3L 1s (3/2-equivalent)

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← 2L 1s⟨3/2⟩	3L 1s (3/2-equivalent)	4L 1s⟨3/2⟩ →
↙ 2L 2s⟨3/2⟩	↓ 3L 2s⟨3/2⟩	4L 2s⟨3/2⟩ ↘

┌╥╥╥┬┐
│║║║││
││││││
└┴┴┴┴┘

Scale structure

Step pattern

LLLs
sLLL

Equave

3/2 (702.0¢)

Period

3/2 (702.0¢)

Generator size^(edf)

Bright

1\4 to 1\3 (175.5¢ to 234.0¢)

Dark

2\3 to 3\4 (468.0¢ to 526.5¢)

Related MOS scales

4L 3s⟨3/2⟩, 3L 4s⟨3/2⟩

Neutralized

2L 2s⟨3/2⟩

2-Flought

7L 1s⟨3/2⟩, 3L 5s⟨3/2⟩

Equal tunings^(edf)

Equalized (L:s = 1:1)

1\4 (175.5¢)

Supersoft (L:s = 4:3)

4\15 (187.2¢)

3\11 (191.4¢)

5\18 (195.0¢)

2\7 (200.6¢)

5\17 (206.5¢)

3\10 (210.6¢)

Superhard (L:s = 4:1)

4\13 (216.0¢)

Collapsed (L:s = 1:0)

1\3 (234.0¢)

3L1s<3/2> is constructed by repeating the fifth-spanning pattern LLLs of the ordinary diatonic mos (5L 2s) at the equave of 3/2. The so-called "Super Ultra Hyper Mega Meta Lydian" is one mode of this mos.

The notation "<3/2>" means the period of the MOS is 3/2, disambiguating it from octave-repeating 3L 1s. The name of the period interval is called the sesquitave (by analogy to the tritave). 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 fifth complement (480 to 514.3 cents).

In the fifth-repeating version of the diatonic scale, each tone has a 3/2 perfect fifth above it. The scale has two major chords and two minor chords.

Angel is a proposed name for this mos. Basic Angel is in 7edf, which is a very good fifth-based equal tuning similar to 12edo.

Notation

There are 4 main ways to notate the angel scale. One method uses a simple sesquitave (fifth) repeating notation consisting of 4 naturals (eg. Do Re Mi Fa, Sol La Si Do). Given that 1-5/4-5/3 is fifth-equivalent to a tone cluster of 1-10/9-5/4, it may be more convenient to notate angel scales as repeating at the double, triple or quadruple sesquitave (major ninth, thirteenth or seventeenth i. e. ~pentave), however it does make navigating the genchain harder. This way, 5/3 is its own pitch class, distinct from 10/9. Notating this way produces a major ninth which is the Aeolian mode of Napoli[6L 2s], a major thirteenth which is the Dorian mode of Bijou[9L 3s] or an ~pentave which is the Mixolydian mode of Hextone[12L 4s]. Since there are exactly 8 naturals in double sesquitave notation, 12 in triple sesquitave notation and 16 in quadruple sesquitave notation, letters A-H (FGABHCDEF) or dozenal or hex digits (0123456789XE0 or D1234567FGACD with flats written C molle or 0123456789ABCDEF0 or G123456789ABCDEFG with flats written F molle) may be used.

