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

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{{Overline|45}}{{Infobox MOS
{{Infobox MOS


| Name = Diatonic/Angel
| Name = Diatonic/Angel
Line 16: Line 15:
| Pattern = LLLs
| Pattern = LLLs
 
 
}}'''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).  
   
   
In the fifth-repeating version of the diatonic scale, each tone has a 3/2 perfect fifth above it. The scale has two major chords and two minor chords.  
In the fifth-repeating version of the diatonic scale, each tone has a 3/2 perfect fifth above it. The scale has two major chords and two minor chords.  
Line 27: Line 23:
==Notation==
==Notation==
   
   
There are 3 main ways to notate the angel scale. One method uses a simple sesquitave (fifth) repeating notation consisting of 4 naturals (eg. Do Re Mi Fa, Sol La Si Do). Given that 1-5/4-5/3 is fifth-equivalent to a tone cluster of 1-10/9-5/4, it may be more convenient to notate diatonic scales as repeating at the double or triple sesquitave (major ninth or thirteenth), however it does make navigating the [[Generator|genchain]] harder. This way, 5/3 is its own pitch class, distinct from 10/9. Notating this way produces a major ninth which is the Aeolian mode of Napoli[6L 2s] or a major thirteenth which is the Dorian mode of Bijou[9L 3s]. Since there are exactly 8 naturals in double sesquitave notation and 12 in triple sesquitave notation, letters A-H (FGABHCDEF) or dozenal digits (0123456789XE0 or D1234567FGACD with flats written C molle) may be used.
There are 4 main ways to notate the angel scale. One method uses a simple sesquitave (fifth) repeating notation consisting of 4 naturals (eg. Do Re Mi Fa, Sol La Si Do). Given that 1-5/4-5/3 is fifth-equivalent to a tone cluster of 1-10/9-5/4, it may be more convenient to notate angel scales as repeating at the double, triple or quadruple sesquitave (major ninth, thirteenth or seventeenth i. e. ~pentave), however it does make navigating the [[Generator|genchain]] harder. This way, 5/3 is its own pitch class, distinct from 10/9. Notating this way produces a major ninth which is the Aeolian mode of Napoli[6L 2s], a major thirteenth which is the Dorian mode of Bijou[9L 3s] or an ~pentave which is the Mixolydian mode of Hextone[12L 4s]. Since there are exactly 8 naturals in double sesquitave notation, 12 in triple sesquitave notation and 16 in quadruple sesquitave notation, letters A-H (FGABHCDEF) or dozenal or hex digits (0123456789XE0 or D1234567FGACD with flats written C molle or 0123456789ABCDEF0 or G123456789ABCDEFG with flats written F molle) may be used.
 
 
{| class="wikitable"
{| class="wikitable"
Line 35: Line 31:
Cents
Cents
 
 
! colspan="3" |Notation
! colspan="4" |Notation
 
 
!Supersoft
!Supersoft
Line 58: Line 54:
 
 
!Bijou
!Bijou
!Hextone
!~15edf
!~15edf
 
 
Line 80: Line 76:
 
 
|0#, D#
|0#, D#
|0#, G#
|1\15
|1\15
 
 
Line 116: Line 112:
 
 
|1b, 1c
|1b, 1c
|1f
|3\15
|3\15
 
 
Line 147: Line 143:
|'''G'''
|'''G'''
 
 
|'''1'''
|'''1'''
|'''1'''
 
 
Line 184: Line 181:
 
 
|1#
|1#
|1#
|5\15
|5\15
 
 
Line 220: Line 217:
 
 
|2b, 2c
|2b, 2c
|2f
|7\15
|7\15
 
 
Line 252: Line 249:
 
 
|2
|2
|2
|8\15
|8\15
 
 
Line 288: Line 285:
 
 
|2#
|2#
|2#
|9\15
|9\15
 
 
Line 324: Line 321:
 
 
|3b, 3c
|3b, 3c
|3f
|10\15
|10\15
 
 
Line 355: Line 352:
|'''Bb'''
|'''Bb'''
 
 
|'''3'''
|'''3'''
|'''3'''
 
 
Line 392: Line 390:
 
 
|3#
|3#
|3#
|12\15
|12\15
 
 
Line 428: Line 426:
 
 
|3x
|3x
|3x
|13\15
|13\15
 
 
Line 462: Line 460:
|Hb
|Hb
| 4b, 4c
| 4b, 4c
|4f
|14\15
|14\15
   
   
Line 488: Line 487:
 
 
!4
!4
!4
!'''15\15'''
!'''15\15'''
 
 
Line 524: Line 523:
 
 
|4#
|4#
|4#
|16\15
|16\15
 
 
Line 560: Line 559:
 
 
|5b, 5c
|5b, 5c
|5
|18\15
|18\15
 
 
Line 591: Line 590:
|'''C'''
|'''C'''
 
 
|'''5'''
|'''5'''
|'''5'''
 
 
Line 628: Line 628:
 
 
| 5#
| 5#
|5#
|20\15
|20\15
 
 
Line 664: Line 664:
 
 
|6b, 6c
|6b, 6c
|6f
|22\15
|22\15
 
 
Line 696: Line 696:
 
 
|6
|6
|6
|23\15
|23\15
 
 
Line 732: Line 732:
 
 
|6#
|6#
|6#
| 24\15
| 24\15
 
 
Line 768: Line 768:
 
 
|7b, 7c
|7b, 7c
|7f
|25\15
|25\15
 
 
Line 799: Line 799:
|'''Eb'''
|'''Eb'''
 
 
|'''7'''
|'''7'''
|'''7'''
 
 
Line 836: Line 837:
 
 
|7#
|7#
|7#
|27\15
|27\15
 
 
Line 872: Line 873:
 
 
|7x
|7x
|7x
|28\15
|28\15
 
 
Line 908: Line 909:
 
 
|8b, Fc
|8b, Fc
|8f
|29\15
|29\15
 
 
Line 929: Line 930:
1270.588…
1270.588…
 
 
| 23\18
| 23\13
 
 
1254.{{Overline|54}}
1254.{{Overline|54}}
Line 940: Line 941:
 
 
! 8, F
! 8, F
!8
! 30\15
! 30\15
 
 
Line 976: Line 977:
 
 
|8#, F#
|8#, F#
|8#
|31\15
|31\15
 
 
Line 1,012: Line 1,013:
 
 
|9b, Gc
|9b, Gc
|9f
|33\15
|33\15
 
 
Line 1,044: Line 1,045:
 
 
|'''9, G'''
|'''9, G'''
|9
|'''34\15'''
|'''34\15'''
 
