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

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#redirect [[3L 1s (3/2-equivalent)]]
'''3L1s<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 [[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 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 [[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 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.
{| class="wikitable"
|+
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\15
46; 6.5
|1\11
63: 6.{{Overline|3}}
|2\18
77; 2, 2.6
| rowspan="2" |1\7
100
|3\17
124; 7.25
|2\10
141; 5.{{Overline|6}}
|3\13
163.{{Overline|63}}
|-
|Reb, Lab
|Gb
|1b, 1c
|1f
|3\15
138; 3.25
|2\11
126; 3.1{{Overline|6}}
|3\18
116; 7.75
|2\17
82; 1.3{{Overline|18}}
|1\10
70; 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\15
230; 1.3
|4\11
252; 1.58{{Overline|3}}
|7\18
270; 1.0{{Overline|3}}
| rowspan="2" |3\7
300
|8\17
331; 29
|5\10
352; 1.0625
|7\13
381.{{Overline|81}}
|-
|Mib, Sib
|Ab
|2b, 2c
|2f
|7\15
323; 13
|5\11
315; 1.2{{Overline|6}}
|8\18
309; 1, 2.1
|7\17
289; 1, 1.9
|4\10
282; 2.8{{Overline|3}}
|5\13
272.{{Overline|72}}
|-
|Mi, Si
|A
|2
|2
|8\15
369; 4.{{Overline|3}}
|6\11
378; 1.0{{Overline|5}}
|10\18
387; 10.{{Overline|3}}
|4\7
400
|10\17
413; 1, 3.8{{Overline|3}}
|6\10
423; 1.{{Overline|8}}
|8\13
436.{{Overline|36}}
|-
|Mi#, Si#
|A#
|2#
|2#
|9\15
415; 2.6
| rowspan="2" |7\11
442; 9.5
|12\18
464; 1.0625
|5\7
500
|13\17
537; 14.5
|8\10
564; 1.41{{Overline|6}}
|11\13
600
|-
|Fab, Dob
|Bbb
|3b, 3c
|3f
|10\15
461; 1, 1.1{{Overline|6}}
|11\18
425; 1.24
|4\7
400
|9\17
372; 2.41{{Overline|6}}
|5\10
352; 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\15
553; 1.{{Overline|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.{{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\10
564; 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\15
2123; 13
|34\11
2147; 2, 1.4
|56\18
2167; 1, 2.875
| rowspan="2" |22\7
2200
|54\17
2234; 2, 14
|32\10
2258; 1, 4.{{Overline|6}}
|42\13
2090.{{Overline|90}}
|-
|Reb, Lab
|Cb
|1b, 1c
|Df
|48\15
2215; 2.6
|35\11
2210; 1.9
|57\18
2206; 2, 4.{{Overline|6}}
|53\17
2193; 9.{{Overline|6}}
|31\10
2188; 4.25
|40\13
2181.{{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\15
2307; 1.{{Overline|4}}
|37\11
2336; 1.1875
|61\18
2361; 3.{{Overline|4}}
| rowspan="2" |24\7
2400
|59\17
2441; 2.{{Overline|63}}
|35\10
2470; 1.7
|46\13
2509.{{Overline|09}}
|-
|Mib, Sib
|Db
|2b, 2c
|Ef
|52\15
2400
|38\11
2400
|62\18
2400
|58\17
2400
|34\10
2400
|44\13
2400
|-
|Mi, Si
|D
|2
|E
|53\15
2446; 6.5
|39\11
2463; 6.{{Overline|3}}
|64\18
2477; 2, 2.6
|25\7
2500
|61\17
2524; 7.25
|36\10
2541; 5.{{Overline|6}}
|47\13
2563.{{Overline|63}}
|-
|Mi#, Si#
|D#
|2#
|E#
|54\15
2492; 3.25
| rowspan="2" |40\11
2526; 3.1
|66\18
2554; 1, 5.2
|26\7
2600
|64\17
2648; 2.625
|38\10
2682; 2.8{{Overline|3}}
|50\13
2727.{{Overline|27}}
|-
|Fab, Dob
|Ebb
|3b, 3c
|Fff
|55\15
2538; 2.1{{Overline|6}}
|65\18
2516; 7.75
|25\7
2500
|60\17
