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

From Xenharmonic Wiki
Jump to navigation Jump to search
← Older editNewer edit →
VisualWikitext
No edit summary
mNo edit summary
Line 18: Line 18:
}}'''3L 1s<3/2>''', is a fifth-repeating MOS scale. The notation "<3/2>" means the period of the MOS is 3/2, disambiguating it from octave-repeating [[3L 1s]]. The name of the period interval is called the '''sesquitave''' (by analogy to the [[tritave]]).  
}}'''3L 1s<3/2>''', is a fifth-repeating MOS scale. The notation "<3/2>" means the period of the MOS is 3/2, disambiguating it from octave-repeating [[3L 1s]]. The name of the period interval is called the '''sesquitave''' (by analogy to the [[tritave]]).  
   
   
 
The generator range is 171.4 to 240 cents, placing it near the [[9/8|diatonic major second]], usually representing a major second of some type. The dark (chroma-negative) generator is, however, its fifth complement (480 to 514.3 cents).  
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).  
 
 
Line 325: Line 325:
|Bbb
|Bbb
 
 
|3bb, 3cc
|3b, 3c
 
 
|10\15
|10\15
Line 357: Line 357:
|'''Bb'''
|'''Bb'''
 
 
|'''3b, 3c'''
|'''3'''
 
 
|'''11\15'''
|'''11\15'''
Line 393: Line 393:
|B
|B
 
 
|3
|3#
 
 
|12\15
|12\15
Line 429: Line 429:
|B#
|B#
 
 
|3#
|3x
 
 
|13\15
|13\15
Line 462: Line 462:
 
 
|Dob, Solb
|Dob, Solb
−
|Hb
|Hb
| 4b, 4c
−
|4b, 4c
−
|14\15
|14\15
   
   
−
646.153…
646.153…
−
|16\18
|16\18
   
   
−
619.354…
619.354…
−
|6\7
|6\7
   
   
−
600
600
−
|14\17
|14\17
   
   
−
579.310…
579.310…
−
|8\10
|8\10
   
   
−
564.705…
564.705…
−
|10\13
|10\13
 
 
Line 556: Line 535:
757.894…
757.894…
 
 
|20\18
| 20\18
 
 
774.193…
774.193…
 
 
| rowspan="2" |8\8
| rowspan="2" | 8\8
 
 
800
800
Line 572: Line 551:
847.058…
847.058…
 
 
|16\13
| 16\13
 
 
872.{{Overline|72}}
872.{{Overline|72}}
Line 592: Line 571:
821.052…
821.052…
 
 
|21\18
| 21\18
 
 
812.903…
812.903…
 
 
|19\17
| 19\17
 
 
786.206…
786.206…
 
 
|11\10
| 11\10
 
 
776.470…
776.470…
 
 
|14\13
| 14\13
 
 
763.{{Overline|63}}
763.{{Overline|63}}
Line 646: Line 625:
|-
|-
 
 
|Re#, La#
| Re#, La#
 
 
|C#
|C#
 
 
|5#
| 5#
 
 
|20\15
|20\15
Line 672: Line 651:
1034.482…
1034.482…
 
 
|15\10
| 15\10
 
 
1058.823…
1058.823…
Line 696: Line 675:
1010.526…
1010.526…
 
 
|26\18
| 26\18
 
 
1006.451…
1006.451…
Line 728: Line 707:
1073.684…
1073.684…
 
 
|28\18
| 28\18
 
 
1083.870…
1083.870…
Line 736: Line 715:
1100
1100
 
 
|27\17
| 27\17
 
 
1117.241…
1117.241…
 
 
|16\10
| 16\10
 
 
1129.411…
1129.411…
 
 
|21\9
| 21\9
 
 
1145.{{Overline|45}}
1145.{{Overline|45}}
Line 752: Line 731:
|Mi#, Si#
|Mi#, Si#
 
 
|D#
| D#
 
 
|6#
|6#
 
 
|24\15
| 24\15
 
 
1107.692…
1107.692…
 
 
| rowspan="2" |18\11
| rowspan="2" | 18\11
 
 
1136.842…
1136.842…
Line 768: Line 747:
1161.290…
1161.290…
 
 
|12\7
| 12\7
 
 
1200
1200
Line 790: Line 769:
|Ebb
|Ebb
 
 
|7bb, 7cc
|7b, 7c
 
 
|25\15
|25\15
Line 800: Line 779:
1122.580…
1122.580…
 
 
|11\7
| 11\7
 
 
1100
1100
Line 822: Line 801:
|'''Eb'''
|'''Eb'''
 
 
|'''7b, 7c'''
|'''7'''
 
 
|'''26\15'''
|'''26\15'''
Line 856: Line 835:
|Fa#, Do#
|Fa#, Do#
 
 
|E
| E
 
 
|7
|7#
 
 
|27\15
|27\15
Line 868: Line 847:
1263.157…
1263.157…
 
 
|33\18
| 33\18
 
 
1277.419…
1277.419…
Line 894: Line 873:
|E#
|E#
 
 
|7#
|7x
 
 
|28\15
|28\15
Line 908: Line 887:
1354.838…
1354.838…
 
 
|14\7
| 14\7
 
 
1400
1400
Line 952: Line 931:
1270.588…
1270.588…
 
 
|23\18
| 23\18
 
 
1254.{{Overline|54}}
1254.{{Overline|54}}
Line 962: Line 941:
!F
!F
 
 
!8, F
! 8, F
 
 
!30\15
! 30\15
 
 
1384.615…
1384.615…
 
 
!22\11
! 22\11
 
 
1389.473…
1389.473…
Line 980: Line 959:
1400
1400
 
 
!34\17
! 34\17
 
 
1406.896…
1406.896…
 
 
!20\10
! 20\10
 
 
1411.764…
1411.764…
Line 1,004: Line 983:
1430.769…
1430.769…
 
 
|23\11
| 23\11
 
 
1452.631…
1452.631…
Line 1,016: Line 995:
1500
1500
 
 
|37\17
| 37\17
 
 
1531.034…
1531.034…
 
 
|22\10
| 22\10
 
 
1552.941…
1552.941…
Line 1,030: Line 1,009:
|-
|-
 
 
|Reb, Lab
| Reb, Lab
 
 
|Gb
|Gb
Line 1,044: Line 1,023:
1515.789…
1515.789…
 
 
|39\18
| 39\18
 
 
1509.677…
1509.677…
Line 1,112: Line 1,091:
1642.105…
1642.105…
 
 
|43\18
| 43\18
 
 
1664.516…
1664.516…
 
 
| rowspan="2" |17\7
| rowspan="2" | 17\7
 
 
1700
1700
Line 1,204: Line 1,183:
|Mi#, Si#
|Mi#, Si#
 
 
|A#
| A#
 
 
|X#, A#
|X#, A#
Line 1,232: Line 1,211:
1976.470…
1976.470…
 
 
|37\13
| 37\13
 
 
2018.{{Overline|18}}
2018.{{Overline|18}}
Line 1,252: Line 1,231:
1819.354…
1819.354…
 
 
|18\7
| 18\7
 
 
1800
1800
Line 1,264: Line 1,243:
1764.705…
1764.705…
 
 
|32\13
| 32\13
 
 
1745.{{Overline|45}}
1745.{{Overline|45}}
Line 1,308: Line 1,287:
|Fa#, Do#
|Fa#, Do#
 
 
|B
| B
 
 
|E, C
|E, C
Line 1,320: Line 1,299:
1957.894…
1957.894…
 
 
|51\18
| 51\18
 
 
1974.193…
1974.193…
Line 1,446: Line 1,425:
|}
|}
   
   
−
{| class="wikitable"
{| class="wikitable"
−
|+Relative cents
|+Relative cents
! colspan="3" | Notation
−
! colspan="3" |Notation
−
!Supersoft
!Supersoft
−
!Soft
!Soft
−
!Semisoft
!Semisoft
−
!Basic
!Basic
−
!Semihard
!Semihard
−
!Hard
!Hard
−
!Superhard
!Superhard
−
|-
|-
! Diatonic
−
!Napoli
!Diatonic
! Bijou
!~15edf
−
!Napoli
−
!Bijou
−
!~15edf
−
!~11edf
!~11edf
−
!~18edf
!~18edf
−
!~7edf
!~7edf
−
!~17edf
!~17edf
−
!~10edf
!~10edf
−
!~13edf
!~13edf
−
|-
|-
−
|Do#, Sol#
|Do#, Sol#
−
|F#
|F#
−
|0#, D#
|0#, D#
−
|1\15
|1\15
   
   
−
''46.{{Overline|6}}''
''46.{{Overline|6}}''
−
|1\11
|1\11
   
   
−
''63.{{Overline|63}}''
''63.{{Overline|63}}''
−
|2\18
|2\18
   
   
−
''77.7̄''
''77.7̄''
−
| rowspan="2" |1\7
| rowspan="2" |1\7
   
   
−
''100''
''100''
| 3\17
   
   
−
|3\17
−
''123.529…''
''123.529…''
| 2\10
   
   
−
|2\10
−
''140''
''140''
−
|3\13
|3\13
   
   
−
''161.538…''
''161.538…''
−
|-
|-
−
|Reb, Lab
|Reb, Lab
| Gb
−
|Gb
−
|1b, 1c
|1b, 1c
−
|3\15
|3\15
   
   
−
''140''
''140''
−
|2\11
|2\11
   
   
−
''127.{{Overline|27}}''
''127.{{Overline|27}}''
−
|3\18
|3\18
   
   
−
''116.{{Overline|6}}''
''116.{{Overline|6}}''
| 2\17
   
   
−
|2\17
−
''82.352…''
''82.352…''
−
|1\10
|1\10
   
   
−
''70''
''70''
−
|1\13
|1\13
   
   
−
''53.846…''
''53.846…''
−
|-
|-
−
|'''Re, La'''
|'''Re, La'''
−
|'''G'''
|'''G'''
−
|'''1'''
|'''1'''
−
|'''4\15'''
|'''4\15'''
   
   
−
'''''186.{{Overline|6}}'''''
'''''186.{{Overline|6}}'''''
−
|'''3\11'''
|'''3\11'''
   
