
| | 224/225
| 224/225
| | [15/14 26/15]
| [15/14 26/15]
| |  
|  
|-
|-
| | 44/27
| 44/27
| | 845.453
| 845.453
| | 13/8, 18/11
| 13/8, 18/11
| | 352/351, 242/243
| 352/351, 242/243
| | [12/11 16/9]
| [12/11 16/9]
| |  
|  
|-
|-
| | 91/54
| 91/54
| | 903.489
| 903.489
| | 22/13
| 22/13
| | 1183/1188
| 1183/1188
| | [9/7 13/6]
| [9/7 13/6]
| |  
|  
|-
|-
| | 130/77
| 130/77
| | 906.698
| 906.698
| | 22/13
| 22/13
| | 845/847
| 845/847
| | [14/13 20/11]
| [14/13 20/11]
| |  
|  
|-
|-
| | 169/100
| 169/100
| | 908.428
| 908.428
| | 22/13
| 22/13
| | 2197/2200
| 2197/2200
| | [20/13 13/5]
| [20/13 13/5]
| |  
|  
|-
|-
| | 392/225
| 392/225
| | 961.114
| 961.114
| | 7/4
| 7/4
| | 224/225
| 224/225
| | [15/14 28/15]
| [15/14 28/15]
| |  
|  
|-
|-
| | 225/128
| 225/128
| | 976.537
| 976.537
| | 7/4
| 7/4
| | 225/224
| 225/224
| | [16/15 15/8]
| [16/15 15/8]
| |  
|  
|-
|-
| | 81/44
| 81/44
| | 1056.502
| 1056.502
| | 11/6, 24/13
| 11/6, 24/13
| | 243/242, 351/352
| 243/242, 351/352
| | [11/9 9/4]
| [11/9 9/4]
| |  
|  
|-
|-
| | 225/121
| 225/121
| | 1073.902
| 1073.902
| | 13/7, 28/15
| 13/7, 28/15
| | 1575/1573, 3375/3388
| 1575/1573, 3375/3388
| | [22/15 30/11]
| [22/15 30/11]
| |  
|  
|-
|-
| | 121/65
| 121/65
| | 1075.795
| 1075.795
| | 13/7, 28/15
| 13/7, 28/15
| | 847/845, 363/364
| 847/845, 363/364
| | [13/11 11/5]
| [13/11 11/5]
| |  
|  
|}
|}


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