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The generator range is 171.4 to 240 cents, placing it on the diatonic major second, usually representing a major second of some type (like [[8/7]]). The bright (chroma-positive) generator is, however, its major sixth complement (685.7 to 720 cents).
The generator range is 171.4 to 240 cents, placing it on the diatonic major second, usually representing a major second of some type (like [[8/7]]). The bright (chroma-positive) generator is, however, its major sixth complement (685.7 to 720 cents).


Because this diatonic is a major sixth-repeating scale, each tone has a 5/3 major sixth above it. The scale has one augmented chord, two major chords, two minor chords. This diatonic also has two dominant 7th chords, making it a warped Neapolitan minor scale.
Because this diatonic is a major sixth-repeating scale, each tone has a major sixth above it. The scale has one augmented chord, two major chords, two minor chords. This diatonic also has two dominant 7th chords, making it a warped Neapolitan minor scale.


[[Basic]] diatonic is in [[9ed5/3]], which is a very good major sixth-based equal tuning similar to [[12edo]].
[[Basic]] diatonic is in [[9ed5/3]], which is a very good major sixth-based equal tuning similar to [[12edo]].
==Notation==
==Notation==


There are 2 main ways to notate the diatonic scale. One method uses a simple sextave (major sixth) repeating notation consisting of 5 naturals (Do, Re, Mi, Fa, Sol or Sol, La, Si, Do, Re). Given that 1-5/4-3/2 is major sixth-equivalent to a tone cluster of 1-10/9-5/4, it may be more convenient to notate these diatonic scales as repeating at the double sextave (augmented eleventh~twelfth), however it does make navigating the [[Generator|genchain]] harder. This way, 3/2 is its own pitch class, distinct from 10\9. Notating this way produces a twelfth which is the Scala Francisci[8L 2s]. Since there are exactly 10 naturals in double sextave notation, Greek numerals 1-10 may be used.
There are 2 main ways to notate the diatonic scale. One method uses a simple sextave (major sixth) repeating notation consisting of 5 naturals (Do, Re, Mi, Fa, Sol;  Fa, Sol, La, Si, Do or Sol, La, Si, Do, Re). Given that 1-5/4-3/2 is major sixth-equivalent to a tone cluster of 1-10/9-5/4, it may be more convenient to notate these diatonic scales as repeating at the double sextave (augmented eleventh~twelfth), however it does make navigating the [[Generator|genchain]] harder. This way, 3/2 is its own pitch class, distinct from 10\9. Notating this way produces a twelfth which is the Scala Francisci[8L 2s]. Since there are exactly 10 naturals in double sextave notation, Greek numerals 1-10 may be used.
