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

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← 3L 1s⟨5/3⟩4L 1s<5/3> 5L 1s⟨5/3⟩ →
↙ 3L 2s⟨5/3⟩ ↓4L 2s⟨5/3⟩ 5L 2s⟨5/3⟩ ↘
┌╥╥╥╥┬┐
│║║║║││
│││││││
└┴┴┴┴┴┘
Scale structure
Step pattern LLLLs
sLLLL
Equave 5/3 (884.4¢)
Period 5/3 (884.4¢)
Generator size(ed5/3)
Bright 1\5 to 1\4 (176.9¢ to 221.1¢)
Dark 3\4 to 4\5 (663.3¢ to 707.5¢)
Related MOS scales
Parent 1L 3s⟨5/3⟩
Sister 1L 4s⟨5/3⟩
Daughters 5L 4s⟨5/3⟩, 4L 5s⟨5/3⟩
Neutralized 3L 2s⟨5/3⟩
2-Flought 9L 1s⟨5/3⟩, 4L 6s⟨5/3⟩
Equal tunings(ed5/3)
Equalized (L:s = 1:1) 1\5 (176.9¢)
Supersoft (L:s = 4:3) 4\19 (186.2¢)
Soft (L:s = 3:2) 3\14 (189.5¢)
Semisoft (L:s = 5:3) 5\23 (192.3¢)
Basic (L:s = 2:1) 2\9 (196.5¢)
Semihard (L:s = 5:2) 5\22 (201.0¢)
Hard (L:s = 3:1) 3\13 (204.1¢)
Superhard (L:s = 4:1) 4\17 (208.1¢)
Collapsed (L:s = 1:0) 1\4 (221.1¢)

4L 1s⟨5/3⟩ is a 5/3-equivalent (non-octave) moment of symmetry scale containing 4 large steps and 1 small step, repeating every interval of 5/3 (884.4¢). Generators that produce this scale range from 176.9¢ to 221.1¢, or from 663.3¢ to 707.5¢. Scales of this form are always proper because there is only one small step.The name of the period interval is called the sextave (by analogy to the tritave).

Because this diatonic is a major sixth-repeating scale, each tone has a 5/3 major 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 major sixth-based equal tuning similar to 12edo.

