User:Moremajorthanmajor/4L 1s (5/3-equivalent): Difference between revisions

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[[Basic]] diatonic is in [[9ed5/3]], which is a very good minor sixth-based equal tuning similar to [[12edo]].
[[Basic]] diatonic is in [[9ed5/3]], 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 (Do, Re, Mi, Fa, Sol or Sol, La, Si, Do, Re). Given that 1-5/4-3/2 is major sixth-equivalent to a tone cluster of 1-10/9-5/4, it may be more convenient to notate these diatonic scales as repeating at the double sextave (augmented eleventh~twelfth), however it does make navigating the [[Generator|genchain]] harder. This way, 3/2 is its own pitch class, distinct from 10\9. Notating this way produces a twelfth which is the Scala Francisci[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 (Do, Re, Mi, Fa, Sol or Sol, La, Si, Do, Re). Given that 1-5/4-3/2 is major sixth-equivalent to a tone cluster of 1-10/9-5/4, it may be more convenient to notate these diatonic scales as repeating at the double sextave (augmented eleventh~twelfth), however it does make navigating the [[Generator|genchain]] harder. This way, 3/2 is its own pitch class, distinct from 10\9. Notating this way produces a twelfth which is the Scala Francisci[8L 2s]. Since there are exactly 10 naturals in double sextave notation, Greek numerals 1-10 may be used.
{| class="wikitable"
{| class="wikitable"
|+
|+
Line 35: Line 35:
|Α#
|Α#
|1\19
|1\19
46.15385
46.154
|1\14
|1\14
63.1579
63.158
|2\23
|2\23
77.41935
77.419
| rowspan="2" |1\9
| rowspan="2" |1\9
100
100
|3\22
|3\22
[[Tel:124.1379|124.1379]]
124.138
|2\13
|2\13
[[Tel:141.1765|141.1765]]
141.1765
|3\17
|3\17
163.{{Overline|63}}
163.{{Overline|63}}
Line 52: Line 52:
|Βb
|Βb
|3\19
|3\19
[[Tel:138.4615|138.4615]]
138.4615
|2\14
|2\14
[[Tel:126.3158|126.3158]]
126.316
|3\23
|3\23
116.129
116.129
|2\22
|2\22
82.7586
82.759
|1\13
|1\13
70.5882
70.588
|1\17
|1\17
54.{{Overline|54}}
54.{{Overline|54}}
Line 67: Line 67:
|'''Β'''
|'''Β'''
|'''4\19'''
|'''4\19'''
'''[[Tel:184.6154|184.6154]]'''
'''184.615'''
|'''3\14'''
|'''3\14'''
'''[[Tel:189.4736|189.4736]]'''
'''189.474'''
|'''5\23'''
|'''5\23'''
'''[[Tel:193.5484|193.5484]]'''
'''193.548'''
|'''2\9'''
|'''2\9'''
'''200'''
'''200'''
|'''5\22'''
|'''5\22'''
'''206.89655'''
'''206.897'''
|'''3\13'''
|'''3\13'''
'''[[Tel:211.7647|211.7647]]'''
'''211.765'''
|'''4\17'''
|'''4\17'''
'''218.{{Overline|18}}'''
'''218.{{Overline|18}}'''
Line 84: Line 84:
|Β#
|Β#
|5\19
|5\19
[[Tel:230.7692|230.7692]]
230.769
|4\14
|4\14
[[Tel:252.6316|252.6316]]
252.632
|7\23
|7\23
[[Tel:270.9677|270.9677]]
270.968
| rowspan="2" |3\9
| rowspan="2" |3\9
300
300
|8\22
|8\22
[[Tel:331.0345|331.0345]]
331.0345
|5\13
|5\13
[[Tel:352.9412|352.9412]]
352.941
|7\17
|7\17
381.{{Overline|81}}
381.{{Overline|81}}
Line 101: Line 101:
|Γb
|Γb
|7\19
|7\19
[[Tel:323.0769|323.0769]]
323.077
|5\14
|5\14
[[Tel:315.7895|315.7895]]
315.7895
|8\23
|8\23
[[Tel:309.6774|309.6774]]
309.677
|7\22
|7\22
[[Tel:289.6552|289.6552]]
289.655
|4\13
|4\13
[[Tel:282.3529|282.3529]]
282.353
|5\17
|5\17
272.{{Overline|72}}
272.{{Overline|72}}
Line 116: Line 116:
|Γ
|Γ
|8\19
|8\19
[[Tel:369.2308|369.2308]]
369.2301
|6\14
|6\14
[[Tel:378.9474|378.9474]]
378.947
|10\23
|10\23
[[Tel:387.0968|387.0968]]
387.097
|4\9
|4\9
400
400
|10\22
|10\22
[[Tel:413.7931|413.7931]]
413.793
|6\13
|6\13
[[Tel:423.5294|423.5294]]
423.529
|8\17
|8\17
436.{{Overline|36}}
436.{{Overline|36}}
Line 134: Line 133:
|Γ#