2482; 1.3{{Overline|18}}
|35\10
2470; 1.7
|45\13
2454.{{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\15
2630; 1.3
|42\11
2652; 1.58{{Overline|3}}
|69\18
2670; 1.0{{Overline|3}}
|27\7
2700
|66\17
2731; 29
|39\10
2752; 1.0625
|51\13
2781.{{Overline|81}}
|-
|Fax, Dox
|E#
|3x
|F#
|58\15
2676; 1.08{{Overline|3}}
| rowspan="2" |43\11
2715; 1.2{{Overline|6}}
|71\18
2748; 2.58{{Overline|3}}
|28\7
2800
|69\17
2855; 4.8
|41\10
2894; 8.5
|54\13
2945.{{Overline|45}}
|-
|Dob, Solb
|Fb
|4b, 4c
|0f, Gf
|59\15
2723; 13
|70\18
2709; 1, 2.1
|27\7
2700
|65\17
2689; 1, 1.9
|38\10
2682; 2.8{{Overline|3}}
|49\13
2672.{{Overline|72}}
|-
!Do, Sol
!F
!4
!0, G
!60\15
2769; 4.'''{{Overline|3}}'''
!44\11
2778; 1.0{{Overline|5}}
!72\18
2787; 3.1
!28\7
2800
!68\17
2813; 1, 3.8{{Overline|3}}
!40\10
2823; 1.{{Overline|8}}
!52\13
2836.{{Overline|36}}
|}
{| class="wikitable"
|+Relative cents<ref name=":02">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\15
''46.{{Overline|6}}''
|1\11
''63.{{Overline|63}}''
|2\18
''77.7̄''
| rowspan="2" |1\7
''100''
|3\17
''123; 1.{{Overline|8}}''
|2\10
''140''
|3\13
''161; 1, 1.1{{Overline|6}}''
|-
|Reb, Lab
|Gb
|1b, 1c
|1f
|3\15
''140''
|2\11
''127.{{Overline|27}}''
|3\18
''116.{{Overline|6}}''
|2\17
''82; 2.8{{Overline|3}}''
|1\10
''70''
|1\13
''53; 1.{{Overline|18}}''
|-
|'''Re, La'''
|'''G'''
|'''1'''
|'''1'''
|'''4\15'''
'''''186.{{Overline|6}}'''''
|'''3\11'''
'''''190.{{Overline|90}}'''''
|'''5\18'''
'''''194.{{Overline|4}}'''''
|'''2\7'''
'''''200'''''
|'''5\17'''
'''''205; 1.1{{Overline|3}}'''''
|'''3\10'''
'''''210'''''
|'''4\13'''
'''''215; 2.6'''''
|-
|Re#, La#
|G#
|1#
|1#
|5\15
''233.{{Overline|3}}''
|4\11
''254.{{Overline|54}}''
|7\18
''272.2̄''
| rowspan="2" |3\7
''300''
|8\17
''329; 2, 2.{{Overline|3}}''
|5\10
''350''
|7\13
''376; 1.08{{Overline|3}}''
|-
|Mib, Sib
|Ab
|2b, 2c
|2f
|7\15
''326.{{Overline|6}}''
|5\11
''318.{{Overline|18}}''
|8\18
''311.{{Overline|1}}''
|7\17
''288; 4.25''
|4\10
''280''
|5\13
''269; 4.{{Overline|3}}''
|-
|Mi, Si
|A
|2
|2
|8\15
''373.{{Overline|3}}''
|6\11
''381.{{Overline|81}}''
|10\18
''388.{{Overline|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''
| rowspan="2" |7\11
''445.{{Overline|45}}''
|12\18
''466.{{Overline|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.{{Overline|6}}''
|11\18
''427.{{Overline|7}}''
|4\7
''400''
|9\17
''370; 1.7''
|5\10
''350''
|6\13
''323; 13''
|-
|'''Fa, Do'''
|'''Bb'''
|'''3'''
|'''3'''
|'''11\15'''
'''''513.{{Overline|3}}'''''
|'''8\11'''
'''''509.{{Overline|09}}'''''
|'''13\18'''
'''''505.{{Overline|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.{{Overline|72}}''
|15\18
''583.{{Overline|3}}''
|6\7
''600''
|15\17
''617; 1.41{{Overline|6}}''
|9\10
''630''
|12\13
''646; 6.5''
|-
|Fax, Dox
|B#
|3x
|3x
|13\15
''606.{{Overline|6}}''
| rowspan="2" |10\11
''636.{{Overline|36}}''
|17\18
''661.{{Overline|1}}''
|7\7
''700''
|18\17
''741; 5.{{Overline|6}}''
|11\10
''770''
|15\13
''807; 1.{{Overline|4}}''
|-
|Dob, Solb
|Hb
|4b, 4c
|4f
