# 14edf

← 13edf | 14edf | 15edf → |

**Division of the just perfect fifth into 14 equal parts** (14EDF) is related to 24 edo, but with the 3/2 rather than the 2/1 being just. The octave is about 3.3514 cents stretched and the step size is about 50.1396 cents. The patent val has a generally sharp tendency for harmonics up to 22, with the exception for 7, 14, and 21.

## Intervals

Degree | |
---|---|

0 | 0 |

1 | 50.1396 |

2 | 100.2793 |

3 | 150.4189 |

4 | 200.5586 |

5 | 250.6982 |

6 | 300.8379 |

7 | 350.9775 |

8 | 401.1171 |

9 | 451.2568 |

10 | 501.3964 |

11 | 551.536 |

12 | 601.6757 |

13 | 651.8154 |

14 | 701.955 |

15 | 752.0946 |

16 | 802.2343 |

17 | 852.3739 |

18 | 902.5136 |

19 | 952.6532 |

20 | 1002.7929 |

21 | 1052.9235 |

22 | 1103.0721 |

23 | 1153.2118 |

24 | 1203.3514 |

25 | 1253.4911 |

26 | 1303.6307 |

27 | 1353.7704 |

28 | 1403.91 |

## 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 "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: 48.97959 cents (4\7/14 = 2\49) to 51.42857 cents (3\5/14 = 3\70)

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.