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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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{{delete}}
{{Infobox MOS}}


{{Infobox MOS|Equalized=2|Equave=8/5|Name=Diatonic|Paucitonic=1|Pattern=LLsLs|nLargeSteps=3|nSmallSteps=2}}
'''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]].  
 
'''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]]).


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 sesquitave 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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1400
1400
|34\19
|34\19
1406.8965f
1406.89655
|20\11
|20\11
1411.7647
1411.7647
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|}
|}
{| class="wikitable"
{| class="wikitable"
|+''ed2\3''
|+''ed8\12 (→ed2\3)''
! colspan="2" |Notation
! colspan="2" |Notation
!Supersoft
!Supersoft
Line 1,016: Line 1,014:
{| class="wikitable"
{| class="wikitable"
!Generators
!Generators
!Sextave notation
!Sixth notation
!Interval category name
!Interval category name
!Generators
!Generators
Line 1,026: Line 1,024:
|0
|0
|La
|La
|sextave (minor sixth)
|perfect sixth (minor sixth)
|0
|0
|La
|La
Line 1,066: Line 1,064:
| -5
| -5
|Lab
|Lab
|diminished sextave
|diminished sixth
|-
|-
|6
|6
Line 1,122: Line 1,120:
|}
|}
==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
Line 1,177: Line 1,175:
|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.
Line 1,188: Line 1,187:
[[Mapping]]: [{{val|1 1 2}}, {{val|0 -1 -3}}]
[[Mapping]]: [{{val|1 1 2}}, {{val|0 -1 -3}}]


[[Vals]]: {{val list|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}}]