Cents^[1]
Notation				Supersoft	Soft	Semisoft	Basic	Semihard	Hard	Superhard
Diatonic	Napoli	Bijou	Hextone	~15edf	~11edf	~18edf	~7edf	~17edf	~10edf	~13edf
Do#, Sol#	F#	0#, D#	0#, G#	1\15 46; 6.5	1\11 63: 6.3	2\18 77; 2, 2.6	1\7 100	3\17 124; 7.25	2\10 141; 5.6	3\13 163.63
Reb, Lab	Gb	1b, 1c	1f	3\15 138; 3.25	2\11 126; 3.16	3\18 116; 7.75	1\7 100	2\17 82; 1.318	1\10 70; 1.7	1\13 54.54
Re, La	G	1	1	4\15 184; 1.625	3\11 189; 2.1	5\18 193; 1, 1, 4.6	2\7 200	5\17 206; 1, 8.6	3\10 211; 1, 3.25	4\13 218.18
Re#, La#	G#	1#	1#	5\15 230; 1.3	4\11 252; 1.583	7\18 270; 1.03	3\7 300	8\17 331; 29	5\10 352; 1.0625	7\13 381.81
Mib, Sib	Ab	2b, 2c	2f	7\15 323; 13	5\11 315; 1.26	8\18 309; 1, 2.1	3\7 300	7\17 289; 1, 1.9	4\10 282; 2.83	5\13 272.72
Mi, Si	A	2	2	8\15 369; 4.3	6\11 378; 1.05	10\18 387; 10.3	4\7 400	10\17 413; 1, 3.83	6\10 423; 1.8	8\13 436.36
Mi#, Si#	A#	2#	2#	9\15 415; 2.6	7\11 442; 9.5	12\18 464; 1.0625	5\7 500	13\17 537; 14.5	8\10 564; 1.416	11\13 600
Fab, Dob	Bbb	3b, 3c	3f	10\15 461; 1, 1.16	7\11 442; 9.5	11\18 425; 1.24	4\7 400	9\17 372; 2.416	5\10 352; 1.0625	6\13 327.27
Fa, Do	Bb	3	3	11\15 507; 1.4	8\11 505; 3.8	13\18 503; 4, 2.3	5\7 500	12\17 496; 1.8125	7\10 494; 8.5	9\13 490.90
Fa#, Do#	B	3#	3#	12\15 553; 1.18	9\11 568; 2.375	15\18 580; 1.55	6\7 600	15\17 620; 1.45	9\10 635; 3.4	12\13 654.54
Fax, Dox	B#	3x	3x	13\15 600	10\11 631; 1.72	17\18 658; 15.5	7\7 700	18\17 744; 1.2083	11\10 776; 2.125	15\13 818.18
Dob, Solb	Hb	4b, 4c	4f	14\15 646; 6.5	10\11 631; 1.72	16\18 619; 2.81	6\7 600	14\17 579; 3.2	8\10 564; 1.416	10\13 545.45
Do, Sol	H	4	4	15\15 692; 3.25	11\11 694; 1, 2.8	18\18 696; 1.2916	7\7 700	17\17 703; 2, 2.16	10\10 705; 1.13	13\13 709.09
Do#, Sol#	Η#	4#	4#	16\15 738; 2.16	12\11 757; 1, 8.5	20\18 774; 5, 6	8\8 800	20\17 827; 1, 1.416	12\10 847; 17	16\13 872.72
Reb, Lab	Cb	5b, 5c	5	18\15 830; 1.3	13\11 821; 19	21\18 812; 1, 9.3	8\8 800	19\17 786; 4.83	11\10 776; 2.125	14\13 763.63
Re, La	C	5	5	19\15 876; 1.083	14\11 884; 4.75	23\18 890; 3.1	9\5 900	22\17 910; 2.9	13\10 917; 1.54	17\13 927.27
Re#, La#	C#	5#	5#	20\15 923: 13	15\11 947; 2, 1.4	25\18 967; 1, 2.875	10\7 1000	25\17 1034; 2, 14	15\10 1058; 1, 4.6	20\13 1090.90
