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{{Infobox MOS
'''Angel''' is a name* proposed by Mason Green for the temperament that tempers out 81:80 (thus, it is a meantone system), and has a period that is a flattened 3:2, four of which make a pentave (5:1), while its generator is an octave. This temperament is very closely related to quarter-comma meantone (which was the standard for most Western classical music); the key difference is that meantone has a period of an octave and a fifth as a generator, whereas the roles of the fifth and octave are reversed in angel. (Due to the period being a fifth, setting the generator to an octave is equivalent to using a perfect fourth or whole tone as the generator instead).
| Name = Angel
| Equave = 3/2
| nLargeSteps = 3
| nSmallSteps = 1
| Equalized = 2
| Paucitonic = 1
| Pattern = LLLs
}}'''Angel''' is a name* proposed by Mason Green for the temperament that tempers out 81:80 (thus, it is a meantone system), and has a period that is a flattened 3:2, four of which make a pentave (5:1), while its generator is an octave. This temperament is very closely related to quarter-comma meantone (which was the standard for most Western classical music); the key difference is that meantone has a period of an octave and a fifth as a generator, whereas the roles of the fifth and octave are reversed in angel. (Due to the period being a fifth, setting the generator to an octave is equivalent to using a perfect fourth or whole tone as the generator instead).


If the pentaves are required to be exactly 5:1, then the fifths will be exactly the same size as the fifths of quarter-comma meantone (namely, the fourth root of five), but the octaves will be slightly flat (by less than half a cent). On the other hand, if the octaves are made perfect (making this a [[31edo|31edo]] temperament), the pentaves will be slightly sharp. Both options are perceptually very close to one another.
If the pentaves are required to be exactly 5:1, then the fifths will be exactly the same size as the fifths of quarter-comma meantone (namely, the fourth root of five), but the octaves will be slightly flat (by less than half a cent). On the other hand, if the octaves are made perfect (making this a [[31edo|31edo]] temperament), the pentaves will be slightly sharp. Both options are perceptually very close to one another.
Line 19: Line 11:
Straight-fretted angel guitars would be a possibility; such guitars would have unequally spaced frets and would need to be tuned in [https://en.wikipedia.org/wiki/All_fifths_tuning all-fifths], since the period is a fifth.
Straight-fretted angel guitars would be a possibility; such guitars would have unequally spaced frets and would need to be tuned in [https://en.wikipedia.org/wiki/All_fifths_tuning all-fifths], since the period is a fifth.


