User:Moremajorthanmajor/3L 1s (perfect fifth-equivalent)

3L 1s<perfect fifth> 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 6 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, Fa Sol La Si, 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, quadruple, quintuple or sextuple sesquitave (major ninth, thirteenth, seventeenth i. e. ~pentave or twenty-first or augmented twenty-fifth), 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], an ~pentave which is the Mixolydian mode of Hextone[12L 4s], a major twenty-first which is the Ionian mode of Guidotonic[15L 5s] or an augmented twenty-fifth which is the Lydian mode of Subdozenal[18L 6s]. Since there are exactly 8 naturals in double sesquitave notation, 12 in triple sesquitave notation, 16 in quadruple sesquitave notation, 20 in quintuple sesquitave notation and 24 in sextuple sesquitave notation, letters A-H (FGABHCDEF), dozenal or hex digits (0123456789XE0 or D1234567FGACD with flats written C molle or 0123456789ABCDEF0 or G123456789ABCDEFG with flats written F molle), the Guidonian names with F as the lowest ut or letters except I and O may be used.

Cents

Notation		Supersoft	Soft	Semisoft	Basic	Semihard	Hard	Superhard
Diatonic	Napoli	~15edf	~11edf	~18edf	~7edf	~17edf	~10edf	~13edf
Do#, Fa#, Sol#	F#	1\15, 46.154	1\11, 63.158	2\18, 77.419	1\7, 100	3\17, 124.138	2\10, 141.176	3\13, 163.636
Reb, Solb, Lab	Gb, Ge	3\15, 138.462	2\11. 126.316	3\18, 116.129		2\17, 82.759	1\10, 70.588	1\13, 54.545
Re, Sol, La	G	4\15, 184.615	3\11, 189.474	5\18, 193.548	2\7, 200	5\17, 206.897	3\10, 211.765	4\13, 218.182
Re#, Sol#, La#	G#	5\15, 230.769	4\11, 252.632	7\18, 270.968	3\7, 300	8\17, 331.034	5\10, 352.941	7\13, 381.818
Mib, Lab, Sib	Ab, Æ	7\15, 323.077	5\11, 315.789	8\18, 309.677		7\17, 289.655	4\10, 282.353	5\13, 272.727
Mi, La, Si	A	8\15, 369.231	6\11, 378.947	10\18, 387.097	4\7, 400	10\17, 413.793	6\10, 423.529	8\13, 436.364
Mi#, La#, Si#	A#	9\15, 415.385	7\11, 442.105	12\18, 464.516	5\7, 500	13\17, 537.069	8\10, 564.706	11\13, 600
Fab, Sibb, Dob	Bbb, Bee	10\15, 461.538		11\18, 425.806	4\7, 400	9\17, 372.414	5\10, 352.941	6\13, 327.273
