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{{URWTC}}
{{Infobox MOS|Equalized=1|Collapsed=1|Pattern=LLLs}}
{{Infobox MOS|Equalized=1|Collapsed=1|Pattern=LLLs}}
{{MOS intro}} The so-called "Super Ultra Hyper Mega Meta Lydian" is one mode of this mos.


'''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&nbsp;1s]]. The name of the period interval is called the '''sesquitave''' (by analogy to the [[tritave]]). The generator range is 171.4 to 240{{c}}, placing it near the [[9/8|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{{c}}).
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 {{dash|1, 5/4, 5/3}} is fifth-equivalent to a tone cluster of {{dash|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 [[Generator|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&nbsp;2s], a major thirteenth which is the Dorian mode of Bijou[9L&nbsp;3s] or an ~pentave which is the Mixolydian mode of Hextone[12L&nbsp;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&nbsp;molle or 0123456789ABCDEF0 or G123456789ABCDEFG with flats written F&nbsp;molle) may be used.
{| class="wikitable"
|+ style="font-size: 105%;" | Cents<ref name=":0">Fractions repeating more than 4 digits written as continued fractions</ref>
|-
! colspan="4" | 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\1546; 6.5
| 1\1163: 6.{{Overline|3}}
| 2\1877; 2, 2.6
| rowspan="2" | 1\7 100
| 3\17124; 7.25
| 2\10141; 5.{{Overline|6}}
| 3\13 163.{{Overline|63}}
|-
| Reb, Lab
| Gb
| 1b, 1c
| 1f
| 3\15138; 3.25
| 2\11126; 3.1{{Overline|6}}
| 3\18116; 7.75
| 2\1782; 1.3{{Overline|18}}
| 1\1070; 1.7
| 1\13 54.{{Overline|54}}
|-
| '''Re, La'''
| '''G'''
| '''1'''
| '''1'''
| '''4\15''''''184; 1.625'''
| '''3\11''''''189; 2.{{Overline|1}}'''
| '''5\18''''''193; 1, 1, 4.{{Overline|6}}'''
| '''2\7''' '''200'''
| '''5\17''''''206; 1, 8.{{Overline|6}}'''
| '''3\10''''''211; 1, 3.25'''
| '''4\13''' '''218.{{Overline|18}}'''
|-
| Re#, La#
| G#
| 1#
| 1#
| 5\15230; 1.3
| 4\11252; 1.58{{Overline|3}}
| 7\18270; 1.0{{Overline|3}}
| rowspan="2" | 3\7 300
| 8\17331; 29
| 5\10352; 1.0625
| 7\13 381.{{Overline|81}}
|-
| Mib, Sib
| Ab
| 2b, 2c
| 2f
| 7\15323; 13
| 5\11315; 1.2{{Overline|6}}
| 8\18309; 1, 2.1
| 7\17289; 1, 1.9
| 4\10282; 2.8{{Overline|3}}
| 5\13 272.{{Overline|72}}
|-
| Mi, Si
| A
| 2
| 2
| 8\15369; 4.{{Overline|3}}
| 6\11378; 1.0{{Overline|5}}
| 10\18387; 10.{{Overline|3}}
| 4\7 400
| 10\17413; 1, 3.8{{Overline|3}}
| 6\10423; 1.{{Overline|8}}
| 8\13 436.{{Overline|36}}
|-
| Mi#, Si#
| A#
| 2#
| 2#