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 Angel 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]].
==Notation==
There are 3 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 diatonic scales as repeating at the double or triple sesquitave (major ninth or thirteenth), 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] or a major thirteenth which is the Dorian mode of Bijou[9L 3s]. Since there are exactly 8 naturals in double sesquitave notation and 12 in triple sesquitave notation, letters A-H (FGABHCDEF) or dozenal digits (0123456789XE0 or D1234567FGACD with flats written C molle) may be used.
{| class="wikitable"
|+
Cents
! colspan="3" |Notation
!Supersoft
!Soft
!Semisoft
!Basic
!Semihard
!Hard
!Superhard
|-
!Angel
!Napoli
!Bijou
!~15edf
!~11edf
!~18edf
!~7edf
!~17edf
!~10edf
!~13edf
|-
|Do#, Sol#
|F#
|0#, D#
|1\15
46.153…
|1\11
63.157…
|2\18
77.419…
| rowspan="2" |1\7
100
|3\17
124.137…
|2\10
141.176…
|3\13
163.{{Overline|63}}
|-
|Reb, Lab
|Gb
|1b, 1c
|3\15
138.461…
|2\11
126.315…
|3\18
116.129…
|2\17
82.758…
|1\10
70.588…
|1\13
54.{{Overline|54}}
|-
|'''Re, La'''
|'''G'''
|'''1'''
|'''4\15'''
'''184.615…'''
|'''3\11'''
'''189.473…'''
|'''5\18'''
'''193.548…'''
|'''2\7'''
'''200'''
|'''5\17'''
'''206.896…'''
|'''3\10'''
'''211.764…'''
|'''4\13'''
'''218.{{Overline|18}}'''
|-
|Re#, La#
|G#
|1#
|5\15
230.769…
|4\11
252.631…
|7\18
270.967…
| rowspan="2" |3\7
300
|8\17
331.034…
|5\10
352.941…
|7\13
381.{{Overline|81}}
|-
|Mib, Sib
|Ab
|2b, 2c
|7\15
323.076…
|5\11
315.789…
|8\18
309.677…
|7\17
289.655…
|4\10
282.352…
|5\13
272.{{Overline|72}}
|-
|Mi, Si
|A
|2
|8\15
369.230…
|6\11
378.947…
|10\18
387.096…
|4\7
400
|10\17
413.793…
|6\10
423.529…
|8\13
436.{{Overline|36}}
|-
|Mi#, Si#
|A#
|2#
|9\15
415.384…
| rowspan="2" |7\11
442.105…
|12\18
464.516…
|5\7
500
|13\17
537.931…
|8\10
564.705…
|11\13
600
|-
|Fab, Dob
|Bbb
|3bb, 3cc
|10\15
461.538…
|11\18
425.806…
|4\7
400
|9\17
372.413…
|5\10
352.941…
|6\13
327.{{Overline|27}}
|-
|'''Fa, Do'''
|'''Bb'''
|'''3b, 3c'''
|'''11\15'''
'''507.692…'''
|'''8\11'''
'''505.263…'''
|'''13\18'''
'''503.225…'''
|'''5\7'''
'''500'''
|'''12\17'''
'''496.551…'''
|'''7\10'''
'''494.117…'''
|'''9\13'''
'''490.{{Overline|90}}'''
|-
|Fa#, Do#
|B
|3
|12\15
553.846…
|9\11
568.421…
|15\18
580.645…
|6\7
600
|15\17
620.689…
|9\10
635.294…
|12\13
654.{{Overline|54}}
|-
|Fax, Dox
|B#
|3#
|13\15
600
| rowspan="2" |10\11
631.578…
|17\18
658.064…
|7\7
700
|18\17
744.827…
|11\10
776.470…
|15\13
818.{{Overline|18}}
|-
|Dob, Solb
|Hb
|4b, 4c
|14\15
646.153…
|16\18
619.354…
|6\7
600
|14\17
579.310…
|8\10
564.705…
|10\13
545.{{Overline|45}}
|-
!Do, Sol
!H
!4
!'''15\15'''
'''692.307…'''
!'''11\11'''
'''694.736…'''
!'''18\18'''
'''696.774…'''
!'''7\7'''
'''700'''
!'''17\17'''
'''703.448…'''
!'''10\10'''
'''705.882…'''
!'''13\13'''