|14\15
''653.{{Overline|3}}''
|16\18
''622.{{Overline|2}}''
|6\7
''600''
|14\17
''576; 2.125''
|8\10
''560''
|10\13
''538; 2.1{{Overline|6}}''
|-
!Do, Sol
!H
!4
!4
! colspan="7" |''700''
|-
|Do#, Sol#
|Η#
|4#
|4#
|16\15
''746.{{Overline|6}}''
|12\11
''763.{{Overline|63}}''
|20\18
''777.{{Overline|7}}''
| rowspan="2" |8\7
''800''
|20\17
''823; 1.{{Overline|8}}''
|12\10
''840''
|16\13
''861; 1, 1.1{{Overline|6}}''
|-
|Reb, Lab
|Cb
|5b, 5c
|5
|18\15
''840''
|13\11
''827.{{Overline|27}}''
|21\18
''816.{{Overline|6}}''
|19\17
''782; 2.8{{Overline|3}}''
|11\10
''770''
|14\13
''753; 1.{{Overline|18}}''
|-
|'''Re, La'''
|'''C'''
|'''5'''
|'''5'''
|'''19\15'''
'''''886.{{Overline|6}}'''''
|'''14\11'''
'''''890.{{Overline|90}}'''''
|'''23\18'''
'''''894.{{Overline|4}}'''''
|'''9\7'''
'''''900'''''
|'''22\17'''
'''''905; 1.1{{Overline|3}}'''''
|'''13\10'''
'''''910'''''
|'''17\13'''
'''''915; 2.6'''''
|-
|Re#, La#
|C#
|5#
|5#
|20\15
''933.{{Overline|3}}''
|15\11
''954.{{Overline|54}}''
|25\18
''972.{{Overline|2}}''
| rowspan="2" |10\7
''1000''
|25\17
''1029; 2, 2.{{Overline|3}}''
|15\10
''1050''
|20\13
''1076; 1.08{{Overline|3}}''
|-
|Mib, Sib
|Db
|6b, 6c
|6f
|22\15
''1026.{{Overline|6}}''
|16\11
''1018.{{Overline|18}}''
|26\18
''1011.{{Overline|1}}''
|24\17
''988; 4.25''
|14\10
''980''
|18\13
''969; 4.{{Overline|3}}''
|-
|Mi, Si
|D
|6
|6
|23\15
''1073.{{Overline|3}}''
|17\11
''1081.{{Overline|81}}''
|28\18
''1088.{{Overline|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''
| rowspan="2" |18\11
''1145.{{Overline|45}}''
|30\18
''1166.{{Overline|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.{{Overline|6}}''
|29\18
''1127.{{Overline|7}}''
|11\7
''1100''
|26\17
''1070; 1.7''
|15\10
''1050''
|19\13
''1023; 13''
|-
|'''Fa, Do'''
|'''Eb'''
|'''7'''
|'''7'''
|'''26\15'''
'''''1213.{{Overline|3}}'''''
|'''19\11'''
'''''1209.{{Overline|09}}'''''
|'''31\18'''
'''''1205.{{Overline|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.{{Overline|72}}''
|33\18
''1283.{{Overline|3}}''
|13\7
''1300''
|32\17
''1317; 1.41{{Overline|6}}''
|19\10
''1330''
|25\13
''1346; 6.5''
|-
|Fax, Dox
|E#
|7x
|7x
|28\15
''1306.{{Overline|6}}''
| rowspan="2" |21\11
''1336.{{Overline|36}}''
|35\18
''1361.{{Overline|1}}''
|14\7
''1400''
|35\17
''1441; 5.{{Overline|6}}''
|21\10
''1470''
|28\13
''1507; 1.{{Overline|4}}''
|-
|Dob, Solb
|Fb
|8b, Fc
|8f
|29\15
''1333.{{Overline|3}}''
|34\18
''1322.{{Overline|2}}''
|13\7
''1300''
|31\17
''1276; 2.125''
|18\10
''1260''
|23\13
''1238; 2.1{{Overline|6}}''
|-
!Do, Sol
!F
!8, F
!8
! colspan="7" |''1400''
|-
|Do#, Sol#
|F#
|8#, F#
|8#
|31\15
''1446.{{Overline|6}}''
|23\11
''1463.{{Overline|63}}''
|38\18
''1477.7̄''
| rowspan="2" |15\7
''1500''
|37\17
''1523; 1.{{Overline|8}}''
|22\10
''1540''
|29\13
''1561; 1, 1.1{{Overline|6}}''
|-
|Reb, Lab
|Gb
|9b, Gc
|9f
|33\15
''1540''
|24\11
''1527.{{Overline|27}}''
|39\18
''1516.{{Overline|6}}''
|36\17
''1482; 2.8{{Overline|3}}''
|21\10
''1470''
|27\13
''1453; 1.{{Overline|18}}''
|-
|'''Re, La'''
