Cubic and octahedral limits: Difference between revisions

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== Octahedral limit ==

The '''octahedral limit'''~~, '''[[Crystal ball|Hahn]] limit''',~~ or '''cardinal limit''' places a limit on the total number of prime factors allowed for a ratio, counting repeats. The '''reduced octahedral limit''' is similar, but ignores the prime 2, allowing for unlimited octave-reduction, and similar metrics can be defined for other prime equaves equaves. For example, 64/63 is (2*2*2*2*2*2)/(3*3*7), which means it is in the 9-octahedral limit. However, six of these prime factors are 2, so it is in the reduced 3-octahedral-limit (where it is equivalent to 1/63).

The '''octahedral limit''' or '''cardinal limit''' places a limit on the total number of prime factors allowed for a ratio, counting repeats. The '''reduced octahedral limit''' is similar, but ignores the prime 2, allowing for unlimited octave-reduction, and similar metrics can be defined for other prime equaves equaves. For example, 64/63 is (2*2*2*2*2*2)/(3*3*7), which means it is in the 9-octahedral limit. However, six of these prime factors are 2, so it is in the reduced 3-octahedral-limit (where it is equivalent to 1/63).

To find the octahedral limit of a ratio, sum up the absolute values of its monzo's entries (excluding the first entry for the reduced octahedral limit). The 2-octahedral-limit is equivalent to the semiprimes, and the 1-octahedral-limit is equivalent to the primes.

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 == Octahedral limit ==
-The '''octahedral limit''', '''[[Crystal ball|Hahn]] limit''', or '''cardinal limit''' places a limit on the total number of prime factors allowed for a ratio, counting repeats. The '''reduced octahedral limit''' is similar, but ignores the prime 2, allowing for unlimited octave-reduction, and similar metrics can be defined for other prime equaves equaves. For example, 64/63 is (2*2*2*2*2*2)/(3*3*7), which means it is in the 9-octahedral limit. However, six of these prime factors are 2, so it is in the reduced 3-octahedral-limit (where it is equivalent to 1/63).
+The '''octahedral limit''' or '''cardinal limit''' places a limit on the total number of prime factors allowed for a ratio, counting repeats. The '''reduced octahedral limit''' is similar, but ignores the prime 2, allowing for unlimited octave-reduction, and similar metrics can be defined for other prime equaves equaves. For example, 64/63 is (2*2*2*2*2*2)/(3*3*7), which means it is in the 9-octahedral limit. However, six of these prime factors are 2, so it is in the reduced 3-octahedral-limit (where it is equivalent to 1/63).
 To find the octahedral limit of a ratio, sum up the absolute values of its monzo's entries (excluding the first entry for the reduced octahedral limit). The 2-octahedral-limit is equivalent to the semiprimes, and the 1-octahedral-limit is equivalent to the primes.