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#REDIRECT [[Moremajorthanmajor/7L 3s (8/3-equivalent)]]
'''7L 3s<perfect eleventh>''' (sometimes called '''Bolivar''' or''' Choralic''') refers to a non-octave [[MOS scale]] family with a period of a perfect eleventh and which has 7 large and 3 small steps. These scales are the sister of '''[[7L 3s (4/1-equivalent)|diaquadic]]''' with the melodic spacing of [[5L 2s|diatonic scales]]. A pathological trait these scales exhibit is that normalization to [[edo]] collapses the range for the [[bright]] [[generator]] to the octave.
==Modes==
The modes contain fundamental chords with notes such that they convert a [[wikipedia:Tritone_substitution|tritone substitution]] into a diatonic chord substitution.
 
*LLLsLLsLLs 9|0 (Lydian ♮11)
*LLsLLLsLLs 8|1 (Major, Ionian)
*LLsLLsLLLs 7|2 (Mixolydian)
*LLsLLsLLsL 6|3 (Mahur)
*LsLLLsLLsL 5|4 (Dorian)
*LsLLsLLLsL 4|5 (Minor, Aeolian)
*LsLLsLLsLL 3|6 (Aeolian b9)
*sLLLsLLsLL 2|7 (Phrygian)
*sLLsLLLsLL 1|8 (Locrian)
*sLLsLLsLLL 0|9 (Locrian b8)
 
==Intervals==
The generator (g) will fall between 480 cents (2\5 - two degrees of [[5edo]]) and 514 cents (2\5 - two degrees of [[5edo]]), hence a perfect fourth.
 
2g, then, will fall between 960 cents (4\5) and 1029 cents (6\7), the range of minor sevenths.
 
The "large step" will fall between 171 cents (1\7) and 240 cents (1\5), the range of major seconds.
 
