Fudging
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Original Wikitext content:
//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|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) || intervals || name ||
|| 242/225 || 126.098 || 15/14 14/13 || 3388/3375 1573/1575 || [15/11 22/15] || || ||
|| [[xenharmonic/27_25|27/25]] || 133.238 || 14/13 13/12 || 351/350 324/325 || [10/9 6/5] [5/3 9/5] || large limma ||
|| 121/112 || 133.810 || 14/13 13/12 || 1573/1568 363/364 || [14/11 11/8] [16/11 11/7] || || ||
|| 225/208 || 136.010 || 14/13 13/12 || 225/224 675/676 || [16/15 15/13] [26/15 15/8] || || ||
|| [[xenharmonic/49_45|49/45]] || 147.428 || 12/11 || 539/540 || [15/14 7/6] [9/7 7/5] [10/7 14/9] [12/7 28/15] || swetismic neutral second ||
|| [[xenharmonic/35_32|35/32]] || 155.140 || 12/11 || 385/384 || [16/15 7/6] [8/7 5/4] [8/5 7/4] [12/7 15/8] || septimal neutral second ||
|| [[xenharmonic/54_49|54/49]] || 168.213 || 11/10 || 540/539 || [7/6 9/7] [14/9 12/7] || Zalzal's mujannab ||
|| 121/105 || 245.541 || 15/13 || 1573/1575 || [14/11 22/15] [15/11 11/7] || || ||
|| 140/121 || 252.504 || 15/13 || 364/363 || [11/10 14/11] [11/7 20/11] || || ||
|| 81/70 || 252.680 || 15/13 || 351/350 || [10/9 9/7] [14/9 9/5] || || ||
|| 75/64 || 274.582 || 7/6 || 225/224 || [16/15 5/4] [8/5 15/8] || || ||
|| 33/28 || 284.447 || 13/11 || 363/364 || [12/11 9/7] [7/6 11/8] [14/11 3/2] [4/3 11/7]
[16/11 12/7] [14/9 11/6] [20/11 15/7] [28/15 11/5] || || ||
|| 32/27 || 294.135 || 13/11 || 352/351 || [9/8 4/3] [3/2 16/9] [9/5 32/15] [15/8 20/9] || || ||
|| 128/105 || 342.905 || 11/9 || 384/385 || [7/4 32/15] [15/8 16/7] || || ||
|| 60/49 || 350.617 || 11/9 || 540/539 || [7/6 10/7] [7/5 12/7] [7/4 15/7] [28/15 16/7] || || ||
|| 49/40 || 351.338 || 11/9 || 441/440 || [8/7 7/5] [10/7 7/4] || || ||
|| 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] [7/5 16/9] [7/4 20/9] [9/5 16/7] || || ||
|| 32/25 || 427.373 || 9/7 || 224/225 || [5/4 8/5] [5/3 32/15] [15/8 12/5] || || ||
|| 35/27 || 449.275 || 13/10 || 350/351 || [6/5 14/9] [9/7 5/3] [12/7 20/9] [9/5 7/3] || || ||
|| 100/77 || 452.484 || 13/10 || 1000/1001 || [11/10 10/7] [7/5 20/11] || || ||
|| 49/36 || 533.742 || 15/11 || 539/540 || [8/7 14/9] [9/7 7/4] [12/7 7/3] || || ||
|| 308/225 || 543.606 || 15/11 11/8 || 3388/3375 224/225 || [15/14 22/15] [15/11 28/15] || || ||
|| 48/35 || 546.815 || 11/8 || 384/385 || [7/6 8/5] [5/4 12/7] [14/9 32/15] [5/3 16/7] [7/4 12/5]
[15/8 18/7] || || ||
|| 112/81 || 561.006 || 18/13 || 728/729 || [9/8 14/9] [9/7 16/9] || || ||
|| 168/121 || 568.145 || 18/13 || 364/363 || [11/7 24/11] [11/6 28/11] || || ||
|| 25/18 || 568.717 || 18/13 || 325/324 || [6/5 5/3] [8/5 20/9] [9/5 5/2] || || ||
|| 36/25 || 631.283 || 13/9 || 324/325 || [10/9 8/5] [5/4 9/5] [5/3 12/5] || || ||
|| 121/84 || 631.855 || 13/9 || 363/364 || [12/11 11/7] [14/11 11/6] || || ||
|| 81/56 || 638.994 || 13/9 || 729/728 || [14/9 9/4] [16/9 18/7] || || ||
|| 35/24 || 653.185 || 16/11 || 385/384 || [16/15 14/9] [8/7 5/3] [6/5 7/4] [9/7 15/8] [8/5 7/3]
[12/7 5/2] || || ||
|| 143/98 || 654.194 || 16/11 || 1573/1568 || [14/13 11/7] [14/11 13/7] || || ||
|| 225/154 || 656.394 || 16/11 22/15 || 225/224 3375/3388 || [22/15 15/7] [28/15 30/11] || || ||
|| 77/50 || 747.516 || 20/13 || 1001/1000 || [10/7 11/5] [20/11 14/5] || || ||
|| 54/35 || 750.725 || 20/13 || 351/350 || [10/9 12/7] [7/6 9/5] [14/9 12/5] [5/3 18/7] || || ||
|| 25/16 || 772.627 || 14/9 || 225/224 || [16/15 5/3] [6/5 15/8] [8/5 5/2] || || ||
|| 63/40 || 786.422 || 11/7 || 441/440 || [10/9 7/4] [8/7 9/5] [10/7 9/4] [16/9 14/5] || || ||
|| 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] [7/4 20/7] || || ||
|| 49/30 || 849.383 || 18/11 || 539/540 || [15/14 7/4] [8/7 28/15] [10/7 7/3] [12/7 14/5] || || ||
|| 105/64 || 857.095 || 18/11 || 385/384 || [16/15 7/4] [8/7 15/8] || || ||
|| 27/16 || 905.865 || 22/13 || 351/352 || [16/15 9/5] [10/9 15/8] [4/3 9/4] [16/9 3] || || ||
|| 56/33 || 915.553 || 22/13 || 364/363 || [15/14 20/11] [11/10 28/15] [9/7 24/11] [11/8 7/3]
[3/2 28/11] [11/7 8/3] [12/7 32/11] [11/6 28/9] || || ||
|| 128/75 || 925.418 || 12/7 || 224/225 || [5/4 32/15] [15/8 16/5] || || ||
|| 140/81 || 947.320 || 26/15 || 350/351 || [9/7 20/9] [9/5 28/9] || || ||
|| 121/70 || 947.496 || 26/15 || 363/364 || [14/11 11/5] [20/11 22/7] || || ||
|| 210/121 || 954.459 || 26/15 || 1575/1573 || [22/15 28/11] [11/7 30/11] || || ||
|| 49/27 || 1031.787 || 20/11 || 539/540 || [9/7 7/3] [12/7 28/9] || || ||
|| 64/35 || 1044.86 || 11/6 || 384/385 || [7/6 32/15] [5/4 16/7] [7/4 16/5] [15/8 24/7] || || ||
|| 90/49 || 1052.572 || 11/6 || 540/539 || [7/6 15/7] [7/5 18/7] [14/9 20/7] [28/15 24/7] || || ||
|| 416/225 || 1063.99 || 24/13 13/7 || 676/675 224/225 || [15/13 32/15] [15/8 52/15] || || ||
|| 224/121 || 1066.19 || 24/13 13/7 || 364/363 1568/1573 || [11/8 28/11] [11/7 32/11] || || ||
|| 50/27 || 1066.762 || 24/13 13/7 || 325/324 350/351 || [6/5 20/9] [9/5 10/3] || || ||
|| 117/110 || 106.806 || 16/15 || 351/352 || [10/9 13/11] [11/9 13/10] [20/13 18/11] [22/13 9/5] || || ||
|| 180/169 || 109.168 || 16/15 || 675/676 || [13/12 15/13] [13/10 18/13] [13/9 20/13]
[26/15 24/13] || || ||
|| [[xenharmonic/77_72|77/72]] || 116.234 || 16/15 15/14 || 385/384 539/540 || [12/11 7/6] [8/7 11/9] [9/7 11/8] [16/11 14/9]
[18/11 7/4] [12/7 11/6] || undecimal secor ||
|| 28/25 || 196.198 || 9/8 || 224/225 || [15/14 6/5] [5/4 7/5] [10/7 8/5] [5/3 28/15] || || ||
|| 224/195 || 240.030 || 15/13 || 224/225 || [15/14 16/13] [13/8 28/15] [13/7 32/15] [15/8 28/13] || || ||
|| 52/45 || 250.304 || 15/13 || 676/675 || [9/8 13/10] [15/13 4/3] [5/4 13/9] [18/13 8/5]
[3/2 26/15] [20/13 16/9] [24/13 32/15] [15/8 13/6] || || ||
|| 77/60 || 431.875 || 9/7 || 539/540 || [15/14 11/8] [12/11 7/5] [8/7 22/15] [15/11 7/4]
[10/7 11/6] [16/11 28/15] [12/7 11/5] [20/11 7/3] || || ||
|| 110/81 || 529.812 || 15/11 || 242/243 || [18/11 20/9] [9/5 22/9] || || ||
|| 160/117 || 541.876 || 15/11 || 352/351 || [9/8 20/13] [13/10 16/9] [13/8 20/9] [9/5 32/13] || || ||
|| 117/80 || 658.124 || 22/15 || 351/352 || [10/9 13/8] [16/13 9/5] [20/13 9/4] [16/9 13/5] || || ||
|| 81/55 || 670.188 || 22/15 || 243/242 || [10/9 18/11] [11/9 9/5] || || ||
|| 120/77 || 768.125 || 14/9 || 540/539 || [11/10 12/7] [7/6 20/11] [11/8 15/7] [7/5 24/11]
[22/15 16/7] [7/4 30/11] [11/6 20/7] [28/15 32/11] || || ||
|| 45/26 || 949.696 || 26/15 || 675/676 || [16/15 24/13] [13/12 15/8] [13/10 9/4] [4/3 30/13]
[13/9 5/2] [8/5 36/13] [26/15 3] [16/9 40/13] || || ||
|| 195/112 || 959.970 || 26/15 || 225/224 || [16/15 13/7] [14/13 15/8] [16/13 15/7] [28/15 13/4] || || ||
|| 25/14 || 1003.802 || 16/9 || 225/224 || [6/5 15/7] [7/5 5/2] [8/5 20/7] [28/15 10/3] || || ||
|| 144/77 || 1083.766 || 28/15 15/8 || 540/539 384/385 || [7/6 24/11] [11/9 16/7] [11/8 18/7] [14/9 32/11]
[7/4 36/11] [11/6 24/7} || || ||
|| 169/90 || 1090.832 || 15/8 || 676/675 || [15/13 13/6] [18/13 13/5] [20/13 26/9] [24/13 52/15] || || ||
|| 220/117 || 1093.194 || 15/8 || 352/351 || [13/11 20/9] [13/10 22/9] [18/11 40/13] [9/5 44/13] || || ||
===Neutral/ambiguous===
|| fudger || cents || approximately || comma(s) || intervals || names ||
|| 130/121 || 124.205 || 15/14 14/13 || 364/363 845/847 || [11/10 13/11] [22/13 20/11] || ||
|| [[xenharmonic/88_81|88/81]] || 143.498 || 13/12 12/11 || 352/351 242/243 || [9/8 11/9] [18/11 16/9] || undecimal subtone ||
