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

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There are 6 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, Fa Sol La Si, 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, quadruple, quintuple or sextuple sesquitave (major ninth, thirteenth, seventeenth i. e. ~pentave or twenty-first or augmented twenty-fifth), 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], an ~pentave which is the Mixolydian mode of Hextone[12L 4s], a major twenty-first which is the Ionian mode of Guidotonic[15L 5s] or an augmented twenty-fifth which is the Lydian mode of Subdozenal[18L 6s]. Since there are exactly 8 naturals in double sesquitave notation, 12 in triple sesquitave notation, 16 in quadruple sesquitave notation, 20 in quintuple sesquitave notation and 24 in sextuple sesquitave notation, letters A-H (FGABHCDEF), dozenal or hex digits (0123456789XE0 or D1234567FGACD with flats written C molle or 0123456789ABCDEF0 or G123456789ABCDEFG with flats written F molle), the Guidonian names with F as the lowest ut or letters except I and O may be used.
There are 6 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, Fa Sol La Si, 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, quadruple, quintuple or sextuple sesquitave (major ninth, thirteenth, seventeenth i. e. ~pentave or twenty-first or augmented twenty-fifth), 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], an ~pentave which is the Mixolydian mode of Hextone[12L 4s], a major twenty-first which is the Ionian mode of Guidotonic[15L 5s] or an augmented twenty-fifth which is the Lydian mode of Subdozenal[18L 6s]. Since there are exactly 8 naturals in double sesquitave notation, 12 in triple sesquitave notation, 16 in quadruple sesquitave notation, 20 in quintuple sesquitave notation and 24 in sextuple sesquitave notation, letters A-H (FGABHCDEF), dozenal or hex digits (0123456789XE0 or D1234567FGACD with flats written C molle or 0123456789ABCDEF0 or G123456789ABCDEFG with flats written F molle), the Guidonian names with F as the lowest ut or letters except I and O may be used.
 
{| class="wikitable"
{| class="wikitable"
 
 
Line 17: Line 17:
Cents
Cents
 
 
! colspan="2" |Notation
! Notation
 
 
!Supersoft
!Supersoft
Line 29: Line 29:
!Semihard
!Semihard
 
 
! Hard
!Hard
 
 
!Superhard
!Superhard
Line 36: Line 36:
 
 
!Diatonic
!Diatonic
! Napoli
!~15edf
!~15edf
 
 
Line 55: Line 53:
 
 
|Do#, Fa#, Sol#
|Do#, Fa#, Sol#
|F#
|1\15, 46.154
|1\15, 46.154
 
 
Line 63: Line 59:
|2\18, 77.419
|2\18, 77.419
 
 
| rowspan="2" |1\7, 100
| rowspan="2" | 1\7, 100
 
 
|3\17, 124.138
|3\17, 124.138
 
 
| 2\10, 141.176
|2\10, 141.176
 
 
|3\13, 163.636
|3\13, 163.636
Line 74: Line 70:
 
 
|Reb, Solb, Lab
|Reb, Solb, Lab
|Gb, Ge
|3\15, 138.462
|3\15, 138.462
 
 
Line 91: Line 85:
 
 
|'''Re, Sol, La'''
|'''Re, Sol, La'''
|'''G'''
|'''4\15,''' '''184.615'''
|'''4\15,''' '''184.615'''
 
 
Line 110: Line 102:
 
 
|Re#, Sol#, La#
|Re#, Sol#, La#
|5\15, 230.769
|G#
| 5\15, 230.769
 
 
|4\11, 252.632
|4\11, 252.632
Line 118: Line 108:
|7\18, 270.968
|7\18, 270.968
 
 
| rowspan="2" |3\7, 300
| rowspan="2" | 3\7, 300
 
 
|8\17, 331.034
|8\17, 331.034
Line 129: Line 119:
 
 
|Mib, Lab, Sib
|Mib, Lab, Sib
|Ab, Æ
|7\15, 323.077
|7\15, 323.077
 
 
|5\11, 315.789
|5\11, 315.789
 
 
| 8\18, 309.677
|8\18, 309.677
 
 
|7\17, 289.655
|7\17, 289.655
 
 
| 4\10, 282.353
|4\10, 282.353
 
 
|5\13, 272.727
|5\13, 272.727
Line 145: Line 133:
|-
|-
 
 
| Mi, La, Si
|Mi, La, Si
|A
|8\15, 369.231
|8\15, 369.231
 
 
|6\11, 378.947
|6\11, 378.947
 
 
| 10\18, 387.097
|10\18, 387.097
 
 
|4\7, 400
|4\7, 400
Line 158: Line 144:
|10\17, 413.793
|10\17, 413.793
 
 
| 6\10, 423.529
|6\10, 423.529
 
 
|8\13, 436.364
|8\13, 436.364
Line 165: Line 151:
 
 
|Mi#, La#, Si#
|Mi#, La#, Si#
|A#
|9\15, 415.385
|9\15, 415.385
 
 
| rowspan="2" |7\11, 442.105
| rowspan="2" | 7\11, 442.105
 
 
|12\18, 464.516
|12\18, 464.516
 
 
| 5\7, 500
|5\7, 500
 
 
|13\17, 537.069
|13\17, 537.069
Line 184: Line 168:
 
 
|Fab, Sibb, Dob
|Fab, Sibb, Dob
|10\15, 461.538
| Bbb, Bee
| 10\15, 461.538
 
 
|11\18, 425.806
|11\18, 425.806
Line 196: Line 178:
|5\10, 352.941
|5\10, 352.941
 
 
| 6\13, 327.273
|6\13, 327.273
 
 
|-
|-
 
 
|'''Fa, Sib, Do'''
|'''Fa, Sib, Do'''
|'''Bb, Be'''
|'''11\15,''' '''507.692'''
|'''11\15,''' '''507.692'''
 
 
Line 220: Line 200:
 
 
|Fa#, Si, Do#
|Fa#, Si, Do#
|B
|12\15, 553.846
|12\15, 553.846
 
 
|9\11, 568.421
|9\11, 568.421
 
 
| 15\18, 580.645
|15\18, 580.645
 
 
|6\7, 600
|6\7, 600
Line 239: Line 217:
|Fax, Si#, Dox
|Fax, Si#, Dox
|B#
|13\15, 600
|13\15, 600
 
 
| rowspan="2" |10\11, 631.579
| rowspan="2" | 10\11, 631.579
 
 
|17\18, 658.064
|17\18, 658.064
Line 258: Line 234:
 
 
|Dob, Fab, Solb
|Dob, Fab, Solb
|Hb, He
|14\15, 646.154
|14\15, 646.154
| 16\18, 619.355
|16\18, 619.355
|6\7, 600
|6\7, 600
|14\17, 579.310
|14\17, 579.310
Line 269: Line 244:
 
 
!Do, Fa, Sol
!Do, Fa, Sol
!H
!'''15\15,''' '''692.308'''
!'''15\15,''' '''692.308'''
 
 
Line 285: Line 258:
!'''13\13,''' '''709.091'''
!'''13\13,''' '''709.091'''
 
 
|-
|}
{| class="wikitable"
 
 
|Do#, Fa#, Sol#
|+
 
 
|Η#
Cents
|16\15, 738.462
!Notation
!Supersoft
 
 
|12\11, 757.895
! Soft
 
 
|20\18, 774.194
!Semisoft
 
 
| rowspan="2" |8\8, 800
!Basic
 
 
|20\17, 827.586
!Semihard
 
 
|12\10, 847.059
! Hard
 
 
|16\13, 872.727
! Superhard
 
 
|-
|-
 
 
|Reb, Solb, Lab
!Napoli
! ~15edf
 
 
|Cb, Ce
! ~11edf
|18\15, 830.769
 
 
|13\11, 821.053
!~18edf
 
 
|21\18, 812.903
!~7edf
 
 
| 19\17, 786.207
!~17edf
 
 
|11\10, 776.471
!~10edf
 
 
|14\13, 763.63
!~13edf
 
 
|-
|-
 
 
|'''Re, Sol, La'''
|F#
|1\15, 46.154
 
 
|'''C'''
|1\11, 63.158
|'''19\15,''' '''876.923'''
 
 
|'''14\11,''' '''884.211'''
| 2\18, 77.419
 
 
|'''23\18,''' '''890.323'''
| rowspan="2" |1\7, 100
 
 
|'''9\5,''' '''900'''
|3\17, 124.138
 
 
|'''22\17,''' '''910.345'''
| 2\10, 141.176
 
 
|'''13\10,''' '''917.647'''
|3\13, 163.636
|'''17\13,''' '''927.273'''
 
 
|-
|-
 
 
|Re#, Sol#, La#
| Gb, Ge
|3\15, 138.462
 
 
| C#
| 2\11. 126.316
|20\15, 923.077
 
 
|15\11, 947.368
|3\18, 116.129
 
 
|25\18, 967.742
|2\17, 82.759
 
 
| rowspan="2" |10\7, 1000
|1\10, 70.588
 
 
|25\17, 1034.483
|1\13, 54.545
|15\10, 1058.824
|20\13, 1090.909
 
 
|-
|-
 
 
|Mib, Lab, Sib
|'''G'''
|'''4\15,''' '''184.615'''
 
 
|Db, De
|'''3\11,''' '''189.474'''
|22\15, 1015.385
|'''5\18,''' '''193.548'''
 
 
|16\11, 1010.526
|'''2\7,''' '''200'''
 
 
|26\18, 1006.452
|'''5\17,''' '''206.897'''
 
 
|24\17, 993.103
|'''3\10,''' '''211.765'''
 
 
|14\10, 988.235
|'''4\13,''' '''218.182'''
|18\13, 981.818
 
 
|-
|-
 
 
|Mi, La, Si
|G#
|5\15, 230.769
 
 
| D
|4\11, 252.632
| 23\15, 1061.538
 
 
|17\11, 1073.684
|7\18, 270.968
 
 
|28\18, 1083.871
| rowspan="2" |3\7, 300
 
 
|11\7, 1100
| 8\17, 331.034
 
 
|27\17, 1117.241
|5\10, 352.941
 
 
| 16\10, 1129.412
|7\13, 381.818
|21\9, 1145.455
 
 
|-
|-
 
 
|Mi#, La#, Si#
|Ab, Æ
|7\15, 323.077
 
 
|D#
|5\11, 315.789
|24\15, 1107.923
 
 
| rowspan="2" |18\11, 1136.842
|8\18, 309.677
 
 
|30\18, 1161.29
|7\17, 289.655
 
 
|12\7, 1200
|4\10, 282.353
 
 
| 30\17, 1241.379
|5\13, 272.727
|18\10, 1270.588
| 24\13, 1309.091
 
 
|-
|-
 
 
| Fab, Sibb, Dob
|A
| 8\15, 369.231
 
 
|Ebb, Eee
|6\11, 378.947
|25\15, 1153.846
 
 
|29\18, 1122.581
|10\18, 387.097
 
 
|11\7, 1100
| 4\7, 400
 
 
|26\17, 1075.862
|10\17, 413.793
 
 
|15\10, 1058.824
|6\10, 423.529
 
 
|19\13, 1036.364
|8\13, 436.364
 
 
|-
|-
 
 
|'''Fa, Sib, Do'''
|A#
| 9\15, 415.385
 
 
|'''Eb, Ee'''
| rowspan="2" |7\11, 442.105
|'''26\15,''' '''1200'''
 
 
|'''19\11,''' '''1200'''
|12\18, 464.516
 
 
|'''31\18,''' '''1200'''
|5\7, 500
 
 
|'''12\7, 1200'''
|13\17, 537.069
 
 
|'''29\17,''' '''1200'''
|8\10, 564.706
 
 
|'''17\10,''' '''1200'''
|11\13, 600
|'''22\13,''' '''1200'''
 
 
|-
|-
 
 
|Fa#, Si, Do#
|Bbb, Bee
|10\15, 461.538
 
 
| E
|11\18, 425.806
| 27\15, 1246.154
 
 
|20\11, 1263.158
|4\7, 400
 
 
|33\18, 1277.419
|9\17, 372.414
 
 
|13\7, 1300
| 5\10, 352.941
 
 
|32\17, 1324.138
|6\13, 327.273
 
 
|19\10, 1341.176
|-
 
 
|25\13, 1363.636
|'''Bb, Be'''
|'''11\15,''' '''507.692'''
 
 
|-
|'''8\11,''' '''505.263'''
 
 
|Fax, Si#, Dox
|'''13\18,''' '''503.226'''
 
 
|E#
|'''5\7, 500'''
| 28\15, 1292.308
 
 
| rowspan="2" | 21\11, 1326.318
|'''12\17,''' '''496.552'''
 
 
| 35\18, 1354.834
|'''7\10,''' '''494.118'''
 
 
|14\7, 1400
|'''9\13,''' '''490.909'''
|35\17, 1448.275
|21\10, 1482.353
|28\13, 1527.273
 
 
|-
|-
 
 
|Dob, Fab, Solb
|B
|12\15, 553.846
 
 
| Fb, Fe
|9\11, 568.421
|29\15, 1338.462
 
 
|34\18, 1316.129
|15\18, 580.645
 
 
|13\7, 1300
|6\7, 600
 
 
|31\17, 1282.759
| 15\17, 620.690
 
 
|18\10, 1270.588
|9\10, 635.294
 
 
|23\13, 1254.545
|12\13, 654.545
 
 
|-
|-
| B#
| 13\15, 600
 
 
!Do, Fa, Sol
| rowspan="2" |10\11, 631.579
 
 
!F
|17\18, 658.064
!30\15, 1384.615
 
 
! 22\11, 1389.473
|7\7, 700
 
 
!36\18, 1393.548
|18\17, 744.828
 
 
!14\7, 1400
|11\10, 776.471
 
 
!34\17, 1406.897
|15\13, 818.182
 
 
!20\10, 1411.765
|-
|Hb, He
|14\15, 646.154
| 16\18, 619.355
|6\7, 600
|14\17, 579.310
|8\10, 564.706
|10\13, 545.455
 
