User:Moremajorthanmajor/3L 2s (minor sixth-equivalent): Difference between revisions

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#REDIRECT [[3L 2s (8/5-equivalent)]]
'''3L 2s<minor sixth>''' (sometimes called '''diatonic'''), is a minor sixth-repeating MOS scale. The notation "<minor sixth>" means the period of the MOS is a minor sixth, 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).
 
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.
 
[[Basic]] diatonic is in [[8ed8/5]], which is a very good minor sixth-based equal tuning similar to [[12edo]].
==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.
{| class="wikitable"
|+
Normalized
! colspan="2" |Notation
!Supersoft
!Soft
!Semisoft
!Basic
!Semihard
!Hard
!Superhard
|-
!Diatonic
!Oriole, Annapolis
!18eds
!13eds
!21eds
!8eds
!19eds
!11eds
!14eds
|-
|La#
|Α#
|1\18
46.154
|1\13
63.158
|2\21
77.419
| rowspan="2" |1\8
100
|3\19
124.138
|2\11
[[Tel:141.1765|141.1765]]
|3\14
163.{{Overline|63}}
|-
|Sib
|Βb
|3\18
138.4615
|2\13
126.316
|3\21
116.129
|2\19
82.759
|1\11
70.588
|1\14
54.{{Overline|54}}
|-
|Si
|Β
|4\18
184.615
|3\13
189.474
|5\21
193.548
|2\8
200
|5\19
206.897
|3\11
211.764
|4\14
218.{{Overline|18}}
|-
|Si#
|Β#
|5\18
230.769
| rowspan="2" |4\13
252.632
|7\21
270.968
|3\8
300
|8\19
331.0345
|5\11
352.941
|7\14
381.{{Overline|81}}
|-
|Dob
|Γb
|6\18
276.923
|6\21
232.258
|2\8
200
|4\19
165.517
|2\11
141.1765
|2\14
109.{{Overline|09}}
|-
|'''Do'''
|'''Γ'''
|'''7\18'''
'''323.076'''
|'''5\13'''
'''315.7895'''
|'''8\21'''
'''309.677'''
|'''3\8'''
'''300'''
|'''7\19'''
'''289.655'''
|'''4\11'''
'''282.353'''
|'''5\14'''
'''272.{{Overline|72}}'''
|-
|Do#
|Γ#
|8\18
369.231
|6\13
378.947
|10\21
387.0968
| rowspan="2" |4\8
400
|10\19
413.793
|6\11
423.529
|8\14
436.{{Overline|36}}
|-
|Reb
|Δb
|10\18
461.5385
|7\13
442.105
|11\21
425.8065
|9\19
372.413
|5\11
352.941
|6\14
327.{{Overline|27}}
|-
|'''Re'''
|'''Δ'''
|'''11\18'''
'''507.692'''
|'''8\13'''
'''505.263'''
|'''13\21'''
'''503.226'''
|'''5\8'''
'''500'''
|'''12\19'''
'''496.551'''
|'''7\11'''
'''494.118'''
|'''9\14'''
'''490.{{Overline|90}}'''
|-
|Re#
|Δ#
|12\18
553.846
|9\13
568.421
|15\21
580.645
| rowspan="2" |6\8
600
|15\19
620.689
|9\11
635.294
|12\14
654.{{Overline|54}}
|-
|Mib
|Εb
|14\18
646.154
|10\13
631.579
|16\21
619.355
|14\19
579.310
|8\11
564.706
|10\14
545.{{Overline|45}}
|-
|Mi
|Ε
|15\18
692.308
|11\13
694.737
|18\21
696.774
|7\8
700
|17\19
703.448
|10\11
705.88235
|13\14
709.{{Overline|09}}
|-
|Mi#
|Ε#
|16\18
738.4615
| rowspan="2" |12\13
757.895
|20\21
774.194
|8\8
800
|20\19
827.586
|12\11
847.059
|16\14
872.{{Overline|72}}
|-
|Lab
|Ϛb/Ϝb
|17\18
784.615
|19\21
735.484
|7\8
700
|16\19
662.069
|9\11
635.294
|11\14
600
|-
!La
!Ϛ/Ϝ
!18\18
830.769
!13\13
821.053
!21\21
812.903
!8\8
 