Fa, Sib, Do	Bb, Be	11\15, 507.692	8\11, 505.263	13\18, 503.226	5\7, 500	12\17, 496.552	7\10, 494.118	9\13, 490.909
Fa#, Si, Do#	B	12\15, 553.846	9\11, 568.421	15\18, 580.645	6\7, 600	15\17, 620.690	9\10, 635.294	12\13, 654.545
Fax, Si#, Dox	B#	13\15, 600	10\11, 631.579	17\18, 658.064	7\7, 700	18\17, 744.828	11\10, 776.471	15\13, 818.182
Dob, Fab, Solb	Hb, He	14\15, 646.154		16\18, 619.355	6\7, 600	14\17, 579.310	8\10, 564.706	10\13, 545.455
Do, Fa, Sol	H	15\15, 692.308	11\11, 694.737	18\18, 696.774	7\7, 700	17\17, 703.448	10\10, 705.882	13\13, 709.091
Do#, Fa#, Sol#	Η#	16\15, 738.462	12\11, 757.895	20\18, 774.194	8\8, 800	20\17, 827.586	12\10, 847.059	16\13, 872.727
Reb, Solb, Lab	Cb, Ce	18\15, 830.769	13\11, 821.053	21\18, 812.903		19\17, 786.207	11\10, 776.471	14\13, 763.63
Re, Sol, La	C	19\15, 876.923	14\11, 884.211	23\18, 890.323	9\5, 900	22\17, 910.345	13\10, 917.647	17\13, 927.273
Re#, Sol#, La#	C#	20\15, 923.077	15\11, 947.368	25\18, 967.742	10\7, 1000	25\17, 1034.483	15\10, 1058.824	20\13, 1090.909
Mib, Lab, Sib	Db, De	22\15, 1015.385	16\11, 1010.526	26\18, 1006.452		24\17, 993.103	14\10, 988.235	18\13, 981.818
Mi, La, Si	D	23\15, 1061.538	17\11, 1073.684	28\18, 1083.871	11\7, 1100	27\17, 1117.241	16\10, 1129.412	21\9, 1145.455
Mi#, La#, Si#	D#	24\15, 1107.923	18\11, 1136.842	30\18, 1161.29	12\7, 1200	30\17, 1241.379	18\10, 1270.588	24\13, 1309.091
Fab, Sibb, Dob	Ebb, Eee	25\15, 1153.846		29\18, 1122.581	11\7, 1100	26\17, 1075.862	15\10, 1058.824	19\13, 1036.364
Fa, Sib, Do	Eb, Ee	26\15, 1200	19\11, 1200	31\18, 1200	12\7, 1200	29\17, 1200	17\10, 1200	22\13, 1200
Fa#, Si, Do#	E	27\15, 1246.154	20\11, 1263.158	33\18, 1277.419	13\7, 1300	32\17, 1324.138	19\10, 1341.176	25\13, 1363.636
Fax, Si#, Dox	E#	28\15, 1292.308	21\11, 1326.318	35\18, 1354.834	14\7, 1400	35\17, 1448.275	21\10, 1482.353	28\13, 1527.273
Dob, Fab, Solb	Fb, Fe	29\15, 1338.462		34\18, 1316.129	13\7, 1300	31\17, 1282.759	18\10, 1270.588	23\13, 1254.545
Do, Fa, Sol	F	30\15, 1384.615	22\11, 1389.473	36\18, 1393.548	14\7, 1400	34\17, 1406.897	20\10, 1411.765	26\13, 1418.182

