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 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]].  


== Temperaments ==
== 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.
[[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


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 [[Marvel family #Char|char]], though it loses accuracy when compared to marvel.
{{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 }}
 
[[Badness]] (Sintel): 0.220


=== Lynchismic ===
=== Lynchismic ===
Line 30: Line 43:
| ⟨ || 0 || 0 || 0 || 0 || 0 || 1 || 0 || ]]
| ⟨ || 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 ==
== See also ==
* [[List of superparticular intervals]]
* [[List of superparticular intervals]]


[[Category:Lynchismic]]
[[Category:Commas named after composers]]
[[Category:Commas named after composers]]
[[Category:Commas named after music theorists]]
[[Category:Commas named after music theorists]]

Latest revision as of 14:40, 21 March 2026

Interval information
Ratio 120/119
Factorization 23 × 3 × 5 × 7-1 × 17-1
Monzo [3 1 1 -1 0 0 -1⟩
Size in cents 14.4874¢
Name lynchisma
Color name 17ury-2, suruyo negative 2nd,
Suruyo comma
FJS name [math]\displaystyle{ \text{d}{-2}^{5}_{7,17} }[/math]
Special properties superparticular,
reduced
Tenney norm (log2 nd) 13.8017
Weil norm (log2 max(n, d)) 13.8138
Wilson norm (sopfr(nd)) 38
Comma size small
S-expressions S15⋅S16,
S18⋅S19⋅S20
Open this interval in xen-calc

120/119, the lynchisma, is a small 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.

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 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

Subgroup-val 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 tunings:

  • WE: ~2 = 1198.9483 ¢, ~3/2 = 702.1229 ¢, ~5/4 = 386.5249 ¢, ~7/4 = 973.6693 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 701.8985 ¢, ~5/4 = 385.5601 ¢, ~7/4 = 973.5597 ¢

Optimal ET sequence: 10, 12, 19, 22, 27g, 31, 41, 53, 198ddggg

Badness (Sintel): 0.220

Lynchismic

Subgroup: 2.3.5.7.11.13.17

Mapping:

[⟨ 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 tunings:

  • WE: ~2 = 1198.9483 ¢, ~3/2 = 702.1229 ¢, ~5/4 = 386.5249 ¢, ~7/4 = 973.6693 ¢, ~11/8 = 554.4584 ¢, ~13/8 = 843.6671 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 701.8985 ¢, ~5/4 = 385.5601 ¢, ~7/4 = 973.5597 ¢, ~11/8 = 553.3440 ¢, ~13/8 = 842.6949 ¢

Optimal ET sequence: 19, 22, 26, 27eg, 31, 41, 49fg, 53, 65d, 84g, 92defg, 106g, 123dfgg, 128dg, 137gg, 145dgg, 171dgg

Badness (Sintel): 0.603

See also

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