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

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Line 159: Line 159:
| -13
| -13
|9w
|9w
|diminished ninth
|diminished tenth
|5L+4s
|5L+4s
|-
|-
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|8^
|8^
|augmented ninth
|augmented ninth
|8L+1s
|7L+1s
| -14
| -14
|2w
|2w
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|diminished ninth
|diminished ninth
|4L+4s
|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==
==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]]:
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" |Generator
!Generator
!Normalized Cents<ref name=":0">Fractions with repeat period 2 or longer in minutes and seconds</ref>
!Cents
!L
!L
!s
!s
Line 199: Line 368:
!Comments
!Comments
|-
|-
|7\10||  ||  ||  ||  ||
|7\10
|
|514.286||1||1||1.000||
|
|514¢17’8”||1||1||1.000||
|-
|-
|  ||  ||  ||  ||  ||40\57
| 40\57
|
|510.000||6||5||1.200||
|
|510||6||5||1.200||
|-
|-
|  ||  ||  ||  ||33\47||
| 33\47
|
|509.091||5||4||1.250||
|
|509¢5’27”||5||4||1.250||
|-
|-
|  ||  ||  ||  ||  ||59\84
| 59\84
|
|508.475||9||7||1.286||
|
|508¢28'34”||9||7||1.286||
|-
|-
|  ||  ||  ||26\37||  ||
| 26\37
|
|507.692||4||3||1.333||
|
|507¢55’23”||4||3||1.333||
|-
|-
| ||  ||  ||  ||  ||71\101
| 45\64
|
|506.667||7||5||1.400||
|
|507¢2’32”||11||8||1.375||
|-
|-
| ||  ||  ||  ||45\64||
| 19\27
|
|505.263||3||2||1.500||L/s = 3/2
|
|506.{{Overline|6}}||7||5||1.400||
|-
|-
| ||  ||19\27||  ||  ||
| 50\71
|
|504.000||8||5||1.600||
|
|505¢15’47”||3||2||1.500||L/s = 3/2
|-
|-
| ||  ||  ||  ||50\71||
| 31\44
|
|503.226||5||3||1.667||
|
|504||8||5||1.600||
|-
|-
| ||  ||  ||31\44||  ||
| 43\61
|
|502.326||7||4||1.750||
|
|503¢13’33”||5||3||1.667||
|-
|-
| ||  ||  ||  ||43\61||
| 55\78
|
|501.818||9||5||1.800||
|
|502¢18’8”||7||4||1.750||
|-
|-
| ||  ||  ||  ||  ||55\78
| 12\17
|
|500.000||2||1||2.000||Basic Bolivar
|
(Generators smaller than this are proper)
|501¢49’5”||9||5||1.800||
|-
|-
|
|101\143
|
|499.010
|
|17
|
|8
|
|2.125
|
|67\95
|
|501¢29’33”
|11
|6
|1.833
|
|
|-
|-
|
|89\126
|
|498.876
|
|15
|
|
|
|
|79\112
|501¢15’57”
|13
|7
|7
|1.857
|2.143
|
|
|-
|-
|  ||12\17||  ||  ||  ||
|
|
|500||2||1||2.000||Basic Bolivar
(Generators smaller than this are proper)
|-
|
|
|
|
|
|
|
|77\109
|77\109
|498¢42’5”
|498.701
|13
|13
|6
|6
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|
|
|-
|-
|
|
|
|
|
|
|65\92
|65\92
|
|498.452
|498¢21’42”
|11
|11
|5
|5
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|
|
|-
|-
|  ||  ||  ||  ||  ||53\75
| 53\75
|
|498.113||9||4||2.250||
|
|498¢6’48”||9||4||2.250||
|-
|-
|  ||  ||  ||  ||41\58||
| 41\58
|
|497.561||7||3||2.333||
|
|497¢33’39”||7||3||2.333||
|-
|-
| ||  ||  ||  ||  ||70\99
| 29\41
|
|496.552||5||2||2.500||
|
|497¢8’34”||12||5||2.400||
|-
|-
|  ||  ||  ||29\41||  ||
| 46\65
|
|495.652||8||3||2.667||
|
|496¢33’6”||5||2||2.500||
|-
|  ||  ||  ||  ||46\65||
|
|
|495¢39’8”||8||3||2.667||
|-
|-
|  ||  ||17\24||  ||  ||
| 17\24
|
|494.118||3||1||3.000||L/s = 3/1
|
|494¢7’4”||3||1||3.000||L/s = 3/1
|-
|-
|
|
|
|
|
|
|73\103
|73\103
|
|493.151
|493¢9’2”
|13
|13
|4
|4
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|
|
|-
|-
|  ||  ||  ||  ||  ||56\79
| 56\79
|
|492.857||10||3||3.333||
|
|492¢51’26”||10||3||3.333||
|-
|-
|  ||  ||  ||  ||39\55||
| 39\55
|
|492.308||7||2||3.500||
|
|492¢18’28”||7||2||3.500||
|-
|-
|  ||  ||  ||  ||  ||61\86
| 61\86
|
|491.803||11||3||3.667||
|
|491¢48’12”||11||3||3.667||
|-
|-
|  ||  ||  ||22\31||  ||
| 22\31
|
|490.909||4||1||4.000||
|
|490¢54’33”||4||1||4.000||
|-
|-
|  ||  ||  ||  ||  ||49\69
| 49\69
|
|489.796||9||2||4.500||
|
|489¢47’45”||9||2||4.500||
|-
|-
|  ||  ||  ||  ||27\38||
| 27\38
|
|488.889||5||1||5.000||
|
|488.{{Overline|8}}||5||1||5.000||
|-
|-
|  ||  ||  ||  ||  ||32\45
| 32\45
|
|487.500||6||1||6.000||
|
|487.5||6||1||6.000||
|-
|-
|5\7||  ||  ||  ||  ||
|5\7
|
|480.000||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.  


Line 421: 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 431: 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 442: 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 />
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