120/119: Difference between revisions

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{{Infobox Interval
{{Infobox Interval
| Icon =
| Name = lynchisma
| Ratio = 120/119
| Color name = 17ury-2, suruyo negative 2nd, <br>Suruyo comma
| Monzo = 3 1 1 -1 0 0 -1
| Comma = yes
| Cents = 14.48740
| 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:Interval ratio]]
== Temperaments ==
[[Category:Superparticular]]
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]]
 
[[Category:Small comma]]
=== 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]]

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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