Fudging

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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]