{| class="wikitable"
{| class="wikitable"
|+Normalized
|+Normalized
! colspan="2" |Notation
!Notation
!Supersoft
!Supersoft
!Soft
!Soft
Line 21: Line 21:
|-
|-
!Diatonic
!Diatonic
!Scala Francisci
!19eds
!19eds
!14eds
!14eds
Line 30: Line 29:
!17eds
!17eds
|-
|-
|Do#, Sol#
|Do#, Fa#, Sol#
|Α#
|1\19, 46.154¢
|1\19, 46.154¢
|1\14, 63.158¢
|1\14, 63.158¢
Line 38: Line 36:
|3\22, 124.138¢
|3\22, 124.138¢
|2\13, 141.176¢
|2\13, 141.176¢
|3\17, 163.{{Overline|63}}¢
|3\17, 163.636¢
|-
|Reb, Solb, Lab
|3\19, 138.462¢
|2\14, 126.316¢
|3\23, 116.129¢
|2\22, 82.759¢
|1\13, 70.588¢
|1\17, 54.545¢
|-
|'''Re, Sol, La'''
|'''4\19,''' '''184.615¢'''
|'''3\14,''' '''189.474¢'''
|'''5\23,''' '''193.548¢'''
|'''2\9,''' '''200¢'''
|'''5\22,''' '''206.897¢'''
|'''3\13,''' '''211.765¢'''
|'''4\17,''' '''218.182¢'''
|-
|Re#, Sol#, La#
|5\19, 230.769¢
|4\14, 252.632¢
|7\23, 270.968¢
| rowspan="2" |3\9, 300¢
|8\22, 331.034¢
|5\13, 352.941¢
|7\17, 381.818¢
|-
|Mib, Lab, Sib
|7\19, 323.077¢
|5\14, 315.789¢
|8\23, 309.677¢
|7\22, 289.655¢
|4\13, 282.353¢
|5\17, 272.727¢
|-
|Mi, La, Si
|8\19, 369.231¢
|6\14, 378.947¢
|10\23, 387.097¢
|4\9, 400¢
|10\22, 413.793¢
|6\13, 423.529¢
|8\17, 436.36&¢
|-
|Mi#, La#, Si#
|9\19, 415.385¢
| rowspan="2" |7\14, 442.105¢
|12\23, 464.516¢
|5\9, 500¢
|13\22, 537.931¢
|8\13, 564.706¢
|11\17, 600¢
|-
|Fab, Sibb, Dob
|10\19, 461.538¢
|11\23, 425.806¢
|4\9, 400¢
|9\22, 372.414¢
|5\13, 352.941¢
|6\17, 327.273¢
|-
|Fa, Sib, Do
|11\19, 507.692¢
|8\14, 505.263¢
|13\23, 503.226¢
|5\9, 500¢
|12\22, 496.552¢
|7\13, 494.118¢
|9\17, 490.909¢
|-
|Fa#, Si, Do#
|12\19, 553.846¢
|9\14, 568.421¢
|15\23, 580.645¢
| rowspan="2" |6\9, 600¢
|15\22, 620.690¢
|9\13, 635.294¢
|12\17, 654.545¢
|-
|Solb, Dob,  Reb
|14\19, 646.154¢
|10\14, 631.579¢
|16\23, 619.355¢
|14\22, 579.310¢
|8\13, 564.706¢
|10\17, 545.455¢
|-
|'''Sol, Do, Re'''
|'''15\19,''' '''692.308¢'''
|'''11\14,''' '''694.737¢'''
|'''18\23,''' '''696.774¢'''
|'''7\9,''' '''700¢'''
|'''17\22,''' '''703.448¢'''
|'''10\13,''' '''705.882¢'''
|'''13\17,''' '''709.091¢'''
|-
|Sol#, Do#, Re#
|16\19, 738.462¢
|12\14, 757.895¢
|20\23, 774.194¢
| rowspan="2" |8\9, 800¢
|20\22, 827.586¢
|12\13, 847.059¢
|16\17, 872.727¢
|-
|Dob, Fab, Solb
|18\19, 830.769¢
|13\14, 821.053¢
|21\23, 812.903¢
|19\22, 786.207¢
|11\13, 776.647¢
|14\17, 763.636¢
|-
!Do, Fa, Sol
!19\19, 876.923¢
!14\14, 884.211¢
!23\23, 890.323¢
!9\9, 900¢
!22\22, 910.345¢
!13\13, 917.647¢
!17\17, 927.273¢
|}
 