   
−
'''''190.{{Overline|90}}'''''
'''''190.{{Overline|90}}'''''
−
|'''5\18'''
|'''5\18'''
   
   
−
'''''194.{{Overline|4}}'''''
'''''194.{{Overline|4}}'''''
−
|'''2\7'''
|'''2\7'''
   
   
−
'''''200'''''
'''''200'''''
−
|'''5\17'''
|'''5\17'''
   
   
−
'''''205.882…'''''
'''''205.882…'''''
−
|'''3\10'''
|'''3\10'''
   
   
−
'''''210'''''
'''''210'''''
−
|'''4\13'''
|'''4\13'''
   
   
−
'''''215.384…'''''
'''''215.384…'''''
−
|-
|-
−
|Re#, La#
|Re#, La#
| G#
−
| 1#
|G#
−
|1#
−
|5\15
|5\15
   
   
−
''233.{{Overline|3}}''
''233.{{Overline|3}}''
−
|4\11
|4\11
   
   
−
''254.{{Overline|54}}''
''254.{{Overline|54}}''
−
|7\18
|7\18
   
   
−
''272.2̄''
''272.2̄''
−
| rowspan="2" |3\7
| rowspan="2" |3\7
   
   
−
''300''
''300''
−
|8\17
|8\17
   
   
−
''329.411…''
''329.411…''
−
|5\10
|5\10
   
   
−
''350''
''350''
−
|7\13
|7\13
   
   
−
''376.923…''
''376.923…''
−
|-
|-
−
|Mib, Sib
|Mib, Sib
−
|Ab
|Ab
−
|2b, 2c
|2b, 2c
−
|7\15
|7\15
   
   
−
''326.{{Overline|6}}''
''326.{{Overline|6}}''
−
|5\11
|5\11
   
   
−
''318.{{Overline|18}}''
''318.{{Overline|18}}''
| 8\18
   
   
−
|8\18
−
''311.{{Overline|1}}''
''311.{{Overline|1}}''
−
|7\17
|7\17
   
   
−
''288.235…''
''288.235…''
| 4\10
   
   
−
|4\10
−
''280''
''280''
−
|5\13
|5\13
   
   
−
''269.230…''
''269.230…''
−
|-
|-
−
|Mi, Si
|Mi, Si
−
|A
|A
| 2
−
|2
−
|8\15
|8\15
   
   
−
''373.{{Overline|3}}''
''373.{{Overline|3}}''
−
|6\11
|6\11
   
   
−
''381.{{Overline|81}}''
''381.{{Overline|81}}''
−
|10\18
|10\18
   
   
−
''388.{{Overline|8}}''
''388.{{Overline|8}}''
−
|4\7
|4\7
   
   
−
''400''
''400''
−
|10\17
|10\17
   
   
−
''411.764…''
''411.764…''
−
|6\10
|6\10
   
   
−
''420''
''420''
−
|8\13
|8\13
   
   
−
''430.769…''
''430.769…''
−
|-
|-
−
|Mi#, Si#
|Mi#, Si#
−
|A#
|A#
−
|2#
|2#
−
|9\15
|9\15
   
   
−
''420''
''420''
−
| rowspan="2" |7\11
| rowspan="2" |7\11
   
   
−
''445.{{Overline|45}}''
''445.{{Overline|45}}''
|12\18
   
   
−
|12\18
−
''466.{{Overline|6}}''
''466.{{Overline|6}}''
−
|5\7
|5\7
   
   
−
''500''
''500''
−
|13\17
|13\17
   
   
−
''535.294…''
''535.294…''
−
|8\10
|8\10
   
   
−
''560''
''560''
−
|11\13
|11\13
   
   
−
''592.307…''
''592.307…''
−
|-
|-
−
|Fab, Dob
|Fab, Dob
−
|Bbb
|Bbb
|3b, 3c
−
|3bb, 3cc
−
|10\15
|10\15
   
   
−
''466.{{Overline|6}}''
''466.{{Overline|6}}''
−
|11\18
|11\18
   
   
−
''427.{{Overline|7}}''
''427.{{Overline|7}}''
−
|4\7
|4\7
   
   
−
''400''
''400''
−
|9\17
|9\17
   
   
−
''370.588…''
''370.588…''
−
|5\10
|5\10
   
   
−
''350''
''350''
−
|6\13
|6\13
   
   
−
''323.076.…''
''323.076.…''
|-
−
|-
−
|'''Fa, Do'''
|'''Fa, Do'''
−
|'''Bb'''
|'''Bb'''
|'''3'''
−
|'''3b, 3c'''
−
|'''11\15'''
|'''11\15'''
   
   
−
'''''513.{{Overline|3}}'''''
'''''513.{{Overline|3}}'''''
−
|'''8\11'''
|'''8\11'''
   
   
−
'''''509.{{Overline|09}}'''''
'''''509.{{Overline|09}}'''''
−
|'''13\18'''
|'''13\18'''
   
   
−
'''''505.{{Overline|5}}'''''
'''''505.{{Overline|5}}'''''
−
|'''5\7'''
|'''5\7'''
   
   
−
'''''500'''''
'''''500'''''
−
|'''12\17'''
|'''12\17'''
   
   
−
'''''494.117…'''''
'''''494.117…'''''
−
|'''7\10'''
|'''7\10'''
   
   
−
'''''490'''''
'''''490'''''
−
|'''9\13'''
|'''9\13'''
   
   
−
'''''484.615…'''''
'''''484.615…'''''
−
|-
|-
−
|Fa#, Do#
|Fa#, Do#
| B
|3#
|12\15
   
   
−
''560''
|B
|9\11
   
   
−
''572.{{Overline|72}}''
|3
| 15\18
   
   
−
|12\15
−
''560''
−
|9\11
−
''572.{{Overline|72}}''
−
|15\18
−
''583.{{Overline|3}}''
''583.{{Overline|3}}''
−
|6\7
|6\7
   
   
−
''600''
''600''
−
|15\17
|15\17
   
   
−
''617.647…''
''617.647…''
−
|9\10
|9\10
   
   
−
''630''
''630''
−
|12\13
|12\13
   
   
−
''646.153…''
''646.153…''
−
|-
|-
| Fax, Dox
−
|Fax, Dox
−
|B#
|B#
|3x
−
|3#
−
|13\15
|13\15
   
   
−
''606. {{Overline|6}}''
''606. {{Overline|6}}''
−
| rowspan="2" |10\11
| rowspan="2" |10\11
   
   
−
''636.{{Overline|36}}''
''636.{{Overline|36}}''
−
|17\18
|17\18
   
   
−
''661.{{Overline|1}}''
''661.{{Overline|1}}''
−
|7\7
|7\7
   
   
−
''700''
''700''
−
|18\17
|18\17
   
   
−
''741.176…''
''741.176…''
−
|11\10
|11\10
   
   
−
''770''
''770''
−
|15\13
|15\13
   
   
−
''807.692…''
''807.692…''
−
|-
|-
−
|Dob, Solb
|Dob, Solb
−
|Hb
|Hb
−
|4b, 4c
|4b, 4c
−
|14\15
|14\15
   
   
−
''653.{{Overline|3}}''
''653.{{Overline|3}}''
−
|16\18
|16\18
   
   
−
''622.{{Overline|2}}''
''622.{{Overline|2}}''
−
|6\7
|6\7
   
   
−
''600''
''600''
| 14\17
   
   
−
|14\17
−
''576.470…''
''576.470…''
−
| 8\10
|8\10
   
   
−
''560''
''560''
−
|10\13
|10\13
   
   
−
''538.461…''
''538.461…''
−
|-
|-
−
!Do, Sol
!Do, Sol
−
!H
!H
−
!4
!4
−
! colspan="7" |''700''
! colspan="7" |''700''
−
|-
|-
−
|Do#, Sol#
|Do#, Sol#
−
|Η#
|Η#
−
|4#
|4#
−
|16\15
|16\15
   
   
−
''746.{{Overline|6}}''
''746.{{Overline|6}}''
−
|12\11
|12\11
   
   
−
''763.{{Overline|63}}''
''763.{{Overline|63}}''
−
|20\18
|20\18
   
   
−
''777.{{Overline|7}}''
''777.{{Overline|7}}''
−
| rowspan="2" |8\7
| rowspan="2" |8\7
   
   
−
''800''
''800''
−
|20\17
|20\17
   
   
−
''823.529…''
''823.529…''
−
|12\10
|12\10
   
   
−
''840''
''840''
−
|16\13
|16\13
   
   
−
''861.538…''
''861.538…''
−
|-
|-
−
|Reb, Lab
|Reb, Lab
−
|Cb
|Cb
−
|5b, 5c
|5b, 5c
−
|18\15
|18\15
   
   
−
''840''
''840''
−
|13\11
|13\11
   
   
−
''827.{{Overline|27}}''
''827.{{Overline|27}}''
−
|21\18
|21\18
   
   
−
''816.{{Overline|6}}''
''816.{{Overline|6}}''
| 19\17
   
   
−
|19\17
−
''782.352…''
''782.352…''
−
|11\10
|11\10
   
   
−
''770''
''770''
−
|14\13
|14\13
   
   
−
''753.846…''
''753.846…''
−
|-
|-
−
|'''Re, La'''
|'''Re, La'''
−
|'''C'''
|'''C'''
−
|'''5'''
|'''5'''
−
|'''19\15'''
|'''19\15'''
   
   
−
'''''886.{{Overline|6}}'''''
'''''886.{{Overline|6}}'''''
−
|'''14\11'''
|'''14\11'''
   
   
−
'''''890.{{Overline|90}}'''''
'''''890.{{Overline|90}}'''''
−
|'''23\18'''
|'''23\18'''
   
   
−
'''''894.{{Overline|4}}'''''
'''''894.{{Overline|4}}'''''
−
|'''9\7'''
|'''9\7'''
   
   
−
'''''900'''''
'''''900'''''
|'''22\17'''
   
   
−
|'''22\17'''
−
'''''905.882…'''''
'''''905.882…'''''
−
|'''13\10'''
|'''13\10'''
   
   
−
'''''910'''''
'''''910'''''
−
|'''17\13'''
|'''17\13'''
   
