3381/3380: Difference between revisions

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 '''3381/3380''', the '''mikkolisma''', is a [[23-limit]] [[superparticular ratio|superparticular comma]] of about 0.51 [[cent]]s. It is the amount by which a stack consisting of [[23/21]] and two [[14/13]]'s succeed [[80/63]].
+== Commatic relationships ==
+This comma is the difference between following superparticular pairs:
+* [[196/195]] and [[208/207]]
+* [[441/440]] and [[507/506]]
+* [[833/832]] and [[1105/1104]]
+* [[2646/2645]] and [[12168/12167]]
+It factors into the following superparticular pairs:
+* [[3888/3887]] and [[25921/25920]]
+* [[3969/3968]] and [[22816/22815]]
+* [[5292/5291]] and [[9361/9360]]
+== Temperaments ==
+Tempering out the comma leads to the rank-8 temperament '''mikkolismic'''. Using the 2.3.5.7.13.23 subgroup leads to the rank-5 temperament '''mikkolic'''.
+=== Mikkolismic ===
+[[Subgroup]]: 2.3.5.7.11.13.17.19.23
+[[Comma list]]: 3381/3380
+[[Mapping]]: <br>
+{| class="right-all"
+| [⟨ || 1 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 2 || ],
+|-
+| ⟨ || 0 || 1 || 0 || 0 || 0 || 0 || 0|| 0 || -1 || ],
+|-
+| ⟨ || 0 || 0 || 1 || 0 || 0 || 0 || 0|| 0 || 1 || ],
+|-
+| ⟨ || 0 || 0 || 0 || 1 || 0 || 0 || 0 || 0 || -2 || ],
+|-
+| ⟨ || 0 || 0 || 0 || 0 || 1 || 0 || 0 || 0 || 0 || ],
+|-
+| ⟨ || 0 || 0 || 0 || 0 || 0 || 1 || 0 || 0 || 2 || ],
+|-
+| ⟨ || 0 || 0 || 0 || 0 || 0 || 0 || 1 || 0 || 0 || ],
+|-
+| ⟨ || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 1 || 0 || ]]
+|}
+[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~3/2 = 701.944, ~5/4 = 386.338, ~7/4 = 968.756, ~11/8 = 551.318, ~13/8 = 840.650, ~17/16 = 104.955, ~19/16 = 297.513
+{{Optimal ET sequence|legend=1| 6fhi, 7deefghi, 7dfghi, 8d, 9, 10e, 10, 10h, 12fi, 16h, 16, 17cg, 17cgh, 19egh, 19, 22fh, 26i, 29g, 38df, 39dfgi, 41i, 46, 58hi, 62, 68, 72hi, 72i, 84, 87, 94, 111, 130, 140, 152fg, 159, 198, 198g, 217, 243e, 270, 282, 311, 373g, 422, 525, 581, 742i, 814, 954h, 1106, 1282, 1323, 1395, 1677e, 1696, 1768gi, 1920, 2874hi, 3202, 3513, 5011di, 6293di }}
+=== Mikkolic ===
+[[Subgroup]]: 2.3.5.7.13.23
+[[Comma list]]: 3381/3380
+[[Mapping]]: <br>
+{| class="right-all"
+| [⟨ || 1 || 0 || 0 || 0 || 0 || 2 || ],
+|-
+| ⟨ || 0 || 1 || 0 || 0 ||  0 || -1 || ],
+|-
+| ⟨ || 0 || 0 || 1 || 0 || 0 || 1 || ],
+|-
+| ⟨ || 0 || 0 || 0 || 1 ||  0 || -2 || ],
+|-
+| ⟨ || 0 || 0 || 0 || 0 || 1 || 2 || ]
+|}
+[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~3/2 = 701.944, ~5/4 = 386.338, ~7/4 = 968.756, ~13/8 = 840.650
+{{Optimal ET sequence|legend=1| 7dfi, 8d, 9, 10, 12fi, 16, 17c, 19, 27, 29, 36, 39dfi, 41i, 45fi, 46, 58i, 68, 84, 103, 111, 113, 130, 198, 224, 243, 311, 373, 441, 1012, 1106, 1236, 1395, 1677, 1920, 2118, 2361, 2631, 3072, 4327 }}
 == Etymology ==