[[Vals]]: {{val list|3ed14/9, 11ed14/9, 13ed14/9}}
==Scale tree==
==Scale tree==
The spectrum looks like this:
The spectrum looks like this:
Line 1,205: Line 1,195:
(bright)
(bright)
! colspan="2" |Normalised
! colspan="2" |Normalised
! colspan="2" |''ed2\3''
! colspan="2" |''ed8\12 (→ed2\3)''
! rowspan="2" |L
! rowspan="2" |L
! rowspan="2" |s
! rowspan="2" |s
Line 1,228: Line 1,218:
|Equalised
|Equalised
|-
|-
|11\18
|17\28
|
|
|
|
|507.692
|510
|323.077
|330
|''488.{{Overline|8}}''
|''485.714''
|''311.{{Overline|1}}''
|''314.286''
|4
|6
|3
|5
|1.333
|1.200
|
|
|-
|-
|
|
|63\103
|48\79
|
|
|507.383
|509.7345
|322.148
|329.2035
|''489.32''
|''486.076''
|''310.68''
|''313.924''
|23
|17
|17
|1.353
|14
|1.214
|
|
|-
|-
|
|
|52\85
|31\51
|
|509.589
|328.767
|''486.2745''
|''313.7255''
|11
|9
|1.222
|
|
|507.317
|-
|321.951
|14\23
|''489.412''
|
|''310.588''
|
|19
|509.{{Overline|09}}
|14
|327.{{Overline|27}}
|1.357
|''486.9565''
|''313.0435''
|5
|4
|1.250
|
|
|-
|-
|
|
|41\67
|39\64
|
|
|507.2165
|508.966
|321.6495
|326.087
|''489.552''
|''487.5''
|''310.448''
|''312.5''
|15
|14
|11
|11
|1.364
|1.273
|
|
|-
|-
|
|
|30\49
|25\41
|
|
|507.062
|508.475
|321.127
|325.424
|''489.796''
|''487.805''
|''310.204''
|''312.195''
|11
|9
|8
|7
|1.375
|1.286
|
|
|-
|-
|
|
|19\31
|36\59
|
|
|506.{{Overline|6}}
|508.235
|320
|324.706
|''490.323''
|''488.136''
|''309.678''
|''311.864''
|7
|13
|5
|10
|1.400
|1.300
|
|
|-
|-
|11\18
|
|
|
|
|46\75
|507.692
|506.422
|323.077
|319.266
|''488.{{Overline|8}}''
|''490.{{Overline|6}}''
|''311.{{Overline|1}}''
|''309.{{Overline|3}}''
|4
|17
|3
|12
|1.333
|1.417
|
|
|-
|-
|
|
|27\44
|63\103
|
|
|506.25
|507.383
|318.75
|322.148
|''490.{{Overline|90}}''
|''489.32''
|''309.{{Overline|09}}''
|''310.68''
|10
|23
|7
|17
|1.429
|1.353
|
|
|-
|-
|
|
|35\57
|52\85
|
|
|506.024
|507.317
|318.072
|321.951
|''491.228''
|''489.412''
|''308.712''
|''310.588''
|13
|19
|9
|14
|1.444
|1.357
|
|
|-
|-
|
|
|43\70
|41\67
|
|
|505.882
|507.2165
|317.647
|321.6495
|''491.429''
|''489.552''
|''308.571''
|''310.448''
|16
|15
|11
|11
|1.4545
|1.364
|
|
|-
|-
|
|
|51\83
|30\49
|
|
|505.785
|507.062
|317.355
|321.127
|''491.566''
|''489.796''
|''308.434''
|''310.204''
|19
|11
|13
|8
|1.4615
|1.375
|
|
|-
|-
|8\13
|
|
|19\31
|
|506.{{Overline|6}}
|320
|''490.323''
|''309.678''
|7
|5
|1.400
|
|
|505.263
|315.79
|''492.308''
|''307.692''
|3
|2
|1.500
|Aeolianic-Meantone starts here
|-
|-
|
|
|45\73
|
|
|504.673
|46\75
|314.019
|506.422
|''493.151''
|319.266
|''306.849''
|''490.{{Overline|6}}''
|''309.{{Overline|3}}''
|17
|17
|11
|12
|1.5455
|1.417
|
|
|-
|-
|
|
|37\60
|27\44
|
|
|504.{{Overline|54}}
|506.25
|313.{{Overline|63}}
|318.75
|''493.{{Overline|3}}''
|''490.{{Overline|90}}''
|''306.{{Overline|6}}''
|''309.{{Overline|09}}''
|14
|10
|7
|1.429
|
|-
|
|35\57
|
|506.024
|318.072
|''491.228''
|''308.712''
|13
|9
|9
|1.556
|1.444
|
|
|-
|-
|
|
|29\47
|43\70
|
|
|504.348
|505.882
|313.043
|317.647
|''493.617''
|''491.429''
|''306.383''
|''308.571''
|16
|11
|11
|7
|1.4545
|1.571
|
|
|-
|-
|
|
|21\34
|51\83
|
|
|504
|505.785
|312
|317.355
|''494.118''
|''491.566''
|''305.882''
|''308.434''
|8
|19
|5
|1.600
|
|-
|
|
|34\55
|503.{{Overline|703}}
|311.{{Overline|1}}
|''494.{{Overline|54}}''
|''305.{{Overline|45}}''
|13
|13
|8
|1.4615
|1.625
|
|
|-
|-
|8\13
|
|
|
|
|47\76
|505.263
|503.571
|315.79
|310.714
|''492.308''
|''494.737''
|''307.692''
|''305.263''
|3
|18
|2
|11
|1.500
|1.636
|Aeolianic-Meantone starts here
|
|-
|-
|
|
|13\21
|45\73
|
|
|503.226
|504.673
|309.678
|314.019
|''495.238''
|''493.151''
|''304.762''
|''306.849''
|5
|17
|3
|11
|1.667
|1.5455
|
|
|-
|-
|
|
|37\60
|
|
|31\50
|504.{{Overline|54}}
|502.{{Overline|702}}
|313.{{Overline|63}}
|308.{{Overline|108}}
|''493.{{Overline|3}}''
|''496''
|''306.{{Overline|6}}''
|''304''
|14
|12
|9
|7
|1.556
|1.714
|
|
|-
|-
|
|
|29\47
|
|
|49\79
|504.348
|502.564
|313.043
|307.692
|''493.617''
|''496.2025''
|''306.383''
|''303.7975''
|19
|11
|11
|1.727
|7
|1.571
|
|
|-
|-
|
|
|18\29
|21\34
|
|
|502.326
