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
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 | 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
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] |
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<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.