'''709.'''{{Overline|09}}
|-
|Do#, Sol#
|Η#
|4#
|16\15
738.461…
|12\11
757.894…
|20\18
774.193…
| rowspan="2" |8\8
800
|20\17
827.586…
|12\10
847.058…
|16\13
872.{{Overline|72}}
|-
|Reb, Lab
|Cb
|5b, 5c
|18\15
830.769…
|13\11
821.052…
|21\18
812.903…
|19\17
786.206…
|11\10
776.470…
|14\13
763.{{Overline|63}}
|-
|'''Re, La'''
|'''C'''
|'''5'''
|'''19\18'''
'''876.923…'''
|'''14\11'''
'''884.210…'''
|'''23\18'''
'''890.322…'''
|'''9\5'''
'''900'''
|'''22\17'''
'''910.344…'''
|'''13\10'''
'''917.647…'''
|'''17\13'''
'''927.{{Overline|27}}'''
|-
|Re#, La#
|C#
|5#
|20\15
923.076…
|15\11
947.368…
|25\18
967.741…
| rowspan="2" |10\7
1000
|25\17
1034.482…
|15\10
1058.823…
|20\13
1090.{{Overline|90}}
|-
|Mib, Sib
|Db
|6b, 6c
|22\15
1015.384…
|16\11
1010.526…
|26\18
1006.451…
|24\17
993.103…
|14\10
988.235…
|18\13
981.{{Overline|81}}
|-
|Mi, Si
|D
|6
|23\15
1061.538…
|17\11
1073.684…
|28\18
1083.870…
|11\7
1100
|27\17
1117.241…
|16\10
1129.411…
|21\9
1145.{{Overline|45}}
|-
|Mi#, Si#
|D#
|6#
|24\15
1107.692…
| rowspan="2" |18\11
1136.842…
|30\18
1161.290…
|12\7
1200
|30\17
1241.379…
|18\10
1270.588…
|24\13
1309.{{Overline|09}}
|-
|Fab, Dob
|Ebb
|7bb, 7cc
|25\15
1153.846…
|29\18
1122.580…
|11\7
1100
|26\17
1075.862…
|15\10
1058.823…
|19\13
1036.{{Overline|36}}
|-
|'''Fa, Do'''
|'''Eb'''
|'''7b, 7c'''
|'''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
|27\15
1246.153…
|20\11
1263.157…
|33\18
1277.419…
|13\7
1300
|32\17
1324.137…
|19\10
1341.176…
|25\13
1363.{{Overline|63}}
|-
|Fax, Dox
|E#
|7#
|28\15
1292.307…
| rowspan="2" |21\11
1326.315…
|35\18
1354.838…
|14\7
1400
|35\17
1448.275…
|21\10
1482.352…
|28\13
1527.{{Overline|27}}
|-
|Dob, Solb
|Fb
|8b, Fc
|29\15
1338.461…
|34\18
1316.129…
|13\7
1300
|31\17
1282.758…
|18\10
1270.588…
|23\18
1254.{{Overline|54}}
|-
!Do, Sol
!F
!8, F
!30\15
1384.615…
!22\11
1389.473…
!36\18
1393.548…
!14\7
1400
!34\17
1406.896…
!20\10
1411.764…
!26\9
1418.{{Overline|18}}
|-
|Do#, Sol#
|F#
|8#, F#
|31\15
1430.769…
|23\11
1452.631…
|38\18
1470.967…
| rowspan="2" |15\7
1500
|37\17
1531.034…
|22\10
1552.941…
|29\13
1581.{{Overline|81}}
|-
|Reb, Lab
|Gb
|9b, Gc
|33\15
1523.076…
|24\11
1515.789…
|39\18
1509.677…
|36\17
1489.655…
|21\10
1482.352…
|27\13
1472.{{Overline|72}}
|-
|'''Re, La'''
|'''G'''
|'''9, G'''
|'''34\15'''
'''1569.230…'''
|'''25\11'''
'''1578.947…'''
|'''41\18'''
'''1587.096…'''
|'''16\7'''
'''1600'''
|'''39\17'''
'''1613.793…'''
|'''23\10'''
'''1623.529…'''
|'''30\13'''
'''1636.{{Overline|36}}'''
|-
|Re#, La#
|G#
|9#, G#
|35\15
1615.384…
|26\11
1642.105…
|43\18
1664.516…
| rowspan="2" |17\7
1700
|42\17
1737.931…
|25\10
1764.705…
|33\13
1800
|-
|Mib, Sib
|Ab
|Xb, Ac
|37\15
1707.692…
|27\11
1705.263…
|44\18
1703.225…
|41\17
1696.551…
|24\10
1694.117…
|31\13
1690.{{Overline|90}}
|-
|Mi, Si
|A
|X, A
|38\15
1753.846…
|28\11
1768.421…
|46\18
1780.645…