Mib, Sib	Db	6b, 6c	6f	22\15 1015; 2.6	16\11 1010; 1.9	26\18 1006; 2, 4.6	10\7 1000	24\17 993; 9.6	14\10 988; 4.25	18\13 981.81
Mi, Si	D	6	6	23\15 1061; 1, 1.16	17\11 1073; 1, 2.16	28\18 1083; 1.148	11\7 1100	27\17 1117; 4, 7	16\10 1129; 2, 2.3	21\9 1145.45
Mi#, Si#	D#	6#	6#	24\15 1107; 1.4	18\11 1136; 1.1875	30\18 1161; 3.4	12\7 1200	30\17 1241; 2.63	18\10 1270; 1.7	24\13 1309.09
Fab, Dob	Ebb	7b, 7c	7f	25\15 1153; 1.18	18\11 1136; 1.1875	29\18 1121; 1, 1, 2.6	11\7 1100	26\17 1075; 1.16	15\10 1058; 1, 4.6	19\13 1036.36
Fa, Do	Eb	7	7	26\15 1200	19\11 1200	31\18 1200	12\7 1200	29\17 1200	17\10 1200	22\13 1200
Fa#, Do#	E	7#	7#	27\15 1246; 6.5	20\11 1263; 6.3	33\18 1277; 2, 2.6	13\7 1300	32\17 1324; 7.25	19\10 1341; 5.6	25\13 1363.63
Fax, Dox	E#	7x	7x	28\15 1292; 3.25	21\11 1326; 3.16	35\18 1354; 1, 5.2	14\7 1400	35\17 1448; 3.625	21\10 1482; 2.83	28\13 1527.27
Dob, Solb	Fb	8b, Fc	8f	29\15 1338; 2.16	21\11 1326; 3.16	34\18 1316; 7.75	13\7 1300	31\17 1282; 1.318	18\10 1270; 1.7	23\13 1254.54
Do, Sol	F	8, F	8	30\15 1384; 1.625	22\11 1389; 2.1	36\18 1393; 1, 1, 4.6	14\7 1400	34\17 1406; 1, 8.6	20\10 1411; 1, 3.25	26\13 1418.18
Do#, Sol#	F#	8#, F#	8#	31\15 1430; 1.3	23\11 1452; 1.583	38\18 1470; 1.03	15\7 1500	37\17 1531; 29	22\10 1552; 1.0625	29\13 1581.81
Reb, Lab	Gb	9b, Gc	9f	33\15 1523; 13	24\11 1515; 1.26	39\18 1509; 1, 2.1	15\7 1500	36\17 1489; 1, 1.9	21\10 1482; 2.83	27\13 1472.72
Re, La	G	9, G	9	34\15 1569; 4.3	25\11 1578; 1.05	41\18 1587; 10.3	16\7 1600	39\17 1613; 1, 3.83	23\10 1623; 1.8	30\13 1636.36
Re#, La#	G#	9#, G#	9#	35\15 1615; 2.6	26\11 1642; 9.5	43\18 1664; 1.0625	17\7 1700	42\17 1737; 14.5	25\10 1764; 1.416	33\13 1800
Mib, Sib	Ab	Xb, Ac	Af	37\15 1707; 1.4	27\11 1705; 3.8	44\18 1703; 4, 2.3	17\7 1700	41\17 1696; 1.8125	24\10 1694; 8.5	31\13 1690.90
Mi, Si	A	X, A	A	38\15 1753; 1.18	28\11 1768; 2.375	46\18 1780; 1.55	18\7 1800	44\17 1820; 1.45	26\10 1835; 3.4	34\13 1854.54
Mi#, Si#	A#	X#, A#	A#	39\15 1800	29\11 1831; 1.72	48\18 1858; 15.5	19\7 1900	47\17 1944; 1.2083	28\10 1976; 2.125	37\13 2018.18
Fab, Dob	Bbb	Ebb, Ccc	Bf	40\15 1846; 6.5	29\11 1831; 1.72	47\18 1819; 2.81	18\7 1800	43\17 1779; 3.2	25\10 1764; 1.416	32\13 1745.45
Fa, Do	Bb	Eb, Cc	B	41\15 1892; 3.25	30\11 1894; 1, 2.8	49\18 1896; 1.2916	19\7 1900	46\17 1903; 2.16	27\10 1905; 1.13	35\13 1909.09
