User:Zeta Function/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.

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

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.

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

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.