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, delta centauri 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.

* Deneb has a ~11/9 generator, three of which give the ~9/5.

* Canopus has a ~15/7 generator, five of which give the ~45/1 (5/1 up two tritaves).

* Miaplacidus has a ~7/5 generator, eight of which give the ~15/1 (5/1 up one tritave).

* Alnilam has a ~55/27 generator, ten of which give the ~1215/1 (5/1 up five tritaves).

* Izar has a ~30375/16807 generator, twelve of which give the ~1215/1 (5/1 up five tritaves).

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=== Polturus ===

This extension of arcturus adds [[polaris]]'s mapping for [[11/9]], mapping it to 5 generators down.

This extension of arcturus adds [[#Polaris|polaris]]'s mapping for [[11/9]], mapping it to 5 generators down.

Subgroup: 3.5.7.11

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Badness (Sintel): 0.177

== Vega ==

[[Subgroup]]: 3.5.7

[[Comma list]]: 6561/6125

: mapping generators: ~3, ~81/35

[[Optimal tuning]]s:

* [[WE]]: ~3 = 1891.9431{{c}}, ~81/35 = 1397.0975{{c}}

: [[error map]]: {{val| -10.012 +7.881 +7.654 }}

* [[CWE]]: ~3 = 1901.9550{{c}}, ~81/35 = 1404.6664{{c}}

: error map: {{val| 0.000 +23.019 +24.995 }}

[[Badness]] (Sintel): 1.34

== Sirius ==

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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): 0.542

== Miaplacidus ==

Miaplacidus tempers out [[5859375/5764801]], the amount by which eight [[7/5]]'s fall short of [[15/1]]. It thus tunes the chord 7:9:15 accurately. Having a generator of ~7/5, it possesses [[mos scale]]s of the families {{mos scalesig|link=1|3L 4s<3/1>}}, {{mos scalesig|link=1|3L 7s<3/1>}}, {{mos scalesig|link=1|3L 10s<3/1>}}, and {{mos scalesig|link=1|13L 3s<3/1>}}. Good tunings for this temperament are [[13edt]], [[16edt]], and [[29edt]].

[[Subgroup]]: 3.5.7

[[Comma list]]: 5859375/5764801

: mapping generators: ~3, ~7/5

[[Optimal tuning]]s:

* [[WE]]: ~3 = 1901.8789{{c}}, ~7/5 = 585.8428{{c}}

: [[error map]]: {{val| -0.076 -1.450 +1.880 }}

* [[CWE]]: ~3 = 1901.9550{{c}}, ~7/5 = 585.8590{{c}}

: error map: {{val| 0.000 -1.396 +1.950 }}

[[Badness]] (Sintel): 1.24

== Izar ==

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= 3.5.11-subgroup temperaments =

== Polaris ==

~~{{Main| Polaris }}~~

Polaris tempers out the comma 177147/171875, and thus equates seven [[5/3]]'s with [[15/11]], or equivalently seven [[9/5]]'s with [[11/9]]. It contains [[mos scale]]s of the families {{mos scalesig|link=1|2L 5s<3/1>}}, {{mos scalesig|link=1|2L 7s<3/1>}}, and {{mos scalesig|link=1|2L 9s<3/1>}}. The hendecatonic scale is generally the most usable of these, with the heptatonic and enneatonic scales usually being lopsided. Good tunings for polaris include [[15edt]], [[17edt]] and [[32edt]].

Polaris tempers out the comma 177147/171875, and thus equates seven [[5/3]]'s with [[15/11]], or equivalently seven [[9/5]]'s with [[11/9]].

[[Subgroup]]: 3.5.11

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[[Badness]] (Sintel): 1.01

; Scales

* [[Polaris9]]

* [[Polaris11]]

== Delta centauri ==

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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 ==

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Named by {{u|CompactStar}} in 2023, catfish tempers out 54925/54043 and 1795625/1771561 in the 5.11.13.17 subgroup.

