11859211/11859210: Difference between revisions

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{{Infobox Interval|Name=tredekisma|Color name=quadno-athoquadlu-azogu 4th, <br>19o^43o1u^4zg4, <br>Quadno-athoquadlu-azogu comma|Sound=|Comma=yes}}11859211/11859210, the ''tredekisma'', is the smallest [[superparticular ratio]] in the [[19-limit]] and [[23-limit]].
{{Infobox Interval
| Name = tredekisma
| Color name = quadno-athoquadlu-azogu 4th, <br>19o^43o1u^4zg4, <br>Quadno-athoquadlu-azogu comma
| Sound =  
| Comma = yes
}}
'''11859211/11859210''', the '''tredekisma''', is the smallest [[superparticular ratio]] in the [[19-limit]] and [[23-limit]].


It can be seen as the difference between a stack of a [[25/16]] plus a [[minisma]] plus a [[131072/130321|Saquadnu comma]], and a [[1024/729|diminished fifth]] plus two [[128/121|Axirabian limmas]]:
It can be seen as the difference between a stack of four [[38/33]]'s and a stack consisting of [[10/7]] and [[16/13]]: (38/33)<sup>4</sup>/((10/7)⋅(16/13)).


(25/16)(4096/4095)(131072/130321)÷(1024/729)(128/121)^2
A possible S-expression of it is (S76⋅S441)/(S77⋅S324).
 
Its S-expression is (S76)(S441)÷(S77)(324)


== Temperaments ==
== Temperaments ==
Only being 0.00014 cents, it is an insanely, unfathomably narrow comma; hundreds of times narrower than well known atomic commas like the [[Chalmersia]] and the [[Kirnberger's atom]]. As such, tempering out or fudging this either way are aurally identical to JI to human ears. It is notably tempered out in [[270edo]] thanks to its very well approximated 2.3.5.7.11.13.19, hence its name; also tempered out in other edos like {{EDOs|94, 118, 152, 364}}, among others.
Only being 0.00014 cents, it is an insanely, unfathomably narrow comma; hundreds of times narrower than well known atomic commas like the [[chalmersia]] and [[Kirnberger's atom]]. As such, tempering out or fudging this alone either way is aurally identical to JI to human ears. It is tempered out in notable edos like {{EDOs| 24, 84, 94, 118, 152, 176, 364, 388, 624, 742, 3125, 8269, 8539, }} … among others. It is also tempered out in any union between these edos, such as [[vishnu]] and [[gariwizmic]].


=== Tredekic ===
=== Tredekic ===
[[Subgroup]]: 2.3.5.7.11.13.19
[[Subgroup]]: 2.3.5.7.11.13.19
{| class="right-all"
{| class="right-all"
|[⟨
| [⟨ || 1 || 0 || 0 || 0 || 0 || 1 || 0 || ],
|1
|0
|0
|0
|0
|1
|0
|],
|-
|-
|⟨
| ⟨ || 0 || 1 || 0 || 0 || 0 || 0 || 1 || ],
|0
|1
|0
|0
|0
|0
|1
|],
|-
|-
|⟨
| ⟨ || 0 || 0 || 1 || 0 || 0 || 1 || 0 || ],
|0
|0
|1
|0
|0
|1
|0
|],
|-
|-
|⟨
| ⟨ || 0 || 0 || 0 || 1 || 0 || 3 || -1 || ],
|0
|0
|0
|1
|0
|3
| -1
|],
|-
|-
|⟨
| ⟨ || 0 || 0 || 0 || 0 || 1 || 0 || 1 || ],
|0
|0
|0
|0
|1
|0
|1
|],
|-
|-
|⟨
| ⟨ || 0 || 0 || 0 || 0 || 0 || -4 || 1 || ]]
|0
|0
|0
|0
|0
| -4
|1
|]]
|}
|}
mapping generators: ~2, ~3, ~5, ~7, ~11, ~133/33
* mapping generators: ~2, ~3, ~5, ~7, ~11, ~133/33


