User:Zeta Function/Half-cubic limit
The half-cubic limit[idiosyncratic term] 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.
| 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.
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, ∞)-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.