936/935: Difference between revisions

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{{Infobox Interval

| Name = ~~ainos~~, ainic comma

| Name = ainisma, ainic comma

| Color name = 17u3o1ug1, sutholugu unison

| Comma = yes

}}

'''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''' 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.~~

== 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]]

~~== Commatic relations ==~~

It factors into the following superparticular pairs:

It factors into ~~two~~ superparticular ~~intervals~~: [[1701/1700]] × [[2080/2079]].

* [[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 temperament is called the '''~~ainic~~ temperament''', or in the 2.3.5.11.13.17 subgroup, the '''~~ainian~~ temperament'''. Both are characterized by the presence of [[Dyadic chord|essentially tempered chord]]s called [[~~ainic~~ chords]].

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 ==

Line 19:

Line 92:

* [[List of superparticular intervals]]

[[Category:~~Ainic~~]]

[[Category:Ainismic]]

[[Category:Commas named by translating something into another language]]

@@ Line 1: / Line 1: @@
 {{Infobox Interval
-| Name = ainos, ainic comma
+| Name = ainisma, ainic comma
+| Color name = 17u3o1ug1, sutholugu unison
 | Comma = yes
 }}
+'''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''' 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.
+== 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]]
-== Commatic relations ==
+It factors into the following superparticular pairs:
-It factors into two superparticular intervals: [[1701/1700]] × [[2080/2079]].
+* [[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 temperament is called the '''ainic temperament''', or in the 2.3.5.11.13.17 subgroup, the '''ainian temperament'''. Both are characterized by the presence of [[Dyadic chord|essentially tempered chord]]s called [[ainic chords]].
+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 ==
@@ Line 19: / Line 92: @@
 * [[List of superparticular intervals]]
-[[Category:Ainic]]
+[[Category:Ainismic]]
+[[Category:Commas named by translating something into another language]]