{| class="wikitable"
|+Normalized
!Notation
!Supersoft
!Soft
!Semisoft
!Basic
!Semihard
!Hard
! Superhard
|-
!Scala Francisci
!19eds
!14eds
!23eds
!9eds
!22eds
!13eds
! 17eds
|-
| Α#
| 1\19, 46.154¢
|1\14, 63.158¢
|2\23, 77.419¢
| rowspan="2" |1\9, 100¢
| 3\22, 124.138¢
|2\13, 141.176¢
|3\17, 163.636¢
|-
|-
|Reb, Lab
|Βb
|Βb
|3\19, 138.462¢
|3\19, 138.462¢
Line 47: Line 195:
|2\22, 82.759¢
|2\22, 82.759¢
|1\13, 70.588¢
|1\13, 70.588¢
|1\17, 54.{{Overline|54}}¢
|1\17, 54.545¢
|-
|-
|'''Re, La'''
|'''Β'''
|'''Β'''
|'''4\19,''' '''184.615¢'''
|'''4\19,''' '''184.615¢'''
Line 57: Line 204:
|'''5\22,''' '''206.897¢'''
|'''5\22,''' '''206.897¢'''
|'''3\13,''' '''211.765¢'''
|'''3\13,''' '''211.765¢'''
|'''4\17,''' '''218.{{Overline|18}}¢'''
|'''4\17,''' '''218.182¢'''
|-
|-
|Re#, La#
|Β#
|Β#
|5\19, 230.769¢
|5\19, 230.769¢
Line 67: Line 213:
|8\22, 331.034¢
|8\22, 331.034¢
|5\13, 352.941¢
|5\13, 352.941¢
|7\17, 381.{{Overline|81}}¢
| 7\17, 381.818¢
|-
|-
|Mib, Sib
|Γb
|Γb
|7\19, 323.077¢
|7\19, 323.077¢
Line 76: Line 221:
|7\22, 289.655¢
|7\22, 289.655¢
|4\13, 282.353¢
|4\13, 282.353¢
|5\17, 272.{{Overline|72}}¢
|5\17, 272.727¢
|-
|-
|Mi, Si
|Γ
|Γ
|8\19, 369.231¢
|8\19, 369.231¢
Line 86: Line 230:
|10\22, 413.793¢
|10\22, 413.793¢
|6\13, 423.529¢
|6\13, 423.529¢
|8\17, 436.{{Overline|36}}¢
|8\17, 436.36&¢
|-
|-
|Mi#, Si#
|Γ#
|Γ#
|9\19, 415.385¢
|9\19, 415.385¢
Line 98: Line 241:
|11\17, 600¢
|11\17, 600¢
|-
|-
|Fab, Dob
|Δb
|Δb
|10\19, 461.538¢
|10\19, 461.538¢
Line 105: Line 247:
|9\22, 372.414¢
|9\22, 372.414¢
|5\13, 352.941¢
|5\13, 352.941¢
|6\17, 327.{{Overline|27}}¢
|6\17, 327.273¢
|-
|-
|Fa, Do
|Δ
|Δ
|11\19, 507.692¢
|11\19, 507.692¢
|8\14, 505.263¢
|8\14, 505.263¢
|13\23, 503.226¢
| 13\23, 503.226¢
|5\9, 500¢
|5\9, 500¢
|12\22, 496.552¢
|12\22, 496.552¢
|7\13, 494.118¢
|7\13, 494.118¢
|9\17, 490.{{Overline|90}}¢
|9\17, 490.909¢
|-
|-
|Fa#, Do#
|Δ#
|Δ#
|12\19, 553.846¢
|12\19, 553.846¢
Line 125: Line 265:
|15\22, 620.690¢
|15\22, 620.690¢
|9\13, 635.294¢
|9\13, 635.294¢
|12\17, 654.{{Overline|54}}¢
|12\17, 654.545¢
|-
|-
|Solb, Reb
|Εb
|Εb
|14\19, 646.154¢
|14\19, 646.154¢
Line 134: Line 273:
|14\22, 579.310¢
|14\22, 579.310¢
|8\13, 564.706¢
|8\13, 564.706¢
|10\17, 545.{{Overline|45}}¢
|10\17, 545.455¢
|-
|-
|'''Sol, Re'''
|'''Ε'''
|'''Ε'''
|'''15\19,''' '''692.308¢'''
|'''15\19,''' '''692.308¢'''
Line 144: Line 282:
|'''17\22,''' '''703.448¢'''
|'''17\22,''' '''703.448¢'''
|'''10\13,''' '''705.882¢'''
|'''10\13,''' '''705.882¢'''
|'''13\17,''' '''709.{{Overline|09}}¢'''
|'''13\17,''' '''709.091¢'''
|-
|-
|Sol#, Re#
|Ε#
|Ε#
|16\19, 738.462¢
|16\19, 738.462¢
Line 152: Line 289:
|20\23, 774.194¢
|20\23, 774.194¢
| rowspan="2" |8\9, 800¢
| rowspan="2" |8\9, 800¢
|20\22, 827.586¢