|18\7
1800
|44\17
1820.689…
|26\10
1835.294…
|34\13
1854.{{Overline|54}}
|-
|Mi#, Si#
|A#
|X#, A#
|39\15
1800
| rowspan="2" |29\11
1831.578…
|48\18
1858.064…
|19\7
1900
|47\17
1944.827…
|28\10
1976.470…
|37\13
2018.{{Overline|18}}
|-
|Fab, Dob
|Bbb
|Ebb, Ccc
|40\15
1846.153…
|47\18
1819.354…
|18\7
1800
|43\17
1779.310…
|25\10
1764.705…
|32\13
1745.4̄5̄
|-
|'''Fa, Do'''
|'''Bb'''
|Eb, Cc
|'''41\15'''
'''1892.307…'''
|'''30\11'''
'''1894.736…'''
|'''49\18'''
'''1896.774…'''
|'''19\7'''
'''1900'''
|'''46\17'''
'''1903.448…'''
|'''27\10'''
'''1905.882…'''
|'''35\13'''
'''1909.{{Overline|09}}'''
|-
|Fa#, Do#
|B
|E, C
|42\15
1938.461…
|31\11
1957.894…
|51\18
1974.193…
|20\7
2000
|49\17
2027.586…
|29\10
1976.470…
|38\13
2072.{{Overline|72}}
|-
|Fax, Dox
|B#
|Ex, Cx
|43\15
1984.615…
| rowspan="2" |32\11
2021.052…
|53\18
2051.612…
|21\7
2100
|52\17
2151.724…
|31\10
2188.235…
|41\13
2236.{{Overline|36}}
|-
|Dob, Solb
|Hb
|0b, Dc
|44\15
2030.769…
|52\18
2012.903…
|20\7
2000
|48\17
1986.206…
|28\10
1967.470…
|36\13
1963.{{Overline|63}}
|-
!Do, Sol
!H
!0, D
!45\15
2076.923…
!33\11
2084.210…
!54\18
2090.322…
!21\7
2100
!51\17
2110.344…
!30\10
2117.647…
!39\13
2127.{{Overline|27}}
|}
{| class="wikitable"
|+Relative cents
! colspan="3" |Notation
!Supersoft
!Soft
!Semisoft
!Basic
!Semihard
!Hard
!Superhard
|-
!Angel
!Napoli
!Bijou
!~15edf
!~11edf
!~18edf
!~7edf
!~17edf
!~10edf
!~13edf
|-
|Do#, Sol#
|F#
|0#, D#
|1\15
''46.{{Overline|6}}''
|1\11
''63.{{Overline|63}}''
|2\18
''77.{{Overline|7}}''
| rowspan="2" |1\7
''100''
|3\17
''123.529…''
|2\10
''140''
|3\13
''161.538…''
|-
|Reb, Lab
|Gb
|1b, 1c
|3\15
''140''
|2\11
''127.{{Overline|27}}''
|3\18
''116.{{Overline|6}}''
|2\17
''82.352…''
|1\10
''70''
|1\13
''53.846…''
|-
|'''Re, La'''
|'''G'''
|'''1'''
|'''4\15'''
'''''186.{{Overline|6}}'''''
|'''3\11'''
'''''190.{{Overline|90}}'''''
|'''5\18'''
'''''194.{{Overline|4}}'''''
|'''2\7'''
'''''200'''''
|'''5\17'''
'''''205.882…'''''
|'''3\10'''
'''''210'''''
|'''4\13'''
'''''215.384…'''''
|-
|Re#, La#
|G#
|1#
|5\15
''233.{{Overline|3}}''
|4\11
''254.{{Overline|54}}''
|7\18
''272.2̄''
| rowspan="2" |3\7
''300''
|8\17
''329.411…''
|5\10
''350''
|7\13
''376.923…''
|-
|Mib, Sib
|Ab
|2b, 2c
|7\15
''326.{{Overline|6}}''
|5\11
''318.{{Overline|18}}''
|8\18
''311.1̄''
|7\17
''288.235…''
|4\10
''280''
|5\13
''269.230…''
|-
|Mi, Si
|A
|2
|8\15
''373.{{Overline|3}}''
|6\11
''381.{{Overline|81}}''
|10\18
''388.{{Overline|8}}''
|4\7
''400''
|10\17
''411.764…''
|6\10
''420''
|8\13
''430.769…''
|-
|Mi#, Si#
|A#
|2#
|9\15
''420''
| rowspan="2" |7\11
''445.{{Overline|45}}''
|12\18
''466.{{Overline|6}}''
|5\7
''500''
|13\17
''535.294…''
|8\10
''560''
|11\13
''592.307…''
|-
|Fab, Dob
|Bbb
|3bb, 3cc
|10\15
''466.{{Overline|6}}''
|11\18
''427.{{Overline|7}}''
|4\7
''400''
|9\17
''370.588…''
|5\10
''350''
|6\13