800
!19\19
786.207
!11\11
776.471
!14\14
 
763.{{Overline|63}}
|-
|La#
|Ϛ#/Ϝ#
|19\18
876.923
|14\13
884.2105
|23\21
890.323
| rowspan="2" |9\8
900
|22\19
910.345
|13\11
917.647
|17\14
927.{{Overline|27}}
|-
|Sib
|Ζb
|21\18
969.231
|15\13
947.368
|24\21
929.032
|21\19
868.9655
|12\11
847.059
|15\14
818.{{Overline|18}}
|-
|Si
|Ζ
|22\18
1015.385
|16\13
1010.526
|26\21
1006.452
|10\8
1000
|24\19
993.1035
|14\11
988.235
|18\14
981.{{Overline|81}}
|-
|Si#
|Ζ#
|23\18
1061.5385
| rowspan="2" |17\13
1071.684
|28\21
1083.871
|11\8
1100
|27\19
1117.241
|16\11
1129.412
|21\14
1145.{{Overline|45}}
|-
|Dob
|Ηb
|24\18
1107.692
|27\21
1045.161
|10\8
1000
|23\19
951.724
|13\11
917.647
|16\14
872.{{Overline|72}}
|-
|'''Do'''
|'''Η'''
|'''25\18'''
'''1153.846'''
|'''18\13'''
'''1136.842'''
|'''29\21'''
'''1122.581'''
|'''11\8'''
'''1100'''
|'''26\19'''
'''1075.862'''
|'''15\11'''
'''1052.8235'''
|'''19\14'''
'''1036.{{Overline|36}}'''
|-
|Do#
|Η#
|26\18
1200
|19\13
1200
|31\21
1200
| rowspan="2" |12\8
1200
|29\19
1200
|17\11
1200
|22\14
1200
|-
|Reb
|Θb
|28\18
1292.308
|20\13
1263.158
|32\21
1238.710
|28\19
1158.621
|16\11
1129.412
|20\14
1090.{{Overline|90}}
|-
|'''Re'''
|'''Θ'''
|'''29\18'''
'''1338.4615'''
|'''21\13'''
'''1326.316'''
|'''34\21'''
'''1316.129'''
|'''13\8'''
'''1300'''
|'''31\19'''
'''1282.759'''
|'''18\11'''
'''1270.588'''
|'''23\14'''
'''1254.{{Overline|54}}'''
|-
|Re#
|Θ#
|30\18
1384.615
|22\13
1389.474
|36\21
1393.548
| rowspan="2" |14\8
1400
|34\19
1406.897
|20\11
1411.765
|26\14
1418.{{Overline|18}}
|-
|Mib
|Ιb
|32\18
1476.923
|23\13
1452.632
|37\21
1432.258
|33\19
1365.517
|19\11
1341.1765
|24\14
1309.{{Overline|09}}
|-
|Mi
|Ι
|33\18
1523.077
|24\13
1515.7895
|39\21
1509.677
|15\8
1500
|36\19
1489.655
|21\11
1482.352
|27\14
1472.{{Overline|72}}
|-
|Mi#
|Ι#
|34\18
1569.231
| rowspan="2" |25\13
1578.947
|41\21
1587.097
|16\8
1600
|39\19
1613.793
|23\11
1623.529
|30\14
1636.{{Overline|36}}
|-
|Lab
|Αb
|35\18
1615.385
|40\21
1548.387
|15\8
1500
|35\19
1448.286
|20\11
1411.765
|25\14
1363.{{Overline|63}}
|-
!La
!Α
!36\18
1661.5385
!26\13
1642.105
!42\21
1625.8065
!16\8
 
