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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}}).  
 
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.  
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.  


[[Basic]] Angel is in [[7edf]], which is a very good fifth-based equal tuning similar to [[12edo]].
'''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==
== 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.
   
   
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"
{| class="wikitable"
|+ style="font-size: 105%;" | Cents<ref name=":0">Fractions repeating more than 4 digits written as continued fractions</ref>
|+
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
|-
|-
! colspan="4" | Notation
!Diatonic
! Supersoft
! Soft
!Napoli
! Semisoft
! Basic
!Bijou
! Semihard
!Hextone
! Hard
!~15edf
! Superhard
|-
!~11edf
! Diatonic  
! Napoli  
!~18edf
! Bijou
! Hextone
!~7edf
! ~15edf  
! ~11edf  
!~17edf
! ~18edf  
! ~7edf  
!~10edf
! ~17edf  
! ~10edf  
!~13edf
! ~13edf  
|-  
|-
| Do#, Sol#  
| F#  
|Do#, Sol#
| 0#, D#
| 0#, G#
|F#
| 1\1546; 6.5  
| 1\1163: 6.{{Overline|3}}  
|0#, D#
| 2\1877; 2, 2.6  
|0#, G#
| rowspan="2" | 1\7 100  
|1\15
| 3\17124; 7.25  
46; 6.5
| 2\10141; 5.{{Overline|6}}  
| 3\13 163.{{Overline|63}}  
|1\11
|-  
63: 6.{{Overline|3}}
| Reb, Lab  
| Gb  
|2\18
| 1b, 1c
77; 2, 2.6
| 1f
| 3\15138; 3.25  
| rowspan="2" |1\7
| 2\11126; 3.1{{Overline|6}}  
| 3\18116; 7.75  
100
| 2\1782; 1.3{{Overline|18}}  
| 1\1070; 1.7  
|3\17
| 1\13 54.{{Overline|54}}  
124; 7.25
|-  
| '''Re, La'''  
|2\10
| '''G'''  
141; 5.{{Overline|6}}
| '''1'''
| '''1'''  
|3\13
| '''4\15''''''184; 1.625'''  
| '''3\11''''''189; 2.{{Overline|1}}'''  
163.{{Overline|63}}
| '''5\18''''''193; 1, 1, 4.{{Overline|6}}'''  
| '''2\7''' '''200'''  
|-
| '''5\17''''''206; 1, 8.{{Overline|6}}'''  
| '''3\10''''''211; 1, 3.25'''  
|Reb, Lab
| '''4\13''' '''218.{{Overline|18}}'''  
|-  
|Gb
| Re#, La#  
| G#  
|1b, 1c
| 1#
|1f
| 1#
|3\15
| 5\15230; 1.3  
138; 3.25
| 4\11252; 1.58{{Overline|3}}  
| 7\18270; 1.0{{Overline|3}}  
|2\11
| rowspan="2" | 3\7 300  
126; 3.1{{Overline|6}}
| 8\17331; 29  
| 5\10352; 1.0625  
|3\18
| 7\13 381.{{Overline|81}}  
116; 7.75
|-  
| Mib, Sib  
|2\17
| Ab  
82; 1.3{{Overline|18}}
| 2b, 2c
| 2f
|1\10
| 7\15323; 13  
70; 1.7
| 5\11315; 1.2{{Overline|6}}  
| 8\18309; 1, 2.1  
|1\13
| 7\17289; 1, 1.9  
| 4\10282; 2.8{{Overline|3}}  
54.{{Overline|54}}
| 5\13 272.{{Overline|72}}  
|-  
|-
| Mi, Si  
| A  
|'''Re, La'''
| 2
| 2
|'''G'''
| 8\15369; 4.{{Overline|3}}  
| 6\11378; 1.0{{Overline|5}}  
|'''1'''
| 10\18387; 10.{{Overline|3}}  
|'''1'''
| 4\7 400
| 10\17413; 1, 3.8{{Overline|3}}  
|'''4\15'''
| 6\10423; 1.{{Overline|8}}  
'''184; 1.625'''
| 8\13 436.{{Overline|36}}  
|-  
|'''3\11'''
| Mi#, Si#  
'''189; 2.{{Overline|1}}'''
| A#  
| 2#
|'''5\18'''
| 2#
'''193; 1, 1, 4.{{Overline|6}}'''
| 9\15415; 2.6  
| rowspan="2" | 7\11442; 9.5  
|'''2\7'''
| 12\18464; 1.0625  
| 5\7 500  
'''200'''
| 13\17537; 14.5  
| 8\10564; 1.41{{Overline|6}}  
|'''5\17'''
| 11\13 600  
'''206; 1, 8.{{Overline|6}}'''
|-  
| Fab, Dob  
|'''3\10'''
| Bbb  
'''211; 1, 3.25'''
| 3b, 3c
| 3f
|'''4\13'''
| 10\15461; 1, 1.1{{Overline|6}}  
| 11\18425; 1.24  
'''218.{{Overline|18}}'''
| 4\7 400  
| 9\17372; 2.41{{Overline|6}}  
|-
| 5\10352; 1.0625  
| 6\13 327.{{Overline|27}}  
|Re#, La#
|-  
| '''Fa, Do'''  
|G#
| '''Bb'''  
| '''3'''
|1#
| '''3'''  
|1#
| '''11\15''''''507; 1.{{Overline|4}}'''  
|5\15
| '''8\11''''''505; 3.8'''  
230; 1.3
| '''13\18''''''503; 4, 2.{{Overline|3}}'''  
| '''5\7''' '''500'''  
|4\11
| '''12\17''''''496; 1.8125'''  