   
−
'''''915.384…'''''
'''''915.384…'''''
−
|-
|-
| Re#, La#
−
|Re#, La#
−
|C#
|C#
−
|5#
|5#
−
|20\15
|20\15
   
   
−
''933.{{Overline|3}}''
''933.{{Overline|3}}''
−
|15\11
|15\11
   
   
−
''954.{{Overline|54}}''
''954.{{Overline|54}}''
−
|25\18
|25\18
   
   
−
''972.{{Overline|2}}''
''972.{{Overline|2}}''
| rowspan="2" | 10\7
   
   
−
| rowspan="2" |10\7
−
''1000''
''1000''
−
|25\17
|25\17
   
   
−
''1029.411…''
''1029.411…''
−
|15\10
|15\10
   
   
−
''1050''
''1050''
−
|20\13
|20\13
   
   
−
''1076.923…''
''1076.923…''
−
|-
|-
−
|Mib, Sib
|Mib, Sib
−
|Db
|Db
−
|6b, 6c
|6b, 6c
−
|22\15
|22\15
   
   
−
''1026.{{Overline|6}}''
''1026.{{Overline|6}}''
−
|16\11
|16\11
   
   
−
''1018.{{Overline|18}}''
''1018.{{Overline|18}}''
−
|26\18
|26\18
   
   
−
''1011. {{Overline|1}}''
''1011. {{Overline|1}}''
−
|24\17
|24\17
   
   
−
''988.235…''
''988.235…''
−
|14\10
|14\10
   
   
−
''980''
''980''
−
|18\13
|18\13
   
   
−
''969.230…''
''969.230…''
−
|-
|-
−
|Mi, Si
|Mi, Si
−
|D
|D
−
|6
|6
| 23\15
   
   
−
|23\15
−
''1073.{{Overline|3}}''
''1073.{{Overline|3}}''
−
|17\11
|17\11
   
   
−
''1081.{{Overline|81}}''
''1081.{{Overline|81}}''
−
|28\18
|28\18
   
   
−
''1088.{{Overline|8}}''
''1088.{{Overline|8}}''
|11\7
   
   
−
|11\7
−
''1100''
''1100''
−
|27\17
|27\17
   
   
−
''1111.764…''
''1111.764…''
−
|16\10
|16\10
   
   
−
''1120''
''1120''
−
|21\13
|21\13
   
   
−
''1130.769…''
''1130.769…''
−
|-
|-
−
|Mi#, Si#
|Mi#, Si#
| D#
−
|D#
−
|6#
|6#
−
|24\15
|24\15
   
   
−
''1120''
''1120''
| rowspan="2" | 18\11
   
   
−
| rowspan="2" |18\11
−
''1145.{{Overline|45}}''
''1145.{{Overline|45}}''
−
|30\18
|30\18
   
   
−
''1166.{{Overline|6}}''
''1166.{{Overline|6}}''
−
|12\7
|12\7
   
   
−
''1200''
''1200''
| 30\17
   
   
−
|30\17
−
''1235.294…''
''1235.294…''
−
|18\10
|18\10
   
   
−
''1260''
''1260''
−
|24\13
|24\13
   
   
−
''1292.307…''
''1292.307…''
−
|-
|-
| Fab, Dob
−
| Ebb
|Fab, Dob
| 7b, 7c
−
|Ebb
−
|7bb, 7cc
−
|25\15
|25\15
   
   
−
''1166.{{Overline|6}}''
''1166.{{Overline|6}}''
−
|29\18
|29\18
   
   
−
''1127.{{Overline|7}}''
''1127.{{Overline|7}}''
−
|11\7
|11\7
   
   
−
''1100''
''1100''
−
|26\17
|26\17
   
   
−
''1070.588…''
''1070.588…''
−
|15\10
|15\10
   
   
−
''1050''
''1050''
−
|19\13
|19\13
   
   
−
''1023.076…''
''1023.076…''
−
|-
|-
−
|'''Fa, Do'''
|'''Fa, Do'''
−
|'''Eb'''
|'''Eb'''
|'''7'''
−
|'''7b, 7c'''
−
|'''26\15'''
|'''26\15'''
   
   
−
'''''1213.{{Overline|3}}'''''
'''''1213.{{Overline|3}}'''''
−
|'''19\11'''
|'''19\11'''
   
   
−
'''''1209.{{Overline|09}}'''''
'''''1209.{{Overline|09}}'''''
|'''31\18'''
   
   
−
|'''31\18'''
−
'''''1205.{{Overline|5}}'''''
'''''1205.{{Overline|5}}'''''
−
|'''12\7'''
|'''12\7'''
   
   
−
'''''1200'''''
'''''1200'''''
−
|'''29\17'''
|'''29\17'''
   
   
−
'''''1194.117…'''''
'''''1194.117…'''''
−
|'''17\10'''
|'''17\10'''
   
   
−
'''''1190'''''
'''''1190'''''
−
|'''22\13'''
|'''22\13'''
   
   
−
'''''1184.615…'''''
'''''1184.615…'''''
−
|-
|-
−
|Fa#, Do#
|Fa#, Do#
−
|E
|E
|7#
−
|7
−
|27\15
|27\15
   
   
−
''1260''
''1260''
−
|20\11
|20\11
   
   
−
''1272.{{Overline|72}}''
''1272.{{Overline|72}}''
| 33\18
   
   
−
|33\18
−
''1283.{{Overline|3}}''
''1283.{{Overline|3}}''
−
|13\7
|13\7
   
   
−
''1300''
''1300''
−
|32\17
|32\17
   
   
−
''1317.647…''
''1317.647…''
−
|19\10
|19\10
   
   
−
''1330''
''1330''
| 25\13
   
   
−
|25\13
−
''1346.153…''
''1346.153…''
−
|-
|-
−
|Fax, Dox
|Fax, Dox
−
|E#
|E#
|7x
−
|7#
−
|28\15
|28\15
   
   
−
''1306.{{Overline|6}}''
''1306.{{Overline|6}}''
−
| rowspan="2" |21\11
| rowspan="2" |21\11
   
   
−
''1336.{{Overline|36}}''
''1336.{{Overline|36}}''
−
|35\18
|35\18
   
   
−
''1361.{{Overline|1}}''
''1361.{{Overline|1}}''
−
|14\7
|14\7
   
   
−
''1400''
''1400''
−
|35\17
|35\17
   
   
−
''1441.176…''
''1441.176…''
−
|21\10
|21\10
   
   
−
''1470''
''1470''
−
|28\13
|28\13
   
   
−
''1507.692…''
''1507.692…''
−
|-
|-
−
|Dob, Solb
|Dob, Solb
−
|Fb
|Fb
|8b, Fc
|29\15
   
   
−
|8b, Fc
−
|29\15
−
''1333.{{Overline|3}}''
''1333.{{Overline|3}}''
−
|34\18
|34\18
   
   
−
''1322.{{Overline|2}}''
''1322.{{Overline|2}}''
−
|13\7
|13\7
   
   
−
''1300''
''1300''
−
|31\17
|31\17
   
   
−
''1276.470…''
''1276.470…''
−
|18\10
|18\10
   
   
−
''1260''
''1260''
−
|23\13
|23\13
   
   
−
''1238.461…''
''1238.461…''
−
|-
|-
−
!Do, Sol
!Do, Sol
−
!F
!F
−
!8, F
!8, F
−
! colspan="7" |''1400''
! colspan="7" |''1400''
−
|-
|-
−
|Do#, Sol#
|Do#, Sol#
−
|F#
|F#
−
|8#, F#
|8#, F#
−
|31\15
|31\15
   
   
−
''1446.{{Overline|6}}''
''1446.{{Overline|6}}''
−
|23\11
|23\11
   
   
−
''1463.{{Overline|63}}''
''1463.{{Overline|63}}''
−
|38\18
|38\18
   
   
−
''1477.7̄''
''1477.7̄''
−
| rowspan="2" |15\7
| rowspan="2" |15\7
   
   
−
''1500''
''1500''
−
|37\17
|37\17
   
   
−
''1523.529…''
''1523.529…''
−
|22\10
|22\10
   
   
−
''1540''
''1540''
| 29\13
   
   
−
|29\13
−
''1561.538…''
''1561.538…''
−
|-
|-
−
|Reb, Lab
|Reb, Lab
−
|Gb
|Gb
| 9b, Gc
−
|9b, Gc
−
|33\15
|33\15
   
   
−
''1540''
''1540''
−
|24\11
|24\11
   
   
−
''1527.{{Overline|27}}''
''1527.{{Overline|27}}''
−
|39\18
|39\18
   
   
−
''1516.{{Overline|6}}''
''1516.{{Overline|6}}''
| 36\17
   
   
−
|36\17
−
''1482.352…''
''1482.352…''
−
|21\10
|21\10
   
   
−
''1470''
''1470''
−
|27\13
|27\13
   
   
−
''1453.846…''
''1453.846…''
−
|-
|-
−
|'''Re, La'''
|'''Re, La'''
−
|'''G'''
|'''G'''
−
|'''9, G'''
|'''9, G'''
−
|'''34\15'''
|'''34\15'''
   
   
−
'''''1586.{{Overline|6}}'''''
'''''1586.{{Overline|6}}'''''
−
|'''25\11'''
|'''25\11'''
   
   
−
'''''1590.{{Overline|90}}'''''
'''''1590.{{Overline|90}}'''''
−
|'''41\18'''
|'''41\18'''
   
   
−
'''''1594.{{Overline|4}}'''''
'''''1594.{{Overline|4}}'''''
−
|'''16\7'''
|'''16\7'''
   
   
−
'''''1600'''''
'''''1600'''''
−
|'''39\17'''
|'''39\17'''
   
   
−
'''''1605.882…'''''
'''''1605.882…'''''
−
|'''23\10'''
|'''23\10'''
   
   
−
'''''1610'''''
'''''1610'''''
−
|'''30\13'''
|'''30\13'''
   
   
−
'''''1615.384…'''''
'''''1615.384…'''''
−
|-
|-
−
|Re#, La#
|Re#, La#
−
|G#
|G#
−
|9#, G#
|9#, G#
−
|35\15
|35\15
   
