Mediant (operation): Difference between revisions

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~~In the context of JI 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).

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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 operation can ~~also~~ be ~~used~~ to ~~find generators and scales in~~ [[~~edo~~]]s ~~representing temperaments. For example, the perfect fifth (3/2) in 12edo is 7 steps out~~ of 12, ~~and the fifth in 19edo is 11 steps out of 19. Hence the perfect fifth in 31edo (= 12 + 19) is (7+11)\(12+19) = 18\31~~.

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

~~[http:~~//~~en.wikipedia~~.~~org/wiki/Mediant_(mathematics) Wikipedia article on the mediant]~~

== Examples ==

The following table shows the mediant ''m'' of some fraction pairs ''f''1, ''f''2.

~~also see:~~

{| class="wikitable center-1 center-2 center-3"

! ''f''1

! ''f''2

! ''m''

! Intermediate Step(s)

|-

| [[3/2]] || [[5/4]] || [[4/3]] || (3 + 5)/(2 + 4) = 8/6

|-

| 3/2 || 4/3 || [[7/5]] || (3 + 4)/(2 + 3)

|-

| 5/4 || [[6/5]] || [[11/9]] || (5 + 6)/(4 + 5)

|-

| [[9/8]] || [[10/9]] || [[19/17]] || (9 + 10)/(8 + 9)

|-

| 9/8 || 19/17 || [[28/25]] || (9 + 19)/(8 + 17)

|-

| 19/17 || 10/9 || [[29/26]] || (19 + 10)/(17 + 9)

|}

[~~http:~~//~~dkeenan~~.~~com~~/~~Music~~/~~NobleMediant~~.~~txt~~ The ~~Noble~~ Mediant: ~~Complex ratios and metastable musical intervals~~]~~, by~~ [[~~Margo_Schulter|Margo Schulter~~]] ~~and~~ [[~~David_Keenan|David Keenan~~]]

== 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.

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'').

The relative error in edo steps is thus

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

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.

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 ==

* [[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

[[Category:Interval]]

[[Category:Elementary math]]

[[Category:Method]]

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-In the context of JI 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 &lt; M &lt; b/d, assuming a/c &lt; b/d).
+{{Interwiki
+| de = Mediant
+| en = Mediant
+| es =
+| ja =
+}}
+{{Wikipedia|Mediant (mathematics)}}
+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'' &lt; ''M'' &lt; ''b''/''d'', assuming ''a''/''c'' &lt; ''b''/''d'').
-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 is 7 steps out of 12, and the fifth in 19edo is 11 steps out of 19. Hence the perfect fifth in 31edo (= 12 + 19) is (7+11)\(12+19) = 18\31.
+The mediant operation can be applied to [[just intonation]] ratios, to [[step]]s of an [[edo]], etc.
-[http://en.wikipedia.org/wiki/Mediant_(mathematics) Wikipedia article on the mediant]
+== Examples ==
+The following table shows the mediant ''m'' of some fraction pairs ''f''<sub>1</sub>, ''f''<sub>2</sub>.
-also see:
+{| class="wikitable center-1 center-2 center-3"
+! ''f''<sub>1</sub>
+! ''f''<sub>2</sub>
+! ''m''
+! Intermediate Step(s)
+|-
+| [[3/2]] || [[5/4]] || [[4/3]] || (3 + 5)/(2 + 4) = 8/6
+|-
+| 3/2 || 4/3 || [[7/5]] || (3 + 4)/(2 + 3)
+|-
+| 5/4 || [[6/5]] || [[11/9]] || (5 + 6)/(4 + 5)
+|-
+| [[9/8]] || [[10/9]] || [[19/17]] || (9 + 10)/(8 + 9)
+|-
+| 9/8 || 19/17 || [[28/25]] || (9 + 19)/(8 + 17)
+|-
+| 19/17 || 10/9 || [[29/26]] || (19 + 10)/(17 + 9)
+|}
-[http://dkeenan.com/Music/NobleMediant.txt The Noble Mediant: Complex ratios and metastable musical intervals], by [[Margo_Schulter|Margo Schulter]] and [[David_Keenan|David Keenan]]
+== 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.
+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'') &minus; ''x'' = (''a'' + ''b'')\(''m'' + ''n'') &minus; ''x''(''m'' + ''n'')\(''m'' + ''n'') = [(''a'' - ''xm'') + (''b'' - ''xn'')]\(''m'' + ''n'').
+The relative error in edo steps is thus
+[(''a'' + ''b'')\(''m'' + ''n'') &minus; ''x''](''m'' + ''n'') = (''a'' - ''xm'') + (''b'' - ''xn''),
+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.
+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 ==
+* [[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
+[[Category:Interval]]
+[[Category:Elementary math]]
+[[Category:Method]]