Direct approximation - Revision history

Dummy index: add interwiki

2025-08-23T12:16:24Z

add interwiki

← Older revision		Revision as of 12:16, 23 August 2025
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			{{interwiki
			\| en = Direct approximation
			\| ja = 直接近似
			}}
	A '''direct approximation''' of an interval in a given [[edo]] is the number of edosteps that most closely approximates it, found by [[rounding]] to the nearest integer the edo number times the [[log2\|binary logarithm]] of the interval:		A '''direct approximation''' of an interval in a given [[edo]] is the number of edosteps that most closely approximates it, found by [[rounding]] to the nearest integer the edo number times the [[log2\|binary logarithm]] of the interval:

Fredg999: Categories

2025-02-21T13:35:35Z

FloraC: A little expansion

2025-02-21T12:00:35Z

A little expansion

← Older revision		Revision as of 12:00, 21 February 2025
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	== Problems ==		== Problems ==
	~~Although~~ direct approximation ~~is perhaps easier to understand than mapping through [[val]]s, it~~ is not always practical in harmony. For example, it is impossible to construct a major triad using the direct approximations of 3/2, 5/4, and 6/5 in [[17edo]] since the step numbers do not add up (5 steps + 4 steps ≠ 10 steps).		Unless one sticks to one or two notes at a time, direct approximation is not always practical in harmony. For example, it is impossible to construct a [[4:5:6\|just major triad]] using the direct approximations of 3/2, 5/4, and 6/5 in 17edo since the step numbers do not add up (5 steps + 4 steps ≠ 10 steps). The closest 3/2 and 5/4 imply the second closest 6/5; the closest 3/2 and 6/5 imply the second closest 5/4; and the closest 5/4 and 6/5 imply the second closest 3/2. We see one of the direct approximations must be given up. This is called [[consistency\|inconsistency]], and chords like this exists in every edo.

			In [[regular temperament theory]], intervals are mapped through [[val]]s. Although more complex, it recognizes the fact that intervals like 3/2, 5/4, and 6/5 are related, as the number of steps of one interval is determined once the other two have been determined. The three situations in the above example correspond to using vals {{val\| 17 27 39 }}, {{val\| 17 27 40 }}, and {{val\| 17 26 39 }}, respectively.

	[[Category:Terms]]		[[Category:Terms]]
	[[Category:Method]]		[[Category:Method]]

FloraC: +"problems" section to show why this isn't always a great idea

2024-05-24T08:54:28Z

+"problems" section to show why this isn't always a great idea

← Older revision		Revision as of 08:54, 24 May 2024
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	for ratio ''i'' in ''n''-edo.		for ratio ''i'' in ''n''-edo.

	== Examples ~~of direct approximations~~ ==		== Examples ==

	{\| class="wikitable center-all"		{\| class="wikitable center-all"
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	Of these intervals, the fifth plays an important role for characterizing [[edo]] systems (as it defines the size of M2, m2, A1). Also, a simple test can show if [[circle-of-fifths notation]] can be applied to a given edo system, because for this the sizes of fifth and octave must be relatively prime.		Of these intervals, the fifth plays an important role for characterizing [[edo]] systems (as it defines the size of M2, m2, A1). Also, a simple test can show if [[circle-of-fifths notation]] can be applied to a given edo system, because for this the sizes of fifth and octave must be relatively prime.

			== Problems ==
			Although direct approximation is perhaps easier to understand than mapping through [[val]]s, it is not always practical in harmony. For example, it is impossible to construct a major triad using the direct approximations of 3/2, 5/4, and 6/5 in [[17edo]] since the step numbers do not add up (5 steps + 4 steps ≠ 10 steps).

	[[Category:Terms]]		[[Category:Terms]]
	[[Category:Method]]		[[Category:Method]]

FloraC: Styling

2022-02-24T22:19:28Z

Styling

← Older revision		Revision as of 22:19, 24 February 2022
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	A '''direct approximation''' of an interval in a given [[~~EDO~~]] is the number of ~~EDO steps~~ that most closely approximates it, found by [[rounding]] to the nearest integer the ~~EDO~~ number times the [[~~Wikipedia: binary logarithm~~\|binary logarithm]] of the interval: <math>~~⌈n_{~~\~~text~~{~~edo~~}}·\log_2(i)⌋</math>.		A '''direct approximation''' of an interval in a given [[edo]] is the number of edosteps that most closely approximates it, found by [[rounding]] to the nearest integer the edo number times the [[log2\|binary logarithm]] of the interval:

			<math>\operatorname {round} (n\log_2(i))</math>

			for ratio ''i'' in ''n''-edo.

