Cubic and octahedral limits
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:
| 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:
| 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 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.
| 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.
| 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.
| 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 |