   
−
''1633.{{Overline|3}}''
''1633.{{Overline|3}}''
−
|26\11
|26\11
   
   
−
''1654.{{Overline|54}}''
''1654.{{Overline|54}}''
−
|43\18
|43\18
   
   
−
''1672.{{Overline|2}}''
''1672.{{Overline|2}}''
−
| rowspan="2" |17\7
| rowspan="2" |17\7
   
   
−
''1700''
''1700''
−
|42\17
|42\17
   
   
−
''1729.411…''
''1729.411…''
−
|25\10
|25\10
   
   
−
''1750''
''1750''
−
|33\13
|33\13
   
   
−
''1776.923…''
''1776.923…''
−
|-
|-
−
|Mib, Sib
|Mib, Sib
| Ab
−
|Ab
−
|Xb, Ac
|Xb, Ac
−
|37\15
|37\15
   
   
−
''1726.{{Overline|6}}''
''1726.{{Overline|6}}''
| 27\11
   
   
−
|27\11
−
''1718.{{Overline|18}}''
''1718.{{Overline|18}}''
−
|44\18
|44\18
   
   
−
''1711.{{Overline|1}}''
''1711.{{Overline|1}}''
−
|41\17
|41\17
   
   
−
''1688.235…''
''1688.235…''
| 24\10
   
   
−
''1680''
|24\10
−
''1680''
−
|31\13
|31\13
   
   
−
''1669.230…''
''1669.230…''
−
|-
|-
−
|Mi, Si
|Mi, Si
−
|A
|A
−
|X, A
|X, A
−
|38\15
|38\15
   
   
−
''1773.{{Overline|3}}''
''1773.{{Overline|3}}''
−
|28\11
|28\11
   
   
−
''1781.{{Overline|81}}''
''1781.{{Overline|81}}''
−
|46\18
|46\18
   
   
−
''1788.{{Overline|8}}''
''1788.{{Overline|8}}''
−
|18\7
|18\7
   
   
−
''1800''
''1800''
| 44\17
   
   
−
|44\17
−
''1811.764…''
''1811.764…''
−
|26\10
|26\10
   
   
−
''1820''
''1820''
−
|34\13
|34\13
   
   
−
''1830.769…''
''1830.769…''
−
|-
|-
−
|Mi#, Si#
|Mi#, Si#
−
|A#
|A#
−
|X#, A#
|X#, A#
−
|39\15
|39\15
   
   
−
''1820''
''1820''
−
| rowspan="2" |29\11
| rowspan="2" |29\11
   
   
−
''1845.{{Overline|45}}''
''1845.{{Overline|45}}''
−
|48\18
|48\18
   
   
−
''1866.{{Overline|6}}''
''1866.{{Overline|6}}''
−
|19\7
|19\7
   
   
−
''1900''
''1900''
−
|47\17
|47\17
   
   
−
''1935.294…''
''1935.294…''
−
|28\10
|28\10
   
   
−
''1960''
''1960''
| 37\13
   
   
−
|37\13
−
''1992.307…''
''1992.307…''
−
|-
|-
−
|Fab, Dob
|Fab, Dob
−
|Bbb
|Bbb
−
|Ebb, Ccc
|Ebb, Ccc
−
|40\15
|40\15
   
   
−
''1866.{{Overline|6}}''
''1866.{{Overline|6}}''
−
|47\18
|47\18
   
   
−
''1827.{{Overline|7}}''
''1827.{{Overline|7}}''
−
|18\7
|18\7
   
   
−
''1800''
''1800''
−
|43\17
|43\17
   
   
−
''1770.588…''
''1770.588…''
−
|25\10
|25\10
   
   
−
''1750''
''1750''
−
|32\13
|32\13
   
   
−
''1723.076…''
''1723.076…''
−
|-
|-
−
|'''Fa, Do'''
|'''Fa, Do'''
−
|'''Bb'''
|'''Bb'''
−
|Eb, Cc
|Eb, Cc
−
|'''41\15'''
|'''41\15'''
   
   
−
'''''1913.{{Overline|3}}'''''
'''''1913.{{Overline|3}}'''''
−
|'''30\11'''
|'''30\11'''
   
   
−
'''''1909.{{Overline|09}}'''''
'''''1909.{{Overline|09}}'''''
−
|'''49\18'''
|'''49\18'''
   
   
−
'''''1905.{{Overline|5}}'''''
'''''1905.{{Overline|5}}'''''
−
|'''19\7'''
|'''19\7'''
   
   
−
'''''1900'''''
'''''1900'''''
−
|'''46\17'''
|'''46\17'''
   
   
−
'''''1894.117…'''''
'''''1894.117…'''''
−
|'''27\10'''
|'''27\10'''
   
   
−
'''''1890'''''
'''''1890'''''
−
|'''35\13'''
|'''35\13'''
   
   
−
'''''1884.615…'''''
'''''1884.615…'''''
−
|-
|-
−
|Fa#, Do#
|Fa#, Do#
−
|B
|B
−
|E, C
|E, C
| 42\15
   
   
−
|42\15
−
''1960''
''1960''
−
|31\11
|31\11
   
   
−
''1972.{{Overline|72}}''
''1972.{{Overline|72}}''
−
|51\18
|51\18
   
   
−
''1983.{{Overline|3}}''
''1983.{{Overline|3}}''
−
|20\7
|20\7
   
   
−
''2000''
''2000''
−
|49\17
|49\17
   
   
−
''2017.647…''
''2017.647…''
−
|29\10
|29\10
   
   
−
''2030''
''2030''
−
|38\13
|38\13
   
   
−
''2046.153…''
''2046.153…''
−
|-
|-
−
|Fax, Dox
|Fax, Dox
−
|B#
|B#
−
|Ex, Cx
|Ex, Cx
−
|43\15
|43\15
   
   
−
''2006.{{Overline|6}}''
''2006.{{Overline|6}}''
−
| rowspan="2" |32\11
| rowspan="2" |32\11
   
   
−
''2036.{{Overline|36}}''
''2036.{{Overline|36}}''
−
|53\18
|53\18
   
   
−
''2061. {{Overline|1}}''
''2061. {{Overline|1}}''
−
|21\7
|21\7
   
   
−
''2100''
''2100''
|52\17
   
   
−
|52\17
−
''2141.176…''
''2141.176…''
−
|31\10
|31\10
   
   
−
''2170''
''2170''
−
|41\13
|41\13
   
   
−
''2207.692…''
''2207.692…''
−
|-
|-
| Dob, Solb
−
|Dob, Solb
−
|Hb
|Hb
−
|0b, Dc
|0b, Dc
−
|44\15
|44\15
   
   
−
''2053.{{Overline|3}}''
''2053.{{Overline|3}}''
−
|52\18
|52\18
   
   
−
''2022.{{Overline|2}}''
''2022.{{Overline|2}}''
−
|20\7
|20\7
   
   
−
''2000''
''2000''
−
|48\17
|48\17
   
   
−
''1976.470…''
''1976.470…''
−
|28\10
|28\10
   
   
−
''1960''
''1960''
| 36\13
   
   
−
|36\13
−
1938.615…
1938.615…
−
|-
|-
−
!Do, Sol
!Do, Sol
!H
−
!H
−
!0, D
!0, D
−
! colspan="7" |2100
! colspan="7" |2100
−
|}
|}
   
   
−
−
==Intervals==
==Intervals==
−
{| class="wikitable"
{| class="wikitable"
−
!Generators
!Generators
! Sesquitave notation
−
!Sesquitave notation
−
!Interval category name
!Interval category name
−
!Generators
!Generators
−
!Notation of 3/2 inverse
!Notation of 3/2 inverse
−
!Interval category name
!Interval category name
−
|-
|-
−
| colspan="6" |The 4-note MOS has the following intervals (from some root):
| colspan="6" |The 4-note MOS has the following intervals (from some root):
−
|-
|-
−
|0
|0
−
|Do, Sol
|Do, Sol
−
|perfect unison
|perfect unison
−
|0
|0
−
|Do, Sol
|Do, Sol
−
|sesquitave (just fifth)
|sesquitave (just fifth)
−
|-
|-
−
|1
|1
−
|Fa, Do
|Fa, Do
−
|perfect fourth
|perfect fourth
−
| -1
| -1
−
|Re, La
|Re, La
−
|perfect second
|perfect second
−
|-
|-
−
|2
|2
−
|Mib, Sib
|Mib, Sib
−
|minor third
|minor third
−
| -2
| -2
−
|Mi, Si
|Mi, Si
−
|major third
|major third
−
|-
|-
−
|3
|3
−
|Reb, Lab
|Reb, Lab
−
|diminished second
|diminished second
−
| -3
| -3
−
|Fa#, Do#
|Fa#, Do#
|augmented fourth
−
|augmented fourth
−
|-
|-
−
| colspan="6" |The chromatic 7-note MOS also has the following intervals (from some root):
| colspan="6" |The chromatic 7-note MOS also has the following intervals (from some root):
−
|-
|-
−
|4
|4
−
|Dob, Solb
|Dob, Solb
−
|diminished sesquitave
|diminished sesquitave
| -4
−
| Do#, Sol#
| -4
−
|Do#, Sol#
−
|augmented unison (chroma)
|augmented unison (chroma)
−
|-
|-
−
|5
|5
−
|Fab, Dob
|Fab, Dob
−
|diminished fourth
|diminished fourth
−
| -5
| -5
−
|Re#, La#
|Re#, La#
−
|augmented second
|augmented second
−
|-
|-
−
|6
|6
| Mibb, Sibb
−
|diminished third
|Mibb, Sibb
| -6
−
|diminished third
−
| -6
−
|Mi#, Si#
|Mi#, Si#
−
|augmented third
|augmented third
|}  
−
|}
−
==Genchain==
==Genchain==
   
   
−
The generator chain for this scale is as follows:
The generator chain for this scale is as follows:
−
{| class="wikitable"
{| class="wikitable"
−
|Mibb
|Mibb
   