	== Examples of direct approximations ==		== Examples of direct approximations ==
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	! [[12edo]] \|\| [[17edo]] \|\| [[19edo]] \|\| [[26edo]]		! [[12edo]] \|\| [[17edo]] \|\| [[19edo]] \|\| [[26edo]]
	\|-		\|-
	! ~~Just perfect~~ fifth, [[3/2]]		! Perfect fifth, [[3/2]]
	\| 7 \|\| 10 \|\| 11 \|\| 15		\| 7 \|\| 10 \|\| 11 \|\| 15
	\|-		\|-
	! Just ~~classic~~ major third, [[5/4]]		! Just major third, [[5/4]]
	\| 4 \|\| 5 \|\| 6 \|\| 8		\| 4 \|\| 5 \|\| 6 \|\| 8
	\|-		\|-
	! Just ~~classic~~ minor third, [[6/5]]		! Just minor third, [[6/5]]
	\| 3 \|\| 4 \|\| 5 \|\| 7		\| 3 \|\| 4 \|\| 5 \|\| 7
	\|-		\|-
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	\|}		\|}

	Of these intervals, the fifth plays an important role for characterizing [[~~EDO~~]] systems (as it defines the size of M2, m2, A1). Also, a simple test can show if [[circle-of-fifths notation]] can be applied to a given ~~EDO~~ system, because for this the sizes of fifth and octave must be relatively prime.		Of these intervals, the fifth plays an important role for characterizing [[edo]] systems (as it defines the size of M2, m2, A1). Also, a simple test can show if [[circle-of-fifths notation]] can be applied to a given edo system, because for this the sizes of fifth and octave must be relatively prime.

	[[Category:Terms]]		[[Category:Terms]]
	[[Category:Method]]		[[Category:Method]]
	~~[[Category:Val]]~~
	~~[[Category:Todo]]~~

Cmloegcmluin: "direct mapping" and "patent interval" → "direct approximation", per discussion (and page move)

2022-02-23T22:15:04Z

"direct mapping" and "patent interval" → "direct approximation", per discussion (and page move)

← Older revision		Revision as of 22:15, 23 February 2022
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	A '''~~patent interval~~''' in a given [[EDO]] is the number of EDO steps ~~needed to reach the best approximation of a given interval – usually, but not necessarily just – in~~ that ~~EDO. The method for calculating patent intervals is referred to as '''direct mapping'''~~, ~~and it involves~~ [[rounding]] the ~~product of~~ the [[Wikipedia: binary logarithm\|binary logarithm]] ~~(''log2'')~~ of the interval ~~ratio~~ (~~''r'') and the EDO number (''nEdo''~~).		A '''direct approximation''' of an interval in a given [[EDO]] is the number of EDO steps that most closely approximates it, found by [[rounding]] to the nearest integer the EDO number times the [[Wikipedia: binary logarithm\|binary logarithm]] of the interval: <math>⌈n_{\text{edo}}·\log_2(i)⌋</math>.

	~~round(log2(r)*nEdo)~~		== Examples of direct approximations ==

	A [[patent val]] is the best mapping of a representative set of intervals (taken to be [[generator]]s for a [[JI subgroup]]) in a given EDO; for the ''p''-[[prime limit]] this set consists of [[prime interval]]s.

	==== Examples of ~~Patent Intervals =~~===
	Of these intervals, the fifth plays an important role for characterizing [[EDO]] systems (as it defines the size of M2, m2, A1). Also, a simple test can show if [[circle-of-fifths notation]] can be applied to a given EDO system, because for this the sizes of fifth and octave must be relatively prime.

	{\| class="wikitable center-all"		{\| class="wikitable center-all"
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	\| 10 \|\| 14 \|\| 15 \|\| 21		\| 10 \|\| 14 \|\| 15 \|\| 21
	\|}		\|}

			Of these intervals, the fifth plays an important role for characterizing [[EDO]] systems (as it defines the size of M2, m2, A1). Also, a simple test can show if [[circle-of-fifths notation]] can be applied to a given EDO system, because for this the sizes of fifth and octave must be relatively prime.