| 20\22, 827.586¢
|12\13, 847.059¢
| 12\13, 847.059¢
|16\14, 872.{{Overline|72}}¢
|16\17, 872.727¢
|-
|-
|Dob, Solb
|Ϛb/Ϝb
|Ϛb/Ϝb
|18\19, 830.769¢
|18\19, 830.769¢
Line 163: Line 299:
|19\22, 786.207¢
|19\22, 786.207¢
|11\13, 776.647¢
|11\13, 776.647¢
|14\17, 763.{{Overline|63}}¢
|14\17, 763.636¢
|-
|-
!Do, Sol
!Ϛ/Ϝ
!Ϛ/Ϝ
!19\19, 876.923¢
!19\19, 876.923¢
Line 173: Line 308:
!22\22, 910.345¢
!22\22, 910.345¢
!13\13, 917.647¢
!13\13, 917.647¢
!17\17, 927.{{Overline|27}}¢
!17\17, 927.273¢
|-
|-
|Do#, Sol#
|Ϛ#/Ϝ#
|Ϛ#/Ϝ#
|20\19, 923.077¢
|20\19, 923.077¢
|15\14, 947.368¢
| 15\14, 947.368¢
|24\23, 929.032¢
|24\23, 929.032¢
| rowspan="2" |10\9, 1000¢
| rowspan="2" |10\9, 1000¢
|25\22, 1034.483¢
|25\22, 1034.483¢
|15\13, 1052.824¢
|15\13, 1052.824¢
|20\17, 1090.{{Overline|90}}¢
|20\17, 1090.909¢
|-
|-
|Reb, Lab
|Ζb
|Ζb
|22\19, 1015.385¢
|22\19, 1015.385¢
Line 192: Line 325:
|24\22, 993.103¢
|24\22, 993.103¢
|14\13, 988.235¢
|14\13, 988.235¢
|18\17, 981.{{Overline|81}}¢
|18\17, 981.818¢
|-
|-
|'''Re, La'''
|'''Ζ'''
|'''Ζ'''
|'''23\19, 1061.538¢'''
|'''23\19, 1061.538¢'''
Line 202: Line 334:
|'''27\22,''' '''1117.241¢'''
|'''27\22,''' '''1117.241¢'''
|'''16\13,,''' '''1129.412¢'''
|'''16\13,,''' '''1129.412¢'''
|'''21\17,''' '''1145.{{Overline|45}}¢'''
|'''21\17,''' '''1145.455¢'''
|-
|-
|Re#, La#
|Ζ#
|Ζ#
|24\19, 1107.692¢
|24\19, 1107.692¢
Line 212: Line 343:
|30\22, 1241.379¢
|30\22, 1241.379¢
|18\13, 1270.588¢
|18\13, 1270.588¢
|24\14, 1309.{{Overline|09}}¢
|24\14, 1309.091¢
|-
|-
|Mib, Sib
|Ηb
|Ηb
|26\19, 1200¢
|26\19, 1200¢
|19\14, 1200¢
|19\14, 1200¢
|31\23,1200¢
|31\23, 1200¢
|29\22, 1200¢
|29\22, 1200¢
|17\13, 1200¢
|17\13, 1200¢
|22\17, 1200¢
|22\17, 1200¢
|-
|-
|Mi, Si
|Η
|Η
|27\19, 1246.154¢
|27\19, 1246.154¢
Line 231: Line 360:
|32\22, 1324.138¢
|32\22, 1324.138¢
|19\13, 1341.176¢
|19\13, 1341.176¢
|25\17, 1363.{{Overline|63}}¢
|25\17, 1363.636¢
|-
|-
|Mi#, Si#
|Η#
|Η#
|28\19, 1292.308¢
|28\19, 1292.308¢
Line 241: Line 369:
|35\22, 1448.276¢
|35\22, 1448.276¢
|21\13, 1482.353¢
|21\13, 1482.353¢
|28\17, 1527.{{Overline|27}}¢
|28\17, 1527.272¢
|-
|-
|Fab, Dob
|Θb
|Θb
|29\19, 1338.462¢
|29\19, 1338.462¢
Line 250: Line 377:
|31\22, 1282.759¢
|31\22, 1282.759¢
|18\13, 1270.588¢
|18\13, 1270.588¢
|23\17, 1254.{{Overline|54}}¢
|23\17, 1254.545¢
|-
|-
|Fa, Do
|Θ
|Θ
|30\19, 1384.615¢
|30\19, 1384.615¢
Line 260: Line 386:
|34\22, 1406.897¢
|34\22, 1406.897¢
|20\13, 1411.765¢
|20\13, 1411.765¢
|26\17, 1418.{{Overline|18}}¢
|26\17, 1418.182¢
|-
|-
|Fa#, Do#
|Θ#
|Θ#
|31\19, 1430.769¢
|31\19, 1430.769¢
Line 268: Line 393:
|38\23, 1470.968¢
|38\23, 1470.968¢
| rowspan="2" |15\9, 1500¢
| rowspan="2" |15\9, 1500¢
|37\22, 1531.0345¢
|37\22, 1531.035¢
|22\13, 1552.941¢
|22\13, 1552.941¢