Fa#, Do#	B	E, C	B#	42\15 1938; 2.16	31\11 1957; 1, 8.5	51\18 1974; 5.16	20\7 2000	49\17 2027; 1, 1.416	29\10 2047; 17	38\13 2072.72
Fax, Dox	B#	Ex, Cx	Bx	43\15 1984; 1.625	32\11 2021; 19	53\18 2051; 1, 1, 1, 1.4	21\7 2100	52\17 2151; 2.625	31\10 2188; 4.25	41\13 2236.36
Dob, Solb	Hb	0b, Dc	Cf	44\15 2030; 1.3	32\11 2021; 19	52\18 2012; 1, 9,3	20\7 2000	48\17 1986; 4.83	28\10 1976; 2.125	36\13 1963.63
Do, Sol	H	0, D	C	45\15 2076; 1.083	33\11 2084; 4.75	54\18 2090; 3.1	21\7 2100	51\17 2110; 2.9	30\10 2117; 1.54	39\13 2127.27
Do#, Sol#	Η#	0#, D#	C#	46\15 2123; 13	34\11 2147; 2, 1.4	56\18 2167; 1, 2.875	22\7 2200	54\17 2234; 2, 14	32\10 2258; 1, 4.6	42\13 2090.90
Reb, Lab	Cb	1b, 1c	Df	48\15 2215; 2.6	35\11 2210; 1.9	57\18 2206; 2, 4.6	22\7 2200	53\17 2193; 9.6	31\10 2188; 4.25	40\13 2181.81
Re, La	C	1	D	49\15 2261; 1, 1.16	36\11 2273; 1, 2.16	59\18 2283; 1.148	23\7 2300	56\17 2317; 4, 7	33\10 2329; 2, 2.3	43\13 2245.45
Re#, La#	C#	1#	D#	50\15 2307; 1.4	37\11 2336; 1.1875	61\18 2361; 3.4	24\7 2400	59\17 2441; 2.63	35\10 2470; 1.7	46\13 2509.09
Mib, Sib	Db	2b, 2c	Ef	52\15 2400	38\11 2400	62\18 2400	24\7 2400	58\17 2400	34\10 2400	44\13 2400
Mi, Si	D	2	E	53\15 2446; 6.5	39\11 2463; 6.3	64\18 2477; 2, 2.6	25\7 2500	61\17 2524; 7.25	36\10 2541; 5.6	47\13 2563.63
Mi#, Si#	D#	2#	E#	54\15 2492; 3.25	40\11 2526; 3.1	66\18 2554; 1, 5.2	26\7 2600	64\17 2648; 2.625	38\10 2682; 2.83	50\13 2727.27
Fab, Dob	Ebb	3b, 3c	Fff	55\15 2538; 2.16	40\11 2526; 3.1	65\18 2516; 7.75	25\7 2500	60\17 2482; 1.318	35\10 2470; 1.7	45\13 2454.54
Fa, Do	Eb	3	Ff	56\15 2584; 1.625	41\11 2589; 2.1	67\18 2593; 1, 1, 4.6	26\7 2600	63\17 2606; 1, 8.6	37\10 2611; 1, 3.25	48\13 2618.18
Fa#, Do#	E	3#	F	57\15 2630; 1.3	42\11 2652; 1.583	69\18 2670; 1.03	27\7 2700	66\17 2731; 29	39\10 2752; 1.0625	51\13 2781.81
Fax, Dox	E#	3x	F#	58\15 2676; 1.083	43\11 2715; 1.26	71\18 2748; 2.583	28\7 2800	69\17 2855; 4.8	41\10 2894; 8.5	54\13 2945.45
Dob, Solb	Fb	4b, 4c	0f, Gf	59\15 2723; 13	43\11 2715; 1.26	70\18 2709; 1, 2.1	27\7 2700	65\17 2689; 1, 1.9	38\10 2682; 2.83	49\13 2672.72
Do, Sol	F	4	0, G	60\15 2769; 4.3	44\11 2778; 1.05	72\18 2787; 3.1	28\7 2800	68\17 2813; 1, 3.83	40\10 2823; 1.8	52\13 2836.36

