Fudging: Difference between revisions

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

'''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===

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[[Category:Just intonation]]

~~{{Todo|review heading structure}}~~

== 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.

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-'''Fudging''', or virtual tempering, is the use of one [[just intonation]] interval to approximate another. 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.
+'''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&lt;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===
@@ Line 1,447: / Line 1,444: @@
 [[Category:Just intonation]]
-{{Todo|review heading structure}}
+== 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&lt;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.