User:Moremajorthanmajor/4L 1s (major sixth-equivalent): Difference between revisions

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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 510: Line 665:
==Temperaments==
==Temperaments==
The most basic rank-2 temperament interpretation of this diatonic is '''Dorianic''', which has pental 4:5:6 or septimal 14:18:21 chords spelled <code>root-(2g)-(p-1g)</code> (p = the major sixth, g = the whole tone). The name "Dorianic" comes from the Dorian major mode having the minor sixth as its characteristic interval.
The most basic rank-2 temperament interpretation of this diatonic is '''Dorianic''', which has pental 4:5:6 or septimal 14:18:21 chords spelled <code>root-(2g)-(p-1g)</code> (p = the major sixth, g = the whole tone). The name "Dorianic" comes from the Dorian major mode having the minor sixth as its characteristic interval.
==='''Dorianic-Meantone'''===
==='''Dorianic[5]-Meantone'''===
[[Subgroup]]: 5/3.4/3.3/2
[[Subgroup]]: 5/3.4/3.3/2


Line 520: Line 675:


[[Optimal ET sequence]]: [[5ed5/3]], [[9ed5/3]], [[14ed5/3]]
[[Optimal ET sequence]]: [[5ed5/3]], [[9ed5/3]], [[14ed5/3]]
==='''Dorianic-Superpyth'''===
==='''Dorianic[5]-Superpyth'''===
[[Subgroup]]: 12/7.4/3.3/2
[[Subgroup]]: 12/7.4/3.3/2


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|-
|5\24
|5\24
|181.{{Overline|81}}
|181.818
|5
|5
|4
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|7\33
|7\33
|186.{{Overline|6}}
|186.667
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|14\65
|14\65
|190.{{Overline|90}}
|190.909
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|9\41
|9\41
|196.{{Overline|36}}
|196.364
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== 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

Latest revision as of 04:49, 20 December 2024

4L 1s<major sixth> (sometimes called diatonic), is a major sixth-repeating MOS scale. The notation "<major sixth>" means the period of the MOS is 5/3, disambiguating it from octave-repeating 4L 1s. The name of the period interval is called the sextave (by analogy to the tritave).

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 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.

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; 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 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.

