# 3edf

← 2edf | 3edf | 4edf → |

(convergent)

**3edf**, if the attempt is made to use it as an actual scale, would divide the just perfect fifth into three equal parts, each of size 233.985 cents, which is to say (3/2)^{1/3} as a frequency ratio. It corresponds to 5.1285 edo. If we want to consider it to be a temperament, it tempers out 16/15, 21/20, 28/27, 81/80, and 256/243 as well as 5edo.

## Factoids about 3edf

3edf's step size is close to the slendric temperament, which tempers out 1029/1024 in the 2.3.7 subgroup.

## Intervals

# | Cents |
---|---|

1 | 233.99 |

2 | 467.97 |

3 | 701.96 |

## Scale tree

EDF scales can be approximated in EDOs by subdividing diatonic fifths. 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 "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 EDOs pop up in our continuum.

Generator range: 228.5714 cents (4\7/3 = 4\21) to 240 cents (3\5/3 = 1\5)

Fifth | Cents | Comments | ||||||
---|---|---|---|---|---|---|---|---|

4\7 | 228.571 | |||||||

27\47 | 229.787 | |||||||

23\40 | 230.000 | |||||||

42\73 | 230.137 | |||||||

19\33 | 230.30 | |||||||

53\92 | 230.435 | |||||||

34\59 | 230.5085 | |||||||

49\85 | 230.588 | |||||||

15\26 | 230.769 | |||||||

56\97 | 230.928 | |||||||

41\71 | 230.986 | |||||||

67\116 | 231.0345 | |||||||

26\45 | 231.1 | Flattone is in this region | ||||||

63\109 | 231.193 | The generator closest to a just 8/7 for EDOs less than 600 | ||||||

37\64 | 231.25 | |||||||

48\83 | 231.325 | |||||||

11\19 | 231.579 | |||||||

51\88 | 231.81 | |||||||

40\69 | 231.884 | |||||||

69\119 | 231.933 | |||||||

29\50 | 232.000 | |||||||

76\131 | 232.061 | Golden meantone (696.2145¢) | ||||||

47\81 | 232.099 | |||||||

65\112 | 232.143 | |||||||

18\31 | 232.258 | Meantone is in this region | ||||||

61\105 | 232.381 | |||||||

43\74 | 232.432 | |||||||

68\117 | 232.479 | |||||||

25\43 | 232.558 | |||||||

57\98 | 232.653 | |||||||

32\55 | 232.72 | |||||||

39\67 | 232.836 | |||||||

7\12 | 233.3 | |||||||

38\65 | 233.846 | |||||||

31\53 | 233.962 | The fifth closest to a just 3/2 for EDOs less than 200 | ||||||

55\94 | 234.043 | Garibaldi / Cassandra | ||||||

24\41 | 234.146 | |||||||

65\111 | 234.234 | |||||||

41\70 | 234.286 | |||||||

58\99 | 234.34 | |||||||

17\29 | 234.483 | |||||||

61\104 | 234.615 | |||||||

44\75 | 234.6 | |||||||

71\121 | 234.711 | Golden neogothic (704.0956¢) | ||||||

27\46 | 234.783 | Neogothic is in this region | ||||||

64\109 | 234.862 | |||||||

37\63 | 234.921 | |||||||

47\80 | 235.000 | |||||||

10\17 | 235.294 | |||||||

43\73 | 235.616 | |||||||

33\56 | 235.714 | |||||||

56\95 | 235.7895 | |||||||

23\39 | 235.897 | |||||||

59\100 | 236.000 | |||||||

36\61 | 236.066 | |||||||

49\83 | 236.145 | |||||||

13\22 | 236.36 | Archy is in this region | ||||||

42\71 | 236.620 | |||||||

29\49 | 236.735 | |||||||

45\76 | 236.842 | |||||||

16\27 | 237.037 | |||||||

35\59 | 237.288 | |||||||

19\32 | 237.500 | |||||||

22\37 | 237.837 | |||||||

3\5 | 240.000 |

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.

Tunings below 7\12 on this chart are called "positive tunings" and they include Pythagorean tuning itself (well approximated by 31\53) as well as superpyth tunings such as 10\17 and 13\22. As these tunings approach 3\5, the majors become sharper and the minors become flatter. Around 13\22 through 16\27, the thirds fall closer to 7-limit than 5-limit intervals: 7:6 and 9:7 as opposed to 6:5 and 5:4.

## Music

- Sequences & Chaos by Bazil Müzik