|Γ#
|9\19
|9\19
[[Tel:415.3846|415.3846]]
415.385
| rowspan="2" |7\14
| rowspan="2" |7\14
[[Tel:442.1053|442.1053]]
442.105
|12\23
|12\23
[[Tel:464.5161|464.5161]]
464.516
|5\9
|5\9
500
500
Line 144: Line 143:
537.931
537.931
|8\13
|8\13
[[Tel:564.7059|564.7059]]
564.706
|11\17
|11\17
600
600
Line 151: Line 150:
|Δb
|Δb
|10\19
|10\19
[[Tel:461.5385|461.5385]]
461.5385
|11\23
|11\23
425.80645
425.8065
|4\9
|4\9
400
400
|9\22
|9\22
[[Tel:372.4138|372.4138]]
372.414
|5\13
|5\13
[[Tel:352.9412|352.9412]]
352.941
|6\17
|6\17
327.{{Overline|27}}
327.{{Overline|27}}
Line 166: Line 165:
|Δ
|Δ
|11\19
|11\19
[[Tel:507.6923|507.6923]]
507.692
|8\14
|8\14
[[Tel:505.2632|505.2632]]
505.263
|13\23
|13\23
[[Tel:503.2259|503.2259]]
503.226
|5\9
|5\9
500
500
|12\22
|12\22
[[Tel:496.5517|496.5517]]
496.552
|7\13
|7\13
494.11765
494.118
|9\17
|9\17
490.{{Overline|90}}
490.{{Overline|90}}
Line 183: Line 182:
|Δ#
|Δ#
|12\19
|12\19
553.84615
553.846
|9\14
|9\14
568.42105
568.421
|15\23
|15\23
[[Tel:580.6452|580.6452]]
580.645
| rowspan="2" |6\9
| rowspan="2" |6\9
600
600
|15\22
|15\22
[[Tel:620.6897|620.6897]]
620.690
|9\13
|9\13
[[Tel:635.2941|635.2941]]
635.294
|12\17
|12\17
654.{{Overline|54}}
654.{{Overline|54}}
Line 200: Line 199:
|Εb
|Εb
|14\19
|14\19
646.15385
646.154
|10\14
|10\14
631.57895
631.579
|16\23
|16\23
[[Tel:619.3548|619.3548]]
619.355
|14\22
|14\22
[[Tel:579.3103|579.3103]]
579.310
|8\13
|8\13
[[Tel:564.7059|564.7059]]
564.706
|10\17
|10\17
545.{{Overline|45}}
545.{{Overline|45}}
Line 215: Line 214:
|'''Ε'''
|'''Ε'''
|'''15\19'''
|'''15\19'''
'''[[Tel:692.3077|692.3077]]'''
'''692.308'''
|'''11\14'''
|'''11\14'''
'''[[Tel:694.7368|694.7368]]'''
'''694.737'''
|'''18\23'''
|'''18\23'''
'''[[Tel:696.7742|696.7742]]'''
'''696.774'''
|'''7\8'''
|'''7\9'''
'''700'''
'''700'''
|'''17\22'''
|'''17\22'''
'''[[Tel:703.4483|703.4483]]'''
'''703.448'''
|'''10\13'''
|'''10\13'''
'''705.88235'''
'''705.882'''
|'''13\17'''
|'''13\17'''
'''709.{{Overline|09}}'''
'''709.{{Overline|09}}'''
Line 232: Line 231:
|Ε#
|Ε#
|16\19
|16\19
[[Tel:738.4615|738.4615]]
738.4615
|12\14
|12\14
[[Tel:757.8947|757.8947]]
757.8947
|20\23
|20\23
774.19355
774.194
| rowspan="2" |8\8
| rowspan="2" |8\9
800
800
|20\22
|20\22
[[Tel:827.5862|827.5862]]
827.586
|12\13
|12\13
[[Tel:847.0588|847.0588]]
847.059
|16\14
|16\14
872.{{Overline|72}}
872.{{Overline|72}}
Line 249: Line 248:
|Ϛb/Ϝb
|Ϛb/Ϝb
|18\19
|18\19
[[Tel:830.7692|830.7692]]
830.769
|13\14
|13\14
[[Tel:821.0526|821.0526]]
821.053
|21\23
|21\23
[[Tel:812.9032|812.9032]]
812.903
|19\22
|19\22
[[Tel:786.2069|786.2069]]
786.207
|11\13
|11\13
[[Tel:776.6471|776.6471]]
776.647
|14\17
|14\17
763.{{Overline|63}}
763.{{Overline|63}}
Line 264: Line 263:
!Ϛ/Ϝ
!Ϛ/Ϝ
!19\19
!19\19
[[Tel:876.9231|876.9231]]
876.923
!14\14
!14\14
[[Tel:884.2105|884.2105]]
884.2105
!23\23
!23\23
[[Tel:890.3226|890.3226]]
890.323
!9\9
!9\9
900
900
!22\22
!22\22
[[Tel:910.3448|910.3448]]
910.345
!13\13
!13\13
[[Tel:917.6471|917.6471]]
917.647
!17\17
!17\17
927.{{Overline|27}}
927.{{Overline|27}}
Line 281: Line 280:
|Ϛ#/Ϝ#
|Ϛ#/Ϝ#
|20\19
|20\19
[[Tel:923.0769|923.0769]]
923.077
|15\14
|15\14
[[Tel:947.3684|947.3684]]
947.368
|24\23
|24\23
[[Tel:929.0323|929.0323]]
929.032
| rowspan="2" |10\9
| rowspan="2" |10\9
1000
1000
|25\22
|25\22
1034.4829
1034.483
|15\13
|15\13
1052.8235
1052.8235
Line 298: Line 297:
|Ζb
|Ζb
|22\19
|22\19
1015.3847
1015.385
|16\14
|16\14
1010.5263
1010.526
|26\23
|26\23
1006.4516
1006.452
|24\22
|24\22
993.10345
993.103
