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 a [[25/16]] plus a [[minisma]] plus a [[131072/130321|undevicesimal diminished comma]], and a [[1024/729|diminished fifth]] plus two [[128/121|Axirabian limmas]]:


(25/16)(4096/4095)(131072/130321)÷(1024/729)(128/121)^2
((25/16)⋅(4096/4095)⋅(131072/130321))/((1024/729)(128/121)<sup>2</sup>)


Its S-expression is (S76)(S441)÷(S77)(S324)
Its S-expression is ((S76)(S441))/((S77)(S324))


== 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 [[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 {{nowrap|{{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]].
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 42:
[[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

Revision as of 12:56, 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 a 25/16 plus a minisma plus a undevicesimal diminished comma, and a diminished fifth plus two Axirabian limmas:

((25/16)⋅(4096/4095)⋅(131072/130321))/((1024/729)(128/121)2)

Its S-expression 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


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

The name tredekisma comes from tredek, which is one step of 270edo, which is an excellent 2.3.5.7.11.13.19 subgroup system that tempers it out.

Retrieved from "https://en.xen.wiki/index.php?title=11859211/11859210&oldid=227690"