   
−
Sibb
Sibb
−
|Fab
|Fab
   
   
−
Dob
Dob
−
|Dob
|Dob
   
   
−
Solb
Solb
−
|Reb
|Reb
   
   
−
Lab
Lab
−
|Mib
|Mib
   
   
−
Sib
Sib
−
|Fa
|Fa
   
   
−
Do
Do
−
|Do
|Do
   
   
−
Sol
Sol
−
|Re
|Re
   
   
−
La
La
−
|Mi
|Mi
   
   
−
Si
Si
−
|Fa#
|Fa#
   
   
−
Do#
Do#
−
|Do#
|Do#
   
   
−
Sol#
Sol#
−
|Re#
|Re#
   
   
−
La#
La#
−
|Mi#
|Mi#
   
   
−
Si#
Si#
−
|-
|-
−
|d3
|d3
−
|d4
|d4
|d5
−
|d6
−
|d2
|d2
| m3
−
|m3
−
|P4
|P4
−
|P1
|P1
−
|P2
|P2
−
|M3
|M3
−
|A4
|A4
| A1
|A2
|A3
|}
==Modes==
   
   
−
|A1
−
|A2
−
|A3
−
|}
−
==Modes==
−
The mode names are based on the species of fifth:
The mode names are based on the species of fifth:
−
{| class="wikitable"
{| class="wikitable"
−
!Mode
!Mode
−
!Scale
!Scale
−
![[Modal UDP Notation|UDP]]
![[Modal UDP Notation|UDP]]
−
! colspan="3" |Interval type
! colspan="3" |Interval type
−
|-
|-
−
!name
!name
−
!pattern
!pattern
−
!notation
!notation
−
!2nd
!2nd
−
!3rd
!3rd
−
!4th
!4th
−
|-
|-
−
|Lydian
|Lydian
|LLLs
−
|LLLs
−
|<nowiki>3|0</nowiki>
|<nowiki>3|0</nowiki>
−
|P
|P
−
|M
|M
| A
−
|A
−
|-
|-
−
|Major
|Major
−
|LLsL
|LLsL
−
|<nowiki>2|1</nowiki>
|<nowiki>2|1</nowiki>
−
|P
|P
−
|M
|M
−
|P
|P
−
|-
|-
| Minor
−
|Minor
−
|LLsL
|LLsL
−
|<nowiki>1|2</nowiki>
|<nowiki>1|2</nowiki>
| P
−
|P
−
|m
|m
−
|P
|P
−
|-
|-
−
|Phrygian
|Phrygian
−
|sLLL
|sLLL
−
|<nowiki>0|3</nowiki>
|<nowiki>0|3</nowiki>
−
|d
|d
−
|m
|m
| P
−
|}  
|P
−
|}
−
==Temperaments==
==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.
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'''===
==='''Napoli-Meantone'''===
   
   
−
[[Subgroup]]: 3/2.6/5.8/5
[[Subgroup]]: 3/2.6/5.8/5
   
   
−
[[Comma]] list: [[81/80]]
 
[[POL2]] generator: ~9/8 = [[Tel:192.6406|192.6406]]
 
[[Mapping]]: [{{val|1 1 2}}, {{val|0 -2 -3}}]
 
[[Vals]]: {{val list|~(7edf, 11edf, 18edf)}}
==='''Napoli-Superpyth'''===
   
   
[[Subgroup]]: 3/2.7/6.14/9
   
   
−
[[Comma]] list: [[64/63]]
[[Comma]] list: [[81/80]]
 