1600
!38\19
1572.414
!22\11
1552.941
!28\14
1527.{{Overline|27}}
|}
{| class="wikitable"
|+''ed8\12 (→ed2\3)''
! colspan="2" |Notation
!Supersoft
!Soft
!Semisoft
!Basic
!Semihard
!Hard
!Superhard
|-
!Diatonic
!Oriole, Annapolis
!18eds
!13eds
!21eds
!8eds
!19eds
!11eds
!14eds
|-
|La#
|Α#
|''1\18''
''44.{{Overline|4}}''
|''1\13''
''61.5385''
|''2\21''
''76.1905''
| rowspan="2" |''1\8''
''100''
|''3\19''
''126.316''
|''2\11''
''145.{{Overline|45}}''
|''3\14''
''171.429''
|-
|Sib
|Βb
|''3\18''
''133.{{Overline|3}}''
|''2\13''
''123.077''
|''3\21''
''114.286''
|''2\19''
''84.2105''
|''1\11''
''72.{{Overline|72}}''
|''1\14''
''57.143''
|-
|Si
|Β
|''4\18''
''177.{{Overline|7}}''
|''3\13''
''184.615''
|''5\21''
''190.476''
|''2\8''
''200''
|''5\19''
''210.526''
|''3\11''
''218.{{Overline|18}}''
|''4\14''
''228.571''
|-
|Si#
|Β#
|''5\18''
''222.{{Overline|2}}''
| rowspan="2" |''4\13''
''246.154''
|''7\21''
''266.{{Overline|6}}''
|''3\8''
''300''
|''8\19''
''336.842''
|''5\11''
''363.{{Overline|63}}''
|''7\14''
''400''
|-
|Dob
|Γb
|''6\18''
''266.{{Overline|6}}''
|''6\21''
''228.571''
|''2\8''
''200''
|''4\19''
''168.421''
|''2\11''
''145.{{Overline|45}}''
|''2\14''
''114.28''
|-
|'''Do'''
|'''Γ'''
|'''''7\18'''''
'''''311.{{Overline|1}}'''''
|'''''5\13'''''
'''''307.692'''''
|'''''8\21'''''
'''''304.762'''''
|'''''3\8'''''
'''''300'''''
|'''''7\19'''''
'''''294.737'''''
|'''''4\11'''''
'''''290.{{Overline|90}}'''''
|'''''5\14'''''
'''''285.714'''''
|-
|Do#
|Γ#
|''8\18''
''355.{{Overline|5}}''
|''6\13''
''369.231''
|''10\21''
''380.952''
| rowspan="2" |''4\8''
''400''
|''10\19''
''421.053''
|''6\11''
''436.{{Overline|36}}''
|''8\14''
''457.143''
|-
|Reb
|Δb
|''10\18''
''444.{{Overline|4}}''
|''7\13''
''430.769''
|''11\21''
''419.048''
|''9\19''
''378.947''
|''5\11''
''363.{{Overline|63}}''
|''6\14''
''342.857''
|-
|'''Re'''
|'''Δ'''
|'''''11\18'''''
'''''488.{{Overline|8}}'''''
|'''''8\13'''''
'''''492.308'''''
|'''''13\21'''''
'''''495.238'''''
|'''''5\8'''''
'''''500'''''
|'''''12\19'''''
'''''505.263'''''
|'''''7\11'''''
'''''509.{{Overline|09}}'''''
|'''''9\14'''''
'''''514.286'''''
|-
|Re#
|Δ#
|''12\18''
''533.{{Overline|3}}''
|''9\13''
''553.846''
|''15\21''
''571.429''
| rowspan="2" |''6\8''
''600''
|''15\19''
''631.421''
|''9\11''
''654.{{Overline|54}}''
|''12\14''
''685.714''
|-
|Mib
|Εb
|''14\18''
''622.{{Overline|2}}''
|''10\13''
''615.385''
|''16\21''
''609.524''
|''14\19''
''589.474''
|''8\11''
''581.{{Overline|81}}''
|''10\14''
''571.424''
|-
|Mi
|Ε
|''15\18''
''666.{{Overline|6}}''
|''11\13''
''676.9231''
|''18\21''
''685.7143''
|''7\8''
''700''
|''17\19''
''715.7895''
|''10\11''
''727.{{Overline|27}}''
|''13\14''
''742.857''
|-
|Mi#
|Ε#
|''16\18''
''711.{{Overline|1}}''
| rowspan="2" |''12\13''
''738.4615''
|''20\21''
''761.905''
|''8\8''
''800''
|''20\19''
''842.105''
|''12\11''
''872.{{Overline|72}}''
|''16\14''
''914.286''
|-
|Lab
|Ϛb/Ϝb
|''17\18''
''755.{{Overline|5}}''
|''19\21''
''723.8095''
|''7\8''
''700''
|''16\19''
''673.684''
|''8\11''
''581.{{Overline|81}}''
|''11\14''
''628.571''
|-
!La
!Ϛ/Ϝ
! colspan="7" |''800''
|-
|La#
|Ϛ#/Ϝ#
|''19\18''
''844.{{Overline|4}}''
|''14\13''
''861.5385''
|''23\21''
''876.1905''
| rowspan="2" |''9\8''
''900''
|''22\19''
''926.316''
|''13\11''
''945.{{Overline|45}}''
|''17\14''
''971.429''
|-
|Sib
|Ζb
|''21\18''
''933.{{Overline|3}}''
|''15\13''
''923.077''
|''24\21''
''914.286''
|''21\19''
''884.2105''
|''12\11''
''872.{{Overline|72}}''
|''15\14''
''857.143''
|-
|Si
|Ζ
|''22\18''
''977.{{Overline|7}}''
|''16\13''
''984.615''
|''26\21''
''990.476''
|''10\8''
''1000''
|''24\19''
''1010.526''
|''14\11''
''1018.{{Overline|18}}''
|''18\14''
''1028.571''
|-
|Si#
|Ζ#
|''23\18''
''1022.{{Overline|2}}''
| rowspan="2" |''17\13''
''1046.154''
|''28\21''
''1066.{{Overline|6}}''
|''11\8''
''1100''
|''27\19''
''1136.8421''
|''16\11''
''1163.{{Overline|63}}''
|''21\14''
''1200''
|-
|Dob
|Ηb
|''24\18''
''1066.{{Overline|6}}''
|''27\21''
''1028.571''
|''10\8''
''1000''
|''23\19''
''968.421''
|''13\11''
''945.{{Overline|45}}''
|''16\14''
''914.286''
|-
|'''Do'''
|'''Η'''
|'''''25\18'''''
'''''1111.{{Overline|1}}'''''
|'''''18\13'''''
'''''1107.692'''''
|'''''29\21'''''
'''''1104.762'''''
|'''''11\8'''''
'''''1100'''''
|'''''26\19'''''
'''''1094.737'''''
|'''''15\11'''''
'''''1090.{{Overline|90}}'''''
|'''''19\14'''''
'''''1085.714'''''
|-
|Do#
|Η#
|''26\18''
''1155.{{Overline|5}}''
|''19\13''
''1169.237''
|''31\21''
''1180.952''
| rowspan="2" |''12\8''
''1200''
|''29\19''
''1221.053''
|''17\11''
''1236.{{Overline|36}}''
|''22\14''
''1257.143''
|-
|Reb
|Θb
|''28\18''
''1244.{{Overline|4}}''
|''20\13''
''1230.769''
|''32\21''
''1219.048''
|''28\19''
''1178.947''
|''16\11''
''1163.{{Overline|63}}''
|''20\14''
''1142.857''
|-
|'''Re'''
|'''Θ'''
|'''''29\18'''''
'''''1288.{{Overline|8}}'''''
|'''''21\13'''''
'''''1292.308'''''
|'''''34\21'''''
'''''1295.238'''''
|'''''13\8'''''
'''''1300'''''
|'''''31\19'''''
'''''1305.263'''''
|'''''18\11'''''
'''''1309.{{Overline|09}}'''''
|'''''23\14'''''
'''''1314.286'''''
|-
|Re#
|Θ#
|''30\18''
''1333.{{Overline|3}}''
|''22\13''
''1187.924''
|''36\21''
''1371.429''
| rowspan="2" |''14\8''
''1400''
|''34\19''
''1431.421''
|''20\11''
''1454.{{Overline|54}}''
|''26\14''
''1485.714''
|-
|Mib
|Ιb
|''32\18''
''1422.{{Overline|2}}''
|''23\13''
''1415.385''
|''37\21''
''1409.524''
|''33\19''
''1389.474''
|''19\11''
''1381.{{Overline|81}}''
|''24\14''
''1371.429''
|-
|Mi
|Ι
|''33\18''
''1466.{{Overline|6}}''
|''24\13''
''1476.923''
|''39\21''
''1485.714''
|''15\8''
''1500''
|''36\19''
''1515.7895''
|''21\11''
''1527.{{Overline|27}}''
|''27\14''
''1542.857''
|-
|Mi#
|Ι#
|''34\18''
''1511.{{Overline|1}}''
| rowspan="2" |''25\13''
''1538.4615''
|''41\21''
''1561.905''
|''16\8''
''1600''
|''39\19''
''1642.105''
|''23\11''
''1672.{{Overline|72}}''
|''30\14''
''1714.286''
|-
|Lab
|Αb
|''35\18''
''1555.{{Overline|5}}''
|''40\21''
''1523.8095''
|''15\8''
''1500''
|''35\19''
''1473.684''
|''20\11''
''1454.{{Overline|54}}''
|''25\14''
''1428.571''
|-
!La
!Α
! colspan="7" |''1600''
|}
 