|14\13
|14\13
[[Tel:988.2353|988.2353]]
988.235
|18\17
|18\17
981.{{Overline|81}}
981.{{Overline|81}}
Line 315: Line 314:
'''1061.5385'''
'''1061.5385'''
|'''17\14'''
|'''17\14'''
'''1071.6842'''
'''1071.684'''
|'''28\23'''
|'''28\23'''
'''1083.871'''
'''1083.871'''
Line 321: Line 320:
'''1100'''
'''1100'''
|'''27\22'''
|'''27\22'''
'''1117.2414'''
'''1117.241'''
|'''16\13'''
|'''16\13'''
'''1129.4118'''
'''1129.412'''
|'''21\17'''
|'''21\17'''
'''1145.{{Overline|45}}'''
'''1145.{{Overline|45}}'''
Line 330: Line 329:
|Ζ#
|Ζ#
|24\19
|24\19
1107.6923
1107.692
|18\14
|18\14
1136.8421
1136.842
|30\23
|30\23
1161.7097
1161.290
| rowspan="2" |12\9
| rowspan="2" |12\9
1200
1200
|30\22
|30\22
1241.3793
1241.379
|18\13
|18\13
1270.5882
1270.588
|24\14
|24\14
1309.{{Overline|09}}
1309.{{Overline|09}}
Line 363: Line 361:
|Η
|Η
|27\19
|27\19
1246.15385
1246.154
|20\14
|20\14
1263.1579
1263.158
|33\23
|33\23
1277.41935
1277.419
|13\9
|13\9
1300
1300
|32\22
|32\22
1324.1379
1324.138
|19\13
|19\13
1341.1765
1341.1765
Line 380: Line 378:
|Η#
|Η#
|28\19
|28\19
1292.3077
1292.308
| rowspan="2" |21\14
| rowspan="2" |21\14
1326.3158
1326.316
|35\23
|35\23
1354.8387
1354.839
|14\9
|14\9
1400
1400
|35\22
|35\22
1448.2759
1448.276
|21\13
|21\13
1482.3529
1482.353
|28\17
|28\17
1527.{{Overline|27}}
1527.{{Overline|27}}
Line 403: Line 401:
1300
1300
|31\22
|31\22
1282.7586
1282.759
|18\13
|18\13
1270.5882
1270.588
|23\17
|23\17
1254.{{Overline|54}}
1254.{{Overline|54}}
Line 412: Line 410:
|Θ
|Θ
|30\19
|30\19
1384.6154
1384.615
|22\14
|22\14
1389.4737
1389.474
|36\23
|36\23
1393.5484
1393.548
|14\9
|14\9
1400
1400
|34\22
|34\22
1406.89655
1406.897
|20\13
|20\13
1411.7647
1411.765
|26\17
|26\17
1418.{{Overline|18}}
1418.{{Overline|18}}
Line 429: Line 427:
|Θ#
|Θ#
|31\19
|31\19
1430.7692
1430.769
|23\14
|23\14
1452.6316
1452.632
|38\23
|38\23
1470.9677
1470.968
| rowspan="2" |15\9
| rowspan="2" |15\9
1500
1500
Line 439: Line 437:
1531.0345
1531.0345
|22\13
|22\13
1552.9412
1552.941
|29\17
|29\17
1581.{{Overline|81}}
1581.{{Overline|81}}
Line 446: Line 444:
|Ιb
|Ιb
|33\19
|33\19
1523.0769
1523.077
|24\14
|24\14
1515.7895
1515.7895
|39\23
|39\23
1509.6774
1509.677
|36\22
|36\22
1489.6551
1489.655
|21\13
|21\13
1482.3529
1482.353
|27\17
|27\17
1472.{{Overline|72}}
1472.{{Overline|72}}
Line 461: Line 459:
|'''Ι'''
|'''Ι'''
|'''34\19'''
|'''34\19'''
'''1569.2308'''
'''1569.231'''
|'''25\14'''
|'''25\14'''
'''1578.9474'''
'''1578.947'''
|'''41\23'''
|'''41\23'''
'''1587.0968'''
'''1587.097'''
|'''16\9'''
|'''16\9'''
'''1600'''
'''1600'''
|'''39\22'''
|'''39\22'''
'''1613.7931'''
'''1613.793'''
|'''23\13'''
|'''23\13'''
'''1623.5294'''
'''1623.529'''
|'''30\17'''
|'''30\17'''
'''1636.{{Overline|36}}'''
'''1636.{{Overline|36}}'''
Line 478: Line 476:
|Ι#
|Ι#
|35\19
|35\19
1615.3846
1615.385
|26\14
|26\14
1642.1053
1642.105
|43\23
|43\23
1664.5161
1664.516
| rowspan="2" |17\9
| rowspan="2" |17\9
1700
1700
Line 488: Line 486:
1737.931
1737.931
|25\13
|25\13
1764.7059
1764.706
|33\17
|33\17
1800
1800
Line 495: Line 493:
|Αb
|Αb
|37\19
|37\19
1707.6923
1707.692
|27\14
|27\14
1705.2632
1705.263
|44\23
|44\23
1703.2258
1703.226
|41\22
|41\22
1696.5517
1696.552
|20\13
|20\13
1694.11765
1694.118
|31\17
|31\17
1490.{{Overline|90}}
1490.{{Overline|90}}
Line 510: Line 508:
!Α
!Α
!38\19
!38\19
1753.84615
1753.846
!28\14
!28\14
1768.42105
1768.421
!46\23
!46\23
1780.6452
1780.645
!18\9
!18\9
1800
1800
!44\22
!44\22
1820.6897
1820.690
!26\13
!26\13
1835.2941
1835.2941

