User:Moremajorthanmajor/2L 1s (perfect fourth-equivalent)

2L 1s<perfect fourth>, is a perfect fourth-repeating MOS scale. The notation "<perfect fourth>" means the period of the MOS is a perfect fourth, disambiguating it from octave-repeating 2L 1s.

The generator range is 171.4 to 240 cents, placing it near the diatonic major second, usually representing a major second of some type. The dark (chroma-negative) generator is, however, its fourth complement (240 to 342.9 cents).

In the fourth-repeating version of the diatonic scale, each tone has a 4/3 perfect fourth above it. The scale has one major chord and two minor chords.

Basic diatonic is in 5ed4/3, which is a very good fourth-based equal tuning similar to 12edo.

Notation

There are 4 main ways to notate this scale. One method uses a simple fourth repeating notation consisting of 3 naturals (eg. Do Re Mi, Sol La Si). Given that 1-5/4-3/2 is fourth-equivalent to a tone cluster of 1-9/8-5/4, it may be more convenient to notate diatonic scales as repeating at the double, triple, quadruple or quintuple fourth (minor seventh, tenth, thirteenth or sixteenth), however it does make navigating the genchain harder. This way, 3/2 is its own pitch class, distinct from 9/8. Notating this way produces a minor tenth which is the Dorian mode of Middletown[6L 3s], also known as the Mahur scale in Persian/Arabic music, a minor thirteenth which is the Aeolian mode of Bijou[8L 4s]; the bastonic chromatic scale, a minor sixteenth which is the Phrygian mode of Hyperionic[10L 5s] or a diminished nineteenth which is the Locrian mode of Subsextal[12L 6s]. Since there are exactly 9 naturals in triple fourth notation, 12 in quadruple fourth, 15 in quintuple fourth notation and 18 in sextuple fourth notation, letters A-G plus J, Q or Q, S (GJABCQDEF or GABCQDSEF, flats written F molle) or dozenal, hex or duohex digits (0123456789XE0 or E1234567GABDE with flats written D molle or 123456789ABCDEF1 or 0123456789XɜABCDEF0 with flats written F molle) may be used.

Cents

Notation		Supersoft	Soft	Semisoft	Basic	Semihard	Hard	Superhard
Fourth	Seventh	~11ed4/3	~8ed4/3	~13ed4/3	~5ed4/3	~12ed4/3	~7ed4\3	~9ed4/3
Do#, Sol#	Sol#	1\11 46; 6.5	1\8 63; 6.3	2\13 77; 2, 2.6	1\5 100	3\12 124; 7.25	2\7 141; 5.6	3\9 163.63
Reb, Lab	Lab	3\11 138; 3.25	2\8 126; 3.16	3\13 116; 7.75		2\12 82; 1.318	1\7 70; 1.7	1\9 54.54
Re, La	La	4\11 184; 1.625	3\8 189; 2.1	5\13 193; 1, 1, 4.6	2\5 200	5\12 206; 1, 8.6	3\7 211; 1, 3.25	4\9 218.18
Re#, La#	La#	5\11 230; 1.3	4\8 252; 1.583	7\13 270; 1.03	3\5 300	8\12 331; 29	5\7 352; 1.0625	7\9 381.81
Mib, Sib	Sib	7\11 323; 13	5\8 315; 1.26	8\13 309; 1, 2.1		7\12 289; 1, 1.9	4\7 282; 2.83	5\9 272.72
Mi, Si	Si	8\11 369; 4.3	6\8 378; 1.05	10\13 387; 10.3	4\5 400	10\12 413; 1, 3.83	6\7 423; 1.8	8\9 436.36
Mi#, Si#	Si#	9\11 415; 2.6	7\8 442; 9.5	12\13 464; 1.9375	5\5 500	13\12 537; 14.5	8\7 564; 1.416	11\9 600
Dob, Solb	Dob	10\11 461; 1, 1.16		11\13 425; 1.24	4\5 400	9\12 372; 2.416	5\7 352; 1.0625	6\9 327.27
Do, Sol	Do	11\11 507; 1.4	8\8 505; 3.8	13\13 503; 4, 2.3	5\5 500	12\12 496; 1.8125	7\7 494; 8.5	9\9 490.90
Do#, Sol#	Do#	12\11 553; 1.18	9\8 568; 2.375	15\13 580; 1.55	6\5 600	15\12 620; 1.45	9\7 635; 3.4	12\9 654.54
Reb, Lab	Reb	14\11 646; 6.5	10\8 631; 1.72	16\13 619; 2.81		14\12 579; 3.2	8\7 564; 1.416	10\9 545.45
Re, La	Re	15\11 692; 3.25	11\8 694; 1, 2.8	18\13 696; 1.2916	7\5 700	17\12 703; 2, 2.16	10\7 705; 1.13	13\9 709.09
Re#, La#	Re#	16\11 738; 2.16	12\8 757; 1, 8.5	20\13 774; 5.16	8\5 800	20\12 827; 1, 1.416	12\7 847; 17	16\9 872.72
Mib, Sib	Mib	18\11 830; 1.3	13\8 821; 19	21\13 812; 1, 9.3		19\12 786; 4.83	11\7 776; 2.125	14\9 763.63
Mi, Si	Mi	19\11 876; 1.083	14\8 884; 4.75	23\13 890; 3.1	9\5 900	22\12 910; 2.9	13\7 917; 1.54	17\9 927.27
Mi#, Si#	Mi#	20\11 923: 13	15\8 947; 2, 1.4	25\13 967; 1, 2.875	10\5 1000	25\12 1034; 2, 14	15\7 1058; 1, 4.6	20\9 1090.90
Dob, Solb	Solb	21\11 969; 4.3		24\13 929; 31	9\5 900	21\12 868; 1, 28	11\7 776; 2.125	15\9 818.18
Do, Sol	Sol	22\11 1015; 2.6	16\8 1010; 1.9	26\13 1006; 2, 4.6	10\5 1000	24\12 993; 9.6	14\7 988; 4.25	18\9 981.81

