User:Moremajorthanmajor/7L 3s (perfect eleventh-equivalent): Difference between revisions

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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|| || || || ||
|
|
|
|
|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||
|
|
|
|
|509¢5’27”||''1193°37’1”''||''506°22'59”''||5||4||1.250||
|509¢5’27”||5||4||1.250||
|-
|-
| || || || || ||59\84
| || || || || ||59\84
|
|
|
|
|508¢28'34”||''1194°2’51”''||''505°58’9”''||9||7||1.286||
|508¢28'34”||9||7||1.286||
|-
|-
| || || ||26\37|| ||
| || || ||26\37|| ||
|
|
|
|
|507¢55’23”||''1194°34’3”''||''505°25’56”''||4||3||1.333||
|507¢55’23”||4||3||1.333||
|-
|-
| || || || || ||71\101
| || || || || ||71\101
|
|
|
|
|507¢2’32”||''1195°2’23”''||''504°57’37”''||11||8||1.375||
|507¢2’32”||11||8||1.375||
|-
|-
| || || || ||45\64||
| || || || ||45\64||
|
|
|
|
|506.{{Overline|6}}||''1195°18’45”''||''504°41’15”''||7||5||1.400||
|506.{{Overline|6}}||7||5||1.400||
|-
|-
| || ||19\27|| || ||
| || ||19\27|| || ||
|
|
|
|
|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||
|
|
|
|
|504||''1197°11‘50”''||''502°48’10”''||8||5||1.600||
|504||8||5||1.600||
|-
|-
| || || ||31\44|| ||
| || || ||31\44|| ||
|
|
|
|
|503¢13’33”||''1197°43’38”''||''502°16’22”''||5||3||1.667||
|503¢13’33”||5||3||1.667||
|-
|-
| || || || ||43\61||
| || || || ||43\61||
|
|
|
|
|502¢18’8”||''1198°21’38”''||''501°39’22”''||7||4||1.750||
|502¢18’8”||7||4||1.750||
|-
|-
| || || || || ||55\78
| || || || || ||55\78
|
|
|
|
|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|| || || ||
|
|
|
|
|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
|
|
|
|
|498¢6’48”||''1201.{{Overline|3}}''||''498.{{Overline|6}}''||9||4||2.250||
|498¢6’48”||9||4||2.250||
|-
|-
| || || || ||41\58||
| || || || ||41\58||
|
|
|
|
|497¢33’39”||''1201°43’27”''||''498°16’33”''||7||3||2.333||
|497¢33’39”||7||3||2.333||
|-
|-
| || || || || ||70\99
| || || || || ||70\99
|
|
|
|
|497¢8’34”||''1202°1’13”''||''497°58’47”''||12||5||2.400||
|497¢8’34”||12||5||2.400||
|-
|-
| || || ||29\41|| ||
| || || ||29\41|| ||
|
|
|
|
|496¢33’6”||''1202°26’21”''||''497°33’39”''||5||2||2.500||
|496¢33’6”||5||2||2.500||
|-
|-
| || || || ||46\65||
| || || || ||46\65||
|
|
|
|
|495¢39’8”||''1203°50’46”''||''496°9’14”''||8||3||2.667||
|495¢39’8”||8||3||2.667||
|-
|-
| || ||17\24|| || ||
| || ||17\24|| || ||
|
|
|
|
|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
|
|
|
|
|492¢51’26”||''1205°3’48”''||''494°56’12”''||10||3||3.333||
|492¢51’26”||10||3||3.333||
|-
|-
| || || || ||39\55||
| || || || ||39\55||
|
|
|
|
|492¢18’28”||''1205°27’16”''||''494°32’44”''||7||2||3.500||
|492¢18’28”||7||2||3.500||
|-
|-
| || || || || ||61\86
| || || || || ||61\86
|
|
|
|
|491¢48’12”||''1205°48’50”''||''494°11’10”''||11||3||3.667||
|491¢48’12”||11||3||3.667||
|-
|-
| || || ||22\31|| ||
| || || ||22\31|| ||
|
|
|
|
|490¢54’33”||''1206°27’6”''||''493°32’54”''||4||1||4.000||
|490¢54’33”||4||1||4.000||
|-
|-
| || || || || ||49\69
| || || || || ||49\69
|
|
|
|
|489¢47’45”||''1207°14’47”''||''492°45’13”''||9||2||4.500||
|489¢47’45”||9||2||4.500||
|-
|-
| || || || ||27\38||
| || || || ||27\38||
|
|
|
|
|488.{{Overline|8}}||''1207°53’41”''||''492°6’19”''||5||1||5.000||
|488.{{Overline|8}}||5||1||5.000||
|-
|-
| || || || || ||32\45
| || || || || ||32\45
|
|
|
|
|487.5||''1208.{{Overline|8}}''||''491.{{Overline|1}}''||6||1||6.000||
|487.5||6||1||6.000||
|-
|-
|5\7|| || || || ||
|5\7|| || || || ||
|
|
|
|
|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]]:

Revision as of 02:33, 19 June 2023

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 diaquadic with the melodic spacing of 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 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.

# 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
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
The chromatic 17-note MOS (either 7L 10s, 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 ninth 5L+4s
14 8^ augmented ninth 8L+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

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:

Generator Normalized Cents[1] L s L/s Comments
7\10 514¢17’8” 1 1 1.000
40\57 510 6 5 1.200
33\47 509¢5’27” 5 4 1.250
59\84 508¢28'34” 9 7 1.286
26\37 507¢55’23” 4 3 1.333
71\101 507¢2’32” 11 8 1.375
45\64 506.6 7 5 1.400
19\27 505¢15’47” 3 2 1.500 L/s = 3/2
50\71 504 8 5 1.600
31\44 503¢13’33” 5 3 1.667
43\61 502¢18’8” 7 4 1.750
55\78 501¢49’5” 9 5 1.800
67\95 501¢29’33” 11 6 1.833
79\112 501¢15’57” 13 7 1.857
12\17 500 2 1 2.000 Basic Bolivar

(Generators smaller than this are proper)

77\109 498¢42’5” 13 6 2.167
65\92 498¢21’42” 11 5 2.200
53\75 498¢6’48” 9 4 2.250
41\58 497¢33’39” 7 3 2.333
70\99 497¢8’34” 12 5 2.400
29\41 496¢33’6” 5 2 2.500
46\65 495¢39’8” 8 3 2.667
17\24 494¢7’4” 3 1 3.000 L/s = 3/1
73\103 493¢9’2” 13 4 3.250
56\79 492¢51’26” 10 3 3.333
39\55 492¢18’28” 7 2 3.500
61\86 491¢48’12” 11 3 3.667
22\31 490¢54’33” 4 1 4.000
49\69 489¢47’45” 9 2 4.500
27\38 488.8 5 1 5.000
32\45 487.5 6 1 6.000
5\7 480 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 14:18:21:28root-3(2g-p)-(2g-p)-(1g) (p = 8/3, g = 2/1) and its minor tetrad 6:7:9:12 or 10:12:15:20 root-2(p-2g)-(2g-p)-(1g) (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: [⟨1 0 -3], ⟨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: [⟨1 0 2], ⟨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:

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 tetrachordal scale.

See also

7L 3s (8/3-equivalent)

  1. ↑ Fractions with repeat period 2 or longer in minutes and seconds
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