	[[Category:Terms]]		[[Category:Terms]]

Xenwolf: Xenwolf moved page Patent interval to Direct approximation: as per suggestion of many users

2022-02-23T22:03:11Z

Xenwolf moved page Patent interval to Direct approximation: as per suggestion of many users

← Older revision	Revision as of 22:03, 23 February 2022
(No difference)

Xenwolf: add names to intervals, links to EDO pages, and some explanation about the "patent fifth"

2022-02-23T16:12:49Z

add names to intervals, links to EDO pages, and some explanation about the "patent fifth"

← Older revision		Revision as of 16:12, 23 February 2022
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	==== Examples of Patent Intervals ====		==== Examples of Patent Intervals ====
			Of these intervals, the fifth plays an important role for characterizing [[EDO]] systems (as it defines the size of M2, m2, A1). Also, a simple test can show if [[circle-of-fifths notation]] can be applied to a given EDO system, because for this the sizes of fifth and octave must be relatively prime.

	{\| class="wikitable center-all"		{\| class="wikitable center-all"
	! \		! Interval, ratio
	! 12edo \|\| 17edo \|\| 19edo \|\| 26edo		! [[12edo]] \|\| [[17edo]] \|\| [[19edo]] \|\| [[26edo]]
	\|-		\|-
	! 3/2		! Just perfect fifth, [[3/2]]
	\| 7 \|\| 10 \|\| 11 \|\| 15		\| 7 \|\| 10 \|\| 11 \|\| 15
	\|-		\|-
	! 5/4		! Just classic major third, [[5/4]]
	\| 4 \|\| 5 \|\| 6 \|\| 8		\| 4 \|\| 5 \|\| 6 \|\| 8
	\|-		\|-
	! 6/5		! Just classic minor third, [[6/5]]
	\| 3 \|\| 4 \|\| 5 \|\| 7		\| 3 \|\| 4 \|\| 5 \|\| 7
	\|-		\|-
	! 7/4		! Harmonic seventh, [[7/4]]
	\| 10 \|\| 14 \|\| 15 \|\| 21		\| 10 \|\| 14 \|\| 15 \|\| 21
	\|}		\|}

Aura: Deleted potentially confusing sentence

2021-12-22T02:58:59Z

Deleted potentially confusing sentence

← Older revision		Revision as of 02:58, 22 December 2021
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	round(log2(r)*nEdo)		round(log2(r)*nEdo)

	A [[patent val]] is the best mapping of a representative set of intervals (taken to be [[generator]]s for a [[JI subgroup]]) in a given EDO; for the ''p''-[[prime limit]] this set consists of [[prime interval]]s~~. Just as the patent val itself can be referred to as the "nearest edomapping", so a patent interval can be referred to as a "direct mapping"~~.		A [[patent val]] is the best mapping of a representative set of intervals (taken to be [[generator]]s for a [[JI subgroup]]) in a given EDO; for the ''p''-[[prime limit]] this set consists of [[prime interval]]s.

	==== Examples of Patent Intervals ====		==== Examples of Patent Intervals ====

Aura at 17:00, 19 January 2021

2021-01-19T17:00:20Z

← Older revision		Revision as of 17:00, 19 January 2021
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	A [[patent val]] is the best mapping of a representative set of intervals (taken to be [[generator]]s for a [[JI subgroup]]) in a given EDO; for the ''p''-[[prime limit]] this set consists of [[prime interval]]s. Just as the patent val itself can be referred to as the "nearest edomapping", so a patent interval can be referred to as a "direct mapping".		A [[patent val]] is the best mapping of a representative set of intervals (taken to be [[generator]]s for a [[JI subgroup]]) in a given EDO; for the ''p''-[[prime limit]] this set consists of [[prime interval]]s. Just as the patent val itself can be referred to as the "nearest edomapping", so a patent interval can be referred to as a "direct mapping".

	=== Examples of Patent Intervals ===		==== Examples of Patent Intervals ====
	{\| class="wikitable center-all"		{\| class="wikitable center-all"
	! \		! \

← Older revision		Revision as of 13:35, 21 February 2025
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	In [[regular temperament theory]], intervals are mapped through [[val]]s. Although more complex, it recognizes the fact that intervals like 3/2, 5/4, and 6/5 are related, as the number of steps of one interval is determined once the other two have been determined. The three situations in the above example correspond to using vals {{val\| 17 27 39 }}, {{val\| 17 27 40 }}, and {{val\| 17 26 39 }}, respectively.		In [[regular temperament theory]], intervals are mapped through [[val]]s. Although more complex, it recognizes the fact that intervals like 3/2, 5/4, and 6/5 are related, as the number of steps of one interval is determined once the other two have been determined. The three situations in the above example correspond to using vals {{val\| 17 27 39 }}, {{val\| 17 27 40 }}, and {{val\| 17 26 39 }}, respectively.

			[[Category:Interval]]
	[[Category:Terms]]		[[Category:Terms]]
	[[Category:Method]]		[[Category:Method]]