Mediant (operation) - Revision history

Fredg999: Degrees -> steps

2025-02-21T13:39:31Z

Degrees -> steps

← Older revision		Revision as of 13:39, 21 February 2025
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	The '''mediant''' ''M'' of two ratios ''a''/''c'' and ''b''/''d'' in lowest terms is ''M'' = (''a'' + ''b'')/(''c'' + ''d''). The result is always between the two ratios (''a''/''c'' < ''M'' < ''b''/''d'', assuming ''a''/''c'' < ''b''/''d'').		The '''mediant''' ''M'' of two ratios ''a''/''c'' and ''b''/''d'' in lowest terms is ''M'' = (''a'' + ''b'')/(''c'' + ''d''). The result is always between the two ratios (''a''/''c'' < ''M'' < ''b''/''d'', assuming ''a''/''c'' < ''b''/''d'').

	The mediant operation can be applied to [[just intonation]] ratios, to [[~~degree~~]]s of an [[edo]], etc.		The mediant operation can be applied to [[just intonation]] ratios, to [[step]]s of an [[edo]], etc.

	== Examples ==		== Examples ==

Fredg999: Improve lead section, rename 2nd section (not really generalizing the concept there), categories (moved :Category:Terms to Mediant, the disambiguation page)

2025-02-21T13:28:34Z

Improve lead section, rename 2nd section (not really generalizing the concept there), categories (moved Category:Terms to Mediant, the disambiguation page)

← Older revision		Revision as of 13:28, 21 February 2025
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	}}		}}
	{{Wikipedia\|Mediant (mathematics)}}		{{Wikipedia\|Mediant (mathematics)}}
	~~In the context of [[just intonation]] [[ratio]]s, the~~ '''mediant''' ''M'' of two ratios ''a''/''c'' and ''b''/''d'' in lowest terms is ''M'' = (''a'' + ''b'')/(''c'' + ''d''). ~~It will~~ always be between the two ratios (''a''/''c'' < ''M'' < ''b''/''d'', assuming ''a''/''c'' < ''b''/''d'').		The '''mediant''' ''M'' of two ratios ''a''/''c'' and ''b''/''d'' in lowest terms is ''M'' = (''a'' + ''b'')/(''c'' + ''d''). The result is always between the two ratios (''a''/''c'' < ''M'' < ''b''/''d'', assuming ''a''/''c'' < ''b''/''d'').

			The mediant operation can be applied to [[just intonation]] ratios, to [[degree]]s of an [[edo]], etc.

	== Examples ==		== Examples ==
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	\|}		\|}

	== ~~Generalization~~ ==		== Applications ==
	The mediant operation can also be used to find generators and scales in [[edo]]s representing [[temperament]]s. For example, the [[3/2\|perfect fifth (3/2)]] in [[12edo]] which [[support]]s [[meantone]] is 7 steps out of 12, and the fifth in [[19edo]], another meantone tuning, is 11 steps out of 19. Hence the perfect fifth in 31edo (which is a meantone tuning because 31 = 12 + 19; more precisely, the 5-limit [[val]] of 31edo is the sum of the 5-limit vals of 12edo and 19edo) is (7 + 11)\(12 + 19) = 18\31, which is in between the sizes of the 12edo fifth and the 19edo one.		The mediant operation can also be used to find generators and scales in [[edo]]s representing [[temperament]]s. For example, the [[3/2\|perfect fifth (3/2)]] in [[12edo]] which [[support]]s [[meantone]] is 7 steps out of 12, and the fifth in [[19edo]], another meantone tuning, is 11 steps out of 19. Hence the perfect fifth in 31edo (which is a meantone tuning because 31 = 12 + 19; more precisely, the 5-limit [[val]] of 31edo is the sum of the 5-limit vals of 12edo and 19edo) is (7 + 11)\(12 + 19) = 18\31, which is in between the sizes of the 12edo fifth and the 19edo one.