[[POL2]] generator: ~8/7 = [[Tel:218.6371|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
[[POL2]] generator: ~9/8 = 192.6406
! rowspan="2" |L
! rowspan="2" |s
−
! rowspan="2" |L/s
! rowspan="2" | Comments
|-
−
!<u>Normalised</u>
[[Mapping]]: [{{val|1 1 2}}, {{val|0 -2 -3}}]
!''ed7\12''
|-
−
| 1\4
|
|
−
|<u>171.428…</u>
[[Vals]]: {{val list|~(7edf, 11edf, 18edf)}}
|''175''
|1
−
|1
==='''Napoli-Superpyth'''===
|1.000
|Equalised
−
|-
[[Subgroup]]: 3/2.7/6.14/9
|6\23
|
−
|
|<u>180</u>
|''182.608…''
−
|6
[[Comma]] list: [[64/63]]
|5
|1.200
−
|
|-
|
−
| 11\42
[[POL2]] generator: ~8/7 = 218.6371
|
|<u>180.821…</u>
−
|''183.{{Overline|3}}''
|11
|9
−
|1.222
[[Mapping]]: [{{val|1 1 2}}, {{val|0 -2 -3}}]
|
|-
−
|5\19
|
|
−
|<u>181.{{Overline|81}}</u>
[[Vals]]: {{val list|~(7edf, 10edf, 13edf, 16edf)}}
|''184.210…''
|5
−
|4
====Scale tree====
|1.250
|
−
|-
The spectrum looks like this:
|
|14\53
−
|
{| class="wikitable"
|<u>182.608…</u>
|''184.905…''
−
|14
! colspan="3" rowspan="2" |Generator
|11
|1.273
−
|
(bright)
|-
|
−
|9\34
! colspan="2" |Cents
|
|<u>183.050…</u>
−
|''185.294…''
! rowspan="2" |L
| 9
|7
−
|1.286
! rowspan="2" |s
|
−
! rowspan="2" |L/s
−
! rowspan="2" |Comments
−
|-
|-
|4\15
−
|
!<u>Normalised</u>  
|
|<u>184.615…</u>
−
|''186.{{Overline|6}}''
!''ed7\12''
|4
|3
−
|1.333
|
|-
|-
|
−
|11\41
|1\4
|
|<u>185.915…</u>
−
|''187.804…''
|
−
|
−
|<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.{{Overline|6}}''
−
|4
−
|3
−
|1.333
−
|
−
|-
−
|
−
|11\41
−
|
−
|<u>185.915…</u>
−
|''187.804…''
−
|11
|11
| 8
−
|8
−
|1.375
|1.375
|
−
|
−
|-
|-
|
−
|
−
|7\26
|7\26
|
−
|
−
|<u>186.{{Overline|6}}</u>
|<u>186.{{Overline|6}}</u>
−
|''188.461…''
|''188.461…''
−
|7
|7
−
|5
|5
−
|1.400
|1.400
|
−
|
−
|-
|-
|
−
|
−
|10\37
|10\37
|
−
|
−
|<u>187.5</u>
|<u>187.5</u>
−
|''189.{{Overline|189}}''
|''189.{{Overline|189}}''
−
|10
|10
| 7
−
| 1.429
|7
|
−
|1.429
−
|
−
|-
|-
|
−
|
−
|13\48
|13\48
−
|
|
−
|<u>187.951…</u>
|<u>187.951…</u>
−
|''189.58{{Overline|3}}''
|''189.58{{Overline|3}}''
−
|13
|13
−
|9
|9
−
|1.444
|1.444
−
|
|
|-
−
|
|-
|16\59
|
−
|<u>188.235…</u>
|
|''189.830…''  
−
|16\59
−
|
−
|<u>188.235…</u>
−
|''189.830…''
−
|16
|16
−
|11
|11
−
|1.4545
|1.4545
−
|
|
−
|-
|-
| 3\11
−
|
|3\11
−
|
|
−
|
−
|<u>189.473…</u>
|<u>189.473…</u>
−
|''190.{{Overline|90}}''
|''190.{{Overline|90}}''
| 3
−
|3
−
|2
|2
−
|1.500
|1.500
−
|Napoli-Meantone starts here
|Napoli-Meantone starts here
−
|-
|-
−
|
−
|17\62
−
|
|
−
|<u>190.654…</u>
−
|''191.935…''
−
|17
−
|11
−
|1.5455
−
|
−
|-
−
|
−
|14\51
|14\51
|
−
|
−
|<u>190.{{Overline|90}}</u>
|<u>190.{{Overline|90}}</u>
−
|''192.156…''
|''192.156…''
−
|14
|14
| 9
−
| 1.556
|9
|
−
|1.556
−
|
−
|-
|-
−
|
|
−
|11\40
|11\40
|
−
|
−
|<u>191.304…</u>
|<u>191.304…</u>
−
|''192.5''
|''192.5''
−
|11
|11
| 7
−
| 1.571
|7
−
|1.571
−
|
|
−
|-
|-
|
−
|
−
|8\29
|8\29
|
−
|
−
|<u>192</u>
|<u>192</u>
−
|''193.103…''
|''193.103…''
| 8
−
| 5
|8
−
|5
−
|1.600
|1.600
−
|
|
−
|-
|-
|
−
|
−
|5\18
|5\18
|  
−
|
−
|<u>193.548…</u>
|<u>193.548…</u>
−
|''194.{{Overline|4}}''
|''194.{{Overline|4}}''
−
|5
|5
−
|3
|3
−
|1.667
|1.667
−
|
|
−
|-
|-
−
|
|
−
|
|
−
|12\43
|12\43
−
|<u>194.{{Overline|594}}</u>
|<u>194.{{Overline|594}}</u>
−
|''195.348…''
|''195.348…''
| 12
−
| 7
|12
−
|7
−
|1.714
|1.714
−
|
|
−
|-
|-
|
−
|
−
|7\25
|7\25
|
−
|
−
|<u>195.348…</u>
|<u>195.348…</u>
−
|''196''
|''196''
−
|7
|7
−
|4
|4
−
|1.750
|1.750
|  
−
|
−
|-
|-
|  
−
|
−
|9\32
|9\32
|
−
|
−
|<u>196.{{Overline|36}}</u>
|<u>196.{{Overline|36}}</u>
−
|''196.875''
|''196.875''
−
|9
|9
−
|5
|5
−
|1.800
|1.800
−
|
|
−
|-
|-
−
|
|
−
|11\39
|11\39
−
|
|
−
|<u>197.014…</u>
|<u>197.014…</u>
−
|''197.435…''
|''197.435…''
−
|11
|11
| 6
−
|6
−
|1.833
|1.833
−
|
|
−
|-
|-
|  
−
|
−
|13\46
|13\46
−
|
|
−
|<u>197.468…</u>
|<u>197.468…</u>
−
|''197.826…''
|''197.826…''
| 13
−
| 7
|13
| 1.857
−
|7
−
|1.857
−
|
|
−
|-
|-
|
−
|
−
|15\53
|15\53
−
|
|
−
|<u>197.802…</u>
|<u>197.802…</u>
−
|''198.113…''
|''198.113…''
−
|15
|15
−
|8
|8
−
|1.875
|1.875
−
|
|
−
|-
|-
−
|
−
|17\60
−
|
|
| 17\60
−
|
|<u>198.058…</u>
|<u>198.058…</u>
−
|''198.{{Overline|3}}''
|''198.{{Overline|3}}''
−
|17
|17
| 9
−
| 1.889
|9
−
|1.889
−
|
|
−
|-
|-
|
−
|
−
|19\67
|19\67
|  
−
|
−
|<u>198.260…</u>
|<u>198.260…</u>
−
|''198.507…''
|''198.507…''
−
|19
|19
| 10
−
|10
−
|1.900
|1.900
−
|
|
−
|-
|-
−
|
|
−
|21\74
|21\74
−
|
|
−
|<u>198.425…</u>
|<u>198.425…</u>
| ''198.{{Overline |''198.''{{Overline|648}}
−
|''198.{{Overline|648}”''
−
|21
|21
| 11
−
|11
−
|1.909
|1.909
|  
−
|
−
|-
|-
−
|
|
−
|23\81
|23\81
−
|
|
−
|<u>198.561…</u>
|<u>198.561…</u>
−
|''198.765…''
|''198.765…''
−
|23
|23
−
|12
|12
| 1.917
−
|1.917
−
|
|
−
|-
|-
−
|
−
|25\88
−
|
|
| 25\88
−
|
|<u>198.675…</u>
|<u>198.675…</u>
−
|''198.8{{Overline|63}}''
|''198.8{{Overline|63}}''
−
|25
|25
| 13
−
|13
−
|1.923
|1.923
−
|
|
−
|-
|-
−
|
|
−
|27\95
|27\95
|  
−
|
−
|<u>198.773…</u>
|<u>198.773…</u>
−
|''198.947…''
|''198.947…''
−
|27
|27
−
|14
|14
−
|1.929
|1.929
−
|
|
−
|-
|-
−
|
|
−
|29\102
|29\102
−
|
|
−
|<u>198.857…</u>
|<u>198.857…</u>
−
|''199.019…''
|''199.019…''
−
|29
|29
−
|15
|15
−
|1.933
|1.933
−
|
|
−
|-
|-
|  
−
| 31\109
|
|
−
|31\109
−
|
−
|<u>198.930…</u>
|<u>198.930…</u>
−
|''199.082…''
|''199.082…''
−
|31
|31
−
|16
|16
−
|1.9375
|1.9375
−
|
|
−
|-
|-
−
|
|
−
|33\116
|33\116
−
|
|
−
|<u>198.994…</u>
|<u>198.994…</u>
−
|''199.137…''
|''199.137…''
−
|33
|33
−
|17
|17
−
|1.941
|1.941
−
|
|
−
|-
|-
|
−
|
−
|35\123
|35\123
−
|
|
−
|<u>199.052…</u>
|<u>199.052…</u>
−
|''199.186…''
|''199.186…''
−
|35
|35
−
|18
|18
−
|1.944
|1.944
−
|
|
−
|-
|-
−
|2\7
|2\7
−
|
|
−
|
|
−
|<u>200</u>
|<u>200</u>
−
|''200''
|''200''
−
|2
|2
−
|1
|1
−
|2.000
|2.000
| Napoli-Meantone ends, Napoli-Pythagorean begins
−
|Napoli-Meantone ends, Napoli-Pythagorean begins
−
|-
|-
−
|
|
−
|19\66
−
|
−
|<u>201.769…</u>
−
|''201.{{Overline|51}}''
−
|19
−
|9
−
|2.111
−
|
−
|-
−
|
−
|17\59
|17\59
−
|
|
−
|<u>201.980…</u>
|<u>201.980…</u>
−
|''201.694…''
|''201.694…''
−
|17
|17
−
|8
|8
−
|2.125
|2.125
|  
−
|
−
|-
|-
−
|
|
| 15\52
−
|  
|15\52
−
|
−
|<u>202.247…</u>
|<u>202.247…</u>
−
|''201.923…''
|''201.923…''
−
|15
|15
−
|7
|7
−
|2.143
|2.143
−
|
|
−
|-
|-
−
|
|
−
|13\45
|13\45
−
|
|
−
|<u>202.597…</u>
|<u>202.597…</u>
−
|''202.{{Overline|2}}''
|''202.{{Overline|2}}''
−
|13
|13
−
|6
|6
−
|2.167
|2.167
−
|
|
−
|-
|-
−
|
|
−
|11\38
|11\38
−
|
|
−
|<u>203.076…</u>
|<u>203.076…</u>
−
|''202.631…''
|''202.631…''
−
|11
|11
−
|5
|5
−
|2.200
|2.200
−
|
|
−
|-
|-
−
|
|
−
|9\31
|9\31
−
|
|
−
|<u>203.773…</u>
|<u>203.773…</u>
−
|''203.225…''
|''203.225…''
−
|9
|9
−
|4
|4
| 2.250
−
|
|2.250
−
|
−
|-
|-
−
|
|
−
|7\24
|7\24
−
|
|
−
|<u>204.878…</u>
|<u>204.878…</u>
−
|''204.1{{Overline|6}}''
|''204.1{{Overline|6}}''
| 7
−
|7
−
|3
|3
−
|2.333
|2.333
−
|
|
−
|-
|-
−
|
−
|
|
|
−
|12\41
|12\41
−
|<u>205.714…</u>
|<u>205.714…</u>
−
|''204.878…''
|''204.878…''
−
|12
|12
−
|5
|5
−
|2.400
|2.400
−
|
|
−
|-
|-
−
|
|
−
|5\17
|5\17
|
−
|
−
|<u>206.896…</u>
|<u>206.896…</u>
−
|''205.882…''
|''205.882…''
−
|5
|5
−
|2
|2
−
|2.500
|2.500
−
|Napoli-Neogothic heartland is from here…
|Napoli-Neogothic heartland is from here…
−
|-
|-
−
|
−
|
|
|
−
|18\61
|18\61
−
|<u>207.692…</u>
|<u>207.692…</u>
−
|''206.557…''
|''206.557…''
−
|18
|18
−
|7
|7
| 2.571
−
|
|2.571
−
|
−
|-
|-
−
|
|
|
−
|
−
|13\44
|13\44
−
|<u>208</u>
|<u>208</u>
|''206.{{Overline|81}}''
−
|''206.8̄1̄''
−
|13
|13
| 5
−
| 2.600
|5
−
|2.600
−
|
|
−
|-
|-
−
|
|
−
|8\27
|8\27
−
|
|
−
|<u>208.695…</u>
|<u>208.695…</u>
|''207.{{Overline|407}}''
−
| 8
|''207.4̄0̄7̄''
−
|8
−
|3
|3
−
|2.667
|2.667
−
|…to here
|…to here
−
|-
|-
|
−
|
−
|11\37
|11\37
−
|
|
−
|<u>209.523…</u>
|<u>209.523…</u>
|''208.{{Overline|108}}''
−
|''208.1̄0̄8̄''
−
|11
|11
−
|4
|4
| 2.750
−
|2.750
−
|
|
−
|-
|-
−
|
|
−
|14\47
|14\47
−
|
|
−
|<u>210</u>
|<u>210</u>
−
|''208.510…''
|''208.510…''
| 14
−
| 5
|14
−
|5
−
|2.800
|2.800
−
|
|
−
|-
|-
|
−
|
−
|17\57
|17\57
−
|
|
−
|<u>210.309…</u>
|<u>210.309…</u>
−
|''208.771…''
|''208.771…''
−
|17
|17
−
|6
|6
−
|2.833
|2.833
|  
−
|
−
|-
|-
−
|
|
| 20\67
−
|20\67
−
|
|
−
|<u>210.526…</u>
|<u>210.526…</u>
−
|''208.955…''
|''208.955…''
| 20
−
|20
−
|7
|7
−
|2.857
|2.857
−
|
|
−
|-
|-
−
|
|
| 23\77
−
|23\77
−
|
|
−
|<u>210.687…</u>
|<u>210.687…</u>
−
|''209.{{Overline|09}}''
|''209.{{Overline|09}}''
−
|23
|23
−
|8
|8
−
|2.875
|2.875
−
|
|
−
|-
|-
−
|3\10
|3\10
|
−
|
−
|
|
−
|<u>211.764…</u>
|<u>211.764…</u>
−
|''210''
|''210''
−
|3
|3
| 1
−
|1
−
|3.000
|3.000
−
|Napoli-Pythagorean ends, Napoli-Superpyth begins
|Napoli-Pythagorean ends, Napoli-Superpyth begins
−
|-
|-
|
−
|
−
|22\73
|22\73
|  
−
|
−
|<u>212.903…</u>
|<u>212.903…</u>
−
|''210.958…''
|''210.958…''
−
|22
|22
−
|7
|7
−
|3.143
|3.143
|  
−
|
−
|-
|-
−
|
|
−
|19\63
|19\63
|  
−
|
−
|<u>213.084…</u>
|<u>213.084…</u>
−
|''211.{{Overline|1}}''
|''211.{{Overline|1}}''
−
|19
|19
−
|6
|6
−
|3.167
|3.167
−
|
|
−
|-
|-
−
|
|
−
|16\53
|16\53
−
|
|
−
|<u>213.{{Overline|3}}</u>
|<u>213.{{Overline|3}}</u>
−
|''211.320…''
|''211.320…''
−
|16
|16
−
|5
|5
−
|3.200
|3.200
−
|
|
−
|-
|-
|
−
|
−
|13\43
|13\43
−
|
|
−
|<u>213.698…</u>
|<u>213.698…</u>
−
|''211.627…''
|''211.627…''
−
|13
|13
−
|4
|4
−
|3.250
|3.250
−
|
|
−
|-
|-
−
|
|
| 10\33
−
|10\33
−
|
|
−
|<u>214.285…</u>
|<u>214.285…</u>
−
|''212.{{Overline|12}}''
|''212.{{Overline|12}}''
−
|10
|10
−
|3
|3
−
|3.333
|3.333
−
|
|
−
|-
|-
−
|
|
−
|7\23
|7\23
−
|
|
−
|<u>215.384…</u>
|<u>215.384…</u>
−
|''213.043…''
|''213.043…''
−
|7
|7
−
|2
|2
−
|3.500
|3.500
−
|
|
−
|-
|-
−
|
|
−
|11\36
|11\36
−
|
|
−
|<u>216.393…</u>
|<u>216.393…</u>
−
|''213.{{Overline|3}}''
|''213.{{Overline|3}}''
| 11
−
|11
−
|3
|3
−
|3.667
|3.667
|
−
|
−
|-
|-
−
|
|
−
|15\49
|15\49
−
|
|
−
|<u>216.867…</u>
|<u>216.867…</u>
−
|''214.285…''
|''214.285…''
−
|15
|15
−
|4
|4
−
|3.750
|3.750
−
|
|
−
|-
|-
−
|4\13
|4\13
−
|
|
−
|
|
−
|<u>218.{{Overline|18}}</u>
|<u>218.{{Overline|18}}</u>
−
|''215.385…''
|''215.385…''
−
|4
|4
−
|1
|1
−
|4.000
|4.000
−
|
|
−
|-
|-
−
|
|
−
|13\42
|13\42
−
|
|
−
|<u>219.718…</u>
|<u>219.718…</u>
−
|''216.{{Overline|6}}''
|''216.{{Overline|6}}''
−
|13
|13
−
|3
|3
−
|4.333
|4.333
−
|
|
−
|-
|-
−
|
|
−
|9\29
|9\29
−
|
|
−
|<u>220.408…</u>
|<u>220.408…</u>
−
|''217.241…''
|''217.241…''
−
|9
|9
−
|2
|2
−
|4.500
|4.500
|  
−
|
−
|-
|-
−
|
|
−
|14\45
|14\45
−
|
|
−
|<u>221.052…</u>
|<u>221.052…</u>
−
|''217.{{Overline|7}}''
|''217.{{Overline|7}}''
−
|14
|14
| 3
−
| 4.667
|3
−
|4.667
−
|
|
−
|-
|-
−
|5\16
|5\16
−
|
|
−
|
|
−
|<u>222.{{Overline|2}}</u>
|<u>222.{{Overline|2}}</u>
−
|''218.75''
|''218.75''
−
|5
|5
| 1
−
|1
−
|5.000
|5.000
−
|Napoli-Superpyth ends
|Napoli-Superpyth ends
−
|-
|-
−
|
|
−
|16\51
|16\51
−
|
|
−
|<u>223.255…</u>
|<u>223.255…</u>
−
|''219.607…''
|''219.607…''
−
|16
|16
−
|3
|3
−
|5.333
|5.333
−
|
|
−
|-
|-
−
|
|
−
|11\35
|11\35
−
|
|
−
|<u>223.728…</u>
|<u>223.728…</u>
−
|''220''
|''220''
−
|11
|11
−
|2
|2
−
|5.500
|5.500
−
|
|
−
|-
|-
−
|
|
−
|17\54
|17\54
−
|
|
−
|<u>224.175…</u>
|<u>224.175…</u>
−
|''220.{{Overline|370}}''
|''220.{{Overline|370}}''
| 17
−
| 3
|17
−
|3
−
|5.667
|5.667
−
|
|
−
|-
|-
−
|6\19
|6\19
|
−
|  
|
−
|
−
|<u>225</u>
|<u>225</u>
−
|''221.052…''
|''221.052…''
−
|6
|6
−
|1
|1
−
|6.000
|6.000
|
−
|
−
|-
|-
−
|1\3
|1\3
|
−
|
|
−
|
−
|<u>240</u>
|<u>240</u>
−
|''233.{{Overline|3}}''
|''233.{{Overline|3}}''
−
|1
|1
−
|0
|0
−
|→ inf
|→ inf
−
|Paucitonic
|Paucitonic
−
|}
|}