|'''G'''
|'''9, G'''
|9
|'''34\15'''
'''''1586.{{Overline|6}}'''''
|'''25\11'''
'''''1590.{{Overline|90}}'''''
|'''41\18'''
'''''1594.{{Overline|4}}'''''
|'''16\7'''
'''''1600'''''
|'''39\17'''
'''''1605; 1.1{{Overline|3}}'''''
|'''23\10'''
'''''1610'''''
|'''30\13'''
'''''1615; 2.6'''''
|-
|Re#, La#
|G#
|9#, G#
|9#
|35\15
''1633.{{Overline|3}}''
|26\11
''1654.{{Overline|54}}''
|43\18
''1672.{{Overline|2}}''
| rowspan="2" |17\7
''1700''
|42\17
''1729; 2, 2.{{Overline|3}}''
|25\10
''1750''
|33\13
''1776; 1.08{{Overline|3}}''
|-
|Mib, Sib
|Ab
|Xb, Ac
|Af
|37\15
''1726.{{Overline|6}}''
|27\11
''1718.{{Overline|18}}''
|44\18
''1711.{{Overline|1}}''
|41\17
''1688; 4.25''
|24\10
''1680''
|31\13
''1669; 4.{{Overline|3}}''
|-
|Mi, Si
|A
|X, A
|A
|38\15
''1773.{{Overline|3}}''
|28\11
''1781.{{Overline|81}}''
|46\18
''1788.{{Overline|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''
| rowspan="2" |29\11
''1845.{{Overline|45}}''
|48\18
''1866.{{Overline|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.{{Overline|6}}''
|47\18
''1827.{{Overline|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.{{Overline|3}}'''''
|'''30\11'''
'''''1909.{{Overline|09}}'''''
|'''49\18'''
'''''1905.{{Overline|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.{{Overline|72}}''
|51\18
''1983.{{Overline|3}}''
|20\7
''2000''
|49\17
''2017; 1.41{{Overline|6}}''
|29\10
''2030''
|38\13
''2046; 6.5''
|-
|Fax, Dox
|B#
|Ex, Cx
|Bx
|43\15
''2006.{{Overline|6}}''
| rowspan="2" |32\11
''2036.{{Overline|36}}''
|53\18
''2061.{{Overline|1}}''
|21\7
''2100''
|52\17
''2141; 5.{{Overline|6}}''
|31\10
''2170''
|41\13
''2207; 1.{{Overline|4}}''
|-
|Dob, Solb
|Hb
|0b, Dc
|Cf
|44\15
''2053.{{Overline|3}}''
|52\18
''2022.{{Overline|2}}''
|20\7
''2000''
|48\17
''1976; 2.125''
|28\10
''1960''
|36\13
''1938; 2.1{{Overline|6}}''
|-
!Do, Sol
!H
!0, D
!C
! colspan="7" |''2100''
|-
|Do#, Sol#
|Η#
|0#, D#
|C#
|46\15
''2146.{{Overline|6}}''
|34\11
''2163.{{Overline|63}}''
|56\18
''2177.{{Overline|7}}''
| rowspan="2" |22\7
''2200''
|54\17
''2223; 1.{{Overline|8}}''
|32\10
''2240''
|42\13
''2261; 1, 1.1{{Overline|6}}''
|-
|Reb, Lab
|Cb
|1b, 1c
|Df
|48\15
''2240''
|35\11
''2227.{{Overline|27}}''
|57\18
''2216.{{Overline|6}}''
|53\17
''2182; 2.8{{Overline|3}}''
|31\10
''2170''
|40\13
''2153; 1.{{Overline|18}}''
|-
|'''Re, La'''
|'''C'''
|'''1'''
|'''D'''
|'''49\15'''
'''''2286.{{Overline|6}}'''''
|'''36\11'''
'''''2290.{{Overline|90}}'''''
|'''59\18'''
'''''2294.{{Overline|4}}'''''
|'''23\7'''
'''''2300'''''
|'''56\17'''
'''''2305; 1.1{{Overline|3}}'''''
|'''33\10'''
'''2310'''
|'''43\13'''
'''''2315; 2.6'''''
|-
|Re#, La#
|C#
|1#
|D#
|50\15
''2223.{{Overline|3}}''
|37\11
''2354.{{Overline|54}}''
|61\18
''2372.''{{Overline|2}}
| rowspan="2" |24\7
''2400''
|59\17
''2429; 2, 2.{{Overline|3}}''
|35\10
''2450''
|46\13
''2476; 1.08{{Overline|3}}''
|-
|Mib, Sib
|Db
|2b, 2c
|Ef
|52\15
''2426.{{Overline|6}}''
|38\11
''2418.{{Overline|18}}''
|62\18
''2411.{{Overline|1}}''
|58\17
''2388; 4.25''
