Mediant (operation): Difference between revisions

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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.

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 the generator of the aL bs tuning of hardness p/q (in lowest terms) and l\n is the 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. 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 ==

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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.
-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 the generator of the aL bs tuning of hardness p/q (in lowest terms) and l\n is the 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. 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 ==