''323.076.…''
|-
|'''Fa, Do'''
|'''Bb'''
|'''3b, 3c'''
|'''11\15'''
'''''513.{{Overline|3}}'''''
|'''8\11'''
'''''509.{{Overline|09}}'''''
|'''13\18'''
'''''505.{{Overline|5}}'''''
|'''5\7'''
'''''500'''''
|'''12\17'''
'''''494.117…'''''
|'''7\10'''
'''''490'''''
|'''9\13'''
'''''484.615…'''''
|-
|Fa#, Do#
|B
|3
|12\15
''560''
|9\11
''572.{{Overline|72}}''
|15\18
''583.{{Overline|3}}''
|6\7
''600''
|15\17
''617.647…''
|9\10
''630''
|12\13
''646.153…''
|-
|Fax, Dox
|B#
|3#
|13\15
''606.{{Overline|6}}''
| rowspan="2" |10\11
''636.{{Overline|36}}''
|17\18
''661.{{Overline|6}}''
|7\7
''700''
|18\17
''741.176…''
|11\10
''770''
|15\13
''807.692…''
|-
|Dob, Solb
|Hb
|4b, 4c
|14\15
''653.{{Overline|3}}''
|16\18
''622.{{Overline|2}}''
|6\7
''600''
|14\17
''576.470…''
|8\10
''560''
|10\13
''538.461…''
|-
!Do, Sol
!H
!4
! colspan="7" |''700''
|-
|Do#, Sol#
|Η#
|4#
|16\15
''746.{{Overline|6}}''
|12\11
''763.{{Overline|63}}''
|20\18
''777.{{Overline|7}}''
| rowspan="2" |8\7
''800''
|20\17
''823.529…''
|12\10
''840''
|16\13
''861.538…''
|-
|Reb, Lab
|Cb
|5b, 5c
|18\15
''840''
|13\11
''827.{{Overline|27}}''
|21\18
''816.{{Overline|6}}''
|19\17
''782.352…''
|11\10
''770''
|14\13
''753.846…''
|-
|'''Re, La'''
|'''C'''
|'''5'''
|'''19\15'''
'''''886.{{Overline|6}}'''''
|'''14\11'''
'''''890.{{Overline|90}}'''''
|'''23\18'''
'''''894.{{Overline|4}}'''''
|'''9\7'''
'''''900'''''
|'''22\17'''
'''''905.882…'''''
|'''13\10'''
'''''910'''''
|'''17\13'''
'''''915.384…'''''
|-
|Re#, La#
|C#
|5#
|20\15
''933.{{Overline|3}}''
|15\11
''954.{{Overline|54}}''
|25\18
''972.2̄''
| rowspan="2" |10\7
''1000''
|25\17
''1029.411…''
|15\10
''1050''
|20\13
''1076.923…''
|-
|Mib, Sib
|Db
|6b, 6c
|22\15
''1026.{{Overline|6}}''
|16\11
''1018.{{Overline|18}}''
|26\18
''1011.{{Overline|1}}''
|24\17
''988.235…''
|14\10
''980''
|18\13
''969.230…''
|-
|Mi, Si
|D
|6
|23\15
''1073.{{Overline|3}}''
|17\11
''1081.{{Overline|81}}''
|28\18
''1088.{{Overline|8}}''
|11\7
''1100''
|27\17
''1111.764…''
|16\10
''1120''
|21\13
''1130.769…''
|-
|Mi#, Si#
|D#
|6#
|24\15
''1120''
| rowspan="2" |18\11
''1145.{{Overline|45}}''
|30\18
''1166.{{Overline|6}}''
|12\7
''1200''
|30\17
''1235.294…''
|18\10
''1260''
|24\13
''1292.307…''
|-
|Fab, Dob
|Ebb
|7bb, 7cc
|25\15
''1166.{{Overline|6}}''
|29\18
''1127.{{Overline|7}}''
|11\7
''1100''
|26\17
''1070.588…''
|15\10
''1050''
|19\13
''1023.076…''
|-
|'''Fa, Do'''
|'''Eb'''
|'''7b, 7c'''
|'''26\15'''
'''''1213.{{Overline|3}}'''''
|'''19\11'''
'''''1209.{{Overline|09}}'''''
|'''31\18'''
'''''1205.{{Overline|5}}'''''
|'''12\7'''
'''''1200'''''
|'''29\17'''
'''''1194.117…'''''
|'''17\10'''
'''''1190'''''
|'''22\13'''
'''''1184.615…'''''
|-
|Fa#, Do#
|E
|7
|27\15
''1260''
|20\11
''1272.{{Overline|72}}''
|33\18
''1283.{{Overline|3}}''
|13\7
''1300''
|32\17
''1317.647…''
|19\10
''1330''
|25\13
''1346.153…''