|34\10
''2380''
|44\13
''2369; 4.{{Overline|3}}''
|-
|Mi, Si
|D
|2
|E
|53\15
''2473,{{Overline|3}}''
|39\11
''2481.{{Overline|81}}''
|64\11
''2488.{{Overline|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''
| rowspan="2" |40\11
''2545.{{Overline|45}}''
|66\18
''2566.{{Overline|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.{{Overline|6}}''
|65\18
''2527.{{Overline|7}}''
|25\7
''2500''
|60\17
''2470; 1.7''
|35\10
''2450''
|45\13
''2423; 13''
|-
|'''Fa, Do'''
|'''Eb'''
|'''3'''
|'''Ff'''
|'''56\15'''
'''''2613.{{Overline|3}}'''''
|'''41\11'''
'''''2609.{{Overline|09}}'''''
|'''67\18'''
'''''2605.{{Overline|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.{{Overline|72}}''
|69\18
''2683.{{Overline|3}}''
|27\7
''2700''
|66\17
''2717; 1.41{{Overline|6}}''
|39\10
''2730''
|51\13
''2746; 6.5''
|-
|Fax, Dox
|E#
|3x
|F#
|58\15
''2706.{{Overline|6}}''
| rowspan="2" |43\11
''2736.{{Overline|36}}''
|71\18
''2761.{{Overline|1}}''
|28\7
''2800''
|69\17
''2841; 5.{{Overline|6}}''
|41\10
''2870''
|54\13
''2907; 1.{{Overline|4}}''
|-
|Dob, Solb
|Fb
|4b, 4c
|0f, Gf
|59\15
''2753.{{Overline|3}}''
|70\18
''2722.{{Overline|2}}''
|27\7
''2700''
|65\17
''2676; 2.125''
|38\10
''2660''
|49\13
''2638; 2.1{{Overline|6}}''
|-
!Do, Sol
!F
!4
!0, G
! colspan="7" |''2800''
|}
==Intervals==
{| class="wikitable"
!Generators
!Sesquitave notation
!Interval category name
!Generators
!Notation of 3/2 inverse
!Interval category name
|-
| colspan="6" |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
|-
| colspan="6" |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:
{| class="wikitable"
|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:
{| class="wikitable"
!Mode
!Scale
![[Modal UDP Notation|UDP]]
! colspan="3" |Interval type
|-
!name
!pattern
!notation
!2nd
!3rd
!4th
|-
|Lydian
|LLLs
|<nowiki>3|0</nowiki>
|P
|M
|A
|-
|Major
|LLsL
|<nowiki>2|1</nowiki>
|P
|M
|P
|-
|Minor
|LLsL
|<nowiki>1|2</nowiki>
|P
|m
|P
|-
|Phrygian
|sLLL
|<nowiki>0|3</nowiki>
|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 <code>root-(p-2g)-(2p-3g)</code> (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 = [[Tel:192.6406|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 = [[Tel:218.6371|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:
{| class="wikitable"
! colspan="3" rowspan="2" |Generator
(bright)
! colspan="2" |Cents
! rowspan="2" |L
! rowspan="2" |s
! rowspan="2" |L/s
! rowspan="2" |Comments
|-
!<u>Normalised<ref name=":03">Fractions repeating more than 4 digits written as continued fractions</ref></u>
!''ed7\12<ref name=":04">Fractions repeating more than 4 digits written as continued fractions</ref>''
|-
|1\4
|
|
|<u>171; 2.{{Overline|3}}</u>
|''175''
|1
|1
|1.000
|Equalised
|-
|6\23
|
|
|<u>180</u>
|''182; 1, 1.{{Overline|5}}''
|6
|5
|1.200
|
|-
|
|11\42
|
|<u>180; 1.21{{Overline|6}}</u>
|''183.{{Overline|3}}''
|11
|9
|1.222
|
|-
|5\19
|
|
|<u>181.{{Overline|81}}</u>
|''184; 4.75''
|5
|4
|1.250
|
|-
|
|14\53
|
|<u>182; 1, 1.5</u>