 
!26\13, 1418.182
|}
{| class="wikitable"
|+Cents
! colspan="2" |Notation
!Supersoft
!Soft
!Semisoft
!Basic
!Semihard
! Hard
!Superhard
|-
|-
!Bijou
!Hextone
! H
! ~15edf
!'''15\15,''' '''692.308'''
!~11edf
!~18edf
!'''11\11,''' '''694.737'''
! ~7edf
! ~17edf
!'''18\18,''' '''696.774'''
!~10edf
!~13edf
! 7\7, 700
|-
|0#, D#
!'''17\17,''' '''703.448'''
|0#, G#
| 1\15, 46.154
!'''10\10,''' '''705.882'''
|1\11, 63.158
|2\18, 77.419
!'''13\13,''' '''709.091'''
| rowspan="2" |1\7, 100
|3\17, 124.138
|2\10, 141.176
|3\13, 163.636
|-
|-
|1b, 1c
|1f
|Η#
|3\15, 138.462
|16\15, 738.462
| 2\11. 126.316
|3\18, 116.129
|12\11, 757.895
|2\17, 82.759
|1\10, 70.588
|20\18, 774.194
| 1\13, 54.545
| rowspan="2" |8\8, 800
|20\17, 827.586
|12\10, 847.059
|16\13, 872.727
|-
|-
|'''1'''
|'''1'''
|Cb, Ce
|'''4\15,''' '''184.615'''
|18\15, 830.769
|'''3\11,''' '''189.474'''
|'''5\18,''' '''193.548'''
|13\11, 821.053
|'''2\7,''' '''200'''
|'''5\17,''' '''206.897'''
|21\18, 812.903
|'''3\10,''' '''211.765'''
|'''4\13,''' '''218.182'''
|19\17, 786.207
|11\10, 776.471
|14\13, 763.63
|-
|-
| 1#
|1#
|'''C'''
|5\15, 230.769
|'''19\15,''' '''876.923'''
| 4\11, 252.632
|7\18, 270.968
|'''14\11,''' '''884.211'''
| rowspan="2" |3\7, 300
|8\17, 331.034
|'''23\18,''' '''890.323'''
|5\10, 352.941
| 7\13, 381.818
|'''9\5,''' '''900'''
|'''22\17,''' '''910.345'''
|'''13\10,''' '''917.647'''
|'''17\13,''' '''927.273'''
|-
|-
|2b, 2c
|2f
|C#
|7\15, 323.077
|20\15, 923.077
|5\11, 315.789
| 8\18, 309.677
|15\11, 947.368
|7\17, 289.655
| 4\10, 282.353
|25\18, 967.742
|5\13, 272.727
| rowspan="2" |10\7, 1000
|25\17, 1034.483
|15\10, 1058.824
|20\13, 1090.909
|-
|-
|2
|2
| Db, De
|8\15, 369.231
|22\15, 1015.385
|6\11, 378.947
|10\18, 387.097
|16\11, 1010.526
|4\7, 400
|10\17, 413.793
|26\18, 1006.452
| 6\10, 423.529
|8\13, 436.364
|24\17, 993.103
|14\10, 988.235
|18\13, 981.818
|-
|-
|2#
|2#
|D
| 9\15, 415.385
|23\15, 1061.538
| rowspan="2" |7\11, 442.105
|12\18, 464.516
|17\11, 1073.684
|5\7, 500
|13\17, 537.069
|28\18, 1083.871
|8\10, 564.706
| 11\13, 600
|11\7, 1100
|27\17, 1117.241
|16\10, 1129.412
|21\9, 1145.455
|-
|-
|3b, 3c
|3f
|D#
|10\15, 461.538
|24\15, 1107.923
|11\18, 425.806
| 4\7, 400
| rowspan="2" |18\11, 1136.842
|9\17, 372.414
|5\10, 352.941
|30\18, 1161.29
|6\13, 327.273
|12\7, 1200
|30\17, 1241.379
|18\10, 1270.588
|24\13, 1309.091
|-
|-
|'''3'''
|'''3'''
|Ebb, Ëe
|'''11\15,''' '''507.692'''
|25\15, 1153.846
|'''8\11,''' '''505.263'''
|'''13\18,''' '''503.226'''
|29\18, 1122.581
|'''5\7, 500'''
|'''12\17,''' '''496.552'''
|11\7, 1100
|'''7\10,''' '''494.118'''
|'''9\13,''' '''490.909'''
|26\17, 1075.862
|15\10, 1058.824
| 19\13, 1036.364
|-
|-
| 3#
|3#
|'''Eb, Ë'''
| 12\15, 553.846
|'''26\15,''' '''1200'''
| 9\11, 568.421
|15\18, 580.645
|'''19\11,''' '''1200'''
| 6\7, 600
|15\17, 620.690
|'''31\18,''' '''1200'''
|9\10, 635.294
|12\13, 654.545
|'''12\7, 1200'''
|-
|3x
|'''29\17,''' '''1200'''
|3x
|13\15, 600
|'''17\10,''' '''1200'''
| rowspan="2" |10\11, 631.579
|17\18, 658.064
|'''22\13,''' '''1200'''
| 7\7, 700
|18\17, 744.828
|11\10, 776.471
|15\13, 818.182
|-
|-
|4b, 4c
|4f
|E
| 14\15, 646.154
|27\15, 1246.154
|16\18, 619.355
| 6\7, 600
|20\11, 1263.158
|14\17, 579.310
|8\10, 564.706
|33\18, 1277.419
|10\13, 545.455
|-
|13\7, 1300
!4
!4
|32\17, 1324.138
!'''15\15,''' '''692.308'''
!'''11\11,''' '''694.737'''
|19\10, 1341.176
!'''18\18,''' '''696.774'''
!7\7, 700
|25\13, 1363.636
!'''17\17,''' '''703.448'''
!'''10\10,''' '''705.882'''
!'''13\13,''' '''709.091'''
|-
|-
|4#
|4#
|E#
|16\15, 738.462
|28\15, 1292.308
|12\11, 757.895
|20\18, 774.194
| rowspan="2" |21\11, 1326.318
| rowspan="2" |8\8, 800
|20\17, 827.586
|35\18, 1354.834
| 12\10, 847.059
|16\13, 872.727
|14\7, 1400
|35\17, 1448.275
| 21\10, 1482.353
|28\13, 1527.273
|-
|-
|5b, 5c
|5
| Fb, Fe
|18\15, 830.769
|29\15, 1338.462
|13\11, 821.053
|21\18, 812.903
|34\18, 1316.129
|19\17, 786.207
| 11\10, 776.471
|13\7, 1300
|14\13, 763.63
|31\17, 1282.759
|18\10, 1270.588
|23\13, 1254.545
|-
|-
|'''5'''
|'''5'''
!F
|'''19\15,''' '''876.923'''
!30\15, 1384.615
|'''14\11,''' '''884.211'''
|'''23\18,''' '''890.323'''
!22\11, 1389.473
|'''9\5,''' '''900'''
|'''22\17,''' '''910.345'''
!36\18, 1393.548
|'''13\10,''' '''917.647'''
|'''17\13,''' '''927.273'''
!14\7, 1400
!34\17, 1406.897
!20\10, 1411.765
!26\13, 1418.182
|}
 