==Intervals==
{| class="wikitable"
!Generators
!Sextave notation
!Interval category name
!Generators
!Notation of sixth inverse
!Interval category name
|-
| colspan="6" |The 5-note MOS has the following intervals (from some root):
|-
|0
|La
|sextave (minor sixth)
|0
|La
|perfect unison
|-
|1
|Re
|perfect fourth
| -1
|Do
|minor third
|-
|2
|Si
|major second
| -2
|Mib
|diminished fifth
|-
|3
|Mi
|perfect fifth
| -3
|Sib
|minor second
|-
|4
|Do#
|major third
| -4
|Reb
|diminished fourth
|-
| colspan="6" |The chromatic 8-note MOS also has the following intervals (from some root):
|-
|5
|La#
|augmented unison (chroma)
| -5
|Lab
|diminished sextave
|-
|6
|Re#
|augmented fourth
| -6
|Dob
|diminished third
|-
|7
|Si#
|augmented second
| -7
|Mibb
|doubly diminished fifth
|}
==Genchain==
The generator chain for this scale is as follows:
{| class="wikitable"
|Sibb
|Mibb
|Dob
|Lab
|Reb
|Sib
|Mib
|Do
|La
|Re
|Si
|Mi
|Do#
|La#
|Re#
|Si#
|Mi#
|-
|d2
|dd5
|d3
|d6
|d4
|m2
|d5
|m3
|P1
|P4
|M2
|P5
|M3
|A1
|A4
|A2
|A5
|}
==Modes==
The mode names are based on the modes of the diatonic scale , in order of size:
{| class="wikitable"
!Mode
!Scale
![[Modal UDP Notation|UDP]]
! colspan="4" |Interval type
|-
!name
!pattern
!notation
!2nd
!3rd
!4th
!5th
|-
|Hindu
|LLsLs
|<nowiki>4|0</nowiki>
|M
|M
|P
|P
|-
|Minor
|LsLLs
|<nowiki>3|1</nowiki>
|M
|m
|P
|P
|-
|Half diminished
|LsLsL
|<nowiki>2|2</nowiki>
|M
|m
|P
|d
|-
|Diminished
|sLLsL
|<nowiki>1|3</nowiki>
|m
|m
|P
|d
|-
|Altered
|sLsLL
|<nowiki>0|4</nowiki>
|m
|m
|d
|d
|}
 
==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.
==='''Aeolianic-Meantone'''===
[[Subgroup]]: 8/5.4/3.3/2
 
[[Comma]] list: [[81/80]]
 
[[POL2]] generator: ~6/5 = 308.3057
 
[[Mapping]]: [{{val|1 1 2}}, {{val|0 -1 -3}}]
 
