1-cubic-limit
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The 1-cubic-limit, 1-Chebyshev-limit or 1-exponential-limit consists of all intervals whose prime factorizations contain a maximum exponent of 1. The reduced 1-cubic-limit contains all intervals whose prime factorizations contain a maximum exponent of 1 once all powers of two are removed. Geometrically, the 1-cubic-limit consists of all intervals in an idealized infinite-dimensional hypercube centered on the unison 1/1 (hence "cubic limit"), or those with a Chebyshev distance of at most 1 from 1/1.
The reduced 1-cubic-limit contains:
- The 7-odd-limit.
- The reduced 1-octahedral-limit.
Below is a list of all octave-reduced 7-limit intervals in the reduced 1-cubic-limit:
| Ratio | Cents | In non-reduced limit? |
|---|---|---|
| 35/24 | 653.18 | No |
| 35/32 | 155.14 | No |
| 105/64 | 857.09 | No |
| 7/6 | 266.87 | Yes |
| 7/4 | 968.83 | No |
| 21/16 | 470.78 | No |
| 28/15 | 1080.56 | No |
| 7/5 | 582.51 | Yes |
| 21/20 | 84.47 | No |
| 5/3 | 884.36 | Yes |
| 5/4 | 386.31 | No |
| 15/8 | 1088.27 | No |
| 4/3 | 498.04 | No |
| 1/1 | 0.00 | Yes |
| 3/2 | 701.96 | Yes |
| 16/15 | 111.73 | No |
| 8/5 | 813.69 | No |
| 5/3 | 884.36 | Yes |
| 40/21 | 1115.53 | No |
| 10/7 | 617.49 | Yes |
| 15/14 | 119.44 | Yes |
| 32/21 | 729.22 | No |
| 8/7 | 231.17 | No |
| 12/7 | 933.13 | No |
| 128/105 | 342.91 | No |
| 64/35 | 1044.86 | No |
| 48/35 | 546.82 | No |
The size of the 1-cubic-limit is infinite, but can be limited by restricting it to a subgroup. The size of a cubic limit n given a subgroup size m is (2n+1)^m.
| Subgroup primes (excluding 2 if reduced) | Reduced size |
|---|---|
| 1 | 3 |
| 2 | 9 |
| 3 | 27 |
| 4 | 81 |
| 5 | 243 |
| 6 | 729 |