|''184; 1, 9.6''
|14
|11
|1.273
|
|-
|
|9\34
|
|<u>183; 19.{{Overline|6}}</u>
|''185; 3.4''
|9
|7
|1.286
|
|-
|4\15
|
|
|<u>184; 1.625</u>
|''186.{{Overline|6}}''
|4
|3
|1.333
|
|-
|
|11\41
|
|<u>185, 1, 10.8{{Overline|3}}</u>
|''187; 1.{{Overline|24}}''
|11
|8
|1.375
|
|-
|
|7\26
|
|<u>186.{{Overline|6}}</u>
|''188; 2.1{{Overline|6}}''
|7
|5
|1.400
|
|-
|
|10\37
|
|<u>187.5</u>
|''189.{{Overline|189}}''
|10
|7
|1.429
|
|-
|
|13\48
|
|<u>187; 1, 19.75</u>
|''189.58{{Overline|3}}''
|13
|9
|1.444
|
|-
|
|16\59
|
|<u>188; 4.25</u>
|''189; 1, 4.9''
|16
|11
|1.455
|
|-
|3\11
|
|
|<u>189; 2.{{Overline|1}}</u>
|''190.{{Overline|90}}''
|3
|2
|1.500
|Napoli-Meantone starts here
|-
|
|17\62
|
|<u>190; 1, 1.{{Overline|12}}</u>
|''191; 1, 14.5''
|17
|11
|1.545
|
|-
|
|14\51
|
|<u>190.{{Overline|90}}</u>
|''192; 8.375''
|14
|9
|1.556
|
|-
|
|11\40
|
|<u>191; 3, 2.{{Overline|3}}</u>
|''192.5''
|11
|7
|1.571
|
|-
|
|8\29
|
|<u>192</u>
|''193; 9.{{Overline|6}}''
|8
|5
|1.600
|
|-
|
|5\18
|
|<u>193; 1, 1, 4.{{Overline|6}}</u>
|''194.{{Overline|4}}''
|5
|3
|1.667
|
|-
|
|
|12\43
|<u>194.{{Overline|594}}</u>
|''195; 2.8{{Overline|6}}''
|12
|7
|1.714
|
|-
|
|7\25
|
|<u>195; 2.8{{Overline|6}}</u>
|''196''
|7
|4
|1.750
|
|-
|
|9\32
|
|<u>196.{{Overline|36}}</u>
|''196.875''
|9
|5
|1.800
|
|-
|
|11\39
|
|<u>197; 67</u>
|''197; 2, 3.4''
|11
|6
|1.833
|
|-
|
|13\46
|
|<u>197; 2.{{Overline|135}}</u>
|''197; 1, 4.75''
|13
|7
|1.857
|
|-
|
|15\53
|
|<u>197; 1, 2, 1, 1.{{Overline|54}}</u>
|''198; 8.8{{Overline|3}}''
|15
|8
|1.875
|
|-
|
|17\60
|
|<u>198; 17.1{{Overline|6}}</u>
|''198.{{Overline|3}}''
|17
|9
|1.889
|
|-
|
|19\67
|
|<u>198: 3, 1, 28</u>
|''198, 1, 1.{{Overline|03}}''
|19
|10
|1.900
|
|-
|
|21\74
|
|<u>198; 2.3{{Overline|518}}</u>
| ''198.{{Overline |''198.{{Overline|648}}''
|21
|11
|1.909
|
|-
|
|23\81
|
|<u>198; 1, 3, 1.7</u>
|''198; 1, 3, 3.8''
|23
|12
|1.917
|
|-
|
|25\88
|
|<u>198; 1, 2, 12.25</u>
|''198.8{{Overline|63}}''
|25
|13
|1.923
|
|-
|
|27\95
|
|<u>198; 1, 3.{{Overline|405}}</u>
|''198; 1.0{{Overline|5}}''
|27
|14
|1.929
|
|-
|
|29\102
|
|<u>198; 1, 1.1{{Overline|6}}</u>
|''199; 51''
|29
|15
|1.933
|
|-
|2\7
|
|
|<u>200</u>
|''200''
|2
|1
|2.000
|Napoli-Meantone ends, Napoli-Pythagorean begins
|-
|
|17\59
|
|<u>201.{{Overline|9801}}</u>
|''201; 1, 2.2{{Overline|7}}''
|17
|8
|2.125
|
|-
|
|15\52
|
|<u>202; 4.0{{Overline|45}}</u>
|''201; 1.08{{Overline|3}}''
|15
|7
|2.143
|
|-
|
|13\45
|
|<u>202; 1, 1, 2.0{{Overline|6}}</u>
|''202.{{Overline|2}}''
|13
|6
|2.167
|
|-
|
|11\38
|
|<u>203; 13</u>
|''202; 1.58{{Overline|3}}''
|11
|5
|2.200
|
|-
|
|9\31
|
|<u>203; 1, 3.41{{Overline|6}}</u>
|''203; 4, 2.{{Overline|3}}''
|9
|4
|2.250
|
|-
|
|7\24
|
|<u>204; 1. 7.2</u>
|''204.1{{Overline|6}}''
|7
|3
|2.333
|
|-
|
|
|12\41
|<u>205; 1.4</u>
|''204; 1. 7.2''
|12
|5
|2.400
|
|-
|
|5\17
|
|<u>206; 1, 8.{{Overline|6}}</u>
|''205; 1.1{{Overline|3}}''
|5
|2
|2.500
|Napoli-Neogothic heartland is from here…
|-
|
|
|18\61
|<u>207; 1.{{Overline|4}}</u>