[[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==
The spectrum looks like this:
{| class="wikitable"
! colspan="3" rowspan="2" |Generator
(bright)
! colspan="2" |Normalised
! colspan="2" |''ed8\12 (→ed2\3)''
! rowspan="2" |L
! rowspan="2" |s
! rowspan="2" |L/s
! rowspan="2" |Comments
|-
!Chroma-positive
!Chroma-negative
!Chroma-positive
!Chroma-negative
|-
|3\5
|
|
|514.286
|342.857
|''480''
|''320''
|1
|1
|1.000
|Equalised
|-
|17\28
|
|
|510
|330
|''485.714''
|''314.286''
|6
|5
|1.200
|
|-
|
|48\79
|
|509.7345
|329.2035
|''486.076''
|''313.924''
|17
|14
|1.214
|
|-
|
|31\51
|
|509.589
|328.767
|''486.2745''
|''313.7255''
|11
|9
|1.222
|
|-
|14\23
|
|
|509.{{Overline|09}}
|327.{{Overline|27}}
|''486.9565''
|''313.0435''
|5
|4
|1.250
|
|-
|
|39\64
|
|508.966
|326.087
|''487.5''
|''312.5''
|14
|11
|1.273
|
|-
|
|25\41
|
|508.475
|325.424
|''487.805''
|''312.195''
|9
|7
|1.286
|
|-
|
|36\59
|
|508.235
|324.706
|''488.136''
|''311.864''
|13
|10
|1.300
|
|-
|11\18
|
|
|507.692
|323.077
|''488.{{Overline|8}}''
|''311.{{Overline|1}}''
|4
|3
|1.333
|
|-
|
|63\103
|
|507.383
|322.148
|''489.32''
|''310.68''
|23
|17
|1.353
|
|-
|
|52\85
|
|507.317
|321.951
|''489.412''
|''310.588''
|19
|14
|1.357
|
|-
|
|41\67
|
|507.2165
|321.6495
|''489.552''
|''310.448''
|15
|11
|1.364
|
|-
|
|30\49
|
|507.062
|321.127
|''489.796''
|''310.204''
|11
|8
|1.375
|
|-
|
|19\31
|
|506.{{Overline|6}}
|320
|''490.323''
|''309.678''
|7
|5
|1.400
|
|-
|
|
|46\75
|506.422
|319.266
|''490.{{Overline|6}}''
|''309.{{Overline|3}}''
|17
|12
|1.417
|
|-
|
|27\44
|
|506.25
|318.75
|''490.{{Overline|90}}''
|''309.{{Overline|09}}''
|10
|7
|1.429
|
|-
|
|35\57
|
|506.024
|318.072
|''491.228''
|''308.712''
|13
|9
|1.444
|
|-
|
|43\70
|
|505.882
|317.647
|''491.429''
|''308.571''
|16
|11
|1.4545
|
|-
|
|51\83
|
|505.785
|317.355
|''491.566''
|''308.434''
|19
|13
|1.4615
|
|-
|8\13
|
|
|505.263
|315.79
|''492.308''
|''307.692''
|3
|2
|1.500
|Aeolianic-Meantone starts here
|-
|
|45\73
|
|504.673
|314.019
|''493.151''
|''306.849''
|17
|11
|1.5455
|
|-
|
|37\60
|
|504.{{Overline|54}}
|313.{{Overline|63}}
|''493.{{Overline|3}}''
|''306.{{Overline|6}}''
|14
|9
|1.556
|
|-
|
|29\47
|
|504.348
|313.043
|''493.617''
|''306.383''
|11
|7
|1.571
|
|-
|
|21\34
|
|504
|312
|''494.118''
|''305.882''
|8
|5
|1.600
|
|-
|
|
|34\55
|503.{{Overline|703}}
|311.{{Overline|1}}
|''494.{{Overline|54}}''
|''305.{{Overline|45}}''
|13
|8
|1.625
|
|-
|
|
|47\76
|503.571
|310.714
|''494.737''
|''305.263''
|18
|11
|1.636
|
|-
|
|13\21
|
|503.226
|309.678
|''495.238''
|''304.762''
|5
|3
|1.667
|
|-
|
|
|31\50
|502.{{Overline|702}}
|308.{{Overline|108}}
|''496''
|''304''
|12
|7
|1.714
|
|-
|
|
|49\79
|502.564
|307.692
|''496.2025''
|''303.7975''
|19
|11
|1.727
|
|-
|
|18\29
|
|502.326
|306.977
|''496.552''
|''303.448''
|7
|4
|1.750
|
|-
|
|23\37
|
|501.{{Overline|81}}
|305.{{Overline|45}}
|''497.{{Overline|297}}''
|''302.{{Overline|702}}''
|9
|5
|1.800
|
|-
|
|28\45
|
|501.492
|304.478
|''497.{{Overline|7}}''
|''302.{{Overline|2}}''
|11
|6
|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
|
|-
|
|
|39\62
|497.872
|293.617
|''503.226''
|''296.774''
|16
|7
|2.286
|
|-
|
|17\27
|
|497.561
|292.683
|''503.{{Overline|703}}''
|''296.{{Overline|296}}''
|7
|3
|2.333
|
|-
|
|
|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
|
|-
|
|
|31\49
|496
|288
|''506.122''
|''293.878''
|13
|5
|2.600
|
|-
|
|
|50\79
|495.868
|287.633
|''506.329''
|''293.671''
|21
|8
|2.625
|
|-
|
|19\30
|
|495.652
|286.957
|''506.{{Overline|6}}''
|''293.{{Overline|3}}''
|8
|3
|2.667
|
|-
|
|26\41
|
|495.238
|285.714
|''507.317''
|''292.683''
|11
|4
|2.750
|
|-
|
|33\52
|
|495
|285
|''507.692''
|''292.308''
|14
|5
|2.800
|
|-
|
|40\63
|
|494.536
|284.536
|''507.9365''
|''292.0635''
|17
|6
|2.833
|
|-
|
|47\74
|
|494.737
|284.211
|''508.{{Overline|108}}''
|''291.{{Overline|891}}''
|20
|7
|2.857
|
|-
|
|54\85
|
|494.6565
|283.9695
|''508.235''
|''291.765''
|23
|8
|2.875
|
|-
|
|61\96
|
|494.{{Overline|594}}
|283.{{Overline|783}}
|''508.{{Overline|3}}''
|''291.{{Overline|6}}''
|26
|9
|2.889
|
|-
|7\11
|
|
|494.118
|282.353
|''509.{{Overline|09}}''
|''290.{{Overline|90}}''
|3
|1
|3.000
|Aeolianic-Pythagorean ends, Aeolianic-Superpyth begins
|-
|
|65\102
|
|493.671
|281.013
|''509.804''
|''290.196''
|28
|9
|3.111
|
|-
|
|58\91
|
|493.617
|280.851
|''509.89''
|''290.11''
|25
|8
|3.125
|
|-
|
|51\80
|
|493.548
|280.645
|''510''
|''290''
|22
|7
|3.143
|
|-
|
|44\69
|
|493.458
|280.374
|''510.145''
|''289.855''
|19
|6
|3.167
|
|-
|
|37\58
|
|493.{{Overline|3}}
|280
|''510.345''
|''289.655''
|16
|5
|3.200
|
|-
|
|30\47
|
|493.151
|279.452
|''510.638''
|''289.362''
|13
|4
|3.250
|
|-
|
|23\36
|
|492.857
|278.571
|''511.{{Overline|1}}''
|''288.{{Overline|8}}''
|10
|3
|3.333
|
|-
|
|16\25
|
|492.308
|276.923
|''512''
|''288''
|7
|2
|3.500
|
|-
|
|25\39
|
|491.803
|275.41
|''512.8205''
|''287.1795''
|11
|3
|3.667
|
|-
|
|34\53
|
|491.566
|274.699
|''513.2075''
|''286.7925''
|15
|4
|3.750
|
|-
|
|43\67
|
|491.429
|274.286
|''513.433''
|''286.567''
|19
|5
|3.800
|
|-
|
|52\81
|
|491.339
|274.016
|''513.58''
|''286.42''
|23
|6
|3.833
|
|-
|
|61\95
|
|491.275
|273.825
|''513.684''
|''286.316''
|27
|7
|3.857
|
|-
|9\14
|
|
|490.{{Overline|90}}
|272.{{Overline|72}}
|''514.286''
|''285.714''
|4
|1
|4.000
|
|-
|
|47\73
|
|490.435
|271.304
|''515.0685''
|''284.3315''
|21
|5
|4.200
|
|-
|
|38\59
|
|490.323
|270.968
|''515.254''
|''284.746''
|17
|4
|4.250
|
|-
|
|29\45
|
|490.141
|270.422
|''515.{{Overline|5}}''
|''284.{{Overline|4}}''
|13
|3
|4.333
|
|-
|
|20\31
|
|489.795
|269.388
|''516.129''
|''283.871''
|9
|2
|4.500
|
|-
|
|31\48
|
|489.474
|268.421
|''516.{{Overline|6}}''
|''283.{{Overline|3}}''
|14
|3
|4.667
|
|-
|
|42\65
|
|489.32
|267.961
|''516.923''
|''283.077''
|19
|4
|4.750
|
|-
|11\17
|
|
|488.{{Overline|8}}
|266.{{Overline|6}}
|''517.647''
|''282.353''
|5
|1
|5.000
|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
|
|
|480
|240
|''533.{{Overline|3}}''
|''266.{{Overline|6}}''
|1
|0
|→ inf
|Paucitonic
|}
==See also==
[[3L 2s (14/9-equivalent)]] - idealized Archytas tuning
 
[[3L 2s (8/5-equivalent)]] - idealized Meantone tuning

Revision as of 04:45, 24 May 2023

3L 2s<minor sixth> (sometimes called diatonic), is a minor sixth-repeating MOS scale. The notation "<minor sixth>" means the period of the MOS is a minor sixth, 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 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.