Relative cents^[2]
Notation				Supersoft	Soft	Semisoft	Basic	Semihard	Hard	Superhard
Diatonic	Napoli	Bijou	Hextone	~15edf	~11edf	~18edf	~7edf	~17edf	~10edf	~13edf
Do#, Sol#	F#	0#, D#	0#, G#	1\15 46.6	1\11 63.63	2\18 77.7̄	1\7 100	3\17 123; 1.8	2\10 140	3\13 161; 1, 1.16
Reb, Lab	Gb	1b, 1c	1f	3\15 140	2\11 127.27	3\18 116.6	1\7 100	2\17 82; 2.83	1\10 70	1\13 53; 1.18
Re, La	G	1	1	4\15 186.6	3\11 190.90	5\18 194.4	2\7 200	5\17 205; 1.13	3\10 210	4\13 215; 2.6
Re#, La#	G#	1#	1#	5\15 233.3	4\11 254.54	7\18 272.2̄	3\7 300	8\17 329; 2, 2.3	5\10 350	7\13 376; 1.083
Mib, Sib	Ab	2b, 2c	2f	7\15 326.6	5\11 318.18	8\18 311.1	3\7 300	7\17 288; 4.25	4\10 280	5\13 269; 4.3
Mi, Si	A	2	2	8\15 373.3	6\11 381.81	10\18 388.8	4\7 400	10\17 411; 1, 3.25	6\10 420	8\13 430; 1.3
Mi#, Si#	A#	2#	2#	9\15 420	7\11 445.45	12\18 466.6	5\7 500	13\17 535; 3.4	8\10 560	11\13 592; 3.25
Fab, Dob	Bbb	3b, 3c	3f	10\15 466.6	7\11 445.45	11\18 427.7	4\7 400	9\17 370; 1.7	5\10 350	6\13 323; 13
Fa, Do	Bb	3	3	11\15 513.3	8\11 509.09	13\18 505.5	5\7 500	12\17 494; 8.5	7\10 490	9\13 484; 1.625
Fa#, Do#	B	3#	3#	12\15 560	9\11 572.72	15\18 583.3	6\7 600	15\17 617; 1.416	9\10 630	12\13 646; 6.5
Fax, Dox	B#	3x	3x	13\15 606.6	10\11 636.36	17\18 661.1	7\7 700	18\17 741; 5.6	11\10 770	15\13 807; 1.4
Dob, Solb	Hb	4b, 4c	4f	14\15 653.3	10\11 636.36	16\18 622.2	6\7 600	14\17 576; 2.125	8\10 560	10\13 538; 2.16
Do, Sol	H	4	4	700
Do#, Sol#	Η#	4#	4#	16\15 746.6	12\11 763.63	20\18 777.7	8\7 800	20\17 823; 1.8	12\10 840	16\13 861; 1, 1.16
Reb, Lab	Cb	5b, 5c	5	18\15 840	13\11 827.27	21\18 816.6	8\7 800	19\17 782; 2.83	11\10 770	14\13 753; 1.18
Re, La	C	5	5	19\15 886.6	14\11 890.90	23\18 894.4	9\7 900	22\17 905; 1.13	13\10 910	17\13 915; 2.6
Re#, La#	C#	5#	5#	20\15 933.3	15\11 954.54	25\18 972.2	10\7 1000	25\17 1029; 2, 2.3	15\10 1050	20\13 1076; 1.083
Mib, Sib	Db	6b, 6c	6f	22\15 1026.6	16\11 1018.18	26\18 1011.1	10\7 1000	24\17 988; 4.25	14\10 980	18\13 969; 4.3
Mi, Si	D	6	6	23\15 1073.3	17\11 1081.81	28\18 1088.8	11\7 1100	27\17 1111; 1, 3.25	16\10 1120	21\13 1130; 1.3
Mi#, Si#	D#	6#	6#	24\15 1120	18\11 1145.45	30\18 1166.6	12\7 1200	30\17 1235; 3.4	18\10 1260	24\13 1292; 3.25
Fab, Dob	Ebb	7b, 7c	7f	25\15 1166.6	18\11 1145.45	29\18 1127.7	11\7 1100	26\17 1070; 1.7	15\10 1050	19\13 1023; 13
Fa, Do	Eb	7	7	26\15 1213.3	19\11 1209.09	31\18 1205.5	12\7 1200	29\17 1194; 8.5	17\10 1190	22\13 1184; 1.625
Fa#, Do#	E	7#	7#	27\15 1260	20\11 1272.72	33\18 1283.3	13\7 1300	32\17 1317; 1.416	19\10 1330	25\13 1346; 6.5
Fax, Dox	E#	7x	7x	28\15 1306.6	21\11 1336.36	35\18 1361.1	14\7 1400	35\17 1441; 5.6	21\10 1470	28\13 1507; 1.4
Dob, Solb	Fb	8b, Fc	8f	29\15 1333.3	21\11 1336.36	34\18 1322.2	13\7 1300	31\17 1276; 2.125	18\10 1260	23\13 1238; 2.16
Do, Sol	F	8, F	8	1400
