Xenharmonic Wiki

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

← Older edit
VisualWikitext
mNo edit summary
 
(10 intermediate revisions by the same user not shown)
Line 1: Line 1:
'''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.
Line 28: Line 28:
!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
Line 133: Line 133:
|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
Line 159: Line 159:
| -13
| -13
|9w
|9w
|diminished ninth
|diminished tenth
|5L+4s
|5L+4s
|-
|-
Line 165: Line 165:
|8^
|8^
|augmented ninth
|augmented ninth
|8L+1s
|7L+1s
| -14
| -14
|2w
|2w
Line 189: Line 189:
|4L+4s
|4L+4s
|}
|}
==Scale tree==
{| class="wikitable"
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]]:
!# generators up
{| class="wikitable center-all"
!Notation (1/1 = G ut, ~8/3 = C sol fa ut)
! colspan="8" rowspan="2" |Generator
!name
!Cents
!In L's and s's
! colspan="2" |''ed17\12''
!# generators up
! rowspan="2" |L
!Notation of twenty-first inverse
! rowspan="2" |s
!name
! rowspan="2" |L/s
!In L's and s's
! rowspan="2" |Comments
|-
|-
!Normalized
| colspan="8" style="text-align:center" |The 20-note MOS has the following intervals (from some root):
<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|| || || || ||
|0
|
|G ut, C sol fa ut
|
|perfect unison, perfect eleventh
|514¢17’8”||''1190°''||''510°''||1||1||1.000||
|0, 7L+3s
|0
|G ut, C sol fa ut
|perfect eleventh, “perfect” minor twenty-first
|7L+3s, 14L+6s
|-
|-
| || || || || ||40\57
|1
|
|G sol fa re ut, C fa
|
|perfect octave, perfect eighteenth
|510||''1192°58’57”''||''507°1’3”''||6||5||1.200||
|5L+2s, 12L+5s
| -1
|C fa ut, F fa ut
|perfect fourth, minor fourteenth
|2L+1s, 9L+4s
|-
|-
| || || || ||33\47||
|2
|
|D sol re ut, G sol fa ut
|
|just fifth, perfect fifteenth
|509¢5’27”||''1193°37’1”''||''506°22'59”''||5||4||1.250||
|3L+1s, 10L+4s
| -2
|F fa ut, B fa
|minor seventh, minor seventeenth
|4L+2s, 11L+5s
|-
|-
| || || || || ||59\84
|3
|
|A re, D sol la re ut
|
|major second, perfect twelfth
|508¢28'34”||''1194°2’51”''||''505°58’9”''||9||7||1.286||
|1L, 8L+3s
| -3
|B fab, E lab
|minor tenth, minor twentieth
|6L+3s, 13L+6s
|-
|-
| || || ||26\37|| ||
|4
|
|A la sol re mi, D sol
|
|major ninth, perfect nineteenth
|507¢55’23”||''1194°34’3”''||''505°25’56”''||4||3||1.333||
|6L+2s, 13L+5s
| -4
|B mib, E la mi reb
|minor third, minor thirteenth
|1L+1s, 8L+4s
|-
|-
| || || || || ||71\101
|5
|
|E la mi re, A la sol re
|
|major sixth, major sixteenth
|507¢2’32”||''1195°2’23”''||''504°57’37”''||11||8||1.375||
|4L+1s, 11L+4s
| -5
|E la mi reb, A la mi reb
|minor sixth, minor sixteenth
|3L+2s, 10L+5s
|-
|-
| || || || ||45\64||
|6
|
|B mi, E la mi re si
|
|major third, major thirteenth
|506.{{Overline|6}}||''1195°18’45”''||''504°41’15”''||7||5||1.400||
|2L, 9L+3s
| -6
|A la sol reb, D  solb
|minor ninth, diminished nineteenth
|5L+3s, 12L+6s
|-
|-
| || ||19\27|| || ||
|7
|
|B si la mi, E la
|
|major tenth, major twentieth
|505¢15’47”||''1196°17’47”''||''503°42’13’''||3||2||1.500||L/s = 3/2
|7L+2s, 14L+5s
| -7
|A reb, D sol la reb
|minor second, diminished twelfth
|1s, 7L+4s
|-
|-
| || || || ||50\71||
|8
|
|F si mi, B si la mi
|
|major seventh, major seventeenth
|504||''1197°11‘50”''||''502°48’10”''||8||5||1.600||
|5L+1s, 12L+4s
| -8
|D sol re utb, G sol fa utb
|diminished fifth, diminished fifteenth
|2L+2s, 9L+5s
|-
|-
| || || ||31\44|| ||
|9
|
|C si, F si mi
|
|augmented fourth, major fourteenth
|503¢13’33”||''1197°43’38”''||''502°16’22”''||5||3||1.667||
|3L, 10L+3s
| -9
|G sol fa utb, C fab
|diminished octave, diminished eighteenth
|4L+3s, 11L+6s
|-
|-
| || || || ||43\61||
|10
|