Cents

Notation		Supersoft	Soft	Semisoft	Basic	Semihard	Hard	Superhard
Bijou	Hextone	~15edf	~11edf	~18edf	~7edf	~17edf	~10edf	~13edf
0#, D#	0#, G#	1\15, 46.154	1\11, 63.158	2\18, 77.419	1\7, 100	3\17, 124.138	2\10, 141.176	3\13, 163.636
1b, 1c	1f	3\15, 138.462	2\11. 126.316	3\18, 116.129		2\17, 82.759	1\10, 70.588	1\13, 54.545
1	1	4\15, 184.615	3\11, 189.474	5\18, 193.548	2\7, 200	5\17, 206.897	3\10, 211.765	4\13, 218.182
1#	1#	5\15, 230.769	4\11, 252.632	7\18, 270.968	3\7, 300	8\17, 331.034	5\10, 352.941	7\13, 381.818
2b, 2c	2f	7\15, 323.077	5\11, 315.789	8\18, 309.677		7\17, 289.655	4\10, 282.353	5\13, 272.727
2	2	8\15, 369.231	6\11, 378.947	10\18, 387.097	4\7, 400	10\17, 413.793	6\10, 423.529	8\13, 436.364
2#	2#	9\15, 415.385	7\11, 442.105	12\18, 464.516	5\7, 500	13\17, 537.069	8\10, 564.706	11\13, 600
3b, 3c	3f	10\15, 461.538		11\18, 425.806	4\7, 400	9\17, 372.414	5\10, 352.941	6\13, 327.273
3	3	11\15, 507.692	8\11, 505.263	13\18, 503.226	5\7, 500	12\17, 496.552	7\10, 494.118	9\13, 490.909
3#	3#	12\15, 553.846	9\11, 568.421	15\18, 580.645	6\7, 600	15\17, 620.690	9\10, 635.294	12\13, 654.545
3x	3x	13\15, 600	10\11, 631.579	17\18, 658.064	7\7, 700	18\17, 744.828	11\10, 776.471	15\13, 818.182
4b, 4c	4f	14\15, 646.154		16\18, 619.355	6\7, 600	14\17, 579.310	8\10, 564.706	10\13, 545.455
4	4	15\15, 692.308	11\11, 694.737	18\18, 696.774	7\7, 700	17\17, 703.448	10\10, 705.882	13\13, 709.091
4#	4#	16\15, 738.462	12\11, 757.895	20\18, 774.194	8\8, 800	20\17, 827.586	12\10, 847.059	16\13, 872.727
5b, 5c	5	18\15, 830.769	13\11, 821.053	21\18, 812.903		19\17, 786.207	11\10, 776.471	14\13, 763.63
5	5	19\15, 876.923	14\11, 884.211	23\18, 890.323	9\5, 900	22\17, 910.345	13\10, 917.647	17\13, 927.273
5#	5#	20\15, 923.077	15\11, 947.368	25\18, 967.742	10\7, 1000	25\17, 1034.483	15\10, 1058.824	20\13, 1090.909
6b, 6c	6f	22\15, 1015.385	16\11, 1010.526	26\18, 1006.452		24\17, 993.103	14\10, 988.235	18\13, 981.818
6	6	23\15, 1061.538	17\11, 1073.684	28\18, 1083.871	11\7, 1100	27\17, 1117.241	16\10, 1129.412	21\9, 1145.455
6#	6#	24\15, 1107.923	18\11, 1136.842	30\18, 1161.290	12\7, 1200	30\17, 1241.379	18\10, 1270.588	24\13, 1309.091
7b, 7c	7f	25\15, 1153.846		29\18, 1122.581	11\7, 1100	26\17, 1075.862	15\10, 1058.824	19\13, 1036.364
7	7	26\15, 1200	19\11, 1200	31\18, 1200	12\7, 1200	29\17, 1200	17\10, 1200	22\13, 1200
7#	7#	27\15, 1246.154	20\11, 1263.158	33\18, 1277.419	13\7, 1300	32\17, 1324.138	19\10, 1341.176	25\13, 1363.636
7x	7x	28\15, 1292.308	21\11, 1326.318	35\18, 1354.834	14\7, 1400	35\17, 1448.275	21\10, 1482.353	28\13, 1527.273
8b, Fc	8f	29\15, 1338.462		34\18, 1316.129	13\7, 1300	31\17, 1282.759	18\10, 1270.588	23\13, 1254.545
8, F	8	30\15, 1384.615	22\11, 1389.473	36\18, 1393.548	14\7, 1400	34\17, 1406.897	20\10, 1411.765	26\13, 1418.182
8#, F#	8#	31\15, 1430.769	23\11, 1452.632	38\18, 1470.968	15\7, 1500	37\17, 1531.034	22\10, 1552.941	29\13, 1581.818