|| 99/91 || 145.874 || 13/12 12/11 || 1188/1183 363/364 || [13/11 9/7] [14/11 18/13] [13/9 11/7] [14/9 22/13] || ||
|| 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] [10/7 22/13] [20/11 28/13] [13/7 11/5] || ||
|| 108/91 || 296.511 || 13/11 || 1188/1183 || [13/12 9/7] [7/6 18/13] [13/9 12/7] [14/9 24/13] || ||
|| 27/22 || 354.547 || 11/9 16/13 || 243/242 351/352 || [10/9 15/11] [11/9 3/2] [4/3 18/11] [22/15 9/5] [16/9 24/11]
[11/6 9/4] || ||
|| 225/182 || 367.184 || 16/13 || 225/224 || [26/15 15/7] [28/15 30/13] || ||
|| 286/225 || 415.308 || 14/11 || 1573/1575 || [15/13 22/15] [15/11 26/15] || ||
|| 225/176 || 425.219 || 14/11 || 225/224 || [16/15 15/11] [22/15 15/8] || ||
|| 135/104 || 451.651 || 13/10 || 675/676 || [16/15 18/13] [13/9 15/8] [26/15 9/4] [16/9 30/13] || ||
|| 220/169 || 456.576 || 13/10 || 2200/2197 || [13/11 20/13] [13/10 22/13] || ||
|| 176/135 || 459.139 || 13/10 || 352/351 || [9/8 22/15] [15/11 16/9] [18/11 32/15] [15/8 22/9] || ||
|| 224/165 || 529.239 || 15/11 || 224/225 || [15/14 16/11] [11/8 28/15] [11/7 32/15] [15/8 28/11] || ||
|| 143/105 || 534.751 || 15/11 || 1573/1575 || [14/13 22/15] [15/13 11/7] [14/11 26/15] [15/11 13/7] || ||
|| 91/66 || 556.081 || 11/8 18/13 || 364/363 1183/1188 || [11/7 13/6] [22/13 7/3] [12/7 26/11] [24/13 28/11] || ||
|| 135/98 || 554.527 || 11/8 || 540/539 || [14/9 15/7] [28/15 18/7] || ||
|| 104/75 || 565.945 || 18/13 || 676/675 || [15/13 8/5] [5/4 26/15] [20/13 32/15]
[15/8 13/5] || ||
|| 45/32 || 590.224 || 7/5 || 225/224 || [16/15 3/2] [4/3 15/8] [8/5 9/4] [16/9 5/2] || ||
|| 64/45 || 609.776 || 10/7 || 224/225 || [9/8 8/5] [5/4 16/9] [3/2 32/15] [15/8 8/3] || ||
|| 75/52 || 634.055 || 675/676 || 13/9 || [16/15 20/13] [13/10 15/8] [8/5 30/13] [26/15 5/2] || ||
|| 132/91 || 643.919 || 13/9 16/11 || 1188/1183 363/364 || [13/12 11/7] [7/6 22/13] [13/11 12/7] [14/11 24/13] || ||
|| 196/135 || 645.473 || 16/11 || 539/540 || [15/14 14/9] [9/7 28/15] || ||
|| 210/143 || 665.249 || 22/15 || 1575/1573 || [22/15 28/13] [11/7 30/13] [26/15 28/11] [13/7 30/11] || ||
|| 165/112 || 670.761 || 22/15 || 225/224 || [16/15 11/7] [14/11 15/8] [16/11 15/7] [28/15 11/4] || ||
|| 135/88 || 740.861 || 20/13 || 351/352 || [16/15 18/11] [11/9 15/8] [22/15 9/4] [16/9 30/11] || ||
|| 169/110 || 743.424 || 20/13 || 2197/2200 || [20/13 26/11] [22/13 13/5] || ||
|| 208/135 || 748.349 || 20/13 || 676/675 || [9/8 26/15] [15/13 16/9] [18/13 32/15] [15/8 26/9] || ||
|| 352/225 || 774.781 || 11/7 || 224/225 || [15/11 32/15] [15/8 44/15] || ||
|| 225/143 || 784.692 || 11/7 || 1575/1573 || [22/15 30/13] [26/15 30/11] || ||
|| 364/225 || 832.816 || 13/8 || 224/225 || [15/14 26/15] [15/13 28/15] || ||
|| 44/27 || 845.453 || 13/8 18/11 || 352/351 242/243 || [12/11 16/9] [9/8 11/6] [15/11 20/9] [3/2 22/9] [18/11 8/3]
[9/5 44/15] || ||
|| 91/54 || 903.489 || 22/13 || 1183/1188 || [9/7 13/6] [18/13 7/3] [12/7 26/9] [24/13 28/9] || ||
|| 130/77 || 906.698 || 22/13 || 845/847 || [14/13 20/11] [11/10 13/7] [7/5 26/11] [22/13 20/7] || ||
|| 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] [16/9 36/11] || ||
|| 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] [20/11 44/13] || ||
===Down-fudge===
|| fudger || cents || approximately || commas || intervals ||
|| 128/117 || 155.562 || 12/11 || 352/351 || [9/8 16/13] [13/8 16/9] ||
|| 169/154 || 160.911 || 11/10 || 845/847 || [14/13 13/11] [22/13 13/7] ||
|| 100/91 || 163.274 || 11/10 || 1000/1001 || [13/10 10/7] [7/5 20/13] ||
|| 182/165 || 169.767 || 11/10 || 364/363 || [15/14 13/11] [15/13 14/11] [11/7 26/15] [22/13 28/15] ||
|| 72/65 || 177.069 || 10/9 || 324/325 || [13/12 6/5] [10/9 16/13] [5/4 18/13] [13/9 8/5] [13/8 9/5] [5/3 24/13] ||
|| 195/176 || 177.478 || 10/9 || 351/352 || [16/15 13/11] [16/13 15/11] [22/15 13/8] [22/13 15/8] ||
|| 49/44 || 186.334 || 10/9 || 441/440 || [8/7 14/11] [11/7 7/4] ||
|| 39/35 || 187.343 || 10/9 || 351/350 || [14/13 6/5] [15/13 9/7] [7/6 13/10] [20/13 12/7] [14/9 26/15] [5/3 13/7] ||
|| 135/121 || 189.543 || 10/9 || 243/242 || [11/9 15/11] [22/15 18/11] ||
|| 121/108 || 196.771 || 9/8 || 242/243 || [12/11 11/9] [18/11 11/6] ||
|| 55/49 || 199.980 || 9/8 || 440/441 || [14/11 10/7] [7/5 11/7] ||
|| 91/81 || 201.534 || 9/8 || 728/729 || [9/7 13/9] [18/13 14/9] ||
|| 169/150 || 206.473 || 9/8 || 676/675 || [15/13 13/10] [20/13 26/15] ||
|| 44/39 || 208.835 || 9/8 || 352/351 || [13/12 11/9] [12/11 16/13] [13/11 4/3] [13/10 22/15] [15/11 20/13] [3/2 22/13]
[13/8 11/6] [18/11 24/13] ||
|| 154/135 || 227.965 || 8/7 || 539/540 || [15/14 11/9] [9/7 22/15] [15/11 14/9] [18/11 28/15] ||
|| 63/55 || 235.104 || 8/7 || 441/440 || [10/9 14/11] [11/9 7/5] [10/7 18/11] [11/7 9/5] ||
|| 55/48 || 235.677 || 8/7 || 385/384 || [16/15 11/9] [12/11 5/4] [6/5 11/8] [16/11 5/3] [8/5 11/6] [18/11 15/8] ||
|| 121/104 || 262.108 || 7/6 || 363/364 || [13/11 11/8] [16/11 22/13] ||
|| 64/55 || 262.368 || 7/6 || 384/385 || [5/4 16/11] [11/8 8/5] [11/6 32/15] [15/8 24/11] ||
|| 90/77 || 270.080 || 7/6 || 540/539 || [11/10 9/7] [7/6 15/11] [11/9 10/7] [7/5 18/11] [22/15 12/7] [14/9 20/11] [11/6 15/7] [28/15 24/11] ||
|| 117/100 || 271.810 || 7/6 || 351/350 || [10/9 13/10] [20/13 9/5] ||
|| 198/169 || 274.173 || 7/6 || 1188/1183 || [13/11 18/13] [13/9 22/13] ||
|| 140/117 || 310.702 || 6/5 || 350/351 || [9/7 20/13] [13/10 14/9] [9/5 28/13] [13/7 20/9] ||
|| 77/64 || 320.144 || 6/5 || 385/384 || [8/7 11/8] [16/11 7/4] ||
|| 65/54 || 320.976 || 6/5 || 325/324 || [6/5 13/9] [18/13 5/3] [9/5 13/6] [24/13 20/9] ||
|| 135/112 || 323.353 || 6/5 || 225/224 || [16/15 9/7] [14/9 15/8] [16/9 15/7] [28/15 9/4] ||
|| 39/32 || 342.483 || 11/9 || 351/352 || [16/15 13/10] [16/13 3/2] [4/3 13/8] [20/13 15/8] [16/9 13/6] [24/13 9/4] ||
|| 56/45 || 378.602 || 5/4 || 224/225 || [15/14 4/3] [9/8 7/5] [5/4 14/9] [9/7 8/5] [10/7 16/9] [3/2 28/15] [12/7 32/15]
[15/8 7/3] ||
|| 81/65 || 380.979 || 5/4 || 324/325 || [10/9 18/13] [13/9 9/5] ||
|| 96/77 || 381.811 || 5/4 || 384/385 || [7/6 16/11] [11/8 12/7] [7/4 24/11] [11/6 16/7] ||
|| 169/135 || 388.877 || 5/4 || 676/675 || [15/13 13/9] [18/13 26/15] ||
|| 33/26 || 412.745 || 14/11 || 363/364 || [13/12 11/8] [12/11 18/13] [13/11 3/2] [4/3 22/13] [13/9 11/6] [16/11 24/13]
[26/15 11/5] [20/11 30/13] ||
|| 143/112 || 423.020 || 14/11 || 1573/1568 || [14/13 11/8] [16/13 11/7] [14/11 13/8] [16/11 13/7] ||
|| 216/169 || 424.810 || 14/11 || 1188/1183 || [13/12 18/13] [13/9 24/13] ||
|| 169/132 || 427.782 || 9/7 || 1183/1188 || [22/13 13/6] [24/13 26/11] ||
|| 50/39 || 430.145 || 9/7 || 350/351 || [6/5 20/13] [13/10 5/3] [26/15 20/9] [9/5 30/13] ||
|| 104/81 || 432.708 || 9/7 || 728/729 || [9/8 13/9] [18/13 16/9] ||
|| 165/128 || 439.587 || 9/7 || 385/384 || [16/15 11/8] [16/11 15/8] ||
|| 156/121 || 439.847 || 9/7 || 364/363 || [22/13 24/11] [11/6 26/11] ||
|| 117/88 || 493.120 || 4/3 || 351/352 || [11/9 13/8] [16/13 18/11] [22/13 9/4] [16/9 26/11] ||
|| 121/91 || 493.282 || 4/3 || 363/364 || [13/11 11/7] [14/11 22/13] ||
|| 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] [7/5 15/8] [8/5 15/7] [28/15 5/2] ||
|| 196/143 || 545.806 || 11/8 || 1568/1573 || [11/7 28/13] [13/7 28/11] ||
|| 169/121 || 578.419 || 7/5 || 845/847 || [22/13 26/11] ||
|| 88/63 || 578.582 || 7/5 || 440/441 || [9/8 11/7] [14/11 16/9] [18/11 16/7] [7/4 22/9] ||
|| 200/143 || 580.782 || 7/5 || 1000/1001 || [11/10 20/13] [13/10 20/11] ||
|| 108/77 || 585.721 || 7/5 || 540/539 || [7/6 18/11] [11/9 12/7] [14/9 24/11] [11/6 18/7] ||