{| class="wikitable"
|+Cents
! Notation
!Supersoft
!Soft
!Semisoft
!Basic
!Semihard
!Hard
! Superhard
|-
|-
|5#
!Bijou
|5#
!~15edf
|20\15, 923.077
!~11edf
| 15\11, 947.368
!~18edf
|25\18, 967.742
!~7edf
| rowspan="2" |10\7, 1000
!~17edf
|25\17, 1034.483
!~10edf
|15\10, 1058.824
!~13edf
|20\13, 1090.909
|-
|-
|6b, 6c
|0#, D#
|6f
|1\15, 46.154
| 22\15, 1015.385
|1\11, 63.158
| 16\11, 1010.526
|2\18, 77.419
|26\18, 1006.452
| rowspan="2" |1\7, 100
|24\17, 993.103
|3\17, 124.138
|14\10, 988.235
|2\10, 141.176
|18\13, 981.818
|3\13, 163.636
|-
|-
|6
|1b, 1c
|6
|3\15, 138.462
|23\15, 1061.538
| 2\11. 126.316
| 17\11, 1073.684
|3\18, 116.129
|28\18, 1083.871
|2\17, 82.759
|11\7, 1100
|1\10, 70.588
|27\17, 1117.241
|1\13, 54.545
| 16\10, 1129.412
|21\9, 1145.455
|-
|-
|6#
|'''1'''
|6#
|'''4\15,''' '''184.615'''
|24\15, 1107.923
|'''3\11,''' '''189.474'''
| rowspan="2" | 18\11, 1136.842
|'''5\18,''' '''193.548'''
|30\18, 1161.290
|'''2\7,''' '''200'''
|12\7, 1200
|'''5\17,''' '''206.897'''
|30\17, 1241.379
|'''3\10,''' '''211.765'''
|18\10, 1270.588
|'''4\13,''' '''218.182'''
|24\13, 1309.091
|-
|-
|7b, 7c
|1#
|7f
|5\15, 230.769
| 25\15, 1153.846
|4\11, 252.632
|29\18, 1122.581
|7\18, 270.968
|11\7, 1100
| rowspan="2" |3\7, 300
|26\17, 1075.862
|8\17, 331.034
| 15\10, 1058.824
|5\10, 352.941
|19\13, 1036.364
|7\13, 381.818
|-
|-
|'''7'''
|2b, 2c
|'''7'''
|7\15, 323.077
|'''26\15,''' '''1200'''
|5\11, 315.789
|'''19\11,''' '''1200'''
| 8\18, 309.677
|'''31\18,''' '''1200'''
| 7\17, 289.655
|'''12\7, 1200'''
|4\10, 282.353
|'''29\17,''' '''1200'''
|5\13, 272.727
|'''17\10,''' '''1200'''
|'''22\13,''' '''1200'''
|-
|-
|7#
|2
|7#
|8\15, 369.231
|27\15, 1246.154
|6\11, 378.947
|20\11, 1263.158
|10\18, 387.097
|33\18, 1277.419
|4\7, 400
|13\7, 1300
|10\17, 413.793
|32\17, 1324.138
|6\10, 423.529
|19\10, 1341.176
|8\13, 436.364
|25\13, 1363.636
|-
|-
|7x
|2#
|7x
| 9\15, 415.385
|28\15, 1292.308
| rowspan="2" |7\11, 442.105
| rowspan="2" |21\11, 1326.318
|12\18, 464.516
|35\18, 1354.834
|5\7, 500
|14\7, 1400
|13\17, 537.069
|35\17, 1448.275
|8\10, 564.706
|21\10, 1482.353
|11\13, 600
|28\13, 1527.273
|-
|-
|8b, Fc
|3b, 3c
|8f
| 10\15, 461.538
|29\15, 1338.462
| 11\18, 425.806
|34\18, 1316.129
|4\7, 400
| 13\7, 1300
|9\17, 372.414
|31\17, 1282.759
|5\10, 352.941
|18\10, 1270.588
|6\13, 327.273
|23\13, 1254.545
|-
|-
!8, F
|'''3'''
!8
|'''11\15,''' '''507.692'''
!30\15, 1384.615
|'''8\11,''' '''505.263'''
!22\11, 1389.473
|'''13\18,''' '''503.226'''
! 36\18, 1393.548
|'''5\7, 500'''
!14\7, 1400
|'''12\17,''' '''496.552'''
!34\17, 1406.897
|'''7\10,''' '''494.118'''
!20\10, 1411.765
|'''9\13,''' '''490.909'''
!26\13, 1418.182
|-
|-
|8#, F#
|3#
|8#
|12\15, 553.846
|31\15, 1430.769
|9\11, 568.421
|23\11, 1452.632
|15\18, 580.645
|38\18, 1470.968
|6\7, 600
| rowspan="2" | 15\7, 1500
|15\17, 620.690
|37\17, 1531.034
|9\10, 635.294
|22\10, 1552.941
|12\13, 654.545
|29\13, 1581.818
|-
|-
|9b, Gc
|3x
|9f
|13\15, 600
|33\15, 1523.077
| rowspan="2" |10\11, 631.579
|24\11, 1515.789
|17\18, 658.064
|39\18, 1509.677
|7\7, 700
|36\17, 1489.655
|18\17, 744.828
| 21\10, 1482.759
|11\10, 776.471
|27\13, 1472.273
|15\13, 818.182
|-
|-
|'''9, G'''
|4b, 4c
|9
|14\15, 646.154
|'''34\15,''' '''1569.231'''
|16\18, 619.355
|'''25\11,''' '''1578.947'''
|6\7, 600
|'''41\18,''' '''1587.097'''
|14\17, 579.310
|'''16\7,''' '''1600'''
|8\10, 564.706
|'''39\17,''' '''1613.793'''
|10\13, 545.455
|'''23\10,''' '''1623.529'''
|'''30\13,''' '''1636.364'''
|-
|-
| 9#, G#
!4
|9#
!'''15\15,''' '''692.308'''
|35\15, 1615.385
!'''11\11,''' '''694.737'''
|26\11, 1642.105
!'''18\18,''' '''696.774'''
|43\18, 1664.516
!7\7, 700
| rowspan="2" |17\7, 1700
!'''17\17,''' '''703.448'''
|42\17, 1737.069
!'''10\10,''' '''705.882'''
| 25\10, 1764.706
!'''13\13,''' '''709.091'''
|33\13, 1800
|-
|-
|Xb, Ac
|4#
|Af
| 16\15, 738.462
|37\15, 1707.692
|12\11, 757.895
|27\11, 1705.263
|20\18, 774.194
|44\18, 1703.226
| rowspan="2" |8\8, 800
| 41\17, 1696.552
|20\17, 827.586
| 24\10, 1694.118
|12\10, 847.059
|31\13, 1690.909
| 16\13, 872.727
|-
|-
| X, A
|5b, 5c
|A
|18\15, 830.769
|38\15, 1753.846
|13\11, 821.053
|28\11, 1768.421
|21\18, 812.903
| 46\18, 1780.645
|19\17, 786.207
|18\7, 1800
|11\10, 776.471
|44\17, 1820.690
|14\13, 763.63
| 26\10, 1835.294
| 34\13, 1854.545
|-
|-
|X#, A#
|'''5'''
|A#
|'''19\15,''' '''876.923'''
|39\15, 1800
|'''14\11,''' '''884.211'''
| rowspan="2" |29\11, 1831.579
|'''23\18,''' '''890.323'''
|48\18, 1858.064
|'''9\5,''' '''900'''
|19\7, 1900
|'''22\17,''' '''910.345'''
|47\17, 1944.828
|'''13\10,''' '''917.647'''
|28\10, 1976.471
|'''17\13,''' '''927.273'''
|37\13, 2018.182
|-
|-
| Ebb, Ccc
|5#
|Ax
|20\15, 923.077
|40\15, 1846.154
|15\11, 947.368
| 47\18, 1819.355
|25\18, 967.742
|18\7, 1800
| rowspan="2" |10\7, 1000
|43\17, 1779.310
|25\17, 1034.483
|25\10, 1764.706
|15\10, 1058.824
|32\13, 1745.545
|20\13, 1090.909
|-
|-
|'''Eb, Cc'''
|6b, 6c
|'''Bf'''
|22\15, 1015.385
|'''41\15,''' '''1892.308'''
|16\11, 1010.526
|'''30\11,''' '''1894.737'''
|26\18, 1006.452
|'''49\18,''' '''1896.774'''
|24\17, 993.103
|'''19\7, 1900'''
|14\10, 988.235
|'''46\17,''' '''1903.448'''
|18\13, 981.818
|'''27\10,''' '''1905.882'''
|'''35\13,''' '''1909.091'''
|-
|-
|E, C
|6
|B
|23\15, 1061.538
|42\15, 1938.462
|17\11, 1073.684
|31\11, 1957.895
| 28\18, 1083.871
| 51\18, 1974.194
|11\7, 1100
|20\7, 2000
|27\17, 1117.241
| 49\17, 2027.586
|16\10, 1129.412
|29\10, 2047.059
|21\9, 1145.455
| 38\13, 2072.727
|-
|-
|Ex, Cx
|6#
|B#
|24\15, 1107.923
| 43\15, 1984.615
| rowspan="2" |18\11, 1136.842
| rowspan="2" |32\11, 2021.053
|30\18, 1161.290
|53\18, 2051.612
|12\7, 1200
|21\7, 2100
|30\17, 1241.379
|52\17, 2151.725
|18\10, 1270.588
|31\10, 2188.235
|24\13, 1309.091
| 41\13, 2236.364
|-
|-
|0b, Dc
| 7b, 7c
|Cf
|25\15, 1153.846
|44\15, 2030.769
|29\18, 1122.581
|52\18, 2012.903
|11\7, 1100
| 20\7, 2000
|26\17, 1075.862
| 48\17, 1986.207
|15\10, 1058.824
|28\10, 1976.471
|19\13, 1036.364
|36\13, 1963.636
|-
|-
!0, D
|'''7'''
!C
|'''26\15,''' '''1200'''
!45\15, 2076.923
|'''19\11,''' '''1200'''
!33\11, 2084.211
|'''31\18,''' '''1200'''
!54\18, 2090.323
|'''12\7, 1200'''
! 21\7, 2100
|'''29\17,''' '''1200'''
!51\17, 2110.345
|'''17\10,''' '''1200'''
!30\10, 2117.647
|'''22\13,''' '''1200'''
!39\13, 2127.273
|-
|-
|0#, D#
|7#
|C#
|27\15, 1246.154
|46\15, 2123.077
|20\11, 1263.158
|34\11, 2147.368
|33\18, 1277.419
|56\15, 2167.742
|13\7, 1300
| rowspan="2" |22\7, 2200
|32\17, 1324.138
|54\17, 2234.483
|19\10, 1341.176
|32\10, 2258.824
|25\13, 1363.636
|42\13, 2090.909
|-
|-
|1b, 1c
|7x
|Df
|28\15, 1292.308
|48\15, 2215.385
| rowspan="2" |21\11, 1326.318
|35\11, 2210.526
|35\18, 1354.834
|57\15, 2206.452
|14\7, 1400
|53\17, 2193.103
|35\17, 1448.275
|31\10, 2188.235
|21\10, 1482.353
|40\13, 2181.818
|28\13, 1527.273
|-
|-
|'''1'''
|8b, Fc
|'''D'''
|29\15, 1338.462
|'''49\15, 2261.538'''
|34\18, 1316.129
|'''36\11, 1073.684'''
|13\7, 1300
|'''59\18, 2283.871'''
|31\17, 1282.759
|'''23\7, 2300'''
|18\10, 1270.588
|'''56\17, 2317.241'''
|23\13, 1254.545
|'''33\10, 2329.412'''
|'''43\13,''' '''2345.455'''
|-
|-
|1#
!8, F
|D#
!30\15, 1384.615
|50\15, 2307.692
!22\11, 1389.473
|37\11, 2336.842
!36\18, 1393.548
|61\18, 2361.290
!14\7, 1400
| rowspan="2" |24\7, 2400
!34\17, 1406.897
|59\17, 2441.379
!20\10, 1411.765
|35\10, 2470.588
!26\13, 1418.182
|46\13, 2509.091
|-
|-
|2b, 2c
|8#, F#
|Ef
|31\15, 1430.769
|52\15, 2400
|23\11, 1452.632
|38\11, 2400
|38\18, 1470.968
|62\18, 2400
| rowspan="2" |15\7, 1500
|58\17, 2400
|37\17, 1531.034
|34\10, 2400
|22\10, 1552.941
|44\13, 2400
|29\13, 1581.818
|-
|-
|2
|9b, Gc
|E
|33\15, 1523.077
|53\15, 2446.154
|24\11, 1515.789
|39\11, 2463.158
|39\18, 1509.677
|64\18, 2477,419
|36\17, 1489.655
|25\7, 2500
|21\10, 1482.759
|61\17, 2524.138
|27\13, 1472.273
|36\10, 2541.176
|47\13, 2563.636
|-
|-
|2#
|'''9, G'''
|E#
|'''34\15,''' '''1569.231'''
|54\15, 2492.308
|'''25\11,''' '''1578.947'''
| rowspan="2" |40\11, 2526.316
|'''41\18,''' '''1587.097'''
|66\18, 2554.838
|'''16\7,''' '''1600'''
|26\7, 2600
|'''39\17,''' '''1613.793'''
|64\17, 2648.275
|'''23\10,''' '''1623.529'''
|38\10, 2682.353
|'''30\13,''' '''1636.364'''
|50\13, 2727.273
|-
|-
|3b, 3c
|9#, G#
|Fff
|35\15, 1615.385
|55\15, 2538.462
|26\11, 1642.105
|65\18, 2516.129
|43\18, 1664.516
|25\7, 2500
| rowspan="2" |17\7, 1700
|60\17, 2482.759
|42\17, 1737.069
|35\10, 2470.588
|25\10, 1764.706
|45\13, 2454.545
|33\13, 1800
|-
|-
|'''3'''
|Xb, Ac
|'''Ff'''
|37\15, 1707.692
|'''56\15, 2584.615'''
|27\11, 1705.263
|'''41\11, 2589.474'''
|44\18, 1703.226
|'''67\18, 2593.548'''
|41\17, 1696.552
|'''26\7, 2600'''
|24\10, 1694.118
|'''63\17, 2606.897'''
|31\13, 1690.909
|'''37\10, 2611.765'''
|'''48\13,''' '''2618.182'''
|-
|-
|3#
|X, A
|F
|38\15, 1753.846
|57\15, 2630.769
|28\11, 1768.421
|42\11, 2652.632
|46\18, 1780.645
|69\18, 2670.968
|18\7, 1800
|27\7, 2700
|44\17, 1820.690
|66\17, 2731.034
|26\10, 1835.294
|39\10, 2752.941
|34\13, 1854.545
|51\13, 2781.818
|-
|-
|3x
|X#, A#
|F#
|39\15, 1800
| rowspan="2" |58\15, 2676.923
| rowspan="2" |29\11, 1831.579
|43\11, 2715.789
|48\18, 1858.064
|71\18, 2748.387
|19\7, 1900
|28\7, 2800
|47\17, 1944.828
|69\17, 2855.172
|28\10, 1976.471
|41\10, 2894.118
|37\13, 2018.182
|54\13, 2945.455
|-
|-
|4bb, 4cc
|Ebb, Ccc
|0ff, Gff
|40\15, 1846.154
|42\11, 2652.632
|47\18, 1819.355
|68\18, 2632.258
|18\7, 1800
|26\7, 2600
|43\17, 1779.310
|62\17, 2565.517
|25\10, 1764.706
|36\10, 2541.176
|32\13, 1745.545
|46\13, 2509.091
|-
|-
|4b, 4c
|'''Eb, Cc'''
|0f, Gf
|'''41\15,''' '''1892.308'''
|59\15, 2723.077
|'''30\11,''' '''1894.737'''
|43\11, 2715.789
|'''49\18,''' '''1896.774'''
|70\18, 2709.677
|'''19\7, 1900'''
|27\7, 2700
|'''46\17,''' '''1903.448'''
|65\17, 2689.552
|'''27\10,''' '''1905.882'''
|38\10, 2682.353
|'''35\13,''' '''1909.091'''
|49\13, 2672.273
|-
|-
!4
|E, C
!0, G
|42\15, 1938.462
!60\15, 2769.231
|31\11, 1957.895
!44\11, 2778.947
|51\18, 1974.194
!72\18, 2787.097
|20\7, 2000
!28\7, 2800
|49\17, 2027.586
!68\17, 2813.793
|29\10, 2047.059
!40\10, 2823.529
|38\13, 2072.727
!52\13, 2836.364
|}
{| class="wikitable"
|+Cents<ref name=":04">Fractions repeating more than 4 digits written as continued fractions</ref>
! colspan="2" |Notation
!Supersoft
!Soft
!Semisoft
! Basic
!Semihard
!Hard
!Superhard
|-
|-
!Guidotonic
|Ex, Cx
!Subdozenal
|43\15, 1984.615
!~15edf
| rowspan="2" |32\11, 2021.053
! ~11edf
|53\18, 2051.612
!~18edf
|21\7, 2100
!~7edf
|52\17, 2151.725
!~17edf
|31\10, 2188.235
!~10edf
|41\13, 2236.364
!~13edf
|-
|-
|F ut#
|0b, Dc
|F#
|44\15, 2030.769
|1\15, 46.154
|52\18, 2012.903
|1\11, 63.158
|20\7, 2000
|2\18, 77.419
|48\17, 1986.207
| rowspan="2" |1\7, 100
|28\10, 1976.471
|3\17, 124.138
|36\13, 1963.636
|2\10, 141.176
|3\13, 163.636
|-
|-
|G reb
! 0, D
| Gb, Ge
!45\15, 2076.923
|3\15, 138.462
!33\11, 2084.211
|2\11. 126.316
!54\18, 2090.323
| 3\18, 116.129
!21\7, 2100
|2\17, 82.759
!51\17, 2110.345
|1\10, 70.588
!30\10, 2117.647
| 1\13, 54.545
!39\13, 2127.273
|}
 
