User:Moremajorthanmajor/4L 1s (major sixth-equivalent)
4L 1s<major sixth> (sometimes called diatonic), is a major sixth-repeating MOS scale. The notation "<major sixth>" 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 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.
| 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, 141.176¢ | 3\17, 163.63¢ |
| Reb, Lab | Βb | 3\19, 138.462¢ | 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, 331.034¢ | 5\13, 352.941¢ | 7\17, 381.81¢ |
| Mib, Sib | Γb | 7\19, 323.077¢ | 5\14, 315.789¢ | 8\23, 309.677¢ | 7\22, 289.655¢ | 4\13, 282.353¢ | 5\17, 272.72¢ | |
| Mi, Si | Γ | 8\19, 369.231¢ | 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, 461.538¢ | 11\23, 425.806¢ | 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, 738.462¢ | 12\14, 757.895¢ | 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, 884.211¢ | 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, 1052.824¢ | 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, 1061.538¢ | 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, 1341.176¢ | 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, 1338.462¢ | 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, 1515.789¢ | 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, 1835.294¢ | 34\17, 1854.54¢ |
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 major mode having the minor sixth as its characteristic interval.
Dorianic-Meantone
Subgroup: 5/3.4/3.3/2
POL2 generator: ~9/8 = 193.8419¢
Mapping: [⟨1 1 1], ⟨0 -2 -1]]
Optimal ET sequence: 5ed5/3, 9ed5/3, 14ed5/3
Dorianic-Superpyth
Subgroup: 12/7.4/3.3/2
POL2 generator: ~9/8 = 216.5781¢
Mapping: [⟨1 1 1], ⟨0 -2 -1]]
Optimal ET sequence: 4ed12/7, 9ed12/7, 13ed12/7, 17ed12/7
Scale tree
The spectrum looks like this:
| Generator
(bright) |
Normalised | L | s | L/s | Comments |
|---|---|---|---|---|---|
| 1\5 | 171.429 | 1 | 1 | 1.000 | Equalised |
| 6\29 | 180.000 | 6 | 5 | 1.200 | |
| 5\24 | 181.81 | 5 | 4 | 1.250 | |
| 14\67 | 182.609 | 14 | 11 | 1.273 | |
| 9\43 | 183.051 | 9 | 7 | 1.286 | |
| 4\19 | 184.615 | 4 | 3 | 1.333 | |
| 11\52 | 185.915 | 11 | 8 | 1.375 | |
| 7\33 | 186.6 | 7 | 5 | 1.400 | |
| 10\47 | 187.5 | 10 | 7 | 1.429 | |
| 3\14 | 189.474 | 3 | 2 | 1.500 | Dorianic-Meantone starts here |
| 14\65 | 190.90 | 14 | 9 | 1.556 | |
| 11\51 | 191.304 | 11 | 7 | 1.571 | |
| 8\37 | 192.000 | 8 | 5 | 1.600 | |
| 5\23 | 193.548 | 5 | 3 | 1.667 | |
| 7\32 | 195.349 | 7 | 4 | 1.750 | |
| 9\41 | 196.36 | 9 | 5 | 1.800 | |
| 11\50 | 197.015 | 11 | 6 | 1.833 | |
| 13\59 | 197.468 | 13 | 7 | 1.857 | |
| 15\68 | 197.802 | 15 | 8 | 1.875 | |
| 17\77 | 198.058 | 17 | 9 | 1.889 | |
| 19\86 | 198.261 | 19 | 10 | 1.900 | |
| 21\95 | 198.425 | 21 | 11 | 1.909 | |
| 23\104 | 198.561 | 23 | 12 | 1.917 | |
| 25\113 | 198.675 | 25 | 13 | 1.923 | |
| 27\122 | 198.773 | 27 | 14 | 1.929 | |
| 29\131 | 198.857 | 29 | 15 | 1.933 | |
| 31\140 | 198.930 | 31 | 16 | 1.9375 | |
| 33\149 | 198.995 | 33 | 17 | 1.941 | |
| 35\158 | 199.052 | 35 | 18 | 1.944 | |
| 2\9 | 200 | 2 | 1 | 2.000 | Dorianic-Meantone ends, Dorianic-Pythagorean begins |
| 17\76 | 201.9801 | 17 | 8 | 2.125 | |
| 15\67 | 202.247 | 15 | 7 | 2.143 | |
| 13\58 | 202.597 | 13 | 6 | 2.167 | |
| 11\49 | 203.076 | 11 | 5 | 2.200 | |
| 9\40 | 203.774 | 9 | 4 | 2.250 | |
| 7\31 | 204.838 | 7 | 3 | 2.333 | |
| 12\53 | 205.714 | 12 | 5 | 2.400 | |
| 5\22 | 206.897 | 5 | 2 | 2.500 | |
| 18\79 | 207.692 | 18 | 7 | 2.571 | |
| 13\57 | 208.000 | 13 | 5 | 2.600 | |
| 8\35 | 208.696 | 8 | 3 | 2.667 | |
| 11\48 | 209.524 | 11 | 4 | 2.750 | |
| 14\61 | 210.000 | 14 | 5 | 2.800 | |
| 3\13 | 211.765 | 3 | 1 | 3.000 | Dorianic-Pythagorean ends, Dorianic-Superpyth begins |
| 22\95 | 212.903 | 22 | 7 | 3.143 | |
| 19\82 | 213.084 | 19 | 6 | 3.167 | |
| 16\69 | 213.3 | 16 | 5 | 3.200 | |
| 13\56 | 213.699 | 13 | 4 | 3.250 | |
| 10\43 | 214.286 | 10 | 3 | 3.333 | |
| 7\30 | 215.385 | 7 | 2 | 3.500 | |
| 11\47 | 216.393 | 11 | 3 | 3.667 | |
| 15\64 | 216.867 | 15 | 4 | 3.750 | |
| 19\81 | 217.143 | 19 | 5 | 3.800 | |
| 4\17 | 218.18 | 4 | 1 | 4.000 | |
| 21\89 | 219.130 | 21 | 5 | 4.200 | |
| 17\72 | 219.355 | 17 | 4 | 4.250 | |
| 13\55 | 219.718 | 13 | 3 | 4.333 | |
| 9\38 | 220.408 | 9 | 2 | 4.500 | |
| 14\59 | 221.053 | 14 | 3 | 4.667 | |
| 5\21 | 222.2 | 5 | 1 | 5.000 | Dorianic-Superpyth ends |
| 11\46 | 223.729 | 11 | 2 | 5.500 | |
| 17\71 | 224.176 | 17 | 3 | 5.667 | |
| 6\25 | 225.000 | 6 | 1 | 6.000 | |
| 1\4 | 240.000 | 1 | 0 | → inf | Paucitonic |
See also
4L 1s (5/3-equivalent) - idealized meantone tuning
4L 1s (22/13-equivalent) - Neogothic tuning
4L 1s (12/7-equivalent) - idealized Archytas tuning
8L 2s ([math]e[/math]-equivalent) - natural tuning
8L 2s (11/4-equivalent) - idealized low tuning, low undecimal tuning
8L 2s (14/5-equivalent) - low septimal tuning
8L 2s (20/7-equivalent) - idealized high tuning, high septimal tuning
8L 2s (32/11-equivalent) - high undecimal tuning