120/119: Difference between revisions

From Xenharmonic Wiki
Jump to navigation Jump to search
← Older edit
VisualWikitext
Fredg999 category edits (talk | contribs)
autopatrolled, Bots, emailconfirmed
4,405 edits
mNo edit summary
 
(10 intermediate revisions by 7 users not shown)
Line 1: Line 1:
{{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 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 22 EDO|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.


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]].
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.


[[Category:17-limit]]
=== Lynchic ===
[[Category:Small commas]]
[[Subgroup]]: 2.3.5.7.17
[[Category:Superparticular]]
 
{{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 ===
[[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]]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 ET sequence|legend=1| 19, 22, 26, 27eg, 31, 41, 49fg, 53, 65d, 84g, 92defg, 106g, 123dfgg, 128dg, 137gg, 145dgg, 171dgg }}
 
[[Badness]] (Sintel): 0.603
 
== See also ==
* [[List of superparticular intervals]]
 
[[Category:Lynchismic]]
[[Category:Commas named after composers]]
[[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

Retrieved from "https://en.xen.wiki/index.php?title=120/119&oldid=226612"