 
Line 1,080: Line 1,081:
 
 
|9#, G#
|9#, G#
|9#
|35\15
|35\15
 
 
Line 1,116: Line 1,117:
 
 
|Xb, Ac
|Xb, Ac
|Af
|37\15
|37\15
 
 
Line 1,148: Line 1,149:
 
 
|X, A
|X, A
|A
|38\15
|38\15
 
 
Line 1,184: Line 1,185:
 
 
|X#, A#
|X#, A#
|A#
|39\15
|39\15
 
 
Line 1,220: Line 1,221:
 
 
|Ebb, Ccc
|Ebb, Ccc
|Bf
|40\15
|40\15
 
 
Line 1,251: Line 1,252:
|'''Bb'''
|'''Bb'''
 
 
|Eb, Cc
|'''Eb, Cc'''
|'''B'''
|'''41\15'''
|'''41\15'''
 
 
Line 1,288: Line 1,289:
 
 
|E, C
|E, C
|B#
|42\15
|42\15
 
 
Line 1,324: Line 1,325:
 
 
|Ex, Cx
|Ex, Cx
|Bx
|43\15
|43\15
 
 
Line 1,360: Line 1,361:
 
 
|0b, Dc
|0b, Dc
|Cf
|44\15
|44\15
 
 
Line 1,392: Line 1,393:
 
 
!0, D
!0, D
!C
!45\15
!45\15
 
 
Line 1,420: Line 1,421:
 
 
2127.{{Overline|27}}
2127.{{Overline|27}}
|}
{| class="wikitable"
|+Relative cents
! colspan="3" | Notation
!Supersoft
!Soft
!Semisoft
!Basic
!Semihard
!Hard
!Superhard
|-
|-
! Diatonic
|Do#, Sol#
!Napoli
|Η#
! Bijou
|0#, D#
!~15edf
|C#
!~11edf
|46\15
!~18edf
2123.076…
!~7edf
|34\11
!~17edf
2147.368…
!~10edf
|56\18
!~13edf
2167.741…
| rowspan="2" |22\7
2200
|54\17
2234.582…
|32\10
2258.823…
|42\13
2090.{{Overline|90}}
|-
|Reb, Lab
|Cb
|1b, 1c
|Df
|48\15
2215.384…
|35\11
2210.526…
|57\18
2206.451…
|53\17
2193.103…
|31\10
2188.235…
|40\13
2181.{{Overline|81}}
|-
|'''Re, La'''
|'''C'''
|'''1'''
|'''D'''
|'''49\15'''
'''2261.538…'''
|'''36\11'''
'''2273.684…'''
|'''59\18'''
'''2283.870…'''
|'''23\7'''
'''2300'''
|'''56\17'''
'''2317.241…'''
|'''33\10'''
'''2329.411…'''
|'''43\13'''
'''2245.{{Overline|45}}'''
|-
|-
|Do#, Sol#
|Re#, La#
|F#
|C#
|0#, D#
|1#
|1\15
|D#
|50\15
''46.{{Overline|6}}''
2307.692…
|1\11
|37\11
2336.842…
''63.{{Overline|63}}''
|61\18
|2\18
2361.290…
| rowspan="2" |24\7
''77.7̄''
2400
| rowspan="2" |1\7
|59\17
2441.379…
''100''
|35\10
| 3\17
2470.588…
|46\13
''123.529…''
2509.{{Overline|09}}
| 2\10
|-
|Mib, Sib
''140''
|Db
|3\13
|2b, 2c
|Ef
''161.538…''
|52\15
2400
|38\11
2400
|62\18
2400
|58\17
2400
|34\10
2400
|44\13
2400
|-
|-
|Reb, Lab
|Mi, Si
| Gb
|D
|1b, 1c
|2
|3\15
|E
|53\15
''140''
2446.153…
|2\11
|39\11
2463.158…
''127.{{Overline|27}}''
|64\18
|3\18
2477.419…
|25\7
''116.{{Overline|6}}''
2500
| 2\17
|61\17
2524.137…
''82.352…''
|36\10
|1\10
2541.176…
|47\13
''70''
2563.{{Overline|63}}
|1\13
|-
|Mi#, Si#
''53.846…''
|D#
|2#
|E#
|54\15
2492.307…
| rowspan="2" |40\11
2526.315…
|66\18
2554.838…
|26\7
2600
|64\17
2648.275…
|38\10
2682.352…
|50\13
2727.{{Overline|27}}
|-
|-
|'''Re, La'''
|Fab, Dob
|'''G'''
|Ebb
|'''1'''
|3b, 3c
|'''4\15'''
|Fff
|55\15
'''''186.{{Overline|6}}'''''
2538.461…
|'''3\11'''
|65\18
2516.129…
'''''190.{{Overline|90}}'''''
|25\7
|'''5\18'''
2500
|60\17
'''''194.{{Overline|4}}'''''
2482.758…
|'''2\7'''
|35\10
2470.588…
'''''200'''''
|45\13
|'''5\17'''
2454.{{Overline|54}}
|-
'''''205.882…'''''
|'''Fa, Do'''
|'''3\10'''
|'''Eb'''
|'''3'''
'''''210'''''
|'''Ff'''
|'''4\13'''
|'''56\15'''
'''2584.615…'''
'''''215.384…'''''
|'''41\11'''
'''2589.473…'''
|'''67\18'''
'''2593.548…'''
|'''26\7'''
'''2600'''
|'''63\17'''
'''2606.896…'''
|'''37\10'''
'''2611.764…'''
|'''48\13'''
'''2618.{{Overline|18}}'''
|-
|Fa#, Do#
|E
|3#
|F
|57\15
2630.769…
|42\11
2652.632…
|69\18
2670.967…
|27\7
2700
|66\17
2731.034…
|39\10
2752.941…
|51\13
2781.{{Overline|81}}
|-
|Fax, Dox
|E#
|3x
|F#
|58\15
2676.923…
| rowspan="2" |43\11
2715.789…
|71\18
2748.387…
|28\7
2800
|69\17
2855.168…
|41\10
2894.117…
|54\13
2945.{{Overline|45}}
|-
|-
|Re#, La#
|Dob, Solb
| G#
|Fb
| 1#
|4b, 4c
|5\15
|0f, Gf
|59\15
''233.{{Overline|3}}''
2723.076…
|4\11
|70\18
2709.677…
''254.{{Overline|54}}''
|27\7
|7\18
2700
|65\17
''272.2̄''
2689.655…
| rowspan="2" |3\7
|38\10
2682.352…
''300''
|49\13
|8\17
2672.{{Overline|72}}
''329.411…''
|5\10
''350''
|7\13
''376.923…''
|-
|-
|Mib, Sib
!Do, Sol
|Ab
!F
|2b, 2c
!4
|7\15
!0, G
!60\15
2769.230…
!44\11
2778.947…
!72\18
2787.097…
!28\7
2800
!68\17
2813.793…
!40\10
2823.529…
!52\113
2836.{{Overline|36}}
|}
   
