No-twos subgroup temperaments: Difference between revisions

* 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.

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

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

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]].

== 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.

[[Comma list]]: 2373046875/2357947691

{{Mapping|legend=3| 1 -5 -5 | 0 9 10 }}

: mapping generators: ~3, ~11/5

* [[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 ===

Comma list: 1875/1859, 14641/14625

{{Mapping|legend=2| 1 -5 -5 -7 | 0 9 10 13 }}

Optimal tunings:  

* 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

* [[Electra7]]

Nearly a [[microtemperament]], alnilam takes a generator of [[~]][[55/27]] and equates nine of them tritave reduced with [[27/11]]. The name was given by {{u|CompactStar}} to continue with the theme of naming no-2's temperaments after proper star names, but also to indirectly reference [[mavila]] as its tritave-complement generator is ~[[81/55]], a flat fifth.

[[Subgroup]]: 3.5.11

[[Comma list]]: {{monzo| -35 9 10 }}

{{Mapping|legend=3| 1 -5 8 | 0 10 -9 }}

: mapping generators: ~3, ~55/27

* [[WE]]: ~3 = 1902.3576{{c}}, ~55/27 = 1229.7879{{c}}

: [[error map]]: {{val| +0.403 -0.222 -0.548 }}

* [[CWE]]: ~3 = 1901.9550{{c}}, ~55/27 = 1229.5321{{c}}

: error map: {{val| 0.000 -0.768 -1.467 }}

[[Optimal ET sequence]]: [[17edt|b17]], [[65edt|b65]], [[82edt|b82]], [[99edt|b99]], [[577edt|b577]], [[676edt|b676]], [[775edt|b775e]], [[874edt|b874e]], [[973edt|b973e]], [[2045edt|b2045ceee]]

[[Badness]] (Sintel): 3.84

 

No-twos temperaments exclude the second harmonic, thus meaning the next lowest prime, 3, is the best remaining option for an equave. If you then also remove the 3rd harmonic, the best remaining option for the equave would be [[5/1]] (also known as the pentane). Of course you can keep removing even more primes making 7th, 11th, etc. harmonics as the equave, but it will probably be impractical.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

== Catfish ==

 

 

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.

 

 

 

 

: [[error map]]: {{val| -1.144 +3.435 -1.869 }}

 

 

 

=== 5.11.13.17 subgroup ===

 

 

 

 

Optimal ET sequence: [[18ed5|c18]], [[41ed5|c41]], [[59ed5|c59]]

 

* [[Catfish7]]

* [[Catfish9]]

[[Category:Catalogs of subgroup temperaments]]