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

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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
! colspan="10" |Generator
!Normalized Cents<ref name=":0">Fractions with repeat period 2 or longer in minutes and seconds</ref>
!Cents<ref name=":0">Fractions with repeat period 2 or longer in minutes and seconds</ref>
!L
!L
!s
!s
Line 202: Line 202:
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|514¢17’8”||1||1||1.000||
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|514.286||1||1||1.000||
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|-
|  ||  ||  ||  ||  ||40\57
|  ||  ||  ||  ||  ||40\57
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|510||6||5||1.200||
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|510.000||6||5||1.200||
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|-
|  ||  ||  ||  ||33\47||
|  ||  ||  ||  ||33\47||
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|509¢5’27”||5||4||1.250||
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|509.{{Overline|09}}||5||4||1.250||
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|-
|  ||  ||  ||  ||  ||59\84
|  ||  ||  ||  ||  ||59\84
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|508¢28'34”||9||7||1.286||
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|508.475||9||7||1.286||
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|-
|  ||  ||  ||26\37||  ||
|  ||  ||  ||26\37||  ||
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|507¢55’23”||4||3||1.333||
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|  ||  ||  ||  ||  ||71\101
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|507¢2’32”||11||8||1.375||
|507.692||4||3||1.333||
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|-
|  ||  ||  ||  ||45\64||
|  ||  ||  ||  ||45\64||
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Line 237: Line 244:
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|505¢15’47”||3||2||1.500||L/s = 3/2
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|505.263||3||2||1.500||L/s = 3/2
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|-
|  ||  ||  ||  ||50\71||
|  ||  ||  ||  ||50\71||
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|504||8||5||1.600||
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|504.000||8||5||1.600||
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|-
|  ||  ||  ||31\44||  ||
|  ||  ||  ||31\44||  ||
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|503¢13’33”||5||3||1.667||
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|503.226||5||3||1.667||
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|-
|  ||  ||  ||  ||43\61||
|  ||  ||  ||  ||43\61||
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|502¢18’8”||7||4||1.750||
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|502.326||7||4||1.750||
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|-
|  ||  ||  ||  ||  ||55\78
|  ||  ||  ||  ||  ||55\78
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|501¢49’5”||9||5||1.800||
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|501.{{Overline|81}}||9||5||1.800||
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|  ||12\17||  ||  ||  ||
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|500.000||2||1||2.000||Basic Bolivar
(Generators smaller than this are proper)
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|67\95
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|501¢29’33”
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|11
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|6
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|1.833
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|101\143
|499.{{Overline|0099}}
|17
|8
|2.125
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|-
|-
Line 280: Line 307:
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|79\112
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|501¢15’57”
|89\126
|13
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|498.876
|15
|7
|7
|1.857
|2.143
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|-
|-
|  ||12\17||  ||  ||  ||
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|500||2||1||2.000||Basic Bolivar
(Generators smaller than this are proper)
|-
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Line 298: Line 323:
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|77\109
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|77\109
|498.701
|498¢42’5”
|13
|13
|6
|6
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|65\92
|65\92
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|498¢21’42”
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|498.452
|11
|11
|5
|5
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|498¢6’48”||9||4||2.250||
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|498.113||9||4||2.250||
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|  ||  ||  ||  ||41\58||
|  ||  ||  ||  ||41\58||
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|497¢33’39”||7||3||2.333||
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|  ||  ||  ||  ||  ||70\99
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|497¢8’34”||12||5||2.400||
|497.561||7||3||2.333||
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|-
|  ||  ||  ||29\41||  ||
|  ||  ||  ||29\41||  ||
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|496¢33’6”||5||2||2.500||
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|496.552||5||2||2.500||
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|-
|  ||  ||  ||  ||46\65||
|  ||  ||  ||  ||46\65||
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|495¢39’8”||8||3||2.667||
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|495.652||8||3||2.667||
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|  ||  ||17\24||  ||  ||
|  ||  ||17\24||  ||  ||
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|494¢7’4”||3||1||3.000||L/s = 3/1
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|494.118||3||1||3.000||L/s = 3/1
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|-
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Line 359: Line 391:
|73\103
|73\103
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|493¢9’2”
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|493.151
|13
|13
|4
|4
Line 368: Line 402:
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|492¢51’26”||10||3||3.333||
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|492.857||10||3||3.333||
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|  ||  ||  ||  ||39\55||
|  ||  ||  ||  ||39\55||
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|492¢18’28”||7||2||3.500||
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|492.308||7||2||3.500||
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|-
|  ||  ||  ||  ||  ||61\86
|  ||  ||  ||  ||  ||61\86
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|491¢48’12”||11||3||3.667||
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|491.803||11||3||3.667||
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|  ||  ||  ||22\31||  ||
|  ||  ||  ||22\31||  ||
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|490¢54’33”||4||1||4.000||
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|490.{{Overline|90}}||4||1||4.000||
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|  ||  ||  ||  ||  ||49\69
|  ||  ||  ||  ||  ||49\69
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|489¢47’45”||9||2||4.500||
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|489.796||9||2||4.500||
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|  ||  ||  ||  ||27\38||
|  ||  ||  ||  ||27\38||
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|487.5||6||1||6.000||
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|487.500||6||1||6.000||
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|5\7||  ||  ||  ||  ||
|5\7||  ||  ||  ||  ||
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|480||1||0||→ inf||
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|480.000||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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