2-octahedral-limit
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The 2-octahedral-limit, 2-taxicab-limit, 2-Manhattan-limit or 2-cardinal-limit consists of all intervals whose prime factorizations contain two elements, counting repeated primes. Equivalently, it contains all the prime and semiprime harmonics and subharmonics, as well as ratios of primes. The reduced 2-octahedral-limit consists of all intervals whose prime factorizations contain two elements, counting repeated primes, but once all powers of 2 are removed. Geometrically, the 2-octahedral-limit consists of all intervals in an idealized infinite-dimensional hyper-octahedron centered on the unison 1/1 (hence "octahedral limit"), or those with a Manhattan distance of at most 2 from 1/1.
The reduced 2-octahedral-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 2-octahedral-limit:
| Ratio | Cents |
|---|---|
| 49/32 | 737.65 |
| 35/32 | 155.14 |
| 21/16 | 470.78 |
| 7/4 | 968.83 |
| 7/6 | 266.87 |
| 7/5 | 582.51 |
| 25/16 | 772.63 |
| 15/8 | 1088.27 |
| 5/4 | 386.31 |
| 5/3 | 884.36 |
| 9/8 | 203.91 |
| 3/2 | 701.96 |
| 1/1 | 0.00 |
| 4/3 | 498.04 |
| 16/9 | 996.09 |
| 6/5 | 315.64 |
| 8/5 | 813.69 |
| 16/15 | 111.73 |
| 32/25 | 427.37 |
| 10/7 | 617.49 |
| 12/7 | 933.13 |
| 8/7 | 231.17 |
| 32/21 | 729.22 |
| 64/35 | 1044.86 |
| 64/49 | 462.35 |
The size of the 2-octahedral-limit is infinite, but can be limited by restricting it to a subgroup. The size of an octahedral limit n given a subgroup size m is the Delannoy number D(n, m).
| Subgroup primes (excluding 2 if reduced) | Reduced size |
|---|---|
| 1 | 5 |
| 2 | 13 |
| 3 | 25 |
| 4 | 41 |
| 5 | 61 |
| 6 | 85 |