9b, Gc	9f	33\15, 1523.077	24\11, 1515.789	39\18, 1509.677		36\17, 1489.655	21\10, 1482.759	27\13, 1472.273
9, G	9	34\15, 1569.231	25\11, 1578.947	41\18, 1587.097	16\7, 1600	39\17, 1613.793	23\10, 1623.529	30\13, 1636.364
9#, G#	9#	35\15, 1615.385	26\11, 1642.105	43\18, 1664.516	17\7, 1700	42\17, 1737.069	25\10, 1764.706	33\13, 1800
Xb, Ac	Af	37\15, 1707.692	27\11, 1705.263	44\18, 1703.226		41\17, 1696.552	24\10, 1694.118	31\13, 1690.909
X, A	A	38\15, 1753.846	28\11, 1768.421	46\18, 1780.645	18\7, 1800	44\17, 1820.690	26\10, 1835.294	34\13, 1854.545
X#, A#	A#	39\15, 1800	29\11, 1831.579	48\18, 1858.064	19\7, 1900	47\17, 1944.828	28\10, 1976.471	37\13, 2018.182
Ebb, Ccc	Ax	40\15, 1846.154		47\18, 1819.355	18\7, 1800	43\17, 1779.310	25\10, 1764.706	32\13, 1745.545
Eb, Cc	Bf	41\15, 1892.308	30\11, 1894.737	49\18, 1896.774	19\7, 1900	46\17, 1903.448	27\10, 1905.882	35\13, 1909.091
E, C	B	42\15, 1938.462	31\11, 1957.895	51\18, 1974.194	20\7, 2000	49\17, 2027.586	29\10, 2047.059	38\13, 2072.727
Ex, Cx	B#	43\15, 1984.615	32\11, 2021.053	53\18, 2051.612	21\7, 2100	52\17, 2151.725	31\10, 2188.235	41\13, 2236.364
0b, Dc	Cf	44\15, 2030.769		52\18, 2012.903	20\7, 2000	48\17, 1986.207	28\10, 1976.471	36\13, 1963.636
0, D	C	45\15, 2076.923	33\11, 2084.211	54\18, 2090.323	21\7, 2100	51\17, 2110.345	30\10, 2117.647	39\13, 2127.273
0#, D#	C#	46\15, 2123.077	34\11, 2147.368	56\15, 2167.742	22\7, 2200	54\17, 2234.483	32\10, 2258.824	42\13, 2090.909
1b, 1c	Df	48\15, 2215.385	35\11, 2210.526	57\15, 2206.452		53\17, 2193.103	31\10, 2188.235	40\13, 2181.818
1	D	49\15, 2261.538	36\11, 1073.684	59\18, 2283.871	23\7, 2300	56\17, 2317.241	33\10, 2329.412	43\13, 2345.455
1#	D#	50\15, 2307.692	37\11, 2336.842	61\18, 2361.290	24\7, 2400	59\17, 2441.379	35\10, 2470.588	46\13, 2509.091
2b, 2c	Ef	52\15, 2400	38\11, 2400	62\18, 2400		58\17, 2400	34\10, 2400	44\13, 2400
2	E	53\15, 2446.154	39\11, 2463.158	64\18, 2477,419	25\7, 2500	61\17, 2524.138	36\10, 2541.176	47\13, 2563.636
2#	E#	54\15, 2492.308	40\11, 2526.316	66\18, 2554.838	26\7, 2600	64\17, 2648.275	38\10, 2682.353	50\13, 2727.273
3b, 3c	Fff	55\15, 2538.462		65\18, 2516.129	25\7, 2500	60\17, 2482.759	35\10, 2470.588	45\13, 2454.545
3	Ff	56\15, 2584.615	41\11, 2589.474	67\18, 2593.548	26\7, 2600	63\17, 2606.897	37\10, 2611.765	48\13, 2618.182
3#	F	57\15, 2630.769	42\11, 2652.632	69\18, 2670.968	27\7, 2700	66\17, 2731.034	39\10, 2752.941	51\13, 2781.818
3x	F#	58\15, 2676.923	43\11, 2715.789	71\18, 2748.387	28\7, 2800	69\17, 2855.172	41\10, 2894.118	54\13, 2945.455
4bb, 4cc	0ff, Gff		42\11, 2652.632	68\18, 2632.258	26\7, 2600	62\17, 2565.517	36\10, 2541.176	46\13, 2509.091
4b, 4c	0f, Gf	59\15, 2723.077	43\11, 2715.789	70\18, 2709.677	27\7, 2700	65\17, 2689.552	38\10, 2682.353	49\13, 2672.273
4	0, G	60\15, 2769.231	44\11, 2778.947	72\18, 2787.097	28\7, 2800	68\17, 2813.793	40\10, 2823.529	52\13, 2836.364

