No-twos subgroup temperaments: Difference between revisions

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Classified by focusing on the mapping of 5th harmonic, similar to [[Rank-2 temperaments by mapping of 3]].

* Arcturus, antares aldebaran and polaris have a 3/1 period and ~5/3 generator. There is one-to-one correspondence between the 3.5 subgroup and mapped intervals.

* Arcturus, antares, aldebaran and polaris have a 3/1 period and ~5/3 generator. There is one-to-one correspondence between the 3.5 subgroup and mapped intervals.

* BPS has a ~9/7 generator, two of which give the ~5/3.

* Sirius has a ~25/21 generator, three of which give the ~5/3.

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By splitting the generator of sirius into three, remus efficiently represents the no-2's 13-limit with [[mos scale]]s of 18, 25, 32, or 39 steps.

This is essentially [[electra]] but with prime 7, or more accurately, electra is the no-7's [[restriction]] of this temperament.

This is essentially [[#Electra|electra]] but with prime 7, or more accurately, electra is the no-7's [[restriction]] of this temperament.

Subgroup: 3.5.7.11

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Badness (Sintel): 0.838

== Electra ==

[[File:Electra chords.wav|thumb|Just 11:13:15 and 143:165:195 triads on A]]

Electra tempers out [[1875/1859]] and [[14641/14625]] in the 3.5.11.13 subgroup, with a generator near [[15/11]] or its tritave complement, [[11/5]]. It possesses [[mos scale]]s of the families {{mos scalesig|link=1|4L 3s<3/1>}}, {{mos scalesig|link=1|7L 4s<3/1>}}, and {{mos scalesig|link=1|7L 11s<3/1>}}.

The 11:13:15 triad is a characteristic feature of many electra scales, and has been suggeested by {{u|CompactStar}} as a fundamental sonority for no-2's music, due to being relatively consonant compared to other no-2's chords. There is also a "minor" version of this triad, 143:165:195. If 11:13:15 is treated as the fundamental sonority of no-2's music, then electra can be viewed as the analogue to [[meantone family #Meantone|meantone]].

The name was given by CompactStar in 2023 after the star {{w|Electra (star)|Electra}}, following a series of non-octave temperaments that were named after stars.

[[Subgroup]]: 3.5.11

[[Comma list]]: 2373046875/2357947691

: mapping generators: ~3, ~11/5

[[Optimal tuning]]s:

* [[WE]]: ~3 = 1901.8666{{c}}, ~11/5 = 1366.1408{{c}}

: [[error map]]: {{val| -0.088 -0.379 +0.757 }}

* [[CWE]]: ~3 = 1901.9550{{c}}, ~11/5 = 1366.1964{{c}}

: error map: {{val| 0.000 -0.321 +0.8712 }}

[[Badness]] (Sintel): 0.656

=== 3.5.11.13 subgroup ===

Subgroup: 3.5.11.13

Comma list: 1875/1859, 14641/14625

Optimal tunings:

* WE: ~3 = 1901.8899{{c}}, ~11/5 = 1365.9871{{c}}

* CWE: ~3 = 1901.9550{{c}}, ~11/5 = 1366.0281{{c}}

Badness (Sintel): 0.405

; Scales

* [[Electra7]]

== Alnilam ==

@@ Line 7: / Line 7: @@
 Classified by focusing on the mapping of 5th harmonic, similar to [[Rank-2 temperaments by mapping of 3]].
-* Arcturus, antares aldebaran and polaris have a 3/1 period and ~5/3 generator. There is one-to-one correspondence between the 3.5 subgroup and mapped intervals.
+* Arcturus, antares, aldebaran and polaris have a 3/1 period and ~5/3 generator. There is one-to-one correspondence between the 3.5 subgroup and mapped intervals.
 * BPS has a ~9/7 generator, two of which give the ~5/3.
 * Sirius has a ~25/21 generator, three of which give the ~5/3.
@@ Line 269: / Line 269: @@
 By splitting the generator of sirius into three, remus efficiently represents the no-2's 13-limit with [[mos scale]]s of 18, 25, 32, or 39 steps.
-This is essentially [[electra]] but with prime 7, or more accurately, electra is the no-7's [[restriction]] of this temperament.
+This is essentially [[#Electra|electra]] but with prime 7, or more accurately, electra is the no-7's [[restriction]] of this temperament.
 Subgroup: 3.5.7.11
@@ Line 760: / Line 760: @@
 Badness (Sintel): 0.838
+== Electra ==
+[[File:Electra chords.wav|thumb|Just 11:13:15 and 143:165:195 triads on A]]
+Electra tempers out [[1875/1859]] and [[14641/14625]] in the 3.5.11.13 subgroup, with a generator near [[15/11]] or its tritave complement, [[11/5]]. It possesses [[mos scale]]s of the families {{mos scalesig|link=1|4L 3s<3/1>}}, {{mos scalesig|link=1|7L 4s<3/1>}}, and {{mos scalesig|link=1|7L 11s<3/1>}}.
+The 11:13:15 triad is a characteristic feature of many electra scales, and has been suggeested by {{u|CompactStar}} as a fundamental sonority for no-2's music, due to being relatively consonant compared to other no-2's chords. There is also a "minor" version of this triad, 143:165:195. If 11:13:15 is treated as the fundamental sonority of no-2's music, then electra can be viewed as the analogue to [[meantone family #Meantone|meantone]].
+The name was given by CompactStar in 2023 after the star {{w|Electra (star)|Electra}}, following a series of non-octave temperaments that were named after stars.
+[[Subgroup]]: 3.5.11
+[[Comma list]]: 2373046875/2357947691
+{{Mapping|legend=3| 1 -5 -5 | 0 9 10 }}
+: mapping generators: ~3, ~11/5
+[[Optimal tuning]]s:
+* [[WE]]: ~3 = 1901.8666{{c}}, ~11/5 = 1366.1408{{c}}
+: [[error map]]: {{val| -0.088 -0.379 +0.757 }}
+* [[CWE]]: ~3 = 1901.9550{{c}}, ~11/5 = 1366.1964{{c}}
+: error map: {{val| 0.000 -0.321 +0.8712 }}
+[[Optimal ET sequence]]: [[32edt|b32]], [[39edt|b39]], [[71edt|b71]], [[1239edt|b1239e]], [[1310edt|b1310e]], …, [[1665edt|b1665e]], [[1736edt|b1736e]]
+[[Badness]] (Sintel): 0.656
+=== 3.5.11.13 subgroup ===
+Subgroup: 3.5.11.13
+Comma list: 1875/1859, 14641/14625
+{{Mapping|legend=2| 1 -5 -5 -7 | 0 9 10 13 }}
+Optimal tunings:
+* WE: ~3 = 1901.8899{{c}}, ~11/5 = 1365.9871{{c}}
+* CWE: ~3 = 1901.9550{{c}}, ~11/5 = 1366.0281{{c}}
+Optimal ET sequence: [[32edt|b32]], [[39edt|b39]], [[71edt|b71]], [[110edt|b110]], [[181edt|b181]]
+Badness (Sintel): 0.405
+; Scales
+* [[Electra7]]
 == Alnilam ==