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{{Infobox Interval
{{Infobox Interval
| Icon =
| Name = ainisma, ainic comma
| Ratio = 936/935
| Color name = 17u3o1ug1, sutholugu unison
| Monzo = 3 2 -1 0 -1 1 -1
| Comma = yes
| Cents = 1.8506
| Name = ainos comma, <br> ainma
| Color name =  
| FJS name =
| Sound =  
}}
}}
'''936/935''', the '''ainisma''' or '''ainic comma''', is a [[17-limit]] [[unnoticeable comma]] with a size of roughly 1.85 [[cent]]s. It arises as the amount by which a stack consisting of [[18/17]] and [[13/11]] exceeds [[5/4]], as well as the amount by which a stack consisting of [[10/9]] and [[17/16]] falls short of 13/11. Moreover, it is also the interval that differentiates the [[273/272|tannisma (273/272)]] from the [[385/384|keenanisma (385/384)]], and the [[256/255|septendecimal kleisma (256/255)]] from the [[352/351|minthma (352/351)]]. Thus, tempering out this comma is a good way to extend minthmic and gentle harmonies to the 17-limit, as well as a good way to bring keenanismic and tannismic harmonies together. See [[#Commatic relations]] for more.


'''936/935''', the '''ainos comma''' or '''ainma''', is an [[unnoticeable comma|unnoticeable]] [[17-limit]] comma with a size of roughly 1.85 cents.  It forms the interval between [[18/17]] and [[55/52]] as well as the interval between [[13/11]] and [[85/72]], the interval between the [[273/272|tannisma (273/272)]] and the [[385/384|keenanisma (385/384)]], and, the interval between the [[256/255|septendecimal kleisma (256/255)]] and the [[352/351|minthma (352/351)]].  Thus, tempering out this comma is a good way to extend Neo-Gothic and Minthmic harmonies to the 17-limit, as well as a good way to bring Keenanismic and Tannismic harmonies together.  Furthermore, it is also the interval that differentiates [[17/13]] from [[72/55]], [[24/17]] from [[55/39]], and [[117/64]] from [[935/512]].  When tempered out in a linearly independent fashion, the resulting temperaments are called "'''[[ainos temperaments]]'''", and are characterized by the presence of [[Dyadic chord|essentially tempered chord]]s called "'''[[ainic chords]]'''".
== Commatic relations ==
This comma is the difference between the following superparticular pairs:
* [[78/77]] and [[85/84]]
* [[144/143]] and [[170/169]]
* [[221/220]] and [[289/288]]
* [[256/255]] and [[352/351]]
* [[273/272]] and [[385/384]]
* [[351/350]] and [[561/560]]
* [[364/363]] and [[595/594]]
* [[375/374]] and [[625/624]]
* [[441/440]] and [[833/832]]
* [[540/539]] and [[1275/1274]]
* [[676/675]] and [[2431/2430]]
* [[715/714]] and [[3025/3024]]


This comma's names comes from the Ancient Greek word "aînos" (meaning "tale", "story" or "fable"<ref>[[Wiktionary: αἶνος #Ancient Greek]]</ref>), which is fitting due to the comma serving as a viable 17-limit extension to minthmic temperaments, among others. Funny enough, this same Greek word is the source of the Ancient Greek word "aínigma" (meaning "riddle"<ref>[[Wiktionary: αἴνιγμα #Ancient Greek]]</ref>), from which we ultimately get our word "enigma", and this is also fitting due to the sheer difficulty that was involved in the initial process of working out both the name and the uses of this comma in a short span of time.
It factors into the following superparticular pairs:
* [[1716/1715]] and [[2058/2057]]
* [[1701/1700]] and [[2080/2079]]
* [[1156/1155]] and [[4914/4913]]
* [[1089/1088]] and [[6656/6655]]
* [[1001/1000]] and [[14400/14399]]
 
== Temperaments ==
When tempered out in the full 17-limit, the resulting rank-6 temperament is called the '''ainismic temperament''', or in the 2.3.5.11.13.17 subgroup, the rank-5 '''ainic temperament'''. Both are characterized by the presence of [[Dyadic chord|essentially tempered chord]]s called [[ainismic chords]].
 