{| class="wikitable"
|+Cents
!Notation
!Supersoft
!Soft
!Semisoft
! Basic
!Semihard
!Hard
!Superhard
|-
|-
|'''G re'''
!Hextone
|'''G'''
!~15edf
|'''4\15,''' '''184.615'''
!~11edf
|'''3\11,''' '''189.474'''
!~18edf
|'''5\18,''' '''193.548'''
!~7edf
|'''2\7,''' '''200'''
!~17edf
|'''5\17,''' '''206.897'''
!~10edf
|'''3\10,''' '''211.765'''
!~13edf
|'''4\13,''' '''218.182'''
|-
|-
| G re#
|0#, G#
|G#
|1\15, 46.154
|1\11, 63.158
|2\18, 77.419
| rowspan="2" |1\7, 100
|3\17, 124.138
|2\10, 141.176
|3\13, 163.636
|-
| 1f
|3\15, 138.462
|2\11. 126.316
|3\18, 116.129
|2\17, 82.759
|1\10, 70.588
|1\13, 54.545
|-
|'''1'''
|'''4\15,''' '''184.615'''
|'''3\11,''' '''189.474'''
|'''5\18,''' '''193.548'''
|'''2\7,''' '''200'''
|'''5\17,''' '''206.897'''
|'''3\10,''' '''211.765'''
|'''4\13,''' '''218.182'''
|-
|1#
|5\15, 230.769
|5\15, 230.769
|4\11, 252.632
|4\11, 252.632
|7\18, 270.968
|7\18, 270.968
| rowspan="2" | 3\7, 300
| rowspan="2" |3\7, 300
|8\17, 331.034
|8\17, 331.034
|5\10, 352.941
|5\10, 352.941
|7\13, 381.818
|7\13, 381.818
|-
|-
| A mib
|2f
|Hb, He
|7\15, 323.077
|7\15, 323.077
|5\11, 315.789
|5\11, 315.789
|8\18, 309.677
|8\18, 309.677
| 7\17, 289.655
|7\17, 289.655
|4\10, 282.353
|4\10, 282.353
|5\13, 272.727
|5\13, 272.727
|-
|-
|A mi
|2
|H
|8\15, 369.231
|8\15, 369.231
| 6\11, 378.947
|6\11, 378.947
|10\18, 387.097
|10\18, 387.097
|4\7, 400
| 4\7, 400
| 10\17, 413.793
|10\17, 413.793
| 6\10, 423.529
|6\10, 423.529
|8\13, 436.364
|8\13, 436.364
|-
|-
|A mi#
|2#
|H#
|9\15, 415.385
|9\15, 415.385
| rowspan="2" | 7\11, 442.105
| rowspan="2" |7\11, 442.105
|12\18, 464.516
|12\18, 464.516
|5\7, 500
|5\7, 500
Line 1,148: Line 1,158:
|11\13, 600
|11\13, 600
|-
|-
|B fa utb
|3f
|Jbb, Jee
| 10\15, 461.538
|10\15, 461.538
|11\18, 425.806
| 11\18, 425.806
|4\7, 400
|4\7, 400
|9\17, 372.414
|9\17, 372.414
|5\10, 352.941
|5\10, 352.941
| 6\13, 327.273
|6\13, 327.273
|-
|-
|'''B fa ut'''
|'''3'''
|'''Jb, Je'''
|'''11\15,''' '''507.692'''
|'''11\15,''' '''507.692'''
|'''8\11,''' '''505.263'''
|'''8\11,''' '''505.263'''
Line 1,167: Line 1,175:
|'''9\13,''' '''490.909'''
|'''9\13,''' '''490.909'''
|-
|-
|B fa ut#
|3#
|J
|12\15, 553.846
|12\15, 553.846
|9\11, 568.421
|9\11, 568.421
Line 1,175: Line 1,182:
|15\17, 620.690
|15\17, 620.690
|9\10, 635.294
|9\10, 635.294
| 12\13, 654.545
|12\13, 654.545
|-
|-
|B fa utx
| 3x
|J#
|13\15, 600
|13\15, 600
| rowspan="2" |10\11, 631.579
| rowspan="2" | 10\11, 631.579
|17\18, 658.064
|17\18, 658.064
|7\7, 700
|7\7, 700
Line 1,187: Line 1,193:
|15\13, 818.182
|15\13, 818.182
|-
|-
|C sol re utb
|4f
|Kb, Ke
| 14\15, 646.154
|14\15, 646.154
|16\18, 619.355
|16\18, 619.355
|6\7, 600
|6\7, 600
|14\17, 579.310
|14\17, 579.310
|8\10, 564.706
|8\10, 564.706
| 10\13, 545.455
|10\13, 545.455
|-
|-
!C sol re ut
!4
!K
!'''15\15,''' '''692.308'''
!'''15\15,''' '''692.308'''
!'''11\11,''' '''694.737'''
!'''11\11,''' '''694.737'''
Line 1,206: Line 1,210:
!'''13\13,''' '''709.091'''
!'''13\13,''' '''709.091'''
|-
|-
|C sol re ut#
| 4#
|K#
|16\15, 738.462
|16\15, 738.462
|12\11, 757.895
|12\11, 757.895
Line 1,216: Line 1,219:
|16\13, 872.727
|16\13, 872.727
|-
|-
|D la mi reb
|5
|Lb, Le
|18\15, 830.769
|18\15, 830.769
|13\11, 821.053
|13\11, 821.053
|21\18, 812.903
|21\18, 812.903
|19\17, 786.207
|19\17, 786.207
|11\10, 776.471
| 11\10, 776.471
|14\13, 763.63
|14\13, 763.63
|-
|-
|'''D la mi re'''
|'''5'''
|'''L'''
|'''19\15,''' '''876.923'''
|'''19\15,''' '''876.923'''
|'''14\11,''' '''884.211'''
|'''14\11,''' '''884.211'''
Line 1,235: Line 1,236:
|'''17\13,''' '''927.273'''
|'''17\13,''' '''927.273'''
|-
|-
|D la mi re#
|5#
|L#
|20\15, 923.077
|20\15, 923.077
|15\11, 947.368
|15\11, 947.368
|25\18, 967.742
| 25\18, 967.742
| rowspan="2" |10\7, 1000
| rowspan="2" |10\7, 1000
|25\17, 1034.483
|25\17, 1034.483
Line 1,245: Line 1,245:
|20\13, 1090.909
|20\13, 1090.909
|-
|-
|E fa mib
|6f
|Mb, Me
|22\15, 1015.385
|22\15, 1015.385
|16\11, 1010.526
|16\11, 1010.526
Line 1,254: Line 1,253:
|18\13, 981.818
|18\13, 981.818
|-
|-
|E fa mi
|6
|M
|23\15, 1061.538
|23\15, 1061.538
|17\11, 1073.684
|17\11, 1073.684
Line 1,264: Line 1,262:
|21\9, 1145.455
|21\9, 1145.455
|-
|-
|E fa mi#
|6#
|M#
|24\15, 1107.923
|24\15, 1107.923
| rowspan="2" |18\11, 1136.842
| rowspan="2" |18\11, 1136.842
|30\18, 1161.29
|30\18, 1161.290
|12\7, 1200
|12\7, 1200
|30\17, 1241.379
|30\17, 1241.379
|18\10, 1270.588
|18\10, 1270.588
| 24\13, 1309.091
|24\13, 1309.091
|-
|-
|F sol fa utb
| 7f
| Nbb, Nee
|25\15, 1153.846
|25\15, 1153.846
|29\18, 1122.581
|29\18, 1122.581
| 11\7, 1100
|11\7, 1100
|26\17, 1075.862
|26\17, 1075.862
|15\10, 1058.824
|15\10, 1058.824
|19\13, 1036.364
|19\13, 1036.364
|-
|-
|'''F sol fa ut'''
|'''7'''
|'''Nb, Ne'''
|'''26\15,''' '''1200'''
|'''26\15,''' '''1200'''
|'''19\11,''' '''1200'''
|'''19\11,''' '''1200'''
Line 1,293: Line 1,288:
|'''22\13,''' '''1200'''
|'''22\13,''' '''1200'''
|-
|-
|F sol fa ut#
|7#
|N
|27\15, 1246.154
|27\15, 1246.154
|20\11, 1263.158
|20\11, 1263.158
Line 1,303: Line 1,297:
|25\13, 1363.636
|25\13, 1363.636
|-
|-
| F sol fa utx
|7x
|N#
|28\15, 1292.308
|28\15, 1292.308
| rowspan="2" |21\11, 1326.318
| rowspan="2" |21\11, 1326.318
Line 1,313: Line 1,306:
|28\13, 1527.273
|28\13, 1527.273
|-
|-
|G la sol reb
|8f
| Pb, Pe
|29\15, 1338.462
|29\15, 1338.462
|34\18, 1316.129
| 34\18, 1316.129
|13\7, 1300
|13\7, 1300
|31\17, 1282.759
|31\17, 1282.759
Line 1,322: Line 1,314:
|23\13, 1254.545
|23\13, 1254.545
|-
|-
!G la sol re
! 8
!P
!30\15, 1384.615
!30\15, 1384.615
!22\11, 1389.473
!22\11, 1389.473
Line 1,332: Line 1,323:
!26\13, 1418.182
!26\13, 1418.182
|-
|-
|G la sol re#
|8#
|P#
|31\15, 1430.769
|31\15, 1430.769
|23\11, 1452.632
|23\11, 1452.632
|38\18, 1470.968
| 38\18, 1470.968
|15\7, 1500
| rowspan="2" |15\7, 1500
|37\17, 1531.034
|37\17, 1531.034
| 22\10, 1552.941
|22\10, 1552.941
|29\13, 1581.818
|29\13, 1581.818
|-
|-
|A la mib
|9f
| Qb, Qe
|33\15, 1523.077
|33\15, 1523.077
|24\11, 1515.789
|24\11, 1515.789
|39\18, 1509.677
|39\18, 1509.677
|15\7, 1500
| 36\17, 1489.655
|36\17, 1489.655
|21\10, 1482.759
|21\10, 1482.759
|27\13, 1472.273
|27\13, 1472.273
|-
|-
|'''A la mi'''
|9
|'''Q'''
|'''34\15,''' '''1569.231'''
|'''34\15,''' '''1569.231'''
|'''25\11,''' '''1578.947'''
|'''25\11,''' '''1578.947'''
Line 1,362: Line 1,349:
|'''30\13,''' '''1636.364'''
|'''30\13,''' '''1636.364'''
|-
|-
|A la mi#
|9#
|Q#
|35\15, 1615.385
|35\15, 1615.385
|26\11, 1642.105
|26\11, 1642.105
Line 1,372: Line 1,358:
|33\13, 1800
|33\13, 1800
|-
|-
|B fa utb
|Af
|Rb, Re
| 37\15, 1707.692
|37\15, 1707.692
| 27\11, 1705.263
|27\11, 1705.263
|44\18, 1703.226
|44\18, 1703.226
|41\17, 1696.552
|41\17, 1696.552
Line 1,381: Line 1,366:
|31\13, 1690.909
|31\13, 1690.909
|-
|-
| B fa ut
|A
|R
| 38\15, 1753.846
|38\15, 1753.846
|28\11, 1768.421
|28\11, 1768.421
|46\18, 1780.645
|46\18, 1780.645
Line 1,391: Line 1,375:
|34\13, 1854.545
|34\13, 1854.545
|-
|-
|B fa ut#
|A#
|R#
| 39\15, 1800
|39\15, 1800
| rowspan="2" |29\11, 1831.579
| rowspan="2" |29\11, 1831.579
|48\18, 1858.064
| 48\18, 1858.064
| 19\7, 1900
|19\7, 1900
|47\17, 1944.828
|47\17, 1944.828
|28\10, 1976.471
|28\10, 1976.471
|37\13, 2018.182
|37\13, 2018.182
|-
|-
|C sol reb
|Ax
|Sbb, See
|40\15, 1846.154
|40\15, 1846.154
|47\18, 1819.355
|47\18, 1819.355
Line 1,410: Line 1,392:
|32\13, 1745.545
|32\13, 1745.545
|-
|-
|'''C sol re'''
|'''Bf'''
|'''Sb, Se'''
|'''41\15,''' '''1892.308'''
|'''41\15,''' '''1892.308'''
|'''30\11,''' '''1894.737'''
|'''30\11,''' '''1894.737'''
Line 1,420: Line 1,401:
|'''35\13,''' '''1909.091'''
|'''35\13,''' '''1909.091'''
|-
|-
|C sol re#
|B
|S#
|42\15, 1938.462
|42\15, 1938.462
|31\11, 1957.895
|31\11, 1957.895
Line 1,427: Line 1,407:
|20\7, 2000
|20\7, 2000
|49\17, 2027.586
|49\17, 2027.586
|29\10, 2047.059
| 29\10, 2047.059
|38\13, 2072.727
|38\13, 2072.727
|-
|-
| C sol rex
|B#
| Sx
|43\15, 1984.615
|43\15, 1984.615
| rowspan="2" |32\11, 2021.053
| rowspan="2" |32\11, 2021.053
Line 1,440: Line 1,419:
|41\13, 2236.364
|41\13, 2236.364
|-
|-
| D la mib
|Cf
|Tb, Te
|44\15, 2030.769
|44\15, 2030.769
|52\18, 2012.903
|52\18, 2012.903
Line 1,449: Line 1,427:
|36\13, 1963.636
|36\13, 1963.636
|-
|-
!D la mi
!C
!T
!45\15, 2076.923
!45\15, 2076.923
!33\11, 2084.211
!33\11, 2084.211
Line 1,459: Line 1,436:
!39\13, 2127.273
!39\13, 2127.273
|-
|-
|D la mi#
|C#
|T#
|46\15, 2123.077
|46\15, 2123.077
|34\11, 2147.368
|34\11, 2147.368
|56\18, 2167.742
|56\15, 2167.742
| rowspan="2" |22\7, 2200
| rowspan="2" |22\7, 2200
| 54\17, 2234.483
|54\17, 2234.483
| 32\10, 2258.824
|32\10, 2258.824
|42\13, 2090.909
|42\13, 2090.909
|-
|-
|E fab
|Df
|Ub, Ue
|48\15, 2215.385
|48\15, 2215.385
|35\11, 2210.526
|35\11, 2210.526
|57\18, 2206.452
|57\15, 2206.452
|53\17, 2193.103
|53\17, 2193.103
|31\10, 2188.235
|31\10, 2188.235
|40\13, 2181.818
|40\13, 2181.818
|-
|-
|'''E fa'''
|'''D'''
|'''U'''
|'''49\15, 2261.538'''
|'''49\15, 2261.538'''
| '''36\11, 1073.684'''
|'''36\11, 1073.684'''
|'''59\18, 2283.871'''
|'''59\18, 2283.871'''
|'''23\7, 2300'''
|'''23\7, 2300'''
| '''56\17, 2317.241'''
|'''56\17, 2317.241'''
|'''33\10, 2329.412'''
|'''33\10, 2329.412'''
|'''43\13,''' '''2345.455'''
|'''43\13,''' '''2345.455'''
|-
|-
|E mi fax
|D#
|U
|50\15, 2307.692
|50\15, 2307.692
| 37\11, 2336.842
|37\11, 2336.842
|61\18, 2361.290
|61\18, 2361.290
| rowspan="2" |24\7, 2400
| rowspan="2" |24\7, 2400
|59\17, 2441.379
|59\17, 2441.379
|35\10, 2470.588
|35\10, 2470.588
| 46\13, 2509.091
|46\13, 2509.091
|-
|-
|F sol fa utb
|Ef
|Vb, Ve
|52\15, 2400
|52\15, 2400
| 38\11, 2400
|38\11, 2400
|62\18, 2400
|62\18, 2400
|58\17, 2400
|58\17, 2400
|34\10, 2400
|34\10, 2400
|44\13, 2400
| 44\13, 2400
|-
|-
|F sol fa ut
|E
|V
|53\15, 2446.154
|53\15, 2446.154
|39\11, 2463.158
| 39\11, 2463.158
|64\18, 2477,419
|64\18, 2477,419
|25\7, 2500
|25\7, 2500
Line 1,517: Line 1,488:
|47\13, 2563.636
|47\13, 2563.636
|-
|-
|F sol fa ut#
|E#
| V#
|54\15, 2492.308
|54\15, 2492.308
| rowspan="2" |40\11, 2526.316
| rowspan="2" |40\11, 2526.316
Line 1,527: Line 1,497:
|50\13, 2727.273
|50\13, 2727.273
|-
|-
|G la mi reb
|Fff
| Wbb, Wee
| 55\15, 2538.462
|55\15, 2538.462
| 65\18, 2516.129
|65\18, 2516.129
|25\7, 2500
|25\7, 2500
|60\17, 2482.759
|60\17, 2482.759
|35\10, 2470.588
|35\10, 2470.588
| 45\13, 2454.545
|45\13, 2454.545
|-
|-
|'''G la mi re'''
|'''Ff'''
|'''Wb, We'''
|'''56\15, 2584.615'''
| '''56\15, 2584.615'''
|'''41\11, 2589.474'''
|'''41\11, 2589.474'''
|'''67\18, 2593.548'''
|'''67\18, 2593.548'''
Line 1,546: Line 1,514:
|'''48\13,''' '''2618.182'''
|'''48\13,''' '''2618.182'''
|-
|-
| G la mi re#
|F
|W
|57\15, 2630.769
|57\15, 2630.769
|42\11, 2652.632
|42\11, 2652.632
|69\18, 2670.968
|69\18, 2670.968
| 27\7, 2700
|27\7, 2700
| 66\17, 2731.034
|66\17, 2731.034
|39\10, 2752.941
|39\10, 2752.941
|51\13, 2781.818
|51\13, 2781.818
|-
|-
|G la mi rex
| F#
|W#
| rowspan="2" |58\15, 2676.923
| rowspan="2" |58\15, 2676.923
|43\11, 2715.789
|43\11, 2715.789
|71\18, 2748.387
|71\18, 2748.387
|28\7, 2800
| 28\7, 2800
|69\17, 2855.172
|69\17, 2855.172
|41\10, 2894.118
|41\10, 2894.118
|54\13, 2945.455
|54\13, 2945.455
|-
|-
|A fa mibb
|0ff, Gff
|Xbb, Xee
|42\11, 2652.632
|42\11, 2652.632
|68\18, 2632.258
|68\18, 2632.258
|26\7, 2600
|26\7, 2600
| 62\17, 2565.517
|62\17, 2565.517
|36\10, 2541.176
|36\10, 2541.176
|46\13, 2509.091
|46\13, 2509.091
|-
|-
|A fa mib
|0f, Gf
|Xb, Xe
|59\15, 2723.077
|59\15, 2723.077
|43\11, 2715.789
|43\11, 2715.789
| 70\18, 2709.677
|70\18, 2709.677
|27\7, 2700
|27\7, 2700
| 65\17, 2689.552
|65\17, 2689.552
|38\10, 2682.353
|38\10, 2682.353
| 49\13, 2672.273
|49\13, 2672.273
|-
|-
!A fa mi
!0, G
!X
!60\15, 2769.231
!60\15, 2769.231
!44\11, 2778.947
!44\11, 2778.947
Line 1,594: Line 1,557:
!40\10, 2823.529
!40\10, 2823.529
!52\13, 2836.364
!52\13, 2836.364
|}
{| class="wikitable"
|+Cents
!Notation
!Supersoft
!Soft
! Semisoft
! Basic
!Semihard
!Hard
!Superhard
|-
|-
|A fa mi#
!Guidotonic
|X#
!~15edf