The "small step" will fall between 0 cents and 171 cents, sometimes sounding like a submajor second, and sometimes sounding like a quartertone or smaller microtone.
{| class="wikitable"
!# generators up
!Notation (1/1 = 0)
!name
!In L's and s's
!# generators up
!Notation of eleventh inverse
!name
!In L's and s's
|-
| colspan="8" style="text-align:center" |The 10-note MOS has the following intervals (from some root):
|-
|0
|0
|perfect unison
|0
|0
|0
|perfect eleventh
|7L+3s
|-
|1
|7
|perfect octave
|5L+2s
| -1
|3
|perfect fourth
|2L+1s
|-
|2
|4
|just fifth
|3L+1s
| -2
|6
|minor seventh
|4L+2s
|-
|3
|1
|major second
|1L
| -3
|9v
|minor tenth
|6L+3s
|-
|4
|8
|major ninth
|6L+2s
| -4
|2v
|minor third
|1L+1s
|-
|5
|5
|major sixth
|4L+1s
| -5
|5v
|minor sixth
|3L+2s
|-
|6
|2
|major third
|2L
| -6
|8v
|minor ninth
|5L+3s
|-
|7
|9
|major tenth
|7L+2s
| -7
|1v
|minor second
|1s
|-
|8
|6^
|major seventh
|5L+1s
| -8
|4v
|diminished fifth
|2L+2s
|-
|9
|3^
|augmented fourth
|3L
| -9
|7v
|diminished octave
|4L+3s
|-
|10
|0^
|augmented unison
|1L-1s
| -10
|0v
|diminished eleventh
|6L+4s
|-
| colspan="8" style="text-align:center" |The chromatic 17-note MOS (either [[7L 10s (perfect eleventh equivalent)|7L 10s]], [[10L 7s (perfect eleventh equivalent)|10L 7s]], or [[17edXI]]) also has the following intervals (from some root):
|-
|11
|7^
|augmented octave
|6L+1s
| -11
|3v
|diminished fourth
|1L+2s
|-
|12
|4^
|augmented fifth
|4L
| -12
|6v
|diminished seventh
|3L+3s
|-
|13
|1^
|augmented second
|2L-1s
| -13
|9w
|diminished tenth
|5L+4s
|-
|14
|8^
|augmented ninth
|7L+1s
| -14
|2w
|diminished third
|2s
|-
|15
|5^
|augmented sixth
|5L
| -15
|5w
|diminished sixth
|2L+3s
|-
|16
|2^
|augmented third
|3L-1s
| -16
|8w
|diminished ninth
|4L+4s
|}
{| class="wikitable"
!# generators up
!Notation (1/1 = G ut, ~8/3 = C sol fa ut)
!name
!In L's and s's
!# generators up
!Notation of twenty-first inverse
!name
!In L's and s's
|-
| colspan="8" style="text-align:center" |The 20-note MOS has the following intervals (from some root):
|-
|0
|G ut, C sol fa ut
|perfect unison, perfect eleventh
|0, 7L+3s
|0
|G ut, C sol fa ut
|perfect eleventh, “perfect” minor twenty-first
|7L+3s, 14L+6s
|-
|1
|G sol fa re ut, C fa
|perfect octave, perfect eighteenth
|5L+2s, 12L+5s
|  -1
|C fa ut, F fa ut
|perfect fourth, minor fourteenth
|2L+1s, 9L+4s
|-
|2
|D sol re ut, G sol fa ut
|just fifth, perfect fifteenth
|3L+1s, 10L+4s
|  -2
|F fa ut, B fa
|minor seventh, minor seventeenth
|4L+2s, 11L+5s
|-
|3
|A re, D sol la re ut
|major second, perfect twelfth
|1L, 8L+3s
|  -3
|B fab, E lab
|minor tenth, minor twentieth
|6L+3s, 13L+6s
|-
|4
|A la sol re mi, D sol
|major ninth, perfect nineteenth
|6L+2s, 13L+5s
|  -4
|B mib, E la mi reb
|minor third, minor thirteenth
|1L+1s, 8L+4s
|-
|5
|E la mi re, A la sol re
|major sixth, major sixteenth
|4L+1s, 11L+4s
|  -5
|E la mi reb, A la mi reb
|minor sixth, minor sixteenth
|3L+2s, 10L+5s
|-
|6
|B mi, E la mi re si
|major third, major thirteenth
|2L, 9L+3s
|  -6
|A la sol reb, D  solb
|minor ninth, diminished nineteenth
|5L+3s, 12L+6s
|-
|7
|B si la mi, E la
|major tenth, major twentieth
|7L+2s, 14L+5s
|  -7
|A reb, D sol la reb
|minor second, diminished twelfth
|1s, 7L+4s
|-
|8
|F si mi, B si la mi
|major seventh, major seventeenth
|5L+1s, 12L+4s
|  -8
|D sol re utb, G sol fa utb
|diminished fifth, diminished fifteenth
|2L+2s, 9L+5s
|-
|9
|C si, F si mi
|augmented fourth, major fourteenth
|3L, 10L+3s
|  -9
|G sol fa utb, C fab
|diminished octave, diminished eighteenth
|4L+3s, 11L+6s
|-
|10
|G ut#, C si
|augmented unison, augmented eleventh
|1L-1s, 8L+2s
|  -10
|G utb, C fa utb
|diminished eleventh, diminished twenty-first
|6L+4s, 13L+7s
|-
| colspan="8" style="text-align:center" |The chromatic 17-note MOS (either [[7L 10s (perfect eleventh equivalent)|14L 20s]], [[10L 7s (perfect eleventh equivalent)|20L 14s]], or [[34edXXI]]) also has the following intervals (from some root):
|-
|11
|G sol re ut#, C si
|augmented octave, augmented eighteenth
|6L+1s, 13L+4s
|  -11
|C fa utb, F fa utb
|diminished fourth, diminished fourteenth
|1L+2s,
8L+5s
|-
|12
|D sol re ut#, G sol fa ut#
|augmented fifth, augmented fifteenth
|4L, 11L+3s
|  -12
|F fa utb, B fab
|diminished seventh, diminished seventeeth
|3L+3s, 10L+6s
|-
|13
|A re#, D sol re ut#
|augmented second, augmented twelfth
|2L-1s, 9L+2s
|  -13
|B fab, E labb
|diminished tenth, diminished twentieth
|5L+4s, 12L+7s
|-
|14
|A la sol re mi#, D sol#
|augmented ninth, augmented nineteenth
|7L+1s, 14L+4s
|  -14
|B mibb, E la mi re sibb
|diminished third, diminished thirteenth
|2s, 7L+5s
|-
|15
|E la mi re#, A la sol re#
|augmented sixth, augmented sixteenth
|5L, 12L 3s
|  -15
|E la mi rebb, A la sol rebb
|diminished sixth, diminished sixteenth
|2L+3s, 9L+6s
|-
|16
|B mi#, E la mi re si#
|augmented third, augmented thirteenth
|3L-1s,
10L+2s
|  -16
|A sol la mi rebb, D la solbb
|diminished ninth, doubly diminished nineteenth
|4L+4s, 11L+7s
|}
==Scale tree==
The generator range reflects two extremes: one where L = s (3\10), and another where s = 0 (2\7). Between these extremes, there is an infinite continuum of possible generator sizes. By taking freshman sums of the two edges (adding the numerators, then adding the denominators), we can fill in this continuum with compatible ~ed8/3s, increasing in number of tones as we continue filling in the in-betweens. Thus, the smallest in-between ~ed8/3 would be (3+2)\(10+7) = 5\17 – five degrees of [[17edXI]]:
{| class="wikitable center-all"
!Generator
!Cents
!L
!s
!L/s
!Comments
|-
|7\10
|514.286||1||1||1.000||
|-
| 40\57
|510.000||6||5||1.200||
|-
| 33\47
|509.091||5||4||1.250||
|-
| 59\84
|508.475||9||7||1.286||
|-
| 26\37
|507.692||4||3||1.333||
|-
| 45\64
|506.667||7||5||1.400||
|-
| 19\27
|505.263||3||2||1.500||L/s = 3/2
|-
| 50\71
|504.000||8||5||1.600||
|-
| 31\44
|503.226||5||3||1.667||
|-
| 43\61
|502.326||7||4||1.750||
|-
| 55\78
|501.818||9||5||1.800||
|-
| 12\17
|500.000||2||1||2.000||Basic Bolivar
(Generators smaller than this are proper)
|-
|101\143
|499.010
|17
|8
|2.125
|
|-
|89\126
|498.876
|15
|7
|2.143
|
|-
|77\109
|498.701
|13
|6
|2.167
|
|-
|65\92
|498.452
|11
|5
|2.200
|
|-
| 53\75
|498.113||9||4||2.250||
|-
| 41\58
|497.561||7||3||2.333||
|-
| 29\41
|496.552||5||2||2.500||
|-
| 46\65
|495.652||8||3||2.667||
|-
| 17\24
|494.118||3||1||3.000||L/s = 3/1
|-
|73\103
|493.151
|13
|4
|3.250
|
|-
| 56\79
|492.857||10||3||3.333||
|-
| 39\55
|492.308||7||2||3.500||
|-
| 61\86
|491.803||11||3||3.667||
|-
| 22\31
|490.909||4||1||4.000||
|-
| 49\69
|489.796||9||2||4.500||
|-
| 27\38
|488.889||5||1||5.000||
|-
| 32\45
|487.500||6||1||6.000||
|-
|5\7
|480.000||1||0||→ inf||
|}The scale produced by stacks of 5\17 is the 12edo diatonic scale.
 
