Punctisma: Difference between revisions

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The '''punctisma''', with a [[ratio]] of '''43904/43875''', is the difference between three [[15/14]] semitones and a [[16/13]] neutral third. It factors into ([[2401/2400]])([[4096/4095]]).

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.

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]]).

== Temperaments ==

Tempering out this comma is characteristic of the 10''n'' family of [[edo]]s, 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 [[41edo]], [[171edo]], and [[~~311edo~~]]. Perhaps surprisingly, [[159edo]] ~~tempers~~ it out as well.

[[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 ==

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[[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 6: / Line 6: @@
 }}
-The '''punctisma''', with a [[ratio]] of '''43904/43875''', is the difference between three [[15/14]] semitones and a [[16/13]] neutral third. It factors into ([[2401/2400]])([[4096/4095]]).
+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.
+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]]).
 == Temperaments ==
-Tempering out this comma is characteristic of the 10''n'' family of [[edo]]s, 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 [[41edo]], [[171edo]], and [[311edo]]. Perhaps surprisingly, [[159edo]] tempers it out as well.
+[[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 ==
@@ Line 17: / Line 59: @@
 [[Category:Commas named for their periods per equave]]
 [[Category:Commas named after their interval size]]
+[[Category:Commas named by translating something into another language]]