|29\17, 1581.{{Overline|81}}¢
|29\17, 1581.182¢
|-
|-
|Solb, Reb
|Ιb
|Ιb
|33\19, 1523.077¢
|33\19, 1523.077¢
Line 279: Line 403:
|36\22, 1489.655¢
|36\22, 1489.655¢
|21\13, 1482.353¢
|21\13, 1482.353¢
|27\17, 1472.{{Overline|72}}¢
|27\17, 1472.727¢
|-
|-
|'''Sol, Re'''
|'''Ι'''
|'''Ι'''
|'''34\19,''' '''1569.231¢'''
|'''34\19,''' '''1569.231¢'''
Line 289: Line 412:
|'''39\22,''' '''1613.793¢'''
|'''39\22,''' '''1613.793¢'''
|'''23\13,''' '''1623.529¢'''
|'''23\13,''' '''1623.529¢'''
|'''30\17,''' '''1636.{{Overline|36}}¢'''
|'''30\17,''' '''1636.363¢'''
|-
|-
|Sol#, Re#
|Ι#
|Ι#
|35\19, 1615.385¢
|35\19, 1615.385¢
Line 301: Line 423:
|33\17, 1800¢
|33\17, 1800¢
|-
|-
|Dob, Solb
|Αb
|Αb
|37\19, 1707.692¢
|37\19, 1707.692¢
Line 308: Line 429:
|41\22, 1696.552¢
|41\22, 1696.552¢
|20\13, 1694.118¢
|20\13, 1694.118¢
|31\17, 1490.{{Overline|90}}¢
|31\17, 1490.909¢
|-
|-
!Do, Sol
!Α
!Α
!38\19, 1753.846¢
!38\19, 1753.846¢
Line 318: Line 438:
!44\22, 1820.690¢
!44\22, 1820.690¢
!26\13, 1835.294¢
!26\13, 1835.294¢
!34\17, 1854.{{Overline|54}}¢
!34\17, 1854.545¢
|}
|}
==Intervals==
==Intervals==
{| class="wikitable"
{| class="wikitable"
Line 332: Line 453:
|-
|-
|0
|0
|Do, Sol
|Do, Fa, Sol
|sextave (major sixth)
|sextave (major sixth)
|0
|0
|Do, Sol
|Do, Fa, Sol
|perfect unison
|perfect unison
|-
|-
|1
|1
|Sol, Re
|Sol, Do, Re
|perfect fifth
|perfect fifth
| -1
| -1
|Re, La
|Re, Sol, La
|major second
|major second
|-
|-
|2
|2
|Fa, Do
|Fa, Sib, Do
|perfect fourth
|perfect fourth
| -2
| -2
|Mi, Si
|Mi, La, Si
|major third
|major third
|-
|-
|3
|3
|Mib, Sib
|Mib, Lab, Sib
|minor third
|minor third
| -3
| -3
|Fa#, Do#
|Fa#, Si, Do#
|augmented fourth
|augmented fourth
|-
|-
|4
|4
|Reb, Lab
|Reb, Solb, Lab
|minor second
| minor second
| -4
| -4
|Sol#, Re#
|Sol#, Do#, Re#
|augmented fifth
|augmented fifth
|-
|-
Line 369: Line 490:
|-
|-
|5
|5
|Dob, Solb
|Dob, Fab, Solb
|diminished sextave
|diminished sextave
| -5
| -5
|Do#, Sol#
|Do#, Fa#, Sol#
|augmented unison (chroma)
|augmented unison (chroma)
|-
|-
|6
|6
|Solb, Reb
|Solb, Dob, Reb
|diminished fifth
| diminished fifth
| -6
| -6
|Re#, La#
|Re#, Sol#, La#
|augmented second
|augmented second
|-
|-
|7
|7
|Fab, Dob
| Fab, Sibb, Dob
|diminished fourth
|diminished fourth
| -7
| -7
|Mi#, Si#
|Mi#, La#, Si#
|augmented third
|augmented third
|-
|-
|8
|8
|Mibb, Sibb
|Mibb, Labb, Sibb
|diminished third
|diminished third
| -8
| -8
|Fax, Dox
|Fax, Si#, Dox
|doubly augmented fourth
| doubly augmented fourth
|}
|}
==Genchain==
==Genchain==
Line 400: Line 521:
{| class="wikitable"
{| class="wikitable"
|Mibb
|Mibb
Labb
Sibb
Sibb
|Fab
|Fab
Sibb
Dob
Dob
|Solb
|Solb
Dob
Reb
Reb
|Dob
|Dob
Fab
Solb
Solb
|Reb
|Reb
Solb
Lab
Lab
|Mib
|Mib
Lab
Sib
Sib
|Fa
|Fa
Sib
Do
| Sol
Do
Do
|Sol
 