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	* [[Mediant hull]]: mediants applied infinitely to generators of several tunings to comprise a range of tunings		* [[Mediant hull]]: mediants applied infinitely to generators of several tunings to comprise a range of tunings

	~~[[Category:Terms]]~~
	[[Category:Interval]]		[[Category:Interval]]
	[[Category:Elementary math]]		[[Category:Elementary math]]
	[[Category:Method]]		[[Category:Method]]

Lériendil: Lériendil moved page Mediant (just intonation) to Mediant (operation): this applies both to JI fractions and to EDO fractions so calling it "(just intonation)" is a bit misleading

2025-02-21T06:08:27Z

Lériendil moved page Mediant (just intonation) to Mediant (operation): this applies both to JI fractions and to EDO fractions so calling it "(just intonation)" is a bit misleading

← Older revision	Revision as of 06:08, 21 February 2025
(No difference)

VectorGraphics: VectorGraphics moved page Mediant to Mediant (just intonation)

2025-02-21T03:23:40Z

VectorGraphics moved page Mediant to Mediant (just intonation)

← Older revision	Revision as of 03:23, 21 February 2025
(No difference)

FloraC: Style

2024-02-07T07:36:23Z

Style

← Older revision		Revision as of 07:36, 7 February 2024
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	}}		}}
	{{Wikipedia\|Mediant (mathematics)}}		{{Wikipedia\|Mediant (mathematics)}}
	In the context of [[just intonation]] ~~ratios~~, the '''mediant''' M of two ratios a/c and b/d in lowest terms is M=(a+b)/(c+d). It will always be between the two ratios (a/c < M < b/d, assuming a/c < b/d).		In the context of [[just intonation]] [[ratio]]s, the '''mediant''' ''M'' of two ratios ''a''/''c'' and ''b''/''d'' in lowest terms is ''M'' = (''a'' + ''b'')/(''c'' + ''d''). It will always be between the two ratios (''a''/''c'' < ''M'' < ''b''/''d'', assuming ''a''/''c'' < ''b''/''d'').

	== Examples ==		== Examples ==
	The following table shows the mediant ''m'' of some fraction pairs ''f1'', ''f2''.		The following table shows the mediant ''m'' of some fraction pairs ''f''<sub>1</sub>, ''f''<sub>2</sub>.

	{\| class="wikitable"		{\| class="wikitable center-1 center-2 center-3"
	! ''f1''		! ''f''<sub>1</sub>
	! ''f2''		! ''f''<sub>2</sub>
	! ''m''		! ''m''
	! ~~intermediate step~~(s)		! Intermediate Step(s)
	\|-		\|-
	\| [[3/2]] \|\| [[5/4]] \|\| [[4/3]] \|\| (3+5)/(2+4) = 8/6		\| [[3/2]] \|\| [[5/4]] \|\| [[4/3]] \|\| (3 + 5)/(2 + 4) = 8/6
	\|-		\|-
	\| [[3/2]] \|\| [[4/3]] \|\| [[7/5]] \|\| (3+4)/(2+3)		\| 3/2 \|\| 4/3 \|\| [[7/5]] \|\| (3 + 4)/(2 + 3)
	\|-		\|-
	\| [[5/4]] \|\| [[6/5]] \|\| [[11/9]] \|\| (5+6)/(4+5)		\| 5/4 \|\| [[6/5]] \|\| [[11/9]] \|\| (5 + 6)/(4 + 5)
	\|-		\|-
	\| [[9/8]] \|\| [[10/9]] \|\| [[19/17]] \|\| (9+10)/(8+9)		\| [[9/8]] \|\| [[10/9]] \|\| [[19/17]] \|\| (9 + 10)/(8 + 9)
	\|-		\|-
	\| 9/8 \|\| 19/17 \|\| [[28/25]] \|\| (9+19)/(8+17)		\| 9/8 \|\| 19/17 \|\| [[28/25]] \|\| (9 + 19)/(8 + 17)
	\|-		\|-
	\| 19/17 \|\| 10/9 \|\| [[29/26]] \|\| (19+10)/(17+9)		\| 19/17 \|\| 10/9 \|\| [[29/26]] \|\| (19 + 10)/(17 + 9)
	\|}		\|}