Basic diatonic is in 8ed8/5, which is a very good minor sixth-based equal tuning similar to 12edo.

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 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.

Normalized
Notation Supersoft Soft Semisoft Basic Semihard Hard Superhard
Diatonic Oriole, Annapolis 18eds 13eds 21eds 8eds 19eds 11eds 14eds
La# Α# 1\18

46.154

1\13

63.158

2\21

77.419

1\8

100

3\19

124.138

2\11

[[1]]

3\14

163.63

Sib Βb 3\18

138.4615

2\13

126.316

3\21

116.129

2\19

82.759

1\11

70.588

1\14

54.54

Si Β 4\18

184.615

3\13

189.474

5\21

193.548

2\8

200

5\19

206.897

3\11

211.764

4\14

218.18

Si# Β# 5\18

230.769

4\13

252.632

7\21

270.968

3\8

300

8\19

331.0345

5\11

352.941

7\14

381.81

Dob Γb 6\18

276.923

6\21

232.258

2\8

200

4\19

165.517

2\11

141.1765

2\14

109.09

Do Γ 7\18

323.076

5\13

315.7895

8\21

309.677

3\8

300

7\19

289.655

4\11

282.353

5\14

272.72

Do# Γ# 8\18

369.231

6\13

378.947

10\21

387.0968

4\8

400

10\19

413.793

6\11

423.529

8\14

436.36

Reb Δb 10\18

461.5385

7\13

442.105

11\21

425.8065

9\19

372.413

5\11

352.941

6\14

327.27

Re Δ 11\18

507.692

8\13

505.263

13\21

503.226

5\8

500

12\19

496.551

7\11

494.118

9\14

490.90

Re# Δ# 12\18

553.846

9\13

568.421

15\21

580.645

6\8

600

15\19

620.689

9\11

635.294

12\14

654.54

Mib Εb 14\18

646.154

10\13

631.579

16\21

619.355

14\19

579.310

8\11

564.706

10\14

545.45

Mi Ε 15\18

692.308

11\13

694.737

18\21

696.774

7\8

700

17\19

703.448

10\11

705.88235

13\14

709.09

Mi# Ε# 16\18

738.4615

12\13

757.895

20\21

774.194

8\8

800

20\19

827.586

12\11

847.059

16\14

872.72

Lab Ϛb/Ϝb 17\18

784.615

19\21

735.484

7\8

700

16\19

662.069

9\11

635.294

11\14

600

La Ϛ/Ϝ 18\18

830.769

13\13

821.053

21\21

812.903

8\8

800

19\19

786.207

11\11

776.471

14\14

763.63

La# Ϛ#/Ϝ# 19\18

876.923

14\13

884.2105

23\21

890.323

9\8

900

22\19

910.345

13\11

917.647

17\14

927.27

Sib Ζb 21\18

969.231

15\13

947.368

24\21

929.032

21\19

868.9655

12\11

847.059

15\14

818.18

Si Ζ 22\18

1015.385

16\13

1010.526

26\21

1006.452

10\8

1000

24\19

993.1035

14\11

988.235

18\14

981.81

Si# Ζ# 23\18

1061.5385

17\13

1071.684

28\21

1083.871

11\8

1100

27\19

1117.241

16\11

1129.412

21\14

1145.45

Dob Ηb 24\18

1107.692

27\21

1045.161

10\8

1000

23\19

951.724

13\11

917.647

16\14

872.72

Do Η 25\18

1153.846

18\13

1136.842

29\21

1122.581

11\8

1100

26\19

1075.862

15\11

1052.8235

19\14

1036.36

Do# Η# 26\18

1200

19\13

1200

31\21

1200

12\8

1200

29\19

1200

17\11

1200

22\14

1200

Reb Θb 28\18

1292.308

20\13

1263.158

32\21

1238.710

28\19

1158.621

16\11

1129.412

20\14

1090.90

Re Θ 29\18

1338.4615

21\13

1326.316

34\21

1316.129

13\8

1300

31\19

1282.759

18\11

1270.588

23\14

1254.54

Re# Θ# 30\18

1384.615

22\13

1389.474

36\21

1393.548

14\8

1400

34\19

1406.897

20\11

1411.765

26\14

1418.18

Mib Ιb 32\18

1476.923

23\13

1452.632

37\21

1432.258

33\19

1365.517

19\11

1341.1765

24\14

1309.09

Mi Ι 33\18

1523.077

24\13

1515.7895

39\21

1509.677

15\8

1500

36\19

1489.655

21\11

1482.352

27\14

1472.72

Mi# Ι# 34\18

1569.231

25\13

1578.947

41\21

1587.097

16\8

1600

39\19

1613.793

23\11

1623.529

30\14

1636.36

Lab Αb 35\18

1615.385

40\21

1548.387

15\8

1500

35\19

1448.286

20\11

1411.765

25\14

1363.63

La Α 36\18

1661.5385

26\13

1642.105

42\21

1625.8065

16\8

1600

38\19

1572.414

22\11

1552.941

28\14

1527.27

ed8\12 (→ed2\3)
Notation Supersoft Soft Semisoft Basic Semihard Hard Superhard
Diatonic Oriole, Annapolis 18eds 13eds 21eds 8eds 19eds 11eds 14eds
La# Α# 1\18