=== Ainic ===
[[Subgroup]]: 2.3.5.11.13.17
 
[[Comma list]]: 936/935
 
[[Sval]] [[mapping]]:<br>
{| class="right-all"
|-
| [⟨ || 1 || 0 || 0 || 0 || 0 || 3 || ],
|-
| ⟨ || 0 || 1 || 0 || 0 || 0 || 2 || ],
|-
| ⟨ || 0 || 0 || 1 || 0 || 0 || -1 || ],
|-
| ⟨ || 0 || 0 || 0 || 1 || 0 || -1 || ],
|-
| ⟨ || 0 || 0 || 0 || 0 || 1 || 1 || ]]
|}
: sval mapping generators: ~2, ~3, ~5, ~11, ~13
 
[[Optimal tuning]]s:
* [[CTE]]: ~2 = 1\1, ~3/2 = 701.7942, ~5/4 = 386.4863, ~11/8 = 551.7011, ~13/8 = 840.0893
* [[CWE]]: ~2 = 1\1, ~3/2 = 701.8763, ~5/4 = 386.5772, ~11/8 = 551.8274, ~13/8 = 840.2857
 
{{Optimal ET sequence|legend=1| 22, 24, 31, 34, 46, 53, 58, 65, 72, 87, 111, 137, 183, 304, 320, 354, 400, 407, 441, 537, 552g, 624 }}
 
=== Ainismic ===
[[Subgroup]]: 2.3.5.7.11.13.17
 
[[Comma list]]: 936/935
 
[[Mapping]]:<br>
{| class="right-all"
|-
| [⟨ || 1 || 0 || 0 || 0 || 0 || 0 || 3 || ],
|-
| ⟨ || 0 || 1 || 0 || 0 || 0 || 0 || 2 || ],
|-
| ⟨ || 0 || 0 || 1 || 0 || 0 || 0 || -1 || ],
|-
| ⟨ || 0 || 0 || 0 || 1 || 0 || 0 || 0 || ],
|-
| ⟨ || 0 || 0 || 0 || 0 || 1 || 0 || -1 || ],
|-
| ⟨ || 0 || 0 || 0 || 0 || 0 || 1 || 1 || ]]
|}
: mapping generators: ~2, ~3, ~5, ~7, ~11, ~13
 
[[Optimal tuning]]s:
* [[CTE]]: ~2 = 1\1, ~3/2 = 701.7942, ~5/4 = 386.4863, ~7/4 = 968.8259, ~11/8 = 551.7011, ~13/8 = 840.0893
* [[CWE]]: ~2 = 1\1, ~3/2 = 701.8763, ~5/4 = 386.5773, ~7/4 = 968.9509, ~11/8 = 551.8275, ~13/8 = 840.2857
 
{{Optimal ET sequence|legend=1| 22, 26, 31, 38df, 41, 46, 58, 72, 103, 111, 121, 149, 159, 171, 183, 217, 224, 270, 354, 400, 441, 552g, 624, 1393ceegg, 1576cdegg }}
 
== Etymology ==
This comma was named by [[Aura]] in 2020. Its names come from the Ancient Greek word [[Wiktionary: αἶνος #Ancient Greek|''aînos'']] ("tale", "story" or "fable"), which is fitting due to the comma serving as a viable 17-limit extension to minthmic temperaments, among others. Funny enough, this same Greek word is the source of the Ancient Greek word [[Wiktionary: αἴνιγμα #Ancient Greek|''aínigma'']] ("riddle"), from which we ultimately get our word "enigma", and this is also fitting due to the sheer difficulty that was involved in the initial process of working out both the name and the uses of this comma in a short span of time.  


== See also ==
== See also ==
* [[Ainic chords]]
* [[Unnoticeable comma]]
* [[Unnoticeable comma]]
* [[List of superparticular intervals]]
* [[List of superparticular intervals]]


== References ==
[[Category:Ainismic]]
<references/>
[[Category:Commas named by translating something into another language]]
 
[[Category:17-limit]]
[[Category:Unnoticeable comma]]
[[Category:Superparticular]]
[[Category:Ainos]]

