Fudging - Revision history

ArrowHead294 at 01:24, 24 November 2024

2024-11-24T01:24:28Z

← Older revision		Revision as of 01:24, 24 November 2024
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	'''Fudging''', or virtual tempering, is the use of one [[just intonation]] interval to approximate another.		'''Fudging''', or virtual tempering, is the use of one [[just intonation]] interval to approximate another.

	==Fudgers==		== 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.		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===		=== 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]]
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	\|}		\|}

	===Down-fudge===		=== Down-fudge ===

	{\| class="wikitable"		{\| class="wikitable"
	\|-		\|-
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	\| [9/7 26/11]		\| [9/7 26/11]
	\|}		\|}

	~~[[Category:15-odd-limit]]~~
	~~[[Category:Method]]~~
	~~[[Category:Terms]]~~
	~~[[Category:Lists of intervals]]~~
	~~[[Category:Just intonation]]~~

	== Limit-raising and limit-lowering fudgers ==		== Limit-raising and limit-lowering fudgers ==
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	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.		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:Terms]]
			[[Category:Lists of intervals]]
			[[Category:Just intonation]]

Fredg999: Fix typo (swapped cells)

2024-08-14T01:01:00Z

Fix typo (swapped cells)

← Older revision		Revision as of 01:01, 14 August 2024
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	\| 75/52		\| 75/52
	\| 634.055		\| 634.055
			\| 13/9
	\| 675/676		\| 675/676
	~~\| 13/9~~
	\| [16/15 20/13]		\| [16/15 20/13]
	\|		\|

BudjarnLambeth: Split the introduction paragraph into multiple subheadings.

2023-04-20T06:50:03Z

Split the introduction paragraph into multiple subheadings.

← Older revision		Revision as of 06:50, 20 April 2023
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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===		===Up-fudge===
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	[[Category:Just intonation]]		[[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.

Fredg999 category edits: Moving from Category:15-limit to Category:15-odd-limit using Cat-a-lot

2023-04-20T02:43:29Z

Moving from Category:15-limit to Category:15-odd-limit using Cat-a-lot

← Older revision		Revision as of 02:43, 20 April 2023
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	\|}		\|}

	[[Category:15-limit]]		[[Category:15-odd-limit]]
	[[Category:Method]]		[[Category:Method]]
	[[Category:Terms]]		[[Category:Terms]]

Fredg999 category edits: Moving from Category:Interval collection to Category:Lists of intervals using Cat-a-lot

2023-04-13T22:04:36Z

Moving from Category:Interval collection to Category:Lists of intervals using Cat-a-lot

← Older revision		Revision as of 22:04, 13 April 2023
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	[[Category:Method]]		[[Category:Method]]
	[[Category:Terms]]		[[Category:Terms]]
	[[Category:~~Interval collection~~]]		[[Category:Lists of intervals]]
	[[Category:Just intonation]]		[[Category:Just intonation]]

	{{Todo\|review heading structure}}		{{Todo\|review heading structure}}

Francium: /* Neutral/ambiguous */

2023-04-09T11:09:12Z

Neutral/ambiguous

← Older revision		Revision as of 11:09, 9 April 2023
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	\|		\|
	\|-		\|-
	\| 256/225		\| [[256/225]]
	\| 223.463		\| 223.463
	\| 8/7		\| 8/7
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	\|		\|
	\|-		\|-
	\| 27/22		\| [[27/22]]
	\| 354.547		\| 354.547
	\| 11/9, 16/13		\| 11/9, 16/13
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	\|		\|
	\|-		\|-
	\| 45/32		\| [[45/32]]
	\| 590.224		\| 590.224
	\| 7/5		\| 7/5
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	\|		\|
	\|-		\|-
	\| 64/45		\| [[64/45]]
	\| 609.776		\| 609.776
	\| 10/7		\| 10/7

Francium: /* Up-fudge */

2023-04-09T11:00:57Z

Up-fudge

Show changes

Keenan Pepper category edits: Moving from Category:Term to Category:Terms using Cat-a-lot

2020-12-04T22:44:58Z

Moving from Category:Term to Category:Terms using Cat-a-lot

← Older revision		Revision as of 22:44, 4 December 2020
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	[[Category:15-limit]]		[[Category:15-limit]]
	[[Category:Method]]		[[Category:Method]]
	[[Category:~~Term~~]]		[[Category:Terms]]
	[[Category:Interval collection]]		[[Category:Interval collection]]
	[[Category:Just intonation]]		[[Category:Just intonation]]

	{{Todo\|review heading structure}}		{{Todo\|review heading structure}}

Xenwolf: lemma bold, redirect lemmas are now: fudger and fudgers

2020-11-27T10:20:03Z

lemma bold, redirect lemmas are now: fudger and fudgers

← Older revision		Revision as of 10:20, 27 November 2020
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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. 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.

	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.		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.

Xenwolf: cleanup, todo added

2020-11-27T10:05:29Z

cleanup, todo added

Show changes