   
''326.{{Overline|6}}''
{| class="wikitable"
|5\11
|+Relative cents
! colspan="4" | Notation
!Supersoft
!Soft
!Semisoft
!Basic
!Semihard
!Hard
!Superhard
|-
! Diatonic
!Napoli
! Bijou
!Hextone
!~15edf
!~11edf
!~18edf
!~7edf
!~17edf
!~10edf
!~13edf
|-
|Do#, Sol#
|F#
|0#, D#
|0#, G#
|1\15
   
   
''318.{{Overline|18}}''
''46.{{Overline|6}}''
| 8\18
|1\11
   
   
''311.{{Overline|1}}''
''63.{{Overline|63}}''
|7\17
|2\18
''77.7̄''
| rowspan="2" |1\7
''100''
| 3\17
   
   
''288.235…''
''123.529…''
| 4\10
| 2\10
   
   
''280''
''140''
|5\13
|3\13
   
   
''269.230…''
''161.538…''
|-
|-
|Mi, Si
|Reb, Lab
|A
| Gb
| 2
|1b, 1c
|8\15
|1f
|3\15
   
   
''373.{{Overline|3}}''
''140''
|6\11
|2\11
   
   
''381.{{Overline|81}}''
''127.{{Overline|27}}''
|10\18
|3\18
   
   
''388.{{Overline|8}}''
''116.{{Overline|6}}''
|4\7
| 2\17
   
   
''400''
''82.352…''
|10\17
|1\10
   
   
''411.764…''
''70''
|6\10
|1\13
   
   
''420''
''53.846…''
|8\13
''430.769…''
|-
|-
|Mi#, Si#
|'''Re, La'''
|A#
|'''G'''
|2#
|'''1'''
|9\15
|'''1'''
|'''4\15'''
   
   
''420''
'''''186.{{Overline|6}}'''''
| rowspan="2" |7\11
|'''3\11'''
   
   
''445.{{Overline|45}}''
'''''190.{{Overline|90}}'''''
|12\18
|'''5\18'''
   
   
''466.{{Overline|6}}''
'''''194.{{Overline|4}}'''''
|5\7
|'''2\7'''
   
   
''500''
'''''200'''''
|13\17
|'''5\17'''
   
   
''535.294…''
'''''205.882…'''''
|8\10
|'''3\10'''
   
   
''560''
'''''210'''''
|11\13
|'''4\13'''
   
   
''592.307…''
'''''215.384…'''''
|-
|-
|Fab, Dob
|Re#, La#
|Bbb
| G#
|3b, 3c
| 1#
|10\15
|1#
|5\15
   
   
''466.{{Overline|6}}''
''233.{{Overline|3}}''
|11\18
|4\11
   
   
''427.{{Overline|7}}''
''254.{{Overline|54}}''
|4\7
|7\18
''272.2̄''
| rowspan="2" |3\7
   
   
''400''
''300''
|9\17
|8\17
   
   
''370.588…''
''329.411…''
|5\10
|5\10
   
   
''350''
''350''
|6\13
|7\13
   
   
''323.076.…''
''376.923…''
|-
|-
|'''Fa, Do'''
|Mib, Sib
|'''Bb'''
|Ab
|'''3'''
|2b, 2c
|'''11\15'''
|2f
|7\15
   
   
'''''513.{{Overline|3}}'''''
''326.{{Overline|6}}''
|'''8\11'''
|5\11
   
   
'''''509.{{Overline|09}}'''''
''318.{{Overline|18}}''
|'''13\18'''
| 8\18
   
   
'''''505.{{Overline|5}}'''''
''311.{{Overline|1}}''
|'''5\7'''
|7\17
   
   
'''''500'''''
''288.235…''
|'''12\17'''
| 4\10
   
   
'''''494.117…'''''
''280''
|'''7\10'''
|5\13
   
   
'''''490'''''
''269.230…''
|'''9\13'''
'''''484.615…'''''
|-
|-
|Fa#, Do#
|Mi, Si
| B
|A
|3#
| 2
|12\15
|2
|8\15
   
   
''560''
''373.{{Overline|3}}''
|9\11
|6\11
   
   
''572.{{Overline|72}}''
''381.{{Overline|81}}''
| 15\18
|10\18
   
   
''583.{{Overline|3}}''
''388.{{Overline|8}}''
|6\7
|4\7
   
   
''600''
''400''
|15\17
|10\17
   
   
''617.647…''
''411.764…''
|9\10
|6\10
   
   
''630''
''420''
|12\13
|8\13
   
   
''646.153…''
''430.769…''
|-
|-
| Fax, Dox
|Mi#, Si#
|B#
|A#
|3x
|2#
|13\15
|2#
|9\15
   
   
''606. {{Overline|6}}''
''420''
| rowspan="2" |10\11
| rowspan="2" |7\11
   
   
''636.{{Overline|36}}''
''445.{{Overline|45}}''
|17\18
|12\18
   
   
''661.{{Overline|1}}''
''466.{{Overline|6}}''
|7\7
|5\7
   
   
''700''
''500''
|18\17
|13\17
   
   
''741.176…''
''535.294…''
|11\10
|8\10
   
   
''770''
''560''
|15\13
|11\13
   
   
''807.692…''
''592.307…''
|-
|-
|Dob, Solb
|Fab, Dob
|Hb
|Bbb
|4b, 4c
|3b, 3c
|14\15
|3f
|10\15
   
   
''653.{{Overline|3}}''
''466.{{Overline|6}}''
|16\18
|11\18
   
   
''622.{{Overline|2}}''
''427.{{Overline|7}}''
|6\7
|4\7
   
   
''600''
''400''
| 14\17
|9\17
   
   
''576.470…''
''370.588…''
| 8\10
|5\10
   
   
''560''
''350''
|10\13
|6\13
   
   
''538.461…''
''323.076.…''
|-
|-
!Do, Sol
|'''Fa, Do'''
!H
|'''Bb'''
!4
|'''3'''
! colspan="7" |''700''
|'''3'''
|-
|'''11\15'''
|Do#, Sol#
|Η#
|4#
|16\15
   