|61\15, 2815.385
!~11edf
|45\11, 2842.105
!~18edf
|74\18, 2864.516
!~7edf
| rowspan="2" |29\7, 2900
!~17edf
|71\17, 2937.069
!~10edf
|42\10, 2964.706
!~13edf
|55\13, 3000
|-
|-
|B sol fab
|F ut#
| Yb, Ye
|1\15, 46.154
|63\15, 2907.692
|1\11, 63.158
|46\11, 2905.263
|2\18, 77.419
|75\18, 2903.226
| rowspan="2" |1\7, 100
| 70\17, 2896.552
|3\17, 124.138
|41\10, 2894.118
|2\10, 141.176
|53\13, 2890.909
|3\13, 163.636
|-
|-
|'''B sol fa'''
|G reb
|'''Y'''
|3\15, 138.462
|'''64\15,''' '''2953.846'''
|2\11. 126.316
|'''47\11,''' '''2968.421'''
|3\18, 116.129
|'''77\18,''' '''2980.645'''
|2\17, 82.759
|'''30\7,''' '''3000'''
|1\10, 70.588
|'''73\17,''' '''3020.690'''
|1\13, 54.545
|'''43\10,''' '''3035.294'''
|'''56\13,''' '''3054.545'''
|-
|-
|B sol fa#
|'''G re'''
|Y#
|'''4\15,''' '''184.615'''
|65\15, 3000
|'''3\11,''' '''189.474'''
| 48\11, 3031.579
|'''5\18,''' '''193.548'''
| 79\18, 3058.064
|'''2\7,''' '''200'''
| rowspan="2" |31\7, 3100
|'''5\17,''' '''206.897'''
|76\17, 3144.828
|'''3\10,''' '''211.765'''
|45\10, 3176.471
|'''4\13,''' '''218.182'''
| 59\13, 3218.182
|-
|-
|C la solb
|G re#
| Zb. Ze
|5\15, 230.769
|67\15, 3092.308
|4\11, 252.632
|49\11, 3094.737
|7\18, 270.968
|80\18, 3096.774
| rowspan="2" |3\7, 300
| 75\17, 3103.448
|8\17, 331.034
|44\10, 3105.882
|5\10, 352.941
|57\13, 3109.091
|7\13, 381.818
|-
|-
| C la sol
|A mib
|Z
|7\15, 323.077
| 68\15, 3138.462
|5\11, 315.789
|50\11, 3157.895
|8\18, 309.677
|82\18, 3174.194
|7\17, 289.655
|32\7, 3200
|4\10, 282.353
|78\17, 3227.586
|5\13, 272.727
|46\10, 3247.059
|60\13, 3272.273
|-
|-
|C la sol#
|A mi
|Z#
|8\15, 369.231
|69\15, 3184.615
| 6\11, 378.947
| rowspan="2" |51\11, 3221.053
|10\18, 387.097
|84\18, 3251.612
|4\7, 400
|33\7, 3300
|10\17, 413.793
| 81\17, 3351.725
|6\10, 423.529
|48\10, 3388.235
|8\13, 436.364
|63\13, 3436.364
|-
|-
|D labb
| A mi#
|Ab, Æ
|9\15, 415.385
|70\15, 3230.769
| rowspan="2" |7\11, 442.105
|83\18, 3212.903
|12\18, 464.516
|32\7, 3200
|5\7, 500
|77\17, 3186.207
|13\17, 537.069
| 45\10, 3176.471
|8\10, 564.706
| 58\13, 3163.636
|11\13, 600
|-
|-
|'''D lab'''
|B fa utb
|'''A'''
|10\15, 461.538
|'''71\15,''' '''3276.923'''
|11\18, 425.806
|'''52\11,''' '''3284.211'''
|4\7, 400
|'''85\18,''' '''3290.323'''
|9\17, 372.414
|'''33\7, 3300'''
|5\10, 352.941
|'''80\17,''' '''3310.345'''
|6\13, 327.273
|'''47\10,''' '''3317.647'''
|'''61\13,''' '''3327.{{Overline|27}}'''
|-
|-
|D la
|'''B fa ut'''
|A#
|'''11\15,''' '''507.692'''
|72\15, 3323.077
|'''8\11,''' '''505.263'''
|53\11, 3347.368
|'''13\18,''' '''503.226'''
|87\18, 3367.742
|'''5\7, 500'''
|34\7, 3400
|'''12\17,''' '''496.552'''
|83\17, 3434.583
|'''7\10,''' '''494.118'''
|49\10, 3458.824
|'''9\13,''' '''490.909'''
|64\13, 3490.909
|-
|-
|D la#
|B fa ut#
|Ax
|12\15, 553.846
|73\15, 3369.231
|9\11, 568.421
| rowspan="2" |54\11, 3410.625
|15\18, 580.645
|89\18, 3445.162
|6\7, 600
|35\7, 3500
|15\17, 620.690
|86\17, 3558.621
|9\10, 635.294
|51\10, 3600
|12\13, 654.545
|67\13, 3654.545
|-
|-
|F utb
|B fa utx
|Bb, Be
| 13\15, 600
| 74\15, 3415.385
| rowspan="2" |10\11, 631.579
|88\18, 3406.452
|17\18, 658.064
|34\7, 3400
|7\7, 700
|82\17, 3393.103
|18\17, 744.828
|48\10, 3388.235
|11\10, 776.471
|62\13, 3381.818
|15\13, 818.182
|-
|-
!F ut
|C sol re utb
! B
| 14\15, 646.154
!75\15, 3461.538
|16\18, 619.355
!55\11, 3473.684
|6\7, 600
! 90\18, 3483.871
|14\17, 579.310
!35\7, 3500
|8\10, 564.706
!85\17, 3517.241
|10\13, 545.455
!50\10, 3529.412
!65\13, 3545.455
|-
|-
|F ut#
!C sol re ut
|B#
!'''15\15,''' '''692.308'''
| 76\15, 3507.692
!'''11\11,''' '''694.737'''
|56\11, 3536.842
!'''18\18,''' '''696.774'''
|92\18, 3561.290
!7\7, 700
| rowspan="2" |36\7, 3600
!'''17\17,''' '''703.448'''
|88\17, 3641.379
!'''10\10,''' '''705.882'''
|52\10, 3670.588
!'''13\13,''' '''709.091'''
|68\13, 3709.091
|-
|-
|G reb
|C sol re ut#
|Cb, Ce
|16\15, 738.462
| 78\15, 3600
|12\11, 757.895
|57\11, 3600
|20\18, 774.194
|93\18, 3600
| rowspan="2" |8\8, 800
|87\17, 3600
|20\17, 827.586
|51\10, 3600
|12\10, 847.059
|66\13, 3600
|16\13, 872.727
|-
|-
|'''G re'''
|D la mi reb
|'''C'''
|18\15, 830.769
|'''79\15,''' '''3646.154'''
|13\11, 821.053
|'''58\11,''' '''3663.158'''
|21\18, 812.903
|'''95\18,''' '''3677.419'''
|19\17, 786.207
|'''37\7,''' '''3700'''
|11\10, 776.471
|'''90\17,''' '''3724.138'''
|14\13, 763.63
|'''53\10,''' '''3741.176'''
|'''69\13,''' '''3763.636'''
|-
|-
|G re#
|'''D la mi re'''
|C#
|'''19\15,''' '''876.923'''
|80\15, 3692.308
|'''14\11,''' '''884.211'''
|59\11, 3726.316
|'''23\18,''' '''890.323'''
|97\18, 3755.838
|'''9\5,''' '''900'''
| rowspan="2" |38\7, 3800
|'''22\17,''' '''910.345'''
|93\17, 3848.275
|'''13\10,''' '''917.647'''
|55\10, 3882.353
|'''17\13,''' '''927.273'''
|72\13, 3927.273
|-
|-
|A mib
|D la mi re#
| Db, De
|20\15, 923.077
|82\15, 3784.615
| rowspan="2" |15\11, 947.368
|60\11, 3789.474
|25\18, 967.742
|98\18, 3793.548
|10\7, 1000
|92\17, 3806.897
|25\17, 1034.483
| 54\10, 3811.765
|15\10, 1058.824
|70\13, 3818.182
|20\13, 1090.909
|-
|-
|A mi
|E fa utb
|D
|21\15, 969.231
|83\15, 3830.769
|24\18, 929.032
|61\11, 3852.632
| 9\5, 900
|100\18, 3870.968
|21\17, 868.966
|39\7, 3900
|12\10, 847.059
|95\17, 3931.03$
|15\13, 818.182
|56\10, 3952.941
| 73\13, 3981.818
|-
|-
|A mi#
|E fa ut
|D#
| 22\15, 1015.385
|84\15, 3876.923
|16\11, 1010.526
| rowspan="2" |62\11, 3915.789
|26\18, 1006.452
|102\18, 3948.387
|10\7, 1000
|40\7, 4000
|24\17, 993.103
|98\17, 4055.172
|14\10, 988.235
|58\10, 4094.118
|18\13, 981.818
|76\13, 4145.455
|-
|-
|B fa utb
|E si mi re
|Ebb, Eee
|23\15, 1061.538
|85\15, 3923.077
|17\11, 1073.684
|101\18, 3909.677
|28\18, 1083.871
|39\7, 3900
|11\7, 1100
|94\17, 3889.552
|27\17, 1117.241
|55\10, 3882.353
|16\10, 1129.412
|71\13, 3872.727
|21\9, 1145.455
|-
|-
|'''B fa ut'''
| E si mi re#
|'''Eb, Ee'''
|24\15, 1107.923
|'''86\15,''' '''3969.231'''
| rowspan="2" |18\11, 1136.842
|'''63\11,''' '''3978.947'''
|30\18, 1161.29
|'''103\18,''' '''3987.097'''
|12\7, 1200
|'''40\7, 4000'''
|30\17, 1241.379
|'''97\17,''' '''4013.793'''
| 18\10, 1270.588
|'''57\10,''' '''4023.529'''
|24\13, 1309.091
|'''74\13,''' '''4036.364'''
|-
|-
|B fa ut#
|F sol fa ut reb
| E
|25\15, 1153.846
|87\15, 4015.385
|29\18, 1122.581
|64\11, 4042.105
|11\7, 1100
|105\18, 4064.516
|26\17, 1075.862
|41\7, 4100
|15\10, 1058.824
| 100\17, 4137.931
|19\13, 1036.364
|59\10, 4164.706
|77\13, 4200
|-
|-
| B fa utx
|'''F sol fa ut re'''
|E#
|'''26\15,''' '''1200'''
|88\15, 4061.583
|'''19\11,''' '''1200'''
| rowspan="2" |65\11, 4105.263
|'''31\18,''' '''1200'''
|107\18, 4141.956
|'''12\7, 1200'''
|42\7, 4200
|'''29\17,''' '''1200'''
|103\17, 4262.069
|'''17\10,''' '''1200'''
|61\10, 4305.882
|'''22\13,''' '''1200'''
|80\13, 4363.636
|-
|-
| C sol reb
|F sol fa ut re#
|Fb, Fe
|27\15, 1246.154
|89\15, 4107.692
|20\11, 1263.158
|106\18, 4103.226
|33\18, 1277.419
| 41\7, 4100
|13\7, 1300
|99\17, 4096.552
|32\17, 1324.138
|58\10, 4094.118
| 19\10, 1341.176
|75\13, 4090.909
| 25\13, 1363.636
|-
|-
!C sol re
|F sol fa ut rex
!F
|28\15, 1292.308
!90\15, 4153.846
| rowspan="2" |21\11, 1326.318
!66\11, 4168.421
|35\18, 1354.834
!108\18, 4180.645
| 14\7, 1400
!42\7, 4200
|35\17, 1448.275
!102\17, 4220.690
|21\10, 1482.353
!60\10, 4235.294
|28\13, 1527.273
!78\13, 4254.545
|}
==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):
|G la sol re mib
| 29\15, 1338.462
|34\18, 1316.129
| 13\7, 1300
|31\17, 1282.759
|18\10, 1270.588
|23\13, 1254.545
|-
|-
|0
!G la sol re mi
|Do, Fa, Sol
!30\15, 1384.615
|perfect unison
!22\11, 1389.473
|0
!36\18, 1393.548
|Do, Fa, Sol
!14\7, 1400
|sesquitave (just fifth)
!34\17, 1406.897
!20\10, 1411.765
!26\13, 1418.182
|-
|-
|1
|G la sol re mi#
|Fa, Sib, Do
|31\15, 1430.769
|perfect fourth
|23\11, 1452.632
| -1
|38\18, 1470.968
|Re, Sol, La
| rowspan="2" |15\7, 1500
|perfect second
|37\17, 1531.034
|22\10, 1552.941
|29\13, 1581.818
|-
|-
|2
|A si la mi fab
|Mib, Lab, Sib
|33\15, 1523.077
|minor third
| 24\11, 1515.789
| -2
|39\18, 1509.677
|Mi, La, Si
|36\17, 1489.655
|major third
|21\10, 1482.759
| 27\13, 1472.273
|-
|-
|3
|'''A si la mi fa'''
|Reb, Solb, Lab
|'''34\15,''' '''1569.231'''
|diminished second
|'''25\11,''' '''1578.947'''
| -3
|'''41\18,''' '''1587.097'''
|Fa#, Si, Do#
|'''16\7,''' '''1600'''
|augmented fourth
|'''39\17,''' '''1613.793'''
|'''23\10,''' '''1623.529'''
|'''30\13,''' '''1636.364'''
|-
|-
| colspan="6" |The chromatic 7-note MOS also has the following intervals (from some root):
|A si la mi fa#
| 35\15, 1615.385
| rowspan="2" |26\11, 1642.105
|43\18, 1664.516
|17\7, 1700
|42\17, 1737.069
| 25\10, 1764.706
|33\13, 1800
|-
|-
|4
|B sol fa utb
|Dob, Fab, Solb
|36\61, 1661.538
|diminished sesquitave
|42\18, 1625.806
| -4
|16\7, 1600
|Do#, Fa#, Sol#
|38\29, 1572.414
|augmented unison (chroma)
|22\10, 1552.941
|28\13, 1527.273
|-
|B sol fa ut
|37\15, 1707.692
|27\11, 1705.263
| 44\18, 1703.226
| 17\7, 1700
|41\17, 1696.552
|24\10, 1694.118
|31\13, 1690.909
|-
|-
|5
|B si
|Fab, Sibb, Dob
|38\15, 1753.846
| diminished fourth
| 28\11, 1768.421
| -5
|46\18, 1780.645
|Re#, Sol#, La#
|18\7, 1800
|augmented second
|44\17, 1820.690
|26\10, 1835.294
|34\13, 1854.545
|-
|-
|6
|B si
|Mibb, Labb, Sibb
|39\15, 1800
|diminished third
| rowspan="2" |29\11, 1831.579
| -6
|48\18, 1858.064
|Mi#, La#, Si#
|19\7, 1900
|augmented third
|47\17, 1944.828
|}
|28\10, 1976.471
|37\13, 2018.182
==Genchain==
|-
|C la sol re utb
The generator chain for this scale is as follows:
|40\15, 1846.154
{| class="wikitable"
|47\18, 1819.355
|Mibb
| 18\7, 1800
Labb
| 43\17, 1779.310
|25\10, 1764.706
Sibb
|32\13, 1745.545
|Fab
|-
Sibb
|'''C la sol re ut'''
|'''41\15,''' '''1892.308'''
Dob
|'''30\11,''' '''1894.737'''
|Dob
|'''49\18,''' '''1896.774'''
Fab
|'''19\7, 1900'''
|'''46\17,''' '''1903.448'''
Solb
|'''27\10,''' '''1905.882'''
|Reb
|'''35\13,''' '''1909.091'''
Solb
|-
|C la sol re ut#
Lab
|42\15, 1938.462
|Mib
|31\11, 1957.895
Lab
|51\18, 1974.194
|20\7, 2000
Sib
|49\17, 2027.586
|Fa
| 29\10, 2047.059
Sib
|38\13, 2072.727
|-
Do
|C la sol re utx
|Do
| rowspan="2" |43\15, 1984.615
Fa
|32\11, 2021.053
|53\18, 2051.612
Sol
|21\7, 2100
|Re
|52\17, 2151.725
Sol
|31\10, 2188.235
|41\13, 2236.364
La
|-
|Mi
|D fa la mi reb
La
|31\11, 1957.895
|50\18, 1935.484
Si
|19\7, 1900
|Fa#
|45\17, 1862.069
Si
|26\10, 1835.294
|33\13, 1800
Do#
|-
|Do#
|D fa la mi re
Fa#
|44\15, 2030.769
|32\11, 2021.053
Sol#
|52\18, 2012.903
|Re#
|20\7, 2000
Sol#
|48\17, 1986.207
|28\10, 1976.471
La#
|36\13, 1963.636
|Mi#
La#
Si#
|-
|-
|d3
!D si la mi re
|d4
!45\15, 2076.923
|d5
!33\11, 2084.211
|d2
!54\18, 2090.323
|m3
!21\7, 2100
|P4
! 51\17, 2110.345
|P1
!30\10, 2117.647
|P2
!39\13, 2127.273
|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
|D si la mi re#
!pattern
|46\15, 2123.077
!notation
| rowspan="2" |34\11, 2147.368
!2nd
|56\18, 2167.742
!3rd
|22\7, 2200
!4th
|54\17, 2234.483
| 32\10, 2258.824
|42\13, 2090.909
|-
|-
|Lydian
|E fab
|LLLs
|47\26, 2169.231
|<nowiki>3|0</nowiki>
|55\16, 2129.032
|P
|21\7, 2100
|M
|50\17, 2068.966
|A
|29\10, 2047.059
|37\13, 2018.182
|-
|E fa
|48\15, 2215.385
|35\11, 2210.526
|57\18, 2206.452
|23\7, 2300
|53\17, 2193.103
|31\10, 2188.235
|40\13, 2181.818
|-
|-
|Major
|E si mi
|LLsL
|49\15, 2261.538
|<nowiki>2|1</nowiki>
|36\11, 1073.684
|P
|59\18, 2283.871
|M
|24\7, 2400
|P
|56\17, 2317.241
|33\10, 2329.412
|43\13, 2345.455
|-
|-
|Minor
|E si mi#
|LLsL
|50\15, 2307.692
|<nowiki>1|2</nowiki>
| rowspan="2" |37\11, 2336.842
|P
|61\18, 2361.290
|m
| rowspan="2" |23\7, 2300
|P
| 59\17, 2441.379
|35\10, 2470.588
|46\13, 2509.091
|-
|-
|Phrygian
|F sol fa utb
|sLLL
|51\15, 2353.846
|<nowiki>0|3</nowiki>
|60\18, 2322.581
|d
|55\17, 2275.862
|m
|32\10, 2258.824
|P
|41\13, 2236.364
|}
|-
|F sol fa ut
==Temperaments==
|52\15, 2400
|38\11, 2400
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.
|62\18, 2400
==='''Napoli-Meantone'''===
|24\7, 2400
|58\17, 2400
[[Subgroup]]: 3/2.6/5.8/5
|34\10, 2400
|44\13, 2400
[[Comma]] list: [[81/80]]
|-
 
