120/119: Difference between revisions

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 {{Infobox Interval
-| Ratio = 120/119
+| Name = lynchisma
-| Monzo = 3 1 1 -1 0 0 -1
+| Color name = 17ury-2, suruyo negative 2nd, <br>Suruyo comma
-| Cents = 14.48740
+| Comma = yes
-| Name = suroyo comma
-| Color name = suruyo negative 2nd, 17ury-2
-| FJS name = d-2<sup>5</sup><sub>7,17</sub>
-| Sound =
 }}
-'''120/119''' is a [[17-limit]] [[superparticular]] ratio of about 14.49 cents. It is the difference between [[20/17]] and [[7/6]], [[17/10]] and [[12/7]], or [[30/17]] and [[7/4]]. Tempering this comma allows you to assign 10:12:15:17 as the inverse of 4:5:6:7, a much simpler version of what would otherwise be 70:84:105:120. [[William Lynch's Thoughts on Septimal Harmony and 22 EDO|William Lynch]] calls this the minor tetrad, and so equating it with the inverse of the major tetrad is quite useful.
+'''120/119''', the '''lynchisma''' is the [[17-limit]] [[superparticular]] comma of about 14.49 cents. It is the difference between [[20/17]] and [[7/6]], [[17/10]] and [[12/7]], or [[30/17]] and [[7/4]]. Tempering this comma allows you to assign 10:12:15:17 as the inverse of 4:5:6:7, a much simpler version of what would otherwise be 70:84:105:120. [[William Lynch's thoughts on septimal harmony and 22edo|William Lynch]] calls this the minor tetrad, and so equating it with the inverse of the major tetrad is quite useful.
-[[Category:17-limit]]
+== Temperaments ==
-[[Category:Small comma]]
+Tempering out this comma in the 17-limit leads to the rank-6 '''lynchismic temperament'''. In the 2.3.5.7.17 subgroup, tempering it out results in the rank-4 '''lynchic temperament'''.
-[[Category:Superparticular]]
+=== Lynchismic ===
+[[Subgroup]]: 2.3.5.7.11.13.17
+[[Mapping]]: <br>
+{| class="right-all"
+|-
+| [⟨ || 1 || 0 || 0 || 0 || 0 || 0 || 3 || ],
+|-
+| ⟨ || 0 || 1 || 0 || 0 || 0 || 0 || 1 || ],
+|-
+| ⟨ || 0 || 0 || 1 || 0 || 0 || 0 || 1 || ],
+|-
+| ⟨ || 0 || 0 || 0 || 1 || 0 || 0 || -1 || ],
+|-
+| ⟨ || 0 || 0 || 0 || 0 || 1 || 0 || 0 || ],
+|-
+| ⟨ || 0 || 0 || 0 || 0 || 0 || 1 || 0 || ]]
+|}
+: Mapping generators: ~2, ~3, ~5, ~7, ~11, ~13
+[[Optimal tuning]]:
+* [[TE]]: ~2 = 1198.953, ~3 = 1901.078, ~5 = 2784.431, ~7 = 3371.578
+* [[CTE]]: ~2 = 1200.000 (1<span style="font-family:'Arial', sans-serif">\</span>1), ~3/2 = 700.835, ~5/4 = 383.910, ~7/4 = 972.340
+=== Lynchic ===
+Subgroup: 2.3.5.7.17
+Mapping: {{mapping| 1 0 0 0 3 | 0 1 0 0 1 | 0 0 1 0 1 | 0 0 0 1 -1 }}
+: Mapping generators: ~2, ~3, ~5, ~7
+Optimal tuning (CTE): ~2 = 1200.000 (1<span style="font-family:'Arial', sans-serif">\</span>1), ~3/2 = 700.835, ~5/4 = 383.910, ~7/4 = 972.340
+{{Optimal ET sequence|legend=0| 10, 12, 19, 22, 26, 31, 41, 53 }}
+== See also ==
+* [[Small comma]]
+* [[List of superparticular intervals]]
+[[Category:Commas named after composers]]
+[[Category:Commas named after music theorists]]