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==Scale tree== | |||
If 4\7 (four degrees of 7EDO) is at one extreme and 3\5 (three degrees of 5EDO) is at the other, all other possible 5L 2s scales exist in a continuum between them. You can chop this continuum up by taking [[Mediant|"freshman sums"]] of the two edges - adding together the numerators, then adding together the denominators (i.e. adding them together as if you would be adding the complex numbers analogous real and imaginary parts). Thus, between 4\7 and 3\5 you have (4+3)\(7+5) = 7\12, seven degrees of 12EDO. | |||
If we carry this freshman-summing out a little further, new, larger [[EDO]]s pop up in our continuum. | |||
Generator range: 48.97959 cents (4\7/14 = 2\49) to 51.42857 cents (3\5/14 = 3\70) | |||
{| class="wikitable center-all" | |||
! colspan="7" |Fifth | |||
!Cents | |||
!Comments | |||
|- | |||
|4\7|| || || || || || ||48.9796 | |||
| | |||
|- | |||
| || || || || || ||27\47||49.2401|| | |||
|- | |||
| || || || || ||23\40|| ||49.2857|| | |||
|- | |||
| || || || || || ||42\73||49.3151|| | |||
|- | |||
| || || || ||19\33|| || ||49.35065|| | |||
|- | |||
| || || || || || ||53\92||49.3789|| | |||
|- | |||
| || || || || ||34\59|| ||49.3947|| | |||
|- | |||
| || || || || || ||49\85||49.4118|| | |||
|- | |||
| || || ||15\26|| || || ||49.45055|| | |||
|- | |||
| || || || || || ||56\97||49.4845|| | |||
|- | |||
| || || || || ||41\71|| ||49.4970|| | |||
|- | |||
| || || || || || ||67\116||49.5074|| | |||
|- | |||
| || || || ||26\45|| || ||49.5238||[[Flattone]] is in this region | |||
|- | |||
| || || || || || ||63\109||49.5413|| | |||
|- | |||
| || || || || ||37\64|| ||49.5535|| | |||
|- | |||
| || || || || || ||48\83||49.5697|| | |||
|- | |||
| || ||11\19|| || || || ||49.6241|| | |||
|- | |||
| || || || || || ||51\88||49.6753|| | |||
|- | |||
| || || || || ||40\69|| ||49.6894|| | |||
|- | |||
| || || || || || ||69\119||49.6999|| | |||
|- | |||
| || || || ||29\50|| || ||49.7143|| | |||
|- | |||
| || || || || || ||76\131||49.7274||[[Golden meantone]] (696.2145¢) | |||
|- | |||
| || || || || ||47\81|| ||49.73545|| | |||
|- | |||
| || || || || || ||65\112||49.7449|| | |||
|- | |||
| || || ||18\31|| || || ||49.7696||[[Meantone]] is in this region | |||
|- | |||
| || || || || || ||61\105||49.7959|| | |||
|- | |||
| || || || || ||43\74|| ||49.8070||The generator closest to a just [[4/3]] for EDOs less than 2800 | |||
|- | |||
| || || || || || ||68\117||49.81685|| | |||
|- | |||
| || || || ||25\43|| || ||49.8339|| | |||
|- | |||
| || || || || || ||57\98||49.8542|| | |||
|- | |||
| || || || || ||32\55|| ||49.8701|| | |||
|- | |||
| || || || || || ||39\67||49.8934|| | |||
|- | |||
| ||7\12|| || || || || ||50.0000|| | |||
|- | |||
| || || || || || ||38\65||50.1099|| | |||
|- | |||
| || || || || ||31\53|| ||50.1348||The fifth closest to a just [[3/2]] for EDOs less than 200 | |||
|- | |||
| || || || || || ||55\94||50.1520||[[Garibaldi]] / [[Cassandra]] | |||
|- | |||
| || || || ||24\41|| || ||50.1742|| | |||
|- | |||
| || || || || || ||65\111||50.19305|| | |||
|- | |||
| || || || || ||41\70|| ||50.2041|| | |||
|- | |||
| || || || || || ||58\99||50.21645|| | |||
|- | |||
| || || ||17\29|| || || ||50.2463|| | |||
|- | |||
| || || || || || ||61\104||50.2747|| | |||
|- | |||
| || || || || ||44\75|| ||50.2857|| | |||
|- | |||
| || || || || || ||71\121||50.2952||Golden neogothic (704.0956¢) | |||
|- | |||
| || || || ||27\46|| || ||50.3106||[[Neogothic]] is in this region | |||
|- | |||
| || || || || || ||64\109||50.32765|| | |||
|- | |||
| || || || || ||37\63|| ||50.3401|| | |||
|- | |||
| || || || || || ||47\80||50.3571|| | |||
|- | |||
| || ||10\17|| || || || ||50.4202|| | |||
|- | |||
| || || || || || ||43\73||50.4892|| | |||
|- | |||
| || || || || ||33\56|| ||50.5102|| | |||
|- | |||
| || || || || || ||56\95||50.5263|| | |||
|- | |||
| || || || ||23\39|| || ||50.54945|| | |||
|- | |||
| || || || || || ||59\100||50.5714|| | |||
|- | |||
| || || || || ||36\61|| ||50.5855|| | |||
|- | |||
| || || || || || ||49\83||50.6024|| | |||
|- | |||
| || || ||13\22|| || || ||50.64935||[[Archy]] is in this region | |||
|- | |||
| || || || || || ||42\71||50.7042|| | |||
|- | |||
| || || || || ||29\49|| ||50.7289|| | |||
|- | |||
| || || || || || ||45\76||50.7519|| | |||
|- | |||
| || || || ||16\27|| || ||50.79365|| | |||
|- | |||
| || || || || || ||35\59||50.8475|| | |||
|- | |||
| || || || || ||19\32|| ||50.8929|| | |||
|- | |||
| || || || || || ||22\37||50.96525|| | |||
|- | |||
|3\5|| || || || || || ||51.4286|| | |||
|}Tunings above 7\12 on this chart are called "negative tunings" (as they lessen the size of the fifth) and include meantone systems such as 1/3-comma (close to 11\19) and 1/4-comma (close to 18\31). As these tunings approach 4\7, the majors become flatter and the minors become sharper. | |||
[[Category:Edf]] | [[Category:Edf]] | ||
[[Category:Edonoi]] | [[Category:Edonoi]] | ||
[[Category:todo:improve synopsis]] | [[Category:todo:improve synopsis]] | ||