CompactStar considers it useful to treat this temperament as [[11/5]]-repeating, which implies a generator approximating [[17/11]] or its inverse 121/85. It can be viewed as the no-2's no-3's analog to [[meantone family #Meantone|meantone]] with 11:13:17 being the substitute for [[4:5:6]], and possesses mos scales of the families {{~~MOS~~ scalesig|2L 3s<11/5>}}, {{~~MOS~~ scalesig|2L 5s<11/5>}}, {{~~MOS~~ scalesig|2L 7s<11/5>}}, and {{~~MOS~~ scalesig|9L 2s<11/5>}}. [[20ed11/5]] is one of the best tunings for this temperament, although it does noticeably flatten the [[13/11]] in the 11:13:17 chord.

CompactStar considers it useful to treat this temperament as [[11/5]]-repeating, which implies a generator approximating [[17/11]] or its inverse 121/85. It can be viewed as the no-2's no-3's analog to [[meantone family #Meantone|meantone]] with 11:13:17 being the substitute for [[4:5:6]], and possesses mos scales of the families {{mos scalesig|2L 3s<11/5>}}, {{mos scalesig|2L 5s<11/5>}}, {{mos scalesig|2L 7s<11/5>}}, and {{mos scalesig|9L 2s<11/5>}}. [[20ed11/5]] is one of the best tunings for this temperament, although it does noticeably flatten the [[13/11]] in the 11:13:17 chord.

[[Subgroup]]: 5.11.13

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* [[Catfish7]]

* [[Catfish9]]

~~= Graphs =~~

~~See: [[Catalog of 3.5.7 subgroup rank two temperaments #Graphs]]~~

~~== Projective tuning space diagrams ==~~

~~See: [[Catalog of 3.5.7 subgroup rank two temperaments #Projective tuning space diagrams]]~~

[[Category:Catalogs of subgroup temperaments]]

[[Category:Catalogs of rank-2 temperaments]]