Do#, Sol#	F#	8#, F#	8#	31\15 1446.6	23\11 1463.63	38\18 1477.7̄	15\7 1500	37\17 1523; 1.8	22\10 1540	29\13 1561; 1, 1.16
Reb, Lab	Gb	9b, Gc	9f	33\15 1540	24\11 1527.27	39\18 1516.6	15\7 1500	36\17 1482; 2.83	21\10 1470	27\13 1453; 1.18
Re, La	G	9, G	9	34\15 1586.6	25\11 1590.90	41\18 1594.4	16\7 1600	39\17 1605; 1.13	23\10 1610	30\13 1615; 2.6
Re#, La#	G#	9#, G#	9#	35\15 1633.3	26\11 1654.54	43\18 1672.2	17\7 1700	42\17 1729; 2, 2.3	25\10 1750	33\13 1776; 1.083
Mib, Sib	Ab	Xb, Ac	Af	37\15 1726.6	27\11 1718.18	44\18 1711.1	17\7 1700	41\17 1688; 4.25	24\10 1680	31\13 1669; 4.3
Mi, Si	A	X, A	A	38\15 1773.3	28\11 1781.81	46\18 1788.8	18\7 1800	44\17 1811; 1, 3.25	26\10 1820	34\13 1830; 1.3
Mi#, Si#	A#	X#, A#	A#	39\15 1820	29\11 1845.45	48\18 1866.6	19\7 1900	47\17 1935; 3.4	28\10 1960	37\13 1992; 3.25
Fab, Dob	Bbb	Ebb, Ccc	Bf	40\15 1866.6	29\11 1845.45	47\18 1827.7	18\7 1800	43\17 1770; 1.7	25\10 1750	32\13 1723; 13
Fa, Do	Bb	Eb, Cc	B	41\15 1913.3	30\11 1909.09	49\18 1905.5	19\7 1900	46\17 1894; 8.5	27\10 1890	35\13 1884; 1.625
Fa#, Do#	B	E, C	B#	42\15 1960	31\11 1972.72	51\18 1983.3	20\7 2000	49\17 2017; 1.416	29\10 2030	38\13 2046; 6.5
Fax, Dox	B#	Ex, Cx	Bx	43\15 2006.6	32\11 2036.36	53\18 2061.1	21\7 2100	52\17 2141; 5.6	31\10 2170	41\13 2207; 1.4
Dob, Solb	Hb	0b, Dc	Cf	44\15 2053.3	32\11 2036.36	52\18 2022.2	20\7 2000	48\17 1976; 2.125	28\10 1960	36\13 1938; 2.16
Do, Sol	H	0, D	C	2100
Do#, Sol#	Η#	0#, D#	C#	46\15 2146.6	34\11 2163.63	56\18 2177.7	22\7 2200	54\17 2223; 1.8	32\10 2240	42\13 2261; 1, 1.16
Reb, Lab	Cb	1b, 1c	Df	48\15 2240	35\11 2227.27	57\18 2216.6	22\7 2200	53\17 2182; 2.83	31\10 2170	40\13 2153; 1.18
Re, La	C	1	D	49\15 2286.6	36\11 2290.90	59\18 2294.4	23\7 2300	56\17 2305; 1.13	33\10 2310	43\13 2315; 2.6
Re#, La#	C#	1#	D#	50\15 2223.3	37\11 2354.54	61\18 2372.2	24\7 2400	59\17 2429; 2, 2.3	35\10 2450	46\13 2476; 1.083
Mib, Sib	Db	2b, 2c	Ef	52\15 2426.6	38\11 2418.18	62\18 2411.1	24\7 2400	58\17 2388; 4.25	34\10 2380	44\13 2369; 4.3
Mi, Si	D	2	E	53\15 2473,3	39\11 2481.81	64\11 2488.8	25\7 2500	61\17 2511; 1, 3.25	36\10 2520	47\13 2530; 1.3
Mi#, Si#	D#	2#	E#	54\15 2520	40\11 2545.45	66\18 2566.6	26\7 2600	64\17 2635; 3.4	38\10 2660	50\13 2692; 3.25
Fab, Dob	Ebb	3b, 3c	Fff	55\15 2566.6	40\11 2545.45	65\18 2527.7	25\7 2500	60\17 2470; 1.7	35\10 2450	45\13 2423; 13
Fa, Do	Eb	3	Ff	56\15 2613.3	41\11 2609.09	67\18 2605.5	26\7 2600	63\17 2594; 8.5	37\10 2590	48\13 2584; 1.625
Fa#, Do#	E	3#	F	57\15 2660	42\11 2672.72	69\18 2683.3	27\7 2700	66\17 2717; 1.416	39\10 2730	51\13 2746; 6.5
Fax, Dox	E#	3x	F#	58\15 2706.6	43\11 2736.36	71\18 2761.1	28\7 2800	69\17 2841; 5.6	41\10 2870	54\13 2907; 1.4
Dob, Solb	Fb	4b, 4c	0f, Gf	59\15 2753.3	43\11 2736.36	70\18 2722.2	27\7 2700	65\17 2676; 2.125	38\10 2660	49\13 2638; 2.16
Do, Sol	F	4	0, G	2800

