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.

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\19 369.231	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\19 461.5385		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\14 757.895	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\14 884.2105	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

ed9\12 (→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 47.368	1\14 64.286	2\23 78.261	1\9 100	3\22 122.72	2\13 138.4615	3\17 158.8235
Reb, Lab	Βb	3\19 142.105	2\14 128.571	3\23 117.391		2\22 81.81	1\13 69.231	1\17 52.941
Re, La	Β	4\19 189.474	3\14 192.857	5\23 195.652	2\9 200	5\22 204.54	3\13 207.692	4\17 211.765
Re#, La#	Β#	5\19 236.842	4\14 257.143	7\23 273.913	3\9 300	8\22 327.27	5\13 346.154	7\17 370.588
Mib, Sib	Γb	7\19 331.579	5\14 321.429	8\23 313.0345		7\22 286.36	4\13 276.923	5\17 264.706
Mi, Si	Γ	8\19 378.947	6\14 385.714	10\23 391.304	4\9 400	10\22 409.09	6\13 415.385	8\17 423.529
Mi#, Si#	Γ#	9\19 426.316	7\14 450	12\23 469.565	5\9 500	13\22 531.81	8\13 553.846	11\17 582.353
Fab, Dob	Δb	10\19 473.684		11\23 430.769	4\9 400	9\22 368.18	5\13 346.154	6\17 317.647
Fa, Do	Δ	11\19 521.053	8\14 514.286	13\23 508.696	5\9 500	12\22 490.90	7\13 484.615	9\17 476.471
Fa#, Do#	Δ#	12\19 568.421	9\14 578.571	15\23 578.9655	6\9 600	15\22 613.63	9\13 623.077	12\17 635.293
Solb, Reb	Εb	14\19 663.158	10\14 642.857	16\23 626.087		14\22 572.72	8\13 553.846	10\17 529.412
Sol, Re	Ε	15\19 710.526	11\14 707.143	18\23 704.348	7\8 700	17\22 695.45	10\13 692.308	13\17 688.235
Sol#, Re#	Ε#	16\19 757.895	12\14 771.429	20\23 782.609	8\8 800	20\22 818.18	12\13 830.769	16\17 847.059
Dob, Solb	Ϛb/Ϝb	18\19 852.632	13\14 835.714	21\23 821.739		19\22 777.27	11\13 761.5385	14\17 741.1765
Do, Sol	Ϛ/Ϝ	900
Do#, Sol#	Ϛ#/Ϝ#	20\19 947.368	15\14 964.286	25\23 978.261	10\9 1000	25\22 1022.72	15\13 1038.4615	20\17 1058.8235
Reb, Lab	Ζb	22\19 1042.105	16\14 1028.571	26\23 1017.391		24\22 981.81	14\13 969.231	18\17 952.941
Re, La	Ζ	23\19 1089.473	17\14 1092.857	28\23 1095.652	11\9 1100	27\22 1104.54	16\13 1107.692	21\17 1111.765
Re#, La#	Ζ#	24\19 1136.842	18\14 1157.143	30\23 1173.913	12\9 1200	30\22 1227.27	18\13 1246.154	24\14 1270.588
Mib, Sib	Ηb	26\19 1231.579	19\14 1221.429	31\23 1213.0345		29\22 1186.36	17\13 1176.923	22\17 1164.706
Mi, Si	Η	27\19 1278.947	20\14 1285.714	33\23 1291.304	13\9 1300	32\22 1309.09	19\13 1315.385	25\17 1323.529
Mi#, Si#	Η#	28\19 1326.316	21\14 1350	35\23 1369.565	14\9 1400	35\22 1431.81	21\13 1453.846	28\17 1482.353
Fab, Dob	Θb	29\19 1373.684		34\23 1330.769	13\9 1300	31\22 1368.18	18\13 1246.154	23\17 1317.647
Fa, Do	Θ	30\19 1421.053	22\14 1414.286	36\23 1408.696	14\9 1400	34\22 1390.90	20\13 1384.615	26\17 1376.471
Fa#, Do#	Θ#	31\19 1468.421	23\14 1478.714	38\23 1487.9655	15\9 1500	37\22 1513.63	22\13 1523.077	29\17 1535.294
Solb, Reb	Ιb	33\19 1563.158	24\14 1542.857	39\23 1526.087		36\22 1472.72	21\13 1453.846	27\17 1429.412
Sol, Re	Ι	34\19 1610.526	25\14 1607.143	41\23 1604.348	16\9 1600	39\22 1595.45	23\13 1592.308	30\17 1588.235
Sol#, Re#	Ι#	35\19 1657.895	26\14 1671.429	43\23 1682.609	17\9 1700	42\22 1718.18	25\13 1730.769	33\17 1747.059
Dob, Solb	Αb	37\19 1752.632	27\14 1735.714	44\23 1721.739		41\22 1677.27	20\13 1661.5385	31\17 1641.1765
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]]

