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| Line 199: |
Line 199: |
| ==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 ~[[17ed8/3]]: |
| {| class="wikitable center-all" | | {{Scale tree}}The scale produced by stacks of 5\17 is the 12edo diatonic scale. |
| ! colspan="8" rowspan="2" |Generator
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| !Cents
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| ! colspan="2" |''ed17\12''
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| ! rowspan="2" |L
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| ! rowspan="2" |s
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| ! rowspan="2" |L/s
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| ! rowspan="2" |Comments
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| |-
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| !Normalized
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| <ref name=":0">Fractions with repeat period 2 or longer in minutes and seconds</ref>
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| !''Chroma-positive<ref name=":0" />''
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| !''Chroma-negative<ref name=":0" />''
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| |-
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| |7\10|| || || || ||
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| |514¢17’8”||''1190°''||''510°''||1||1||1.000||
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| |-
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| | || || || || ||40\57
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| |510||''1192°58’57”''||''507°1’3”''||6||5||1.200||
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| |-
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| | || || || ||33\47||
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| |509¢5’27”||''1193°37’1”''||''506°22'59”''||5||4||1.250||
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| |-
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| | || || || || ||59\84
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| |508¢28'34”||''1194°2’51”''||''505°58’9”''||9||7||1.286||
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| |-
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| | || || ||26\37|| ||
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| |507¢55’23”||''1194°34’3”''||''505°25’56”''||4||3||1.333||
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| |-
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| | || || || || ||71\101
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| |507¢2’32”||''1195°2’23”''||''504°57’37”''||11||8||1.375||
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| |-
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| | || || || ||45\64||
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| |506.{{Overline|6}}||''1195°18’45”''||''504°41’15”''||7||5||1.400||
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| |-
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| | || ||19\27|| || ||
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| |505¢15’47”||''1196°17’47”''||''503°42’13’''||3||2||1.500||L/s = 3/2
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| |-
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| | || || || ||50\71||
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| |504||''1197°11‘50”''||''502°48’10”''||8||5||1.600||
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| |-
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| | || || ||31\44|| ||
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| |503¢13’33”||''1197°43’38”''||''502°16’22”''||5||3||1.667||
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| |-
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| | || || || ||43\61||
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| |502¢18’8”||''1198°21’38”''||''501°39’22”''||7||4||1.750||
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| |-
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| | || || || || ||55\78
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| |501¢49’5”||''1198°43’4”''||''501°16’56”''||9||5||1.800||
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| |-
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| |67\95
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| |501¢29’33”
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| |''1198°56’51”''
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| |''501°3’9”''
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| |11
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| |6
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| |1.833
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| |-
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| |79\112
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| |501¢15’57”
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| |''1199°6’26”''
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| |''500°53’34”''
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| |13
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| |7
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| |1.857
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| |-
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| | ||12\17|| || || ||
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| |500||''1200°''||''500°''||2||1||2.000||Basic Bolivar
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| (Generators smaller than this are proper)
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| |-
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| |77\109
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| |498¢42’5”
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| |''1200°54’30”''
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| |''499°5’30”''
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| |13
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| |6
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| |2.167
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| |-
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| |65\92
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| |498¢21’42”
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| |''1201°5’13”''
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| |''498°54’47”''
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| |11
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| |5
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| |2.200
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| |-
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| | || || || || ||53\75
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| |498¢6’48”||''1201.{{Overline|3}}''||''498.{{Overline|6}}''||9||4||2.250||
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| |-
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| | || || || ||41\58||
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| |497¢33’39”||''1201°43’27”''||''498°16’33”''||7||3||2.333||
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| |-
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| | || || || || ||70\99
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| |497¢8’34”||''1202°1’13”''||''497°58’47”''||12||5||2.400||
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| |-
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| | || || ||29\41|| ||
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| |496¢33’6”||''1202°26’21”''||''497°33’39”''||5||2||2.500||
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| |-
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| | || || || ||46\65||
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| |495¢39’8”||''1203°50’46”''||''496°9’14”''||8||3||2.667||
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| |-
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| | || ||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
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| |-
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| |73\103
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| |493¢9’2”
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| |''1204°51’16”''
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| |''495°8’44”''
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| |13
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| |4
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| |3.250
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| |-
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| | || || || || ||56\79
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| |492¢51’26”||''1205°3’48”''||''494°56’12”''||10||3||3.333||
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| |-
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| | || || || ||39\55||
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| |492¢18’28”||''1205°27’16”''||''494°32’44”''||7||2||3.500||
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| |-
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| | || || || || ||61\86
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| |491¢48’12”||''1205°48’50”''||''494°11’10”''||11||3||3.667||
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| |-
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| | || || ||22\31|| ||
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| | 490¢54’33”||''1206°27’6”''||''493°32’54”''||4||1||4.000||
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| |-
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| | || || || || ||49\69
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| | 489¢47’45”||''1207°14’47”''||''492°45’13”''||9||2||4.500||
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| |-
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| | || || || ||27\38||
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| | 488.{{Overline|8}}||''1207°53’41”''||''492°6’19”''||5||1||5.000||
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| |-
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| | || || || || ||32\45
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| | 487.5||''1208.{{Overline|8}}''||''491.{{Overline|1}}''||6||1||6.000||
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| |-
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| |5\7|| || || || ||
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| | 480||''1214°17’8”''||''485°42’52’''||1||0||→ inf||
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| |}The scale produced by stacks of 5\17 is the 12edo diatonic scale.
| |
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| Other compatible ~ed8/3s include: ~37ed8/3, ~27ed8/3, ~44ed8/3, ~41ed8/3, ~24ed8/3, ~31ed8/3. | | Other compatible ~ed8/3s include: ~37ed8/3, ~27ed8/3, ~44ed8/3, ~41ed8/3, ~24ed8/3, ~31ed8/3. |