Notation

There are 2 main ways to notate the diatonic scale. One method uses a simple sextave (major 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 19ed5/3 14ed5/3 23ed5/3 9ed5/3 22ed5/3 13ed5/3 17ed5/3
Do#, Sol# 1\19, 46.545 1\14, 63.168 2\23, 76.901 1\9, 98.262 3\22, 120.594 2\13, 136.055 3\17, 156.063
Reb, Lab 3\19, 139.636 2\14, 126.337 3\23, 115.351 2\22, 80.396 1\13, 68.028 1\17, 52.021
Re, La 4\19, 186.181 3\14, 189.505 5\23, 192.252 2\9, 196.524 5\22, 200.991 3\13, 204.083 4\17, 208.084
Re#, La# 5\19, 232.726 4\14, 252.673 7\23, 269.153 3\9, 294.786 8\22, 321.585 5\13, 340.138 7\17, 364.148
Mib, Sib 7\19, 325.816 5\14, 315.842 8\23, 307.603 7\22, 281.387 4\13, 272.110 5\17, 260.106
Mi, Si 8\19, 372.362 6\14, 379.011 10\23, 384.504 4\9, 393.048 10\22, 401.981 6\13, 408.166 8\17, 416.169
Mi#, Si# 9\19, 418.906 7\14, 442.179 12\23, 461.405 5\9, 491.310 13\22, 522.576 8\13, 544.221 11\17, 572.232
Fab, Dob 10\19, 465.452 11\23, 422.954 4\9, 393.048 9\22, 361.783 5\13, 340.138 6\17, 312.127
Fa, Do 11\19, 511.997 8\14, 505.348 13\23, 499.855 5\9, 491.310 12\22, 482.377 7\13, 476.193 9\17, 468.190
Fa#, Do# 12\19, 558.542 9\14, 568.516 15\23, 576.756 6\9, 589.572 15\22, 602.972 9\13, 612.248 12\17, 624.253
Solb, Reb 14\19, 651.632 10\14, 631.685 16\23, 615.206 14\22, 562.773 8\13, 544.221 10\17, 520.211
Sol, Re 15\19, 698.178 11\14, 694.853 18\23, 692.107 7\9, 687.835 17\22, 683.368 10\13, 680.276 13\17, 676.274
Sol#, Re# 16\19, 744.723 12\14, 758.022 20\23, 769.008 8\9, 786.096 20\22, 803.962 12\13, 816.331 16\14, 832.338
Dob, Solb 18\19, 837.814 13\14, 821.190 21\23, 809.458 19\22, 763.764 11\13, 748.304 14\17, 728.295
Do, Sol 19\19, 884.359 14\14, 884.359 23\23, 884.359 9\9, 884.359 22\22, 884.359 13\13, 884.359 17\17, 884.359
Normalized
Notation Supersoft Soft Semisoft Basic Semihard Hard Superhard
Scala Francisci 19ed5/3 14ed5/3 23ed5/3 9ed5/3 22ed5/3 13ed5/3 17ed5/3
Α# 1\19, 46.545 1\14, 63.168 2\23, 76.901 1\9, 98.262 3\22, 120.594 2\13, 136.055 3\17, 156.063
Βb 3\19, 139.636 2\14, 126.337 3\23, 115.351 2\22, 80.396 1\13, 68.028 1\17, 52.021
Β 4\19, 186.181 3\14, 189.505 5\23, 192.252 2\9, 196.524 5\22, 200.991 3\13, 204.083 4\17, 208.084
Β# 5\19, 232.726 4\14, 252.673 7\23, 269.153 3\9, 294.786 8\22, 321.585 5\13, 340.138 7\17, 364.148
Γb 7\19, 325.816 5\14, 315.842 8\23, 307.603 7\22, 281.387 4\13, 272.110 5\17, 260.106
Γ 8\19, 372.362 6\14, 379.011 10\23, 384.504 4\9, 393.048 10\22, 401.981 6\13, 408.166 8\17, 416.169
Γ# 9\19, 418.906 7\14, 442.179 12\23, 461.405 5\9, 491.310 13\22, 522.576 8\13, 544.221 11\17, 572.232
Δb 10\19, 465.452 11\23, 422.954 4\9, 393.048 9\22, 361.783 5\13, 340.138 6\17, 312.127
Δ 11\19, 511.997 8\14, 505.348 13\23, 499.855 5\9, 491.310 12\22, 482.377 7\13, 476.193 9\17, 468.190
Δ# 12\19, 558.542 9\14, 568.516 15\23, 576.756 6\9, 589.572 15\22, 602.972 9\13, 612.248 12\17, 624.253
Εb 14\19, 651.632 10\14, 631.685 16\23, 615.206 14\22, 562.773 8\13, 544.221 10\17, 520.211
Ε 15\19, 698.178 11\14, 694.853 18\23, 692.107 7\9, 687.835 17\22, 683.368 10\13, 680.276 13\17, 676.274
Ε# 16\19, 744.723 12\14, 758.022 20\23, 769.008 8\9, 786.096 20\22, 803.962 12\13, 816.331 16\14, 832.338
Ϛb/Ϝb 18\19, 837.814 13\14, 821.190 21\23, 809.458 19\22, 763.764 11\13, 748.304 14\17, 728.295
Ϛ/Ϝ 19\19, 884.359 14\14, 884.359 23\23, 884.359 9\9, 884.359 22\22, 884.359 13\13, 884.359 17\17, 884.359
Ϛ#/Ϝ# 20\19, 930.903 15\14, 947.527 24\23, 922.806 10\9, 982.621 25\22, 1004.953 15\13, 1020.413 20\17, 1040.422
Ζb 22\19, 1023.994 16\14, 1010.696 26\23, 999.710 24\22, 964.755 14\13, 952.386 18\17, 936.380
Ζ 23\19, 1070.539 17\14, 1073.864 28\23, 1076.611 11\9, 1080.882 27\22, 1085.349 16\13, 1088.441 21\17, 1092.442
Ζ# 24\19, 1117.085 18\14, 1137.033 30\23, 1153.511 12\9, 1179.145 30\22, 1205.944 18\13, 1224.497 24\14, 1248.506
Ηb 26\19, 1210.175 19\14, 1200.201 31\23, 1191.952 29\22, 1165.745 17\13, 1156.469 22\17, 1144.464
Η 27\19, 1256.720 20\14, 1263.370 33\23, 1268.863 13\9, 1277.407 32\22, 1286.340 19\13, 1292.524 25\17, 1300.528
Η# 28\19, 1303.265 21\14, 1326.538 35\23, 1345.763 14\9, 1375.669 35\22, 1406.934 21\13, 1428.579 28\17, 1456.591
Θb 29\19, 1349.811 34\23, 1307.313 13\9, 1277.407 31\22, 1246.142 18\13, 1224.497 23\17, 1196.485
Θ 30\19, 1396.356 22\14, 1389.707 36\23, 1384.214 14\9, 1375.669 34\22, 1366.736 20\13, 1360.552 26\17, 1352.549
Θ# 31\19, 1442.901 23\14, 1452.875 38\23, 1461.114 15\9, 1473.931 37\22, 1487.331 22\13, 1496.606 29\17, 1508.612
Ιb 33\19, 1535.991 24\14, 1516.044 39\23, 1499.565 36\22, 1447.132 21\13, 1428.579 27\17, 1404.570
Ι 34\19, 1582.537 25\14, 1579.212 41\23, 1576.466 16\9, 1572.193 39\22, 1567.723 23\13, 1564.635 30\17, 1560.633
Ι# 35\19, 1629.081 26\14, 1642.380 43\23, 1653.366 17\9, 1670.455 42\22, 1688.321 25\13, 1700.690 33\17, 1664.675
Αb 37\19, 1722.172 27\14, 1705.549 44\23, 1691.817 41\22, 1648.123 24\13, 1632.662 31\17, 1612.654
Α 38\19, 1768.717 28\14, 1768.717 46\23, 1768.717 18\9, 1768.717 44\22, 1768.717 26\13, 1768.717 34\17, 1768.717