|| 77/54 || 614.279 || 10/7 || 539/540 || [12/11 14/9] [9/7 11/6] [18/11 7/3] [12/7 22/9] ||
|| 143/100 || 619.218 || 10/7 || 1001/1000 || [20/13 11/5] [20/11 13/5] ||
|| 63/44 || 621.418 || 10/7 || 441/440 || [8/7 18/11] [11/9 7/4] [11/7 9/4] [16/9 28/11] ||
|| 242/169 || 621.581 || 10/7 || 847/845 || [13/11 22/13] ||
|| 72/49 || 666.258 || 22/15 || 540/539 || [7/6 12/7] [14/9 16/7] [7/4 18/7] ||
|| 112/75 || 694.243 || 3/2 || 224/225 || [15/14 8/5] [5/4 28/15] [10/7 32/15] [15/8 14/5] ||
|| 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] [22/13 28/11] ||
|| 176/117 || 706.880 || 3/2 || 352/351 || [9/8 22/13] [13/11 16/9] [13/8 22/9] [18/11 32/13] ||
|| 121/78 || 760.153 || 14/9 || 363/364 || [12/11 22/13] [13/11 11/6] ||
|| 256/165 || 760.413 || 14/9 || 384/385 || [11/8 32/15] [15/8 32/11] ||
|| 81/52 || 767.292 || 14/9 || 729/728 || [13/9 9/4] [16/9 36/13] ||
|| 39/25 || 769.855 || 14/9 || 351/350 || [10/9 26/15] [15/13 9/5] [20/13 12/5] [5/3 13/5] ||
|| 264/169 || 772.218 || 14/9 || 1188/1183 || [13/12 22/13] [13/11 24/13] ||
|| 169/108 || 775.190 || 11/7 || 1183/1188 || [18/13 13/6] [24/13 26/9] ||
|| 224/143 || 776.980 || 11/7 || 1568/1573 || [11/8 28/13] [11/7 32/13] [13/8 28/11] [13/7 32/11] ||
|| 52/33 || 787.255 || 11/7 || 364/363 || [11/10 26/15] [15/13 20/11] [11/8 13/6] [18/13 24/11] [3/2 26/11] [22/13 8/3]
[11/6 26/9] [24/13 32/11] ||
|| 270/169 || 811.123 || 8/5 || 675/676 || [13/9 30/13] [26/15 36/13] ||
|| 77/48 || 818.189 || 8/5 || 385/384 || [12/11 7/4] [8/7 11/6] [16/11 7/3] [12/7 11/4] ||
|| 130/81 || 819.021 || 8/5 || 325/324 || [18/13 20/9] [9/5 26/9] ||
|| 45/28 || 821.398 || 8/5 || 225/224 || [16/15 12/7] [7/6 15/8] [4/3 15/7] [7/5 9/4] [14/9 5/2] [8/5 18/7] [16/9 20/7] [28/15 3] ||
|| 64/39 || 857.517 || 18/11 || 352/351 || [13/12 16/9] [9/8 24/13] [13/10 32/15] [3/2 32/13] [13/8 8/3] [15/8 40/13] ||
|| 224/135 || 876.647 || 5/3 || 224/225 || [15/14 16/9] [9/8 28/15] [9/7 32/15] [15/8 28/9] ||
|| 108/65 || 879.024 || 5/3 || 324/325 || [13/12 9/5] [10/9 24/13] [13/9 12/5] [5/3 36/13] ||
|| 128/77 || 879.856 || 5/3 || 384/385 || [11/8 16/7] [7/4 32/11] ||
|| 117/70 || 889.298 || 5/3 || 351/350 || [14/13 9/5] [10/9 13/7] [20/13 18/7] [14/9 13/5] ||
|| 169/99 || 925.827 || 12/7 || 1183/1188 || [18/13 26/11] [22/13 26/9] ||
|| 200/117 || 928.190 || 12/7 || 350/351 || [13/10 20/9] [9/5 40/13] ||
|| 77/45 || 929.920 || 12/7 || 539/540 || [15/14 11/6] [12/11 28/15] [9/7 11/5] [15/11 7/3] [10/7 22/9] [18/11 14/5] [12/7 44/15] [20/11 28/9] ||
|| 55/32 || 937.632 || 12/7 || 385/384 || [16/15 11/6] [12/11 15/8] [16/11 5/2] [8/5 11/4] ||
|| 208/121 || 937.892 || 12/7 || 364/363 || [11/8 26/11] [22/13 32/11] ||
|| 96/55 || 964.323 || 7/4 || 384/385 || [11/9 32/15] [5/4 24/11] [11/8 12/5] [5/3 32/11] [11/6 16/5] [15/8 36/11] ||
|| 110/63 || 964.896 || 7/4 || 440/441 || [14/11 20/9] [7/5 22/9] [18/11 20/7] [9/5 22/7] ||
|| 135/77 || 972.035 || 7/4 || 540/539 || [11/9 15/7] [22/15 18/7] [14/9 30/11] [28/15 36/11] ||
|| 39/22 || 991.165 || 16/9 || 351/352 || [11/9 13/6] [16/13 24/11] [4/3 26/11] [22/15 13/5] [20/13 30/11] [22/13 3] [11/6 13/4] [24/13 36/11] ||
|| 300/169 || 993.527 || 16/9 || 675/676 || [13/10 30/13] [26/15 40/13] ||
|| 162/91 || 998.466 || 16/9 || 729/728 || [13/9 18/7] [14/9 36/13] ||
|| 98/55 || 1000.02 || 16/9 || 441/440 || [10/7 28/11] [11/7 14/5] ||
|| 216/121 || 1003.229 || 16/9 || 243/242 || [11/9 24/11] [11/6 36/11] ||
|| 242/135 || 1010.457 || 9/5 || 242/243 || [15/11 22/9] [18/11 44/15] ||
|| 70/39 || 1012.657 || 9/5 || 350/351 || [6/5 28/13] [9/7 30/13] [13/10 7/3] [12/7 40/13] [26/15 28/9] [13/7 10/3] ||
|| 88/49 || 1013.666 || 9/5 || 440/441 || [14/11 16/7] [7/4 22/7] ||
|| 352/195 || 1022.522 || 9/5 || 352/351 || [13/11 32/15] [15/11 32/13] [13/8 44/15] [15/8 44/13] ||
|| 65/36 || 1022.931 || 9/5 || 325/324 || [6/5 13/6] [16/13 20/9] [18/13 5/2] [8/5 26/9] [9/5 13/4] [24/13 10/3] ||
|| 165/91 || 1030.233 || 20/11 || 363/364 || [13/11 15/7] [14/11 30/13] [26/15 22/7] [28/15 44/13] ||
|| 91/50 || 1036.726 || 20/11 || 1001/1000 || [10/7 13/5] [20/13 14/5] ||
|| 308/169 || 1039.089 || 20/11 || 847/845 || [13/11 28/13] [13/7 44/13] ||
|| 117/64 || 1044.438 || 11/6 || 351/352 || [16/13 9/4] [16/9 13/4] ||
|| 182/99 || 1054.126 || 11/6 24/13 || 364/363 1183/1188 || [9/7 26/11] [18/13 28/11] [11/7 26/9] [22/13 28/9] ||Original HTML content:
<html><head><title>Fudging</title></head><body><em>Fudging</em>, or virtual tempering, is the use of one <a class="wiki_link" href="/just%20intonation">just intonation</a> interval to approximate another. Below are listed fudging intervals ("fudgers") which are all ratios between two 15-limit <a class="wiki_link" href="/tonality%20diamond">tonality diamond</a> 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.<br />
<br />
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 <a class="wiki_link" href="/13_10">13/10</a>, 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.<br />
<br />
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.<br />
<br />
<!-- ws:start:WikiTextHeadingRule:0:<h2> --><h2 id="toc0"><a name="x-Fudging Intervals"></a><!-- ws:end:WikiTextHeadingRule:0 -->Fudging Intervals</h2>
<!-- ws:start:WikiTextHeadingRule:2:<h3> --><h3 id="toc1"><a name="x-Fudging Intervals-Up-fudge"></a><!-- ws:end:WikiTextHeadingRule:2 -->Up-fudge</h3>
<table class="wiki_table">
<tr>
<td>fudger<br />
</td>
<td>cents<br />
</td>
<td>approximately<br />
</td>
<td>comma(s)<br />
</td>
<td>intervals<br />
</td>
<td>name<br />
</td>
</tr>
<tr>
<td>242/225<br />
</td>
<td>126.098<br />
</td>
<td>15/14 14/13<br />
</td>
<td>3388/3375 1573/1575<br />
</td>
<td>[15/11 22/15]<br />
</td>
<td><br />
</td>
<td><br />
</td>
</tr>
<tr>
<td><a class="wiki_link" href="http://xenharmonic.wikispaces.com/27_25">27/25</a><br />
</td>
<td>133.238<br />
</td>
<td>14/13 13/12<br />
</td>
<td>351/350 324/325<br />
</td>
<td>[10/9 6/5] [5/3 9/5]<br />
</td>
<td>large limma<br />
</td>
</tr>
<tr>
<td>121/112<br />
</td>
<td>133.810<br />
</td>
<td>14/13 13/12<br />
</td>
<td>1573/1568 363/364<br />
</td>
<td>[14/11 11/8] [16/11 11/7]<br />
</td>
<td><br />
</td>
<td><br />
</td>
</tr>
<tr>
<td>225/208<br />
</td>
<td>136.010<br />
</td>
<td>14/13 13/12<br />
</td>
<td>225/224 675/676<br />
</td>
<td>[16/15 15/13] [26/15 15/8]<br />
</td>
<td><br />
</td>
<td><br />
</td>
</tr>
<tr>
<td><a class="wiki_link" href="http://xenharmonic.wikispaces.com/49_45">49/45</a><br />
</td>
<td>147.428<br />
</td>
<td>12/11<br />
</td>
<td>539/540<br />
</td>
<td>[15/14 7/6] [9/7 7/5] [10/7 14/9] [12/7 28/15]<br />
</td>
<td>swetismic neutral second<br />
</td>
</tr>
<tr>
<td><a class="wiki_link" href="http://xenharmonic.wikispaces.com/35_32">35/32</a><br />
</td>
<td>155.140<br />
</td>
<td>12/11<br />
</td>
<td>385/384<br />
</td>
<td>[16/15 7/6] [8/7 5/4] [8/5 7/4] [12/7 15/8]<br />
</td>
<td>septimal neutral second<br />
</td>
</tr>
<tr>
<td><a class="wiki_link" href="http://xenharmonic.wikispaces.com/54_49">54/49</a><br />
</td>
<td>168.213<br />
</td>
<td>11/10<br />
</td>
<td>540/539<br />
</td>
<td>[7/6 9/7] [14/9 12/7]<br />
</td>
<td>Zalzal's mujannab<br />
</td>
</tr>
<tr>
<td>121/105<br />
</td>
<td>245.541<br />
</td>
<td>15/13<br />
</td>
<td>1573/1575<br />
</td>
<td>[14/11 22/15] [15/11 11/7]<br />
</td>
<td><br />
</td>
<td><br />
</td>
</tr>
<tr>
<td>140/121<br />
</td>
<td>252.504<br />
</td>
<td>15/13<br />
</td>
<td>364/363<br />
</td>