Notation				Supersoft	Soft	Semisoft	Basic	Semihard	Hard	Superhard
Mahur	Bijou	Hyperionic	Subsextal	~11ed4/3	~8ed4/3	~13ed4/3	~5ed4/3	~12ed4/3	~7ed4\3	~9ed4/3
G#	0#, E#	1#	0#	1\11 46; 6.5	1\8 63; 6.3	2\13 77; 2, 2.6	1\5 100	3\12 124; 7.25	2\7 141; 5.6	3\9 163.63
Jf, Af	1b, 1d	2f	1f	3\11 138; 3.25	2\8 126; 3.16	3\13 116; 7.75		2\12 82; 1.318	1\7 70; 1.7	1\9 54.54
J, A	1	2	1	4\11 184; 1.625	3\8 189; 2.1	5\13 193; 1, 1, 4.6	2\5 200	5\12 206; 1, 8.6	3\7 211; 1, 3.25	4\9 218.18
J#, A#	1#	2#	1#	5\11 230; 1.3	4\8 252; 1.583	7\13 270; 1.03	3\5 300	8\12 331; 29	5\7 352; 1.0625	7\9 381.81
Af, Bf	2b, 2d	3f	2f	7\11 323; 13	5\8 315; 1.26	8\13 309; 1, 2.1		7\12 289; 1, 1.9	4\7 282; 2.83	5\9 272.72
A, B	2	3	2	8\11 369; 4.3	6\8 378; 1.05	10\13 387; 10.3	4\5 400	10\12 413; 1, 3.83	6\7 423; 1.8	8\9 436.36
A#, B#	2#	3#	2#	9\11 415; 2.6	7\8 442; 9.5	12\13 464; 1.9375	5\5 500	13\12 537; 14.5	8\7 564; 1.416	11\9 600
Bb, Cf	3b, 3d	4f	3f	10\11 461; 1, 1.16		11\13 425; 1.24	4\5 400	9\12 372; 2.416	5\7 352; 1.0625	6\9 327.27
B, C	3	4	3	11\11 507; 1.4	8\8 505; 3.8	13\13 503; 4, 2.3	5\5 500	12\12 496; 1.8125	7\7 494; 8.5	9\9 490.90
B#, C#	3#	4#	3#	12\11 553; 1.18	9\8 568; 2.375	15\13 580; 1.55	6\5 600	15\12 620; 1.45	9\7 635; 3.4	12\9 654.54
Cf, Qf	4b, 4d	5f	4f	14\11 646; 6.5	10\8 631; 1.72	16\13 619; 2.81		14\12 579; 3.2	8\7 564; 1.416	10\9 545.45
C, Q	4	5	4	15\11 692; 3.25	11\8 694; 1, 2.8	18\13 696; 1.2916	7\5 700	17\12 703; 2, 2.16	10\7 705; 1.13	13\9 709.09
C#, Q#	4#	5#	4#	16\11 738; 2.16	12\8 757; 1, 8.5	20\13 774; 5.16	8\5 800	20\12 827; 1, 1.416	12\7 847; 17	16\9 872.72
Qf, Df	5b, 5d	6f	5f	18\11 830; 1.3	13\8 821; 19	21\13 812; 1, 9.3		19\12 786; 4.83	11\7 776; 2.125	14\9 763.63
Q, D	5	6	5	19\11 876; 1.083	14\8 884; 4.75	23\13 890; 3.1	9\5 900	22\12 910; 2.9	13\7 917; 1.54	17\9 927.27
Q#, D#	5#	6#	5#	20\11 923: 13	15\8 947; 2, 1.4	25\13 967; 1, 2.875	10\5 1000	25\12 1034; 2, 14	15\7 1058; 1, 4.6	20\9 1090.90
Df, Sf	6b, 6d	7f	6f	21\11 969; 4.3		24\13 929; 31	9\5 900	21\12 868; 1, 28	11\7 776; 2.125	15\9 818.18
D, S	6	7	6	22\11 1015; 2.6	16\8 1010; 1.9	26\13 1006; 2, 4.6	10\5 1000	24\12 993; 9.6	14\7 988; 4.25	18\9 981.81