	== Generalization ==		== Generalization ==
	The mediant operation can also be used to find generators and scales in [[edo]]s representing ~~temperaments~~. For example, the perfect fifth ([[3/2]]) in 12edo which [[support]]s [[meantone]] is 7 steps out of 12, and the fifth in [[19edo]], another meantone tuning, is 11 steps out of 19. Hence the perfect fifth in 31edo (which is a meantone tuning because 31 = 12 + 19; more precisely, the 5-limit [[val]] of 31edo is the sum of the 5-limit vals of 12edo and 19edo) is (7+11)\(12+19) = 18\31, which is in between the sizes of the 12edo fifth and the 19edo one.		The mediant operation can also be used to find generators and scales in [[edo]]s representing [[temperament]]s. For example, the [[3/2\|perfect fifth (3/2)]] in [[12edo]] which [[support]]s [[meantone]] is 7 steps out of 12, and the fifth in [[19edo]], another meantone tuning, is 11 steps out of 19. Hence the perfect fifth in 31edo (which is a meantone tuning because 31 = 12 + 19; more precisely, the 5-limit [[val]] of 31edo is the sum of the 5-limit vals of 12edo and 19edo) is (7 + 11)\(12 + 19) = 18\31, which is in between the sizes of the 12edo fifth and the 19edo one.

	Given a target interval x (written logarithmically in octaves), the [[relative error]] of the mediant of two edo approximations a\m and b\n to x is the sum of the respective relative errors of a\m and b\n. Since x is exactly equal to xm\m in m-edo and xn\n in n-edo, the absolute error of the approximation (a+b)\(m+n) is		Given a target interval ''x'' (written logarithmically in octaves), the [[relative error]] of the mediant of two edo approximations ''a''\''m'' and ''b''\''n'' to ''x'' is the sum of the respective relative errors of ''a''\''m'' and ''b''\''n''. Since ''x'' is exactly equal to ''xm''\''m'' in ''m''-edo and ''xn''\''n'' in ''n''-edo, the absolute error of the approximation (''a'' + ''b'')\(''m'' + ''n'') is

	(a+b)\(m+n) − x = (a+b)\(m+n) − x(m+n)\(m+n) = [(a-xm)+(b-xn)]\(m+n).		(''a'' + ''b'')\(''m'' + ''n'') − ''x'' = (''a'' + ''b'')\(''m'' + ''n'') − ''x''(''m'' + ''n'')\(''m'' + ''n'') = [(''a'' - ''xm'') + (''b'' - ''xn'')]\(''m'' + ''n'').

	The relative error in edo steps is thus		The relative error in edo steps is thus

	[(a+b)\(m+n) − x](m+n) = (a-xm) + (b-xn),		[(''a'' + ''b'')\(''m'' + ''n'') − ''x''](''m'' + ''n'') = (''a'' - ''xm'') + (''b'' - ''xn''),

	which is the sum of the relative errors in m- and n-edo.		which is the sum of the relative errors in m- and n-edo.

	Edos admitting a [[5L 2s]] diatonic ~~MOS~~ subscale can be generated by taking mediants of 4\7 (the fifth is too flat and 5L 2s equalizes (L = s) into [[7edo]]) and 3\5 (the fifth is too sharp and 5L 2s collapses (s = 0) into [[5edo]]), the first generation being the 12edo diatonic generator 7\12, the second generation being 10\17 and 11\19 fifths, and so on.		Edos admitting a [[5L 2s]] diatonic mos subscale can be generated by taking mediants of 4\7 (the fifth is too flat and 5L 2s equalizes (L = s) into [[7edo]]) and 3\5 (the fifth is too sharp and 5L 2s collapses (s = 0) into [[5edo]]), the first generation being the 12edo diatonic generator 7\12, the second generation being 10\17 and 11\19 fifths, and so on.

