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Fudging, or virtual tempering, is the use of one just intonation interval to approximate another. Below are listed fudging intervals ("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.

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. 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.

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

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

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
675/676
13/9
[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

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]