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3L 2s (8/5-equivalent): Difference between revisions

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Why is there a 14/9-repeating temperament??
 
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{{Infobox MOS|Equalized=3|Equave=8/5|Name=Diatonic|Collapsed=2|Pattern=LLsLs|nLargeSteps=3|nSmallSteps=2}}
{{Infobox MOS}}


'''3L 2s<8/5>''' (sometimes called '''diatonic'''), is a minor sixth-repeating MOS scale. The notation "<8/5>" means the period of the MOS is 8/5, disambiguating it from octave-repeating [[3L 2s]]. The name of the period interval is called the '''sextave''' (by analogy to the [[tritave]]).
'''3L 2s<8/5>''' is a minor sixth-repeating MOS scale. The notation "<8/5>" means the period of the MOS is 8/5, disambiguating it from octave-repeating [[3L 2s]].  


The generator range is 240 to 342.9 cents, placing it on the [[6/5|diatonic minor third]], usually representing a minor third of some type (like [[6/5]]). The bright (chroma-positive) generator is, however, its minor sixth complement (480 to 514.3 cents).  
The generator range is 240 to 342.9 cents, placing it on the [[6/5|diatonic minor third]], usually representing a minor third of some type (like [[6/5]]). The bright (chroma-positive) generator is, however, its minor sixth complement (480 to 514.3 cents).  


Because this diatonic is a minor sixth-repeating scale, each tone has an 8/5 minor sixth above it. The scale has one major chord, one minor chord and three diminished chords. This diatonic also has two diminished 7th chords, making it a warped melodic minor scale.
Because this is a minor sixth-repeating scale, each tone has an 8/5 minor sixth above it. The scale has one major chord, one minor chord and three diminished chords. This scale also has two diminished 7th chords, making it a warped melodic minor scale.


[[Basic]] diatonic is in [[8ed8/5]], which is a very good minor sixth-based equal tuning similar to [[12edo]].
[[Basic]] 3L 2s<8/5> is in [[8ed8/5]], which is a very good minor sixth-based equal tuning similar to [[12edo]].
==Notation==
==Notation==
There are 2 main ways to notate the diatonic scale. One method uses a simple sextave (minor sixth) repeating notation consisting of 5 naturals (La, Si, Do, Re, Mi). Given that 1-7/6-3/2 is minor sixth-equivalent to a tone cluster of 1-16/15-7/6, it may be more convenient to notate these diatonic scales as repeating at the double sextave (diminished eleventh~tenth), however it does make navigating the [[Generator|genchain]] harder. This way, 3/2 is its own pitch class, distinct from 16\15. Notating this way produces a tenth which is the Dorian mode of Annapolis[6L 4s] or Oriole[6L 4s]. Since there are exactly 10 naturals in double sextave notation, Greek numerals 1-10 may be used.
There are 2 main ways to notate this scale. One method uses a simple sixth repeating notation consisting of 5 naturals (La, Si, Do, Re, Mi). Given that 1-7/6-3/2 is minor sixth-equivalent to a tone cluster of 1-16/15-7/6, it may be more convenient to notate these diatonic scales as repeating at the double sixth (diminished eleventh~tenth), however it does make navigating the [[Generator|genchain]] harder. This way, 3/2 is its own pitch class, distinct from 16\15. Notating this way produces a tenth which is the Dorian mode of Annapolis[6L 4s] or Oriole[6L 4s]. Since there are exactly 10 naturals in double sixth notation, Greek numerals 1-10 may be used.
{| class="wikitable"
{| class="wikitable"
|+
|+
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{| class="wikitable"
{| class="wikitable"
!Generators
!Generators
!Sextave notation
!Sixth notation
!Interval category name
!Interval category name
!Generators
!Generators
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|0
|0
|La
|La
|sextave (minor sixth)
|perfect sixth (minor sixth)
|0
|0
|La
|La
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| -5
| -5
|Lab
|Lab
|diminished sextave
|diminished sixth
|-
|-
|6
|6
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|}
|}
==Modes==
==Modes==
The mode names are based on the major satellites of Uranus, in order of size:
The mode names are based on the modes of the diatonic scale , in order of size:
{| class="wikitable"
{| class="wikitable"
!Mode
!Mode
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|d
|d
|}
|}
==Temperaments==
==Temperaments==
The most basic rank-2 temperament interpretation of this diatonic is '''Aeolianic''', which has septimal 6:7:9 or pental 10:12:15 chords spelled <code>root-(p-1g)-(3g)</code> (p = the minor sixth, g = the approximate 4/3). The name "Aeolianic" comes from the Aeolian minor mode having the minor sixth as its characteristic interval.
The most basic rank-2 temperament interpretation of this diatonic is '''Aeolianic''', which has septimal 6:7:9 or pental 10:12:15 chords spelled <code>root-(p-1g)-(3g)</code> (p = the minor sixth, g = the approximate 4/3). The name "Aeolianic" comes from the Aeolian minor mode having the minor sixth as its characteristic interval.
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[[Optimal ET sequence]]: 5ed8/5, 8ed8/5, 13ed8/5
[[Optimal ET sequence]]: 5ed8/5, 8ed8/5, 13ed8/5
===='''Aeolianic-Superpyth'''====
[[Subgroup]]: 14/9.4/3.3/2
[[Comma]] list: [[64/63]]
[[POL2]] generator: ~7/6 = 276.0795
[[Mapping]]:  [{{val|1 1 2}}, {{val|0 -1 -3}}]


[[Optimal ET sequence]]: 3ed14/9, 11ed14/9, 14ed14/9
==Scale tree==
==Scale tree==
The spectrum looks like this:
The spectrum looks like this:
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