| 9\15415; 2.6
| rowspan="2" | 7\11442; 9.5
| 12\18464; 1.0625
| 5\7 500
| 13\17537; 14.5
| 8\10564; 1.41{{Overline|6}}
| 11\13 600
|-
| Fab, Dob
| Bbb
| 3b, 3c
| 3f
| 10\15461; 1, 1.1{{Overline|6}}
| 11\18425; 1.24
| 4\7 400
| 9\17372; 2.41{{Overline|6}}
| 5\10352; 1.0625
| 6\13 327.{{Overline|27}}
|-
| '''Fa, Do'''
| '''Bb'''
| '''3'''
| '''3'''
| '''11\15''''''507; 1.{{Overline|4}}'''
| '''8\11''''''505; 3.8'''
| '''13\18''''''503; 4, 2.{{Overline|3}}'''
| '''5\7''' '''500'''
| '''12\17''''''496; 1.8125'''
| '''7\10''''''494; 8.5'''
| '''9\13''' '''490.{{Overline|90}}'''
|-
| Fa#, Do#
| B
| 3#
| 3#
| 12\15553; 1.{{Overline|18}}
| 9\11568; 2.375
| 15\18580; 1.55
| 6\7 600
| 15\17620; 1.45
| 9\10635; 3.4
| 12\13 654.{{Overline|54}}
|-
| Fax, Dox
| B#
| 3x
| 3x
| 13\15 600
| rowspan="2" | 10\11 631; 1.{{Overline|72}}
| 17\18 658; 15.5
| 7\7 700
| 18\17 744; 1.208{{Overline|3}}
| 11\10 776; 2.125
| 15\13 818.{{Overline|18}}
|-
| Dob, Solb
| Hb
| 4b, 4c
| 4f
| 14\15 646; 6.5
| 16\18 619; 2.{{Overline|81}}
| 6\7 600
| 14\17 579; 3.{{Overline|2}}
| 8\10564; 1.41{{Overline|6}}
| 10\13 545.{{Overline|45}}
|-
! Do, Sol
! H
! 4
! 4
! '''15\15''' '''692; 3.25'''
! '''11\11''' '''694; 1, 2.8'''
! '''18\18''' '''696; 1.291'''{{Overline|6}}
! '''7\7''' '''700'''
! '''17\17''' '''703; 2, 2.1'''{{Overline|6}}
! '''10\10''' '''705; 1.1'''{{Overline|3}}
! '''13\13''' '''709.'''{{Overline|09}}
|-
| Do#, Sol#
| Η#
| 4#
| 4#
| 16\15 738; 2.1{{Overline|6}}
| 12\11 757; 1, 8.5
| 20\18 774; 5, 6
| rowspan="2" | 8\8 800
| 20\17 827; 1, 1.41{{Overline|6}}
| 12\10 847; 17
| 16\13 872.{{Overline|72}}
|-
| Reb, Lab
| Cb
| 5b, 5c
| 5
| 18\15 830; 1.3
| 13\11 821; 19
| 21\18 812; 1, 9.{{Overline|3}}
| 19\17 786; 4.8{{Overline|3}}
| 11\10 776; 2.125
| 14\13 763.{{Overline|63}}
|-
| '''Re, La'''
| '''C'''
| '''5'''
| '''5'''
| '''19\15''' '''876; 1.08{{Overline|3}}'''
| '''14\11''' '''884; 4.75'''
| '''23\18''' '''890; 3.1'''
| '''9\5''' '''900'''
| '''22\17''' '''910; 2.9'''
| '''13\10''' '''917; 1.{{Overline|54}}'''
| '''17\13''' '''927.{{Overline|27}}'''
|-
| Re#, La#
| C#
| 5#
| 5#
| 20\15 923: 13
| 15\11 947; 2, 1.4
| 25\18 967; 1, 2.875
| rowspan="2" | 10\7 1000
| 25\17 1034; 2, 14
| 15\10 1058; 1, 4.{{Overline|6}}
| 20\13 1090.{{Overline|90}}
|-
| Mib, Sib
| Db
| 6b, 6c
| 6f
| 22\15 1015; 2.6
| 16\11 1010; 1.9
| 26\18 1006; 2, 4.{{Overline|6}}
| 24\17 993; 9.{{Overline|6}}
| 14\10 988; 4.25
| 18\13 981.{{Overline|81}}
|-
| Mi, Si
| D
| 6
| 6
| 23\15 1061; 1, 1.1{{Overline|6}}
| 17\11 1073; 1, 2.1{{Overline|6}}