D#, S#	6#	7#	6#	23\11 1061; 1, 1.16	17\8 1073; 1, 2.16	28\13 1083; 1.148	11\5 1100	27\12 1117; 4, 7	16\7 1129; 2, 2.3	24\9 1309.09
Ef	7b, 7d	8f	7f	25\11 1153; 1.18	18\8 1136; 1.1875	29\13 1122; 1.72		26\12 1075; 1.16	15\7 1058; 1, 4.6	19\9 1036.36
E	7	8	7	26\11 1200	19\8 1200	31\13 1200	12\5 1200	29\12 1200	17\7 1200	22\9 1200
E#	7#	8#	7#	27\11 1246; 6,5	20\8 1263; 6.3	33\13 1277; 2, 2.6	13\5 1300	32\12 1324; 7.25	19\7 1341; 5.6	25\9 1363.63
Ff	8b, Gd	9f	8f	29\11 1338; 3.25	21\8 1326; 3.16̄	34\13 1316; 7.75		31\12 1282; 1.318	18\7 1270; 1.7	23\9 1254.54
F	8, G	9	8	30\11 1384; 1.625	22\8 1389; 2.1̄	36\13 1393; 1, 1, 4.6	14\5 1400	34\12 1406; 1, 8.6	20\7 1411; 1, 3.25	26\9 1418.18
F#	8#, G#	9#	8#	31\11 1430; 1.3	23\8 1452; 1.583	38\13 1470; 1.03	15\5 1500	37\12 1531; 29	22\7 1552; 1.0625	29\9 1581.81
Gf	9b, Ad	Af	9f	32\11 1476; 1.083		37\13 1432: 3.875	14\5 1400	33\12 1365; 1.93	19\7 1341; 5.3	24\9 1309.09
G	9, A	A	9	33\11 1523; 13	24\8 1515; 1.26	39\13 1509; 1, 2.1	15\5 1500	36\12 1489; 1, 1.9	21\7 1482; 2.83	27\9 1472.72
G#	9#, A#	A#	9#	34\11 1569; 4.3	25\8 1578; 1.05̄	41\13 1587; 10.3	16\5 1600	39\12 1613; 1, 3.83	23\7 1623; 1.8	30\9 1636.36
Jf, Af	Xb, Bd	Bf	Xb	36\11 1661; 1, 1.16	26\8 1642; 9.5	42\13 1625; 1.24		38\12 1572; 29	22\7 1552; 1.0625	28\9 1527.27
J, A	X, B	B	X	37\11 1707; 1.4	27\8 1705; 3.8	44\13 1703; 4, 2.3̄	17\5 1700	41\12 1696; 1.8125	24\7 1694; 8.5	31\9 1690.90
J#, A#	X#, B#	B#	X#	38\11 1753; 1.18	28\8 1768; 2.375	46\13 1780; 1.55	18\5 1800	44\12 1820; 1.45	26\7 1835; 3,4	34\9 1854.54
Af, Bf	Eb, Dd	Cf	ɛf	40\11 1846; 6.5	29\8 1831; 1.72	47\13 1819; 2.81		43\12 1779; 3.2	25\7 1764; 1, 3.25	32\9 1745.45
A, B	E, D	C	ɛ	41\11 1892; 3.25	30\8 1894; 1, 2.8	49\13 1896; 1.2916	19\5 1900	46\12 1903; 2, 2.16	27\7 1905; 1, 7.5	35\9 1909.09
A#, B#	E#, D#	C#	ɛ#	42\11 1938; 2.16	31\8 1957; 1, 8.5	51\13 1974; 5.16	20\5 2000	49\12 2027; 1, 1.416	29\7 2047; 17	38\9 2072.72
Bb, Cf	0b, Ed	Df	Af	43\11 1984; 1.625		50\13 1935; 2.06	19\5 1900	45\12 1862; 14.5	26\7 1835; 3,4	33\9 1800
B, C	0, E	D	A	44\11 2030; 1.3	32\8 2021; 19	52\13 2012; 1, 9.3	20\5 2000	48\12 1986; 4.83	28\7 1976; 2.125	36\9 1963.63