	This holds for any mos pattern ~~aL bs~~, with a-edo and (a+b)-edo for collapsed and equalized tunings of ~~aL bs~~, corresponding to hardness 1/1 and 1/0. If gcd(k, m) = gcd(l, n) = 1, and k\m of m-edo is the [[bright]] generator of the ~~aL bs~~ tuning of hardness p/q (in lowest terms) and l\n of n-edo is the bright generator of the tuning of hardness r/s (in lowest terms), the mediant generator (k+l)\(m+n) generates ~~aL bs~~ with hardness (p+r)/(q+s). See [[5L 2s#Scale tree]]. The preceding paragraph also applies to non-octave equave scales, where edo tunings are replaced with the appropriate ed-equave tunings.		This holds for any mos pattern ''a''L ''b''s, with ''a''-edo and (''a'' + ''b'')-edo for collapsed and equalized tunings of ''a''L ''b''s, corresponding to hardness 1/1 and 1/0. If gcd(''k'', ''m'') = gcd(''l'', ''n'') = 1, and ''k''\''m'' of ''m''-edo is the [[bright]] generator of the ''a''L ''b''s tuning of hardness ''p''/''q'' (in lowest terms) and ''l''\''n'' of ''n''-edo is the bright generator of the tuning of hardness ''r''/''s'' (in lowest terms), the mediant generator (''k'' + ''l'')\(''m'' + ''n'') generates ''a''L ''b''s with hardness (''p'' + ''r'')/(''q'' + ''s''). See [[5L 2s #Scale tree]]. The preceding paragraph also applies to non-octave equave scales, where edo tunings are replaced with the appropriate ed-equave tunings.

	== See also ==		== See also ==

Fredg999 category edits: Removing from Category:Theory using Cat-a-lot

2023-03-04T20:26:17Z

Removing from Category:Theory using Cat-a-lot

← Older revision		Revision as of 20:26, 4 March 2023
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	* [[Mediant hull]]: mediants applied infinitely to generators of several tunings to comprise a range of tunings		* [[Mediant hull]]: mediants applied infinitely to generators of several tunings to comprise a range of tunings

	~~[[Category:Theory]]~~
	[[Category:Terms]]		[[Category:Terms]]
	[[Category:Interval]]		[[Category:Interval]]
	[[Category:Elementary math]]		[[Category:Elementary math]]
	[[Category:Method]]		[[Category:Method]]

Cmloegcmluin: /* See also */ explain link

2023-02-21T21:07:51Z

Inthar at 01:29, 20 February 2023

2023-02-20T01:29:12Z

← Older revision		Revision as of 01:29, 20 February 2023
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	Edos admitting a [[5L 2s]] diatonic MOS subscale can be generated by taking mediants of 4\7 (the fifth is too flat and 5L 2s equalizes (L = s) into [[7edo]]) and 3\5 (the fifth is too sharp and 5L 2s collapses (s = 0) into [[5edo]]), the first generation being the 12edo diatonic generator 7\12, the second generation being 10\17 and 11\19 fifths, and so on.		Edos admitting a [[5L 2s]] diatonic MOS subscale can be generated by taking mediants of 4\7 (the fifth is too flat and 5L 2s equalizes (L = s) into [[7edo]]) and 3\5 (the fifth is too sharp and 5L 2s collapses (s = 0) into [[5edo]]), the first generation being the 12edo diatonic generator 7\12, the second generation being 10\17 and 11\19 fifths, and so on.

	This holds for any mos pattern aL bs, with a-edo and (a+b)-edo for collapsed and equalized tunings of aL bs, corresponding to hardness 1/1 and 1/0. If gcd(k, m) = gcd(l, n) = 1, and k\m is the [[bright]] generator of the aL bs tuning of hardness p/q (in lowest terms) and l\n is the bright generator of the tuning of hardness r/s (in lowest terms), the mediant generator (k+l)\(m+n) generates aL bs with hardness (p+r)/(q+s). See [[5L 2s#Scale tree]].		This holds for any mos pattern aL bs, with a-edo and (a+b)-edo for collapsed and equalized tunings of aL bs, corresponding to hardness 1/1 and 1/0. If gcd(k, m) = gcd(l, n) = 1, and k\m of m-edo is the [[bright]] generator of the aL bs tuning of hardness p/q (in lowest terms) and l\n of n-edo is the bright generator of the tuning of hardness r/s (in lowest terms), the mediant generator (k+l)\(m+n) generates aL bs with hardness (p+r)/(q+s). See [[5L 2s#Scale tree]]. The preceding paragraph also applies to non-octave equave scales, where edo tunings are replaced with the appropriate ed-equave tunings.