|-
|Fax, Dox
|E#
|7#
|28\15
''1306.{{Overline|6}}''
| rowspan="2" |21\11
''1336.{{Overline|36}}''
|35\18
''1361.{{Overline|1}}''
|14\7
''1400''
|35\17
''1441.176…''
|21\10
''1470''
|28\13
''1507.692…''
|-
|Dob, Solb
|Fb
|8b, Fc
|29\15
''1333.{{Overline|3}}''
|34\18
''1322.{{Overline|2}}''
|13\7
''1300''
|31\17
''1276.470…''
|18\10
''1260''
|23\13
''1238.461…''
|-
!Do, Sol
!F
!8, F
! colspan="7" |''1400''
|-
|Do#, Sol#
|F#
|8#, F#
|31\15
''1446.{{Overline|6}}''
|23\11
''1463.{{Overline|63}}''
|38\18
''1477.7̄''
| rowspan="2" |15\7
''1500''
|37\17
''1523.529…''
|22\10
''1540''
|29\13
''1561.538…''
|-
|Reb, Lab
|Gb
|9b, Gc
|33\15
''1540''
|24\11
''1527.{{Overline|27}}''
|39\18
''1516.{{Overline|6}}''
|36\17
''1482.352…''
|21\10
''1470''
|27\13
''1453.846…''
|-
|'''Re, La'''
|'''G'''
|'''9, G'''
|'''34\15'''
'''''1586.{{Overline|6}}'''''
|'''25\11'''
'''''1590.{{Overline|90}}'''''
|'''41\18'''
'''''1594.{{Overline|4}}'''''
|'''16\7'''
'''''1600'''''
|'''39\17'''
'''''1605.882…'''''
|'''23\10'''
'''''1610'''''
|'''30\13'''
'''''1615.384…'''''
|-
|Re#, La#
|G#
|9#, G#
|35\15
''1633.{{Overline|3}}''
|26\11
''1654.{{Overline|54}}''
|43\18
''1672.{{Overline|2}}''
| rowspan="2" |17\7
''1700''
|42\17
''1729.411…''
|25\10
''1750''
|33\13
''1776.923…''
|-
|Mib, Sib
|Ab
|Xb, Ac
|37\15
''1726.{{Overline|6}}''
|27\11
''1718.{{Overline|18}}''
|44\18
''1711.{{Overline|1}}''
|41\17
''1688.235…''
|24\10
''1680''
|31\13
''1669.230…''
|-
|Mi, Si
|A
|X, A
|38\15
''1773.{{Overline|3}}''
|28\11
''1781.{{Overline|81}}''
|46\18
''1788.{{Overline|8}}''
|18\7
''1800''
|44\17
''1811.764…''
|26\10
''1820''
|34\13
''1830.769…''
|-
|Mi#, Si#
|A#
|X#, A#
|39\15
''1820''
| rowspan="2" |29\11
''1845.{{Overline|45}}''
|48\18
''1866.{{Overline|6}}''
|19\7
''1900''
|47\17
''1935.294…''
|28\10
''1960''
|37\13
''1992.307…''
|-
|Fab, Dob
|Bbb
|Ebb, Ccc
|40\15
''1866.{{Overline|6}}''
|47\18
''1827.{{Overline|7}}''
|18\7
''1800''
|43\17
''1770.588…''
|25\10
''1750''
|32\13
''1723.076…''
|-
|'''Fa, Do'''
|'''Bb'''
|Eb, Cc
|'''41\15'''
'''''1913.{{Overline|3}}'''''
|'''30\11'''
'''''1909.{{Overline|09}}'''''
|'''49\18'''
'''''1905.{{Overline|5}}'''''
|'''19\7'''
'''''1900'''''
|'''46\17'''
'''''1894.117…'''''
|'''27\10'''
'''''1890'''''
|'''35\13'''
'''''1884.615…'''''
|-
|Fa#, Do#
|B
|E, C
|42\15
''1960''
|31\11
''1972.{{Overline|72}}''
|51\18
''1983.{{Overline|3}}''
|20\7
''2000''
|49\17
''2017.647…''
|29\10
''2030''
|38\13
''2046.153…''
|-
|Fax, Dox
|B#
|Ex, Cx
|43\15
''2006.{{Overline|6}}''
| rowspan="2" |32\11
''2036.''{{Overline|36}}
|53\18
''2061.{{Overline|1}}''
|21\7
''2100''
|52\17
''2141.176…''
|31\10
''2170''
|41\13
''2207.692…''
|-
|Dob, Solb
|Hb
|0b, Dc
|44\15
''2053.{{Overline|3}}''
|52\18
''2022.{{Overline|2}}''
|20\7
''2000''
|48\17
''1976.470…''
|28\10
''1960''
|''36\13''
''1938.615…''
|-
!Do, Sol
!H
