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

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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 diatonic is a minor sixth-repeating scale, each tone has a 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]].
[[Basic]] diatonic is in [[8ed8/5]], which is a very good minor sixth-based equal tuning similar to [[12edo]].
Line 9: Line 9:




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 the diatonic scale. One method uses a simple sextave (minor sixth) repeating notation consisting of 5 naturals (La, Si, Do, Re, Mi; Mi, Fa, Sol, La, Si). 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"
{| class="wikitable"
|+
|+
Normalized
Normalized
! colspan="2" |Notation
! colspan="3" |Notation
!Supersoft
!Supersoft
!Soft
!Soft
Line 22: Line 22:
!Superhard
!Superhard
|-
|-
!Diatonic
!Aeolian
!Phrygian
!Oriole, Annapolis
!Oriole, Annapolis
!18eds
!18eds
Line 33: Line 34:
|-
|-
|La#
|La#
|Mi#
|Α#
|Α#
|1\18, 46.154
|1\18, 46.154
Line 43: Line 45:
|-
|-
|Sib
|Sib
|Fa
|Βb
|Βb
|3\18, 138.462
|3\18, 138.462
Line 52: Line 55:
|-
|-
|Si
|Si
|Fa#
|Β
|Β
|4\18, 184.615
|4\18, 184.615
Line 62: Line 66:
|-
|-
|Si#
|Si#
|Fax
|Β#
|Β#
|5\18, 230.769
|5\18, 230.769
Line 72: Line 77:
|-
|-
|Dob
|Dob
|Solb
|Γb
|Γb
|6\18, 276.923
|6\18, 276.923
Line 81: Line 87:
|-
|-
|'''Do'''
|'''Do'''
|'''Sol'''
|'''Γ'''
|'''Γ'''
|'''7\18,''' '''323.076'''
|'''7\18,''' '''323.076'''
Line 91: Line 98:
|-
|-
|Do#
|Do#
|Sol#
|Γ#
|Γ#
|8\18, 369.231
|8\18, 369.231
Line 101: Line 109:
|-
|-
|Reb
|Reb
|Lab
|Δb
|Δb
|10\18, 461.538
|10\18, 461.538
Line 110: Line 119:
|-
|-
|'''Re'''
|'''Re'''
|'''La'''
|'''Δ'''
|'''Δ'''
|'''11\18,''' '''507.692'''
|'''11\18,''' '''507.692'''
Line 120: Line 130:
|-
|-
|Re#
|Re#
|La#
|Δ#
|Δ#
|12\18, 553.846
|12\18, 553.846
Line 130: Line 141:
|-
|-
|Mib
|Mib
|Sib
|Εb
|Εb
|14\18, 646.154
|14\18, 646.154
Line 139: Line 151:
|-
|-
|Mi
|Mi
|Si
|Ε
|Ε
|15\18, 692.308
|15\18, 692.308
Line 149: Line 162:
|-
|-
|Mi#
|Mi#
|Si#
|Ε#
|Ε#
|16\18, 738.462
|16\18, 738.462
Line 159: Line 173:
|-
|-
|Lab
|Lab
|Mib
|Ϛb/Ϝb
|Ϛb/Ϝb
|17\18, 784.615
|17\18, 784.615
Line 168: Line 183:
|-
|-
!La
!La
!Mi
!Ϛ/Ϝ
!Ϛ/Ϝ
!18\18, 830.769
!18\18, 830.769
Line 178: Line 194:
|-
|-
|La#
|La#
|Mi#
|Ϛ#/Ϝ#
|Ϛ#/Ϝ#
|19\18, 876.923
|19\18, 876.923
Line 188: Line 205:
|-
|-
|Sib
|Sib
|Fa
|Ζb
|Ζb
|21\18, 969.231
|21\18, 969.231
Line 197: Line 215:
|-
|-
|Si
|Si
|Fa#
|Ζ
|Ζ
|22\18, 1015.385
|22\18, 1015.385