252; 1.58{{Overline|3}}
| '''7\10''''''494; 8.5'''  
| '''9\13''' '''490.{{Overline|90}}'''  
|7\18
|-  
270; 1.0{{Overline|3}}
| Fa#, Do#  
| B  
| rowspan="2" |3\7
| 3#
| 3#
300
| 12\15553; 1.{{Overline|18}}  
| 9\11568; 2.375  
|8\17
| 15\18580; 1.55  
331; 29
| 6\7 600  
| 15\17620; 1.45  
|5\10
| 9\10635; 3.4  
352; 1.0625
| 12\13 654.{{Overline|54}}  
|-
|7\13
| Fax, Dox  
| B#  
381.{{Overline|81}}
| 3x
| 3x
|-
| 13\15 600  
| rowspan="2" | 10\11 631; 1.{{Overline|72}}  
|Mib, Sib
| 17\18 658; 15.5  
| 7\7 700  
|Ab
| 18\17 744; 1.208{{Overline|3}}  
| 11\10 776; 2.125  
|2b, 2c
| 15\13 818.{{Overline|18}}  
|2f
|-  
|7\15
| Dob, Solb
323; 13
| Hb
| 4b, 4c
|5\11
| 4f
315; 1.2{{Overline|6}}
| 14\15 646; 6.5
| 16\18 619; 2.{{Overline|81}}
|8\18
| 6\7 600
309; 1, 2.1
| 14\17 579; 3.{{Overline|2}}
| 8\10564; 1.41{{Overline|6}}
|7\17
| 10\13 545.{{Overline|45}}  
289; 1, 1.9
|-  
! Do, Sol  
|4\10
! H  
282; 2.8{{Overline|3}}
! 4
! 4
|5\13
! '''15\15''' '''692; 3.25'''  
! '''11\11''' '''694; 1, 2.8'''  
272.{{Overline|72}}
! '''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}}  
|Mi, Si
! '''13\13''' '''709.'''{{Overline|09}}  
|-  
|A
| Do#, Sol#  
| Η#  
|2
| 4#
|2
| 4#
|8\15
| 16\15 738; 2.1{{Overline|6}}  
369; 4.{{Overline|3}}
| 12\11 757; 1, 8.5  
| 20\18 774; 5, 6  
|6\11
| rowspan="2" | 8\8 800  
378; 1.0{{Overline|5}}
| 20\17 827; 1, 1.41{{Overline|6}}  
| 12\10 847; 17  
|10\18
| 16\13 872.{{Overline|72}}  
387; 10.{{Overline|3}}
|-  
| Reb, Lab  
|4\7
| Cb  
| 5b, 5c
400
| 5
| 18\15 830; 1.3  
|10\17
| 13\11 821; 19  
413; 1, 3.8{{Overline|3}}
| 21\18 812; 1, 9.{{Overline|3}}  
| 19\17 786; 4.8{{Overline|3}}  
|6\10
| 11\10 776; 2.125  
423; 1.{{Overline|8}}
| 14\13 763.{{Overline|63}}  
|-  
|8\13
| '''Re, La'''  
| '''C'''  
436.{{Overline|36}}
| '''5'''
| '''5'''  
|-
| '''19\15''' '''876; 1.08{{Overline|3}}'''  
| '''14\11''' '''884; 4.75'''  
|Mi#, Si#
| '''23\18''' '''890; 3.1'''  
| '''9\5''' '''900'''  
|A#
| '''22\17''' '''910; 2.9'''  
| '''13\10''' '''917; 1.{{Overline|54}}'''  
|2#
| '''17\13''' '''927.{{Overline|27}}'''  
|2#
|-  
|9\15
| Re#, La#  
415; 2.6
| C#  
| 5#
| 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#
|5#
| 20\15 923: 13
|20\15
| 15\11 947; 2, 1.4
| 25\18 967; 1, 2.875
923: 13
| rowspan="2" | 10\7 1000
| 25\17 1034; 2, 14
|15\11
| 15\10 1058; 1, 4.{{Overline|6}}
| 20\13 1090.{{Overline|90}}
947; 2, 1.4
|-
| Mib, Sib
|25\18
| Db
| 6b, 6c
967; 1, 2.875
| 6f
| 22\15 1015; 2.6
| rowspan="2" |10\7
| 16\11 1010; 1.9
| 26\18 1006; 2, 4.{{Overline|6}}
1000
| 24\17 993; 9.{{Overline|6}}
| 14\10 988; 4.25
|25\17
| 18\13 981.{{Overline|81}}
|-
1034; 2, 14
| Mi, Si
| D
| 15\10
| 6
| 6
1058; 1, 4.{{Overline|6}}
| 23\15 1061; 1, 1.1{{Overline|6}}
| 17\11 1073; 1, 2.1{{Overline|6}}
|20\13
| 28\18 1083; 1.{{Overline|148}}
| 11\7 1100
1090.{{Overline|90}}
| 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#
|Mib, Sib
| Η#
| 0#, D#
|Db
| C#
| 46\152123; 13
|6b, 6c
| 34\112147; 2, 1.4
|6f
| 56\182167; 1, 2.875
|22\15
| rowspan="2" | 22\72200
| 54\172234; 2, 14
1015; 2.6
| 32\102258; 1, 4.{{Overline|6}}
| 42\132090.{{Overline|90}}
|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
| Reb, Lab
| Cb
|Gb
| 1b, 1c
| Df
|9b, Gc
| 48\152215; 2.6
|9f
| 35\112210; 1.9
|33\15
| 57\182206; 2, 4.{{Overline|6}}
| 53\172193; 9.{{Overline|6}}
1523; 13
| 31\10 2188; 4.25
| 40\132181.{{Overline|81}}
|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'''
|'''Re, La'''
| '''C'''
| '''1'''
|'''G'''
| '''D'''
| '''49\15''''''2261; 1, 1.1{{Overline|6}}'''
|'''9, G'''
| '''36\11''''''2273; 1, 2.1{{Overline|6}}'''
|9
| '''59\18''''''2283; 1.{{Overline|148}}'''
