120/119: Difference between revisions

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'''120/119''', the '''lynchisma''', is a [[small comma|small]] [[17-limit]] [[superparticular]] [[comma]] of about 14.49 [[cent]]s. It is the difference between [[20/17]] and [[7/6]], [[17/10]] and [[12/7]], or [[30/17]] and [[7/4]].

'''~~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.

== Temperaments ==

[[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. In either case, it allows you to assign [[10:12:15:17]] as the inverse of [[4:5:6:7]], an otonal chord that 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.

Since 120/119 factors as ([[225/224]])⋅([[256/255]]), it would make sense to temper them both out, so lynchic can be further tempered to a simple extension of [[marvel]] that adds prime 17 known as [[char]], though it loses accuracy when compared to marvel.

=== Lynchic ===

[[Subgroup]]: 2.3.5.7.17

{{Mapping|legend=2| 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]]s:

* [[WE]]: ~2 = 1198.9483{{c}}, ~3/2 = 702.1229{{c}}, ~5/4 = 386.5249{{c}}, ~7/4 = 973.6693{{c}}

* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 701.8985{{c}}, ~5/4 = 385.5601{{c}}, ~7/4 = 973.5597{{c}}

{{Optimal ET sequence|legend=1| 10, 12, 19, 22, 27g, 31, 41, 53, 198ddggg }}

~~== Temperaments ==~~

[[Badness]] (Sintel): 0.220

~~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'''~~.

=== Lynchismic ===

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| ⟨ || 0 || 0 || 0 || 0 || 0 || 1 || 0 || ]]

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: ~~Mapping~~ generators: ~2, ~3, ~5, ~7, ~11, ~13

: 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]]s:

* [[WE]]: ~2 = 1198.9483{{c}}, ~3/2 = 702.1229{{c}}, ~5/4 = 386.5249{{c}}, ~7/4 = 973.6693{{c}}, ~11/8 = 554.4584{{c}}, ~13/8 = 843.6671{{c}}

* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 701.8985{{c}}, ~5/4 = 385.5601{{c}}, ~7/4 = 973.5597{{c}}, ~11/8 = 553.3440{{c}}, ~13/8 = 842.6949{{c}}

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=1| 19, 22, 26, 27eg, 31, 41, 49fg, 53, 65d, 84g, 92defg, 106g, 123dfgg, 128dg, 137gg, 145dgg, 171dgg }}

~~{{Optimal ET sequence|legend=~~0~~| 10, 12, 19, 22, 26, 31, 41, 53 }}~~

[[Badness]] (Sintel): 0.603

== See also ==

* [[Small comma]]

* [[List of superparticular intervals]]

[[Category:Lynchismic]]

[[Category:Commas named after composers]]

[[Category:Commas named after music theorists]]

@@ Line 4: / Line 4: @@
 | Comma = yes
 }}
+'''120/119''', the '''lynchisma''', is a [[small comma|small]] [[17-limit]] [[superparticular]] [[comma]] of about 14.49 [[cent]]s. It is the difference between [[20/17]] and [[7/6]], [[17/10]] and [[12/7]], or [[30/17]] and [[7/4]].
-'''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.
+== Temperaments ==
+[[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. In either case, it allows you to assign [[10:12:15:17]] as the inverse of [[4:5:6:7]], an otonal chord that 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.
+Since 120/119 factors as ([[225/224]])⋅([[256/255]]), it would make sense to temper them both out, so lynchic can be further tempered to a simple extension of [[marvel]] that adds prime 17 known as [[char]], though it loses accuracy when compared to marvel.
+=== Lynchic ===
+[[Subgroup]]: 2.3.5.7.17
+{{Mapping|legend=2| 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]]s:
+* [[WE]]: ~2 = 1198.9483{{c}}, ~3/2 = 702.1229{{c}}, ~5/4 = 386.5249{{c}}, ~7/4 = 973.6693{{c}}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 701.8985{{c}}, ~5/4 = 385.5601{{c}}, ~7/4 = 973.5597{{c}}
+{{Optimal ET sequence|legend=1| 10, 12, 19, 22, 27g, 31, 41, 53, 198ddggg }}
-== Temperaments ==
+[[Badness]] (Sintel): 0.220
-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'''.
 === Lynchismic ===
@@ Line 28: / Line 43: @@
 | ⟨ || 0 || 0 || 0 || 0 || 0 || 1 || 0 || ]]
 |}
-: Mapping generators: ~2, ~3, ~5, ~7, ~11, ~13
+: 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]]s:
+* [[WE]]: ~2 = 1198.9483{{c}}, ~3/2 = 702.1229{{c}}, ~5/4 = 386.5249{{c}}, ~7/4 = 973.6693{{c}}, ~11/8 = 554.4584{{c}}, ~13/8 = 843.6671{{c}}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 701.8985{{c}}, ~5/4 = 385.5601{{c}}, ~7/4 = 973.5597{{c}}, ~11/8 = 553.3440{{c}}, ~13/8 = 842.6949{{c}}
-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=1| 19, 22, 26, 27eg, 31, 41, 49fg, 53, 65d, 84g, 92defg, 106g, 123dfgg, 128dg, 137gg, 145dgg, 171dgg }}
-{{Optimal ET sequence|legend=0| 10, 12, 19, 22, 26, 31, 41, 53 }}
+[[Badness]] (Sintel): 0.603
 == See also ==
-* [[Small comma]]
 * [[List of superparticular intervals]]
+[[Category:Lynchismic]]
 [[Category:Commas named after composers]]
 [[Category:Commas named after music theorists]]