Line 207: Line 226:
|-
|-
|Si#
|Si#
|Fax
|Ζ#
|Ζ#
|23\18, 1061.538
|23\18, 1061.538
Line 217: Line 237:
|-
|-
|Dob
|Dob
|Solb
|Ηb
|Ηb
|24\18, 1107.692
|24\18, 1107.692
Line 226: Line 247:
|-
|-
|'''Do'''
|'''Do'''
|'''Sol'''
|'''Η'''
|'''Η'''
|'''25\18,''' '''1153.846'''
|'''25\18,''' '''1153.846'''
Line 236: Line 258:
|-
|-
|Do#
|Do#
|Sol#
|Η#
|Η#
|26\18, 1200
|26\18, 1200
Line 246: Line 269:
|-
|-
|Reb
|Reb
|Lab
|Θb
|Θb
|28\18, 1292.308
|28\18, 1292.308
Line 255: Line 279:
|-
|-
|'''Re'''
|'''Re'''
|'''La'''
|'''Θ'''
|'''Θ'''
|'''29\18,''' '''1338.462'''
|'''29\18,''' '''1338.462'''
Line 265: Line 290:
|-
|-
|Re#
|Re#
|La#
|Θ#
|Θ#
|30\18, 1384.615
|30\18, 1384.615
Line 275: Line 301:
|-
|-
|Mib
|Mib
|Sib
|Ιb
|Ιb
|32\18, 1476.923
|32\18, 1476.923
Line 284: Line 311:
|-
|-
|Mi
|Mi
|Si
|Ι
|Ι
|33\18, 1523.077
|33\18, 1523.077
Line 294: Line 322:
|-
|-
|Mi#
|Mi#
|Si#
|Ι#
|Ι#
|34\18, 1569.231
|34\18, 1569.231
Line 304: Line 333:
|-
|-
|Lab
|Lab
|Mib
|Αb
|Αb
|35\18, 1615.385
|35\18, 1615.385
Line 313: Line 343:
|-
|-
!La
!La
!Mi
!Α
!Α
!36\18, 1661.538
!36\18, 1661.538
Line 336: Line 367:
|-
|-
|0
|0
|La
|La, Mi
|sextave (minor sixth)
|sextave (minor sixth)
| 0
| 0
|La
|La, Mi
| perfect unison
| perfect unison
|-
|-
|1
|1
|Re
|Re, La
|perfect fourth
|perfect fourth
| -1
| -1
|Do
|Do, Sol
|minor third
|minor third
|-
|-
|2
|2
|Si
|Si, Fa#
|major second
|major second
| -2
| -2
|Mib
|Mib, Sib
|diminished fifth
|diminished fifth
|-
|-
|3
|3
|Mi
|Mi, Si
|perfect fifth
|perfect fifth
| -3
| -3
|Sib
|Sib, Fa
|minor second
|minor second
|-
|-
|4
|4
|Do#
|Do#, Sol#
|major third
|major third
| -4
| -4
|Reb
|Reb, Lb
|diminished fourth
|diminished fourth
|-
|-
Line 373: Line 404:
|-
|-
|5
|5
|La#
|La#, Mi#
|augmented unison (chroma)
|augmented unison (chroma)
| -5
| -5
| Lab
| Lab, Mib
|diminished sextave
|diminished sextave
|-
|-
|6
|6
| Re#
| Re#, La#
|augmented fourth
|augmented fourth
| -6
| -6
|Dob
|Dob, Solb
|diminished third
|diminished third
|-
|-
|7
|7
|Si#
|Si#, Fax
|augmented second
|augmented second
| -7
| -7
|Mibb
|Mibb, Sibb
|doubly diminished fifth
|doubly diminished fifth
|}
|}
Line 397: Line 428:
{| class="wikitable"
{| class="wikitable"
|Sibb
|Sibb
Fab
|Mibb
|Mibb
Sibb
|Dob
|Dob
Solb
|Lab
|Lab
Mib
|Reb
|Reb
Lab
|Sib
|Sib
Fa
|Mib
|Mib
Sib
|Do
|Do
Sol
|La
|La
Mi
|Re
|Re
La
|Si
|Si
Fa#
|Mi
|Mi
Si
|Do#
|Do#
Sol#
|La#
|La#
Mi#
|Re#
|Re#
La#
|Si#
|Si#
Fax
|Mi#
|Mi#
Si#
|-
|-
|d2
|d2
Line 782: Line 830:


[[3L 2s (11/7-equivalent)]] and [[3L 2s (π/2-equivalent)|3L 2s ([math]π[/math]/2-equivalent)]] - Neogothic tuning
[[3L 2s (11/7-equivalent)]] and [[3L 2s (π/2-equivalent)|3L 2s ([math]π[/math]/2-equivalent)]] - Neogothic tuning
[[3L 2s (128/81-equivalent)]] - Pythagorean tuning


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


[[6L 4s (5/2-equivalent)]] - Annapolis Meantone tuning
[[6L 4s (5/2-equivalent)]] - Annapolis Meantone tuning
[[6L 4s (81/32-equivalent)]] - Annapolis Pythagorean tuning


[[6L 4s (28/11-equivalent)]] - Annapolis Neogothic tuning
[[6L 4s (28/11-equivalent)]] - Annapolis Neogothic tuning


[[6L 4s (18/7-equivalent)]] - Annapolis Archytas tuning
[[6L 4s (18/7-equivalent)]] - Annapolis Archytas tuning

Latest revision as of 20:31, 20 October 2024

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 a 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; Mi, Fa, Sol, La, Si). 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
Aeolian Phrygian Oriole, Annapolis 18eds 13eds 21eds 8eds 19eds 11eds 14eds
La# Mi# Α# 1\18, 46.154 1\13, 63.158 2\21, 77.419 1\8, 100 3\19, 124.138 2\11, 141.176 3\14, 163.636
Sib Fa Βb 3\18, 138.462 2\13, 126.316 3\21, 116.129 2\19, 82.759 1\11, 70.588 1\14, 54.545
Si Fa# 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, 218.182
Si# Fax Β# 5\18, 230.769 4\13, 252.632 7\21, 270.968 3\8, 300 8\19, 331.034 5\11 352.941 7\14, 381.818
Dob Solb Γb 6\18, 276.923 6\21, 232.258 2\8, 200 4\19, 165.517 2\11, 141.176 2\14, 109.091
Do Sol Γ 7\18, 323.076 5\13, 315.789 8\21, 309.677 3\8, 300 7\19, 289.655 4\11. 282.353 5\14, 272.727
Do# Sol# Γ# 8\18, 369.231 6\13, 378.947 10\21, 387.097 4\8, 400 10\19, 413.793 6\11, 423.529 8\14, 436.364
Reb Lab Δb 10\18, 461.538 7\13, 442.105 11\21, 425.806 9\19, 372.413 5\11 352.941 6\14, 327.272
Re La 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, 490.909
Re# La# 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, 654.545
Mib Sib Εb 14\18, 646.154 10\13, 631.579 16\21, 619.355 14\19, 579.310 8\11, 564.706 10\14, 545.455
Mi Si Ε 15\18, 692.308 11\13, 694.737 18\21, 696.774 7\8, 700 17\19, 703.448 10\11, 705.882 13\14, 709.091
Mi# Si# Ε# 16\18, 738.462 12\13, 757.895 20\21, 774.194 8\8, 800 20\19, 827.586 12\11, 847.059 16\14, 872.727
Lab Mib 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 Mi 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, 763.636
La# Mi# Ϛ#/Ϝ# 19\18, 876.923 14\13, 884.211 23\21, 890.323 9\8, 900 22\19, 910.345 13\11, 917.647 17\14, 927.273
Sib Fa Ζb 21\18, 969.231 15\13, 947.368 24\21, 929.032 21\19, 868.966 12\11, 847.059 15\14, 818.182
Si Fa# Ζ 22\18, 1015.385 16\13, 1010.526 26\21, 1006.452 10\8, 1000 24\19, 993.103 14\11, 988.235 18\14, 981.81
Si# Fax Ζ# 23\18, 1061.538 17\13, 1071.684 28\21, 1083.871 11\8, 1100 27\19, 1117.241 16\11, 1129.412 21\14, 1145.455
Dob Solb Ηb 24\18, 1107.692 27\21, 1045.161 10\8, 1000 23\19, 951.724 13\11, 917.647 16\14, 872.727
Do Sol Η 25\18, 1153.846 18\13, 1136.842 29\21, 1122.581 11\8, 1100 26\19, 1075.862 15\11, 1058.824 19\14, 1036.364
Do# Sol# Η# 26\18, 1200 19\13, 1200 31\21, 1200 12\8, 1200 29\19, 1200 17\11, 1200 22\14, 1200
Reb Lab Θb 28\18, 1292.308 20\13, 1263.158 32\21, 1238.710 28\19, 1158.621 16\11, 1129.412 20\14, 1090.909
Re La Θ 29\18, 1338.462 21\13, 1326.316 34\21, 1316.129 13\8, 1300 31\19, 1282.759 18\11, 1270.588 23\14, 1254.545
Re# La# 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, 1418.182
Mib Sib Ιb 32\18, 1476.923 23\13, 1452.632 37\21, 1432.258 33\19, 1365.517 19\11, 1341.176 24\14, 1309.091
Mi Si Ι 33\18, 1523.077 24\13, 1515.789 39\21, 1509.677 15\8, 1500 36\19, 1489.655 21\11, 1482.352 27\14, 1472.727
Mi# Si# 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, 1636.364
Lab Mib Αb 35\18, 1615.385 40\21, 1548.387 15\8, 1500 35\19, 1448.286 20\11, 1411.765 25\14, 1363.636
La Mi Α 36\18, 1661.538 26\13, 1642.105 42\21, 1625.806 16\8, 1600 38\19, 1572.414 22\11, 1552.941 28\14, 1527.273