   
''746.{{Overline|6}}''
'''''513.{{Overline|3}}'''''
|12\11
|'''8\11'''
   
   
''763.{{Overline|63}}''
'''''509.{{Overline|09}}'''''
|20\18
|'''13\18'''
   
   
''777.{{Overline|7}}''
'''''505.{{Overline|5}}'''''
| rowspan="2" |8\7
|'''5\7'''
   
   
''800''
'''''500'''''
|20\17
|'''12\17'''
   
   
''823.529…''
'''''494.117…'''''
|12\10
|'''7\10'''
   
   
''840''
'''''490'''''
|16\13
|'''9\13'''
   
   
''861.538…''
'''''484.615…'''''
|-
|-
|Reb, Lab
|Fa#, Do#
|Cb
| B
|5b, 5c
|3#
|18\15
|3#
|12\15
   
   
''840''
''560''
|13\11
|9\11
''572.{{Overline|72}}''
| 15\18
   
   
''827.{{Overline|27}}''
''583.{{Overline|3}}''
|21\18
|6\7
   
   
''816.{{Overline|6}}''
''600''
| 19\17
|15\17
   
   
''782.352…''
''617.647…''
|11\10
|9\10
   
   
''770''
''630''
|14\13
|12\13
   
   
''753.846…''
''646.153…''
|-
|-
|'''Re, La'''
| Fax, Dox
|'''C'''
|B#
|'''5'''
|3x
|'''19\15'''
|3x
   
|13\15
'''''886.{{Overline|6}}'''''
   
|'''14\11'''
''606. {{Overline|6}}''
| rowspan="2" |10\11
   
   
'''''890.{{Overline|90}}'''''
''636.{{Overline|36}}''
|'''23\18'''
|17\18
   
   
'''''894.{{Overline|4}}'''''
''661.{{Overline|1}}''
|'''9\7'''
|7\7
   
   
'''''900'''''
''700''
|'''22\17'''
|18\17
   
   
'''''905.882…'''''
''741.176…''
|'''13\10'''
|11\10
   
   
'''''910'''''
''770''
|'''17\13'''
|15\13
   
   
'''''915.384…'''''
''807.692…''
|-
|-
| Re#, La#
|Dob, Solb
|C#
|Hb
|5#
|4b, 4c
|20\15
|4f
|14\15
   
   
''933.{{Overline|3}}''
''653.{{Overline|3}}''
|15\11
|16\18
   
   
''954.{{Overline|54}}''
''622.{{Overline|2}}''
|25\18
|6\7
   
   
''972.{{Overline|2}}''
''600''
| rowspan="2" | 10\7
| 14\17
   
   
''1000''
''576.470…''
|25\17
| 8\10
   
   
''1029.411…''
''560''
|15\10
|10\13
   
   
''1050''
''538.461…''
|20\13
''1076.923…''
|-
|-
|Mib, Sib
!Do, Sol
|Db
!H
|6b, 6c
!4
|22\15
!4
! colspan="7" |''700''
|-
|Do#, Sol#
|Η#
|4#
|4#
|16\15
   
   
''1026.{{Overline|6}}''
''746.{{Overline|6}}''
|16\11
|12\11
   
   
''1018.{{Overline|18}}''
''763.{{Overline|63}}''
|26\18
|20\18
   
   
''1011. {{Overline|1}}''
''777.{{Overline|7}}''
|24\17
| rowspan="2" |8\7
   
   
''988.235…''
''800''
|14\10
|20\17
   
   
''980''
''823.529…''
|18\13
|12\10
   
   
''969.230…''
''840''
|16\13
''861.538…''
|-
|-
|Mi, Si
|Reb, Lab
|D
|Cb
|6
|5b, 5c
| 23\15
|5
|18\15
   
   
''1073.{{Overline|3}}''
''840''
|17\11
|13\11
   
   
''1081.{{Overline|81}}''
''827.{{Overline|27}}''
|28\18
|21\18
   
   
''1088.{{Overline|8}}''
''816.{{Overline|6}}''
|11\7
| 19\17
   
   
''1100''
''782.352…''
|27\17
|11\10
   
   
''1111.764…''
''770''
|16\10
|14\13
   
   
''1120''
''753.846…''
|21\13
|-
|'''Re, La'''
|'''C'''
|'''5'''
|'''5'''
|'''19\15'''
   
   
''1130.769…''
'''''886.{{Overline|6}}'''''
|-
|'''14\11'''
|Mi#, Si#
| D#
|6#
|24\15
   
   
''1120''
'''''890.{{Overline|90}}'''''
| rowspan="2" | 18\11
|'''23\18'''
   
   
''1145.{{Overline|45}}''
'''''894.{{Overline|4}}'''''
|30\18
|'''9\7'''
   
   
''1166.{{Overline|6}}''
'''''900'''''
|12\7
|'''22\17'''
   
   
''1200''
'''''905.882…'''''
| 30\17
|'''13\10'''
   
   
''1235.294…''
'''''910'''''
|18\10
|'''17\13'''
   
   
''1260''
'''''915.384…'''''
|24\13
|-
| Re#, La#
|C#
|5#
|5#
|20\15
   
   
''1292.307…''
''933.{{Overline|3}}''
|-
|15\11
| Fab, Dob
| Ebb
| 7b, 7c
|25\15
   
   
''1166.{{Overline|6}}''
''954.{{Overline|54}}''
|29\18
|25\18
   
   
''1127.{{Overline|7}}''
''972.{{Overline|2}}''
|11\7
| rowspan="2" | 10\7
   
   
''1100''
''1000''
|26\17
|25\17
   
   
''1070.588…''
''1029.411…''
|15\10
|15\10
   
   
''1050''
''1050''
|19\13
|20\13
   
   
''1023.076…''
''1076.923…''
|-
|-
|'''Fa, Do'''
|Mib, Sib
|'''Eb'''
|Db
|'''7'''
|6b, 6c
|'''26\15'''
|6f
|22\15
   