|''206; 1.{{Overline|259}}''
|18
|7
|2.571
|
|-
|
|
|13\44
|<u>208</u>
|''206.{{Overline|81}}''
|13
|5
|2.600
|
|-
|
|8\27
|
|<u>208; 1.4375</u>
|''207.{{Overline|407}}''
|8
|3
|2.667
|…to here
|-
|
|11\37
|
|<u>209; 1.{{Overline|90}}</u>
|''208.{{Overline|108}}''
|11
|4
|2.750
|
|-
|
|14\47
|
|<u>210</u>
|''208; 1.958{{Overline|3}}''
|14
|5
|2.800
|
|-
|
|17\57
|
|<u>210; 3.2{{Overline|3}}</u>
|''208; 1.29{{Overline|54}}''
|17
|6
|2.833
|
|-
|
|20\67
|
|<u>210; 1.9</u>
|''208; 1, 21.{{Overline|3}}''
|20
|7
|2.857
|
|-
|
|23\77
|
|<u>210; 1.4{{Overline|5}}</u>
|''209.{{Overline|09}}''
|23
|8
|2.875
|
|-
|3\10
|
|
|<u>211; 1, 3.25</u>
|''210''
|3
|1
|3.000
|Napoli-Pythagorean ends, Napoli-Archy begins
|-
|
|22\73
|
|<u>212; 1, 9.{{Overline|3}}</u>
|''210; 1, 23.{{Overline|3}}''
|22
|7
|3.143
|
|-
|
|19\63
|
|<u>213; 11.{{Overline|8}}</u>
|''211.{{Overline|1}}''
|19
|6
|3.167
|
|-
|
|16\53
|
|<u>213.{{Overline|3}}</u>
|''211; 3, 8.5''
|16
|5
|3.200
|
|-
|
|13\43
|
|<u>213; 1, 2.3{{Overline|18}}</u>
|''211; 1.{{Overline|592}}''
|13
|4
|3.250
|
|-
|
|10\33
|
|<u>214; 3.5</u>
|''212.{{Overline|12}}''
|10
|3
|3.333
|
|-
|
|7\23
|
|<u>215; 2.6</u>
|''213; 23''
|7
|2
|3.500
|
|-
|
|11\36
|
|<u>216; 2.541{{Overline|6}}</u>
|''213.{{Overline|3}}''
|11
|3
|3.667
|
|-
|
|15\49
|
|<u>216; 1.152{{Overline|7}}</u>
|''214; 3.5''
|15
|4
|3.750
|
|-
|
|19\62
|
|<u>217; 7</u>
|''214; 1.9375''
|19
|5
|3.800
|
|-
|4\13
|
|
|<u>218.{{Overline|18}}</u>
|''215; 2.6''
|4
|1
|4.000
|
|-
|
|13\42
|
|<u>219; 1, 2.55</u>
|''216.{{Overline|6}}''
|13
|3
|4.333
|
|-
|
|9\29
|
|<u>220; 2.45</u>
|''217; 4, 7''
|9
|2
|4.500
|
|-
|
|14\45
|
|<u>221; 19</u>
|''217.{{Overline|7}}''
|14
|3
|4.667
|
|-
|5\16
|
|
|<u>222.{{Overline|2}}</u>
|''218.75''
|5
|1
|5.000
|Napoli-Archy ends
|-
|
|16\51
|
|<u>223; 3.{{Overline|90}}</u>
|''219; 1, 1.55''
|16
|3
|5.333
|
|-
|
|11\35
|
|<u>223; 1, 2.6875</u>
|''220''
|11
|2
|5.500
|
|-
|
|17\54
|
|<u>224; 5.7{{Overline|2}}</u>
|''220.{{Overline|370}}''
|17
|3
|5.667
|
|-
|6\19
|
|
|<u>225</u>
|''221; 19''
|6
|1
|6.000
|
|-
|1\3
|
|
|<u>240</u>
|''233.{{Overline|3}}''
|1
|0
|→ inf
|Paucitonic
|}
 
== See also ==
[[3L 1s (3/2-equivalent)]] - idealized tuning<references />

Revision as of 05:41, 26 May 2023

3L1s<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 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

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

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

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

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

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\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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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 = [[1]]

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 = [[2]]

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
Normalised[3] ed7\12[4]
1\4 171; 2.3 175 1 1 1.000 Equalised
6\23 180 182; 1, 1.5 6 5 1.200
