1-octahedral-limit
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The 1-octahedral-limit, 1-taxicab-limit, 1-Manhattan-limit or 1-cardinal-limit consists of all intervals whose prime factorizations contain one element, counting repeated primes. Equivalently, it contains all the prime harmonics and subharmonics. The reduced 1-octahedral-limit consists of all intervals whose prime factorizations contain one element, counting repeated primes, but once all powers of 2 are removed. Geometrically, the 1-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 1 from 1/1.
The reduced 1-octahedral-limit contains:
Below is a list of all octave-reduced 7-limit intervals in the reduced 1-octahedral-limit:
| Ratio | Cents | In non-reduced limit? |
|---|---|---|
| 7/4 | 968.83 | No |
| 5/4 | 386.31 | No |
| 3/2 | 701.96 | No |
| 1/1 | 0.00 | Yes |
| 4/3 | 498.04 | No |
| 8/5 | 813.69 | No |
| 8/7 | 231.17 | No |
The size of the 1-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 | 3 |
| 2 | 5 |
| 3 | 7 |
| 4 | 9 |
| 5 | 11 |
| 6 | 13 |