833/832
| Ratio | 833/832 |
| Factorization | 2-6 × 72 × 13-1 × 17 |
| Monzo | [-6 0 0 2 0 -1 1⟩ |
| Size in cents | 2.0795607¢ |
| Names | horizma, horizon comma |
| Color name | 17o3uzz2, sothuzozo 2nd, Sothuzozo comma |
| FJS name | [math]\text{m2}^{7,7,17}_{13}[/math] |
| Special properties | superparticular, reduced |
| Tenney height (log2 nd) | 19.4026 |
| Weil height (log2 max(n, d)) | 19.4043 |
| Wilson height (sopfr(nd)) | 56 |
| Harmonic entropy (Shannon, [math]\sqrt{nd}[/math]) |
~1.23934 bits |
| Comma size | unnoticeable |
| S-expressions | S14 / S16, S49 × S50 × S51 |
| open this interval in xen-calc | |
833/832, the horizma or horizon comma, is a 17-limit (also 2.7.13.17 subgroup) unnoticeable comma, the difference between 52/49 and 17/16. It is also the difference between 49/48 and 52/51.
Commatic relations
This comma identifies itself as the difference between the following superparticular pairs:
- 105/104 and 120/119
- 196/195 and 256/255
- 289/288 and 442/441
- 385/384 and 715/714
- 441/440 and 936/935
- 561/560 and 1716/1715
- 595/594 and 2080/2079
- 625/624 and 2500/2499
- 729/728 and 5832/5831
It factors into the following superparticular pairs:
It also factors into the product of the following ultraparticulars:
- 43904/43875 and 57375/57344.
Temperaments
Tempering out this comma in the 17-limit results in the rank-6 horizmic temperament, or in the 2.7.13.17 subgroup, the rank-3 horizon temperament.
Horizon
Subgroup: 2.7.13.17
Comma list: 833/832
Sval mapping: [⟨1 0 0 6], ⟨0 1 0 -2], ⟨0 0 1 1]]
- sval mapping generators: ~2, ~7, ~11
Optimal ET sequence: 10, 21, 26, 36, 46, 47, 57, 93, 150, 207, 357, 704g, 854g, 911dg, 1061dg, 1268dg, 1418dgg
Horizmic
Subgroup: 2.3.5.7.11.13.17
Comma list: 833/832
| [⟨ | 1 | 0 | 0 | 0 | 0 | 0 | 6 | ], |
| ⟨ | 0 | 1 | 0 | 0 | 0 | 0 | 0 | ], |
| ⟨ | 0 | 0 | 1 | 0 | 0 | 0 | 0 | ], |
| ⟨ | 0 | 0 | 0 | 1 | 0 | 0 | -2 | ], |
| ⟨ | 0 | 0 | 0 | 0 | 1 | 0 | 0 | ], |
| ⟨ | 0 | 0 | 0 | 0 | 0 | 1 | 1 | ]] |
- mapping generators: ~2, ~3, ~5, ~7, ~11, ~13
- CTE: ~2 = 1\1, ~3/2 = 701.9550, ~5/4 = 386.3137, ~7/4 = 968.2966, ~11/8 = 551.3179, ~13/8 = 840.9875
- CWE: ~2 = 1\1, ~3/2 = 701.8313, ~5/4 = 386.1324, ~7/4 = 968.1967, ~11/8 = 551.0479, ~13/8 = 840.5950
Optimal ET sequence: 22f, 26, 31, 41, 46, 58, 72, 103, 130, 140, 171, 212g, 217, 243e, 289, 301, 311, 414, 460, 684g, 771, 1004dg, 1144degg, 1558cdegg, 1775ddgg
Etymology
The horizma was named by Jerdle in 2021.