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'''7L 3s<perfect eleventh>''' (sometimes called '''Bolivar''' or''' Choralic''') refers to a non-octave [[MOS scale]] family with a period of an [[8/3]] 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.
'''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==
==Modes==
The modes contain fundamental chords with notes such that they convert a [[wikipedia:Tritone_substitution|tritone substitution]] into a diatonic chord substitution.
The modes contain fundamental chords with notes such that they convert a [[wikipedia:Tritone_substitution|tritone substitution]] into a diatonic chord substitution.
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!In L's and s's
!In L's and s's
!# generators up
!# generators up
!Notation of 8/3 inverse
!Notation of eleventh inverse
!name
!name
!In L's and s's
!In L's and s's
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|6L+4s
|6L+4s
|-
|-
| colspan="8" style="text-align:center" |The chromatic 17-note MOS (either [[7L 10s (eleventh equivalent)|7L 10s]], [[10L 7s (eleventh equivalent)|10L 7s]], or ~[[17ed8/3]]) also has the following intervals (from some root):
| 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
|11
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|}
|}
==Scale tree==
==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 ~[[17ed8/3]]:
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"
{| class="wikitable center-all"
! colspan="8" rowspan="2" |Generator
! colspan="8" |Generator
!Cents
!Normalized Cents<ref name=":0">Fractions with repeat period 2 or longer in minutes and seconds</ref>
! colspan="2" |''ed17\12''
!L
! rowspan="2" |L
!s
! rowspan="2" |s
!L/s
! rowspan="2" |L/s
!Comments
! rowspan="2" |Comments
|-
|-
!Normalized
|7\10|| || || || ||
<ref name=":0">Fractions with repeat period 2 or longer in minutes and seconds</ref>
!''Chroma-positive<ref name=":0" />''
!''Chroma-negative<ref name=":0" />''
|-
|7\10|| || || || ||
|
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|514¢17’8”||''1190°''||''510°''||1||1||1.000||
|514¢17’8”||1||1||1.000||
|-
|-
| || || || || ||40\57
| || || || || ||40\57
|
|
|
|
|510||''1192°58’57”''||''507°1’3”''||6||5||1.200||
|510||6||5||1.200||
|-
|-
| || || || ||33\47||
| || || || ||33\47||
|
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|509¢5’27”||''1193°37’1”''||''506°22'59”''||5||4||1.250||
|509¢5’27”||5||4||1.250||
|-
|-
| || || || || ||59\84
| || || || || ||59\84
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|508¢28'34”||''1194°2’51”''||''505°58’9”''||9||7||1.286||
|508¢28'34”||9||7||1.286||
|-
|-
| || || ||26\37|| ||
| || || ||26\37|| ||
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|507¢55’23”||''1194°34’3”''||''505°25’56”''||4||3||1.333||
|507¢55’23”||4||3||1.333||
|-
|-
| || || || || ||71\101
| || || || || ||71\101
|
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|507¢2’32”||''1195°2’23”''||''504°57’37”''||11||8||1.375||
|507¢2’32”||11||8||1.375||
|-
|-
| || || || ||45\64||
| || || || ||45\64||
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|506.{{Overline|6}}||''1195°18’45”''||''504°41’15”''||7||5||1.400||
|506.{{Overline|6}}||7||5||1.400||
|-
|-
| || ||19\27|| || ||
| || ||19\27|| || ||
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|505¢15’47”||''1196°17’47”''||''503°42’13’''||3||2||1.500||L/s = 3/2
|505¢15’47”||3||2||1.500||L/s = 3/2
|-
|-
| || || || ||50\71||
| || || || ||50\71||
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|504||''1197°11‘50”''||''502°48’10”''||8||5||1.600||
|504||8||5||1.600||
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|-
| || || ||31\44|| ||
| || || ||31\44|| ||
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|503¢13’33”||''1197°43’38”''||''502°16’22”''||5||3||1.667||
|503¢13’33”||5||3||1.667||
|-
|-
| || || || ||43\61||
| || || || ||43\61||
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|502¢18’8”||''1198°21’38”''||''501°39’22”''||7||4||1.750||
|502¢18’8”||7||4||1.750||
|-
|-
| || || || || ||55\78
| || || || || ||55\78
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|501¢49’5”||''1198°43’4”''||''501°16’56”''||9||5||1.800||
|501¢49’5”||9||5||1.800||
|-
|-
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|501¢29’33”
|501¢29’33”
|''1198°56’51”''
|''501°3’9”''
|11
|11
|6
|6
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|79\112
|79\112
|501¢15’57”
|501¢15’57”
|''1199°6’26”''
|''500°53’34”''
|13
|13
|7
|7
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|-
|-
| ||12\17|| || || ||
| ||12\17|| || || ||
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|500||''1200°''||''500°''||2||1||2.000||Basic Bolivar
|500||2||1||2.000||Basic Bolivar
(Generators smaller than this are proper)
(Generators smaller than this are proper)
