833/832

Interval information

Ratio	833/832
Factorization	2-6 × 72 × 13-1 × 17
Monzo	[-6 0 0 2 0 -1 1⟩
Size in cents	2.079561¢
Names	horizma, horizon comma
Color name	17o3uzz2, sothuzozo 2nd, Sothuzozo comma
FJS name	[math]\displaystyle{ \text{m2}^{7,7,17}_{13} }[/math]
Special properties	superparticular, reduced
Tenney norm (log2 nd)	19.4026
Weil norm (log2 max(n, d))	19.4043
Wilson norm (sopfr(nd))	56
Comma size	unnoticeable
S-expressions	S14/S16, S49⋅S50⋅S51
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833/832, the horizma or horizon comma, is an unnoticeable 17-limit (also 2.7.13.17-subgroup) comma with a size of roughly 2.08 cents. It is the difference between 17/13 and a stack of two 8/7's. It is also the difference between 52/49 and 17/16, and 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:

• 1275/1274 and 2401/2400
• 1225/1224 and 2601/2600

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. In either case, it enables horizmic chords.

Horizon

Subgroup: 2.7.13.17

Subgroup-val mapping: [⟨1 0 0 6], ⟨0 1 0 -2], ⟨0 0 1 1]]

mapping generators: ~2, ~7, ~13

Optimal tunings:

• WE: ~2 = 1200.1274 ¢, ~7/4 = 968.2363 ¢, ~13/8 = 840.4362 ¢
• CWE: ~2 = 1200.0000 ¢, ~7/4 = 968.1967 ¢, ~13/8 = 840.5950 ¢

Optimal ET sequence: 10, 21, 26, 36, 46, 47, 57, 93, 150, 207, 357, 704g, 854g, 911dg, 1061dg, 1268dg, 1418dgg

Badness (Sintel): 0.0198

Horizmic

Subgroup: 2.3.5.7.11.13.17

Mapping:

[⟨	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

Optimal tunings:

• WE: ~2 = 1200.1274 ¢, ~3/2 = 701.8276 ¢, ~5/4 = 386.0589 ¢, ~7/4 = 968.2363 ¢, ~11/8 = 550.9357 ¢, ~13/8 = 840.4362 ¢
• CWE: ~2 = 1200.0000 ¢, ~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

Badness (Sintel): 1.14

Etymology

The horizma was named by Jerdle in 2021.

See also

• List of superparticular intervals