Cents

Notation		Supersoft	Soft	Semisoft	Basic	Semihard	Hard	Superhard
Guidotonic	Subdozenal	~15edf	~11edf	~18edf	~7edf	~17edf	~10edf	~13edf
F ut#	F#	1\15, 46.154	1\11, 63.158	2\18, 77.419	1\7, 100	3\17, 124.138	2\10, 141.176	3\13, 163.636
G reb	Gb, Ge	3\15, 138.462	2\11. 126.316	3\18, 116.129		2\17, 82.759	1\10, 70.588	1\13, 54.545
G re	G	4\15, 184.615	3\11, 189.474	5\18, 193.548	2\7, 200	5\17, 206.897	3\10, 211.765	4\13, 218.182
G re#	G#	5\15, 230.769	4\11, 252.632	7\18, 270.968	3\7, 300	8\17, 331.034	5\10, 352.941	7\13, 381.818
A mib	Hb, He	7\15, 323.077	5\11, 315.789	8\18, 309.677		7\17, 289.655	4\10, 282.353	5\13, 272.727
A mi	H	8\15, 369.231	6\11, 378.947	10\18, 387.097	4\7, 400	10\17, 413.793	6\10, 423.529	8\13, 436.364
A mi#	H#	9\15, 415.385	7\11, 442.105	12\18, 464.516	5\7, 500	13\17, 537.069	8\10, 564.706	11\13, 600
B fa utb	Jbb, Jee	10\15, 461.538		11\18, 425.806	4\7, 400	9\17, 372.414	5\10, 352.941	6\13, 327.273
B fa ut	Jb, Je	11\15, 507.692	8\11, 505.263	13\18, 503.226	5\7, 500	12\17, 496.552	7\10, 494.118	9\13, 490.909
B fa ut#	J	12\15, 553.846	9\11, 568.421	15\18, 580.645	6\7, 600	15\17, 620.690	9\10, 635.294	12\13, 654.545
B fa utx	J#	13\15, 600	10\11, 631.579	17\18, 658.064	7\7, 700	18\17, 744.828	11\10, 776.471	15\13, 818.182
C sol re utb	Kb, Ke	14\15, 646.154		16\18, 619.355	6\7, 600	14\17, 579.310	8\10, 564.706	10\13, 545.455
C sol re ut	K	15\15, 692.308	11\11, 694.737	18\18, 696.774	7\7, 700	17\17, 703.448	10\10, 705.882	13\13, 709.091
C sol re ut#	K#	16\15, 738.462	12\11, 757.895	20\18, 774.194	8\8, 800	20\17, 827.586	12\10, 847.059	16\13, 872.727
D la mi reb	Lb, Le	18\15, 830.769	13\11, 821.053	21\18, 812.903		19\17, 786.207	11\10, 776.471	14\13, 763.63
D la mi re	L	19\15, 876.923	14\11, 884.211	23\18, 890.323	9\5, 900	22\17, 910.345	13\10, 917.647	17\13, 927.273
D la mi re#	L#	20\15, 923.077	15\11, 947.368	25\18, 967.742	10\7, 1000	25\17, 1034.483	15\10, 1058.824	20\13, 1090.909
E fab	Mbb, Mee	21\15, 969.231		24\18, 929.032	9\5, 900	21\17, 868.966	12\10, 847.059	15\13, 818.182
E fa	Mb, Me	22\15, 1015.385	16\11, 1010.526	26\18, 1006.452	10\7, 1000	24\17, 993.103	14\10, 988.235	18\13, 981.818
E sí mi	M	23\15, 1061.538	17\11, 1073.684	28\18, 1083.871	11\7, 1100	27\17, 1117.241	16\10, 1129.412	21\9, 1145.455
E si mi#	M#	24\15, 1107.923	18\11, 1136.842	30\18, 1161.29	12\7, 1200	30\17, 1241.379	18\10, 1270.588	24\13, 1309.091
F sol fa utb	Nbb, Nee	25\15, 1153.846		29\18, 1122.581	11\7, 1100	26\17, 1075.862	15\10, 1058.824	19\13, 1036.364
F sol fa ut	Nb, Ne	26\15, 1200	19\11, 1200	31\18, 1200	12\7, 1200	29\17, 1200	17\10, 1200	22\13, 1200
F sol fa ut#	N	27\15, 1246.154	20\11, 1263.158	33\18, 1277.419	13\7, 1300	32\17, 1324.138	19\10, 1341.176	25\13, 1363.636
F sol fa utx	N#	28\15, 1292.308	21\11, 1326.318	35\18, 1354.834	14\7, 1400	35\17, 1448.275	21\10, 1482.353	28\13, 1527.273
G la sol reb	Pb, Pe	29\15, 1338.462		34\18, 1316.129	13\7, 1300	31\17, 1282.759	18\10, 1270.588	23\13, 1254.545
G la sol re	P	30\15, 1384.615	22\11, 1389.473	36\18, 1393.548	14\7, 1400	34\17, 1406.897	20\10, 1411.765	26\13, 1418.182
