3L 1s (3/2-equivalent): Difference between revisions

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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 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
}}

Latest revision as of 17:57, 3 March 2025

This page on a regular temperament, temperament collection, or aspect of regular temperament theory is being revised for clarity as part of WikiProject TempClean.
← 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
Parent 1L 2s⟨3/2⟩
Sister 1L 3s⟨3/2⟩
Daughters 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 ¢)
Soft (L:s = 3:2) 3\11 (191.4 ¢)
Semisoft (L:s = 5:3) 5\18 (195.0 ¢)
Basic (L:s = 2:1) 2\7 (200.6 ¢)
Semihard (L:s = 5:2) 5\17 (206.5 ¢)
Hard (L:s = 3:1) 3\10 (210.6 ¢)
Superhard (L:s = 4:1) 4\13 (216.0 ¢)
Collapsed (L:s = 1:0) 1\3 (234.0 ¢)

3L 1s⟨3/2⟩ is a 3/2-equivalent (fifth-equivalent) moment of symmetry scale containing 3 large steps and 1 small step, repeating every interval of 3/2 (702.0 ¢). Generators that produce this scale range from 175.5 ¢ to 234 ¢, or from 468 ¢ to 526.5 ¢. Scales of this form are always proper because there is only one small step. 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 ¢, 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 ¢).

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\1546; 6.5 1\1163: 6.3 2\1877; 2, 2.6 1\7 100 3\17124; 7.25 2\10141; 5.6 3\13 163.63
Reb, Lab Gb 1b, 1c 1f 3\15138; 3.25 2\11126; 3.16 3\18116; 7.75 2\1782; 1.318 1\1070; 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\15230; 1.3 4\11252; 1.583 7\18270; 1.03 3\7 300 8\17331; 29 5\10352; 1.0625 7\13 381.81
Mib, Sib Ab 2b, 2c 2f 7\15323; 13 5\11315; 1.26 8\18309; 1, 2.1 7\17289; 1, 1.9 4\10282; 2.83 5\13 272.72
Mi, Si A 2 2 8\15369; 4.3 6\11378; 1.05 10\18387; 10.3 4\7 400 10\17413; 1, 3.83 6\10423; 1.8 8\13 436.36
Mi#, Si# A# 2# 2# 9\15415; 2.6 7\11442; 9.5 12\18464; 1.0625 5\7 500 13\17537; 14.5 8\10564; 1.416 11\13 600
Fab, Dob Bbb 3b, 3c 3f 10\15461; 1, 1.16 11\18425; 1.24 4\7 400 9\17372; 2.416 5\10352; 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\15553; 1.18 9\11568; 2.375 15\18580; 1.55 6\7 600 15\17620; 1.45 9\10635; 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 16\18 619; 2.81 6\7 600 14\17 579; 3.2 8\10564; 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 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 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 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 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 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 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 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 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\152123; 13 34\112147; 2, 1.4 56\182167; 1, 2.875 22\72200 54\172234; 2, 14 32\102258; 1, 4.6 42\132090.90
Reb, Lab Cb 1b, 1c Df 48\152215; 2.6 35\112210; 1.9 57\182206; 2, 4.6 53\172193; 9.6 31\10 2188; 4.25 40\132181.81
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\152307; 1.4 37\112336; 1.1875 61\182361; 3.4 24\72400 59\172441; 2.63 35\102470; 1.7 46\132509.09
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; 6.3 64\182477; 2, 2.6 25\72500 61\172524; 7.25 36\102541; 5.6 47\132563.63
Mi#, Si# D# 2# E# 54\152492; 3.25 40\112526; 3.1 66\182554; 1, 5.2 26\72600 64\172648; 2.625 38\102682; 2.83 50\132727.27
Fab, Dob Ebb 3b, 3c Fff 55\152538; 2.16 65\182516; 7.75 25\72500 60\172482; 1.318 35\102470; 1.7 45\132454.54
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\152630; 1.3 42\112652; 1.583 69\182670; 1.03 27\72700 66\172731; 29 39\102752; 1.0625 51\132781.81
Fax, Dox E# 3x F# 58\152676; 1.083 43\112715; 1.26 71\182748; 2.583 28\72800 69\172855; 4.8 41\102894; 8.5 54\132945.45
Dob, Solb Fb 4b, 4c 0f, Gf 59\152723; 13 70\182709; 1, 2.1 27\72700 65\172689; 1, 1.9 38\102682; 2.83 49\132672.72
Do, Sol F 4 0, G 60\152769; 4.3 44\112778; 1.05 72\182787; 3.1 28\72800 68\172813; 1, 3.83 40\102823; 1.8 52\132836.36

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 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)
Bright Dark L:s Hardness
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⟩
  1. ↑ Fractions repeating more than 4 digits written as continued fractions
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