<td>[11/10 14/11] [11/7 20/11]<br />
</td>
<td><br />
</td>
<td><br />
</td>
</tr>
<tr>
<td>81/70<br />
</td>
<td>252.680<br />
</td>
<td>15/13<br />
</td>
<td>351/350<br />
</td>
<td>[10/9 9/7] [14/9 9/5]<br />
</td>
<td><br />
</td>
<td><br />
</td>
</tr>
<tr>
<td>75/64<br />
</td>
<td>274.582<br />
</td>
<td>7/6<br />
</td>
<td>225/224<br />
</td>
<td>[16/15 5/4] [8/5 15/8]<br />
</td>
<td><br />
</td>
<td><br />
</td>
</tr>
<tr>
<td>33/28<br />
</td>
<td>284.447<br />
</td>
<td>13/11<br />
</td>
<td>363/364<br />
</td>
<td>[12/11 9/7] [7/6 11/8] [14/11 3/2] [4/3 11/7]<br />
[16/11 12/7] [14/9 11/6] [20/11 15/7] [28/15 11/5]<br />
</td>
<td><br />
</td>
<td><br />
</td>
</tr>
<tr>
<td>32/27<br />
</td>
<td>294.135<br />
</td>
<td>13/11<br />
</td>
<td>352/351<br />
</td>
<td>[9/8 4/3] [3/2 16/9] [9/5 32/15] [15/8 20/9]<br />
</td>
<td><br />
</td>
<td><br />
</td>
</tr>
<tr>
<td>128/105<br />
</td>
<td>342.905<br />
</td>
<td>11/9<br />
</td>
<td>384/385<br />
</td>
<td>[7/4 32/15] [15/8 16/7]<br />
</td>
<td><br />
</td>
<td><br />
</td>
</tr>
<tr>
<td>60/49<br />
</td>
<td>350.617<br />
</td>
<td>11/9<br />
</td>
<td>540/539<br />
</td>
<td>[7/6 10/7] [7/5 12/7] [7/4 15/7] [28/15 16/7]<br />
</td>
<td><br />
</td>
<td><br />
</td>
</tr>
<tr>
<td>49/40<br />
</td>
<td>351.338<br />
</td>
<td>11/9<br />
</td>
<td>441/440<br />
</td>
<td>[8/7 7/5] [10/7 7/4]<br />
</td>
<td><br />
</td>
<td><br />
</td>
</tr>
<tr>
<td>100/81<br />
</td>
<td>364.807<br />
</td>
<td>16/13<br />
</td>
<td>325/324<br />
</td>
<td>[9/5 20/9]<br />
</td>
<td><br />
</td>
<td><br />
</td>
</tr>
<tr>
<td>121/98<br />
</td>
<td>364.984<br />
</td>
<td>16/13<br />
</td>
<td>1573/1568<br />
</td>
<td>[14/11 11/7]<br />
</td>
<td><br />
</td>
<td><br />
</td>
</tr>
<tr>
<td>80/63<br />
</td>
<td>413.578<br />
</td>
<td>14/11<br />
</td>
<td>440/441<br />
</td>
<td>[9/8 10/7] [7/5 16/9] [7/4 20/9] [9/5 16/7]<br />
</td>
<td><br />
</td>
<td><br />
</td>
</tr>
<tr>
<td>32/25<br />
</td>
<td>427.373<br />
</td>
<td>9/7<br />
</td>
<td>224/225<br />
</td>
<td>[5/4 8/5] [5/3 32/15] [15/8 12/5]<br />
</td>
<td><br />
</td>
<td><br />
</td>
</tr>
<tr>
<td>35/27<br />
</td>
<td>449.275<br />
</td>
<td>13/10<br />
</td>
<td>350/351<br />
</td>
<td>[6/5 14/9] [9/7 5/3] [12/7 20/9] [9/5 7/3]<br />
</td>
<td><br />
</td>
<td><br />
</td>
</tr>
<tr>
<td>100/77<br />
</td>
<td>452.484<br />
</td>
<td>13/10<br />
</td>
<td>1000/1001<br />
</td>
<td>[11/10 10/7] [7/5 20/11]<br />
</td>
<td><br />
</td>
<td><br />
</td>
</tr>
<tr>
<td>49/36<br />
</td>
<td>533.742<br />
</td>
<td>15/11<br />
</td>
<td>539/540<br />
</td>
<td>[8/7 14/9] [9/7 7/4] [12/7 7/3]<br />
</td>
<td><br />
</td>
<td><br />
</td>
</tr>
<tr>
<td>308/225<br />
</td>
<td>543.606<br />
</td>
<td>15/11 11/8<br />
</td>
<td>3388/3375 224/225<br />
</td>
<td>[15/14 22/15] [15/11 28/15]<br />
</td>
<td><br />
</td>
<td><br />
</td>
</tr>
<tr>
<td>48/35<br />
</td>
<td>546.815<br />
</td>
<td>11/8<br />
</td>
<td>384/385<br />
</td>
<td>[7/6 8/5] [5/4 12/7] [14/9 32/15] [5/3 16/7] [7/4 12/5]<br />
[15/8 18/7]<br />
</td>
<td><br />
</td>
<td><br />
</td>
</tr>
<tr>
<td>112/81<br />
</td>
<td>561.006<br />
</td>
<td>18/13<br />
</td>
<td>728/729<br />
</td>
<td>[9/8 14/9] [9/7 16/9]<br />
</td>
<td><br />
</td>
<td><br />
</td>
</tr>
<tr>
<td>168/121<br />
</td>
<td>568.145<br />
</td>
<td>18/13<br />
</td>
<td>364/363<br />
</td>
<td>[11/7 24/11] [11/6 28/11]<br />
</td>
<td><br />
</td>
<td><br />
</td>
</tr>
<tr>
<td>25/18<br />
</td>
<td>568.717<br />
</td>
<td>18/13<br />
</td>
<td>325/324<br />
</td>
<td>[6/5 5/3] [8/5 20/9] [9/5 5/2]<br />
</td>
<td><br />
</td>
<td><br />
</td>
</tr>
<tr>
<td>36/25<br />
</td>
<td>631.283<br />
</td>
<td>13/9<br />
</td>
<td>324/325<br />
</td>
<td>[10/9 8/5] [5/4 9/5] [5/3 12/5]<br />
</td>
<td><br />
</td>
<td><br />
</td>
</tr>
<tr>
<td>121/84<br />
</td>
<td>631.855<br />
</td>
<td>13/9<br />
</td>
<td>363/364<br />
</td>
<td>[12/11 11/7] [14/11 11/6]<br />
</td>
<td><br />
</td>
<td><br />
</td>
</tr>
<tr>
<td>81/56<br />
</td>
<td>638.994<br />
</td>
<td>13/9<br />
</td>
<td>729/728<br />
</td>
<td>[14/9 9/4] [16/9 18/7]<br />
</td>
<td><br />
</td>
<td><br />
</td>
</tr>
<tr>
<td>35/24<br />
</td>
<td>653.185<br />
</td>
<td>16/11<br />
</td>
<td>385/384<br />
</td>
<td>[16/15 14/9] [8/7 5/3] [6/5 7/4] [9/7 15/8] [8/5 7/3]<br />
[12/7 5/2]<br />
</td>
<td><br />
</td>
<td><br />
</td>
</tr>
<tr>
<td>143/98<br />
</td>
<td>654.194<br />
</td>
<td>16/11<br />
</td>
<td>1573/1568<br />
</td>
<td>[14/13 11/7] [14/11 13/7]<br />
</td>
<td><br />
</td>
<td><br />
</td>
</tr>
<tr>
<td>225/154<br />
</td>
<td>656.394<br />
</td>
<td>16/11 22/15<br />
</td>
<td>225/224 3375/3388<br />
</td>
<td>[22/15 15/7] [28/15 30/11]<br />
</td>
<td><br />
</td>
<td><br />
</td>
</tr>
<tr>
<td>77/50<br />
</td>
<td>747.516<br />
</td>
<td>20/13<br />
</td>
<td>1001/1000<br />
</td>
<td>[10/7 11/5] [20/11 14/5]<br />
</td>
<td><br />
</td>
<td><br />
</td>
</tr>
<tr>
<td>54/35<br />
</td>
<td>750.725<br />
</td>
<td>20/13<br />
</td>
<td>351/350<br />
</td>
<td>[10/9 12/7] [7/6 9/5] [14/9 12/5] [5/3 18/7]<br />
</td>
<td><br />
</td>
<td><br />
</td>
</tr>
<tr>
<td>25/16<br />
</td>
<td>772.627<br />
</td>
<td>14/9<br />
</td>
<td>225/224<br />
</td>
<td>[16/15 5/3] [6/5 15/8] [8/5 5/2]<br />
</td>
<td><br />
</td>
<td><br />
</td>
</tr>
<tr>
<td>63/40<br />
</td>
<td>786.422<br />
</td>
<td>11/7<br />
</td>
<td>441/440<br />
</td>
<td>[10/9 7/4] [8/7 9/5] [10/7 9/4] [16/9 14/5]<br />
</td>
<td><br />
</td>
<td><br />
</td>
</tr>
<tr>
<td>196/121<br />
</td>
<td>835.016<br />
</td>
<td>13/8<br />
</td>
<td>1568/1573<br />
</td>
<td>[11/7 28/11]<br />
</td>
<td><br />
</td>
<td><br />
</td>
</tr>
<tr>
<td>81/50<br />
</td>
<td>835.193<br />
</td>
<td>13/8<br />
</td>
<td>324/325<br />
</td>
<td>[10/9 9/5]<br />
</td>
<td><br />
</td>
<td><br />
</td>
</tr>
<tr>
<td>80/49<br />
</td>
<td>848.662<br />
</td>
<td>18/11<br />
</td>
<td>440/441<br />
</td>
<td>[7/5 16/7] [7/4 20/7]<br />
</td>
<td><br />
</td>
<td><br />
</td>
</tr>
<tr>
<td>49/30<br />
</td>
<td>849.383<br />
</td>
<td>18/11<br />
</td>
<td>539/540<br />
</td>
<td>[15/14 7/4] [8/7 28/15] [10/7 7/3] [12/7 14/5]<br />
</td>
<td><br />
</td>
<td><br />
</td>
</tr>
<tr>
<td>105/64<br />
</td>
<td>857.095<br />
</td>
<td>18/11<br />
</td>
<td>385/384<br />
</td>
<td>[16/15 7/4] [8/7 15/8]<br />
</td>
<td><br />
</td>
<td><br />
</td>
</tr>
<tr>
<td>27/16<br />
</td>
<td>905.865<br />
</td>
<td>22/13<br />
</td>
<td>351/352<br />
</td>
<td>[16/15 9/5] [10/9 15/8] [4/3 9/4] [16/9 3]<br />
</td>
<td><br />
</td>
<td><br />
</td>
</tr>
<tr>
<td>56/33<br />
</td>
<td>915.553<br />
</td>
<td>22/13<br />
</td>
<td>364/363<br />
</td>
<td>[15/14 20/11] [11/10 28/15] [9/7 24/11] [11/8 7/3]<br />
[3/2 28/11] [11/7 8/3] [12/7 32/11] [11/6 28/9]<br />
</td>
<td><br />
</td>
<td><br />
</td>
</tr>
<tr>
<td>128/75<br />
</td>
<td>925.418<br />
</td>
<td>12/7<br />
</td>
<td>224/225<br />
</td>
<td>[5/4 32/15] [15/8 16/5]<br />
</td>
<td><br />
</td>
<td><br />
</td>
</tr>
<tr>
<td>140/81<br />
</td>
<td>947.320<br />
</td>
<td>26/15<br />
</td>
<td>350/351<br />
</td>
<td>[9/7 20/9] [9/5 28/9]<br />
</td>
<td><br />
</td>
<td><br />
</td>
</tr>
<tr>
<td>121/70<br />
</td>
<td>947.496<br />
</td>
<td>26/15<br />
</td>
<td>363/364<br />
</td>
<td>[14/11 11/5] [20/11 22/7]<br />
</td>
<td><br />
</td>