Re
Re
|Do
|Do
Fa
Sol
Sol
|Re
|Re
Sol
La
La
|Mi
|Mi
La
Si
Si
|Fa#
|Fa#
Si
Do#
Do#
|Sol#
|Sol#
Do#
Re#
Re#
|Do#
|Do#
Fa#
Sol#
Sol#
|Re#
|Re#
Sol#
La#
La#
|Mi#
|Mi#
La#
Si#
Si#
|Fax
|Fax
Si#
Dox
Dox
|-
|-
Line 991: Line 1,146:
|}
|}


== See also ==
==See also==
[[4L 1s (5/3-equivalent)]] - idealized meantone tuning
[[4L 1s (5/3-equivalent)]] - idealized meantone tuning
[[4L 1s (27/16-equivalent)]] - Pythagorean tuning


[[4L 1s (22/13-equivalent)]] - Neogothic tuning
[[4L 1s (22/13-equivalent)]] - Neogothic tuning
Line 999: Line 1,156:


[[8L 2s (e-equivalent)|8L 2s ([math]e[/math]-equivalent)]] - natural tuning
[[8L 2s (e-equivalent)|8L 2s ([math]e[/math]-equivalent)]] - natural tuning
[[8L 2s (2000/729-equivalent)]] - 1/2 comma meantone tuning


[[8L 2s (11/4-equivalent)]] - idealized low tuning, low undecimal tuning
[[8L 2s (11/4-equivalent)]] - idealized low tuning, low undecimal tuning


[[8L 2s (14/5-equivalent)]] - low septimal tuning
[[8L 2s (45/16-equivalent)]] - 1/6 comma meantone tuning
 
[[8L 2s (14/5-equivalent)]] - low septimal (meantone) tuning
 
[[8L 2s (729/256-equivalent)]] - Pythagorean tuning


[[8L 2s (20/7-equivalent)]] - idealized high tuning, high septimal tuning
[[8L 2s (20/7-equivalent)]] - idealized high tuning, high septimal tuning
[[8L 2s (81/28-equivalent)]] - 1/6 comma Archytas tuning


[[8L 2s (32/11-equivalent)]] - high undecimal tuning
[[8L 2s (32/11-equivalent)]] - high undecimal tuning
[[8L 2s (2000/729-equivalent)|8L 2s (1024/343-equivalent)]] - 1/2 comma Archytas tuning
[[8L 2s (3/1-equivalent)]] - warped Pythagorean tuning
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