Revision as of 07:07, 7 March 2023

Lua error in Module:MOS at line 28: attempt to index local 'equave' (a nil value).

4L 1s<5/3> (sometimes called diatonic), is a minor sixth-repeating MOS scale. The notation "<5/3>" means the period of the MOS is 5/3, disambiguating it from octave-repeating 4L 1s. The name of the period interval is called the sextave (by analogy to the tritave).

The generator range is 171.4 to 240 cents, placing it on the diatonic major second, usually representing a major second of some type (like 8/7). The bright (chroma-positive) generator is, however, its major sixth complement (685.7 to 720 cents).

Because this diatonic is a major sixth-repeating scale, each tone has a 5/3 minor sixth above it. The scale has one augmented chord, two major chords, two minor chords. This diatonic also has two dominant 7th chords, making it a warped Neapolitan minor scale.

Basic diatonic is in 9ed5/3, 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 (Do, Re, Mi, Fa, Sol or Sol, La, Si, Do, Re). Given that 1-5/4-3/2 is major sixth-equivalent to a tone cluster of 1-10/9-5/4, it may be more convenient to notate these diatonic scales as repeating at the double sextave (augmented eleventh~twelfth), however it does make navigating the genchain harder. This way, 3/2 is its own pitch class, distinct from 10\9. Notating this way produces a twelfth which is the Scala Francisci[8L 2s]. 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 Scala Francisci 19eds 14eds 23eds 9eds 22eds 13eds 17eds
Do#, Sol# Α# 1\19