44.4

1\13

61.5385

2\21

76.1905

1\8

100

3\19

126.316

2\11

145.45

3\14

171.429

Sib Βb 3\18

133.3

2\13

123.077

3\21

114.286

2\19

84.2105

1\11

72.72

1\14

57.143

Si Β 4\18

177.7

3\13

184.615

5\21

190.476

2\8

200

5\19

210.526

3\11

218.18

4\14

228.571

Si# Β# 5\18

222.2

4\13

246.154

7\21

266.6

3\8

300

8\19

336.842

5\11

363.63

7\14

400

Dob Γb 6\18

266.6

6\21

228.571

2\8

200

4\19

168.421

2\11

145.45

2\14

114.28

Do Γ 7\18

311.1

5\13

307.692

8\21

304.762

3\8

300

7\19

294.737

4\11

290.90

5\14

285.714

Do# Γ# 8\18

355.5

6\13

369.231

10\21

380.952

4\8

400

10\19

421.053

6\11

436.36

8\14

457.143

Reb Δb 10\18

444.4

7\13

430.769

11\21

419.048

9\19

378.947

5\11

363.63

6\14

342.857

Re Δ 11\18

488.8

8\13

492.308

13\21

495.238

5\8

500

12\19

505.263

7\11

509.09

9\14

514.286

Re# Δ# 12\18

533.3

9\13

553.846

15\21

571.429

6\8

600

15\19

631.421

9\11

654.54

12\14

685.714

Mib Εb 14\18

622.2

10\13

615.385

16\21

609.524

14\19

589.474

8\11

581.81

10\14

571.424

Mi Ε 15\18

666.6

11\13

676.9231

18\21

685.7143

7\8

700

17\19

715.7895

10\11

727.27

13\14

742.857

Mi# Ε# 16\18

711.1

12\13

738.4615

20\21

761.905

8\8

800

20\19

842.105

12\11

872.72

16\14

914.286

Lab Ϛb/Ϝb 17\18

755.5

19\21

723.8095

7\8

700

16\19

673.684

8\11

581.81

11\14

628.571

La Ϛ/Ϝ 800
La# Ϛ#/Ϝ# 19\18

844.4

14\13

861.5385

23\21

876.1905

9\8

900

22\19

926.316

13\11

945.45

17\14

971.429

Sib Ζb 21\18

933.3

15\13

923.077

24\21

914.286

21\19

884.2105

12\11

872.72

15\14

857.143

Si Ζ 22\18

977.7

16\13

984.615

26\21

990.476

10\8

1000

24\19

1010.526

14\11

1018.18

18\14

1028.571

Si# Ζ# 23\18

1022.2

17\13

1046.154

28\21

1066.6

11\8

1100

27\19

1136.8421

16\11

1163.63

21\14

1200

Dob Ηb 24\18

1066.6

27\21

1028.571

10\8

1000

23\19

968.421

13\11

945.45

16\14

914.286

Do Η 25\18

1111.1

18\13

1107.692

29\21

1104.762

11\8

1100

26\19

1094.737

15\11

1090.90

19\14

1085.714

Do# Η# 26\18

1155.5

19\13

1169.237

31\21

1180.952

12\8

1200

29\19

1221.053

17\11

1236.36

22\14

1257.143

Reb Θb 28\18

1244.4

20\13

1230.769

32\21

1219.048

28\19

1178.947

16\11

1163.63

20\14

1142.857

Re Θ 29\18

1288.8

21\13

1292.308

34\21

1295.238

13\8

1300

31\19

1305.263

18\11

1309.09

23\14

1314.286

Re# Θ# 30\18

1333.3

22\13

1187.924

36\21

1371.429

14\8

1400

34\19

1431.421

20\11

1454.54

26\14

1485.714

Mib Ιb 32\18

1422.2

23\13

1415.385

37\21

1409.524

33\19

1389.474

19\11

1381.81

24\14

1371.429

Mi Ι 33\18

1466.6

24\13

1476.923

39\21

1485.714

15\8

1500

36\19

1515.7895

21\11

1527.27

27\14

1542.857

Mi# Ι# 34\18

1511.1

25\13

1538.4615

41\21

1561.905

16\8

1600

39\19

1642.105

23\11

1672.72

30\14

1714.286

Lab Αb 35\18

1555.5

40\21

1523.8095

15\8

1500

35\19

1473.684

20\11

1454.54

25\14

1428.571

La Α 1600

Intervals

Generators Sextave notation Interval category name Generators Notation of sixth inverse Interval category name
The 5-note MOS has the following intervals (from some root):
0 La sextave (minor sixth) 0 La perfect unison
1 Re perfect fourth -1 Do minor third
2 Si major second -2 Mib diminished fifth
3 Mi perfect fifth -3 Sib minor second
4 Do# major third -4 Reb diminished fourth
The chromatic 8-note MOS also has the following intervals (from some root):
5 La# augmented unison (chroma) -5 Lab diminished sextave
6 Re# augmented fourth -6 Dob diminished third
7 Si# augmented second -7 Mibb doubly diminished fifth

Genchain

The generator chain for this scale is as follows:

Sibb Mibb Dob Lab Reb Sib Mib Do La Re Si Mi Do# La# Re# Si# Mi#
d2 dd5 d3 d6 d4 m2 d5 m3 P1 P4 M2 P5 M3 A1 A4 A2 A5

Modes

The mode names are based on the modes of the diatonic scale , in order of size:

Mode Scale UDP Interval type
name pattern notation 2nd 3rd 4th 5th
Hindu LLsLs 4|0 M M P P
Minor LsLLs 3|1 M m P P
Half diminished LsLsL 2|2 M m P d
Diminished sLLsL 1|3 m m P d
Altered sLsLL 0|4 m m d d

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 root-(p-1g)-(3g) (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.