G la sol re#	P#	31\15, 1430.769	23\11, 1452.632	38\18, 1470.968	15\7, 1500	37\17, 1531.034	22\10, 1552.941	29\13, 1581.818
A si la mib	Qb, Qe	33\15, 1523.077	24\11, 1515.789	39\18, 1509.677		36\17, 1489.655	21\10, 1482.759	27\13, 1472.273
A si la mi	Q	34\15, 1569.231	25\11, 1578.947	41\18, 1587.097	16\7, 1600	39\17, 1613.793	23\10, 1623.529	30\13, 1636.364
A si la mi#	Q#	35\15, 1615.385	26\11, 1642.105	43\18, 1664.516	17\7, 1700	42\17, 1737.069	25\10, 1764.706	33\13, 1800
B fa utb	Rb, Re	36\61, 1661.538		42\18, 1625.806	16\7, 1600	38\29, 1572.414	22\10, 1552.941	28\13, 1527.273
B fa ut	R	37\15, 1707.692	27\11, 1705.263	44\18, 1703.226	17\7, 1700	41\17, 1696.552	24\10, 1694.118	31\13, 1690.909
B si	R#	38\15, 1753.846	28\11, 1768.421	46\18, 1780.645	18\7, 1800	44\17, 1820.690	26\10, 1835.294	34\13, 1854.545
B si	R#	39\15, 1800	29\11, 1831.579	48\18, 1858.064	19\7, 1900	47\17, 1944.828	28\10, 1976.471	37\13, 2018.182
C sol re utb	Sb, Se	40\15, 1846.154		47\18, 1819.355	18\7, 1800	43\17, 1779.310	25\10, 1764.706	32\13, 1745.545
C sol re ut	S	41\15, 1892.308	30\11, 1894.737	49\18, 1896.774	19\7, 1900	46\17, 1903.448	27\10, 1905.882	35\13, 1909.091
C sol re ut#	S#	42\15, 1938.462	31\11, 1957.895	51\18, 1974.194	20\7, 2000	49\17, 2027.586	29\10, 2047.059	38\13, 2072.727
C sol re utx	Sx	43\15, 1984.615	32\11, 2021.053	53\18, 2051.612	21\7, 2100	52\17, 2151.725	31\10, 2188.235	41\13, 2236.364
D la mi reb	Tb, Te	44\15, 2030.769		52\18, 2012.903	20\7, 2000	48\17, 1986.207	28\10, 1976.471	36\13, 1963.636
D la mi re	T	45\15, 2076.923	33\11, 2084.211	54\18, 2090.323	21\7, 2100	51\17, 2110.345	30\10, 2117.647	39\13, 2127.273
D la mi re#	T#	46\15, 2123.077	34\11, 2147.368	56\18, 2167.742	22\7, 2200	54\17, 2234.483	32\10, 2258.824	42\13, 2090.909
E fab	Ub, Üe	47\26, 2169.231		55\16, 2129.032	21\7, 2100	50\17, 2068.966	29\10, 2047.059	37\13, 2018.182
E fa	Ub, Ü	48\15, 2215.385	35\11, 2210.526	57\18, 2206.452	23\7, 2300	53\17, 2193.103	31\10, 2188.235	40\13, 2181.818
E si mi	U	49\15, 2261.538	36\11, 1073.684	59\18, 2283.871	24\7, 2400	56\17, 2317.241	33\10, 2329.412	43\13, 2345.455
E si mi#	U#	50\15, 2307.692	37\11, 2336.842	61\18, 2361.290	23\7, 2300	59\17, 2441.379	35\10, 2470.588	46\13, 2509.091
F sol fa utb	Vb, Ve	51\15, 2353.846		60\18, 2322.581		55\17, 2275.862	32\10, 2258.824	41\13, 2236.364
F sol fa ut	V	52\15, 2400	38\11, 2400	62\18, 2400	24\7, 2400	58\17, 2400	34\10, 2400	44\13, 2400
F sol fa ut#	V#	53\15, 2446.154	39\11, 2463.158	64\18, 2477,419	25\7, 2500	61\17, 2524.138	36\10, 2541.176	47\13, 2563.636
G la sol reb	Wb, We	55\15, 2538.462	40\11, 2526.316	65\18, 2516.129		60\17, 2482.759	35\10, 2470.588	45\13, 2454.545
G la sol re	Wb	56\15, 2584.615	41\11, 2589.474	67\18, 2593.548	26\7, 2600	63\17, 2606.897	37\10, 2611.765	48\13, 2618.182
G la sol re#	W#	57\15, 2630.769	42\11, 2652.632	69\18, 2670.968	27\7, 2700	66\17, 2731.034	39\10, 2752.941	51\13, 2781.818
A si la mib	Xb, Xe	59\15, 2723.077	43\11, 2715.789	70\18, 2709.677		65\17, 2689.552	38\10, 2682.353	49\13, 2672.273
A si la mi	X	60\15, 2769.231	44\11, 2778.947	72\18, 2787.097	28\7, 2800	68\17, 2813.793	40\10, 2823.529	52\13, 2836.364