|-
|-
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|77\109
|77\109
|498¢42’5”
|498¢42’5”
|''1200°54’30”''
|''499°5’30”''
|13
|13
|6
|6
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|498¢21’42”
|498¢21’42”
|''1201°5’13”''
|''498°54’47”''
|11
|11
|5
|5
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|-
| || || || || ||53\75
| || || || || ||53\75
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|498¢6’48”||''1201.{{Overline|3}}''||''498.{{Overline|6}}''||9||4||2.250||
|498¢6’48”||9||4||2.250||
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|-
| || || || ||41\58||
| || || || ||41\58||
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|497¢33’39”||''1201°43’27”''||''498°16’33”''||7||3||2.333||
|497¢33’39”||7||3||2.333||
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|-
| || || || || ||70\99
| || || || || ||70\99
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|497¢8’34”||''1202°1’13”''||''497°58’47”''||12||5||2.400||
|497¢8’34”||12||5||2.400||
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| || || ||29\41|| ||
| || || ||29\41|| ||
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|496¢33’6”||''1202°26’21”''||''497°33’39”''||5||2||2.500||
|496¢33’6”||5||2||2.500||
|-
|-
| || || || ||46\65||
| || || || ||46\65||
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|495¢39’8”||''1203°50’46”''||''496°9’14”''||8||3||2.667||
|495¢39’8”||8||3||2.667||
|-
|-
| || ||17\24|| || ||
| || ||17\24|| || ||
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|494¢7’4”||''1204.1{{Overline|6}}''||''495.8{{Overline|3}}''||3||1||3.000||L/s = 3/1
|494¢7’4”||3||1||3.000||L/s = 3/1
|-
|-
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|493¢9’2”
|493¢9’2”
|''1204°51’16”''
|''495°8’44”''
|13
|13
|4
|4
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|-
|-
| || || || || ||56\79
| || || || || ||56\79
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|492¢51’26”||''1205°3’48”''||''494°56’12”''||10||3||3.333||
|492¢51’26”||10||3||3.333||
|-
|-
| || || || ||39\55||
| || || || ||39\55||
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|492¢18’28”||''1205°27’16”''||''494°32’44”''||7||2||3.500||
|492¢18’28”||7||2||3.500||
|-
|-
| || || || || ||61\86
| || || || || ||61\86
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|491¢48’12”||''1205°48’50”''||''494°11’10”''||11||3||3.667||
|491¢48’12”||11||3||3.667||
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|-
| || || ||22\31|| ||
| || || ||22\31|| ||
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|490¢54’33”||''1206°27’6”''||''493°32’54”''||4||1||4.000||
|490¢54’33”||4||1||4.000||
|-
|-
| || || || || ||49\69
| || || || || ||49\69
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|489¢47’45”||''1207°14’47”''||''492°45’13”''||9||2||4.500||
|489¢47’45”||9||2||4.500||
|-
|-
| || || || ||27\38||
| || || || ||27\38||
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|488.{{Overline|8}}||''1207°53’41”''||''492°6’19”''||5||1||5.000||
|488.{{Overline|8}}||5||1||5.000||
|-
|-
| || || || || ||32\45
| || || || || ||32\45
|
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|487.5||''1208.{{Overline|8}}''||''491.{{Overline|1}}''||6||1||6.000||
|487.5||6||1||6.000||
|-
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|5\7|| || || || ||
|5\7|| || || || ||
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|480||''1214°17’8”''||''485°42’52’''||1||0||→ inf||
|480||1||0||→ inf||
|}The scale produced by stacks of 5\17 is the 12edo diatonic scale.  
|}The scale produced by stacks of 5\17 is the 12edo diatonic scale.  


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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.
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==
==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.
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'''===
==='''Bolivar-Meantone'''===
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[[Mapping]]: [{{val|1 0 -3}}, {{val|0 1 6}}]
[[Mapping]]: [{{val|1 0 -3}}, {{val|0 1 6}}]


[[Optimal ET sequence]]: ~(17ed8/3, 27ed8/3, 44ed8/3)
[[Optimal ET sequence]]: 17ed8/3, 27ed8/3, 44ed8/3
==='''Bolivar-Superpyth'''===
==='''Bolivar-Superpyth'''===
[[Subgroup]]: 8/3.2.7/6
[[Subgroup]]: 8/3.2.7/6
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[[Mapping]]:  [{{val|1 0 2}}, {{val|0 1 -4}}]
[[Mapping]]:  [{{val|1 0 2}}, {{val|0 1 -4}}]


[[Optimal ET sequence]]: ~(17ed8/3, 24ed8/3, 31ed8/3, 38ed8/3)
[[Optimal ET sequence]]: 17ed8/3, 24ed8/3, 31ed8/3, 38ed8/3
==7-note subsets==
==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]]:
If you stop the chain at 7 tones, you have a heptatonic scale of the form [[3L 4s (eleventh equivalent)|3L 4s]]:
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