B#, C#	0#, E#	D#	A#	45\11 2076; 1.083	33\8 2084; 4.75	54\13 2090; 3.1	21\5 2100	51\12 2110; 2.9	30\7 2117; 1.54	39\9 2127.27
Cf, Qf	1b, 1d	Ef	Bf	47\11 2169; 4.3	34\8 2147; 2, 1.4	55\13 2129; 31		50\12 2068; 1, 28	29\7 2047; 17	37\9 2018.18
C, Q	1	E	B	48\11 2215; 2.6	35\8 2210; 1.9	57\13 2206; 2, 4.6	22\5 2200	53\12 2193; 9.6	31\7 2188; 4.25	40\9 2181.81
C#, Q#	1#	E#	B#	49\11 2261; 1, 1.16	36\8 2273; 1, 2.16	59\13 2083; 1.148	23\5 2300	56\12 2327; 4, 7	33\7 2329; 2, 2.3	43\9 2345.45
Qf, Df	2b, 2d	Ff	Cf	51\11 2353; 1.18	37\8 2336; 1.1875	61\13 2322; 1.72		55\12 2275; 1.16	32\7 2258; 1, 4.6	41\9 2236.36
Q, D	2	F	C	52\11 2400	38\8 2400	62\13 2400	24\5 2400	58\12 2400	34\7 2400	44\9 2400
Q#, D#	2#	F#	C#	53\11 2446; 6.5	39\8 2463; 6.3	64\13 2477; 2, 2.6	25\5 2500	61\12 2524; 7.25	36\7 2541; 5.6	47/9 2563.63
Df, Sf	3b, 3d	1f	Df	54\11 2492; 3.25		63\13 2438; 1.136	24\5 2400	57\12 2358; 1.61̄	33\7 2329; 2, 2.3	42\9 2390.90
D, S	3	1	D	55\11 2538; 2.16	40\8 2526; 3.16	65\13 2516; 7.75	25\5 2500	60\12 2482; 1.318	35\7 2470; 1.7	45\9 2454.54
D#, S#	3#	1#	D#	56\11 2584; 1.625	41\8 2589; 2.1̄	67\13 2593; 1, 1, 4.6	26\5 2600	63\12 2606; 1, 8.6	37\7 2611; 1, 3.25	48\9 2618.18
Ef	4b, 4d	2f	Ef	58\11 2676; 1.083	42\8 2652; 1.583	69\13 2670; 1.03		62\12 2565; 1.93	36\7 2541; 5.6	46\9 2509.09
E	4	2	E	59\11 2723; 13	43\8 2715; 1.26	70\13 2709; 1, 2.1	27\5 2700	65\12 2689; 1, 1.9	38\7 2682; 2.83	49\9 2672.72
E#	4#	2#	E#	60\11 2769; 4.3	44\8 2778; 1.05̄	72\13 2787; 10.3	28\5 2800	68\12 2813; 1, 3.83	40\7 2823; 1.8	52\9 2836.36
Ff	5b, 5d	3f	Ff	62\11 2861; 1, 1.16	45\8 2842; 9.5	73\13 2825; 1.24		67\12 2772; 29	39\7 2752; 1.0625	50\9 2727.27
F	5	3	F	63\11 2907; 1.4	46\8 2905; 3.8	75\13 2903; 4, 2.3̄	29\5 2900	70\12 2896; 1.8125	41\7 2894; 8.5	53\9 2890.90
F#	5#	3#	F#	64\11 2953; 1.18	47\8 2968; 2.375	77\13 2980; 1.55	30\5 3000	73\12 3020; 1.45	43\7 3035; 3,4	55\9 3000
Gf	6b, 6d	4f	0f	65\11 3000		76\13 2941; 1, 14.5	29\5 2900	69\29 2855; 5.8	40\7 2823; 1.8	52\9 2836.36
G	6	4	0	66\11 3046; 6.5	48\8 3031; 1.72	78\13 3019; 2.81	30\5 3000	72\12 2979; 3.2	42\7 2964; 1, 3.25	54\9 2945.45