A si la mi#	X#	61\15, 2815.385	45\11, 2842.105	74\18, 2864.516	29\7, 2900	71\17, 2937.069	42\10, 2964.706	55\13, 3000
B fab	Ybb, Yee	62\15, 2861.538		73\18, 2825.806	28\7, 2800	67\17, 2772.414	39\10, 2752.941	50\13, 2727.273
B fa	Yb, Ye	63\15, 2907.692	46\11, 2905.263	75\18, 2903.226	29\7, 2900	70\17, 2896.552	41\10, 2894.118	53\13, 2890.909
B si	Y	64\15, 2953.846	47\11, 2968.421	77\18, 2980.645	30\7, 3000	73\17, 3020.690	43\10, 3035.294	56\13, 3054.545
B si#	Y#	65\15, 3000	48\11, 3031.579	79\18, 3058.064	31\7, 3100	76\17, 3144.828	45\10, 3176.471	59\13, 3218.182
C solb	Zb. Ze	67\15, 3092.308	49\11, 3094.737	80\18, 3096.774		75\17, 3103.448	44\10, 3105.882	57\13, 3109.091
C sol	Z	68\15, 3138.462	50\11, 3157.895	82\18, 3174.194	32\7, 3200	78\17, 3227.586	46\10, 3247.059	60\13, 3272.273
C sol#	Z#	69\15, 3184.615	51\11, 3221.053	84\18, 3251.612	33\7, 3300	81\17, 3351.725	48\10, 3388.235	63\13, 3436.364
D labb	Ab, Æ	70\15, 3230.769		83\18, 3212.903	32\7, 3200	77\17, 3186.207	45\10, 3176.471	58\13, 3163.636
D lab	A	71\15, 3276.923	52\11, 3284.211	85\18, 3290.323	33\7, 3300	80\17, 3310.345	47\10, 3317.647	61\13, 3327.27
D la	A#	72\15, 3323.077	53\11, 3347.368	87\18, 3367.742	34\7, 3400	83\17, 3434.583	49\10, 3458.824	64\13, 3490.909
D la#	Ax	73\15, 3369.231	54\11, 3410.625	89\18, 3445.162	35\7, 3500	86\17, 3558.621	51\10, 3600	67\13, 3654.545
F utb	Bb, Be	74\15, 3415.385		88\18, 3406.452	34\7, 3400	82\17, 3393.103	48\10, 3388.235	62\13, 3381.818
F ut	B	75\15, 3461.538	55\11, 3473.684	90\18, 3483.871	35\7, 3500	85\17, 3517.241	50\10, 3529.412	65\13, 3545.455
F ut#	B#	76\15, 3507.692	56\11, 3536.842	92\18, 3561.290	36\7, 3600	88\17, 3641.379	52\10, 3670.588	68\13, 3709.091
G reb	Cb, Ce	78\15, 3600	57\11, 3600	93\18, 3600		87\17, 3600	51\10, 3600	66\13, 3600
G re	C	79\15, 3646.154	58\11, 3663.158	95\18, 3677.419	37\7, 3700	90\17, 3724.138	53\10, 3741.176	69\13, 3763.636
G re#	C#	80\15, 3692.308	59\11, 3726.316	97\18, 3755.838	38\7, 3800	93\17, 3848.275	55\10, 3882.353	72\13, 3927.273
A mib	Db, De	82\15, 3784.615	60\11, 3789.474	98\18, 3793.548		92\17, 3806.897	54\10, 3811.765	70\13, 3818.182
A mi	D	83\15, 3830.769	61\11, 3852.632	100\18, 3870.968	39\7, 3900	95\17, 3931.03$	56\10, 3952.941	73\13, 3981.818
A mi#	D#	84\15, 3876.923	62\11, 3915.789	102\18, 3948.387	40\7, 4000	98\17, 4055.172	58\10, 4094.118	76\13, 4145.455
B fa utb	Ebb, Eee	85\15, 3923.077		101\18, 3909.677	39\7, 3900	94\17, 3889.552	55\10, 3882.353	71\13, 3872.727
B fa ut	Eb, Ee	86\15, 3969.231	63\11, 3978.947	103\18, 3987.097	40\7, 4000	97\17, 4013.793	57\10, 4023.529	74\13, 4036.364
B fa ut#	E	87\15, 4015.385	64\11, 4042.105	105\18, 4064.516	41\7, 4100	100\17, 4137.931	59\10, 4164.706	77\13, 4200
B fa utx	E#	88\15, 4061.583	65\11, 4105.263	107\18, 4141.956	42\7, 4200	103\17, 4262.069	61\10, 4305.882	80\13, 4363.636
C sol reb	Fb, Fe	89\15, 4107.692		106\18, 4103.226	41\7, 4100	99\17, 4096.552	58\10, 4094.118	75\13, 4090.909
C sol re	F	90\15, 4153.846	66\11, 4168.421	108\18, 4180.645	42\7, 4200	102\17, 4220.690	60\10, 4235.294	78\13, 4254.545