Aeolianic-Meantone

Subgroup: 8/5.4/3.3/2

Comma list: 81/80

POL2 generator: ~6/5 = 308.3057

Mapping: [⟨1 1 2], ⟨0 -1 -3]]

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: [⟨1 1 2], ⟨0 -1 -3]]

Optimal ET sequence: 3ed14/9, 11ed14/9, 14ed14/9

Scale tree

The spectrum looks like this:

Generator

(bright)

Normalised ed8\12 (→ed2\3) L s L/s Comments
Chroma-positive Chroma-negative Chroma-positive Chroma-negative
3\5 514.286 342.857 480 320 1 1 1.000 Equalised
17\28 510 330 485.714 314.286 6 5 1.200
48\79 509.7345 329.2035 486.076 313.924 17 14 1.214
31\51 509.589 328.767 486.2745 313.7255 11 9 1.222
14\23 509.09 327.27 486.9565 313.0435 5 4 1.250
39\64 508.966 326.087 487.5 312.5 14 11 1.273
25\41 508.475 325.424 487.805 312.195 9 7 1.286
36\59 508.235 324.706 488.136 311.864 13 10 1.300
11\18 507.692 323.077 488.8 311.1 4 3 1.333
63\103 507.383 322.148 489.32 310.68 23 17 1.353
52\85 507.317 321.951 489.412 310.588 19 14 1.357
41\67 507.2165 321.6495 489.552 310.448 15 11 1.364
30\49 507.062 321.127 489.796 310.204 11 8 1.375
19\31 506.6 320 490.323 309.678 7 5 1.400
46\75 506.422 319.266 490.6 309.3 17 12 1.417
27\44 506.25 318.75 490.90 309.09 10 7 1.429
35\57 506.024 318.072 491.228 308.712 13 9 1.444
43\70 505.882 317.647 491.429 308.571 16 11 1.4545
51\83 505.785 317.355 491.566 308.434 19 13 1.4615
8\13 505.263 315.79 492.308 307.692 3 2 1.500 Aeolianic-Meantone starts here
45\73 504.673 314.019 493.151 306.849 17 11 1.5455
37\60 504.54 313.63 493.3 306.6 14 9 1.556
29\47 504.348 313.043 493.617 306.383 11 7 1.571
21\34 504 312 494.118 305.882 8 5 1.600
34\55 503.703 311.1 494.54 305.45 13 8 1.625
47\76 503.571 310.714 494.737 305.263 18 11 1.636
13\21 503.226 309.678 495.238 304.762 5 3 1.667
31\50 502.702 308.108 496 304 12 7 1.714
49\79 502.564 307.692 496.2025 303.7975 19 11 1.727
18\29 502.326 306.977 496.552 303.448 7 4 1.750
23\37 501.81 305.45 497.297 302.702 9 5 1.800
28\45 501.492 304.478 497.7 302.2 11 6 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
39\62 497.872 293.617 503.226 296.774 16 7 2.286
17\27 497.561 292.683 503.703 296.296 7 3 2.333
29\46 497.143 291.429 504.348 295.652 12 5 2.400
41\65 496.96 290.90 504.615 295.385 17 7 2.429
12\19 496.552 289.655 505.263 294.737 5 2 2.500
31\49 496 288 506.122 293.878 13 5 2.600
50\79 495.868 287.633 506.329 293.671 21 8 2.625
19\30 495.652 286.957 506.6 293.3 8 3 2.667
26\41 495.238 285.714 507.317 292.683 11 4 2.750
33\52 495 285 507.692 292.308 14 5 2.800
40\63 494.536 284.536 507.9365 292.0635 17 6 2.833
47\74 494.737 284.211 508.108 291.891 20 7 2.857
54\85 494.6565 283.9695 508.235 291.765 23 8 2.875
61\96 494.594 283.783 508.3 291.6 26 9 2.889
7\11 494.118 282.353 509.09 290.90 3 1 3.000 Aeolianic-Pythagorean ends, Aeolianic-Superpyth begins
65\102 493.671 281.013 509.804 290.196 28 9 3.111
58\91 493.617 280.851 509.89 290.11 25 8 3.125
51\80 493.548 280.645 510 290 22 7 3.143
44\69 493.458 280.374 510.145 289.855 19 6 3.167
37\58 493.3 280 510.345 289.655 16 5 3.200
30\47 493.151 279.452 510.638 289.362 13 4 3.250
23\36 492.857 278.571 511.1 288.8 10 3 3.333
16\25 492.308 276.923 512 288 7 2 3.500
25\39 491.803 275.41 512.8205 287.1795 11 3 3.667
34\53 491.566 274.699 513.2075 286.7925 15 4 3.750
43\67 491.429 274.286 513.433 286.567 19 5 3.800
52\81 491.339 274.016 513.58 286.42 23 6 3.833
61\95 491.275 273.825 513.684 286.316 27 7 3.857
9\14 490.90 272.72 514.286 285.714 4 1 4.000
47\73 490.435 271.304 515.0685 284.3315 21 5 4.200
38\59 490.323 270.968 515.254 284.746 17 4 4.250
29\45 490.141 270.422 515.5 284.4 13 3 4.333
20\31 489.795 269.388 516.129 283.871 9 2 4.500
31\48 489.474 268.421 516.6 283.3 14 3 4.667
42\65 489.32 267.961 516.923 283.077 19 4 4.750
11\17 488.8 266.6 517.647 282.353 5 1 5.000 Aeolianic-Superpyth ends
35\54 488.372 265.116 518.518 281.481 16 3 5.333
24\37 488.136 264.407 518.918 281.081 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\3 480 240 533.3 266.6 1 0 → inf Paucitonic

See also

3L 2s (14/9-equivalent) - idealized Archytas tuning

3L 2s (8/5-equivalent) - idealized Meantone tuning

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