   
'''''1213.{{Overline|3}}'''''
''1026.{{Overline|6}}''
|'''19\11'''
|16\11
   
   
'''''1209.{{Overline|09}}'''''
''1018.{{Overline|18}}''
|'''31\18'''
|26\18
   
   
'''''1205.{{Overline|5}}'''''
''1011.{{Overline|1}}''
|'''12\7'''
|24\17
   
   
'''''1200'''''
''988.235…''
|'''29\17'''
|14\10
   
   
'''''1194.117…'''''
''980''
|'''17\10'''
|18\13
   
   
'''''1190'''''
''969.230…''
|'''22\13'''
|-
|Mi, Si
|D
|6
|6
| 23\15
   
   
'''''1184.615…'''''
''1073.{{Overline|3}}''
|-
|17\11
|Fa#, Do#
|E
|7#
|27\15
   
   
''1260''
''1081.{{Overline|81}}''
|20\11
|28\18
   
   
''1272.{{Overline|72}}''
''1088.{{Overline|8}}''
| 33\18
|11\7
   
   
''1283.{{Overline|3}}''
''1100''
|13\7
|27\17
   
   
''1300''
''1111.764…''
|32\17
|16\10
   
   
''1317.647…''
''1120''
|19\10
|21\13
   
   
''1330''
''1130.769…''
| 25\13
''1346.153…''
|-
|-
|Fax, Dox
|Mi#, Si#
|E#
| D#
|7x
|6#
|28\15
|6#
|24\15
   
   
''1306.{{Overline|6}}''
''1120''
| rowspan="2" |21\11
| rowspan="2" | 18\11
   
   
''1336.{{Overline|36}}''
''1145.{{Overline|45}}''
|35\18
|30\18
   
   
''1361.{{Overline|1}}''
''1166.{{Overline|6}}''
|14\7
|12\7
   
   
''1400''
''1200''
|35\17
| 30\17
   
   
''1441.176…''
''1235.294…''
|21\10
|18\10
   
   
''1470''
''1260''
|28\13
|24\13
   
   
''1507.692…''
''1292.307…''
|-
|-
|Dob, Solb
| Fab, Dob
|Fb
| Ebb
|8b, Fc
| 7b, 7c
|29\15
|7f
|25\15
   
   
''1333.{{Overline|3}}''
''1166.{{Overline|6}}''
|34\18
|29\18
   
   
''1322.{{Overline|2}}''
''1127.{{Overline|7}}''
|13\7
|11\7
   
   
''1300''
''1100''
|31\17
|26\17
   
   
''1276.470…''
''1070.588…''
|18\10
|15\10
   
   
''1260''
''1050''
|23\13
|19\13
   
   
''1238.461…''
''1023.076…''
|-
|-
!Do, Sol
|'''Fa, Do'''
!F
|'''Eb'''
!8, F
|'''7'''
! colspan="7" |''1400''
|'''7'''
|-
|'''26\15'''
|Do#, Sol#
|F#
|8#, F#
|31\15
   
   
''1446.{{Overline|6}}''
'''''1213.{{Overline|3}}'''''
|23\11
|'''19\11'''
   
   
''1463.{{Overline|63}}''
'''''1209.{{Overline|09}}'''''
|38\18
|'''31\18'''
   
   
''1477.7̄''
'''''1205.{{Overline|5}}'''''
| rowspan="2" |15\7
|'''12\7'''
   
   
''1500''
'''''1200'''''
|37\17
|'''29\17'''
   
   
''1523.529…''
'''''1194.117…'''''
|22\10
|'''17\10'''
   
   
''1540''
'''''1190'''''
| 29\13
|'''22\13'''
   
   
''1561.538…''
'''''1184.615…'''''
|-
|-
|Reb, Lab
|Fa#, Do#
|Gb
|E
| 9b, Gc
|7#
|33\15
|7#
|27\15
   
   
''1540''
''1260''
|24\11
|20\11
   
   
''1527.{{Overline|27}}''
''1272.{{Overline|72}}''
|39\18
| 33\18
   
   
''1516.{{Overline|6}}''
''1283.{{Overline|3}}''
| 36\17
|13\7
   
   
''1482.352…''
''1300''
|21\10
|32\17
''1317.647…''
|19\10
   
   
''1470''
''1330''
|27\13
| 25\13
   
   
''1453.846…''
''1346.153…''
|-
|-
|'''Re, La'''
|Fax, Dox
|'''G'''
|E#
|'''9, G'''
|7x
|'''34\15'''
|7x
|28\15
   
   
'''''1586.{{Overline|6}}'''''
''1306.{{Overline|6}}''
|'''25\11'''
| rowspan="2" |21\11
   
   
'''''1590.{{Overline|90}}'''''
''1336.{{Overline|36}}''
|'''41\18'''
|35\18
   
   
'''''1594.{{Overline|4}}'''''
''1361.{{Overline|1}}''
|'''16\7'''
|14\7
   
   
'''''1600'''''
''1400''
|'''39\17'''
|35\17
   
   
'''''1605.882…'''''
''1441.176…''
|'''23\10'''
|21\10
   
   
'''''1610'''''
''1470''
|'''30\13'''
|28\13
   
   
'''''1615.384…'''''
''1507.692…''
|-
|-
|Re#, La#
|Dob, Solb
|G#
|Fb
|9#, G#
|8b, Fc
|35\15
|8f
|29\15
   
   
''1633.{{Overline|3}}''
''1333.{{Overline|3}}''
|26\11
|34\18
   
   
''1654.{{Overline|54}}''
''1322.{{Overline|2}}''
|43\18
|13\7
   
   
''1672.{{Overline|2}}''
''1300''
| rowspan="2" |17\7
|31\17
   
   
''1700''
''1276.470…''
|42\17
|18\10
   
   
''1729.411…''
''1260''
|25\10
|23\13
   
   
''1750''
''1238.461…''
|33\13
|-
!Do, Sol
''1776.923…''
!F
!8, F
!8
! colspan="7" |''1400''
|-
|-
|Mib, Sib
|Do#, Sol#
| Ab
|F#
|Xb, Ac
|8#, F#
|37\15
|8#
|31\15
   
   
''1726.{{Overline|6}}''
''1446.{{Overline|6}}''
| 27\11
|23\11
   
   
''1718.{{Overline|18}}''
''1463.{{Overline|63}}''
|44\18
|38\18
''1477.7̄''
| rowspan="2" |15\7
   
   
''1711.{{Overline|1}}''
''1500''
|41\17
|37\17
   
   
''1688.235…''
''1523.529…''
| 24\10
|22\10
   
   
''1680''
''1540''
|31\13
| 29\13
   
   
''1669.230…''
''1561.538…''
|-
|-
|Mi, Si
|Reb, Lab
|A
|Gb
|X, A
| 9b, Gc
|38\15
|9f
|33\15
   