Revision as of 03:25, 16 July 2022

Lua error in Module:MOS at line 46: attempt to index local 'equave' (a nil value).3L 1s<3/2>, is a fifth-repeating MOS scale. The notation "<3/2>" means the period of the MOS is 3/2, disambiguating it from octave-repeating 3L 1s. The name of the period interval is called the sesquitave (by analogy to the tritave).


The generator range is 171.4 to 240 cents, placing it near the diatonic major second, usually representing a major second of some type. The dark (chroma-negative) generator is, however, its fifth complement (480 to 514.3 cents).


In the fifth-repeating version of the diatonic scale, each tone has a 3/2 perfect fifth above it. The scale has two major chords and two minor chords. P

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 diatonic 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 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.

Cents
Notation Supersoft Soft Semisoft Basic Semihard Hard Superhard
Diatonic 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…

1\7

100

3\17

124.137…

2\10

141.176…

3\13

163.63

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

54.54

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

218.18

Re#, La# G# 1# 5\15

230.769…

4\11

252.631…

7\18

270.967…

3\7

300

8\17

331.034…

5\10

352.941…

7\13

381.81

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

272.72

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

436.36

Mi#, Si# A# 2# 9\15

415.384…

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 3b, 3c 10\15

461.538…

11\18

425.806…

4\7

400

9\17

372.413…

5\10

352.941…

6\13

327.27

Fa, Do Bb 3 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

490.90

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

654.54

Fax, Dox B# 3x 13\15

600

10\11

631.578…

17\18

658.064…

7\7

700

18\17

744.827…

11\10

776.470…

15\13

818.18

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

545.45

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

709.09

Do#, Sol# Η# 4# 16\15

738.461…

12\11

757.894…

20\18

774.193…

8\8

800

20\17

827.586…

12\10

847.058…

16\13

872.72

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

763.63

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

927.27

Re#, La# C# 5# 20\15

923.076…

15\11

947.368…

25\18

967.741…

10\7

1000

25\17

1034.482…

15\10

1058.823…

20\13

1090.90

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

981.81

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

1145.45

Mi#, Si# D# 6# 24\15

1107.692…

18\11

1136.842…

30\18

1161.290…

12\7

1200

30\17

1241.379…

18\10

1270.588…

24\13

1309.09

Fab, Dob Ebb 7b, 7c 25\15

1153.846…

29\18

1122.580…

11\7

1100

26\17

1075.862…

15\10

1058.823…

19\13

1036.36

Fa, Do Eb 7 26\15

1200

19\11

1200

31\18

1200

12\7

1200

29\17

1200

17\10

1200

22\13

1200

Fa#, Do# E 7# 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

1363.63

Fax, Dox E# 7x 28\15

1292.307…

21\11

1326.315…

35\18

1354.838…

14\7

1400

35\17

1448.275…

21\10

1482.352…

28\13

1527.27

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

1254.54

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\13

1418.18

Do#, Sol# F# 8#, F# 31\15

1430.769…

23\11

1452.631…

38\18

1470.967…

15\7

1500

37\17

1531.034…

22\10

1552.941…

29\13

1581.81

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

1472.72

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

1636.36

Re#, La# G# 9#, G# 35\15

1615.384…

26\11

1642.105…

43\18

1664.516…

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

1690.90

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

1854.54

Mi#, Si# A# X#, A# 39\15

1800

29\11

1831.578…

48\18

1858.064…

19\7

1900

47\17

1944.827…

28\10

1976.470…

37\13

2018.18

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

1745.45

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

1909.09

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

2072.72

Fax, Dox B# Ex, Cx 43\15

1984.615…

32\11

2021.052…

53\18

2051.612…

21\7

2100

52\17

2151.724…

31\10

2188.235…

41\13

2236.36

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

1963.63

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

2127.27

Relative cents
Notation Supersoft Soft Semisoft Basic Semihard Hard Superhard
Diatonic Napoli Bijou ~15edf ~11edf ~18edf ~7edf ~17edf ~10edf ~13edf
Do#, Sol# F# 0#, D# 1\15