!0, D
! colspan="7" |''2100''
|}
==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
|d6
|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
|LsLL
|<nowiki>0|3</nowiki>
|d
|m
|P
|}
==Temperaments==
The most basic rank-2 temperament interpretation of diatonic 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 = 192.6406
[[Mapping]]: [{{val|1 1 2}}, {{val|0 -2 -3}}]
[[Vals]]: {{val list|~(7edf, 11edf, 18edf)}}
==='''Napoli-Superpyth'''===
[[Subgroup]]: 3/2.7/6.14/9
[[Comma]] list: [[64/63]]
[[POL2]] generator: ~8/7 = 218.6371
[[Mapping]]: [{{val|1 1 2}}, {{val|0 -2 -3}}]
[[Vals]]: {{val list|~(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</u>
!''ed7\12''
|-
|1\4
|
|
|<u>171.428…</u>
|''175''
|1
|1
|1.000
|Equalised
|-
|6\23
|
|
|<u>180</u>
|''182.608…''
|6
|5
|1.200
|
|-
|
|11\42
|
|<u>180.821…</u>
|''183.{{Overline|3}}''
|11
|9
|1.222
|
|-
|5\19
|
|
|<u>181.{{Overline|81}}</u>
|''184.210…''
|5
|4
|1.250
|
|-
|
|14\53
|
|<u>182.608…</u>
|''184.905…''
|14
|11
|1.273
|
|-
|
|9\34
|
|<u>183.050…</u>
|''185.294…''
|9
|7
|1.286
|
|-
|4\15
|
|
|<u>184.615…</u>
|''186.6̄''
|4
|3
|1.333
|
|-
|
|11\41
|
|<u>185.915…</u>
|''187.804…''
|11
|8
|1.375
|
|-
|
|7\26
|
|<u>186.{{Overline|6}}</u>
|''188.461…''
|7
|5
|1.400
|
|-
|
|10\37
|
|<u>187.5</u>
|''189.{{Overline|189}}''
|10
|7
|1.429
|
|-
|
|13\48
|
|<u>187.951…</u>
|''189.58{{Overline|3}}''
|13
|9
|1.444
|
|-
|
|16\59
|
|<u>188.235…</u>
|''189.830…''
|16
|11
|1.4545
|
|-
|3\11
|
|
|<u>189.473…</u>
|''190.{{Overline|90}}''
|3
|2
|1.500
|Napoli-Meantone starts here
|-
|
|17\62
|
|<u>190.654…</u>
|''191.935…''
|17
|11
|1.5455
|
|-
|
|14\51
|
|<u>190.{{Overline|90}}</u>
|''192.156…''
|14
|9
|1.556
|
|-
|
|11\40
|
|<u>191.304…</u>
|''192.5''
|11
|7
|1.571
|
|-
|
|8\29
|
|<u>192</u>
|''193.103…''
|8
|5
|1.600
|
|-
|
|5\18
|
|<u>193.548…</u>
|''194.{{Overline|4}}''
|5
|3
|1.667
|
|-
|
|
|12\43
|<u>194.{{Overline|594}}</u>
|''195.348…''
|12
|7
|1.714
|
|-
|
|7\25
|
|<u>195.348…</u>
|''196''
|7
|4
|1.750
|
|-
|
|9\32
|
|<u>196.{{Overline|36}}</u>
|''196.875''
|9
|5
|1.800
|
|-
|
|11\39
|
|<u>197.014…</u>
|''197.435…''
|11
|6
|1.833
|
|-
|
|13\46
|
|<u>197.468…</u>
|''197.826…''
|13
|7
|1.857
|
|-
|
|15\53
|
|<u>197.802…</u>
|''198.113…''
|15
|8
|1.875
|
|-
|
|17\60
|
|<u>198.058…</u>
|''198.3̄''
|17
|9
|1.889
|
|-
|
|19\67
|
|<u>198.260…</u>
|''198.507…''
|19
|10
|1.900
|
|-
|
|21\74
|
|<u>198.425…</u>
|''198.{{Overline|648}}''
|21
|11
|1.909
|
|-
|
|23\81
|
|<u>198.561…</u>
|''198.765…''
|23
|12
|1.917
|
|-
|
|25\88
|
|<u>198.675…</u>
|''198.8{{Overline|63}}''
|25
|13
|1.923
|
|-
|
|27\95
|
|<u>198.773…</u>
|''198.947…''
|27
|14
|1.929
|
|-
|
|29\102
|
|<u>198.857…</u>
|''199.019…''
|29
|15
|1.933
|
|-
|
|31\109
|
|<u>198.930…</u>
|''199.082…''
|31
|16
|1.9375
|
|-
|
|33\116
|
|<u>198.994…</u>
|''199.137…''
|33
|17
|1.941
|