|G ut#, C si
|
|augmented unison, augmented eleventh
|502¢18’8”||''1198°21’38”''||''501°39’22”''||7||4||1.750||
|1L-1s, 8L+2s
| -10
|G utb, C fa utb
|diminished eleventh, diminished twenty-first
|6L+4s, 13L+7s
|-
|-
| || || || || ||55\78
| 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):
|
|
|501¢49’5”||''1198°43’4”''||''501°16’56”''||9||5||1.800||
|-
|-
|
|
|
|
|
|
|67\95
|
|501¢29’33”
|''1198°56’51”''
|''501°3’9”''
|11
|11
|6
|G sol re ut#, C si
|1.833
|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
|-
|-
|
|
|
|
|
|
|
|79\112
|501¢15’57”
|''1199°6’26”''
|''500°53’34”''
|13
|13
|7
|A re#, D sol re ut#
|1.857
|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|| || || ||
| 12\17
|
|500.000||2||1||2.000||Basic Bolivar
|
|500||''1200°''||''500°''||2||1||2.000||Basic Bolivar
(Generators smaller than this are proper)
(Generators smaller than this are proper)
|-
|-
|101\143
|499.010
|17
|8
|2.125
|
|
|-
|89\126
|498.876
|15
|7
|2.143
|
|
|
|-
|
|
|
|
|77\109
|77\109
|498¢42’5”
|498.701
|''1200°54’30”''
|''499°5’30”''
|13
|13
|6
|6
Line 319: Line 426:
|
|
|-
|-
|
|
|
|
|
|
|65\92
|65\92
|
|498.452
|498¢21’42”
|''1201°5’13”''
|''498°54’47”''
|11
|11
|5
|5
Line 335: Line 433:
|
|
|-
|-
| || || || || ||53\75
| 53\75
|
|498.113||9||4||2.250||
|
|498¢6’48”||''1201.{{Overline|3}}''||''498.{{Overline|6}}''||9||4||2.250||
|-
|-
| || || || ||41\58||
| 41\58
|
|497.561||7||3||2.333||
|
|497¢33’39”||''1201°43’27”''||''498°16’33”''||7||3||2.333||
|-
|-
| || || || || ||70\99
| 29\41
|
|496.552||5||2||2.500||
|
|497¢8’34”||''1202°1’13”''||''497°58’47”''||12||5||2.400||
|-
|-
| || || ||29\41|| ||
| 46\65
|
|495.652||8||3||2.667||
|
|496¢33’6”||''1202°26’21”''||''497°33’39”''||5||2||2.500||
|-
|-
| || || || ||46\65||
| 17\24
|
|494.118||3||1||3.000||L/s = 3/1
|
|495¢39’8”||''1203°50’46”''||''496°9’14”''||8||3||2.667||
|-
| || ||17\24|| || ||
|
|
|494¢7’4”||''1204.1{{Overline|6}}''||''495.8{{Overline|3}}''||3||1||3.000||L/s = 3/1
|-
|-
|
|
|
|
|
|
|73\103
|73\103
|
|493.151
|493¢9’2”
|''1204°51’16”''
|''495°8’44”''
|13
|13
|4
|4
Line 381: Line 455:
|
|
|-
|-
| || || || || ||56\79
| 56\79
|
|492.857||10||3||3.333||
|
|492¢51’26”||''1205°3’48”''||''494°56’12”''||10||3||3.333||
|-
|-
| || || || ||39\55||
| 39\55
|
|492.308||7||2||3.500||
|
|492¢18’28”||''1205°27’16”''||''494°32’44”''||7||2||3.500||
|-
|-
| || || || || ||61\86
| 61\86
|
|491.803||11||3||3.667||
|
|491¢48’12”||''1205°48’50”''||''494°11’10”''||11||3||3.667||
|-
|-
| || || ||22\31|| ||
| 22\31
|
|490.909||4||1||4.000||
|
|490¢54’33”||''1206°27’6”''||''493°32’54”''||4||1||4.000||
|-
|-
| || || || || ||49\69
| 49\69
|
|489.796||9||2||4.500||
|
|489¢47’45”||''1207°14’47”''||''492°45’13”''||9||2||4.500||
|-
|-
| || || || ||27\38||
| 27\38
|
|488.889||5||1||5.000||
|
|488.{{Overline|8}}||''1207°53’41”''||''492°6’19”''||5||1||5.000||
|-
|-
| || || || || ||32\45
| 32\45
|
|487.500||6||1||6.000||
|
|487.5||''1208.{{Overline|8}}''||''491.{{Overline|1}}''||6||1||6.000||
|-
|-
|5\7|| || || || ||
|5\7
|
|480.000||1||0||→ inf||
|
|480||''1214°17’8”''||''485°42’52’''||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.  


Line 426: Line 484:
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'''===
Line 437: Line 495:
[[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
Line 447: Line 505:
[[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]]:
Line 458: Line 516:


== See also ==
== See also ==
[[7L 3s (8/3-equivalent)]]<references />
[[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 />
Retrieved from "https://en.xen.wiki/w/User:Moremajorthanmajor/7L_3s_(perfect_eleventh-equivalent)"