User:Zeta Function/Half-cubic limit - Revision history

Zeta Function: Zeta Function moved page Half-cubic limit to User:Zeta Function/Half-cubic limit: Idea that needs some work

2026-07-30T18:58:32Z

Zeta Function moved page Half-cubic limit to User:Zeta Function/Half-cubic limit: Idea that needs some work

← Older revision	Revision as of 18:58, 30 July 2026
(No difference)

Fredg999: Idiosyncratic term

2026-07-30T18:52:56Z

Idiosyncratic term

← Older revision		Revision as of 18:52, 30 July 2026
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	The half-cubic limit for a prime limit is defined to be the set of all intervals within a certain prime limit whose numerators and denominators both cannot contain more than two non-2 prime factors (including multiplicity), where all intervals are octave-reduced. In this way, it is analogous to the reduced 2-cubic limit but is significantly more restricted, which explains the reason for its naming as the half-cubic limit.		The '''half-cubic limit'''{{idio}} for a [[prime limit]] is defined to be the set of all intervals within a certain prime limit whose numerators and denominators both cannot contain more than two non-2 prime factors (including multiplicity), where all intervals are octave-reduced. In this way, it is analogous to the reduced 2-cubic limit but is significantly more restricted, which explains the reason for its naming as the half-cubic limit.

	Here is the set of all intervals in the 3-half-cubic limit.		Here is the set of all intervals in the 3-half-cubic limit.

Zeta Function: Fix error

2026-07-30T18:43:27Z

Fix error

← Older revision		Revision as of 18:43, 30 July 2026
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	The half-cubic limit for a prime limit is defined to be the set of all intervals within a certain prime limit whose numerators and denominators both cannot contain more than two non-2 prime factors (including multiplicity), where all intervals are octave-reduced. In this way, it is analogous to the reduced 1-cubic limit but is significantly more restricted, which explains the reason for its naming as the half-cubic limit.		The half-cubic limit for a prime limit is defined to be the set of all intervals within a certain prime limit whose numerators and denominators both cannot contain more than two non-2 prime factors (including multiplicity), where all intervals are octave-reduced. In this way, it is analogous to the reduced 2-cubic limit but is significantly more restricted, which explains the reason for its naming as the half-cubic limit.

	Here is the set of all intervals in the 3-half-cubic limit.		Here is the set of all intervals in the 3-half-cubic limit.

Zeta Function: /* Generalization */ Fix formatting

2026-07-30T18:43:01Z

Generalization: Fix formatting

← Older revision		Revision as of 18:43, 30 July 2026
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	== Generalization ==		== Generalization ==
	In general, the (p, q, a)-half-cubic limit is the set of octave-reduced intervals in the p-prime limit and a-odd limit whose numerators and denominators do not contain more than q prime factors (including multiplicity). Thus, the previously defined limit would correspond to a (p, 2, ~~{{math\|∞}}~~)-half cubic limit. This generalization is designed to give more flexibility during composing than the first framework. A good example of such a limit might be the (13, 2, 65)-half cubic limit, which is suited to 13-limit composing.		In general, the (p, q, a)-half-cubic limit is the set of octave-reduced intervals in the p-prime limit and a-odd limit whose numerators and denominators do not contain more than q prime factors (including multiplicity). Thus, the previously defined limit would correspond to a (p, 2, ∞)-half cubic limit. This generalization is designed to give more flexibility during composing than the first framework. A good example of such a limit might be the (13, 2, 65)-half cubic limit, which is suited to 13-limit composing.
	[[Category:Limit]]		[[Category:Limit]]

Zeta Function: Add generalization

2026-07-30T18:41:31Z

Add generalization

← Older revision		Revision as of 18:41, 30 July 2026
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	The half-cubic limit for a prime limit is defined to be the set of all intervals within a certain prime limit whose numerators and denominators both cannot contain more than two non-2 prime factors, where all intervals are octave-reduced. In this way, it is analogous to the reduced 1-cubic limit but is significantly more restricted, which explains the reason for its naming as the half-cubic limit.		The half-cubic limit for a prime limit is defined to be the set of all intervals within a certain prime limit whose numerators and denominators both cannot contain more than two non-2 prime factors (including multiplicity), where all intervals are octave-reduced. In this way, it is analogous to the reduced 1-cubic limit but is significantly more restricted, which explains the reason for its naming as the half-cubic limit.

	Here is the set of all intervals in the 3-half-cubic limit.		Here is the set of all intervals in the 3-half-cubic limit.
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	\|2		\|2
	\|}		\|}
	This limit was specifically designed to be useful for composition and scale-building purposes, as it attempts to chart a middle path between the restrictiveness of the odd limit and the excessive complexity of intervals that is acceptable with prime-limits. ~~If one wants to make it more useful~~, ~~they can~~ set a ~~high~~-~~ish~~ odd limit such ~~as 41 or 99~~ to ~~filter out even more intervals~~.		This limit was specifically designed to be useful for composition and scale-building purposes, as it attempts to chart a middle path between the restrictiveness of the odd limit and the excessive complexity of intervals that is acceptable with prime-limits.

