Punctisma: Difference between revisions

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{{Infobox Interval

~~| Icon =~~

| Ratio = 43904/43875

~~| Monzo = 7 -3 -3 3 0 -1~~

| Name = punctisma

~~| Cents = 1.143913~~

| Color name = s3uz3g33, sathutrizogu 3rd

| Name =

| Comma = yes

| Color name =

| ~~FJS name =~~

~~| Sound~~ =

}}

43904/43875 is the difference between 3 [[15/14]] semitones and a [[16/13]] neutral third~~. It's tempering out is characteristic of the 10n family of edos, which approximate both of these intervals within a fraction of a cent~~.

The '''punctisma''', with a [[ratio]] of '''43904/43875''', is an [[unnoticeable comma|unnoticeable]] [[13-limit]] (specifically [[2.3.5.7.13 subgroup|2.3.5.7.13-]][[subgroup]]) [[comma]] measuring about 1.14 [[cent]]s. It is the difference between three [[15/14]] semitones and a [[16/13]] neutral third.

[[~~Category~~:13-~~limit~~]]

In terms of other commas, it is the difference between [[196/195]] and [[225/224]]. It factors into ([[2401/2400]])⋅([[4096/4095]]). In the [[19-limit]] it further factors into ([[2926/2925]])⋅([[3136/3135]]).

[[Category:~~Interval ratio~~]]

[[Category:~~Comma~~]]

== Temperaments ==

[[Category:~~Unnoticeable comma~~]]

[[Tempering out]] this comma in the full 13-limit leads to the rank-5 '''punctismic''' temperament, or in the 2.3.5.17.13 subgroup, the rank-4 '''punctic''' temperament. It is characteristic of the [[10edo|10''n'']] family of [[edo]]s, including [[decoid]] tunings [[130edo]], [[140edo]], and [[270edo]], which approximate both 15/14 and 16/13 to within a fraction of a cent. It is also tempered out by any temperament which tempers out both 2401/2400 and 4096/4095, such as [[31edo]], [[41edo]], [[99edo]], [[171edo]], [[311edo]], and [[441edo]]. Perhaps surprisingly, [[111edo]] and [[159edo]] temper it out as well.

=== Punctic ===

[[Subgroup]]: 2.3.5.7.13

{{Mapping|legend=2| 1 0 0 0 7 | 0 1 0 0 -3 | 0 0 1 0 -3 | 0 0 0 1 1 }}

: mapping generators: ~2, ~3, ~5, ~7

[[Optimal tuning]]s:

: [[WE]]: ~2 = 1199.9609{{c}}, ~3/2 = 702.0362{{c}}, ~5/4 = 386.4823{{c}}, ~7/4 = 968.7720{{c}}

: [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 702.0380{{c}}, ~5/4 = 386.4679{{c}}, ~7/4 = 968.7382{{c}}

{{Optimal ET sequence|legend=1| 12f, 19, 31, 41, 50, 60, 80, 99, 111, 121, 130, 140, 171, 251, 270, 441, 1152, 1453, 1593, 2034, 2164, 2605, 3046, 4179cf, 4620cf, 5191cff }}

[[Badness]] (Sintel): 0.467

=== Punctismic ===

[[Subgroup]]: 2.3.5.7.11.13

[[Mapping]]:

{| class="right-all"

|-

| [⟨ || 1 || 0 || 0 || 0 || 0 || 7 || ],

|-

| ⟨ || 0 || 1 || 0 || 0 || 0 || -3 || ],

|-

| ⟨ || 0 || 0 || 1 || 0 || 0 || -3 || ],

|-

| ⟨ || 0 || 0 || 0 || 1 || 0 || 3 || ],

|-

| ⟨ || 0 || 0 || 0 || 0 || 1 || 1 || ]]

|}

: mapping generators: ~2, ~3, ~5, ~7, ~11

[[Optimal tuning]]s:

: [[WE]]: ~2 = 1199.9609{{c}}, ~3/2 = 702.0362{{c}}, ~5/4 = 386.4823{{c}}, ~7/4 = 968.7720{{c}}, ~11/8 = 551.4353{{c}}

: [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 702.0380{{c}}, ~5/4 = 386.4679{{c}}, ~7/4 = 968.7382{{c}}, ~11/8 = 551.3948{{c}}

{{Optimal ET sequence|legend=1| 12f, …, 19, 22f, 29, 31, 41, 60e, 72f, 80, 99e, 109, 111, 121, 130, 190, 270, 552, 581, 711, 742, 882, 1012, 1323, 1593, 2605, 4198cf, 4750cef }}

[[Badness]] (Sintel): 1.77

== Etymology ==

This comma was named by [[User:Jerdle|Jerdle]] and [[Aura]] in 2023, from Latin [[Wiktionary: punctum #Latin|''pūnctum'']] ("point") based on both its relation to the 10''n'' family of edos as well as its small size.