46.154

1\14

63.158

2\23

77.419

1\9

100

3\22

124.138

2\13

141.1765

3\17

163.63

Reb, Lab Βb 3\19

138.4615

2\14

126.316

3\23

116.129

2\22

82.759

1\13

70.588

1\17

54.54

Re, La Β 4\19

184.615

3\14

189.474

5\23

193.548

2\9

200

5\22

206.897

3\13

211.765

4\17

218.18

Re#, La# Β# 5\19

230.769

4\14

252.632

7\23

270.968

3\9

300

8\22

331.0345

5\13

352.941

7\17

381.81

Mib, Sib Γb 7\19

323.077

5\14

315.7895

8\23

309.677

7\22

289.655

4\13

282.353

5\17

272.72

Mi, Si Γ 8\19

369.2301

6\14

378.947

10\23

387.097

4\9

400

10\22

413.793

6\13

423.529

8\17

436.36

Mi#, Si# Γ# 9\19

415.385

7\14

442.105

12\23

464.516

5\9

500

13\22

537.931

8\13

564.706

11\17

600

Fab, Dob Δb 10\19

461.5385

11\23

425.8065

4\9

400

9\22

372.414

5\13

352.941

6\17

327.27

Fa, Do Δ 11\19

507.692

8\14

505.263

13\23

503.226

5\9

500

12\22

496.552

7\13

494.118

9\17

490.90

Fa#, Do# Δ# 12\19

553.846

9\14

568.421

15\23

580.645

6\9

600

15\22

620.690

9\13

635.294

12\17

654.54

Solb, Reb Εb 14\19

646.154

10\14

631.579

16\23

619.355

14\22

579.310

8\13

564.706

10\17

545.45

Sol, Re Ε 15\19

692.308

11\14

694.737

18\23

696.774

7\9

700

17\22

703.448

10\13

705.882

13\17

709.09

Sol#, Re# Ε# 16\19

738.4615

12\14

757.8947

20\23

774.194

8\9

800

20\22

827.586

12\13

847.059

16\14

872.72

Dob, Solb Ϛb/Ϝb 18\19

830.769

13\14

821.053

21\23

812.903

19\22

786.207

11\13

776.647

14\17

763.63

Do, Sol Ϛ/Ϝ 19\19

876.923

14\14

884.2105

23\23

890.323

9\9

900

22\22

910.345

13\13

917.647

17\17

927.27

Do#, Sol# Ϛ#/Ϝ# 20\19

923.077

15\14

947.368

24\23

929.032

10\9

1000

25\22

1034.483

15\13

1052.8235

20\17

1090.90

Reb, Lab Ζb 22\19

1015.385

16\14

1010.526

26\23

1006.452

24\22

993.103

14\13

988.235

18\17

981.81

Re, La Ζ 23\19

1061.5385

17\14

1071.684

28\23

1083.871

11\9

1100

27\22

1117.241

16\13

1129.412

21\17

1145.45

Re#, La# Ζ# 24\19

1107.692

18\14

1136.842

30\23

1161.290

12\9

1200

30\22

1241.379

18\13

1270.588

24\14

1309.09

Mib, Sib Ηb 26\19

1200

19\14

1200

31\23

1200

29\22

1200

17\13

1200

22\17

1200

Mi, Si Η 27\19

1246.154

20\14

1263.158

33\23

1277.419

13\9

1300

32\22

1324.138

19\13

1341.1765

25\17

1363.63

Mi#, Si# Η# 28\19

1292.308

21\14

1326.316

35\23

1354.839

14\9

1400

35\22

1448.276

21\13

1482.353

28\17

1527.27

Fab, Dob Θb 29\19

1338.4615

34\23

1316.129

13\9

1300

31\22

1282.759

18\13

1270.588

23\17

1254.54

Fa, Do Θ 30\19

1384.615

22\14

1389.474

36\23

1393.548

14\9

1400