<td><br />
</td>
</tr>
<tr>
<td>210/121<br />
</td>
<td>954.459<br />
</td>
<td>26/15<br />
</td>
<td>1575/1573<br />
</td>
<td>[22/15 28/11] [11/7 30/11]<br />
</td>
<td><br />
</td>
<td><br />
</td>
</tr>
<tr>
<td>49/27<br />
</td>
<td>1031.787<br />
</td>
<td>20/11<br />
</td>
<td>539/540<br />
</td>
<td>[9/7 7/3] [12/7 28/9]<br />
</td>
<td><br />
</td>
<td><br />
</td>
</tr>
<tr>
<td>64/35<br />
</td>
<td>1044.86<br />
</td>
<td>11/6<br />
</td>
<td>384/385<br />
</td>
<td>[7/6 32/15] [5/4 16/7] [7/4 16/5] [15/8 24/7]<br />
</td>
<td><br />
</td>
<td><br />
</td>
</tr>
<tr>
<td>90/49<br />
</td>
<td>1052.572<br />
</td>
<td>11/6<br />
</td>
<td>540/539<br />
</td>
<td>[7/6 15/7] [7/5 18/7] [14/9 20/7] [28/15 24/7]<br />
</td>
<td><br />
</td>
<td><br />
</td>
</tr>
<tr>
<td>416/225<br />
</td>
<td>1063.99<br />
</td>
<td>24/13 13/7<br />
</td>
<td>676/675 224/225<br />
</td>
<td>[15/13 32/15] [15/8 52/15]<br />
</td>
<td><br />
</td>
<td><br />
</td>
</tr>
<tr>
<td>224/121<br />
</td>
<td>1066.19<br />
</td>
<td>24/13 13/7<br />
</td>
<td>364/363 1568/1573<br />
</td>
<td>[11/8 28/11] [11/7 32/11]<br />
</td>
<td><br />
</td>
<td><br />
</td>
</tr>
<tr>
<td>50/27<br />
</td>
<td>1066.762<br />
</td>
<td>24/13 13/7<br />
</td>
<td>325/324 350/351<br />
</td>
<td>[6/5 20/9] [9/5 10/3]<br />
</td>
<td><br />
</td>
<td><br />
</td>
</tr>
<tr>
<td>117/110<br />
</td>
<td>106.806<br />
</td>
<td>16/15<br />
</td>
<td>351/352<br />
</td>
<td>[10/9 13/11] [11/9 13/10] [20/13 18/11] [22/13 9/5]<br />
</td>
<td><br />
</td>
<td><br />
</td>
</tr>
<tr>
<td>180/169<br />
</td>
<td>109.168<br />
</td>
<td>16/15<br />
</td>
<td>675/676<br />
</td>
<td>[13/12 15/13] [13/10 18/13] [13/9 20/13]<br />
[26/15 24/13]<br />
</td>
<td><br />
</td>
<td><br />
</td>
</tr>
<tr>
<td><a class="wiki_link" href="http://xenharmonic.wikispaces.com/77_72">77/72</a><br />
</td>
<td>116.234<br />
</td>
<td>16/15 15/14<br />
</td>
<td>385/384 539/540<br />
</td>
<td>[12/11 7/6] [8/7 11/9] [9/7 11/8] [16/11 14/9]<br />
[18/11 7/4] [12/7 11/6]<br />
</td>
<td>undecimal secor<br />
</td>
</tr>
<tr>
<td>28/25<br />
</td>
<td>196.198<br />
</td>
<td>9/8<br />
</td>
<td>224/225<br />
</td>
<td>[15/14 6/5] [5/4 7/5] [10/7 8/5] [5/3 28/15]<br />
</td>
<td><br />
</td>
<td><br />
</td>
</tr>
<tr>
<td>224/195<br />
</td>
<td>240.030<br />
</td>
<td>15/13<br />
</td>
<td>224/225<br />
</td>
<td>[15/14 16/13] [13/8 28/15] [13/7 32/15] [15/8 28/13]<br />
</td>
<td><br />
</td>
<td><br />
</td>
</tr>
<tr>
<td>52/45<br />
</td>
<td>250.304<br />
</td>
<td>15/13<br />
</td>
<td>676/675<br />
</td>
<td>[9/8 13/10] [15/13 4/3] [5/4 13/9] [18/13 8/5]<br />
[3/2 26/15] [20/13 16/9] [24/13 32/15] [15/8 13/6]<br />
</td>
<td><br />
</td>
<td><br />
</td>
</tr>
<tr>
<td>77/60<br />
</td>
<td>431.875<br />
</td>
<td>9/7<br />
</td>
<td>539/540<br />
</td>
<td>[15/14 11/8] [12/11 7/5] [8/7 22/15] [15/11 7/4]<br />
[10/7 11/6] [16/11 28/15] [12/7 11/5] [20/11 7/3]<br />
</td>
<td><br />
</td>
<td><br />
</td>
</tr>
<tr>
<td>110/81<br />
</td>
<td>529.812<br />
</td>
<td>15/11<br />
</td>
<td>242/243<br />
</td>
<td>[18/11 20/9] [9/5 22/9]<br />
</td>
<td><br />
</td>
<td><br />
</td>
</tr>
<tr>
<td>160/117<br />
</td>
<td>541.876<br />
</td>
<td>15/11<br />
</td>
<td>352/351<br />
</td>
<td>[9/8 20/13] [13/10 16/9] [13/8 20/9] [9/5 32/13]<br />
</td>
<td><br />
</td>
<td><br />
</td>
</tr>
<tr>
<td>117/80<br />
</td>
<td>658.124<br />
</td>
<td>22/15<br />
</td>
<td>351/352<br />
</td>
<td>[10/9 13/8] [16/13 9/5] [20/13 9/4] [16/9 13/5]<br />
</td>
<td><br />
</td>
<td><br />
</td>
</tr>
<tr>
<td>81/55<br />
</td>
<td>670.188<br />
</td>
<td>22/15<br />
</td>
<td>243/242<br />
</td>
<td>[10/9 18/11] [11/9 9/5]<br />
</td>
<td><br />
</td>
<td><br />
</td>
</tr>
<tr>
<td>120/77<br />
</td>
<td>768.125<br />
</td>
<td>14/9<br />
</td>
<td>540/539<br />
</td>
<td>[11/10 12/7] [7/6 20/11] [11/8 15/7] [7/5 24/11]<br />
[22/15 16/7] [7/4 30/11] [11/6 20/7] [28/15 32/11]<br />
</td>
<td><br />
</td>
<td><br />
</td>
</tr>
<tr>
<td>45/26<br />
</td>
<td>949.696<br />
</td>
<td>26/15<br />
</td>
<td>675/676<br />
</td>
<td>[16/15 24/13] [13/12 15/8] [13/10 9/4] [4/3 30/13]<br />
[13/9 5/2] [8/5 36/13] [26/15 3] [16/9 40/13]<br />
</td>
<td><br />
</td>
<td><br />
</td>
</tr>
<tr>
<td>195/112<br />
</td>
<td>959.970<br />
</td>
<td>26/15<br />
</td>
<td>225/224<br />
</td>
<td>[16/15 13/7] [14/13 15/8] [16/13 15/7] [28/15 13/4]<br />
</td>
<td><br />
</td>
<td><br />
</td>
</tr>
<tr>
<td>25/14<br />
</td>
<td>1003.802<br />
</td>
<td>16/9<br />
</td>
<td>225/224<br />
</td>
<td>[6/5 15/7] [7/5 5/2] [8/5 20/7] [28/15 10/3]<br />
</td>
<td><br />
</td>
<td><br />
</td>
</tr>
<tr>
<td>144/77<br />
</td>
<td>1083.766<br />
</td>
<td>28/15 15/8<br />
</td>
<td>540/539 384/385<br />
</td>
<td>[7/6 24/11] [11/9 16/7] [11/8 18/7] [14/9 32/11]<br />
[7/4 36/11] [11/6 24/7}<br />
</td>
<td><br />
</td>
<td><br />
</td>
</tr>
<tr>
<td>169/90<br />
</td>
<td>1090.832<br />
</td>
<td>15/8<br />
</td>
<td>676/675<br />
</td>
<td>[15/13 13/6] [18/13 13/5] [20/13 26/9] [24/13 52/15]<br />
</td>
<td><br />
</td>
<td><br />
</td>
</tr>
<tr>
<td>220/117<br />
</td>
<td>1093.194<br />
</td>
<td>15/8<br />
</td>
<td>352/351<br />
</td>
<td>[13/11 20/9] [13/10 22/9] [18/11 40/13] [9/5 44/13]<br />
</td>
<td><br />
</td>
<td><br />
</td>
</tr>
</table>
<!-- ws:start:WikiTextHeadingRule:4:<h3> --><h3 id="toc2"><a name="x-Fudging Intervals-Neutral/ambiguous"></a><!-- ws:end:WikiTextHeadingRule:4 -->Neutral/ambiguous</h3>
<table class="wiki_table">
<tr>
<td>fudger<br />
</td>
<td>cents<br />
</td>
<td>approximately<br />
</td>
<td>comma(s)<br />
</td>
<td>intervals<br />
</td>
<td>names<br />
</td>
</tr>
<tr>
<td>130/121<br />
</td>
<td>124.205<br />
</td>
<td>15/14 14/13<br />
</td>
<td>364/363 845/847<br />
</td>
<td>[11/10 13/11] [22/13 20/11]<br />
</td>
<td><br />
</td>
</tr>
<tr>
<td><a class="wiki_link" href="http://xenharmonic.wikispaces.com/88_81">88/81</a><br />
</td>
<td>143.498<br />
</td>
<td>13/12 12/11<br />
</td>
<td>352/351 242/243<br />
</td>
<td>[9/8 11/9] [18/11 16/9]<br />
</td>
<td>undecimal subtone<br />
</td>
</tr>
<tr>
<td>99/91<br />
</td>
<td>145.874<br />
</td>
<td>13/12 12/11<br />
</td>
<td>1188/1183 363/364<br />
</td>
<td>[13/11 9/7] [14/11 18/13] [13/9 11/7] [14/9 22/13]<br />
</td>
<td><br />
</td>
</tr>
<tr>
<td>256/225<br />
</td>
<td>223.463<br />
</td>
<td>8/7<br />
</td>
<td>224/225<br />
</td>
<td>[15/8 32/15]<br />
</td>
<td><br />
</td>
</tr>
<tr>
<td>225/196<br />
</td>
<td>238.886<br />
</td>
<td>8/7<br />
</td>
<td>225/224<br />
</td>
<td>[28/15 15/7]<br />
</td>
<td><br />
</td>
</tr>
<tr>
<td>200/169<br />
</td>
<td>291.572<br />
</td>
<td>13/11<br />
</td>
<td>2200/2197<br />
</td>
<td>[13/10 20/13]<br />
</td>
<td><br />
</td>
</tr>
<tr>
<td>77/65<br />
</td>
<td>293.302<br />
</td>
<td>13/11<br />
</td>
<td>847/845<br />
</td>
<td>[13/11 7/5] [10/7 22/13] [20/11 28/13] [13/7 11/5]<br />
</td>
<td><br />
</td>
</tr>
<tr>
<td>108/91<br />
</td>
<td>296.511<br />
</td>
<td>13/11<br />
</td>
<td>1188/1183<br />
</td>
<td>[13/12 9/7] [7/6 18/13] [13/9 12/7] [14/9 24/13]<br />
</td>
<td><br />
</td>
</tr>
<tr>
<td>27/22<br />
</td>
<td>354.547<br />
</td>
<td>11/9 16/13<br />
</td>
<td>243/242 351/352<br />
</td>
<td>[10/9 15/11] [11/9 3/2] [4/3 18/11] [22/15 9/5] [16/9 24/11]<br />
[11/6 9/4]<br />
</td>
<td><br />
</td>
</tr>