|504
|306.977
|312
|''496.552''
|''494.118''
|''303.448''
|''305.882''
|7
|8
|4
|5
|1.750
|1.600
|
|
|-
|-
|
|
|23\37
|
|
|501.{{Overline|81}}
|34\55
|305.{{Overline|45}}
|503.{{Overline|703}}
|''497.{{Overline|297}}''
|311.{{Overline|1}}
|''302.{{Overline|702}}''
|''494.{{Overline|54}}''
|9
|''305.{{Overline|45}}''
|5
|13
|1.800
|8
|1.625
|
|
|-
|-
|
|
|28\45
|
|
|501.492
|47\76
|304.478
|503.571
|''497.{{Overline|7}}''
|310.714
|''302.{{Overline|2}}''
|''494.737''
|''305.263''
|18
|11
|11
|6
|1.636
|1.833
|
|
|-
|-
|
|
|33\53
|13\21
|
|
|501.265
|503.226
|303.797
|309.678
|''498.113''
|''495.238''
|''301.887''
|''304.762''
|13
|5
|7
|3
|1.857
|1.667
|
|
|-
|-
|
|
|38\61
|
|
|501.09
|31\50
|303.297
|502.{{Overline|702}}
|''498.361''
|308.{{Overline|108}}
|''301.639''
|''496''
|15
|''304''
|8
|12
|1.875
|7
|1.714
|
|
|-
|-
|
|
|43\69
|
|
|500.971
|49\79
|302.913
|502.564
|''498.551''
|307.692
|''301.449''
|''496.2025''
|17
|''303.7975''
|9
|19
|1.889
|11
|1.727
|
|
|-
|-
|
|
|48\77
|18\29
|
|
|500.87
|502.326
|302.609
|306.977
|''498.701''
|''496.552''
|''301.299''
|''303.448''
|19
|7
|10
|4
|1.900
|1.750
|
|
|-
|-
|5\8
|
|
|23\37
|
|
|500
|501.{{Overline|81}}
|300
|305.{{Overline|45}}
|''500''
|''497.{{Overline|297}}''
|''300''
|''302.{{Overline|702}}''
|2
|1
|2.000
|Aeolianic-Meantone ends, Aeolianic-Pythagorean begins
|-
|
|47\75
|
|499.115
|297.345
|''501.{{Overline|3}}''
|''298.{{Overline|3}}''
|19
|9
|9
|2.111
|5
|1.800
|
|
|-
|-
|
|
|42\67
|28\45
|
|
|499.01
|501.492
|297.03
|304.478
|''501.4925''
|''497.{{Overline|7}}''
|''298.5075''
|''302.{{Overline|2}}''
|17
|11
|8
|6
|2.125
|1.833
|
|-
|
|33\53
|
|501.265
|303.797
|''498.113''
|''301.887''
|13
|7
|1.857
|
|-
|
|38\61
|
|501.09
|303.297
|''498.361''
|''301.639''
|15
|8
|1.875
|
|-
|
|43\69
|
|500.971
|302.913
|''498.551''
|''301.449''
|17
|9
|1.889
|
|-
|5\8
|
|
|500
|300
|''500''
|''300''
|2
|1
|2.000
|Aeolianic-Meantone ends, Aeolianic-Pythagorean begins
|-
|
|42\67
|
|499.01
|297.03
|''501.4925''
|''298.5075''
|17
|8
|2.125
|
|-
|
|37\59
|
|498.876
|296.629
|''501.695''
|''298.305''
|15
|7
|2.143
|
|-
|
|32\51
|
|498.701
|296.104
|''501.961''
|''298.039''
|13
|6
|2.167
|
|-
|
|27\43
|
|498.461
|295.385
|''502.326''
|''297.674''
|11
|5
|2.200
|
|-
|
|22\35
|
|498.113
|294.34
|''502.857''
|''297.143''
|9
|4
|2.250
|
|
|-
|-
|
|
|37\59
|
|
|498.876
|39\62
|296.629
|497.872
|''501.695''
|293.617
|''298.305''
|''503.226''
|15
|''296.774''
|16
|7
|7
|2.143
|2.286
|
|
|-
|-
|
|
|32\51
|17\27
|
|
|498.701
|497.561
|296.104
|292.683
|''501.961''
|''503.{{Overline|703}}''
|''298.039''
|''296.{{Overline|296}}''
|13
|7
|6
|3
|2.167
|2.333
|
|
|-
|-
|
|
|27\43
|
|
|498.461
|29\46
|295.385
|497.143
|''502.326''
|291.429
|''297.674''
|''504.348''
|11
|''295.652''
|12
|5
|5
|2.200
|2.400
|
|
|-
|-
|
|
|22\35
|
|
|498.113
|41\65
|294.34
|496.{{Overline|96}}
|''502.857''
|290.{{Overline|90}}
|''297.143''
|''504.615''
|9
|''295.385''
|4
|17
|2.250
|
|-
|
|
|39\62
|497.872
|293.617
|''503.226''
|''296.774''
|16
|7
|7
|2.286
|2.429
|
|
|-
|-
|
|
|17\27
|12\19
|
|
|497.561
|496.552
|292.683
|289.655
|''503.{{Overline|703}}''
|''505.263''
|''296.{{Overline|296}}''
|''294.737''
|7
|5
|3
|2
|2.333
|2.500
|
|-
|
|
|29\46
|497.143
|291.429
|''504.348''
|''295.652''
|12
|5
|2.400
|
|-
|
|
|41\65
|496.{{Overline|96}}
|290.{{Overline|90}}
|''504.615''
|''295.385''
|17
|7
|2.429
|
|-
|
|12\19
|
|496.552
|289.655
|''505.263''
|''294.737''
|5
|2
|2.500
|
|
|-
|-
Line 2,005: Line 2,055:
|21
|21
|5
|5
|R.200
|4.200
|
|
|-
|-
Line 2,079: Line 2,129:
|5.000
|5.000
|Aeolianic-Superpyth ends
|Aeolianic-Superpyth ends
|-
|
|35\54
|
|488.372
|265.116
|''518.{{Overline|518}}''
|''281.{{Overline|481}}''
|16
|3
|5.333
|
|-
|
|24\37
|
|488.136
|264.407
|''518.{{Overline|918}}''
|''281.{{Overline|081}}''
|11
|2
|5.500
|
|-
|
|37\57
|
|487.912
|263.736
|''519.298''
|''280.702''
|17
|3
|5.667
|
|-
|13\20
|
|
|487.5
|262.2
|''520''
|''280''
|6
|1
|6.000
|
|-
|-
|2\3
|2\3
Retrieved from "https://en.xen.wiki/w/3L_2s_(8/5-equivalent)"