| 28\18 1083; 1.{{Overline|148}}
| 11\7 1100
| 27\17 1117; 4, 7
| 16\10 1129; 2, 2.{{Overline|3}}
| 21\9 1145.{{Overline|45}}
|-
| Mi#, Si#
| D#
| 6#
| 6#
| 24\15 1107; 1.{{Overline|4}}
| rowspan="2" | 18\11 1136; 1.1875
| 30\18 1161; 3.{{Overline|4}}
| 12\7 1200
| 30\17 1241; 2.{{Overline|63}}
| 18\10 1270; 1.7
| 24\13 1309.{{Overline|09}}
|-
| Fab, Dob
| Ebb
| 7b, 7c
| 7f
| 25\15 1153; 1.{{Overline|18}}
| 29\18 1121; 1, 1, 2.6
| 11\7 1100
| 26\17 1075; 1.16
| 15\10 1058; 1, 4.{{Overline|6}}
| 19\13 1036.{{Overline|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.{{Overline|3}}
| 33\18 1277; 2, 2.6
| 13\7 1300
| 32\17 1324; 7.25
| 19\10 1341; 5.{{Overline|6}}
| 25\13 1363.{{Overline|63}}
|-
| Fax, Dox
| E#
| 7x
| 7x
| 28\15 1292; 3.25
| rowspan="2" | 21\11 1326; 3.1{{Overline|6}}
| 35\18 1354; 1, 5.2
| 14\7 1400
| 35\17 1448; 3.625
| 21\10 1482; 2.8{{Overline|3}}
| 28\13 1527.{{Overline|27}}
|-
| Dob, Solb
| Fb
| 8b, Fc
| 8f
| 29\15 1338; 2.1{{Overline|6}}
| 34\18 1316; 7.75
| 13\7 1300
| 31\17 1282; 1.3{{Overline|18}}
| 18\10 1270; 1.7
| 23\13 1254.{{Overline|54}}
|-
! Do, Sol
! F
! 8, F
! 8
! 30\15 1384; 1.625
! 22\11 1389; 2.{{Overline|1}}
! 36\18 1393; 1, 1, 4.{{Overline|6}}
! 14\7 1400
! 34\17 1406; 1, 8.{{Overline|6}}
! 20\10 1411; 1, 3.25
! 26\13 1418.{{Overline|18}}
|-
| Do#, Sol#
| F#
| 8#, F#
| 8#
| 31\15 1430; 1.3
| 23\11 1452; 1.58{{Overline|3}}
| 38\18 1470; 1.0{{Overline|3}}
| rowspan="2" | 15\7 1500
| 37\17 1531; 29
| 22\10 1552; 1.0625
| 29\13 1581.{{Overline|81}}
|-
| Reb, Lab
| Gb
| 9b, Gc
| 9f
| 33\15 1523; 13
| 24\11 1515; 1.2{{Overline|6}}
| 39\18 1509; 1, 2.1
| 36\17 1489; 1, 1.9
| 21\10 1482; 2.8{{Overline|3}}
| 27\13 1472.{{Overline|72}}
|-
| '''Re, La'''
| '''G'''
| '''9, G'''
| 9
| '''34\15''' '''1569; 4.{{Overline|3}}'''
| '''25\11''' '''1578; 1.0{{Overline|5}}'''
| '''41\18''' '''1587; 10.{{Overline|3}}'''
| '''16\7''' '''1600'''
| '''39\17''' '''1613; 1, 3.8{{Overline|3}}'''
| '''23\10''' '''1623; 1.{{Overline|8}}'''
| '''30\13''' '''1636.{{Overline|36}}'''
|-
| Re#, La#
| G#
| 9#, G#
| 9#
| 35\15 1615; 2.6
| 26\11 1642; 9.5
| 43\18 1664; 1.0625
| rowspan="2" | 17\7 1700
| 42\17 1737; 14.5
| 25\10 1764; 1.41{{Overline|6}}
| 33\13 1800
|-
| Mib, Sib
| Ab
| Xb, Ac
| Af
| 37\15 1707; 1.{{Overline|4}}
| 27\11 1705; 3.8
| 44\18 1703; 4, 2.{{Overline|3}}
| 41\17 1696; 1.8125
| 24\10 1694; 8.5
| 31\13 1690.{{Overline|90}}
|-
| Mi, Si
| A
| X, A
| A
| 38\15 1753; 1.{{Overline|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.{{Overline|54}}