|F sol fa ut#
[[POL2]] generator: ~9/8 = 192.6406¢
|53\15, 2446.154
 
|39\11, 2463.158
[[Mapping]]: [{{val|1 1 2}}, {{val|0 -2 -3}}]
|64\18, 2477,419
 
| rowspan="2" |25\7, 2500
[[Optimal ET sequence]]: ~([[7edf]], [[11edf]], [[18edf]])
|61\17, 2524.138
==='''Napoli-Archy'''===
|36\10, 2541.176
|47\13, 2563.636
[[Subgroup]]: 3/2.7/6.14/9
[[Comma]] list: [[64/63]]
 
[[POL2]] generator: ~8/7 = 218.6371¢
 
[[Mapping]]: [{{val|1 1 2}}, {{val|0 -2 -3}}]
 
[[Optimal ET sequence]]: ~([[7edf]], [[10edf]], [[13edf]], [[16edf]])
===Scale tree===
The spectrum looks like this:
{| class="wikitable"
!Generator
(bright)
!Cents
!L
!s
!L/s
!Comments
|-
|-
|1\4
|G la sol reb
|171.429
|55\15, 2538.462
|1
|40\11, 2526.316
|1
|65\18, 2516.129
|1.000
|60\17, 2482.759
|Equalised
|35\10, 2470.588
|45\13, 2454.545
|-
|-
|6\23
|'''G la sol re'''
|180.000
|'''56\15, 2584.615'''
|6
|'''41\11, 2589.474'''
|5
|'''67\18, 2593.548'''
|1.200
|'''26\7, 2600'''
|
|'''63\17, 2606.897'''
|'''37\10, 2611.765'''
|'''48\13,''' '''2618.182'''
|-
|-
|5\19
|G la sol re#
|181.818
|57\15, 2630.769
|5
|42\11, 2652.632
|4
|69\18, 2670.968
|1.250
| rowspan="2" |27\7, 2700
|
|66\17, 2731.034
|39\10, 2752.941
|51\13, 2781.818
|-
|-
|14\53
|A si la mib
|182.609
|59\15, 2723.077
|14
|43\11, 2715.789
|11
|70\18, 2709.677
|1.273
|65\17, 2689.552
|
|38\10, 2682.353
|49\13, 2672.273
|-
|-
|9\34
!A si la mi
| 183.051
!60\15, 2769.231
|9
!44\11, 2778.947
|7
!72\18, 2787.097
| 1.286
!28\7, 2800
|
!68\17, 2813.793
!40\10, 2823.529
!52\13, 2836.364
|-
|-
|4\15
|A si la mi#
|184.615
|61\15, 2815.385
|4
| rowspan="2" |45\11, 2842.105
|3
| 74\18, 2864.516
|1.333
|29\7, 2900
|
|71\17, 2937.069
|42\10, 2964.706
|55\13, 3000
|-
|-
|11\41
|B fab
|185.915
|62\15, 2861.538
|11
|73\18, 2825.806
|8
| 28\7, 2800
|1.375
|67\17, 2772.414
|
|39\10, 2752.941
|50\13, 2727.273
|-
|-
|7\26
|B fa
|186.667
|63\15, 2907.692
|7
|46\11, 2905.263
|5
|75\18, 2903.226
|1.400
|29\7, 2900
|
|70\17, 2896.552
|41\10, 2894.118
|53\13, 2890.909
|-
|-
|10\37
|'''B si'''
|187.5
|'''64\15, 2953.846'''
|10
|'''47\11, 2968.421'''
| 7
|'''77\18, 2980.645'''
| 1.429
|'''30\7, 3000'''
|
|'''73\17, 3020.690'''
|'''43\10, 3035.294'''
|'''56\13, 3054.545'''
|-
|-
|13\48
|B si#
| 187.952
|65\15, 3000
|13
|48\11, 3031.579
|9
|79\18, 3058.064
|1.444
| rowspan="2" |31\7, 3100
|
|76\17, 3144.828
|45\10, 3176.471
|59\13, 3218.182
|-
|-
|16\59
|C solb
|188.253
|67\15, 3092.308
|16
|49\11, 3094.737
|11
|80\18, 3096.774
|1.455
|75\17, 3103.448
|
|44\10, 3105.882
|57\13, 3109.091
|-
|-
|3\11
|C sol
|189.474
|68\15, 3138.462
|3
|50\11, 3157.895
|2
| 82\18, 3174.194
|1.500
|32\7, 3200
| Napoli-Meantone starts here
|78\17, 3227.586
| 46\10, 3247.059
|60\13, 3272.273
|-
|-
|14\51
|C sol#
|190.909
| 69\15, 3184.615
|14
| rowspan="2" |51\11, 3221.053
|9
|84\18, 3251.612
|1.556
|33\7, 3300
|
|81\17, 3351.725
|48\10, 3388.235
|63\13, 3436.364
|-
|-
|11\40
|D labb
|191.304
|70\15, 3230.769
|11
|83\18, 3212.903
|7
|32\7, 3200
| 1.571
|77\17, 3186.207
|
|45\10, 3176.471
|58\13, 3163.636
|-
|-
|8\29
|'''D lab'''
|192.000
|'''71\15,''' '''3276.923'''
|8
|'''52\11,''' '''3284.211'''
|5
|'''85\18,''' '''3290.323'''
|1.600
|'''33\7, 3300'''
|
|'''80\17,''' '''3310.345'''
|'''47\10,''' '''3317.647'''
|'''61\13,''' '''3327.{{Overline|27}}'''
|-
|-
|5\18
|D la
|193.548
|72\15, 3323.077
|5
|53\11, 3347.368
|3
|87\18, 3367.742
|1.667
|34\7, 3400
|
|83\17, 3434.583
|49\10, 3458.824
|64\13, 3490.909
|-
|-
|12\43
|D la#
|194.595
|73\15, 3369.231
|12
| rowspan="2" |54\11, 3410.625
|7
|89\18, 3445.162
|1.714
|35\7, 3500
|
|86\17, 3558.621
|51\10, 3600
|67\13, 3654.545
|-
|-
|7\25
|F utb
|195.348
|74\15, 3415.385
|7
|88\18, 3406.452
|4
|34\7, 3400
|1.750
|82\17, 3393.103
|
|48\10, 3388.235
|62\13, 3381.818
|-
|-
|9\32
!F ut
|196.364
!75\15, 3461.538
|9
!55\11, 3473.684
|5
!90\18, 3483.871
|1.800
!35\7, 3500
|
!85\17, 3517.241
!50\10, 3529.412
!65\13, 3545.455
|}
 