Other compatible ~ed8/3s include: ~37ed8/3, ~27ed8/3, ~44ed8/3, ~41ed8/3, ~24ed8/3, ~31ed8/3.
 
You can also build this scale by equally dividing frequency ratio 8:3 which is not a member of an edo or stacking frequency ratio 4:3 which is not a member of an equal division of it within it.
==Rank-2 temperaments==
The '''Bolivar''' rank-2 temperament spells its major tetrad 4:5:6:8 or 14:18:21:28<code>root-3(2g-p)-(2g-p)-(1g)</code> (p = 8/3, g = 2/1) and its minor tetrad 6:7:9:12 or 10:12:15:20 <code>root-2(p-2g)-(2g-p)-(1g)</code> (p = 8/3, g = 2/1). Basic 17ed8/3 fits both interpretations.
 
==='''Bolivar-Meantone'''===
[[Subgroup]]: 8/3.2.5/4
 
[[Comma]] list: [[81/80]]
 
[[POL2]] generator: ~2/1 = 1196.3254
 
[[Mapping]]: [{{val|1 0 -3}}, {{val|0 1 6}}]
 
[[Optimal ET sequence]]: [[17ed8/3]], [[27ed8/3]], [[44ed8/3]]
==='''Bolivar-Superpyth'''===
[[Subgroup]]: 8/3.2.7/6
 
[[Comma]] list: [[64/63]]
 
[[POL2]] generator: ~2/1 = 1206.6167
 
[[Mapping]]:  [{{val|1 0 2}}, {{val|0 1 -4}}]
 
[[Optimal ET sequence]]: [[17ed8/3]], [[24ed8/3]], [[31ed8/3]], [[38ed8/3]]
==7-note subsets==
If you stop the chain at 7 tones, you have a heptatonic scale of the form [[3L 4s (eleventh equivalent)|3L 4s]]:
 
L s s L s L s
 
The large steps here consist of L+s of the 10-tone system, and the small step is the same as L.
==Tetrachordal structure==
Due to the frequency of perfect fourths and fifths in this scale, it can also be analyzed as a [[tetrachord|tetrachordal scale]].
 
== See also ==
[[7L 3s (8/3-equivalent)]] - idealized tuning
 
[[14L 6s (7/1-equivalent)]] - Guidotonic dominant Archytas tuning
 
[[14L 6s (78/11-equivalent)]] - Guidotonic dominant Neogothic tuning
 
[[14L 6s (36/5-equivalent)]] - Guidotonic dominant Meantone tuning<references />
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