833/832: Difference between revisions

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 {{Infobox Interval
-| Icon =
+| Name = horizma, horizon comma
-| Ratio = 833/832
+| Color name = 17o3uzz2, sothuzozo 2nd,<br>Sothuzozo comma
-| Monzo = -6 0 0 2 0 -1 1
+| Comma = yes
-| Cents = 2.07956
-| Name = horizon comma
-| Color name = sothuzozo 2nd, 17o3uzz2
-| FJS name = m2<sup>7,7,17</sup><sub>13</sub>
-<!--| Neutral FJS name = sd2<sup>7,7,17</sup><sub>13</sub>-->
 }}
+'''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 '''horizon comma''', is a [[17-limit]] [[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:
+* [[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
+{{Mapping|legend=2| 1 0 0 6 | 0 1 0 -2 | 0 0 1 1 }}
+: mapping generators: ~2, ~7, ~13
+[[Optimal tuning]]s:
+* [[WE]]: ~2 = 1200.1274{{c}}, ~7/4 = 968.2363{{c}}, ~13/8 = 840.4362{{c}}
+* [[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 }}
+[[Badness]] (Sintel): 0.0198
+=== Horizmic ===
+[[Subgroup]]: 2.3.5.7.11.13.17
+[[Mapping]]:<br>
+{| class="right-all"
+|-
+| [⟨ || 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 tuning]]s:
+* [[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{{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 }}
+[[Badness]] (Sintel): 1.14
+== Etymology ==
+The horizma was named by [[User:Jerdle|Jerdle]] in 2021.
 == See also ==
-* [[Unnoticeable comma]]
 * [[List of superparticular intervals]]
-[[Category:17-limit]]
+[[Category:Horizmic]]
-[[Category:Unnoticeable comma]]
+[[Category:Commas with unknown etymology]]