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 classical modes:

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 mode having the major 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]]

Optimal ET sequence: 5ed5/3, 9ed5/3, 14ed5/3

Scale tree

The spectrum looks like this:

Scale Tree and Tuning Spectrum of 4L 1s⟨5/3⟩
Generator(ed5/3) Cents Step Ratio Comments(always proper)
Bright Dark L:s Hardness
1\5 176.872 707.487 1:1 1.000 Equalized 4L 1s⟨5/3⟩
6\29 182.971 701.388 6:5 1.200
5\24 184.241 700.117 5:4 1.250
9\43 185.098 699.260 9:7 1.286
4\19 186.181 698.178 4:3 1.333 Supersoft 4L 1s⟨5/3⟩
11\52 187.076 697.283 11:8 1.375
7\33 187.591 696.767 7:5 1.400
10\47 188.161 696.197 10:7 1.429
3\14 189.505 694.853 3:2 1.500 Soft 4L 1s⟨5/3⟩
11\51 190.744 693.615 11:7 1.571
8\37 191.213 693.146 8:5 1.600
13\60 191.611 692.748 13:8 1.625
5\23 192.252 692.107 5:3 1.667 Semisoft 4L 1s⟨5/3⟩
12\55 192.951 691.408 12:7 1.714
7\32 193.453 690.905 7:4 1.750
9\41 194.128 690.231 9:5 1.800
2\9 196.524 687.835 2:1 2.000 Basic 4L 1s⟨5/3⟩
9\40 198.981 685.378 9:4 2.250
7\31 199.694 684.665 7:3 2.333
12\53 200.232 684.127 12:5 2.400
5\22 200.991 683.368 5:2 2.500 Semihard 4L 1s⟨5/3⟩
13\57 201.696 682.663 13:5 2.600
8\35 202.139 682.220 8:3 2.667
11\48 202.666 681.693 11:4 2.750
3\13 204.083 680.276 3:1 3.000 Hard 4L 1s⟨5/3⟩
10\43 205.665 678.694 10:3 3.333
7\30 206.350 678.008 7:2 3.500
11\47 206.978 677.381 11:3 3.667
4\17 208.084 676.274 4:1 4.000 Superhard 4L 1s⟨5/3⟩
9\38 209.453 674.905 9:2 4.500
5\21 210.562 673.797 5:1 5.000
6\25 212.246 672.113 6:1 6.000
1\4 221.090 663.269 1:0 → ∞ Collapsed 4L 1s⟨5/3⟩

See also

8L 2s (72/25-equivalent) - 8/1 complement of Scala Francisci