			== Generalization ==
			In general, the (p, q, a)-half-cubic limit is the set of octave-reduced intervals in the p-prime limit and a-odd limit whose numerators and denominators do not contain more than q prime factors (including multiplicity). Thus, the previously defined limit would correspond to a (p, 2, {{math\|∞}})-half cubic limit. This generalization is designed to give more flexibility during composing than the first framework. A good example of such a limit might be the (13, 2, 65)-half cubic limit, which is suited to 13-limit composing.
	[[Category:Limit]]		[[Category:Limit]]

Zeta Function: Add comment on motivation

2026-07-30T18:31:21Z

Add comment on motivation

← Older revision		Revision as of 18:31, 30 July 2026
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	\|2		\|2
	\|}		\|}
			This limit was specifically designed to be useful for composition and scale-building purposes, as it attempts to chart a middle path between the restrictiveness of the odd limit and the excessive complexity of intervals that is acceptable with prime-limits. If one wants to make it more useful, they can set a high-ish odd limit such as 41 or 99 to filter out even more intervals.
	[[Category:Limit]]		[[Category:Limit]]

Zeta Function: Factually incorrect sentence

2026-07-30T18:27:50Z

Factually incorrect sentence

← Older revision		Revision as of 18:27, 30 July 2026
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	The half-cubic limit for a prime limit is defined to be the set of all intervals within a certain prime limit whose numerators and denominators both cannot contain more than two non-2 prime factors, where all intervals are octave-reduced. In this way, it is analogous to the reduced 1-cubic limit but is significantly more restricted, which explains the reason for its naming as the half-cubic limit.		The half-cubic limit for a prime limit is defined to be the set of all intervals within a certain prime limit whose numerators and denominators both cannot contain more than two non-2 prime factors, where all intervals are octave-reduced. In this way, it is analogous to the reduced 1-cubic limit but is significantly more restricted, which explains the reason for its naming as the half-cubic limit.

	~~The monzos of such intervals must take the form where one prime factor can have up to a maximum of 2, the other prime factor can have up to a maximum of -2, and all other non-2 primes are 0s.~~

	Here is the set of all intervals in the 3-half-cubic limit.		Here is the set of all intervals in the 3-half-cubic limit.

Zeta Function: Fix categorization

2026-07-30T18:25:34Z

Fix categorization

← Older revision		Revision as of 18:25, 30 July 2026
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	\|2		\|2
	\|}		\|}
	[[Category:~~Limits~~]]		[[Category:Limit]]

Zeta Function: Create new page

2026-07-30T18:25:00Z

Create new page

New page

The half-cubic limit for a prime limit is defined to be the set of all intervals within a certain prime limit whose numerators and denominators both cannot contain more than two non-2 prime factors, where all intervals are octave-reduced. In this way, it is analogous to the reduced 1-cubic limit but is significantly more restricted, which explains the reason for its naming as the half-cubic limit.

The monzos of such intervals must take the form where one prime factor can have up to a maximum of 2, the other prime factor can have up to a maximum of -2, and all other non-2 primes are 0s.

Here is the set of all intervals in the 3-half-cubic limit.
{| class="wikitable"
|+
!Interval
!Threes
!Twos
|-
|1/1
|0
|0
|-
|3/2
|1
| -1
|-
|4/3
| -1
|2
|-
|9/8
|2
| -3
|-
|16/9
| -2
|4
|}
Here is the set of all intervals in the 5-half-cubic limit.
{| class="wikitable"
|+
!Interval
!Fives
!Threes
!Twos
|-
|1/1
|0
|0
|0
|-
|3/2
|0
|1
| -1
|-
|4/3
|0
| -1
|2
|-
|5/4
|1
|0
| -2
|-
|8/5
| -1
|0
|3
|-
|5/3
|1
| -1
|0
|-
|6/5
| -1
|1
|1
|-
|9/8
|0
|2
| -3
|-
|16/9
|0
| -2
|4
|-
|9/5
| -1
|2
|0
|-
|10/9
|1
| -2
|1
|-
|15/8
|1
|1
| -3
|-
|16/15
| -1
| -1
|4
|-
|25/16
|2
|0
| -4
|-
|32/25
| -2
|0
|5
|-
|25/24
|2
| -1
| -3
|-
|48/25
| -2
|1
|4
|-
|25/18
|2
| -2
| -1
|-
|36/25
| -2
|2
|2
|}
[[Category:Limits]]