833/832: Difference between revisions

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'''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]].
'''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 ==
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* [[1275/1274]] and [[2401/2400]]
* [[1275/1274]] and [[2401/2400]]
* [[1225/1224]] and [[2601/2600]]
* [[1225/1224]] and [[2601/2600]]
It also factors into the product of the following ultraparticulars:
It also factors into the product of the following ultraparticulars:
* [[43904/43875]] and [[57375/57344]].
* [[43904/43875]] and [[57375/57344]].


== 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'''.
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 ===
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[[Optimal tuning]]s:
[[Optimal tuning]]s:
* [[CTE]]: ~2 = 1\1, ~7/4 = 968.2966, ~13/8 = 840.9875
* [[CTE]]: ~2 = 1200.0000, ~7/4 = 968.2966, ~13/8 = 840.9875
* [[CWE]]: ~2 = 1\1, ~7/4 = 968.1967, ~13/8 = 840.5950
* [[CWE]]: ~2 = 1200.0000, ~7/4 = 968.1967, ~13/8 = 840.5950


{{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 }}
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[[Optimal tuning]]s:
[[Optimal tuning]]s:
* [[CTE]]: ~2 = 1\1, ~3/2 = 701.9550, ~5/4 = 386.3137, ~7/4 = 968.2966, ~11/8 = 551.3179, ~13/8 = 840.9875
* [[CTE]]: ~2 = 1200.0000, ~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
* [[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|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 }}
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== See also ==
== See also ==
* [[Unnoticeable comma]]
* [[List of superparticular intervals]]
* [[List of superparticular intervals]]


[[Category:Horizmic]]
[[Category:Horizmic]]
[[Category:Commas with unknown etymology]]

Latest revision as of 15:44, 8 February 2025

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 height (log2 nd) 19.4026
Weil height (log2 max(n, d)) 19.4043
Wilson height (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. 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

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 tunings:

  • CTE: ~2 = 1200.0000, ~7/4 = 968.2966, ~13/8 = 840.9875
  • 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

Horizmic

Subgroup: 2.3.5.7.11.13.17

Comma list: 833/832

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:

  • CTE: ~2 = 1200.0000, ~3/2 = 701.9550, ~5/4 = 386.3137, ~7/4 = 968.2966, ~11/8 = 551.3179, ~13/8 = 840.9875
  • 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

Etymology

The horizma was named by Jerdle in 2021.

See also

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