833/832: Difference between revisions

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'''833/832''', the '''horizma''' or '''horizon comma''', is an [[unnoticeable comma|unnoticeable]] [[17-limit]] (also 2.7.13.17-[[subgroup]]) [[comma]] with a size of roughly 2.08 [[cent]]s. 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]].  
'''833/832''', the '''horizma''' or '''horizon comma''', is an [[unnoticeable comma|unnoticeable]] [[17-limit]] (also 2.7.13.17 [[subgroup]]) [[comma]]. 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 ==
== Commatic relations ==
Line 27: Line 26:


== Temperaments ==
== 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]].  
[[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 ===
=== Horizon ===
[[Subgroup]]: 2.7.13.17
[[Subgroup]]: 2.7.13.17
[[Comma list]]: 833/832


{{Mapping|legend=2| 1 0 0 6 | 0 1 0 -2 | 0 0 1 1 }}
{{Mapping|legend=2| 1 0 0 6 | 0 1 0 -2 | 0 0 1 1 }}
 
: mapping generators: ~2, ~7, ~13
: sval mapping generators: ~2, ~7, ~11


[[Optimal tuning]]s:
[[Optimal tuning]]s:
* [[CTE]]: ~2 = 1200.0000, ~7/4 = 968.2966, ~13/8 = 840.9875
* [[WE]]: ~2 = 1200.1274{{c}}, ~7/4 = 968.2363{{c}}, ~13/8 = 840.4362{{c}}
* [[CWE]]: ~2 = 1200.0000, ~7/4 = 968.1967, ~13/8 = 840.5950
* [[CWE]]: ~2 = 1200.0000{{c}}, ~7/4 = 968.1967{{c}}, ~13/8 = 840.5950{{c}}


{{Optimal ET sequence|legend=1| 10, 21, 26, 36, 46, 47, 57, 93, 150, 207, 357, 704g, 854g, 911dg, 1061dg, 1268dg, 1418dgg }}
{{Optimal ET sequence|legend=1| 10, 21, 26, 36, 46, 47, 57, 93, 150, 207, 357, 704g, 854g, 911dg, 1061dg, 1268dg, 1418dgg }}
[[Badness]] (Sintel): 0.0198


=== Horizmic ===
=== Horizmic ===
[[Subgroup]]: 2.3.5.7.11.13.17
[[Subgroup]]: 2.3.5.7.11.13.17
[[Comma list]]: 833/832


[[Mapping]]:<br>
[[Mapping]]:<br>
Line 67: Line 63:


[[Optimal tuning]]s:
[[Optimal tuning]]s:
* [[CTE]]: ~2 = 1200.0000, ~3/2 = 701.9550, ~5/4 = 386.3137, ~7/4 = 968.2966, ~11/8 = 551.3179, ~13/8 = 840.9875
* [[WE]]: ~2 = 1200.1274{{c}}, ~3/2 = 701.8276{{c}}, ~5/4 = 386.0589{{c}}, ~7/4 = 968.2363{{c}}, ~11/8 = 550.9357{{c}}, ~13/8 = 840.4362{{c}}
* [[CWE]]: ~2 = 1200.0000, ~3/2 = 701.8313, ~5/4 = 386.1324, ~7/4 = 968.1967, ~11/8 = 551.0479, ~13/8 = 840.5950
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 701.8313{{c}}, ~5/4 = 386.1324{{c}}, ~7/4 = 968.1967{{c}}, ~11/8 = 551.0479{{c}}, ~13/8 = 840.5950{{c}}


{{Optimal ET sequence|legend=1| 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 }}
{{Optimal ET sequence|legend=1| 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 ==
== Etymology ==

Latest revision as of 15:24, 22 March 2026

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
Open this interval in xen-calc

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:

It factors into the following superparticular pairs:

It also factors into the product of the following ultraparticulars:

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

Retrieved from "https://en.xen.wiki/index.php?title=833/832&oldid=226661"