{| class="wikitable"
|+Cents
!Notation
!Supersoft
!Soft
! Semisoft
!Basic
!Semihard
!Hard
!Superhard
|-
|-
|11\39
!Subdozenal
|197.015
!~15edf
|11
!~11edf
|6
!~18edf
|1.833
!~7edf
|
!~17edf
!~10edf
!~13edf
|-
|-
|13\46
|F#
|197.468
|1\15, 46.154
|13
|1\11, 63.158
|7
|2\18, 77.419
|1.857
| rowspan="2" |1\7, 100
|
|3\17, 124.138
|2\10, 141.176
|3\13, 163.636
|-
|-
|15\53
|Gb, Ge
|197.802
|3\15, 138.462
|15
|2\11. 126.316
|8
|3\18, 116.129
|1.875
|2\17, 82.759
|
|1\10, 70.588
|1\13, 54.545
|-
|-
|17\60
|'''G'''
|198.058
|'''4\15,''' '''184.615'''
|17
|'''3\11,''' '''189.474'''
|9
|'''5\18,''' '''193.548'''
|1.889
|'''2\7,''' '''200'''
|
|'''5\17,''' '''206.897'''
|'''3\10,''' '''211.765'''
|'''4\13,''' '''218.182'''
|-
|-
|19\67
|G#
|198.261
|5\15, 230.769
|19
|4\11, 252.632
|10
|7\18, 270.968
|1.900
| rowspan="2" |3\7, 300
|
|8\17, 331.034
|5\10, 352.941
|7\13, 381.818
|-
|-
|21\74
|Hb, He
|198.425
|7\15, 323.077
|21
|5\11, 315.789
|11
|8\18, 309.677
|1.909
|7\17, 289.655
|
|4\10, 282.353
|5\13, 272.727
|-
|-
|23\81
|H
|198.561
|8\15, 369.231
|23
|6\11, 378.947
|12
|10\18, 387.097
|1.917
|4\7, 400
|
|10\17, 413.793
|6\10, 423.529
|8\13, 436.364
|-
|-
|25\88
|H#
|198.675
|9\15, 415.385
|25
| rowspan="2" |7\11, 442.105
|13
|12\18, 464.516
|1.923
|5\7, 500
|
|13\17, 537.069
|8\10, 564.706
|11\13, 600
|-
|-
|27\95
|Jbb, Jee
|198.773
|10\15, 461.538
|27
|11\18, 425.806
|14
|4\7, 400
|1.929
|9\17, 372.414
|
|5\10, 352.941
|6\13, 327.273
|-
|-
|29\102
|'''Jb, Je'''
| 198.857
|'''11\15,''' '''507.692'''
|29
|'''8\11,''' '''505.263'''
|15
|'''13\18,''' '''503.226'''
|1.933
|'''5\7, 500'''
|
|'''12\17,''' '''496.552'''
|'''7\10,''' '''494.118'''
|'''9\13,''' '''490.909'''
|-
|-
|31\109
|J
|198.930
|12\15, 553.846
| 31
|9\11, 568.421
|16
|15\18, 580.645
|1.9375
|6\7, 600
|
|15\17, 620.690
|9\10, 635.294
|12\13, 654.545
|-
|-
|33\116
|J#
|198.995
|13\15, 600
|33
| rowspan="2" |10\11, 631.579
| 17
|17\18, 658.064
|1.941
|7\7, 700
|
|18\17, 744.828
|11\10, 776.471
|15\13, 818.182
|-
|-
|35\123
|Kb, Ke
|199.009
|14\15, 646.154
| 35
|16\18, 619.355
|18
|6\7, 600
|1.944
|14\17, 579.310
|
|8\10, 564.706
|10\13, 545.455
|-
|-
|2\7
!K
|200
!'''15\15,''' '''692.308'''
|2
!'''11\11,''' '''694.737'''
|1
!'''18\18,''' '''696.774'''
|2.000
!7\7, 700
|Napoli-Meantone ends, Napoli-Pythagorean begins
!'''17\17,''' '''703.448'''
!'''10\10,''' '''705.882'''
!'''13\13,''' '''709.091'''
|-
|-
|17\59
|K#
| 201.980
|16\15, 738.462
|17
|12\11, 757.895
|8
|20\18, 774.194
|2.125
| rowspan="2" |8\8, 800
|
|20\17, 827.586
|12\10, 847.059
|16\13, 872.727
|-
|-
|15\52
|Lb, Le
|202.247
|18\15, 830.769
|15
|13\11, 821.053
|7
|21\18, 812.903
|2.143
|19\17, 786.207
|
|11\10, 776.471
|14\13, 763.63
|-
|-
|13\45
|'''L'''
|202.597
|'''19\15,''' '''876.923'''
|13
|'''14\11,''' '''884.211'''
|6
|'''23\18,''' '''890.323'''
|2.167
|'''9\5,''' '''900'''
|
|'''22\17,''' '''910.345'''
|'''13\10,''' '''917.647'''
|'''17\13,''' '''927.273'''
|-
|-
|11\38
|L#
|203.077
|20\15, 923.077
|11
| rowspan="2" |15\11, 947.368
|5
|25\18, 967.742
|2.200
|10\7, 1000
|
|25\17, 1034.483
|15\10, 1058.824
|20\13, 1090.909
|-
|-
|9\31
|Mbb, Mee
|203.774
|21\15, 969.231
|9
|24\18, 929.032
|4
|9\5, 900
|2.250
|21\17, 868.966
|
|12\10, 847.059
|15\13, 818.182
|-
|-
|7\24
|Mb, Me
|204.878
|22\15, 1015.385
|7
|16\11, 1010.526
|3
|26\18, 1006.452
|2.333
|10\7, 1000
|
|24\17, 993.103
|14\10, 988.235
|18\13, 981.818
|-
|-
|12\41
|M
|205.714
|23\15, 1061.538
|12
|17\11, 1073.684
| 5
|28\18, 1083.871
|2.400
|11\7, 1100
|
|27\17, 1117.241
|16\10, 1129.412
|21\9, 1145.455
|-
|-
|5\17
|M#
|206.897
|24\15, 1107.923
|5
| rowspan="2" |18\11, 1136.842
|2
|30\18, 1161.29
|2.500
|12\7, 1200
|Napoli-Neogothic heartland is from here…
|30\17, 1241.379
|18\10, 1270.588
|24\13, 1309.091
|-
|-
|18\61
|Nbb, Nee
|207.693
|25\15, 1153.846
|18
|29\18, 1122.581
|7
|11\7, 1100
|2.571
|26\17, 1075.862
|
|15\10, 1058.824
|19\13, 1036.364
|-
|-
|13\44
|'''Nb, Ne'''
|208.000
|'''26\15,''' '''1200'''
|13
|'''19\11,''' '''1200'''
|5
|'''31\18,''' '''1200'''
|2.600
|'''12\7, 1200'''
|
|'''29\17,''' '''1200'''
|'''17\10,''' '''1200'''
|'''22\13,''' '''1200'''
|-
|-
|8\27
|N
|208.696
|27\15, 1246.154
|8
|20\11, 1263.158
|3
|33\18, 1277.419
|2.667
|13\7, 1300
|…to here
|32\17, 1324.138
|19\10, 1341.176
|25\13, 1363.636
|-
|-
|11\37
|N#
|209.524
|28\15, 1292.308
| 11
| rowspan="2" |21\11, 1326.318
| 4
|35\18, 1354.834
|2.750
|14\7, 1400
|
|35\17, 1448.275
|21\10, 1482.353
|28\13, 1527.273
|-
|-
|14\47
|Pb, Pe
|210.000
|29\15, 1338.462
|14
|34\18, 1316.129
|5
|13\7, 1300
|2.800
|31\17, 1282.759
|
|18\10, 1270.588
|23\13, 1254.545
|-
|-
| 3\10
!P
|211.765
!30\15, 1384.615
| 3
!22\11, 1389.473
|1
!36\18, 1393.548
|3.000
!14\7, 1400
|Napoli-Pythagorean ends, Napoli-Archy begins
!34\17, 1406.897
!20\10, 1411.765
!26\13, 1418.182
|-
|-
|22\73
|P#
|212.903
|31\15, 1430.769
|22
|23\11, 1452.632
|7
|38\18, 1470.968
|3.143
| rowspan="2" |15\7, 1500
|
|37\17, 1531.034
|22\10, 1552.941
|29\13, 1581.818
|-
|-
|19\63
|Qb, Qe
|213.084
|33\15, 1523.077
|19
|24\11, 1515.789
|6
|39\18, 1509.677
|3.167
|36\17, 1489.655
|
|21\10, 1482.759
|27\13, 1472.273
|-
|-
|16\53
|'''Q'''
|213.333
|'''34\15,''' '''1569.231'''
|16
|'''25\11,''' '''1578.947'''
|5
|'''41\18,''' '''1587.097'''
|3.200
|'''16\7,''' '''1600'''
|
|'''39\17,''' '''1613.793'''
|'''23\10,''' '''1623.529'''
|'''30\13,''' '''1636.364'''
|-
|-
|13\43
|Q#
|213.699
|35\15, 1615.385
|13
| rowspan="2" |26\11, 1642.105
|4
|43\18, 1664.516
|3.250
|17\7, 1700
|
|42\17, 1737.069
|25\10, 1764.706
|33\13, 1800
|-
|-
|10\33
|Rb, Re
|214.286
|36\61, 1661.538
| 10
|42\18, 1625.806
|3
|16\7, 1600
|3.333
|38\29, 1572.414
|
|22\10, 1552.941
|28\13, 1527.273
|-
|-
|7\23
|R
|215.385
|37\15, 1707.692
|7
|27\11, 1705.263
|2
|44\18, 1703.226
|3.500
|17\7, 1700
|
|41\17, 1696.552
|24\10, 1694.118
|31\13, 1690.909
|-
|-
|11\36
|R#
|216.393
|38\15, 1753.846
|11
|28\11, 1768.421
|3
|46\18, 1780.645
| 3.667
|18\7, 1800
|
|44\17, 1820.690
|26\10, 1835.294
|34\13, 1854.545
|-
|-
|15\49
|R#
| 216.867
|39\15, 1800
|15
| rowspan="2" |29\11, 1831.579
|4
|48\18, 1858.064
| 3.750
|19\7, 1900
|
|47\17, 1944.828
|28\10, 1976.471
|37\13, 2018.182
|-
|-
|19\62
|Sb, Se
| 217.143
|40\15, 1846.154
|19
|47\18, 1819.355
|5
|18\7, 1800
|3.800
|43\17, 1779.310
|
|25\10, 1764.706
|32\13, 1745.545
|-
|-
|4\13
|'''S'''
|218.182
|'''41\15,''' '''1892.308'''
|4
|'''30\11,''' '''1894.737'''
|1
|'''49\18,''' '''1896.774'''
|4.000
|'''19\7, 1900'''
|
|'''46\17,''' '''1903.448'''
|'''27\10,''' '''1905.882'''
|'''35\13,''' '''1909.091'''
|-
|-
|13\42
|S#
|219.718
|42\15, 1938.462
|13
|31\11, 1957.895
|3
|51\18, 1974.194
|4.333
|20\7, 2000
|
|49\17, 2027.586
|29\10, 2047.059
|38\13, 2072.727
|-
|-
|9\29
|Sx
|220.408
|43\15, 1984.615
|9
| rowspan="2" |32\11, 2021.053
|2
|53\18, 2051.612
|4.500
|21\7, 2100
|
|52\17, 2151.725
|31\10, 2188.235
|41\13, 2236.364
|-
|-
|14\45
|Tb, Te
|221.053
|44\15, 2030.769
|14
|52\18, 2012.903
|3
|20\7, 2000
|4.667
|48\17, 1986.207
|
|28\10, 1976.471
|36\13, 1963.636
|-
|-
|5\16
!T
| 222.222
!45\15, 2076.923
| 5
!33\11, 2084.211
|1
!54\18, 2090.323
|5.000
!21\7, 2100
|Napoli-Archy ends
!51\17, 2110.345
!30\10, 2117.647
!39\13, 2127.273
|-
|-
|11\35
|T#
|223.728
|46\15, 2123.077
|11
| rowspan="2" |34\11, 2147.368
|2
|56\18, 2167.742
|5.500
|22\7, 2200
|
|54\17, 2234.483
|32\10, 2258.824
|42\13, 2090.909
|-
|-
|17\54
|Ub, Üe
|224.176
|47\26, 2169.231
|17
|55\16, 2129.032
|3
|21\7, 2100
| 5.667
|50\17, 2068.966
|
|29\10, 2047.059
|37\13, 2018.182
|-
|-
|6\19
|Ub, Ü
|225.000
|48\15, 2215.385
|6
|35\11, 2210.526
|1
|57\18, 2206.452
|6.000
|23\7, 2300
|53\17, 2193.103
|31\10, 2188.235
|40\13, 2181.818
|-
|U
|49\15, 2261.538
|36\11, 1073.684
|59\18, 2283.871
|24\7, 2400
|56\17, 2317.241
|33\10, 2329.412
|43\13, 2345.455
|-
|U#
|50\15, 2307.692
| rowspan="2" |37\11, 2336.842
|61\18, 2361.290
| rowspan="2" |23\7, 2300
|59\17, 2441.379
|35\10, 2470.588
|46\13, 2509.091
|-
|Vb, Ve
|51\15, 2353.846
|60\18, 2322.581
|55\17, 2275.862
|32\10, 2258.824
|41\13, 2236.364
|-
|V
|52\15, 2400
|38\11, 2400
|62\18, 2400
|24\7, 2400
|58\17, 2400
|34\10, 2400
|44\13, 2400
|-
|V#
|53\15, 2446.154
|39\11, 2463.158
|64\18, 2477,419
| rowspan="2" |25\7, 2500
|61\17, 2524.138
|36\10, 2541.176
|47\13, 2563.636
|-
|Wb, We
|55\15, 2538.462
|40\11, 2526.316
|65\18, 2516.129
|60\17, 2482.759
|35\10, 2470.588
|45\13, 2454.545
|-
|'''Wb'''
|'''56\15, 2584.615'''
|'''41\11, 2589.474'''
|'''67\18, 2593.548'''
|'''26\7, 2600'''
|'''63\17, 2606.897'''
|'''37\10, 2611.765'''
|'''48\13,''' '''2618.182'''
|-
|W#
|57\15, 2630.769
|42\11, 2652.632
|69\18, 2670.968
| rowspan="2" |27\7, 2700
|66\17, 2731.034
|39\10, 2752.941
|51\13, 2781.818
|-
|Xb, Xe
|59\15, 2723.077
|43\11, 2715.789
|70\18, 2709.677
|65\17, 2689.552
|38\10, 2682.353
|49\13, 2672.273
|-
!X
!60\15, 2769.231
!44\11, 2778.947
!72\18, 2787.097
!28\7, 2800
!68\17, 2813.793
!40\10, 2823.529
!52\13, 2836.364
|-
|X#
|61\15, 2815.385
| rowspan="2" |45\11, 2842.105
|74\18, 2864.516
|29\7, 2900
|71\17, 2937.069
|42\10, 2964.706
|55\13, 3000
|-
|Ybb, Yee
|62\15, 2861.538
|73\18, 2825.806
|28\7, 2800
|67\17, 2772.414
|39\10, 2752.941
|50\13, 2727.273
|-
|Yb, Ye
|63\15, 2907.692
|46\11, 2905.263
|75\18, 2903.226
|29\7, 2900
|70\17, 2896.552
|41\10, 2894.118
|53\13, 2890.909
|-
|'''Y'''
|'''64\15, 2953.846'''
|'''47\11, 2968.421'''
|'''77\18, 2980.645'''
|'''30\7, 3000'''
|'''73\17, 3020.690'''
|'''43\10, 3035.294'''
|'''56\13, 3054.545'''
|-
|Y#
|65\15, 3000
|48\11, 3031.579
|79\18, 3058.064
| rowspan="2" |31\7, 3100
|76\17, 3144.828
|45\10, 3176.471
|59\13, 3218.182
|-
|Zb. Ze
|67\15, 3092.308
|49\11, 3094.737
|80\18, 3096.774
|75\17, 3103.448
|44\10, 3105.882
|57\13, 3109.091
|-
|Z
|68\15, 3138.462
|50\11, 3157.895
|82\18, 3174.194
|32\7, 3200
|78\17, 3227.586
|46\10, 3247.059
|60\13, 3272.273
|-
|Z#
|69\15, 3184.615
| rowspan="2" |51\11, 3221.053
|84\18, 3251.612
|33\7, 3300
|81\17, 3351.725
|48\10, 3388.235
|63\13, 3436.364
|-
|Ab, Æ
|70\15, 3230.769
|83\18, 3212.903
|32\7, 3200
|77\17, 3186.207
|45\10, 3176.471
|58\13, 3163.636
|-
|'''A'''
|'''71\15,''' '''3276.923'''
|'''52\11,''' '''3284.211'''
|'''85\18,''' '''3290.323'''
|'''33\7, 3300'''
|'''80\17,''' '''3310.345'''
|'''47\10,''' '''3317.647'''
|'''61\13,''' '''3327.{{Overline|27}}'''
|-
|A#
|72\15, 3323.077
|53\11, 3347.368
|87\18, 3367.742
|34\7, 3400
|83\17, 3434.583
|49\10, 3458.824
|64\13, 3490.909
|-
|Ax
|73\15, 3369.231
| rowspan="2" |54\11, 3410.625
|89\18, 3445.162
|35\7, 3500
|86\17, 3558.621
|51\10, 3600
|67\13, 3654.545
|-
|Bb, Be
|74\15, 3415.385
|88\18, 3406.452
|34\7, 3400
|82\17, 3393.103
|48\10, 3388.235
|62\13, 3381.818
|-
!B
!75\15, 3461.538
!55\11, 3473.684
!90\18, 3483.871
!35\7, 3500
!85\17, 3517.241
!50\10, 3529.412
!65\13, 3545.455
|-
|B#
|76\15, 3507.692
|56\11, 3536.842
|92\18, 3561.290
| rowspan="2" |36\7, 3600
|88\17, 3641.379
|52\10, 3670.588
|68\13, 3709.091
|-
|Cb, Ce
|78\15, 3600
|57\11, 3600
|93\18, 3600
|87\17, 3600
|51\10, 3600
|66\13, 3600
|-
|'''C'''
|'''79\15,''' '''3646.154'''
|'''58\11,''' '''3663.158'''
|'''95\18,''' '''3677.419'''
|'''37\7,''' '''3700'''
|'''90\17,''' '''3724.138'''
|'''53\10,''' '''3741.176'''
|'''69\13,''' '''3763.636'''
|-
|C#
|80\15, 3692.308
|59\11, 3726.316
|97\18, 3755.838
| rowspan="2" |38\7, 3800
|93\17, 3848.275
|55\10, 3882.353
|72\13, 3927.273
|-
|Db, De
|82\15, 3784.615
|60\11, 3789.474
|98\18, 3793.548
|92\17, 3806.897
|54\10, 3811.765
|70\13, 3818.182
|-
|D
|83\15, 3830.769
|61\11, 3852.632
|100\18, 3870.968
|39\7, 3900
|95\17, 3931.03$
|56\10, 3952.941
|73\13, 3981.818
|-
|D#
|84\15, 3876.923
| rowspan="2" |62\11, 3915.789
|102\18, 3948.387
|40\7, 4000
|98\17, 4055.172
|58\10, 4094.118
|76\13, 4145.455
|-
|Ebb, Ëe
|85\15, 3923.077
|101\18, 3909.677
|39\7, 3900
|94\17, 3889.552
|55\10, 3882.353
|71\13, 3872.727
|-
|'''Eb, Ë'''
|'''86\15,''' '''3969.231'''
|'''63\11,''' '''3978.947'''
|'''103\18,''' '''3987.097'''
|'''40\7, 4000'''
|'''97\17,''' '''4013.793'''
|'''57\10,''' '''4023.529'''
|'''74\13,''' '''4036.364'''
|-
|E
|87\15, 4015.385
|64\11, 4042.105
|105\18, 4064.516
|41\7, 4100
|100\17, 4137.931
|59\10, 4164.706
|77\13, 4200
|-
|E#
|88\15, 4061.583
| rowspan="2" |65\11, 4105.263
|107\18, 4141.956
|42\7, 4200
|103\17, 4262.069
|61\10, 4305.882
|80\13, 4363.636
|-
|Fb, Fe
|89\15, 4107.692
|106\18, 4103.226
|41\7, 4100
|99\17, 4096.552
|58\10, 4094.118
|75\13, 4090.909
|-
!F
!90\15, 4153.846
!66\11, 4168.421
!108\18, 4180.645
!42\7, 4200
!102\17, 4220.690
!60\10, 4235.294
!78\13, 4254.545
|}
==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, Fa, Sol
|perfect unison
|0
|Do, Fa, Sol
|sesquitave (just fifth)
|-
|1
|Fa, Sib, Do
|perfect fourth
| -1
|Re, Sol, La
|perfect second
|-
|2
|Mib, Lab, Sib
|minor third
| -2
|Mi, La, Si
|major third
|-
|3
|Reb, Solb, Lab
|diminished second
| -3
|Fa#, Si, Do#
|augmented fourth
|-
| colspan="6" |The chromatic 7-note MOS also has the following intervals (from some root):
|-
|4
|Dob, Fab, Solb
|diminished sesquitave
| -4
|Do#, Fa#, Sol#
|augmented unison (chroma)
|-
|5
|Fab, Sibb, Dob
|diminished fourth
| -5
|Re#, Sol#, La#
|augmented second
|-
|6
|Mibb, Labb, Sibb
|diminished third
| -6
|Mi#, La#, Si#
|augmented third
|}
==Genchain==
The generator chain for this scale is as follows:
{| class="wikitable"
|Mibb
Labb
Sibb
|Fab
Sibb
Dob
|Dob
Fab
Solb
|Reb
Solb
Lab
|Mib
Lab
Sib
|Fa
Sib
Do
|Do
Fa
Sol
|Re
Sol
La
|Mi
La
Si
|Fa#
Si
Do#
|Do#
Fa#
Sol#
|Re#
Sol#
La#
|Mi#
La#
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
|LsLL
|<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 (Hex meantone)'''===
[[Subgroup]]: 3/2.6/5.8/5 (5.2.3)
[[Comma]] list: [[81/80]]
 