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, Mi sextave (minor sixth) 0 La, Mi perfect unison
1 Re, La perfect fourth -1 Do, Sol minor third
2 Si, Fa# major second -2 Mib, Sib diminished fifth
3 Mi, Si perfect fifth -3 Sib, Fa minor second
4 Do#, Sol# major third -4 Reb, Lb diminished fourth
The chromatic 8-note MOS also has the following intervals (from some root):
5 La#, Mi# augmented unison (chroma) -5 Lab, Mib diminished sextave
6 Re#, La# augmented fourth -6 Dob, Solb diminished third
7 Si#, Fax augmented second -7 Mibb, Sibb doubly diminished fifth

Genchain

The generator chain for this scale is as follows:

Sibb

Fab

Mibb

Sibb

Dob

Solb

Lab

Mib

Reb

Lab

Sib

Fa

Mib

Sib

Do

Sol

La

Mi

Re

La

Si

Fa#

Mi

Si

Do#

Sol#

La#

Mi#

Re#

La#

Si#

Fax

Mi#

Si#

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, 8ed14/9, 11ed14/9, 14ed14/9

Scale tree

The spectrum looks like this:

Generator

(bright)

Normalised L s L/s Comments
3\5 514.286 1 1 1.000 Equalised
17\28 510.000 6 5 1.200
14\23 509.091 5 4 1.250
25\41 508.475 9 7 1.286
11\18 507.692 4 3 1.333
35\57 506.024 13 9 1.444
8\13 505.263 3 2 1.500 Aeolianic-Meantone starts here
21\34 504.000 8 5 1.600
13\21 503.226 5 3 1.667
18\29 502.326 7 4 1.750
23\37 501.818 9 5 1.800
28\45 501.492 11 6 1.833
33\53 501.265 13 7 1.857
38\61 501.09 15 8 1.875
43\69 500.971 17 9 1.889
5\8 500.000 2 1 2.000 Aeolianic-Meantone ends, Aeolianic-Pythagorean begins
42\67 499.010 17 8 2.125
37\59 498.876 15 7 2.143
32\51 498.701 13 6 2.167
27\43 498.461 11 5 2.200
22\35 498.113 9 4 2.250
17\27 497.561 7 3 2.333
12\19 496.552 5 2 2.500
19\30 495.652 8 3 2.667
26\41 495.238 11 4 2.750
33\52 495.000 14 5 2.800
7\11 494.118 3 1 3.000 Aeolianic-Pythagorean ends, Aeolianic-Superpyth begins
30\47 493.151 13 4 3.250
23\36 492.857 10 3 3.333
16\25 492.308 7 2 3.500
25\39 491.803 11 3 3.667
9\14 490.909 4 1 4.000
20\31 489.795 9 2 4.500
11\17 488.889 5 1 5.000 Aeolianic-Superpyth ends
13\20 487.500 6 1 6.000
2\3 480.000 1 0 → inf Paucitonic

See also

3L 2s (13/8-equivalent) and 3L 2s ([math]φ[/math]-equivalent) - Harmonic and Golden tuning

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

3L 2s (11/7-equivalent) and 3L 2s ([math]π[/math]/2-equivalent) - Neogothic tuning

3L 2s (128/81-equivalent) - Pythagorean tuning

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

6L 4s (5/2-equivalent) - Annapolis Meantone tuning

6L 4s (81/32-equivalent) - Annapolis Pythagorean tuning

6L 4s (28/11-equivalent) - Annapolis Neogothic tuning

6L 4s (18/7-equivalent) - Annapolis Archytas tuning

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