<tr>
<td>225/182<br />
</td>
<td>367.184<br />
</td>
<td>16/13<br />
</td>
<td>225/224<br />
</td>
<td>[26/15 15/7] [28/15 30/13]<br />
</td>
<td><br />
</td>
</tr>
<tr>
<td>286/225<br />
</td>
<td>415.308<br />
</td>
<td>14/11<br />
</td>
<td>1573/1575<br />
</td>
<td>[15/13 22/15] [15/11 26/15]<br />
</td>
<td><br />
</td>
</tr>
<tr>
<td>225/176<br />
</td>
<td>425.219<br />
</td>
<td>14/11<br />
</td>
<td>225/224<br />
</td>
<td>[16/15 15/11] [22/15 15/8]<br />
</td>
<td><br />
</td>
</tr>
<tr>
<td>135/104<br />
</td>
<td>451.651<br />
</td>
<td>13/10<br />
</td>
<td>675/676<br />
</td>
<td>[16/15 18/13] [13/9 15/8] [26/15 9/4] [16/9 30/13]<br />
</td>
<td><br />
</td>
</tr>
<tr>
<td>220/169<br />
</td>
<td>456.576<br />
</td>
<td>13/10<br />
</td>
<td>2200/2197<br />
</td>
<td>[13/11 20/13] [13/10 22/13]<br />
</td>
<td><br />
</td>
</tr>
<tr>
<td>176/135<br />
</td>
<td>459.139<br />
</td>
<td>13/10<br />
</td>
<td>352/351<br />
</td>
<td>[9/8 22/15] [15/11 16/9] [18/11 32/15] [15/8 22/9]<br />
</td>
<td><br />
</td>
</tr>
<tr>
<td>224/165<br />
</td>
<td>529.239<br />
</td>
<td>15/11<br />
</td>
<td>224/225<br />
</td>
<td>[15/14 16/11] [11/8 28/15] [11/7 32/15] [15/8 28/11]<br />
</td>
<td><br />
</td>
</tr>
<tr>
<td>143/105<br />
</td>
<td>534.751<br />
</td>
<td>15/11<br />
</td>
<td>1573/1575<br />
</td>
<td>[14/13 22/15] [15/13 11/7] [14/11 26/15] [15/11 13/7]<br />
</td>
<td><br />
</td>
</tr>
<tr>
<td>91/66<br />
</td>
<td>556.081<br />
</td>
<td>11/8 18/13<br />
</td>
<td>364/363 1183/1188<br />
</td>
<td>[11/7 13/6] [22/13 7/3] [12/7 26/11] [24/13 28/11]<br />
</td>
<td><br />
</td>
</tr>
<tr>
<td>135/98<br />
</td>
<td>554.527<br />
</td>
<td>11/8<br />
</td>
<td>540/539<br />
</td>
<td>[14/9 15/7] [28/15 18/7]<br />
</td>
<td><br />
</td>
</tr>
<tr>
<td>104/75<br />
</td>
<td>565.945<br />
</td>
<td>18/13<br />
</td>
<td>676/675<br />
</td>
<td>[15/13 8/5] [5/4 26/15] [20/13 32/15]<br />
[15/8 13/5]<br />
</td>
<td><br />
</td>
</tr>
<tr>
<td>45/32<br />
</td>
<td>590.224<br />
</td>
<td>7/5<br />
</td>
<td>225/224<br />
</td>
<td>[16/15 3/2] [4/3 15/8] [8/5 9/4] [16/9 5/2]<br />
</td>
<td><br />
</td>
</tr>
<tr>
<td>64/45<br />
</td>
<td>609.776<br />
</td>
<td>10/7<br />
</td>
<td>224/225<br />
</td>
<td>[9/8 8/5] [5/4 16/9] [3/2 32/15] [15/8 8/3]<br />
</td>
<td><br />
</td>
</tr>
<tr>
<td>75/52<br />
</td>
<td>634.055<br />
</td>
<td>675/676<br />
</td>
<td>13/9<br />
</td>
<td>[16/15 20/13] [13/10 15/8] [8/5 30/13] [26/15 5/2]<br />
</td>
<td><br />
</td>
</tr>
<tr>
<td>132/91<br />
</td>
<td>643.919<br />
</td>
<td>13/9 16/11<br />
</td>
<td>1188/1183 363/364<br />
</td>
<td>[13/12 11/7] [7/6 22/13] [13/11 12/7] [14/11 24/13]<br />
</td>
<td><br />
</td>
</tr>
<tr>
<td>196/135<br />
</td>
<td>645.473<br />
</td>
<td>16/11<br />
</td>
<td>539/540<br />
</td>
<td>[15/14 14/9] [9/7 28/15]<br />
</td>
<td><br />
</td>
</tr>
<tr>
<td>210/143<br />
</td>
<td>665.249<br />
</td>
<td>22/15<br />
</td>
<td>1575/1573<br />
</td>
<td>[22/15 28/13] [11/7 30/13] [26/15 28/11] [13/7 30/11]<br />
</td>
<td><br />
</td>
</tr>
<tr>
<td>165/112<br />
</td>
<td>670.761<br />
</td>
<td>22/15<br />
</td>
<td>225/224<br />
</td>
<td>[16/15 11/7] [14/11 15/8] [16/11 15/7] [28/15 11/4]<br />
</td>
<td><br />
</td>
</tr>
<tr>
<td>135/88<br />
</td>
<td>740.861<br />
</td>
<td>20/13<br />
</td>
<td>351/352<br />
</td>
<td>[16/15 18/11] [11/9 15/8] [22/15 9/4] [16/9 30/11]<br />
</td>
<td><br />
</td>
</tr>
<tr>
<td>169/110<br />
</td>
<td>743.424<br />
</td>
<td>20/13<br />
</td>
<td>2197/2200<br />
</td>
<td>[20/13 26/11] [22/13 13/5]<br />
</td>
<td><br />
</td>
</tr>
<tr>
<td>208/135<br />
</td>
<td>748.349<br />
</td>
<td>20/13<br />
</td>
<td>676/675<br />
</td>
<td>[9/8 26/15] [15/13 16/9] [18/13 32/15] [15/8 26/9]<br />
</td>
<td><br />
</td>
</tr>
<tr>
<td>352/225<br />
</td>
<td>774.781<br />
</td>
<td>11/7<br />
</td>
<td>224/225<br />
</td>
<td>[15/11 32/15] [15/8 44/15]<br />
</td>
<td><br />
</td>
</tr>
<tr>
<td>225/143<br />
</td>
<td>784.692<br />
</td>
<td>11/7<br />
</td>
<td>1575/1573<br />
</td>
<td>[22/15 30/13] [26/15 30/11]<br />
</td>
<td><br />
</td>
</tr>
<tr>
<td>364/225<br />
</td>
<td>832.816<br />
</td>
<td>13/8<br />
</td>
<td>224/225<br />
</td>
<td>[15/14 26/15] [15/13 28/15]<br />
</td>
<td><br />
</td>
</tr>
<tr>
<td>44/27<br />
</td>
<td>845.453<br />
</td>
<td>13/8 18/11<br />
</td>
<td>352/351 242/243<br />
</td>
<td>[12/11 16/9] [9/8 11/6] [15/11 20/9] [3/2 22/9] [18/11 8/3]<br />
[9/5 44/15]<br />
</td>
<td><br />
</td>
</tr>
<tr>
<td>91/54<br />
</td>
<td>903.489<br />
</td>
<td>22/13<br />
</td>
<td>1183/1188<br />
</td>
<td>[9/7 13/6] [18/13 7/3] [12/7 26/9] [24/13 28/9]<br />
</td>
<td><br />
</td>
</tr>
<tr>
<td>130/77<br />
</td>
<td>906.698<br />
</td>
<td>22/13<br />
</td>
<td>845/847<br />
</td>
<td>[14/13 20/11] [11/10 13/7] [7/5 26/11] [22/13 20/7]<br />
</td>
<td><br />
</td>
</tr>
<tr>
<td>169/100<br />
</td>
<td>908.428<br />
</td>
<td>22/13<br />
</td>
<td>2197/2200<br />
</td>
<td>[20/13 13/5]<br />
</td>
<td><br />
</td>
</tr>
<tr>
<td>392/225<br />
</td>
<td>961.114<br />
</td>
<td>7/4<br />
</td>
<td>224/225<br />
</td>
<td>[15/14 28/15]<br />
</td>
<td><br />
</td>
</tr>
<tr>
<td>225/128<br />
</td>
<td>976.537<br />
</td>
<td>7/4<br />
</td>
<td>225/224<br />
</td>
<td>[16/15 15/8]<br />
</td>
<td><br />
</td>
</tr>
<tr>
<td>81/44<br />
</td>
<td>1056.502<br />
</td>
<td>11/6 24/13<br />
</td>
<td>243/242 351/352<br />
</td>
<td>[11/9 9/4] [16/9 36/11]<br />
</td>
<td><br />
</td>
</tr>
<tr>
<td>225/121<br />
</td>
<td>1073.902<br />
</td>
<td>13/7 28/15<br />
</td>
<td>1575/1573 3375/3388<br />
</td>
<td>[22/15 30/11]<br />
</td>
<td><br />
</td>
</tr>
<tr>
<td>121/65<br />
</td>
<td>1075.795<br />
</td>
<td>13/7 28/15<br />
</td>
<td>847/845 363/364<br />
</td>
<td>[13/11 11/5] [20/11 44/13]<br />
</td>
<td><br />
</td>
</tr>
</table>
<!-- ws:start:WikiTextHeadingRule:6:<h3> --><h3 id="toc3"><a name="x-Fudging Intervals-Down-fudge"></a><!-- ws:end:WikiTextHeadingRule:6 -->Down-fudge</h3>
<table class="wiki_table">
<tr>
<td>fudger<br />
</td>
<td>cents<br />
</td>
<td>approximately<br />
</td>
<td>commas<br />
</td>
<td>intervals<br />
</td>
</tr>
<tr>
<td>128/117<br />
</td>
<td>155.562<br />
</td>
<td>12/11<br />
</td>
<td>352/351<br />
</td>
<td>[9/8 16/13] [13/8 16/9]<br />
</td>
</tr>
<tr>
<td>169/154<br />
</td>
<td>160.911<br />
</td>
<td>11/10<br />
</td>
<td>845/847<br />
</td>
<td>[14/13 13/11] [22/13 13/7]<br />
</td>
</tr>
<tr>
<td>100/91<br />
</td>
<td>163.274<br />
</td>
<td>11/10<br />
</td>
<td>1000/1001<br />
</td>
<td>[13/10 10/7] [7/5 20/13]<br />
</td>
</tr>
<tr>
<td>182/165<br />
</td>
<td>169.767<br />
</td>
<td>11/10<br />
</td>
<td>364/363<br />
</td>
<td>[15/14 13/11] [15/13 14/11] [11/7 26/15] [22/13 28/15]<br />
</td>
</tr>
<tr>
<td>72/65<br />
</td>
<td>177.069<br />
</td>
<td>10/9<br />
</td>
<td>324/325<br />
</td>
<td>[13/12 6/5] [10/9 16/13] [5/4 18/13] [13/9 8/5] [13/8 9/5] [5/3 24/13]<br />
</td>
</tr>
<tr>
<td>195/176<br />
</td>
<td>177.478<br />
</td>
<td>10/9<br />
</td>
<td>351/352<br />
</td>
<td>[16/15 13/11] [16/13 15/11] [22/15 13/8] [22/13 15/8]<br />
</td>
</tr>
<tr>
<td>49/44<br />
</td>
<td>186.334<br />
</td>
<td>10/9<br />
</td>
<td>441/440<br />
</td>
<td>[8/7 14/11] [11/7 7/4]<br />
</td>
</tr>
<tr>
<td>39/35<br />
</td>
<td>187.343<br />
</td>
<td>10/9<br />
</td>