|-
|
|35\123
|
|<u>199.052…</u>
|''199.186…''
|35
|18
|1.944
|
|-
|2\7
|
|
|<u>200</u>
|''200''
|2
|1
|2.000
|Napoli-Meantone ends, Napoli-Pythagorean begins
|-
|
|19\66
|
|<u>201.769…</u>
|''201.{{Overline|51}}''
|19
|9
|2.111
|
|-
|
|17\59
|
|<u>201.980…</u>
|''201.694…''
|17
|8
|2.125
|
|-
|
|15\52
|
|<u>202.247…</u>
|''201.923…''
|15
|7
|2.143
|
|-
|
|13\45
|
|<u>202.597…</u>
|''202.{{Overline|2}}''
|13
|6
|2.167
|
|-
|
|11\38
|
|<u>203.076…</u>
|''202.631…''
|11
|5
|2.200
|
|-
|
|9\31
|
|<u>203.773…</u>
|''203.225…''
|9
|4
|2.250
|
|-
|
|7\24
|
|<u>204.878…</u>
|''204.1{{Overline|6}}''
|7
|3
|2.333
|
|-
|
|
|12\41
|<u>205.714…</u>
|''204.878…''
|12
|5
|2.400
|
|-
|
|5\17
|
|<u>206.896…</u>
|''205.882…''
|5
|2
|2.500
|Napoli-Neogothic heartland is from here…
|-
|
|
|18\61
|<u>207.692…</u>
|''206.557…''
|18
|7
|2.571
|
|-
|
|
|13\44
|<u>208</u>
|''206.{{Overline|81}}''
|13
|5
|2.600
|
|-
|
|8\27
|
|<u>208.695…</u>
|''207.{{Overline|407}}''
|8
|3
|2.667
|…to here
|-
|
|11\37
|
|<u>209.523…</u>
|''208.{{Overline|108}}''
|11
|4
|2.750
|
|-
|
|14\47
|
|<u>210</u>
|''208.510…''
|14
|5
|2.800
|
|-
|
|17\57
|
|<u>210.309…</u>
|''208.771…''
|17
|6
|2.833
|
|-
|
|20\67
|
|<u>210.526…</u>
|''208.955…''
|20
|7
|2.857
|
|-
|
|23\77
|
|<u>210.687…</u>
|''209.{{Overline|09}}''
|23
|8
|2.875
|
|-
|3\10
|
|
|<u>211.764…</u>
|''210''
|3
|1
|3.000
|Napoli-Pythagorean ends, Napoli-Superpyth begins
|-
|
|22\73
|
|<u>212.903…</u>
|''210.958…''
|22
|7
|3.143
|
|-
|
|19\63
|
|<u>213.084…</u>
|''211.{{Overline|1}}''
|19
|6
|3.167
|
|-
|
|16\53
|
|<u>213.{{Overline|3}}</u>
|''211.320…''
|16
|5
|3.200
|
|-
|
|13\43
|
|<u>213.698…</u>
|''211.627…''
|13
|4
|3.250
|
|-
|
|10\33
|
|<u>214.285…</u>
|''212.{{Overline|12}}''
|10
|3
|3.333
|
|-
|
|7\23
|
|<u>215.384…</u>
|''213.043…''
|7
|2
|3.500
|
|-
|
|11\36
|
|<u>216.393…</u>
|''213.{{Overline|3}}''
|11
|3
|3.667
|
|-
|
|15\49
|
|<u>216.867…</u>
|''214.285…''
|15
|4
|3.750
|
|-
|4\13
|
|
|<u>218.{{Overline|18}}</u>
|''215.385…''
|4
|1
|4.000
|
|-
|
|13\42
|
|<u>219.718…</u>
|''216.{{Overline|6}}''
|13
|3
|4.333
|
|-
|
|9\29
|
|<u>220.408…</u>
|''217.241…''
|9
|2
|4.500
|
|-
|
|14\45
|
|<u>221.052…</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>223.255…</u>
|''219.607…''
|16
|3
|5.333
|
|-
|
|11\35
|
|<u>223.728…</u>
|''220''
|11
|2
|5.500
|
|-
|
|17\54
|
|<u>224.175…</u>
|''220.{{Overline|370}}''
|17
|3
|5.667
|
|-
|6\19
|
|
|<u>225</u>
|''221.052…''
|6
|1
|6.000
|
|-
|1\3
|
|
|<u>240</u>
|''233.{{Overline|3}}''
|1
|0
|→ inf
|Paucitonic
|}
<nowiki />* Because this temperament almost seems too good to be true.
<nowiki />* Because this temperament almost seems too good to be true.
[[Category:31edo]]
[[Category:31edo]]
[[Category:meantone]]
[[Category:meantone]]
[[Category:pentave]]
[[Category:pentave]]
Retrieved from "https://en.xen.wiki/w/Angel"