Normalized
Notation Supersoft Soft Semisoft Basic Semihard Hard Superhard
Diatonic 19eds 14eds 23eds 9eds 22eds 13eds 17eds
Do#, Fa#, Sol# 1\19, 46.154¢ 1\14, 63.158¢ 2\23, 77.419¢ 1\9, 100¢ 3\22, 124.138¢ 2\13, 141.176¢ 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¢ 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¢ 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¢ 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¢ 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¢
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¢ 1\9, 100¢ 3\22, 124.138¢ 2\13, 141.176¢ 3\17, 163.636¢
Βb 3\19, 138.462¢ 2\14, 126.316¢ 3\23, 116.129¢ 2\22, 82.759¢ 1\13, 70.588¢ 1\17, 54.545¢
Β 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¢
Β# 5\19, 230.769¢ 4\14, 252.632¢ 7\23, 270.968¢ 3\9, 300¢ 8\22, 331.034¢ 5\13, 352.941¢ 7\17, 381.818¢
Γb 7\19, 323.077¢ 5\14, 315.789¢ 8\23, 309.677¢ 7\22, 289.655¢ 4\13, 282.353¢ 5\17, 272.727¢
Γ 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&¢
Γ# 9\19, 415.385¢ 7\14, 442.105¢ 12\23, 464.516¢ 5\9, 500¢ 13\22, 537.931¢ 8\13, 564.706¢ 11\17, 600¢
Δb 10\19, 461.538¢ 11\23, 425.806¢ 4\9, 400¢ 9\22, 372.414¢ 5\13, 352.941¢ 6\17, 327.273¢
Δ 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¢
Δ# 12\19, 553.846¢ 9\14, 568.421¢ 15\23, 580.645¢ 6\9, 600¢ 15\22, 620.690¢ 9\13, 635.294¢ 12\17, 654.545¢
Εb 14\19, 646.154¢ 10\14, 631.579¢ 16\23, 619.355¢ 14\22, 579.310¢ 8\13, 564.706¢ 10\17, 545.455¢
Ε 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¢
Ε# 16\19, 738.462¢ 12\14, 757.895¢ 20\23, 774.194¢ 8\9, 800¢ 20\22, 827.586¢ 12\13, 847.059¢ 16\17, 872.727¢
Ϛb/Ϝb 18\19, 830.769¢ 13\14, 821.053¢ 21\23, 812.903¢ 19\22, 786.207¢ 11\13, 776.647¢ 14\17, 763.636¢
Ϛ/Ϝ 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¢
Ϛ#/Ϝ# 20\19, 923.077¢ 15\14, 947.368¢ 24\23, 929.032¢ 10\9, 1000¢ 25\22, 1034.483¢ 15\13, 1052.824¢ 20\17, 1090.909¢
Ζb 22\19, 1015.385¢ 16\14, 1010.526¢ 26\23, 1006.452¢ 24\22, 993.103¢ 14\13, 988.235¢ 18\17, 981.818¢
Ζ 23\19, 1061.538¢ 17\14, 1071.684¢ 28\23, 1083.871¢ 11\9, 1100¢ 27\22, 1117.241¢ 16\13,, 1129.412¢ 21\17, 1145.455¢
Ζ# 24\19, 1107.692¢ 18\14, 1136.842¢ 30\23, 1161.290¢ 12\9, 1200¢ 30\22, 1241.379¢ 18\13, 1270.588¢ 24\14, 1309.091¢
Ηb 26\19, 1200¢ 19\14, 1200¢ 31\23, 1200¢ 29\22, 1200¢ 17\13, 1200¢ 22\17, 1200¢
Η 27\19, 1246.154¢ 20\14, 1263.158¢ 33\23, 1277.419¢ 13\9, 1300¢ 32\22, 1324.138¢ 19\13, 1341.176¢ 25\17, 1363.636¢
Η# 28\19, 1292.308¢ 21\14, 1326.316¢ 35\23, 1354.839¢ 14\9, 1400¢ 35\22, 1448.276¢ 21\13, 1482.353¢ 28\17, 1527.272¢
Θb 29\19, 1338.462¢ 34\23, 1316.129¢ 13\9, 1300¢ 31\22, 1282.759¢ 18\13, 1270.588¢ 23\17, 1254.545¢
Θ 30\19, 1384.615¢ 22\14, 1389.474¢ 36\23, 1393.548¢ 14\9, 1400¢ 34\22, 1406.897¢ 20\13, 1411.765¢ 26\17, 1418.182¢
Θ# 31\19, 1430.769¢ 23\14, 1452.632¢ 38\23, 1470.968¢ 15\9, 1500¢ 37\22, 1531.035¢ 22\13, 1552.941¢ 29\17, 1581.182¢
Ιb 33\19, 1523.077¢ 24\14, 1515.789¢ 39\23, 1509.677¢ 36\22, 1489.655¢ 21\13, 1482.353¢ 27\17, 1472.727¢
Ι 34\19, 1569.231¢ 25\14, 1578.947¢ 41\23, 1587.097¢ 16\9, 1600¢ 39\22, 1613.793¢ 23\13, 1623.529¢ 30\17, 1636.363¢
Ι# 35\19, 1615.385¢ 26\14, 1642.105¢ 43\23, 1664.516¢ 17\9, 1700¢ 42\22, 1737.931¢ 25\13, 1764.706¢ 33\17, 1800¢
Αb 37\19, 1707.692¢ 27\14, 1705.263¢ 44\23, 1703.226¢ 41\22, 1696.552¢ 20\13, 1694.118¢ 31\17, 1490.909¢
Α 38\19, 1753.846¢ 28\14, 1768.421¢ 46\23, 1780.645¢ 18\9, 1800¢ 44\22, 1820.690¢ 26\13, 1835.294¢ 34\17, 1854.545¢

Intervals

Generators Sextave notation Interval category name Generators Notation of sixth inverse Interval category name
The 5-note MOS has the following intervals (from some root):
0 Do, Fa, Sol sextave (major sixth) 0 Do, Fa, Sol perfect unison
1 Sol, Do, Re perfect fifth -1 Re, Sol, La major second
2 Fa, Sib, Do perfect fourth -2 Mi, La, Si major third
3 Mib, Lab, Sib minor third -3 Fa#, Si, Do# augmented fourth
4 Reb, Solb, Lab minor second -4 Sol#, Do#, Re# augmented fifth
The chromatic 9-note MOS also has the following intervals (from some root):
5 Dob, Fab, Solb diminished sextave -5 Do#, Fa#, Sol# augmented unison (chroma)
6 Solb, Dob, Reb diminished fifth -6 Re#, Sol#, La# augmented second
7 Fab, Sibb, Dob diminished fourth -7 Mi#, La#, Si# augmented third
8 Mibb, Labb, Sibb diminished third -8 Fax, Si#, Dox doubly augmented fourth

Genchain

The generator chain for this scale is as follows:

Mibb

Labb

Sibb

Fab

Sibb

Dob

Solb

Dob

Reb

Dob

Fab

Solb

Reb

Solb

Lab

Mib

Lab

Sib

Fa

Sib

Do

Sol

Do

Re

Do

Fa

Sol

Re

Sol

La

Mi

La

Si

Fa#

Si

Do#

Sol#

Do#

Re#

Do#

Fa#

Sol#

Re#

Sol#

La#

Mi#

La#

Si#

Fax

Si#

Dox

d3 d4 d5 d6 m2 m3 P4 P5 P1 M2 M3 A4 A5 A1 A2 A3 AA4

Modes

The mode names are based on the classical modes:

