Cubic and octahedral limits: Difference between revisions
Created page with "The cubic and octahedral limits are alternative ways to limit the complexity of intervals intervals compared to the odd limit. == Cubic limit == The '''cubic limit''' or '''exponential limit''' places a limit on the exponents allowed in the prime factorization of a number. The '''reduced cubic limit''' is similar, but ignores the prime 2, allowing for unlimited octave-reduction, and similar metrics can be defined for other prime equaves. For example, 64/63, since i..." |
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== Octahedral limit == | == Octahedral limit == | ||
The '''octahedral limit''' | 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. | 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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[[Category:Regular temperament theory]] | |||
[[Category:Odd limit]] | |||
[[Category:Limit]] | |||
[[Category:Terms]] | |||