[[POL2]] generator: ~9/8 = 192.6406¢
 
[[Mapping]]: [{{val|1 1 2}}, {{val|0 -2 -3}}]
 
[[Optimal ET sequence]]: *[[28ed5]], [[44ed5]], [[72ed5]] ≈ [[7edf]], [[11edf]], [[18edf]]
==='''Napoli-Archy (Hex Archytas)'''===
[[Subgroup]]: 3/2.7/6.14/9 (36/7.2.3)
[[Comma]] list: [[64/63]]
 
[[POL2]] generator: ~8/7 = 218.6371¢
 
[[Mapping]]: [{{val|1 1 2}}, {{val|0 -2 -3}}]
 
[[Optimal ET sequence]]: *[[28ed36/7]], [[40ed36/7]], [[52ed36/7]] ≈ [[7edf]], [[10edf]], [[13edf]]
===Scale tree===
The spectrum looks like this:
{| class="wikitable"
!Generator
(bright)
!Cents
!L
!s
!L/s
!Comments
|-
|1\4
|171.429
|1
|1
|1.000
|Equalised
|-
|6\23
|180.000
|6
|5
|1.200
|
|-
|5\19
|181.818
|5
|4
|1.250
|
|-
|14\53
|182.609
|14
|11
|1.273
|
|
|-
|-
|1\3
|9\34
|240.000
|183.051
|1
|9
|0
|7
|→ inf
|1.286
|Paucitonic
|
|}
|-
 
|4\15
==See also==
|184.615
[[3L 1s (3/2-equivalent)]] - idealized tuning
|4
|3
|1.333
|
|-
|11\41
|185.915
|11
|8
|1.375
|
|-
|7\26
|186.667
|7
|5
|1.400
|
|-
|10\37
|187.5
|10
|7
|1.429
|
|-
|13\48
|187.952
|13
|9
|1.444
|
|-
|16\59
|188.253
|16
|11
|1.455
|
|-
|3\11
|189.474
|3
|2
|1.500
|Napoli-Meantone starts here
|-
|14\51
|190.909
|14
|9
|1.556
|
|-
|11\40
|191.304
|11
|7
|1.571
|
|-
|8\29
|192.000
|8
|5
|1.600
|
|-
|5\18
|193.548
|5
|3
|1.667
|
|-
|12\43
|194.595
|12
|7
|1.714
|
|-
|7\25
|195.348
|7
|4
|1.750
|
|-
|9\32
|196.364
|9
|5
|1.800
|
|-
|11\39
|197.015
|11
|6
|1.833
|
|-
|13\46
|197.468
|13
|7
|1.857
|
|-
|15\53
|197.802
|15
|8
|1.875
|
|-
|17\60
|198.058
|17
|9
|1.889
|
|-
|19\67
|198.261
|19
|10
|1.900
|
|-
|21\74
|198.425
|21
|11
|1.909
|
|-
|23\81
|198.561
|23
|12
|1.917
|
|-
|25\88
|198.675
|25
|13
|1.923
|
|-
|27\95
|198.773
|27
|14
|1.929
|
|-
|29\102
|198.857
|29
|15
|1.933
|
|-
|31\109
|198.930
|31
|16
|1.9375
|
|-
|33\116
|198.995
|33
|17
|1.941
|
|-
|35\123
|199.009
|35
|18
|1.944
|
|-
|2\7
|200
|2
|1
|2.000
|Napoli-Meantone ends, Napoli-Pythagorean begins
|-
|17\59
|201.980
|17
|8
|2.125
|
|-
|15\52
|202.247
|15
|7
|2.143
|
|-
|13\45
|202.597
|13
|6
|2.167
|
|-
|11\38
|203.077
|11
|5
|2.200
|
|-
|9\31
|203.774
|9
|4
|2.250
|
|-
|7\24
|204.878
|7
|3
|2.333
|
|-
|12\41
|205.714
|12
|5
|2.400
|
|-
|5\17
|206.897
|5
|2
|2.500
|Napoli-Neogothic heartland is from here…
|-
|18\61
|207.693
|18
|7
|2.571
|
|-
|13\44
|208.000
|13
|5
|2.600
|
|-
|8\27
|208.696
|8
|3
|2.667
|…to here
|-
|11\37
|209.524
|11
|4
|2.750
|
|-
|14\47
|210.000
|14
|5
|2.800
|
|-
|3\10
|211.765
|3
|1
|3.000
|Napoli-Pythagorean ends, Napoli-Archy begins
|-
|22\73
|212.903
|22
|7
|3.143
|
|-
|19\63
|213.084
|19
|6
|3.167
|
|-
|16\53
|213.333
|16
|5
|3.200
|
|-
|13\43
|213.699
|13
|4
|3.250
|
|-
|10\33
|214.286
|10
|3
|3.333
|
|-
|7\23
|215.385
|7
|2
|3.500
|
|-
|11\36
|216.393
|11
|3
|3.667
|
|-
|15\49
|216.867
|15
|4
|3.750
|
|-
|19\62
|217.143
|19
|5
|3.800
|
|-
|4\13
|218.182
|4
|1
|4.000
|
|-
|13\42
|219.718
|13
|3
|4.333
|
|-
|9\29
|220.408
|9
|2
|4.500
|
|-
|14\45
|221.053
|14
|3
|4.667
|
|-
|5\16
|222.222
|5
|1
|5.000
|Napoli-Archy ends
|-
|11\35
|223.728
|11
|2
|5.500
|
|-
|17\54
|224.176
|17
|3
|5.667
|
|-
|6\19
|225.000
|6
|1
|6.000
|
|-
|1\3
|240.000
|1
|0
|→ inf
|Paucitonic
|}
 
==See also==
[[3L 1s (3/2-equivalent)]] - idealized tuning
 
[[6L 2s (20/9-equivalent)]] - Neapolitan 1/2-comma meantone
 
[[6L 2s (88/39-equivalent)]] - Neapolitan gentle temperament
 
[[6L 2s (16/7-equivalent)]] - Neapolitan 1/2-comma archy
 
[[9L 3s (10/3-equivalent)]] - Bijou 1/3-comma meantone
 
[[9L 3s (44/13-equivalent)]] - Bijou gentle temperament
 
[[9L 3s (24/7-equivalent)]] - Bijou 1/3-comma archy
 
[[12L 4s (5/1-equivalent)]] - Hex meantone
 
[[12L 4s (56/11-equivalent)]] - Hextone gentle temperament
 
[[12L 4s (36/7-equivalent)]] - Hextone 1/4-comma archy
 
[[15L 5s (15/2-equivalent)]] - Guidotonic major 1/5-comma meantone
 
[[15L 5s (84/11-equivalent)]] - Guidotonic major gentle temperament
 
[[15L 5s (54/7-equivalent)]] - Guidotonic major 1/5-comma archy
 
[[18L 6s (11/1-equivalent)]] - Subdozenal harmonic tuning


[[6L 2s (20/9-equivalent)]] - Neapolitan 1/2-comma meantone
[[18L 6s (56/5-equivalent)]] - Subdozenal low septimal (meantone) tuning


[[6L 2s (88/39-equivalent)]] - Neapolitan gentle temperament
[[18L 6s (512/45-equivalent)]] - Subdozenal 1/6-comma meantone  
 
[[6L 2s (16/7-equivalent)]] - Neapolitan 1/2-comma archy
 
[[9L 3s (10/3-equivalent)]] - Bijou 1/3-comma meantone
 
[[9L 3s (44/13-equivalent)]] - Bijou gentle temperament
 
[[9L 3s (24/7-equivalent)]] - Bijou 1/3-comma archy
 
[[12L 4s (5/1-equivalent)]] - Hex meantone
 
[[12L 4s (56/11-equivalent)]] - Hextone gentle temperament
 
[[12L 4s (36/7-equivalent)]] - Hextone 1/4-comma archy
 
[[15L 5s (15/2-equivalent)]] - Guidotonic major 1/5-comma meantone
 
[[15L 5s (84/11-equivalent)]] - Guidotonic major gentle temperament
 
[[15L 5s (54/7-equivalent)]] - Guidotonic major 1/5-comma archy
 
[[18L 6s (11/1-equivalent)]] - Subdozenal harmonic tuning


[[18L 6s (56/5-equivalent)]] - Subdozenal low septimal meantone tuning
[[18L 6s (80/7-equivalent)]] - Subdozenal high septimal tuning


[[18L 6s (80/7-equivalent)]] - Subdozenal high septimal meantone tuning
[[18L 6s (128/11-equivalent)]] - Subdozenal subharmonic tuning


[[18L 6s (128/11-equivalent)]] - Subdozenal subharmonic tuning<references />
[[18L 6s (11/1-equivalent)|18L 6s (12/1-equivalent)]] - Warped Pythagorean tuning
Retrieved from "https://en.xen.wiki/w/User:Moremajorthanmajor/3L_1s_(perfect_fifth-equivalent)"