Intervals

Generators	Fourth notation	Interval category name	Generators	Notation of 4/3 inverse	Interval category name
The 3-note MOS has the following intervals (from some root):
0	Do, Sol	perfect unison	0	Do, Sol	perfect fourth
1	Mib, Sib	diminished third	-1	Re, La	perfect second
2	Reb, Lab	diminished second	-2	Mi, Si	perfect third
The chromatic 5-note MOS also has the following intervals (from some root):
3	Dob, Solb	diminished fourth	-3	Do#, Sol#	augmented unison (chroma)
4	Mibb, Sibb	doubly diminished third	-4	Re#, La#	augmented second

Genchain

The generator chain for this scale is as follows:

Mibb Sibb	Dob Solb	Reb Lab	Mib Sib	Do Sol	Re La	Mi Si	Do# Sol#	Re# La#	Mi# Si#
dd3	d4	d2	d3	P1	P2	P3	A1	A2	A3

Modes

The mode names are based on the species of fourth:

Mode	Scale	UDP	Interval type
name	pattern	notation	2nd	3rd
Major	LLs	2|0	P	P
Minor	LsL	1|1	P	d
Phrygian	LsLL	0|2	d	d

Temperaments

The most basic rank-2 temperament interpretation of diatonic is Mahuric. The name "Mahuric" comes from the “Mahur” scale in Persian and Arabic music. The major triad is spelled root-2g-(p+g) (p = 4/3, g = the whole tone) and approximates 4:5:6 in pental interpretations or 14:18:21 in septimal ones. Basic ~5ed4/3 fits both interpretations.

Mahuric-Meantone

Subgroup: 4/3.5/4.3/2

Comma list: 81/80

POL2 generator: ~9/8 = 193.6725¢

Mapping: [⟨1 0 1], ⟨0 2 1]]

Optimal ET sequence: ~(5ed4/3, 8ed4/3, 13ed4/3)

Mahuric-Superpyth

Subgroup: 4/3.9/7.3/2

Comma list: 64/63

POL2 generator: ~8/7 = 216.7325¢

Mapping: [⟨1 0 1], ⟨0 2 1]]

Optimal ET sequence: ~(5ed4/3, 7ed4/3, 9ed4/3, 11ed4/3)

Scale tree

The spectrum looks like this:

Generator (bright)			Cents	L	s	L/s	Comments
1\3			171; 2.3	1	1	1.000	Equalised
6\17			180	6	5	1.200
	11\31		180; 1.216	11	9	1.222
5\14			181.81	5	4	1.250
	14\39		182; 1, 1.5	14	11	1.273
	9\25		183; 19.6	9	7	1.286
4\11			184; 1.625	4	3	1.333
	15\41		185; 1.763	15	11	1.364
	11\30		185, 1, 10.83	11	8	1.375
	7\19		186.6	7	5	1.400
	10\27		187.5	10	7	1.429
	13\35		187; 1, 19.75	13	9	1.444
	16\43		188; 4.25	16	11	1.4545
3\8			189; 2.1	3	2	1.500	Mahuric-Meantone starts here
	14\37		190.90	14	9	1.556
	11\29		191; 3, 2.3	11	7	1.571
	8\21		192	8	5	1.600
	5\13		193; 1, 1, 4.6	5	3	1.667
		12\31	194.594	12	7	1.714
	7\18		195; 2.86	7	4	1.750
	9\23		196.36	9	5	1.800
	11\28		197; 67	11	6	1.833
	13\33		197; 2.135	13	7	1.857
	15\38		197; 1, 2, 1, 1.54	15	8	1.875
	17\43		198; 17.16	17	9	1.889
	19\48		198: 3, 1, 28	19	10	1.900
	21\53		198; 2.3518	21	11	1.909
	23\58		198; 1, 3, 1.7	23	12	1.917
	25\63		198; 1, 2, 12.25	25	13	1.923
	27\68		198; 1, 3.405	27	14	1.929
	29\73		198; 1, 1.16	29	15	1.933
	31\78		198; 1, 12, 2.8	31	16	1.9375
	33\83		198; 1.005	33	17	1.941
	35\88		199; 19.18	35	18	1.944
2\5			200	2	1	2.000	Mahuric-Meantone ends, Mahuric-Pythagorean begins
	17\42		201.9801	17	8	2.125
	15\37		202; 4.045	15	7	2.143
	13\32		202; 1, 1, 2.06	13	6	2.167
	11\27		203; 13	11	5	2.200
	9\22		203; 1, 3.416	9	4	2.250
	7\17		204; 1. 7.2	7	3	2.333
		12\29	205; 1.4	12	5	2.400
	5\12		206; 1, 8.6	5	2	2.500	Mahuric-Neogothic heartland is from here…
		18\43	207; 1.4	18	7	2.571
		13\31	208	13	5	2.600
	8\19		208; 1.4375	8	3	2.667	…to here
	11\26		209; 1.90	11	4	2.750
	14\33		210	14	5	2.800
3\7			211; 1, 3.25	3	1	3.000	Mahuric-Pythagorean ends, Mahuric-Superpyth begins
	22\51		212; 1, 9.3	22	7	3.143
	19\44		213; 11.8	19	6	3.167
	16\37		213.3̄	16	5	3.200
	13\30		213; 1, 2.318	13	4	3.250
	10\23		214; 3.5	10	3	3.333
	7\16		215; 2.6	7	2	3.500
	11\25		216; 2.5416	11	3	3.667
	15\34		216; 1.1527	15	4	3.750
	19\43		217; 7	19	5	3.800
4\9			218.18	4	1	4.000
	13\29		219; 1, 2.55	13	3	4.333
	9\20		220; 2.45	9	2	4.500
	14\31		221; 19	14	3	4.667
5\11			222.2	5	1	5.000	Mahuric-Superpyth ends
	11\24		223; 1, 2.6875	11	2	5.500
	17\37		224; 5.72	17	3	5.667
6\13			225	6	1	6.000
1\3			240	1	0	→ inf	Paucitonic

See also

2L 1s (4/3-equivalent) - idealized tuning

4L 2s (7/4-equivalent) - Mixolydian Archytas temperament

4L 2s (39/22-equivalent) - Mixolydian Neogothic temperament

4L 2s (9/5-equivalent) - Mixolydian Meantone temperament

6L 3s (7/3-equivalent) - Mahuric-Archytas temperament

6L 3s (26/11-equivalent) - Mahuric-Neogothic temperament

6L 3s (12/5-equivalent) - Mahuric-Meantone temperament

8L 4s (28/9-equivalent) - Bijou Archytas temperament

8L 4s (22/7-equivalent) - Bijou Neogothic temperament

8L 4s (16/5-equivalent) - Bijou Meantone temperament

10L 5s (112/27-equivalent) - Hyperionic Archytas temperament

10L 5s (88/21-equivalent) - Hyperionic Neogothic temperament

10L 5s (30/7-equivalent) - Hyperionic Meantone temperament

12L 6s (11/2-equivalent) - Low undecimal Subsextal temperament

12L 6s (28/5-equivalent) - Low septimal Subsextal temperament

12L 6s (80/7-equivalent) - High septimal Subsextal temperament

12L 6s (64/11-equivalent) - High undecimal Subsextal temperament