   
''1773.{{Overline|3}}''
''1540''
|28\11
|24\11
   
   
''1781.{{Overline|81}}''
''1527.{{Overline|27}}''
|46\18
|39\18
   
   
''1788.{{Overline|8}}''
''1516.{{Overline|6}}''
|18\7
| 36\17
   
   
''1800''
''1482.352…''
| 44\17
|21\10
   
   
''1811.764…''
''1470''
|26\10
|27\13
   
   
''1820''
''1453.846…''
|34\13
''1830.769…''
|-
|-
|Mi#, Si#
|'''Re, La'''
|A#
|'''G'''
|X#, A#
|'''9, G'''
|39\15
|9
|'''34\15'''
   
   
''1820''
'''''1586.{{Overline|6}}'''''
| rowspan="2" |29\11
|'''25\11'''
   
   
''1845.{{Overline|45}}''
'''''1590.{{Overline|90}}'''''
|48\18
|'''41\18'''
   
   
''1866.{{Overline|6}}''
'''''1594.{{Overline|4}}'''''
|19\7
|'''16\7'''
   
   
''1900''
'''''1600'''''
|47\17
|'''39\17'''
   
   
''1935.294…''
'''''1605.882…'''''
|28\10
|'''23\10'''
   
   
''1960''
'''''1610'''''
| 37\13
|'''30\13'''
   
   
''1992.307…''
'''''1615.384…'''''
|-
|-
|Fab, Dob
|Re#, La#
|Bbb
|G#
|Ebb, Ccc
|9#, G#
|40\15
|9#
|35\15
   
   
''1866.{{Overline|6}}''
''1633.{{Overline|3}}''
|47\18
|26\11
   
   
''1827.{{Overline|7}}''
''1654.{{Overline|54}}''
|18\7
|43\18
''1672.{{Overline|2}}''
| rowspan="2" |17\7
   
   
''1800''
''1700''
|43\17
|42\17
   
   
''1770.588…''
''1729.411…''
|25\10
|25\10
   
   
''1750''
''1750''
|32\13
|33\13
   
   
''1723.076…''
''1776.923…''
|-
|-
|'''Fa, Do'''
|Mib, Sib
|'''Bb'''
| Ab
|Eb, Cc
|Xb, Ac
|'''41\15'''
|Af
|37\15
   
   
'''''1913.{{Overline|3}}'''''
''1726.{{Overline|6}}''
|'''30\11'''
| 27\11
   
   
'''''1909.{{Overline|09}}'''''
''1718.{{Overline|18}}''
|'''49\18'''
|44\18
   
   
'''''1905.{{Overline|5}}'''''
''1711.{{Overline|1}}''
|'''19\7'''
|41\17
   
   
'''''1900'''''
''1688.235…''
|'''46\17'''
| 24\10
   
   
'''''1894.117…'''''
''1680''
|'''27\10'''
|31\13
   
   
'''''1890'''''
''1669.230…''
|'''35\13'''
'''''1884.615…'''''
|-
|-
|Fa#, Do#
|Mi, Si
|B
|A
|E, C
|X, A
| 42\15
|A
|38\15
   
   
''1960''
''1773.{{Overline|3}}''
|31\11
|28\11
   
   
''1972.{{Overline|72}}''
''1781.{{Overline|81}}''
|51\18
|46\18
   
   
''1983.{{Overline|3}}''
''1788.{{Overline|8}}''
|20\7
|18\7
   
   
''2000''
''1800''
|49\17
| 44\17
   
   
''2017.647…''
''1811.764…''
|29\10
|26\10
   
   
''2030''
''1820''
|38\13
|34\13
   
   
''2046.153…''
''1830.769…''
|-
|-
|Fax, Dox
|Mi#, Si#
|B#
|A#
|Ex, Cx
|X#, A#
|43\15
|A#
|39\15
   
   
''2006.{{Overline|6}}''
''1820''
| rowspan="2" |32\11
| rowspan="2" |29\11
   
   
''2036.{{Overline|36}}''
''1845.{{Overline|45}}''
|53\18
|48\18
   
   
''2061. {{Overline|1}}''
''1866.{{Overline|6}}''
|21\7
|19\7
   
   
''2100''
''1900''
|52\17
|47\17
   
   
''2141.176…''
''1935.294…''
|31\10
|28\10
   
   
''2170''
''1960''
|41\13
| 37\13
   
   
''2207.692…''
''1992.307…''
|-
|-
| Dob, Solb
|Fab, Dob
|Hb
|Bbb
|0b, Dc
|Ebb, Ccc
|44\15
|Bf
|40\15
   