Intervals

Generators	Sesquitave notation	Interval category name	Generators	Notation of 3/2 inverse	Interval category name
The 4-note MOS has the following intervals (from some root):
0	Do, Sol	perfect unison	0	Do, Sol	sesquitave (just fifth)
1	Fa, Do	perfect fourth	-1	Re, La	perfect second
2	Mib, Sib	minor third	-2	Mi, Si	major third
3	Reb, Lab	diminished second	-3	Fa#, Do#	augmented fourth
The chromatic 7-note MOS also has the following intervals (from some root):
4	Dob, Solb	diminished sesquitave	-4	Do#, Sol#	augmented unison (chroma)
5	Fab, Dob	diminished fourth	-5	Re#, La#	augmented second
6	Mibb, Sibb	diminished third	-6	Mi#, Si#	augmented third

Genchain

The generator chain for this scale is as follows:

Mibb Sibb	Fab Dob	Dob Solb	Reb Lab	Mib Sib	Fa Do	Do Sol	Re La	Mi Si	Fa# Do#	Do# Sol#	Re# La#	Mi# Si#
d3	d4	d5	d2	m3	P4	P1	P2	M3	A4	A1	A2	A3

Modes

The mode names are based on the species of fifth:

Mode	Scale	UDP	Interval type
name	pattern	notation	2nd	3rd	4th
Lydian	LLLs	3\|0	P	M	A
Major	LLsL	2\|1	P	M	P
Minor	LLsL	1\|2	P	m	P
Phrygian	sLLL	0\|3	d	m	P

Temperaments

The most basic rank-2 temperament interpretation of angel is Napoli. The name "Napoli" comes from the “Neapolitan” sixth triad spelled root-(p-2g)-(2p-3g) (p = 3/2, g = the whole tone) which serves as its minor triad approximating 5:6:8 in pental interpretations or 18:21:28 in septimal ones. Basic ~7edf fits both interpretations.

