120/119: Difference between revisions

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{{Infobox Interval
{{Infobox Interval
| Ratio = 120/119
| Monzo = 3 1 1 -1 0 0 -1
| Cents = 14.48740
| Name = lynchisma
| Name = lynchisma
| Color name = suruyo negative 2nd, 17ury-2
| Color name = 17ury-2, suruyo negative 2nd, <br>Suruyo comma
| FJS name = d-2<sup>5</sup><sub>7,17</sub>
| Comma = yes
| Sound =  
}}
}}


'''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 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.


By tempering it out is defined the '''lynchismic temperament''', which enables the [[lynchismic chords]]. EDOs supporting this temperament includes {{EDOs|10, 12, 19, 22, 26, 31, 41}} and [[53edo|53]].
== 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'''.


[[Category:17-limit]]
=== Lynchismic ===
[[Category:Small commas]]
[[Subgroup]]: 2.3.5.7.11.13.17
[[Category:Superparticular]]
 
[[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]]

Latest revision as of 22:34, 20 April 2025

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 height (log2 nd) 13.8017
Weil height (log2 max(n, d)) 13.8138
Wilson height (sopfr(nd)) 38
Comma size small
S-expressions S15 × S16,
S18 × S19 × S20
Open this interval in xen-calc

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 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.

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 tuning:

  • TE: ~2 = 1198.953, ~3 = 1901.078, ~5 = 2784.431, ~7 = 3371.578
  • CTE: ~2 = 1200.000 (1\1), ~3/2 = 700.835, ~5/4 = 383.910, ~7/4 = 972.340

Lynchic

Subgroup: 2.3.5.7.17

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\1), ~3/2 = 700.835, ~5/4 = 383.910, ~7/4 = 972.340

Optimal ET sequence: 10, 12, 19, 22, 26, 31, 41, 53

See also

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