Mediant (operation): Difference between revisions
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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 be applied to [[just intonation]] ratios, to [[step]]s of an [[edo]], etc. | |||
== Examples == | |||
The following table shows the mediant ''m'' of some fraction pairs ''f''<sub>1</sub>, ''f''<sub>2</sub>. | |||
{| 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) | |||
|} | |||
== 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]] | |||
Latest revision as of 13:39, 21 February 2025
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 steps of an edo, etc.
Examples
The following table shows the mediant m of some fraction pairs f1, f2.
| f1 | f2 | 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) |
Applications
The mediant operation can also be used to find generators and scales in edos representing temperaments. For example, the perfect fifth (3/2) in 12edo which supports 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 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
- 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