|'''34\15'''
| '''23\7''''''2300'''
| '''56\17''''''2317; 4, 7'''
'''1569; 4.{{Overline|3}}'''
| '''33\10''''''2329; 2, 2.{{Overline|3}}'''
| '''43\13''''''2245.{{Overline|45}}'''
|'''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#
|Re#, La#
| C#
| 1#
|G#
| D#
| 50\152307; 1.{{Overline|4}}
|9#, G#
| 37\112336; 1.1875
|9#
| 61\182361; 3.{{Overline|4}}
|35\15
| rowspan="2" | 24\72400
| 59\172441; 2.{{Overline|63}}
1615; 2.6
| 35\102470; 1.7
| 46\132509.{{Overline|09}}
|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
|Mib, Sib
| Db
| 2b, 2c
|Ab
| Ef
| 52\152400
|Xb, Ac
| 38\112400
|Af
| 62\182400
|37\15
| 58\172400
| 34\102400
1707; 1.{{Overline|4}}
| 44\132400
|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
|Mi, Si
| D
|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
|2
| E
|8\15
| 53\152446; 6.5
| 39\112463; 6.{{Overline|3}}
''373.{{Overline|3}}''
| 64\182477; 2, 2.6
|6\11
| 25\72500
| 61\172524; 7.25
''381.{{Overline|81}}''
| 36\102541; 5.{{Overline|6}}
|10\18
| 47\132563.{{Overline|63}}
''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
| Mi#, Si#
!H
| D#
!4
| 2#
!4
| E#
! colspan="7" |''700''
| 54\152492; 3.25
|-
| rowspan="2" | 40\112526; 3.1
|Do#, Sol#
| 66\182554; 1, 5.2
|Η#
| 26\72600
|4#
| 64\172648; 2.625
|4#
| 38\102682; 2.8{{Overline|3}}
|16\15
| 50\132727.{{Overline|27}}
''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
| Fab, Dob
| Ebb
| Ebb
| 7b, 7c
| 3b, 3c
|7f
| Fff
|25\15
| 55\152538; 2.1{{Overline|6}}
| 65\182516; 7.75
''1166.{{Overline|6}}''
| 25\72500
|29\18
| 60\172482; 1.3{{Overline|18}}
| 35\102470; 1.7
''1127.{{Overline|7}}''
| 45\132454.{{Overline|54}}
|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#
| '''Fa, Do'''
|G#
| '''Eb'''
|9#, G#
| '''3'''
|9#
| '''Ff'''
|35\15
| '''56\15''''''2584; 1.625'''
| '''41\11''''''2589; 2.{{Overline|1}}'''
''1633.{{Overline|3}}''
| '''67\18''''''2593; 1, 1, 4.{{Overline|6}}'''
|26\11
| '''26\7''''''2600'''
| '''63\17''''''2606; 1, 8.{{Overline|6}}'''
''1654.{{Overline|54}}''
| '''37\10''''''2611; 1, 3.25'''
|43\18
| '''48\13''''''2618.{{Overline|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
| Fa#, Do#
| Ab
| E
|Xb, Ac
| 3#
|Af
| F
|37\15
| 57\152630; 1.3
| 42\112652; 1.58{{Overline|3}}
''1726.{{Overline|6}}''
| 69\182670; 1.0{{Overline|3}}
| 27\11
| 27\72700
| 66\172731; 29
''1718.{{Overline|18}}''
| 39\102752; 1.0625
|44\18
| 51\132781.{{Overline|81}}
''1711.{{Overline|1}}''
|41\17
''1688; 4.25''
| 24\10
''1680''
|31\13
''1669; 4.{{Overline|3}}''
|-
|-
|Mi, Si
| Fax, Dox
|A
| E#
|X, A
| 3x
|A
| F#
|38\15
| 58\152676; 1.08{{Overline|3}}
| rowspan="2" | 43\112715; 1.2{{Overline|6}}
''1773.{{Overline|3}}''
| 71\182748; 2.58{{Overline|3}}
|28\11
| 28\72800
| 69\172855; 4.8
''1781.{{Overline|81}}''
| 41\102894; 8.5
|46\18
| 54\132945.{{Overline|45}}
''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
| Dob, Solb
|Hb
| Fb
|0b, Dc
| 4b, 4c
|Cf
| 0f, Gf
|44\15
| 59\152723; 13
| 70\182709; 1, 2.1
''2053.{{Overline|3}}''
| 27\72700
|52\18
| 65\172689; 1, 1.9
| 38\102682; 2.8{{Overline|3}}
''2022.{{Overline|2}}''
| 49\132672.{{Overline|72}}
|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
! Do, Sol
!F
! F
!4
! 4
!0, G
! 0, G
! colspan="7" |''2800''
! 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}}
|}
|}
==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==
== Modes ==
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:
The mode names are based on the species of fifth:
{| class="wikitable"
{{MOS modes
!Mode
| Mode Names=
!Scale
Lydian $
![[Modal UDP Notation|UDP]]
Minor $
! colspan="3" |Interval type
Major $
|-
Phrygian $
!name
|}
!pattern
 