Latest revision as of 01:54, 16 November 2024

Interval information
Ratio 936/935
Factorization 23 × 32 × 5-1 × 11-1 × 13 × 17-1
Monzo [3 2 -1 0 -1 1 -1⟩
Size in cents 1.850598¢
Names ainisma,
ainic comma
Color name 17u3o1ug1, sutholugu unison
FJS name [math]\displaystyle{ \text{P1}^{13}_{5,11,17} }[/math]
Special properties superparticular,
reduced
Tenney height (log2 nd) 19.7392
Weil height (log2 max(n, d)) 19.7407
Wilson height (sopfr(nd)) 58
Comma size unnoticeable
S-expression S52 × S53 × S54
Open this interval in xen-calc

936/935, the ainisma or ainic comma, is a 17-limit unnoticeable comma with a size of roughly 1.85 cents. It arises as the amount by which a stack consisting of 18/17 and 13/11 exceeds 5/4, as well as the amount by which a stack consisting of 10/9 and 17/16 falls short of 13/11. Moreover, it is also the interval that differentiates the tannisma (273/272) from the keenanisma (385/384), and the septendecimal kleisma (256/255) from the minthma (352/351). Thus, tempering out this comma is a good way to extend minthmic and gentle harmonies to the 17-limit, as well as a good way to bring keenanismic and tannismic harmonies together. See #Commatic relations for more.

Commatic relations

This comma is the difference between the following superparticular pairs:

It factors into the following superparticular pairs:

Temperaments

When tempered out in the full 17-limit, the resulting rank-6 temperament is called the ainismic temperament, or in the 2.3.5.11.13.17 subgroup, the rank-5 ainic temperament. Both are characterized by the presence of essentially tempered chords called ainismic chords.

Ainic

Subgroup: 2.3.5.11.13.17

Comma list: 936/935

Sval mapping:

[⟨ 1 0 0 0 0 3 ],
⟨ 0 1 0 0 0 2 ],
⟨ 0 0 1 0 0 -1 ],
⟨ 0 0 0 1 0 -1 ],
⟨ 0 0 0 0 1 1 ]]
sval mapping generators: ~2, ~3, ~5, ~11, ~13

Optimal tunings:

  • CTE: ~2 = 1\1, ~3/2 = 701.7942, ~5/4 = 386.4863, ~11/8 = 551.7011, ~13/8 = 840.0893
  • CWE: ~2 = 1\1, ~3/2 = 701.8763, ~5/4 = 386.5772, ~11/8 = 551.8274, ~13/8 = 840.2857

Optimal ET sequence: 22, 24, 31, 34, 46, 53, 58, 65, 72, 87, 111, 137, 183, 304, 320, 354, 400, 407, 441, 537, 552g, 624

Ainismic

Subgroup: 2.3.5.7.11.13.17

Comma list: 936/935

Mapping:

[⟨ 1 0 0 0 0 0 3 ],
⟨ 0 1 0 0 0 0 2 ],
⟨ 0 0 1 0 0 0 -1 ],
⟨ 0 0 0 1 0 0 0 ],
⟨ 0 0 0 0 1 0 -1 ],
⟨ 0 0 0 0 0 1 1 ]]
mapping generators: ~2, ~3, ~5, ~7, ~11, ~13

Optimal tunings:

  • CTE: ~2 = 1\1, ~3/2 = 701.7942, ~5/4 = 386.4863, ~7/4 = 968.8259, ~11/8 = 551.7011, ~13/8 = 840.0893
  • CWE: ~2 = 1\1, ~3/2 = 701.8763, ~5/4 = 386.5773, ~7/4 = 968.9509, ~11/8 = 551.8275, ~13/8 = 840.2857

Optimal ET sequence: 22, 26, 31, 38df, 41, 46, 58, 72, 103, 111, 121, 149, 159, 171, 183, 217, 224, 270, 354, 400, 441, 552g, 624, 1393ceegg, 1576cdegg

Etymology

This comma was named by Aura in 2020. Its names come from the Ancient Greek word aînos ("tale", "story" or "fable"), which is fitting due to the comma serving as a viable 17-limit extension to minthmic temperaments, among others. Funny enough, this same Greek word is the source of the Ancient Greek word aínigma ("riddle"), from which we ultimately get our word "enigma", and this is also fitting due to the sheer difficulty that was involved in the initial process of working out both the name and the uses of this comma in a short span of time.

See also

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