[[Optimal tuning|Optimal tunings]]: TBA
[[Optimal tuning]]s: TBA


{{Optimal ET sequence|TBA|legend=1}}
{{Optimal ET sequence||legend=1|TBA }}


[[Badness]] (Sintel): TBA
[[Badness]] (Sintel): TBA
Line 84: Line 40:
[[Subgroup]]: 2.3.5.7.11.13.17.19
[[Subgroup]]: 2.3.5.7.11.13.17.19
{| class="right-all"
{| class="right-all"
|[⟨
| [⟨ || 1 || 0 || 0 || 0 || 0 || 1 || 0 || 0 |],
|1
|0
|0
|0
|0
|1
|0
|0
|],
|-
|-
|⟨
| ⟨ || 0 || 1 || 0 || 0 || 0 || 0 || 0 || 1 || ],
|0
|1
|0
|0
|0
|0
|0
|1
|],
|-
|-
|⟨
| ⟨ || 0 || 0 || 1 || 0 || 0 || 1 || 0 || 0 || ],
|0
|0
|1
|0
|0
|1
|0
|0
|],
|-
|-
|⟨
| ⟨ || 0 || 0 || 0 || 1 || 0 || 3 || 0 || -1 || ],
|0
|0
|0
|1
|0
|3
|0
| -1
|],
|-
|-
|⟨
| ⟨ || 0 || 0 || 0 || 0 || 1 || 0 || 0 || 1 || ],
|0
|0
|0
|0
|1
|0
|0
|1
|],
|-
|-
|⟨
| ⟨ || 0 || 0 || 0 || 0 || 0 || -4 || 0 || 1 || ],
|0
|0
|0
|0
|0
| -4
|0
|1
|],
|-
|-
|⟨
| ⟨ || 0 || 0 || 0 || 0 || 0 || 0 || 1 || 0 || ]]
|0
|0
|0
|0
|0
|0
|1
|0
|]]
|}
|}
mapping generators: ~2, ~3, ~5, ~7, ~11, ~17, ~133/33
* mapping generators: ~2, ~3, ~5, ~7, ~11, ~17, ~133/33


[[Optimal tuning|Optimal tunings]]: TBA
[[Optimal tuning]]s: TBA


{{Optimal ET sequence|TBA|legend=1}}
{{Optimal ET sequence|legend=1|TBA}}


[[Badness]] (Sintel): TBA
[[Badness]] (Sintel): TBA
== Etymology ==
This comma was named by [[Eufalesio]] in 2026. The name ''tredekisma'' comes from ''tredek'', which is the name of one step of [[270edo]] by [[Joseph Monzo]], and 270edo is an excellent 2.3.5.7.11.13.19-subgroup system that tempers it out.
[[Category:Commas named for their regular temperament properties]]

Latest revision as of 13:27, 12 April 2026

Interval information
Ratio 11859211/11859210
Factorization 2-1 × 3-4 × 5-1 × 7 × 11-4 × 13 × 194
Monzo [-1 -4 -1 1 -4 1 0 4⟩
Size in cents 0.0001459822¢
Name tredekisma
Color name quadno-athoquadlu-azogu 4th,
19o^43o1u^4zg4,
Quadno-athoquadlu-azogu comma
FJS name [math]\displaystyle{ \text{ddd4}^{7,13,19,19,19,19}_{5,11,11,11,11} }[/math]
Special properties superparticular,
reduced
Tenney norm (log2 nd) 46.999
Weil norm (log2 max(n, d)) 46.999
Wilson norm (sopfr(nd)) 159
Comma size unnoticeable
Open this interval in xen-calc

11859211/11859210, the tredekisma, is the smallest superparticular ratio in the 19-limit and 23-limit.

It can be seen as the difference between a stack of four 38/33's and a stack consisting of 10/7 and 16/13: (38/33)4/((10/7)⋅(16/13)).

A possible S-expression of it is (S76⋅S441)/(S77⋅S324).

Temperaments

Only being 0.00014 cents, it is an insanely, unfathomably narrow comma; hundreds of times narrower than well known atomic commas like the chalmersia and Kirnberger's atom. As such, tempering out or fudging this alone either way is aurally identical to JI to human ears. It is tempered out in notable edos like 24, 84, 94, 118, 152, 176, 364, 388, 624, 742, 3125, 8269, 8539, … among others. It is also tempered out in any union between these edos, such as vishnu and gariwizmic.

Tredekic

Subgroup: 2.3.5.7.11.13.19

[⟨ 1 0 0 0 0 1 0 ],
⟨ 0 1 0 0 0 0 1 ],
⟨ 0 0 1 0 0 1 0 ],
⟨ 0 0 0 1 0 3 -1 ],
⟨ 0 0 0 0 1 0 1 ],
⟨ 0 0 0 0 0 -4 1 ]]
  • mapping generators: ~2, ~3, ~5, ~7, ~11, ~133/33

Optimal tunings: TBA

Optimal ET sequence: 

Badness (Sintel): TBA

Tredekismic

Subgroup: 2.3.5.7.11.13.17.19

[⟨ 1 0 0 0 0 1 0 ],
⟨ 0 1 0 0 0 0 0 1 ],
⟨ 0 0 1 0 0 1 0 0 ],
⟨ 0 0 0 1 0 3 0 -1 ],
⟨ 0 0 0 0 1 0 0 1 ],
⟨ 0 0 0 0 0 -4 0 1 ],
⟨ 0 0 0 0 0 0 1 0 ]]
  • mapping generators: ~2, ~3, ~5, ~7, ~11, ~17, ~133/33

Optimal tunings: TBA

Optimal ET sequence: TBA

Badness (Sintel): TBA

Etymology

This comma was named by Eufalesio in 2026. The name tredekisma comes from tredek, which is the name of one step of 270edo by Joseph Monzo, and 270edo is an excellent 2.3.5.7.11.13.19-subgroup system that tempers it out.

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