VectorGraphics at 02:02, 28 April 2025

2025-04-28T02:02:19Z

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	== Octahedral limit ==		== 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 (222222)/(33*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 (222222)/(33*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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2025-04-25T02:48:35Z

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			[[Category:Regular temperament theory]]
			[[Category:Odd limit]]
			[[Category:Limit]]
			[[Category:Terms]]

VectorGraphics: 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..."

2025-04-24T23:44:15Z

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..."

New page

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 its monzo is [6 -2 0 -1⟩, is in the 6-cubic-limit including 2, but in the reduced 2-cubic-limit (where it is equivalent to 1/63).

To find the cubic limit of a ratio, find the largest entry in its monzo by absolute value value (excluding the first entry for the reduced cubic limit). The set of intervals in a given cubic limit ''n'' is defined by [a b c d ...⟩ where each value can range from -n to n. Given a prime subgroup with p primes (or, for the reduced cubic limit, a prime subgroup with p odd primes), the number of intervals in a cubic limit c is (2c+1)^p.

Here is the set of all octave-reduced intervals in the 5-limit reduced 1-cubic limit. Bolded intervals are also in the full 5-limit 1-cubic-limit:
{| class="wikitable"
!Interval
!Fives
!Threes
!Twos
|-
|16/15
| -1
| -1
|4
|-
|4/3
|0
| -1
|2
|-
|'''5/3'''
|1
| -1
|0
|-
|8/5
| -1
|0
|3
|-
|'''1/1'''
|0
|0
|0
|-
|5/4
|1
|0
| -2
|-
|'''6/5'''
| -1
|1
|1
|-
|'''3/2'''
|0
|1
| -1
|-
|15/8
|1
|1
| -3
|}
Here is the set of all octave-reduced intervals in the 5-limit reduced 2-cubic limit. Bolded intervals are also in the full 5-limit limit 2-cubic-limit:
{| class="wikitable"
|+
!Interval
!Fives
!Threes
!Twos
|-
|256/225
| -2
| -2
|8
|-
|64/45
| -1
| -2
|6
|-
|16/9
|0
| -2
|4
|-
|'''10/9'''
|1
| -2
|1
|-
|'''25/18'''
|2
| -2
| -1
|-
|128/75
| -2
| -1
|7
|-
|16/15
| -1
| -1
|4
|-
|'''4/3'''
|0
| -1
|2
|-
|'''5/3'''
|1
| -1
|0
|-
|25/24
|2
| -1
| -3
|-
|32/25
| -2
|0
|5
|-
|8/5
| -1
|0
|3
|-
|'''1/1'''
|0
|0
|0
|-
|'''5/4'''
|1
|0
| -2
|-
|25/16
|2
|0
| -4
|-
|48/25
| -2
|1
|4
|-
|'''6/5'''
| -1
|1
|1
|-
|'''3/2'''
|0
|1
| -1
|-
|15/8
|1
|1
| -3
|-
|75/64
|2
|1
| -6
|-
|'''36/25'''
| -2
|2
|2
|-
|'''9/5'''
| -1
|2
|0
|-
|9/8
|0
|2
| -3
|-
|45/32
|1
|2
| -5
|-
|225/128
|2
|2
| -7
|}

== 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).

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.

Here is the set of all octave-reduced intervals intervals in the 5-limit reduced 1-octahedral-limit. Bolded intervals are also in the full 5-limit 2-octahedral-limit.
{| class="wikitable"
!Interval
!Fives
!Threes
!Twos
!Sum (without twos)
!Sum (with twos twos)
|-
|4/3
|0
| -1
|2
|1
|3
|-
|8/5
| -1
|0
|3
|1
|4
|-
|'''1/1'''
|0
|0
|0
|0
|0
|-
|5/4
|1
|0
| -2
|1
|3
|-
|'''3/2'''
|0
|1
| -1
|1
|2
|}

Here is the set of all octave-reduced intervals in the 5-limit reduced 2-octahedral limit. Bolded intervals are also in the full 5-limit 3-octahedral-limit.
{| class="wikitable"
!Interval
!Fives
!Threes
!Twos
!Sum (without twos)
!Sum (with twos twos)
|-
|16/9
|0
| -2
|4
|2
|6
|-
|16/15
| -1
| -1
|4
|2
|6
|-
|'''4/3'''
|0
| -1
|2
|1
|3
|-
|'''5/3'''
|1
| -1
|0
|2
|2
|-
|32/25
| -2
|0
|5
|2
|7
|-
|8/5
| -1
|0
|3
|1
|4
|-
|'''1/1'''
|0
|0
|0
|0
|0
|-
|'''5/4'''
|1
|0
| -2
|1
|3
|-
|25/16
|2
|0
| -4
|2
|6
|-
|'''6/5'''
| -1
|1
|1
|2
|3
|-
|'''3/2'''
|0
|1
| -1
|1
|2
|-
|15/8
|1
|1
| -3
|2
|5
|-
|9/8
|0
|2
| -3
|2
|5
|}

Here is the set set of all octave-reduced intervals in the 5-limit reduced 3-octahedral limit. Bolded intervals are also in the full 5-limit 4-octahedral-limit.
{| class="wikitable"
!Interval
!Fives
!Threes
!Twos
!Sum (without twos)
!Sum (with twos twos)
|-
|32/27
|0
| -3
|5
|3
|8
|-
|64/45
| -1
| -2
|6
|3
|9
|-
|16/9
|0
| -2
|4
|2
|6
|-
|'''10/9'''
|1
| -2
|1
|3
|4
|-
|128/75
| -2
| -1
|7
|3
|10
|-
|16/15
| -1
| -1
|4
|2
|6
|-
|'''4/3'''
|0
| -1
|2
|1
|3
|-
|'''5/3'''
|1
| -1
|0
|2
|2
|-
|25/24
|2
| -1
| -3
|3
|6
|-
|128/125
| -3
|0
|7
|3
|10
|-
|32/25
| -2
|0
|5
|2
|7
|-
|'''8/5'''
| -1
|0
|3
|1
|4
|-
|'''1/1'''
|0
|0
|0
|0
|0
|-
|'''5/4'''
|1
|0
| -2
|1
|3
|-
|25/16
|2
|0
| -4
|2
|6
|-
|125/64
|3
|0
| -6
|3
|9
|-
|48/25
| -2
|1
|4
|3
|7
|-
|'''6/5'''
| -1
|1
|1
|2
|3
|-
|'''3/2'''
|0
|1
| -1
|1
|2
|-
|15/8
|1
|1
| -3
|2
|5
|-
|75/64
|2
|1
| -6
|3
|9
|-
|'''9/5'''
| -1
|2
|0
|3
|3
|-
|9/8
|0
|2
| -3
|2
|5
|-
|45/32
|1
|2
| -5
|3
|8
|-
|27/16
|0
|3
| -4
|3
|7
|}

Cubic and octahedral limits - Revision history

VectorGraphics at 02:02, 28 April 2025

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