34\22

1406.897

20\13

1411.765

26\17

1418.18

Fa#, Do# Θ# 31\19

1430.769

23\14

1452.632

38\23

1470.968

15\9

1500

37\22

1531.0345

22\13

1552.941

29\17

1581.81

Solb, Reb Ιb 33\19

1523.077

24\14

1515.7895

39\23

1509.677

36\22

1489.655

21\13

1482.353

27\17

1472.72

Sol, Re Ι 34\19

1569.231

25\14

1578.947

41\23

1587.097

16\9

1600

39\22

1613.793

23\13

1623.529

30\17

1636.36

Sol#, Re# Ι# 35\19

1615.385

26\14

1642.105

43\23

1664.516

17\9

1700

42\22

1737.931

25\13

1764.706

33\17

1800

Dob, Solb Αb 37\19

1707.692

27\14

1705.263

44\23

1703.226

41\22

1696.552

20\13

1694.118

31\17

1490.90

Do, Sol Α 38\19

1753.846

28\14

1768.421

46\23

1780.645

18\9

1800

44\22

1820.690

26\13

1835.2941

34\17

1854.54

ed3\4
Notation Supersoft Soft Semisoft Basic Semihard Hard Superhard
Diatonic Scala Francisci 19eds 14eds 23eds 9eds 22eds 13eds 17eds
Do#, Sol# Α# 1\19

47.3684

1\14

64.2857

2\23

78.2609

1\9

100

3\22

122.72

2\13

[[1]]

3\17

[[2]]

Reb, Lab Βb 3\19

[[3]]

2\14

[[4]]

3\23

[[5]]

2\22

81.81

1\13

69.2308

1\17

52.9412

Re, La Β 4\19

[[6]]

3\14

[[7]]

5\23

[[8]]

2\9

200

5\22

204.54

3\13

[[9]]

4\17

[[10]]

Re#, La# Β# 5\19

[[11]]

4\14

[[12]]

7\23

273.913

3\9

300

8\22

327.27

5\13

346.15385

7\17

[[13]]

Mib, Sib Γb 7\19

331.57895

5\14

[[14]]

8\23

[[15]]

7\22

286.36

4\13

[[16]]

5\17

[[17]]

Mi, Si Γ 8\19

[[18]]

6\14

[[19]]

10\23

391.304

4\9

400

10\22

409.09

6\13

[[20]]

8\17

[[21]]

Mi#, Si# Γ# 9\19

[[22]]

7\14

450

12\23

[[23]]

5\9

500

13\22

531.81

8\13

553.84615

11\17

[[24]]

Fab, Dob Δb 10\19

[[25]]

11\23

[[26]]

4\9

400

9\22

368.18

5\13

346.15385

6\17

[[27]]

Fa, Do Δ 11\19

[[28]]

8\14

[[29]]

13\23

508.696

5\9

500

12\22

490.90

7\13

[[30]]

9\17

[[31]]

Fa#, Do# Δ# 12\19

568.42105

9\14

[[32]]

15\23

[[33]]

6\9

600

15\22

613.63

9\13

[[34]]

12\17

[[35]]

Solb, Reb Εb 14\19

[[36]]

10\14

[[37]]

16\23

626.087

14\22

572.72

8\13

553.84615

10\17

[[38]]

Sol, Re Ε 15\19

[[39]]

11\14

[[40]]

18\23

[[41]]

7\8

700

17\22

695.45

10\13

[[42]]

13\17

[[43]]

Sol#, Re# Ε# 16\19

[[44]]

12\14

[[45]]

20\23

[[46]]

8\8

800

20\22

818.18

12\13

[[47]]

16\14

[[48]]

Dob, Solb Ϛb/Ϝb 18\19

[[49]]

13\14

[[50]]

21\23

[[51]]

19\22

777.27

11\13

[[52]]

14\17

[[53]]

Do, Sol Ϛ/Ϝ 900
Do#, Sol# Ϛ#/Ϝ# 20\19

[[54]]

15\14

[[55]]

25\23

[[56]]

10\9

1000

25\22

1022.72

15\13

1038.4615

20\17

1058.8235

Reb, Lab Ζb 22\19

1042.1053

16\14

1028.5714

26\23

1017.3913

24\22

981.81

14\13

[[57]]

18\17

[[58]]

