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 ==
@@ Line 22: / Line 22: @@
 * [[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'''.
+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
+{{Mapping|legend=2| 1 0 0 6 | 0 1 0 -2 | 0 0 1 1 }}
+: sval mapping generators: ~2, ~7, ~11
+[[Optimal tuning]]s:
+* [[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|legend=1| 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]]:<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:
+* [[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|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 }}
 == Etymology ==
@@ Line 30: / Line 76: @@
 == See also ==
-* [[Unnoticeable comma]]
 * [[List of superparticular intervals]]
 [[Category:Horizmic]]
+[[Category:Commas with unknown etymology]]