[[Category:Punctismic]]

[[Category:Commas named for their periods per equave]]

[[Category:Commas named after their interval size]]

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

@@ Line 1: / Line 1: @@
 {{Infobox Interval
-| Icon =
 | Ratio = 43904/43875
-| Monzo = 7 -3 -3 3 0 -1
+| Name = punctisma
-| Cents = 1.143913
+| Color name = s3uz<sup>3</sup>g<sup>3</sup>3, sathutrizogu 3rd
-| Name =
+| Comma = yes
-| Color name =
-| FJS name =
-| Sound =
 }}
-/43875 is the difference between 3 [[15/14]] semitones and a [[16/13]] neutral third. It's tempering out is characteristic of the 10n family of edos, which approximate both of these intervals within a fraction of a cent.
+The '''punctisma''', with a [[ratio]] of '''43904/43875''', is an [[unnoticeable comma|unnoticeable]] [[13-limit]] (specifically [[2.3.5.7.13 subgroup|2.3.5.7.13-]][[subgroup]]) [[comma]] measuring about 1.14 [[cent]]s. It is the difference between three [[15/14]] semitones and a [[16/13]] neutral third.
-[[Category:13-limit]]
+In terms of other commas, it is the difference between [[196/195]] and [[225/224]]. It factors into ([[2401/2400]])⋅([[4096/4095]]). In the [[19-limit]] it further factors into ([[2926/2925]])⋅([[3136/3135]]).
-[[Category:Interval ratio]]
-[[Category:Comma]]
+== Temperaments ==
-[[Category:Unnoticeable comma]]
+[[Tempering out]] this comma in the full 13-limit leads to the rank-5 '''punctismic''' temperament, or in the 2.3.5.17.13 subgroup, the rank-4 '''punctic''' temperament. It is characteristic of the [[10edo|10''n'']] family of [[edo]]s, including [[decoid]] tunings [[130edo]], [[140edo]], and [[270edo]], which approximate both 15/14 and 16/13 to within a fraction of a cent. It is also tempered out by any temperament which tempers out both 2401/2400 and 4096/4095, such as [[31edo]], [[41edo]], [[99edo]], [[171edo]], [[311edo]], and [[441edo]]. Perhaps surprisingly, [[111edo]] and [[159edo]] temper it out as well.
+=== Punctic ===
+[[Subgroup]]: 2.3.5.7.13
+{{Mapping|legend=2| 1 0 0 0 7 | 0 1 0 0 -3 | 0 0 1 0 -3 | 0 0 0 1 1 }}
+: mapping generators: ~2, ~3, ~5, ~7
+[[Optimal tuning]]s:
+: [[WE]]: ~2 = 1199.9609{{c}}, ~3/2 = 702.0362{{c}}, ~5/4 = 386.4823{{c}}, ~7/4 = 968.7720{{c}}
+: [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 702.0380{{c}}, ~5/4 = 386.4679{{c}}, ~7/4 = 968.7382{{c}}
+{{Optimal ET sequence|legend=1| 12f, 19, 31, 41, 50, 60, 80, 99, 111, 121, 130, 140, 171, 251, 270, 441, 1152, 1453, 1593, 2034, 2164, 2605, 3046, 4179cf, 4620cf, 5191cff }}
+[[Badness]] (Sintel): 0.467
+=== Punctismic ===
+[[Subgroup]]: 2.3.5.7.11.13
+[[Mapping]]: <br>
+{| class="right-all"
+|-
+| [⟨ || 1 || 0 || 0 || 0 || 0 || 7 || ],
+|-
+| ⟨ || 0 || 1 || 0 || 0 || 0 || -3 || ],
+|-
+| ⟨ || 0 || 0 || 1 || 0 || 0 || -3 || ],
+|-
+| ⟨ || 0 || 0 || 0 || 1 || 0 || 3 || ],
+|-
+| ⟨ || 0 || 0 || 0 || 0 || 1 || 1 || ]]
+|}
+: mapping generators: ~2, ~3, ~5, ~7, ~11
+[[Optimal tuning]]s:
+: [[WE]]: ~2 = 1199.9609{{c}}, ~3/2 = 702.0362{{c}}, ~5/4 = 386.4823{{c}}, ~7/4 = 968.7720{{c}}, ~11/8 = 551.4353{{c}}
+: [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 702.0380{{c}}, ~5/4 = 386.4679{{c}}, ~7/4 = 968.7382{{c}}, ~11/8 = 551.3948{{c}}
+{{Optimal ET sequence|legend=1| 12f, …, 19, 22f, 29, 31, 41, 60e, 72f, 80, 99e, 109, 111, 121, 130, 190, 270, 552, 581, 711, 742, 882, 1012, 1323, 1593, 2605, 4198cf, 4750cef }}
+[[Badness]] (Sintel): 1.77
+== Etymology ==
+This comma was named by [[User:Jerdle|Jerdle]] and [[Aura]] in 2023, from Latin [[Wiktionary: punctum #Latin|''pūnctum'']] ("point") based on both its relation to the 10''n'' family of edos as well as its small size.
+[[Category:Punctismic]]
+[[Category:Commas named for their periods per equave]]
+[[Category:Commas named after their interval size]]
+[[Category:Commas named by translating something into another language]]