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

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]]

Optimal ET sequence: 4ed14/9, 13ed14/9, 17ed14/9

Scale tree

The spectrum looks like this:

Generator (bright)			Normalised	ed9\12 (→ed3\4)	L	s	L/s	Comments
1\5			171.429	180	1	1	1.000	Equalised
6\29			180	186.207	6	5	1.200
5\24			181.81	187.5	5	4	1.250
	14\67		182.609	188.06	14	11	1.273
	9\43		183.051	188.372	9	7	1.286
4\19			184.615	189.474	4	3	1.333
	11\52		185.915	190.385	11	8	1.375
	7\33		186.6	190.90	7	5	1.400
	10\47		187.5	191.498	10	7	1.429
3\14			189.474	192.857	3	2	1.500	Dorianic-Meantone starts here
	14\65		190.90	193.846	14	9	1.556
	11\51		191.304	194.118	11	7	1.571
	8\37		192	194.594	8	5	1.600
	5\23		193.548	195.652	5	3	1.667
	7\32		195.349	196.875	7	4	1.750
	9\41		196.36	197.561	9	5	1.800
	11\50		197.015	198	11	6	1.833
	13\59		197.468	198.305	13	7	1.857
	15\68		197.802	198.529	15	8	1.875
	17\77		198.058	198.701	17	9	1.889
	19\86		198.261	198.837	19	10	1.900
	21\95		198.425	198.947	21	11	1.909
	23\104		198.561	199.039	23	12	1.917
2\9			200	200	2	1	2.000	Dorianic-Meantone ends, Dorianic-Pythagorean begins
	21\94		201.6	201.064	21	10	2.100
	19\85		201.77	201.1765	19	9	2.111
	17\76		201.98	201.316	17	8	2.125
	15\67		202.247	201.4925	15	7	2.143
	13\58		202.597	201.724	13	6	2.167
	11\49		203.076	202.041	11	5	2.200
	9\40		203.774	202.5	9	4	2.250
	7\31		204.838	203.226	7	3	2.333
		12\53	205.714	203.774	12	5	2.400
	5\22		206.897	204.54	5	2	2.500
		18\79	207.692	205.063	18	7	2.571
	8\35		208.696	205.714	8	3	2.667
	11\48		209.524	206.25	11	4	2.750
	14\61		210	206.557	14	5	2.800
3\13			211.765	207.692	3	1	3.000	Dorianic-Pythagorean ends, Dorianic-Superpyth begins
	22\95		212.903	208.421	22	7	3.143
	19\82		213.084	208.5365	19	6	3.167
	16\69		213.3	208.696	16	5	3.200
	13\56		213.699	208.929	13	4	3.250
	10\43		214.286	209.322	10	3	3.333
	7\30		215.385	210	7	2	3.500
	11\47		216.393	210.638	11	3	3.667
	15\64		216.867	210.9375	15	4	3.750
	19\81		217.143	211.1	19	5	3.800
4\17			218.18	211.765	4	1	4.000
	21\89		219.13	212.36	21	5	R.200
	17\72		219.355	212.5	17	4	4.250
	13\55		219.718	212.72	13	3	4.333
	9\38		220.408	213.158	9	2	4.500
	14\59		221.053	213.559	14	3	4.667
5\21			222.2	214.286	5	1	5.000	Dorianic-Superpyth ends
	11\46		223.729	215.217	11	2	5.500
	17\71		224.176	215.492	17	3	5.667
6\25			225	216	6	1	6.000
1\4			240	225	1	0	→ inf	Paucitonic

See also

4L 1s (5/3-equivalent) - idealized meantone tuning

4L 1s (12/7-equivalent) - idealized Archytas tuning