11\42 180; 1.216 183.3 11 9 1.222
5\19 181.81 184; 4.75 5 4 1.250
14\53 182; 1, 1.5 184; 1, 9.6 14 11 1.273
9\34 183; 19.6 185; 3.4 9 7 1.286
4\15 184; 1.625 186.6 4 3 1.333
11\41 185, 1, 10.83 187; 1.24 11 8 1.375
7\26 186.6 188; 2.16 7 5 1.400
10\37 187.5 189.189 10 7 1.429
13\48 187; 1, 19.75 189.583 13 9 1.444
16\59 188; 4.25 189; 1, 4.9 16 11 1.455
3\11 189; 2.1 190.90 3 2 1.500 Napoli-Meantone starts here
17\62 190; 1, 1.12 191; 1, 14.5 17 11 1.545
14\51 190.90 192; 8.375 14 9 1.556
11\40 191; 3, 2.3 192.5 11 7 1.571
8\29 192 193; 9.6 8 5 1.600
5\18 193; 1, 1, 4.6 194.4 5 3 1.667
12\43 194.594 195; 2.86 12 7 1.714
7\25 195; 2.86 196 7 4 1.750
9\32 196.36 196.875 9 5 1.800
11\39 197; 67 197; 2, 3.4 11 6 1.833
13\46 197; 2.135 197; 1, 4.75 13 7 1.857
15\53 197; 1, 2, 1, 1.54 198; 8.83 15 8 1.875
17\60 198; 17.16 198.3 17 9 1.889
19\67 198: 3, 1, 28 198, 1, 1.03 19 10 1.900
21\74 198; 2.3518 198.648 21 11 1.909
23\81 198; 1, 3, 1.7 198; 1, 3, 3.8 23 12 1.917
25\88 198; 1, 2, 12.25 198.863 25 13 1.923
27\95 198; 1, 3.405 198; 1.05 27 14 1.929
29\102 198; 1, 1.16 199; 51 29 15 1.933
2\7 200 200 2 1 2.000 Napoli-Meantone ends, Napoli-Pythagorean begins
17\59 201.9801 201; 1, 2.27 17 8 2.125
15\52 202; 4.045 201; 1.083 15 7 2.143
13\45 202; 1, 1, 2.06 202.2 13 6 2.167
11\38 203; 13 202; 1.583 11 5 2.200
9\31 203; 1, 3.416 203; 4, 2.3 9 4 2.250
7\24 204; 1. 7.2 204.16 7 3 2.333
12\41 205; 1.4 204; 1. 7.2 12 5 2.400
5\17 206; 1, 8.6 205; 1.13 5 2 2.500 Napoli-Neogothic heartland is from here…
18\61 207; 1.4 206; 1.259 18 7 2.571
13\44 208 206.81 13 5 2.600
8\27 208; 1.4375 207.407 8 3 2.667 …to here
11\37 209; 1.90 208.108 11 4 2.750
14\47 210 208; 1.9583 14 5 2.800
17\57 210; 3.23 208; 1.2954 17 6 2.833
20\67 210; 1.9 208; 1, 21.3 20 7 2.857
23\77 210; 1.45 209.09 23 8 2.875
3\10 211; 1, 3.25 210 3 1 3.000 Napoli-Pythagorean ends, Napoli-Archy begins
22\73 212; 1, 9.3 210; 1, 23.3 22 7 3.143
19\63 213; 11.8 211.1 19 6 3.167
16\53 213.3 211; 3, 8.5 16 5 3.200
13\43 213; 1, 2.318 211; 1.592 13 4 3.250
10\33 214; 3.5 212.12 10 3 3.333
7\23 215; 2.6 213; 23 7 2 3.500
11\36 216; 2.5416 213.3 11 3 3.667
15\49 216; 1.1527 214; 3.5 15 4 3.750
19\62 217; 7 214; 1.9375 19 5 3.800
4\13 218.18 215; 2.6 4 1 4.000
13\42 219; 1, 2.55 216.6 13 3 4.333
9\29 220; 2.45 217; 4, 7 9 2 4.500
14\45 221; 19 217.7 14 3 4.667
5\16 222.2 218.75 5 1 5.000 Napoli-Archy ends
16\51 223; 3.90 219; 1, 1.55 16 3 5.333
11\35 223; 1, 2.6875 220 11 2 5.500
17\54 224; 5.72 220.370 17 3 5.667
6\19 225 221; 19 6 1 6.000
1\3 240 233.3 1 0 → inf Paucitonic

See also

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

  1. ↑ Fractions repeating more than 4 digits written as continued fractions
  2. ↑ Fractions repeating more than 4 digits written as continued fractions
  3. ↑ Fractions repeating more than 4 digits written as continued fractions
  4. ↑ Fractions repeating more than 4 digits written as continued fractions
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