<td>351/350<br />
</td>
<td>[14/13 6/5] [15/13 9/7] [7/6 13/10] [20/13 12/7] [14/9 26/15] [5/3 13/7]<br />
</td>
</tr>
<tr>
<td>135/121<br />
</td>
<td>189.543<br />
</td>
<td>10/9<br />
</td>
<td>243/242<br />
</td>
<td>[11/9 15/11] [22/15 18/11]<br />
</td>
</tr>
<tr>
<td>121/108<br />
</td>
<td>196.771<br />
</td>
<td>9/8<br />
</td>
<td>242/243<br />
</td>
<td>[12/11 11/9] [18/11 11/6]<br />
</td>
</tr>
<tr>
<td>55/49<br />
</td>
<td>199.980<br />
</td>
<td>9/8<br />
</td>
<td>440/441<br />
</td>
<td>[14/11 10/7] [7/5 11/7]<br />
</td>
</tr>
<tr>
<td>91/81<br />
</td>
<td>201.534<br />
</td>
<td>9/8<br />
</td>
<td>728/729<br />
</td>
<td>[9/7 13/9] [18/13 14/9]<br />
</td>
</tr>
<tr>
<td>169/150<br />
</td>
<td>206.473<br />
</td>
<td>9/8<br />
</td>
<td>676/675<br />
</td>
<td>[15/13 13/10] [20/13 26/15]<br />
</td>
</tr>
<tr>
<td>44/39<br />
</td>
<td>208.835<br />
</td>
<td>9/8<br />
</td>
<td>352/351<br />
</td>
<td>[13/12 11/9] [12/11 16/13] [13/11 4/3] [13/10 22/15] [15/11 20/13] [3/2 22/13]<br />
[13/8 11/6] [18/11 24/13]<br />
</td>
</tr>
<tr>
<td>154/135<br />
</td>
<td>227.965<br />
</td>
<td>8/7<br />
</td>
<td>539/540<br />
</td>
<td>[15/14 11/9] [9/7 22/15] [15/11 14/9] [18/11 28/15]<br />
</td>
</tr>
<tr>
<td>63/55<br />
</td>
<td>235.104<br />
</td>
<td>8/7<br />
</td>
<td>441/440<br />
</td>
<td>[10/9 14/11] [11/9 7/5] [10/7 18/11] [11/7 9/5]<br />
</td>
</tr>
<tr>
<td>55/48<br />
</td>
<td>235.677<br />
</td>
<td>8/7<br />
</td>
<td>385/384<br />
</td>
<td>[16/15 11/9] [12/11 5/4] [6/5 11/8] [16/11 5/3] [8/5 11/6] [18/11 15/8]<br />
</td>
</tr>
<tr>
<td>121/104<br />
</td>
<td>262.108<br />
</td>
<td>7/6<br />
</td>
<td>363/364<br />
</td>
<td>[13/11 11/8] [16/11 22/13]<br />
</td>
</tr>
<tr>
<td>64/55<br />
</td>
<td>262.368<br />
</td>
<td>7/6<br />
</td>
<td>384/385<br />
</td>
<td>[5/4 16/11] [11/8 8/5] [11/6 32/15] [15/8 24/11]<br />
</td>
</tr>
<tr>
<td>90/77<br />
</td>
<td>270.080<br />
</td>
<td>7/6<br />
</td>
<td>540/539<br />
</td>
<td>[11/10 9/7] [7/6 15/11] [11/9 10/7] [7/5 18/11] [22/15 12/7] [14/9 20/11] [11/6 15/7] [28/15 24/11]<br />
</td>
</tr>
<tr>
<td>117/100<br />
</td>
<td>271.810<br />
</td>
<td>7/6<br />
</td>
<td>351/350<br />
</td>
<td>[10/9 13/10] [20/13 9/5]<br />
</td>
</tr>
<tr>
<td>198/169<br />
</td>
<td>274.173<br />
</td>
<td>7/6<br />
</td>
<td>1188/1183<br />
</td>
<td>[13/11 18/13] [13/9 22/13]<br />
</td>
</tr>
<tr>
<td>140/117<br />
</td>
<td>310.702<br />
</td>
<td>6/5<br />
</td>
<td>350/351<br />
</td>
<td>[9/7 20/13] [13/10 14/9] [9/5 28/13] [13/7 20/9]<br />
</td>
</tr>
<tr>
<td>77/64<br />
</td>
<td>320.144<br />
</td>
<td>6/5<br />
</td>
<td>385/384<br />
</td>
<td>[8/7 11/8] [16/11 7/4]<br />
</td>
</tr>
<tr>
<td>65/54<br />
</td>
<td>320.976<br />
</td>
<td>6/5<br />
</td>
<td>325/324<br />
</td>
<td>[6/5 13/9] [18/13 5/3] [9/5 13/6] [24/13 20/9]<br />
</td>
</tr>
<tr>
<td>135/112<br />
</td>
<td>323.353<br />
</td>
<td>6/5<br />
</td>
<td>225/224<br />
</td>
<td>[16/15 9/7] [14/9 15/8] [16/9 15/7] [28/15 9/4]<br />
</td>
</tr>
<tr>
<td>39/32<br />
</td>
<td>342.483<br />
</td>
<td>11/9<br />
</td>
<td>351/352<br />
</td>
<td>[16/15 13/10] [16/13 3/2] [4/3 13/8] [20/13 15/8] [16/9 13/6] [24/13 9/4]<br />
</td>
</tr>
<tr>
<td>56/45<br />
</td>
<td>378.602<br />
</td>
<td>5/4<br />
</td>
<td>224/225<br />
</td>
<td>[15/14 4/3] [9/8 7/5] [5/4 14/9] [9/7 8/5] [10/7 16/9] [3/2 28/15] [12/7 32/15]<br />
[15/8 7/3]<br />
</td>
</tr>
<tr>
<td>81/65<br />
</td>
<td>380.979<br />
</td>
<td>5/4<br />
</td>
<td>324/325<br />
</td>
<td>[10/9 18/13] [13/9 9/5]<br />
</td>
</tr>
<tr>
<td>96/77<br />
</td>
<td>381.811<br />
</td>
<td>5/4<br />
</td>
<td>384/385<br />
</td>
<td>[7/6 16/11] [11/8 12/7] [7/4 24/11] [11/6 16/7]<br />
</td>
</tr>
<tr>
<td>169/135<br />
</td>
<td>388.877<br />
</td>
<td>5/4<br />
</td>
<td>676/675<br />
</td>
<td>[15/13 13/9] [18/13 26/15]<br />
</td>
</tr>
<tr>
<td>33/26<br />
</td>
<td>412.745<br />
</td>
<td>14/11<br />
</td>
<td>363/364<br />
</td>
<td>[13/12 11/8] [12/11 18/13] [13/11 3/2] [4/3 22/13] [13/9 11/6] [16/11 24/13]<br />
[26/15 11/5] [20/11 30/13]<br />
</td>
</tr>
<tr>
<td>143/112<br />
</td>
<td>423.020<br />
</td>
<td>14/11<br />
</td>
<td>1573/1568<br />
</td>
<td>[14/13 11/8] [16/13 11/7] [14/11 13/8] [16/11 13/7]<br />
</td>
</tr>
<tr>
<td>216/169<br />
</td>
<td>424.810<br />
</td>
<td>14/11<br />
</td>
<td>1188/1183<br />
</td>
<td>[13/12 18/13] [13/9 24/13]<br />
</td>
</tr>
<tr>
<td>169/132<br />
</td>
<td>427.782<br />
</td>
<td>9/7<br />
</td>
<td>1183/1188<br />
</td>
<td>[22/13 13/6] [24/13 26/11]<br />
</td>
</tr>
<tr>
<td>50/39<br />
</td>
<td>430.145<br />
</td>
<td>9/7<br />
</td>
<td>350/351<br />
</td>
<td>[6/5 20/13] [13/10 5/3] [26/15 20/9] [9/5 30/13]<br />
</td>
</tr>
<tr>
<td>104/81<br />
</td>
<td>432.708<br />
</td>
<td>9/7<br />
</td>
<td>728/729<br />
</td>
<td>[9/8 13/9] [18/13 16/9]<br />
</td>
</tr>
<tr>
<td>165/128<br />
</td>
<td>439.587<br />
</td>
<td>9/7<br />
</td>
<td>385/384<br />
</td>
<td>[16/15 11/8] [16/11 15/8]<br />
</td>
</tr>
<tr>
<td>156/121<br />
</td>
<td>439.847<br />
</td>
<td>9/7<br />
</td>
<td>364/363<br />
</td>
<td>[22/13 24/11] [11/6 26/11]<br />
</td>
</tr>
<tr>
<td>117/88<br />
</td>
<td>493.120<br />
</td>
<td>4/3<br />
</td>
<td>351/352<br />
</td>
<td>[11/9 13/8] [16/13 18/11] [22/13 9/4] [16/9 26/11]<br />
</td>
</tr>
<tr>
<td>121/91<br />
</td>
<td>493.282<br />
</td>
<td>4/3<br />
</td>
<td>363/364<br />
</td>
<td>[13/11 11/7] [14/11 22/13]<br />
</td>
</tr>
<tr>
<td>225/169<br />
</td>
<td>495.482<br />
</td>
<td>4/3<br />
</td>
<td>675/676<br />
</td>
<td>[26/15 30/13]<br />
</td>
</tr>
<tr>
<td>162/121<br />
</td>
<td>505.184<br />
</td>
<td>4/3<br />
</td>
<td>243/242<br />
</td>
<td>[11/9 18/11]<br />
</td>
</tr>
<tr>
<td>75/56<br />
</td>
<td>505.757<br />
</td>
<td>4/3<br />
</td>
<td>225/224<br />
</td>
<td>[16/15 10/7] [7/5 15/8] [8/5 15/7] [28/15 5/2]<br />
</td>
</tr>
<tr>
<td>196/143<br />
</td>
<td>545.806<br />
</td>
<td>11/8<br />
</td>
<td>1568/1573<br />
</td>
<td>[11/7 28/13] [13/7 28/11]<br />
</td>
</tr>
<tr>
<td>169/121<br />
</td>
<td>578.419<br />
</td>
<td>7/5<br />
</td>
<td>845/847<br />
</td>
<td>[22/13 26/11]<br />
</td>
</tr>
<tr>
<td>88/63<br />
</td>
<td>578.582<br />
</td>
<td>7/5<br />
</td>
<td>440/441<br />
</td>
<td>[9/8 11/7] [14/11 16/9] [18/11 16/7] [7/4 22/9]<br />
</td>
</tr>
<tr>
<td>200/143<br />
</td>
<td>580.782<br />
</td>
<td>7/5<br />
</td>
<td>1000/1001<br />
</td>
<td>[11/10 20/13] [13/10 20/11]<br />
</td>
</tr>
<tr>
<td>108/77<br />
</td>
<td>585.721<br />
</td>
<td>7/5<br />
</td>
<td>540/539<br />
</td>
<td>[7/6 18/11] [11/9 12/7] [14/9 24/11] [11/6 18/7]<br />
</td>
</tr>
<tr>
<td>77/54<br />
</td>
<td>614.279<br />
</td>
<td>10/7<br />
</td>
<td>539/540<br />
</td>
<td>[12/11 14/9] [9/7 11/6] [18/11 7/3] [12/7 22/9]<br />
</td>
</tr>
<tr>
<td>143/100<br />
</td>
<td>619.218<br />
</td>
<td>10/7<br />
</td>
<td>1001/1000<br />
</td>
<td>[20/13 11/5] [20/11 13/5]<br />
</td>
</tr>
<tr>
<td>63/44<br />
</td>
<td>621.418<br />
</td>
<td>10/7<br />
</td>
<td>441/440<br />
</td>
<td>[8/7 18/11] [11/9 7/4] [11/7 9/4] [16/9 28/11]<br />
</td>
</tr>
<tr>
<td>242/169<br />
</td>
<td>621.581<br />
</td>
<td>10/7<br />
</td>
<td>847/845<br />
</td>
<td>[13/11 22/13]<br />
</td>
</tr>
<tr>
<td>72/49<br />
</td>
<td>666.258<br />
</td>
<td>22/15<br />
</td>
<td>540/539<br />
</td>