	== See also ==		== See also ==

Inthar: /* Generalization */

2023-02-20T01:19:09Z

Generalization

← Older revision		Revision as of 01:19, 20 February 2023
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	Edos admitting a [[5L 2s]] diatonic MOS subscale can be generated by taking mediants of 4\7 (the fifth is too flat and 5L 2s equalizes (L = s) into [[7edo]]) and 3\5 (the fifth is too sharp and 5L 2s collapses (s = 0) into [[5edo]]), the first generation being the 12edo diatonic generator 7\12, the second generation being 10\17 and 11\19 fifths, and so on.		Edos admitting a [[5L 2s]] diatonic MOS subscale can be generated by taking mediants of 4\7 (the fifth is too flat and 5L 2s equalizes (L = s) into [[7edo]]) and 3\5 (the fifth is too sharp and 5L 2s collapses (s = 0) into [[5edo]]), the first generation being the 12edo diatonic generator 7\12, the second generation being 10\17 and 11\19 fifths, and so on.

	This holds for any mos pattern aL bs, with a-edo and (a+b)-edo for collapsed and equalized tunings of aL bs. If gcd(k, m) = gcd(l, n) = 1, and k\m is the [[bright]] generator of the aL bs tuning of hardness p/q (in lowest terms) and l\n is the bright generator of the tuning of hardness r/s (in lowest terms), the mediant generator (k+l)\(m+n) generates aL bs with hardness (p+r)/(q+s). See [[5L 2s#Scale tree]].		This holds for any mos pattern aL bs, with a-edo and (a+b)-edo for collapsed and equalized tunings of aL bs, corresponding to hardness 1/1 and 1/0. If gcd(k, m) = gcd(l, n) = 1, and k\m is the [[bright]] generator of the aL bs tuning of hardness p/q (in lowest terms) and l\n is the bright generator of the tuning of hardness r/s (in lowest terms), the mediant generator (k+l)\(m+n) generates aL bs with hardness (p+r)/(q+s). See [[5L 2s#Scale tree]].

	== See also ==		== See also ==

Inthar: /* Generalization */

2023-02-20T01:16:46Z

Generalization

← Older revision		Revision as of 01:16, 20 February 2023
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	Given a target interval x (written logarithmically in octaves), the [[relative error]] of the mediant of two edo approximations a\m and b\n to x is the sum of the respective relative errors of a\m and b\n. Since x is exactly equal to xm\m in m-edo and xn\n in n-edo, the absolute error of the approximation (a+b)\(m+n) is		Given a target interval x (written logarithmically in octaves), the [[relative error]] of the mediant of two edo approximations a\m and b\n to x is the sum of the respective relative errors of a\m and b\n. Since x is exactly equal to xm\m in m-edo and xn\n in n-edo, the absolute error of the approximation (a+b)\(m+n) is

	[(a+b)\(m+n) − x~~](m+n)~~ = (a+b)\(m+n) − x(m+n)\(m+n) = [(a-xm)+(b-xn)]\(m+n).		(a+b)\(m+n) − x = (a+b)\(m+n) − x(m+n)\(m+n) = [(a-xm)+(b-xn)]\(m+n).

	The relative error in edo steps is thus		The relative error in edo steps is thus

← Older revision		Revision as of 21:07, 21 February 2023
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	== See also ==		== See also ==
	* [[Merciful intonation]]		* [[Merciful intonation]]: uses noble (phi-weighted) mediants of just intoned intervals
	* [[Mediant hull]] — mediants applied infinitely to generators of several tunings to comprise a range of tunings		* [[Mediant hull]]: mediants applied infinitely to generators of several tunings to comprise a range of tunings

	[[Category:Theory]]		[[Category:Theory]]