46.6

1\11

63.63

2\18

77.7̄

1\7

100

3\17

123.529…

2\10

140

3\13

161.538…

Reb, Lab Gb 1b, 1c 3\15

140

2\11

127.27

3\18

116.6

2\17

82.352…

1\10

70

1\13

53.846…

Re, La G 1 4\15

186.6

3\11

190.90

5\18

194.4

2\7

200

5\17

205.882…

3\10

210

4\13

215.384…

Re#, La# G# 1# 5\15

233.3

4\11

254.54

7\18

272.2̄

3\7

300

8\17

329.411…

5\10

350

7\13

376.923…

Mib, Sib Ab 2b, 2c 7\15

326.6

5\11

318.18

8\18

311.1

7\17

288.235…

4\10

280

5\13

269.230…

Mi, Si A 2 8\15

373.3

6\11

381.81

10\18

388.8

4\7

400

10\17

411.764…

6\10

420

8\13

430.769…

Mi#, Si# A# 2# 9\15

420

7\11

445.45

12\18

466.6

5\7

500

13\17

535.294…

8\10

560

11\13

592.307…

Fab, Dob Bbb 3b, 3c 10\15

466.6

11\18

427.7

4\7

400

9\17

370.588…

5\10

350

6\13

323.076.…

Fa, Do Bb 3 11\15

513.3

8\11

509.09

13\18

505.5

5\7

500

12\17

494.117…

7\10

490

9\13

484.615…

Fa#, Do# B 3# 12\15

560

9\11

572.72

15\18

583.3

6\7

600

15\17

617.647…

9\10

630

12\13

646.153…

Fax, Dox B# 3x 13\15

606. 6

10\11

636.36

17\18

661.1

7\7

700

18\17

741.176…

11\10

770

15\13

807.692…

Dob, Solb Hb 4b, 4c 14\15

653.3

16\18

622.2

6\7

600

14\17

576.470…

8\10

560

10\13

538.461…

Do, Sol H 4 700
Do#, Sol# Η# 4# 16\15

746.6

12\11

763.63

20\18

777.7

8\7

800

20\17

823.529…

12\10

840

16\13

861.538…

Reb, Lab Cb 5b, 5c 18\15

840

13\11

827.27

21\18

816.6

19\17

782.352…

11\10

770

14\13

753.846…

Re, La C 5 19\15

886.6

14\11

890.90

23\18

894.4

9\7

900

22\17

905.882…

13\10

910

17\13

915.384…

Re#, La# C# 5# 20\15

933.3

15\11

954.54

25\18

972.2

10\7

1000

25\17

1029.411…

15\10

1050

20\13

1076.923…

Mib, Sib Db 6b, 6c 22\15

1026.6

16\11

1018.18

26\18

1011. 1

24\17

988.235…

14\10

980

18\13

969.230…

Mi, Si D 6 23\15

1073.3

17\11

1081.81

28\18

1088.8

11\7

1100

27\17

1111.764…

16\10

1120

21\13

1130.769…

Mi#, Si# D# 6# 24\15

1120

18\11

1145.45

30\18

1166.6

12\7

1200

30\17

1235.294…

18\10

1260

24\13

1292.307…

Fab, Dob Ebb 7b, 7c 25\15

1166.6

29\18

1127.7

11\7

1100

26\17

1070.588…

15\10

1050

19\13

1023.076…

Fa, Do Eb 7 26\15

1213.3

19\11

1209.09

31\18

1205.5

12\7

1200

29\17

1194.117…

17\10

1190

22\13

1184.615…

Fa#, Do# E 7# 27\15

1260

20\11

1272.72

33\18

1283.3

13\7

1300

32\17

1317.647…

19\10

1330

25\13

1346.153…

Fax, Dox E# 7x 28\15

1306.6

21\11

1336.36

35\18

1361.1

14\7

1400

35\17

1441.176…

21\10

1470

28\13

1507.692…

Dob, Solb Fb 8b, Fc 29\15

1333.3

34\18

1322.2

13\7

1300

31\17

1276.470…

18\10

1260

23\13

1238.461…

Do, Sol F 8, F 1400
Do#, Sol# F# 8#, F# 31\15

1446.6

23\11

1463.63

38\18

1477.7̄

15\7

1500

37\17

1523.529…

22\10

1540

29\13

1561.538…

Reb, Lab Gb 9b, Gc 33\15

1540

24\11

1527.27

39\18

1516.6

36\17

1482.352…

21\10

1470

27\13

1453.846…

Re, La G 9, G 34\15

1586.6

25\11

1590.90

41\18

1594.4

16\7

1600

39\17

1605.882…

23\10

1610

30\13

1615.384…

Re#, La# G# 9#, G# 35\15

1633.3

26\11

1654.54

43\18

1672.2

17\7

1700

42\17

1729.411…

25\10

1750

33\13

1776.923…

Mib, Sib Ab Xb, Ac 37\15

1726.6

27\11

1718.18

44\18

1711.1

41\17

1688.235…

24\10

1680

31\13

1669.230…

Mi, Si A X, A 38\15

1773.3

28\11

1781.81

46\18

1788.8

18\7

1800

44\17

1811.764…

26\10

1820

34\13

1830.769…

Mi#, Si# A# X#, A# 39\15

1820

29\11

1845.45

48\18

1866.6

19\7

1900

47\17

1935.294…

28\10

1960

37\13

1992.307…

Fab, Dob Bbb Ebb, Ccc 40\15

1866.6

47\18

1827.7

18\7

1800

43\17

1770.588…

25\10

1750

32\13

1723.076…

Fa, Do Bb Eb, Cc 41\15

1913.3

30\11

1909.09

49\18

1905.5

19\7

1900

46\17

1894.117…

27\10

1890

35\13

1884.615…

Fa#, Do# B E, C 42\15

1960

31\11

1972.72

51\18

1983.3

20\7

2000

49\17

2017.647…

29\10

2030

38\13

2046.153…

Fax, Dox B# Ex, Cx 43\15

2006.6

32\11

2036.36

53\18

2061. 1

21\7

2100

52\17

2141.176…

31\10

2170

41\13

2207.692…

Dob, Solb Hb 0b, Dc 44\15

2053.3

52\18

2022.2

20\7

2000

48\17

1976.470…

28\10

1960

36\13

1938.615…

Do, Sol H 0, D 2100

Intervals

Generators Sesquitave notation Interval category name Generators Notation of 3/2 inverse Interval category name
The 4-note MOS has the following intervals (from some root):
0 Do, Sol perfect unison 0 Do, Sol sesquitave (just fifth)
1 Fa, Do perfect fourth -1 Re, La perfect second
2 Mib, Sib minor third -2 Mi, Si major third
3 Reb, Lab diminished second -3 Fa#, Do# augmented fourth
The chromatic 7-note MOS also has the following intervals (from some root):
4 Dob, Solb diminished sesquitave -4 Do#, Sol# augmented unison (chroma)
5 Fab, Dob diminished fourth -5 Re#, La# augmented second
6 Mibb, Sibb diminished third -6 Mi#, Si# augmented third

Genchain

The generator chain for this scale is as follows:

Mibb

Sibb

Fab

Dob

Dob

Solb

Reb

Lab

Mib

Sib

Fa

Do

Do

Sol

Re

La

Mi

Si

Fa#

Do#

Do#

Sol#

Re#

La#

Mi#

Si#

d3 d4 d5 d2 m3 P4 P1 P2 M3 A4 A1 A2 A3

Modes

The mode names are based on the species of fifth:

Mode Scale UDP Interval type
name pattern notation 2nd 3rd 4th
Lydian LLLs 3|0 P M A
Major LLsL 2|1 P M P
Minor LLsL 1|2 P m P
Phrygian sLLL 0|3 d m P

Temperaments

The most basic rank-2 temperament interpretation of diatonic is Napoli. The name "Napoli" comes from the “Neapolitan” sixth triad spelled root-(p-2g)-(2p-3g) (p = 3/2, g = the whole tone) which serves as its minor triad approximating 5:6:8 in pental interpretations or 18:21:28 in septimal ones. Basic ~7edf fits both interpretations.

Napoli-Meantone

Subgroup: 3/2.6/5.8/5

Comma list: 81/80

POL2 generator: ~9/8 = [[1]]

Mapping: [⟨1 1 2], ⟨0 -2 -3]]

Vals: Template:Val list

Napoli-Superpyth

Subgroup: 3/2.7/6.14/9

Comma list: 64/63

POL2 generator: ~8/7 = [[2]]

Mapping: [⟨1 1 2], ⟨0 -2 -3]]

Vals: Template:Val list

Scale tree

The spectrum looks like this:

Generator

(bright)

Cents L s L/s Comments
Normalised ed7\12
1\4 171.428… 175 1 1 1.000 Equalised
6\23 180 182.608… 6 5 1.200
11\42 180.821… 183.3 11 9 1.222
5\19 181.81 184.210… 5 4 1.250
14\53 182.608… 184.905… 14 11 1.273
9\34 183.050… 185.294… 9 7 1.286
4\15 184.615… 186.6 4 3 1.333
11\41 185.915… 187.804… 11 8 1.375
7\26 186.6 188.461… 7 5 1.400
10\37 187.5 189.189 10 7 1.429
13\48 187.951… 189.583 13 9 1.444
16\59 188.235… 189.830… 16 11 1.4545
3\11 189.473… 190.90 3 2 1.500 Napoli-Meantone starts here
14\51 190.90 192.156… 14 9 1.556
11\40 191.304… 192.5 11 7 1.571
8\29 192 193.103… 8 5 1.600
5\18 193.548… 194.4 5 3 1.667
12\43 194.594 195.348… 12 7 1.714
7\25 195.348… 196 7 4 1.750
9\32 196.36 196.875 9 5 1.800
11\39 197.014… 197.435… 11 6 1.833
13\46 197.468… 197.826… 13 7 1.857
15\53 197.802… 198.113… 15 8 1.875
17\60 198.058… 198.3 17 9 1.889
19\67 198.260… 198.507… 19 10 1.900
21\74 198.425… 198.648 21 11 1.909
23\81 198.561… 198.765… 23 12 1.917
25\88 198.675… 198.863 25 13 1.923
27\95 198.773… 198.947… 27 14 1.929
29\102 198.857… 199.019… 29 15 1.933
31\109 198.930… 199.082… 31 16 1.9375
33\116 198.994… 199.137… 33 17 1.941
35\123 199.052… 199.186… 35 18 1.944
2\7 200 200 2 1 2.000 Napoli-Meantone ends, Napoli-Pythagorean begins
17\59 201.980… 201.694… 17 8 2.125
15\52 202.247… 201.923… 15 7 2.143
13\45 202.597… 202.2 13 6 2.167
11\38 203.076… 202.631… 11 5 2.200
9\31 203.773… 203.225… 9 4 2.250
7\24 204.878… 204.16 7 3 2.333
12\41 205.714… 204.878… 12 5 2.400
5\17 206.896… 205.882… 5 2 2.500 Napoli-Neogothic heartland is from here…
18\61 207.692… 206.557… 18 7 2.571
13\44 208 206.81 13 5 2.600
8\27 208.695… 207.407 8 3 2.667 …to here
11\37 209.523… 208.108 11 4 2.750
14\47 210 208.510… 14 5 2.800
17\57 210.309… 208.771… 17 6 2.833
20\67 210.526… 208.955… 20 7 2.857
23\77 210.687… 209.09 23 8 2.875
3\10 211.764… 210 3 1 3.000 Napoli-Pythagorean ends, Napoli-Superpyth begins
22\73 212.903… 210.958… 22 7 3.143
19\63 213.084… 211.1 19 6 3.167
16\53 213.3 211.320… 16 5 3.200
13\43 213.698… 211.627… 13 4 3.250
10\33 214.285… 212.12 10 3 3.333
7\23 215.384… 213.043… 7 2 3.500
11\36 216.393… 213.3 11 3 3.667
15\49 216.867… 214.285… 15 4 3.750
4\13 218.18 215.385… 4 1 4.000
13\42 219.718… 216.6 13 3 4.333
9\29 220.408… 217.241… 9 2 4.500
14\45 221.052… 217.7 14 3 4.667
5\16 222.2 218.75 5 1 5.000 Napoli-Superpyth ends
16\51 223.255… 219.607… 16 3 5.333
11\35 223.728… 220 11 2 5.500
17\54 224.175… 220.370 17 3 5.667
6\19 225 221.052… 6 1 6.000
1\3 240 233.3 1 0 → inf Paucitonic
Retrieved from "https://en.xen.wiki/index.php?title=User:Moremajorthanmajor/3L_1s_(perfect_fifth-equivalent)&oldid=93464"