|-
| Mi#, Si#
| A#
| X#, A#
| A#
| 39\15 1800
| rowspan="2" | 29\11 1831; 1.{{Overline|72}}
| 48\18 1858; 15.5
| 19\7 1900
| 47\17 1944; 1.208{{Overline|3}}
| 28\10 1976; 2.125
| 37\13 2018.{{Overline|18}}
|-
| Fab, Dob
| Bbb
| Ebb, Ccc
| Bf
| 40\15 1846; 6.5
| 47\18 1819; 2.{{Overline|81}}
| 18\7 1800
| 43\17 1779; 3.{{Overline|2}}
| 25\10 1764; 1.41{{Overline|6}}
| 32\13 1745.{{Overline|45}}
|-
| '''Fa, Do'''
| '''Bb'''
| '''Eb, Cc'''
| '''B'''
| '''41\15''' '''1892; 3.25'''
| '''30\11''' '''1894; 1, 2.8'''
| '''49\18''' '''1896; 1.291{{Overline|6}}'''
| '''19\7''' '''1900'''
| '''46\17''' '''1903; 2.1{{Overline|6}}'''
| '''27\10''' '''1905; 1.1{{Overline|3}}'''
| '''35\13''' '''1909.{{Overline|09}}'''
|-
| Fa#, Do#
| B
| E, C
| B#
| 42\15 1938; 2.1{{Overline|6}}
| 31\11 1957; 1, 8.5
| 51\18 1974; 5.1{{Overline|6}}
| 20\7 2000
| 49\17 2027; 1, 1.41{{Overline|6}}
| 29\10 2047; 17
| 38\13 2072.{{Overline|72}}
|-
| Fax, Dox
| B#
| Ex, Cx
| Bx
| 43\15 1984; 1.625
| rowspan="2" | 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.{{Overline|36}}
|-
| Dob, Solb
| Hb
| 0b, Dc
| Cf
| 44\15 2030; 1.3
| 52\18 2012; 1, 9,{{Overline|3}}
| 20\7 2000
| 48\17 1986; 4.8{{Overline|3}}
| 28\10 1976; 2.125
| 36\13 1963.{{Overline|63}}
|-
! Do, Sol
! H
! 0, D
! C
! 45\15 2076; 1.08'''{{Overline|3}}'''
! 33\11 2084; 4.75
! 54\18 2090; 3.1
! 21\7 2100
! 51\17 2110; 2.9
! 30\10 2117; 1.{{Overline|54}}
! 39\13 2127.{{Overline|27}}
|-
| Do#, Sol#
| Η#
| 0#, D#
| C#
| 46\152123; 13
| 34\112147; 2, 1.4
| 56\182167; 1, 2.875
| rowspan="2" | 22\72200
| 54\172234; 2, 14
| 32\102258; 1, 4.{{Overline|6}}
| 42\132090.{{Overline|90}}
|-
| Reb, Lab
| Cb
| 1b, 1c
| Df
| 48\152215; 2.6
| 35\112210; 1.9
| 57\182206; 2, 4.{{Overline|6}}
| 53\172193; 9.{{Overline|6}}
| 31\10 2188; 4.25
| 40\132181.{{Overline|81}}
|-
| '''Re, La'''
| '''C'''
| '''1'''
| '''D'''
| '''49\15''''''2261; 1, 1.1{{Overline|6}}'''
| '''36\11''''''2273; 1, 2.1{{Overline|6}}'''
| '''59\18''''''2283; 1.{{Overline|148}}'''
| '''23\7''''''2300'''
| '''56\17''''''2317; 4, 7'''
| '''33\10''''''2329; 2, 2.{{Overline|3}}'''
| '''43\13''''''2245.{{Overline|45}}'''
|-
| Re#, La#
| C#
| 1#
| D#
| 50\152307; 1.{{Overline|4}}
| 37\112336; 1.1875
| 61\182361; 3.{{Overline|4}}
| rowspan="2" | 24\72400
| 59\172441; 2.{{Overline|63}}
| 35\102470; 1.7
| 46\132509.{{Overline|09}}
|-
| Mib, Sib
| Db
| 2b, 2c
| Ef
| 52\152400
| 38\112400
| 62\182400
| 58\172400
| 34\102400
| 44\132400
|-
| Mi, Si
| D
| 2
| E
| 53\152446; 6.5