Re, La Ζ 23\19

1089.4737

17\14

1092.8571

28\23

1095.6522

11\9

1100

27\22

1104.54

16\13

1107.6923

21\17

1111.7647

Re#, La# Ζ# 24\19

1136.8421

18\14

1157.1429

30\23

1173.913

12\9

1200

30\22

1227.27

18\13

1246.15385

24\14

1270.5882

Mib, Sib Ηb 26\19

1231.57895

19\14

1221.4286

31\23

1213.0345

29\22

1186.36

17\13

1176.9231

22\17

1164.7059

Mi, Si Η 27\19

1278.9474

20\14

1285.7143

33\23

1291.304

13\9

1300

32\22

1309.09

19\13

1315.3846

25\17

1323.5294

Mi#, Si# Η# 28\19

1326.3158

21\14

1350

35\23

1369.5652

14\9

1400

35\22

1431.81

21\13

1453.15385

28\17

1482.3529

Fab, Dob Θb 29\19

1373.6842

34\23

1330.7692

13\9

1300

31\22

1368.18

18\13

1346.84615

23\17

1317.6471

Fa, Do Θ 30\19

1421.0526

22\14

1414.2857

36\23

1408.696

14\9

1400

34\22

1390.90

20\13

1384.6154

26\17

1376.4706

Fa#, Do# Θ# 31\19

1468.42105

23\14

1478.7143

38\23

1487.9655

15\9

1500

37\22

1513.63

22\13

1523.0769

29\17

1581.81

Solb, Reb Ιb 33\19

1563.1579

24\14

1542.8571

39\23

1526.087

36\22

1472.72

21\13

1453.15385

27\17

1429.4118

Sol, Re Ι 34\19

1610.5263

25\14

1607.1429

41\23

1604.3478

16\9

1600

39\22

1595.45

23\13

1592.3077

30\17

1588.2353

Sol#, Re# Ι# 35\19

1657.8947

26\14

1671.4286

43\23

1682.6087

17\9

1700

42\22

1718.18

25\13

1730.7692

33\17

1747.0588

Dob, Solb Αb 37\19

1752.6316

27\14

1735.7143

44\23

1721.7391

41\22

1677.27

20\13

1661.5385

31\17

1641.1761

Do, Sol Α 1800

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 Do, Sol sextave (major sixth) 0 Do, Sol perfect unison
1 Sol, Re perfect fifth -1 Re, La major second
2 Fa, Do perfect fourth -2 Mi, Si major third
3 Mib, Sib minor third -3 Fa#, Do# augmented fourth
4 Reb, Lab minor second -4 Sol#, Re# augmented fifth
The chromatic 9-note MOS also has the following intervals (from some root):
5 Dob, Solb diminished sextave -5 Do#, Sol# augmented unison (chroma)
6 Solb, Reb diminished fifth -6 Re#, La# augmented second
7 Fab, Dob diminished fourth -7 Mi#, Si# augmented third
8 Mibb, Sibb diminished third -8 Fax, Dox doubly augmented fourth

Genchain

The generator chain for this scale is as follows:

Mibb

Sibb

Fab

Dob

Solb

Reb

Dob

Solb

Reb

Lab

Mib

Sib

Fa

Do

Sol

Re

Do

Sol

Re

La

Mi

Si

Fa#

Do#

Sol#

Re#

Do#

Sol#

Re#

La#

Mi#

Si#

Fax

Dox

d3 d4 d5 d6 m2 m3 P4 P5 P1 M2 M3 A4 A5 A1 A2 A3 AA4

Modes

The mode names are based on the major satellites of Uranus, in order of size:

Mode Scale UDP Interval type
name pattern notation 2nd 3rd 4th 5th
Lydian Augmented LLLLs 4|0 M M A A
Lydian LLLsL 3|1 M M A P
Major LLsLL 2|2 M M P P
Dorian LsLLL 1|3 M m P P
Neapolitan sLLLL 0|4 m m P P

Temperaments

The most basic rank-2 temperament interpretation of this diatonic is Dorianic, which has pental 4:5:6 or septimal 14:18:21 chords spelled root-(2g)-(p-1g) (p = the major sixth, g = the whole tone). The name "Dorianic" comes from the Dorian major mode having the minor sixth as its characteristic interval.

Dorianic-Meantone

Subgroup: 5/3.4/3.3/2

Comma list: 81/80

POL2 generator: ~9/8 = 193.8419

Mapping: [⟨1 1 1], ⟨0 -2 -1]]

Vals: Template:Val list

Dorianic-Superpyth

Subgroup: 12/7.4/3.3/2

Comma list: 64/63

POL2 generator: ~9/8 = 216.5781

Mapping: [⟨1 1 1], ⟨0 -2 -1]]

Vals: Template:Val list

Scale tree

The spectrum looks like this:

Generator

(bright)