Napoli-Meantone

Subgroup: 3/2.6/5.8/5

Comma list: 81/80

POL2 generator: ~9/8 = 192.6406

Mapping: [⟨1 1 2], ⟨0 -2 -3]]

Optimal ET sequence: ~(7edf, 11edf, 18edf)

Napoli-Archy

Subgroup: 3/2.7/6.14/9

Comma list: 64/63

POL2 generator: ~8/7 = 218.6371

Mapping: [⟨1 1 2], ⟨0 -2 -3]]

Optimal ET sequence: ~(7edf, 10edf, 13edf, 16edf)

Scale tree

The spectrum looks like this:

Scale Tree and Tuning Spectrum of 3L 1s⟨3/2⟩
Generator^(edf)						Cents		Step ratio		Comments^{(always proper)}
Generator^(edf)						Bright	Dark	L:s	Hardness	Comments^{(always proper)}
1\4						175.489	526.466	1:1	1.000	Equalized 3L 1s⟨3/2⟩
					6\23	183.119	518.836	6:5	1.200
				5\19		184.725	517.230	5:4	1.250
					9\34	185.812	516.143	9:7	1.286
			4\15			187.188	514.767	4:3	1.333	Supersoft 3L 1s⟨3/2⟩
					11\41	188.329	513.626	11:8	1.375
				7\26		188.988	512.967	7:5	1.400
					10\37	189.718	512.237	10:7	1.429
		3\11				191.442	510.513	3:2	1.500	Soft 3L 1s⟨3/2⟩ Napoli-Meantone starts here
					11\40	193.038	508.917	11:7	1.571
				8\29		193.643	508.312	8:5	1.600
					13\47	194.158	507.797	13:8	1.625
			5\18			194.988	506.968	5:3	1.667	Semisoft 3L 1s⟨3/2⟩
					12\43	195.894	506.061	12:7	1.714
				7\25		196.547	505.408	7:4	1.750
					9\32	197.425	504.530	9:5	1.800
	2\7					200.559	501.396	2:1	2.000	Basic 3L 1s⟨3/2⟩ Napoli-Meantone ends, Napoli-Pythagorean begins
					9\31	203.793	498.162	9:4	2.250
				7\24		204.737	497.218	7:3	2.333
					12\41	205.450	496.505	12:5	2.400
			5\17			206.457	495.498	5:2	2.500	Semihard 3L 1s⟨3/2⟩ Napoli-Neogothic heartland is from here...
					13\44	207.396	494.559	13:5	2.600
				8\27		207.987	493.968	8:3	2.667	...to here
					11\37	208.689	493.266	11:4	2.750
		3\10				210.587	491.369	3:1	3.000	Hard 3L 1s⟨3/2⟩ Napoli-Pythagorean ends, Napoli-Archy begins
					10\33	212.714	489.241	10:3	3.333
				7\23		213.638	488.317	7:2	3.500
					11\36	214.486	487.469	11:3	3.667
			4\13			215.986	485.969	4:1	4.000	Superhard 3L 1s⟨3/2⟩
					9\29	217.848	484.107	9:2	4.500
				5\16		219.361	482.594	5:1	5.000	Napoli-Archy ends
					6\19	221.670	480.285	6:1	6.000
1\3						233.985	467.970	1:0	→ ∞	Collapsed 3L 1s⟨3/2⟩

↑ Fractions repeating more than 4 digits written as continued fractions
↑ Fractions repeating more than 4 digits written as continued fractions

[:0-1] Fractions repeating more than 4 digits written as continued fractions

[:02-2] Fractions repeating more than 4 digits written as continued fractions

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