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

Genchain

The generator chain for this scale is as follows:

Mibb Labb Sibb

Fab Sibb Dob

Dob Fab Solb

Reb Solb Lab

Mib Lab Sib

Fa Sib Do

Do Fa Sol

Re Sol La

Mi La Si

Fa# Si Do#

Do# Fa# Sol#

Re# Sol# La#

Mi# La# 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	LsLL	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:

Generator (bright)	Cents	L	s	L/s	Comments
1\4	171.429	1	1	1.000	Equalised
6\23	180.000	6	5	1.200
5\19	181.818	5	4	1.250
14\53	182.609	14	11	1.273
9\34	183.051	9	7	1.286
4\15	184.615	4	3	1.333
11\41	185.915	11	8	1.375
7\26	186.667	7	5	1.400
10\37	187.5	10	7	1.429
13\48	187.952	13	9	1.444
16\59	188.253	16	11	1.455
3\11	189.474	3	2	1.500	Napoli-Meantone starts here
14\51	190.909	14	9	1.556
11\40	191.304	11	7	1.571
8\29	192.000	8	5	1.600
5\18	193.548	5	3	1.667
12\43	194.595	12	7	1.714
7\25	195.348	7	4	1.750
9\32	196.364	9	5	1.800
11\39	197.015	11	6	1.833
13\46	197.468	13	7	1.857
15\53	197.802	15	8	1.875
17\60	198.058	17	9	1.889
19\67	198.261	19	10	1.900
21\74	198.425	21	11	1.909
23\81	198.561	23	12	1.917
25\88	198.675	25	13	1.923
27\95	198.773	27	14	1.929
29\102	198.857	29	15	1.933
31\109	198.930	31	16	1.9375
33\116	198.995	33	17	1.941
35\123	199.009	35	18	1.944
2\7	200	2	1	2.000	Napoli-Meantone ends, Napoli-Pythagorean begins
17\59	201.980	17	8	2.125
15\52	202.247	15	7	2.143
13\45	202.597	13	6	2.167
11\38	203.077	11	5	2.200
9\31	203.774	9	4	2.250
7\24	204.878	7	3	2.333
12\41	205.714	12	5	2.400
5\17	206.897	5	2	2.500	Napoli-Neogothic heartland is from here…
18\61	207.693	18	7	2.571
13\44	208.000	13	5	2.600
8\27	208.696	8	3	2.667	…to here
11\37	209.524	11	4	2.750
14\47	210.000	14	5	2.800
3\10	211.765	3	1	3.000	Napoli-Pythagorean ends, Napoli-Archy begins
22\73	212.903	22	7	3.143
19\63	213.084	19	6	3.167
16\53	213.333	16	5	3.200
13\43	213.699	13	4	3.250
10\33	214.286	10	3	3.333
7\23	215.385	7	2	3.500
11\36	216.393	11	3	3.667
15\49	216.867	15	4	3.750
19\62	217.143	19	5	3.800
4\13	218.182	4	1	4.000
13\42	219.718	13	3	4.333
9\29	220.408	9	2	4.500
14\45	221.053	14	3	4.667
5\16	222.222	5	1	5.000	Napoli-Archy ends
11\35	223.728	11	2	5.500
17\54	224.176	17	3	5.667
6\19	225.000	6	1	6.000
1\3	240.000	1	0	→ inf	Paucitonic

See also

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

6L 2s (20/9-equivalent) - Neapolitan 1/2-comma meantone

6L 2s (88/39-equivalent) - Neapolitan gentle temperament

6L 2s (16/7-equivalent) - Neapolitan 1/2-comma archy

9L 3s (10/3-equivalent) - Bijou 1/3-comma meantone

9L 3s (44/13-equivalent) - Bijou gentle temperament

9L 3s (24/7-equivalent) - Bijou 1/3-comma archy

12L 4s (5/1-equivalent) - Hex meantone

12L 4s (56/11-equivalent) - Hextone gentle temperament

12L 4s (36/7-equivalent) - Hextone 1/4-comma archy

15L 5s (15/2-equivalent) - Guidotonic major 1/5-comma meantone

15L 5s (84/11-equivalent) - Guidotonic major gentle temperament

15L 5s (54/7-equivalent) - Guidotonic major 1/5-comma archy

18L 6s (11/1-equivalent) - Subdozenal harmonic tuning

18L 6s (56/5-equivalent) - Subdozenal low septimal (meantone) tuning

18L 6s (512/45-equivalent) - Subdozenal 1/6-comma meantone

18L 6s (80/7-equivalent) - Subdozenal high septimal tuning

18L 6s (128/11-equivalent) - Subdozenal subharmonic tuning

18L 6s (12/1-equivalent) - Warped Pythagorean tuning