   
''2053.{{Overline|3}}''
''1866.{{Overline|6}}''
|52\18
|47\18
   
   
''2022.{{Overline|2}}''
''1827.{{Overline|7}}''
|20\7
|18\7
   
   
''2000''
''1800''
|48\17
|43\17
   
   
''1976.470…''
''1770.588…''
|28\10
|25\10
   
   
''1960''
''1750''
| 36\13
|32\13
   
   
1938.615…
''1723.076…''
|-
|-
!Do, Sol
|'''Fa, Do'''
!H
|'''Bb'''
|Eb, Cc
|'''B'''
|'''41\15'''
'''''1913.{{Overline|3}}'''''
|'''30\11'''
'''''1909.{{Overline|09}}'''''
|'''49\18'''
'''''1905.{{Overline|5}}'''''
|'''19\7'''
'''''1900'''''
|'''46\17'''
'''''1894.117…'''''
|'''27\10'''
'''''1890'''''
|'''35\13'''
'''''1884.615…'''''
|-
|Fa#, Do#
|B
|E, C
|B#
| 42\15
''1960''
|31\11
''1972.{{Overline|72}}''
|51\18
''1983.{{Overline|3}}''
|20\7
''2000''
|49\17
''2017.647…''
|29\10
''2030''
|38\13
''2046.153…''
|-
|Fax, Dox
|B#
|Ex, Cx
|Bx
|43\15
''2006.{{Overline|6}}''
| rowspan="2" |32\11
''2036.{{Overline|36}}''
|53\18
''2061.{{Overline|1}}''
|21\7
''2100''
|52\17
''2141.176…''
|31\10
''2170''
|41\13
''2207.692…''
|-
| Dob, Solb
|Hb
|0b, Dc
|Cf
|44\15
''2053.{{Overline|3}}''
|52\18
''2022.{{Overline|2}}''
|20\7
''2000''
|48\17
''1976.470…''
|28\10
''1960''
| 36\13
''1938.615…''
|-
!Do, Sol
!H
!0, D
!0, D
! colspan="7" |2100
!C
! colspan="7" |''2100''
|-
|Do#, Sol#
|Η#
|0#, D#
|C#
|46\15
''2146.{{Overline|6}}''
|34\11
''2163.{{Overline|63}}''
|56\18
''2177.{{Overline|7}}''
| rowspan="2" |22\7
''2200''
|54\17
''2223.529…''
|32\10
''2240''
|42\13
''2261.538…''
|-
|Reb, Lab
|Cb
|1b, 1c
|Df
|48\15
''2240''
|35\11
''2227.{{Overline|27}}''
|57\18
''2216.{{Overline|6}}''
|53\17
''2182.352…''
|31\10
''2170''
|40\13
''2153.846…''
|-
|'''Re, La'''
|'''C'''
|'''1'''
|'''D'''
|'''49\15'''
'''''2286.{{Overline|6}}'''''
|'''36\11'''
'''''2290.{{Overline|90}}'''''
|'''59\18'''
'''''2294.{{Overline|4}}'''''
|'''23\7'''
'''''2300'''''
|'''56\17'''
'''''2305.882…'''''
|'''33\10'''
'''2310'''
|'''43\13'''
'''''2315.384…'''''
|-
|Re#, La#
|C#
|1#
|D#
|50\15
''2223.{{Overline|3}}''
|37\11
''2354.{{Overline|54}}''
|61\18
''2372.''{{Overline|2}}
| rowspan="2" |24\7
''2400''
|59\17
''2429.411…''
|35\10
''2450''
|46\13
''2476.923…''
|-
|Mib, Sib
|Db
|2b, 2c
|Ef
|52\15
''2426.{{Overline|6}}''
|38\11
''2418.{{Overline|18}}''
|62\18
''2411.{{Overline|1}}''
|58\17
''2388.235…''
|34\10
''2380''
|44\13
''2369.230…''
|-
|Mi, Si
|D
|2
|E
|53\15
''2473,{{Overline|3}}''
|39\11
''2481.{{Overline|81}}''
|64\11
''2488.{{Overline|8}}''
|25\7
''2500''
|61\17
''2511.764…''
|36\10
''2520''
|47\13
''2530.769…''
|-
|Mi#, Si#
|D#
|2#
|E#
|54\15
''2520''
| rowspan="2" |40\11
''2545.{{Overline|45}}''
|66\18
''2566.{{Overline|6}}''
|26\7
''2600''
|64\17
''2635.294…''
|38\10
''2660''
|50\13
''2692.307…''
|-
|Fab, Dob
|Ebb
|3b, 3c
|Fff
|55\15
''2566.{{Overline|6}}''
|65\18
''2527.{{Overline|7}}''
|25\7
''2500''
|60\17
''2470.588…''
|35\10
''2450''
|45\13
''2423.076…''
|-
|'''Fa, Do'''
|'''Eb'''
|'''3'''
|'''Ff'''
|'''56\15'''
'''''2613.{{Overline|3}}'''''
|'''41\11'''
'''''2609.{{Overline|09}}'''''
|'''67\18'''
'''''2605.{{Overline|5}}'''''
|'''26\7'''
'''''2600'''''
|'''63\17'''
'''''2594.117…'''''
|'''37\10'''
'''''2590'''''
|'''48\13'''
'''''2584.615…'''''
|-
|Fa#, Do#
|E
|3#
|F
|57\15
''2660''
|42\11
''2672.{{Overline|72}}''
|69\18
''2683.{{Overline|3}}''
|27\7
''2700''
|66\17
''2717.647…''
|39\10
''2730''
|51\13
''2746.153…''
|-
|Fax, Dox
|E#
|3x
|F#
|58\15
''2706.{{Overline|6}}''
| rowspan="2" |43\11
''2736.{{Overline|36}}''
|71\18
''2761.{{Overline|1}}''
|28\7
''2800''
|69\17
''2841.176…''
|41\10
''2870''
|54\13
''2907.692…''
|-
|Dob, Solb
|Fb
|4b, 4c
|0f, Gf
|59\15
''2753.{{Overline|3}}''
|70\18
''2722.{{Overline|2}}''
|27\7
''2700''
|65\17
''2676.470…''
|38\10
''2660''
|49\13
''2638.615…''
|-
!Do, Sol
!F
!4
!0, G
! colspan="7" |''2800''
|}
|}
   
   

Revision as of 03:21, 17 July 2022

45Lua 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.

Basic Angel is in 7edf, which is a very good fifth-based equal tuning similar to 12edo.

Notation

There are 4 main ways to notate the angel scale. One method uses a simple sesquitave (fifth) repeating notation consisting of 4 naturals (eg. Do Re Mi Fa, Sol La Si Do). Given that 1-5/4-5/3 is fifth-equivalent to a tone cluster of 1-10/9-5/4, it may be more convenient to notate angel scales as repeating at the double, triple or quadruple sesquitave (major ninth, thirteenth or seventeenth i. e. ~pentave), however it does make navigating the genchain harder. This way, 5/3 is its own pitch class, distinct from 10/9. Notating this way produces a major ninth which is the Aeolian mode of Napoli[6L 2s], a major thirteenth which is the Dorian mode of Bijou[9L 3s] or an ~pentave which is the Mixolydian mode of Hextone[12L 4s]. Since there are exactly 8 naturals in double sesquitave notation, 12 in triple sesquitave notation and 16 in quadruple sesquitave notation, letters A-H (FGABHCDEF) or dozenal or hex digits (0123456789XE0 or D1234567FGACD with flats written C molle or 0123456789ABCDEF0 or G123456789ABCDEFG with flats written F molle) may be used.

