Perfect balance: Difference between revisions

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The only perfectly balanced scale within a [[prime EDO]] ''N'' is the equally spaced scale containing every tone in ''N''-EDO. For a composite EDO ''N'', one can construct new perfectly balanced scales by superimposing equally spaced scales of size ''k'' where ''k'' is a divisor of ''N'', such that the scales are transposed so that no pitches coincide. For example, in [[18edo]] the 3edo {0, 6, 12} and the [[2edo]] {1, 10} may be combined to form the perfectly balanced scale {0, 1, 6, 10, 12}.

This leads to the question of whether every perfectly balanced scale within an EDO is decomposable into a union of one or more equally spaced disjoint scales. This turns out to be false, and counterexamples can occur when ''N'' has three or more distinct prime factors. The smallest EDO with three distinct prime factors is [[30edo]], and ~~indeed~~ it has six ~~such~~ scales up to transposition:

This leads to the question of whether every perfectly balanced scale within an EDO is decomposable into a union of one or more equally spaced disjoint scales. This turns out to be false, and counterexamples can occur when ''N'' has three or more distinct prime factors. The smallest EDO with three distinct prime factors is [[30edo]], and it has six scales up to transposition:

{0, 1, 7, 13, 19, 20}

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{0, 1, 2, 8, 12, 14, 18, 20, 24}

Searching for ~~these~~ scales is nontrivial. Milne et al. [http://www.dynamictonality.com/perfect_balance_files/ computed all such patterns] for products of three distinct primes up to ''N'' = 102.

These scales along with the evenly spaced scales of [[2edo]], [[3edo]], and [[5edo]] are the full set of "minimal" perfectly balanced scales in 30edo, which cannot be expressed as the union of two disjoint perfectly balanced scales.

Searching for minimal perfectly balanced scales is nontrivial. Milne et al. [http://www.dynamictonality.com/perfect_balance_files/ computed all such patterns] for products of three distinct primes up to ''N'' = 102.

== Outside EDOs ==

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 The only perfectly balanced scale within a [[prime EDO]] ''N'' is the equally spaced scale containing every tone in ''N''-EDO. For a composite EDO ''N'', one can construct new perfectly balanced scales by superimposing equally spaced scales of size ''k'' where ''k'' is a divisor of ''N'', such that the scales are transposed so that no pitches coincide. For example, in [[18edo]] the 3edo {0, 6, 12} and the [[2edo]] {1, 10} may be combined to form the perfectly balanced scale {0, 1, 6, 10, 12}.
-This leads to the question of whether every perfectly balanced scale within an EDO is decomposable into a union of one or more equally spaced disjoint scales. This turns out to be false, and counterexamples can occur when ''N'' has three or more distinct prime factors. The smallest EDO with three distinct prime factors is [[30edo]], and indeed it has six such scales up to transposition:
+This leads to the question of whether every perfectly balanced scale within an EDO is decomposable into a union of one or more equally spaced disjoint scales. This turns out to be false, and counterexamples can occur when ''N'' has three or more distinct prime factors. The smallest EDO with three distinct prime factors is [[30edo]], and it has six scales up to transposition:
      {0, 1, 7, 13, 19, 20}
@@ Line 20: / Line 20: @@
      {0, 1, 2, 8, 12, 14, 18, 20, 24}
-Searching for these scales is nontrivial. Milne et al. [http://www.dynamictonality.com/perfect_balance_files/ computed all such patterns] for products of three distinct primes up to ''N'' = 102.
+These scales along with the evenly spaced scales of [[2edo]], [[3edo]], and [[5edo]] are the full set of "minimal" perfectly balanced scales in 30edo, which cannot be expressed as the union of two disjoint perfectly balanced scales.
+Searching for minimal perfectly balanced scales is nontrivial. Milne et al. [http://www.dynamictonality.com/perfect_balance_files/ computed all such patterns] for products of three distinct primes up to ''N'' = 102.
 == Outside EDOs ==