Schismic rank-3 family: Difference between revisions

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Orthoschismic extends rank-3 schismic to the 11-limit in one of the most efficient ways possible.

Orthoschismic can be notated with [[chain-of-fifths notation]] with two additional set of accidentals, one for the generic comma step (one should choose whether this step represents the syntonic comma, which is more characteristic in schismic, or the septimal comma, which is more characteristic in [[olympic]]), and the other for the generic aberschisma step which stands in for the [[~~garischisma~~]] and the [[aberschisma]].

Orthoschismic can be notated with [[chain-of-fifths notation]] with two additional set of accidentals, one for the generic comma step (one should choose whether this step represents the syntonic comma, which is more characteristic in schismic, or the septimal comma, which is more characteristic in [[olympic]]), and the other for the generic aberschisma step which stands in for the [[septischisma]] and the [[aberschisma]].

It was named by [[Flora Canou]] in 2023, using the Greek prefix [[wiktionary: ortho- #English|ortho-]] to signify the additional rank.

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Comma list: 540/539, 729/728, 4096/4095

~~Mapping:~~ {{~~mapping~~| 1 0 15 0 17 -3 | 0 1 -8 0 -5 6 | 0 0 0 1 -2 -1 }}

{{Mapping|legend=0| 1 0 15 0 17 -3 | 0 1 -8 0 -5 6 | 0 0 0 1 -2 -1 }}

Optimal tunings:

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Badness (Sintel): 0.779

== Altierran ==

Named by [[Aura]] in 2021, altierran tempers out both the schisma and the [[quartisma]].

[[Subgroup]]: 2.3.5.7.11

[[Comma list]]: 32805/32768, 161280/161051

: mapping generators: ~2, ~3, ~33/2

[[Optimal tuning]]s:

* [[WE]]: ~2 = 1200.0326{{c}}, ~3/2 = 701.7490{{c}}, ~33/32 = 53.3904{{c}}

: [[error map]]: {{val| +0.033 -0.173 -0.077 -0.060 +0.454 }}

* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 701.7278{{c}}, ~33/32 = 53.3946{{c}}

: error map: {{val| 0.000 -0.227 -0.136 -0.125 +0.349 }}

{{Optimal ET sequence|legend=1| 24, 46c, 65d, 89, 135, 159, 224, 383, 472, 696, 1168, 1327, 1551, 2023e }}

[[Badness]] (Sintel): 5.48

=== Tenierian ===

Subgroup: 2.3.5.7.11.13

Comma list: 10985/10976, 32805/32768, 161280/161051

{{Mapping|legend=0| 1 2 -1 3 3 5 | 0 -3 24 -3 3 -11 | 0 0 0 5 1 5 }}

: mapping generators: ~2, ~208/189, ~33/32

Optimal tunings:

* WE: ~2 = 1200.0411{{c}}, ~208/189 = 166.0935{{c}}, ~33/32 = 53.4151{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~208/189 = 166.0886{{c}}, ~33/32 = 53.4211{{c}}

{{Optimal ET sequence|legend=0| 65d, 159, 224, 383, 607, 1257, 1481e, 1705e }}

Badness (Sintel): 15.8

[[Category:Schismic rank-3 family| ]]

[[Category:Temperament families]]

[[Category:Catalogs of rank-3 temperaments]]

@@ Line 22: / Line 22: @@
 Orthoschismic extends rank-3 schismic to the 11-limit in one of the most efficient ways possible.
-Orthoschismic can be notated with [[chain-of-fifths notation]] with two additional set of accidentals, one for the generic comma step (one should choose whether this step represents the syntonic comma, which is more characteristic in schismic, or the septimal comma, which is more characteristic in [[olympic]]), and the other for the generic aberschisma step which stands in for the [[garischisma]] and the [[aberschisma]].
+Orthoschismic can be notated with [[chain-of-fifths notation]] with two additional set of accidentals, one for the generic comma step (one should choose whether this step represents the syntonic comma, which is more characteristic in schismic, or the septimal comma, which is more characteristic in [[olympic]]), and the other for the generic aberschisma step which stands in for the [[septischisma]] and the [[aberschisma]].
 It was named by [[Flora Canou]] in 2023, using the Greek prefix [[wiktionary: ortho- #English|ortho-]] to signify the additional rank.
@@ Line 47: / Line 47: @@
 Comma list: 540/539, 729/728, 4096/4095
-Mapping: {{mapping| 1 0 15 0 17 -3 | 0 1 -8 0 -5 6 | 0 0 0 1 -2 -1 }}
+{{Mapping|legend=0| 1 0 15 0 17 -3 | 0 1 -8 0 -5 6 | 0 0 0 1 -2 -1 }}
 Optimal tunings:
@@ Line 56: / Line 56: @@
 Badness (Sintel): 0.779
+== Altierran ==
+Named by [[Aura]] in 2021, altierran tempers out both the schisma and the [[quartisma]].
+[[Subgroup]]: 2.3.5.7.11
+[[Comma list]]: 32805/32768, 161280/161051
+{{Mapping|legend=1| 1 0 15 1 5 | 0 1 -8 1 -1 | 0 0 0 5 1 }}
+: mapping generators: ~2, ~3, ~33/2
+[[Optimal tuning]]s:
+* [[WE]]: ~2 = 1200.0326{{c}}, ~3/2 = 701.7490{{c}}, ~33/32 = 53.3904{{c}}
+: [[error map]]: {{val| +0.033 -0.173 -0.077 -0.060 +0.454 }}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 701.7278{{c}}, ~33/32 = 53.3946{{c}}
+: error map: {{val| 0.000 -0.227 -0.136 -0.125 +0.349 }}
+{{Optimal ET sequence|legend=1| 24, 46c, 65d, 89, 135, 159, 224, 383, 472, 696, 1168, 1327, 1551, 2023e }}
+[[Badness]] (Sintel): 5.48
+=== Tenierian ===
+Subgroup: 2.3.5.7.11.13
+Comma list: 10985/10976, 32805/32768, 161280/161051
+{{Mapping|legend=0| 1 2 -1 3 3 5 | 0 -3 24 -3 3 -11 | 0 0 0 5 1 5 }}
+: mapping generators: ~2, ~208/189, ~33/32
+Optimal tunings:
+* WE: ~2 = 1200.0411{{c}}, ~208/189 = 166.0935{{c}}, ~33/32 = 53.4151{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~208/189 = 166.0886{{c}}, ~33/32 = 53.4211{{c}}
+{{Optimal ET sequence|legend=0| 65d, 159, 224, 383, 607, 1257, 1481e, 1705e }}
+Badness (Sintel): 15.8
 [[Category:Schismic rank-3 family| ]] <!-- main article -->
 [[Category:Temperament families]]
 [[Category:Catalogs of rank-3 temperaments]]