936/935: Difference between revisions

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| Comma = yes

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'''936/935''', the '''ainisma''' or '''ainic comma''', is an [[unnoticeable comma|unnoticeable]] [[17-limit]] [[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|major 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 '''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.

== Commatic relations ==

This comma is the difference between the following superparticular pairs:

* [[52/51]] and [[55/54]]

* [[78/77]] and [[85/84]]

* [[144/143]] and [[170/169]]

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== Temperaments ==

When tempered out in the full 17-limit, the resulting temperament is called the '''ainismic ~~temperament~~''', or in the 2.3.5.11.13.17 subgroup, the '''ainic ~~temperament~~'''. Both are characterized by the presence of [[~~Dyadic chord|~~essentially tempered chord]]s called [[ainismic 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 [[essentially tempered chord]]s called [[ainismic chords]].

=== Ainic ===

[[Subgroup]]: 2.3.5.11.13.17

[[Subgroup val|Subgroup-val]] [[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 || ]]

|}

: mapping generators: ~2, ~3, ~5, ~11, ~13

[[Optimal tuning]]s:

* [[WE]]: ~2 = 1199.9169{{c}}, ~3/2 = 701.8989{{c}}, ~5/4 = 386.6292{{c}}, ~11/8 = 551.8987{{c}}, ~13/8 = 840.3976{{c}}

* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 701.8763{{c}}, ~5/4 = 386.5772{{c}}, ~11/8 = 551.8274{{c}}, ~13/8 = 840.2857{{c}}

{{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 }}

[[Badness]] (Sintel): 0.119

=== Ainismic ===

[[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 || 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:

* [[WE]]: ~2 = 1199.9169{{c}}, ~3/2 = 701.8989{{c}}, ~5/4 = 386.6292{{c}}, ~7/4 = 968.9921{{c}}, ~11/8 = 551.8987{{c}}, ~13/8 = 840.3976{{c}}

* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 701.8763{{c}}, ~5/4 = 386.5773{{c}}, ~7/4 = 968.9509{{c}}, ~11/8 = 551.8275{{c}}, ~13/8 = 840.2857{{c}}

{{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 }}

[[Badness]] (Sintel): 0.322

== Etymology ==

Line 36:

Line 90:

== See also ==

* [[Unnoticeable comma]]

* [[List of superparticular intervals]]

[[Category:Ainismic]]

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

@@ Line 4: / Line 4: @@
 | Comma = yes
 }}
+'''936/935''', the '''ainisma''' or '''ainic comma''', is an [[unnoticeable comma|unnoticeable]] [[17-limit]] [[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|major 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 '''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.
 == Commatic relations ==
 This comma is the difference between the following superparticular pairs:
+* [[52/51]] and [[55/54]]
 * [[78/77]] and [[85/84]]
 * [[144/143]] and [[170/169]]
@@ Line 30: / Line 30: @@
 == Temperaments ==
-When tempered out in the full 17-limit, the resulting temperament is called the '''ainismic temperament''', or in the 2.3.5.11.13.17 subgroup, the '''ainic temperament'''. Both are characterized by the presence of [[Dyadic chord|essentially tempered chord]]s called [[ainismic 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 [[essentially tempered chord]]s called [[ainismic chords]].
+=== Ainic ===
+[[Subgroup]]: 2.3.5.11.13.17
+[[Subgroup val|Subgroup-val]] [[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 || ]]
+|}
+: mapping generators: ~2, ~3, ~5, ~11, ~13
+[[Optimal tuning]]s:
+* [[WE]]: ~2 = 1199.9169{{c}}, ~3/2 = 701.8989{{c}}, ~5/4 = 386.6292{{c}}, ~11/8 = 551.8987{{c}}, ~13/8 = 840.3976{{c}}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 701.8763{{c}}, ~5/4 = 386.5772{{c}}, ~11/8 = 551.8274{{c}}, ~13/8 = 840.2857{{c}}
+{{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 }}
+[[Badness]] (Sintel): 0.119
+=== Ainismic ===
+[[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 || 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:
+* [[WE]]: ~2 = 1199.9169{{c}}, ~3/2 = 701.8989{{c}}, ~5/4 = 386.6292{{c}}, ~7/4 = 968.9921{{c}}, ~11/8 = 551.8987{{c}}, ~13/8 = 840.3976{{c}}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 701.8763{{c}}, ~5/4 = 386.5773{{c}}, ~7/4 = 968.9509{{c}}, ~11/8 = 551.8275{{c}}, ~13/8 = 840.2857{{c}}
+{{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 }}
+[[Badness]] (Sintel): 0.322
 == Etymology ==
@@ Line 36: / Line 90: @@
 == See also ==
-* [[Unnoticeable comma]]
 * [[List of superparticular intervals]]
 [[Category:Ainismic]]
+[[Category:Commas named by translating something into another language]]