Normalised ed3\4 L s L/s Comments
Chroma-positive Chroma-negative Chroma-positive Chroma-negative
1\5 171.429 685.714 180 720 1 1 1.000 Equalised
6\29 180 690 186.207 713.793 6 5 1.200
5\24 181.81 490.90 187.5 712.5 5 4 1.250
14\67 182.609 691.304 188.06 711.94 14 11 1.273
9\43 183.051 691.525 188.372 711.628 9 7 1.286
4\19 184.615 692.308 189.474 710.526 4 3 1.333
11\52 185.915 692.958 190.385 709.615 11 8 1.375
7\33 186.6 693.3 190.90 709.09 7 5 1.400
10\47 187.5 693.75 191.498 708.519 10 7 1.429
3\14 189.474 694.737 192.857 707.143 3 2 1.500 Dorianic-Meantone starts here
14\65 190.90 695.45 193.846 706.154 14 9 1.556
11\51 191.304 695.652 194.118 705.882 11 7 1.571
8\37 192 696 194.594 705.495 8 5 1.600
13\60 192.692 696.296 195 705 13 8 1.625
5\23 193.548 696.774 195.652 704.348 5 3 1.667
12\55 194.594 697.297 196.36 703.63 12 7 1.714
7\32 195.349 697.674 196.875 703.125 7 4 1.750
9\41 196.36 698.18 197.561 702.439 9 5 1.800
11\50 197.015 698.507 198 702 11 6 1.833
13\59 197.468 698.734 198.305 701.695 13 7 1.857
15\68 197.802 698.901 198.529 701.471 15 8 1.875
17\77 198.058 699.029 198.701 701.299 17 9 1.889
19\86 198.261 699.13 198.837 701.163 19 10 1.900
21\95 198.425 699.213 198.947 701.053 21 11 1.909
23\104 198.561 699.281 199.039 700.961 23 12 1.917
2\9 200 700 200 700 2 1 2.000 Dorianic-Meantone ends, Dorianic-Pythagorean begins
23\103 201.46 700.73 200.971 699.029 23 11 2.091
21\94 201.6 700.8 201.064 698.936 21 10 2.100
19\85 201.77 700.885 [[59]] [[60]] 19 9 2.111
17\76 201.98 700.99 201.316 698.684 17 8 2.125
15\67 202.247 701.123 [[61]] [[62]] 15 7 2.143
13\58 202.597 701.299 201.724 698.276 13 6 2.167
11\49 203.076 701.538 202.041 697.959 11 5 2.200
9\40 203.774 701.887 202.5 697.5 9 4 2.250
7\31 204.838 702.439 203.226 696.774 7 3 2.333
12\53 205.714 702.858 203.774 696.226 12 5 2.400
5\22 206.897 703.448 204.54 695.45 5 2 2.500
18\79 207.692 703.847 205.063 694.937 18 7 2.571
8\35 208.696 704.348 205.714 694.286 8 3 2.667
11\48 209.524 704.762 206.25 693.75 11 4 2.750
14\61 210 705 206.557 693.443 14 5 2.800
3\13 211.765 705.882 207.692 692.308 3 1 3.000 Dorianic-Pythagorean ends, Dorianic-Superpyth begins
22\95 212.903 706.452 208.421 691.579 22 7 3.143
19\82 213.084 706.542 [[63]] [[64]] 19 6 3.167
16\69 213.3 706.6 208.696 691.304 16 5 3.200
13\56 213.699 706.849 208.929 691.071 13 4 3.250
10\43 214.286 707.143 209.322 690.678 10 3 3.333
7\30 215.385 707.692 210 690 7 2 3.500
11\47 216.393 708.192 210.638 689.362 11 3 3.667
15\64 216.867 708.434 [[65]] [[66]] 15 4 3.750
19\81 217.143 708.571 211.1 688.8 19 5 3.800
4\17 218.18 709.09 211.765 688.235 4 1 4.000
21\89 219.13 709.565 212.36 687.64 21 5 R.200
17\72 219.355 709.677 212.5 687.5 17 4 4.250
13\55 219.718 709.859 212.72 687.27 13 3 4.333
9\38 220.408 710.204 213.158 686.842 9 2 4.500
14\59 221.053 710.526 213.559 686.441 14 3 4.667
5\21 222.2 711.1 214.286 685.714 5 1 5.000 Dorianic-Superpyth ends
16\67 223.256 711.628 214.925 685.075 16 3 5.333
11\46 223.729 711.864 215.217 684.783 11 2 5.500
17\71 224.176 712.088 215.492 215.508 17 3 5.667
6\25 225 712.5 216 684 6 1 6.000
1\4 240 720 225 675 1 0 → inf Paucitonic
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