!notation
== Temperaments ==
!2nd
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.
!3rd
 
!4th
=== Napoli-Meantone ===
|-
|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
[[Subgroup]]: 3/2.6/5.8/5
   
   
Line 2,958: Line 667:
[[Mapping]]: [{{val|1 1 2}}, {{val|0 -2 -3}}]
[[Mapping]]: [{{val|1 1 2}}, {{val|0 -2 -3}}]


[[Vals]]: {{val list|~(7edf, 11edf, 18edf)}}
[[Optimal ET sequence]]: ~(7edf, 11edf, 18edf)
==='''Napoli-Superpyth'''===
 
=== Napoli-Archy ===
[[Subgroup]]: 3/2.7/6.14/9
[[Subgroup]]: 3/2.7/6.14/9
   
   
Line 2,969: Line 678:
[[Mapping]]: [{{val|1 1 2}}, {{val|0 -2 -3}}]
[[Mapping]]: [{{val|1 1 2}}, {{val|0 -2 -3}}]


[[Vals]]: {{val list|~(7edf, 10edf, 13edf, 16edf)}}
[[Optimal ET sequence]]: ~(7edf, 10edf, 13edf, 16edf)
===Scale tree===
 
=== Scale tree ===
The spectrum looks like this:
The spectrum looks like this:
{| class="wikitable"
{{MOS tuning spectrum
! colspan="3" rowspan="2" |Generator
| 3/2 = Napoli-Meantone starts here
| 2/1 = Napoli-Meantone ends, Napoli-Pythagorean begins
(bright)
| 5/2 = Napoli-Neogothic heartland is from here...
! colspan="2" |Cents
| 8/3 = ...to here
! rowspan="2" |L
| 3/1 = Napoli-Pythagorean ends, Napoli-Archy begins
! rowspan="2" |s
| 5/1 = Napoli-Archy ends
! 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-Superpyth 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-Superpyth 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
|}

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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