| 39\112463; 6.{{Overline|3}}
| 64\182477; 2, 2.6
| 25\72500
| 61\172524; 7.25
| 36\102541; 5.{{Overline|6}}
| 47\132563.{{Overline|63}}
|-
| Mi#, Si#
| D#
| 2#
| E#
| 54\152492; 3.25
| rowspan="2" | 40\112526; 3.1
| 66\182554; 1, 5.2
| 26\72600
| 64\172648; 2.625
| 38\102682; 2.8{{Overline|3}}
| 50\132727.{{Overline|27}}
|-
| Fab, Dob
| Ebb
| 3b, 3c
| Fff
| 55\152538; 2.1{{Overline|6}}
| 65\182516; 7.75
| 25\72500
| 60\172482; 1.3{{Overline|18}}
| 35\102470; 1.7
| 45\132454.{{Overline|54}}
|-
| '''Fa, Do'''
| '''Eb'''
| '''3'''
| '''Ff'''
| '''56\15''''''2584; 1.625'''
| '''41\11''''''2589; 2.{{Overline|1}}'''
| '''67\18''''''2593; 1, 1, 4.{{Overline|6}}'''
| '''26\7''''''2600'''
| '''63\17''''''2606; 1, 8.{{Overline|6}}'''
| '''37\10''''''2611; 1, 3.25'''
| '''48\13''''''2618.{{Overline|18}}'''
|-
| Fa#, Do#
| E
| 3#
| F
| 57\152630; 1.3
| 42\112652; 1.58{{Overline|3}}
| 69\182670; 1.0{{Overline|3}}
| 27\72700
| 66\172731; 29
| 39\102752; 1.0625
| 51\132781.{{Overline|81}}
|-
| Fax, Dox
| E#
| 3x
| F#
| 58\152676; 1.08{{Overline|3}}
| rowspan="2" | 43\112715; 1.2{{Overline|6}}
| 71\182748; 2.58{{Overline|3}}
| 28\72800
| 69\172855; 4.8
| 41\102894; 8.5
| 54\132945.{{Overline|45}}
|-
| Dob, Solb
| Fb
| 4b, 4c
| 0f, Gf
| 59\152723; 13
| 70\182709; 1, 2.1
| 27\72700
| 65\172689; 1, 1.9
| 38\102682; 2.8{{Overline|3}}
| 49\132672.{{Overline|72}}
|-
! Do, Sol
! F
! 4
! 0, G
! 60\152769; 4.'''{{Overline|3}}'''
! 44\112778; 1.0{{Overline|5}}
! 72\182787; 3.1
! 28\72800
! 68\172813; 1, 3.8{{Overline|3}}
! 40\102823; 1.{{Overline|8}}
! 52\132836.{{Overline|36}}
|}
== Modes ==
The mode names are based on the species of fifth:
{{MOS modes
| Mode Names=
Lydian $
Minor $
Major $
Phrygian $
|}
 
== Temperaments ==
The most basic rank-2 temperament interpretation of angel is '''Napoli'''. The name "Napoli" comes from the “Neapolitan” sixth triad spelled <code>root-(p-2g)-(2p-3g)</code> ({{nowrap|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]]: [{{val|1 1 2}}, {{val|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]]: [{{val|1 1 2}}, {{val|0 -2 -3}}]
 
[[Optimal ET sequence]]: ~(7edf, 10edf, 13edf, 16edf)
 
=== Scale tree ===
The spectrum looks like this:
{{MOS tuning spectrum
| 3/2 = Napoli-Meantone starts here
| 2/1 = Napoli-Meantone ends, Napoli-Pythagorean begins
| 5/2 = Napoli-Neogothic heartland is from here...
| 8/3 = ...to here
| 3/1 = Napoli-Pythagorean ends, Napoli-Archy begins
| 5/1 = Napoli-Archy ends
}}
Retrieved from "https://en.xen.wiki/w/3L_1s_(3/2-equivalent)"