Mode Scale UDP Interval type
name pattern notation 2nd 3rd 4th 5th
Lydian Augmented LLLLs 4|0 M M A A
Lydian LLLsL 3|1 M M A P
Major LLsLL 2|2 M M P P
Dorian LsLLL 1|3 M m P P
Neapolitan sLLLL 0|4 m m P P

Temperaments

The most basic rank-2 temperament interpretation of this diatonic is Dorianic, which has pental 4:5:6 or septimal 14:18:21 chords spelled root-(2g)-(p-1g) (p = the major sixth, g = the whole tone). The name "Dorianic" comes from the Dorian major mode having the minor sixth as its characteristic interval.

Dorianic[5]-Meantone

Subgroup: 5/3.4/3.3/2

Comma list: 81/80

POL2 generator: ~9/8 = 193.8419¢

Mapping: [⟨1 1 1], ⟨0 -2 -1]]

Optimal ET sequence: 5ed5/3, 9ed5/3, 14ed5/3

Dorianic[5]-Superpyth

Subgroup: 12/7.4/3.3/2

Comma list: 64/63

POL2 generator: ~9/8 = 216.5781¢

Mapping: [⟨1 1 1], ⟨0 -2 -1]]

Optimal ET sequence: 4ed12/7, 9ed12/7, 13ed12/7, 17ed12/7

Scale tree

The spectrum looks like this:

Generator

(bright)

Normalised L s L/s Comments
1\5 171.429 1 1 1.000 Equalised
6\29 180.000 6 5 1.200
5\24 181.818 5 4 1.250
14\67 182.609 14 11 1.273
9\43 183.051 9 7 1.286
4\19 184.615 4 3 1.333
11\52 185.915 11 8 1.375
7\33 186.667 7 5 1.400
10\47 187.5 10 7 1.429
3\14 189.474 3 2 1.500 Dorianic-Meantone starts here
14\65 190.909 14 9 1.556
11\51 191.304 11 7 1.571
8\37 192.000 8 5 1.600
5\23 193.548 5 3 1.667
7\32 195.349 7 4 1.750
9\41 196.364 9 5 1.800
11\50 197.015 11 6 1.833
13\59 197.468 13 7 1.857
15\68 197.802 15 8 1.875
17\77 198.058 17 9 1.889
19\86 198.261 19 10 1.900
21\95 198.425 21 11 1.909
23\104 198.561 23 12 1.917
25\113 198.675 25 13 1.923
27\122 198.773 27 14 1.929
29\131 198.857 29 15 1.933
31\140 198.930 31 16 1.9375
33\149 198.995 33 17 1.941
35\158 199.052 35 18 1.944
2\9 200 2 1 2.000 Dorianic-Meantone ends, Dorianic-Pythagorean begins
17\76 201.980 17 8 2.125
15\67 202.247 15 7 2.143
13\58 202.597 13 6 2.167
11\49 203.076 11 5 2.200
9\40 203.774 9 4 2.250
7\31 204.838 7 3 2.333
12\53 205.714 12 5 2.400
5\22 206.897 5 2 2.500
18\79 207.692 18 7 2.571
13\57 208.000 13 5 2.600
8\35 208.696 8 3 2.667
11\48 209.524 11 4 2.750
14\61 210.000 14 5 2.800
3\13 211.765 3 1 3.000 Dorianic-Pythagorean ends, Dorianic-Superpyth begins
22\95 212.903 22 7 3.143
19\82 213.084 19 6 3.167
16\69 213.333 16 5 3.200
13\56 213.699 13 4 3.250
10\43 214.286 10 3 3.333
7\30 215.385 7 2 3.500
11\47 216.393 11 3 3.667
15\64 216.867 15 4 3.750
19\81 217.143 19 5 3.800
4\17 218.182 4 1 4.000
21\89 219.130 21 5 4.200
17\72 219.355 17 4 4.250
13\55 219.718 13 3 4.333
9\38 220.408 9 2 4.500
14\59 221.053 14 3 4.667
5\21 222.222 5 1 5.000 Dorianic-Superpyth ends
11\46 223.729 11 2 5.500
17\71 224.176 17 3 5.667
6\25 225.000 6 1 6.000
1\4 240.000 1 0 → inf Paucitonic

See also

4L 1s (5/3-equivalent) - idealized meantone tuning

4L 1s (27/16-equivalent) - Pythagorean tuning

4L 1s (22/13-equivalent) - Neogothic tuning

4L 1s (12/7-equivalent) - idealized Archytas tuning

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 (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 (81/28-equivalent) - 1/6 comma Archytas tuning

8L 2s (32/11-equivalent) - high undecimal tuning

8L 2s (1024/343-equivalent) - 1/2 comma Archytas tuning

8L 2s (3/1-equivalent) - warped Pythagorean tuning

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