@@ 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, delta centauri 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.
 * Deneb has a ~11/9 generator, three of which give the ~9/5.
 * Canopus has a ~15/7 generator, five of which give the ~45/1 (5/1 up two tritaves).
+* Miaplacidus has a ~7/5 generator, eight of which give the ~15/1 (5/1 up one tritave).
 * Alnilam has a ~55/27 generator, ten of which give the ~1215/1 (5/1 up five tritaves).
 * Izar has a ~30375/16807 generator, twelve of which give the ~1215/1 (5/1 up five tritaves).
@@ Line 43: / Line 44: @@
 === Polturus ===
-This extension of arcturus adds [[polaris]]'s mapping for [[11/9]], mapping it to 5 generators down.
+This extension of arcturus adds [[#Polaris|polaris]]'s mapping for [[11/9]], mapping it to 5 generators down.
 Subgroup: 3.5.7.11
@@ Line 191: / Line 192: @@
 Badness (Sintel): 0.177
+== Vega ==
+[[Subgroup]]: 3.5.7
+[[Comma list]]: 6561/6125
+{{Mapping|legend=3| 1 0 4 | 0 2 -3 }}
+: mapping generators: ~3, ~81/35
+[[Optimal tuning]]s:
+* [[WE]]: ~3 = 1891.9431{{c}}, ~81/35 = 1397.0975{{c}}
+: [[error map]]: {{val| -10.012 +7.881 +7.654 }}
+* [[CWE]]: ~3 = 1901.9550{{c}}, ~81/35 = 1404.6664{{c}}
+: error map: {{val| 0.000 +23.019 +24.995 }}
+[[Optimal ET sequence]]: [[4edt|b4]], [[11edt|b11d]], [[15edt|b15]], [[19edt|b19]], [[23edt|b23]], [[42edt|b42d]], [[65edt|b65cd]]
+[[Badness]] (Sintel): 1.34
 == Sirius ==
@@ Line 269: / Line 288: @@
 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 456: / Line 475: @@
 Badness (Sintel): 0.542
+== Miaplacidus ==
+Miaplacidus tempers out [[5859375/5764801]], the amount by which eight [[7/5]]'s fall short of [[15/1]]. It thus tunes the chord 7:9:15 accurately. Having a generator of ~7/5, it possesses [[mos scale]]s of the families {{mos scalesig|link=1|3L 4s<3/1>}}, {{mos scalesig|link=1|3L 7s<3/1>}}, {{mos scalesig|link=1|3L 10s<3/1>}}, and {{mos scalesig|link=1|13L 3s<3/1>}}. Good tunings for this temperament are [[13edt]], [[16edt]], and [[29edt]].
+[[Subgroup]]: 3.5.7
+[[Comma list]]: 5859375/5764801
+{{Mapping|legend=3| 1 -1 -1 | 0 8 9 }}
+: mapping generators: ~3, ~7/5
+[[Optimal tuning]]s:
+* [[WE]]: ~3 = 1901.8789{{c}}, ~7/5 = 585.8428{{c}}
+: [[error map]]: {{val| -0.076 -1.450 +1.880 }}
+* [[CWE]]: ~3 = 1901.9550{{c}}, ~7/5 = 585.8590{{c}}
+: error map: {{val| 0.000 -1.396 +1.950 }}
+[[Optimal ET sequence]]: [[13edt|b13]], [[120edt|b120]], [[133edt|b133]], …, [[211edt|b211]], [[224edt|b224]]
+[[Badness]] (Sintel): 1.24
 == Izar ==
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 = 3.5.11-subgroup temperaments =
 == Polaris ==
-{{Main| Polaris }}
+Polaris tempers out the comma 177147/171875, and thus equates seven [[5/3]]'s with [[15/11]], or equivalently seven [[9/5]]'s with [[11/9]].  It contains [[mos scale]]s of the families {{mos scalesig|link=1|2L 5s<3/1>}}, {{mos scalesig|link=1|2L 7s<3/1>}}, and {{mos scalesig|link=1|2L 9s<3/1>}}. The hendecatonic scale is generally the most usable of these, with the heptatonic and enneatonic scales usually being lopsided. Good tunings for polaris include [[15edt]], [[17edt]] and [[32edt]].
-Polaris tempers out the comma 177147/171875, and thus equates seven [[5/3]]'s with [[15/11]], or equivalently seven [[9/5]]'s with [[11/9]].
 [[Subgroup]]: 3.5.11
@@ Line 633: / Line 670: @@
 [[Badness]] (Sintel): 1.01
+; Scales
+* [[Polaris9]]
+* [[Polaris11]]
 == Delta centauri ==
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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|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 ==
@@ Line 1,095: / Line 1,180: @@
 Named by {{u|CompactStar}} in 2023, catfish tempers out 54925/54043 and 1795625/1771561 in the 5.11.13.17 subgroup.
-CompactStar considers it useful to treat this temperament as [[11/5]]-repeating, which implies a generator approximating [[17/11]] or its inverse 121/85. It can be viewed as the no-2's no-3's analog to [[meantone family #Meantone|meantone]] with 11:13:17 being the substitute for [[4:5:6]], and possesses mos scales of the families {{MOS scalesig|2L 3s<11/5>}}, {{MOS scalesig|2L 5s<11/5>}}, {{MOS scalesig|2L 7s<11/5>}}, and {{MOS scalesig|9L 2s<11/5>}}. [[20ed11/5]] is one of the best tunings for this temperament, although it does noticeably flatten the [[13/11]] in the 11:13:17 chord.
+CompactStar considers it useful to treat this temperament as [[11/5]]-repeating, which implies a generator approximating [[17/11]] or its inverse 121/85. It can be viewed as the no-2's no-3's analog to [[meantone family #Meantone|meantone]] with 11:13:17 being the substitute for [[4:5:6]], and possesses mos scales of the families {{mos scalesig|2L 3s<11/5>}}, {{mos scalesig|2L 5s<11/5>}}, {{mos scalesig|2L 7s<11/5>}}, and {{mos scalesig|9L 2s<11/5>}}. [[20ed11/5]] is one of the best tunings for this temperament, although it does noticeably flatten the [[13/11]] in the 11:13:17 chord.
 [[Subgroup]]: 5.11.13
@@ Line 1,133: / Line 1,218: @@
 * [[Catfish7]]
 * [[Catfish9]]
-= Graphs =
-See: [[Catalog of 3.5.7 subgroup rank two temperaments #Graphs]]
-== Projective tuning space diagrams ==
-See: [[Catalog of 3.5.7 subgroup rank two temperaments #Projective tuning space diagrams]]
 [[Category:Catalogs of subgroup temperaments]]
 [[Category:Catalogs of rank-2 temperaments]]