# Fudging

(Redirected from Virtual tempering)

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]