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

46.153…

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 1f 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 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# 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 2f 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 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# 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 3f 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 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# 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 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 4f 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 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# 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 5 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 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# 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 6f 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 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# 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 7f 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 7 26\15

1200

19\11

1200

31\18

1200

12\7

1200

29\17

1200

17\10

1200

22\13

1200

Fa#, Do# E 7# 7# 27\15

1246.153…

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 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 8f 29\15

1338.461…

34\18

1316.129…

13\7

1300

31\17

1282.758…

18\10

1270.588…

23\13

1254.54

Do, Sol F 8, F 8 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# 8# 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 9f 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 9 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# 9# 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 Af 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 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# 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 Bf 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 B 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 B# 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 Bx 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 Cf 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 C 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

Do#, Sol# Η# 0#, D# C# 46\15

2123.076…

34\11

2147.368…

56\18

2167.741…

22\7

2200

54\17

2234.582…

32\10

2258.823…

42\13

2090.90

Reb, Lab Cb 1b, 1c Df 48\15

2215.384…

35\11

2210.526…

57\18

2206.451…

53\17

2193.103…

31\10

2188.235…

40\13

2181.81

Re, La C 1 D 49\15

2261.538…

36\11

2273.684…

59\18

2283.870…

23\7

2300

56\17

2317.241…

33\10

2329.411…

43\13

2245.45

Re#, La# C# 1# D# 50\15

2307.692…

37\11

2336.842…

61\18

2361.290…

24\7

2400

59\17

2441.379…

35\10

2470.588…

46\13

2509.09

Mib, Sib Db 2b, 2c Ef 52\15

2400

38\11

2400

62\18

2400

58\17

2400

34\10

2400

44\13

2400

Mi, Si D 2 E 53\15

2446.153…

39\11

2463.158…

64\18

2477.419…

25\7

2500

61\17

2524.137…

36\10

2541.176…

47\13

2563.63

Mi#, Si# D# 2# E# 54\15

2492.307…

40\11

2526.315…

66\18

2554.838…

26\7

2600

64\17

2648.275…

38\10

2682.352…

50\13

2727.27

Fab, Dob Ebb 3b, 3c Fff 55\15

2538.461…

65\18

2516.129…

25\7

2500

60\17

2482.758…

35\10

2470.588…

45\13

2454.54

Fa, Do Eb 3 Ff 56\15

2584.615…

41\11

2589.473…

67\18

2593.548…

26\7

2600

63\17

2606.896…

37\10

2611.764…

48\13

2618.18

Fa#, Do# E 3# F 57\15

2630.769…

42\11

2652.632…

69\18

2670.967…

27\7

2700

66\17

2731.034…

39\10

2752.941…

51\13

2781.81

Fax, Dox E# 3x F# 58\15

2676.923…

43\11

2715.789…

71\18

2748.387…

28\7

2800

69\17

2855.168…

41\10

2894.117…

54\13

2945.45

Dob, Solb Fb 4b, 4c 0f, Gf 59\15

2723.076…

70\18

2709.677…

27\7

2700

65\17

2689.655…

38\10

2682.352…

49\13

2672.72

Do, Sol F 4 0, G 60\15

2769.230…

44\11

2778.947…

72\18

2787.097…

28\7

2800

68\17

2813.793…

40\10

2823.529…

52\113

2836.36

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

46.6

1\11

63.63

2\18

77.7̄

1\7

100

3\17

123.529…

2\10

140

3\13

161.538…

Reb, Lab Gb 1b, 1c 1f 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 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# 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 2f 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 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# 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 3f 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 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# 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 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 4f 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 4 700
Do#, Sol# Η# 4# 4# 16\15

746.6

12\11

763.63

20\18

777.7

8\7

800

20\17

823.529…

12\10

840

16\13

861.538…

Reb, Lab Cb 5b, 5c 5 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 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# 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 6f 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 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# 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 7f 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 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# 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 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 8f 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 8 1400
Do#, Sol# F# 8#, F# 8# 31\15

1446.6

23\11

1463.63

38\18

1477.7̄

15\7

1500

37\17

1523.529…

22\10

1540

29\13

1561.538…

Reb, Lab Gb 9b, Gc 9f 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 9 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# 9# 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 Af 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 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# 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 Bf 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 B 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 B# 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 Bx 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 Cf 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 C 2100
Do#, Sol# Η# 0#, D# C# 46\15

2146.6

34\11

2163.63

56\18

2177.7

22\7

2200

54\17

2223.529…

32\10

2240

42\13

2261.538…

Reb, Lab Cb 1b, 1c Df 48\15

2240

35\11

2227.27

57\18

2216.6

53\17

2182.352…

31\10

2170

40\13

2153.846…

Re, La C 1 D 49\15

2286.6

36\11

2290.90

59\18

2294.4

23\7

2300

56\17

2305.882…

33\10

2310

43\13

2315.384…

Re#, La# C# 1# D# 50\15

2223.3

37\11

2354.54

61\18

2372.2

24\7

2400

59\17

2429.411…

35\10

2450

46\13

2476.923…

Mib, Sib Db 2b, 2c Ef 52\15

2426.6

38\11

2418.18

62\18

2411.1

58\17

2388.235…

34\10

2380

44\13

2369.230…

Mi, Si D 2 E 53\15

2473,3

39\11

2481.81

64\11

2488.8

25\7

2500

61\17

2511.764…

36\10

2520

47\13

2530.769…

Mi#, Si# D# 2# E# 54\15

2520

40\11

2545.45

66\18

2566.6

26\7

2600

64\17

2635.294…

38\10

2660

50\13

2692.307…

Fab, Dob Ebb 3b, 3c Fff 55\15

2566.6

65\18

2527.7

25\7

2500

60\17

2470.588…

35\10

2450

45\13

2423.076…

Fa, Do Eb 3 Ff 56\15

2613.3

41\11

2609.09

67\18

2605.5

26\7

2600

63\17

2594.117…

37\10

2590

48\13

2584.615…

Fa#, Do# E 3# F 57\15

2660

42\11

2672.72

69\18

2683.3

27\7

2700

66\17

2717.647…

39\10

2730

51\13

2746.153…

Fax, Dox E# 3x F# 58\15

2706.6

43\11

2736.36

71\18

2761.1

28\7

2800

69\17

2841.176…

41\10

2870

54\13

2907.692…

Dob, Solb Fb 4b, 4c 0f, Gf 59\15

2753.3

70\18

2722.2

27\7

2700

65\17

2676.470…

38\10

2660

49\13

2638.615…

Do, Sol F 4 0, G 2800

Intervals

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

Genchain

The generator chain for this scale is as follows:

Mibb

Sibb

Fab

Dob

Dob

Solb

Reb

Lab

Mib

Sib

Fa

Do

Do

Sol

Re

La

Mi

Si

Fa#

Do#

Do#

Sol#

Re#

La#

Mi#

Si#

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

Modes

The mode names are based on the species of fifth:

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

Temperaments

The most basic rank-2 temperament interpretation of 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 = 192.6406

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 = 218.6371

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
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