<td>[7/6 12/7] [14/9 16/7] [7/4 18/7]<br />
</td>
</tr>
<tr>
<td>112/75<br />
</td>
<td>694.243<br />
</td>
<td>3/2<br />
</td>
<td>224/225<br />
</td>
<td>[15/14 8/5] [5/4 28/15] [10/7 32/15] [15/8 14/5]<br />
</td>
</tr>
<tr>
<td>121/81<br />
</td>
<td>694.816<br />
</td>
<td>3/2<br />
</td>
<td>242/243<br />
</td>
<td>[18/11 22/9]<br />
</td>
</tr>
<tr>
<td>338/225<br />
</td>
<td>704.518<br />
</td>
<td>3/2<br />
</td>
<td>676/675<br />
</td>
<td>[15/13 26/15]<br />
</td>
</tr>
<tr>
<td>182/121<br />
</td>
<td>706.718<br />
</td>
<td>3/2<br />
</td>
<td>364/363<br />
</td>
<td>[11/7 26/11] [22/13 28/11]<br />
</td>
</tr>
<tr>
<td>176/117<br />
</td>
<td>706.880<br />
</td>
<td>3/2<br />
</td>
<td>352/351<br />
</td>
<td>[9/8 22/13] [13/11 16/9] [13/8 22/9] [18/11 32/13]<br />
</td>
</tr>
<tr>
<td>121/78<br />
</td>
<td>760.153<br />
</td>
<td>14/9<br />
</td>
<td>363/364<br />
</td>
<td>[12/11 22/13] [13/11 11/6]<br />
</td>
</tr>
<tr>
<td>256/165<br />
</td>
<td>760.413<br />
</td>
<td>14/9<br />
</td>
<td>384/385<br />
</td>
<td>[11/8 32/15] [15/8 32/11]<br />
</td>
</tr>
<tr>
<td>81/52<br />
</td>
<td>767.292<br />
</td>
<td>14/9<br />
</td>
<td>729/728<br />
</td>
<td>[13/9 9/4] [16/9 36/13]<br />
</td>
</tr>
<tr>
<td>39/25<br />
</td>
<td>769.855<br />
</td>
<td>14/9<br />
</td>
<td>351/350<br />
</td>
<td>[10/9 26/15] [15/13 9/5] [20/13 12/5] [5/3 13/5]<br />
</td>
</tr>
<tr>
<td>264/169<br />
</td>
<td>772.218<br />
</td>
<td>14/9<br />
</td>
<td>1188/1183<br />
</td>
<td>[13/12 22/13] [13/11 24/13]<br />
</td>
</tr>
<tr>
<td>169/108<br />
</td>
<td>775.190<br />
</td>
<td>11/7<br />
</td>
<td>1183/1188<br />
</td>
<td>[18/13 13/6] [24/13 26/9]<br />
</td>
</tr>
<tr>
<td>224/143<br />
</td>
<td>776.980<br />
</td>
<td>11/7<br />
</td>
<td>1568/1573<br />
</td>
<td>[11/8 28/13] [11/7 32/13] [13/8 28/11] [13/7 32/11]<br />
</td>
</tr>
<tr>
<td>52/33<br />
</td>
<td>787.255<br />
</td>
<td>11/7<br />
</td>
<td>364/363<br />
</td>
<td>[11/10 26/15] [15/13 20/11] [11/8 13/6] [18/13 24/11] [3/2 26/11] [22/13 8/3]<br />
[11/6 26/9] [24/13 32/11]<br />
</td>
</tr>
<tr>
<td>270/169<br />
</td>
<td>811.123<br />
</td>
<td>8/5<br />
</td>
<td>675/676<br />
</td>
<td>[13/9 30/13] [26/15 36/13]<br />
</td>
</tr>
<tr>
<td>77/48<br />
</td>
<td>818.189<br />
</td>
<td>8/5<br />
</td>
<td>385/384<br />
</td>
<td>[12/11 7/4] [8/7 11/6] [16/11 7/3] [12/7 11/4]<br />
</td>
</tr>
<tr>
<td>130/81<br />
</td>
<td>819.021<br />
</td>
<td>8/5<br />
</td>
<td>325/324<br />
</td>
<td>[18/13 20/9] [9/5 26/9]<br />
</td>
</tr>
<tr>
<td>45/28<br />
</td>
<td>821.398<br />
</td>
<td>8/5<br />
</td>
<td>225/224<br />
</td>
<td>[16/15 12/7] [7/6 15/8] [4/3 15/7] [7/5 9/4] [14/9 5/2] [8/5 18/7] [16/9 20/7] [28/15 3]<br />
</td>
</tr>
<tr>
<td>64/39<br />
</td>
<td>857.517<br />
</td>
<td>18/11<br />
</td>
<td>352/351<br />
</td>
<td>[13/12 16/9] [9/8 24/13] [13/10 32/15] [3/2 32/13] [13/8 8/3] [15/8 40/13]<br />
</td>
</tr>
<tr>
<td>224/135<br />
</td>
<td>876.647<br />
</td>
<td>5/3<br />
</td>
<td>224/225<br />
</td>
<td>[15/14 16/9] [9/8 28/15] [9/7 32/15] [15/8 28/9]<br />
</td>
</tr>
<tr>
<td>108/65<br />
</td>
<td>879.024<br />
</td>
<td>5/3<br />
</td>
<td>324/325<br />
</td>
<td>[13/12 9/5] [10/9 24/13] [13/9 12/5] [5/3 36/13]<br />
</td>
</tr>
<tr>
<td>128/77<br />
</td>
<td>879.856<br />
</td>
<td>5/3<br />
</td>
<td>384/385<br />
</td>
<td>[11/8 16/7] [7/4 32/11]<br />
</td>
</tr>
<tr>
<td>117/70<br />
</td>
<td>889.298<br />
</td>
<td>5/3<br />
</td>
<td>351/350<br />
</td>
<td>[14/13 9/5] [10/9 13/7] [20/13 18/7] [14/9 13/5]<br />
</td>
</tr>
<tr>
<td>169/99<br />
</td>
<td>925.827<br />
</td>
<td>12/7<br />
</td>
<td>1183/1188<br />
</td>
<td>[18/13 26/11] [22/13 26/9]<br />
</td>
</tr>
<tr>
<td>200/117<br />
</td>
<td>928.190<br />
</td>
<td>12/7<br />
</td>
<td>350/351<br />
</td>
<td>[13/10 20/9] [9/5 40/13]<br />
</td>
</tr>
<tr>
<td>77/45<br />
</td>
<td>929.920<br />
</td>
<td>12/7<br />
</td>
<td>539/540<br />
</td>
<td>[15/14 11/6] [12/11 28/15] [9/7 11/5] [15/11 7/3] [10/7 22/9] [18/11 14/5] [12/7 44/15] [20/11 28/9]<br />
</td>
</tr>
<tr>
<td>55/32<br />
</td>
<td>937.632<br />
</td>
<td>12/7<br />
</td>
<td>385/384<br />
</td>
<td>[16/15 11/6] [12/11 15/8] [16/11 5/2] [8/5 11/4]<br />
</td>
</tr>
<tr>
<td>208/121<br />
</td>
<td>937.892<br />
</td>
<td>12/7<br />
</td>
<td>364/363<br />
</td>
<td>[11/8 26/11] [22/13 32/11]<br />
</td>
</tr>
<tr>
<td>96/55<br />
</td>
<td>964.323<br />
</td>
<td>7/4<br />
</td>
<td>384/385<br />
</td>
<td>[11/9 32/15] [5/4 24/11] [11/8 12/5] [5/3 32/11] [11/6 16/5] [15/8 36/11]<br />
</td>
</tr>
<tr>
<td>110/63<br />
</td>
<td>964.896<br />
</td>
<td>7/4<br />
</td>
<td>440/441<br />
</td>
<td>[14/11 20/9] [7/5 22/9] [18/11 20/7] [9/5 22/7]<br />
</td>
</tr>
<tr>
<td>135/77<br />
</td>
<td>972.035<br />
</td>
<td>7/4<br />
</td>
<td>540/539<br />
</td>
<td>[11/9 15/7] [22/15 18/7] [14/9 30/11] [28/15 36/11]<br />
</td>
</tr>
<tr>
<td>39/22<br />
</td>
<td>991.165<br />
</td>
<td>16/9<br />
</td>
<td>351/352<br />
</td>
<td>[11/9 13/6] [16/13 24/11] [4/3 26/11] [22/15 13/5] [20/13 30/11] [22/13 3] [11/6 13/4] [24/13 36/11]<br />
</td>
</tr>
<tr>
<td>300/169<br />
</td>
<td>993.527<br />
</td>
<td>16/9<br />
</td>
<td>675/676<br />
</td>
<td>[13/10 30/13] [26/15 40/13]<br />
</td>
</tr>
<tr>
<td>162/91<br />
</td>
<td>998.466<br />
</td>
<td>16/9<br />
</td>
<td>729/728<br />
</td>
<td>[13/9 18/7] [14/9 36/13]<br />
</td>
</tr>
<tr>
<td>98/55<br />
</td>
<td>1000.02<br />
</td>
<td>16/9<br />
</td>
<td>441/440<br />
</td>
<td>[10/7 28/11] [11/7 14/5]<br />
</td>
</tr>
<tr>
<td>216/121<br />
</td>
<td>1003.229<br />
</td>
<td>16/9<br />
</td>
<td>243/242<br />
</td>
<td>[11/9 24/11] [11/6 36/11]<br />
</td>
</tr>
<tr>
<td>242/135<br />
</td>
<td>1010.457<br />
</td>
<td>9/5<br />
</td>
<td>242/243<br />
</td>
<td>[15/11 22/9] [18/11 44/15]<br />
</td>
</tr>
<tr>
<td>70/39<br />
</td>
<td>1012.657<br />
</td>
<td>9/5<br />
</td>
<td>350/351<br />
</td>
<td>[6/5 28/13] [9/7 30/13] [13/10 7/3] [12/7 40/13] [26/15 28/9] [13/7 10/3]<br />
</td>
</tr>
<tr>
<td>88/49<br />
</td>
<td>1013.666<br />
</td>
<td>9/5<br />
</td>
<td>440/441<br />
</td>
<td>[14/11 16/7] [7/4 22/7]<br />
</td>
</tr>
<tr>
<td>352/195<br />
</td>
<td>1022.522<br />
</td>
<td>9/5<br />
</td>
<td>352/351<br />
</td>
<td>[13/11 32/15] [15/11 32/13] [13/8 44/15] [15/8 44/13]<br />
</td>
</tr>
<tr>
<td>65/36<br />
</td>
<td>1022.931<br />
</td>
<td>9/5<br />
</td>
<td>325/324<br />
</td>
<td>[6/5 13/6] [16/13 20/9] [18/13 5/2] [8/5 26/9] [9/5 13/4] [24/13 10/3]<br />
</td>
</tr>
<tr>
<td>165/91<br />
</td>
<td>1030.233<br />
</td>
<td>20/11<br />
</td>
<td>363/364<br />
</td>
<td>[13/11 15/7] [14/11 30/13] [26/15 22/7] [28/15 44/13]<br />
</td>
</tr>
<tr>
<td>91/50<br />
</td>
<td>1036.726<br />
</td>
<td>20/11<br />
</td>
<td>1001/1000<br />
</td>
<td>[10/7 13/5] [20/13 14/5]<br />
</td>
</tr>
<tr>
<td>308/169<br />
</td>
<td>1039.089<br />
</td>
<td>20/11<br />
</td>
<td>847/845<br />
</td>
<td>[13/11 28/13] [13/7 44/13]<br />
</td>
</tr>
<tr>
<td>117/64<br />
</td>
<td>1044.438<br />
</td>
<td>11/6<br />
</td>
<td>351/352<br />
</td>
<td>[16/13 9/4] [16/9 13/4]<br />
</td>
</tr>
<tr>
<td>182/99<br />
</td>
<td>1054.126<br />
</td>
<td>11/6 24/13<br />
</td>
<td>364/363 1183/1188<br />
</td>
<td>[9/7 26/11] [18/13 28/11] [11/7 26/9] [22/13 28/9]<br />
</td>
</tr>
</table>
</body></html>