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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.
'''Fudging''', or virtual tempering, is the use of one [[just intonation]] interval to approximate another.


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.
== 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.
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===


=== Up-fudge ===
{| class="wikitable"
{| class="wikitable"
|-
|-
! fudger
! Fudger
! cents
! Cents
! approximately
! Approximately
! comma(s)
! Comma(s)
! (example) intervals
! (Example) intervals
! name
! Name
|-
|-
| 242/225
| [[242/225]]
| 126.098
| 126.098
| 15/14, 14/13
| 15/14, 14/13
Line 81: Line 77:
|  
|  
|-
|-
| 81/70
| [[81/70]]
| 252.680
| 252.680
| 15/13
| 15/13
Line 88: Line 84:
|  
|  
|-
|-
| 75/64
| [[75/64]]
| 274.582
| 274.582
| 7/6
| 7/6
Line 102: Line 98:
|  
|  
|-
|-
| 32/27
| [[32/27]]
| 294.135
| 294.135
| 13/11
| 13/11
Line 116: Line 112:
|  
|  
|-
|-
| 60/49
| [[60/49]]
| 350.617
| 350.617
| 11/9
| 11/9
Line 123: Line 119:
|  
|  
|-
|-
| 49/40
| [[49/40]]
| 351.338
| 351.338
| 11/9
| 11/9
Line 130: Line 126:
|  
|  
|-
|-
| 100/81
| [[100/81]]
| 364.807
| 364.807
| 16/13
| 16/13
Line 144: Line 140:
|  
|  
|-
|-
| 80/63
| [[80/63]]
| 413.578
| 413.578
| 14/11
| 14/11
Line 151: Line 147:
|  
|  
|-
|-
| 32/25
| [[32/25]]
| 427.373
| 427.373
| 9/7
| 9/7
Line 158: Line 154:
|  
|  
|-
|-
| 35/27
| [[35/27]]
| 449.275
| 449.275
| 13/10
| 13/10
Line 172: Line 168:
|  
|  
|-
|-
| 49/36
| [[49/36]]
| 533.742
| 533.742
| 15/11
| 15/11
Line 186: Line 182:
|  
|  
|-
|-
| 48/35
| [[48/35]]
| 546.815
| 546.815
| 11/8
| 11/8
Line 207: Line 203:
|  
|  
|-
|-
| 25/18
| [[25/18]]
| 568.717
| 568.717
| 18/13
| 18/13
Line 214: Line 210:
|  
|  
|-
|-
| 36/25
| [[36/25]]
| 631.283
| 631.283
| 13/9
| 13/9
Line 235: Line 231:
|  
|  
|-
|-
| 35/24
| [[35/24]]
| 653.185
| 653.185
| 16/11
| 16/11
Line 263: Line 259:
|  
|  
|-
|-
| 54/35
| [[54/35]]
| 750.725
| 750.725
| 20/13
| 20/13
Line 270: Line 266:
|  
|  
|-
|-
| 25/16
| [[25/16]]
| 772.627
| 772.627
| 14/9
| 14/9
Line 305: Line 301:
|  
|  
|-
|-
| 49/30
| [[49/30]]
| 849.383
| 849.383
| 18/11
| 18/11
Line 312: Line 308:
|  
|  
|-
|-
| 105/64
| [[105/64]]
| 857.095
| 857.095
| 18/11
| 18/11
| 385/384
| 385/384
| [16/15 7/4] ]
| [16/15 7/4]  
|  
|  
|-
|-
| 27/16
| [[27/16]]
| 905.865
| 905.865
| 22/13
| 22/13
Line 333: Line 329:
|  
|  
|-
|-
| 128/75
| [[128/75]]
| 925.418
| 925.418
| 12/7
| 12/7
Line 340: Line 336:
|  
|  
|-
|-
| 140/81
| [[140/81]]
| 947.320
| 947.320
| 26/15
| 26/15
Line 368: Line 364:
|  
|  
|-
|-
| 64/35
| [[64/35]]
| 1044.86
| 1044.86
| 11/6
| 11/6
Line 424: Line 420:
| undecimal secor
| undecimal secor
|-
|-
| 28/25
| [[28/25]]
| 196.198
| 196.198
| 9/8
| 9/8
Line 431: Line 427:
|  
|  
|-
|-
| 224/195
| [[224/195]]
| 240.030
| 240.030
| 15/13
| 15/13
Line 445: Line 441:
|  
|  
|-
|-
| 77/60
| [[77/60]]
| 431.875
| 431.875
| 9/7
| 9/7
Line 452: Line 448:
|  
|  
|-
|-
| 110/81
| [[110/81]]
| 529.812
| 529.812
| 15/11
| 15/11
Line 473: Line 469:
|  
|  
|-
|-
| 81/55
| [[81/55]]
| 670.188
| 670.188
| 22/15
| 22/15
Line 501: Line 497:
|  
|  
|-
|-
| 25/14
| [[25/14]]
| 1003.802
| 1003.802
| 16/9
| 16/9
Line 562: Line 558:
|  
|  
|-
|-
| 256/225
| [[256/225]]
| 223.463
| 223.463
| 8/7
| 8/7
Line 597: Line 593:
|  
|  
|-
|-
| 27/22
| [[27/22]]
| 354.547
| 354.547
| 11/9, 16/13
| 11/9, 16/13
Line 681: Line 677:
|  
|  
|-
|-
| 45/32
| [[45/32]]
| 590.224
| 590.224
| 7/5
| 7/5
Line 688: Line 684:
|  
|  
|-
|-
| 64/45
| [[64/45]]
| 609.776
| 609.776
| 10/7
| 10/7
Line 697: Line 693:
| 75/52
| 75/52
| 634.055
| 634.055
| 13/9
| 675/676
| 675/676
| 13/9
| [16/15 20/13]
| [16/15 20/13]
|  
|  
Line 836: Line 832:
|}
|}


===Down-fudge===
=== Down-fudge ===
 
{| class="wikitable"
{| class="wikitable"
|-
|-
Line 1,441: Line 1,436:
|}
|}


[[Category:15-limit]]
== 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.
 
[[Category:15-odd-limit]]
[[Category:Method]]
[[Category:Method]]
[[Category:Term]]
[[Category:Terms]]
[[Category:Interval collection]]
[[Category:Lists of intervals]]
[[Category:Just intonation]]
[[Category:Just intonation]]
{{Todo|review heading structure}}
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