# 65edo

# 65 tone equal temperament

**65edo** divides the octave into 65 equal parts of 18.4615 cents each. It can be characterized as the temperament which tempers out the schisma, 32805/32768, the sensipent comma, 78732/78125, and the wuerschmidt comma. In the 7-limit, there are two different maps; the first is <65 103 151 182|, tempering out 126/125, 245/243 and 686/675, so that 65edo supports sensi temperament, and the second is <65 103 151 183|, tempering out 225/224, 3125/3097, 4000/3969 and 5120/5103, so that 65edo supports garibaldi temperament. In both cases, the tuning privileges the 5-limit over the 7-limit, as the 5-limit of 65 is quite accurate. The same can be said for the two different versions of 7-limit wuerschmidt temperament (wurschmidt and worschmidt) these two mappings provide.

65edo approximates the intervals 3/2, 5/4, 11/8 and 19/16 well, so that it does a good job representing the 2.3.5.11.19 just intonation subgroup. To this one may want to add 13/8 and 17/16, giving the 19-limit no-sevens subgroup 2.3.5.11.13.17.19. Also of interest is the 19-limit 2*65 subgroup 2.3.5.49.11.91.119.19, on which 65 has the same tuning and commas as 130edo.

65edo contains 13edo as a subset. The offset between a just perfect fifth at 702 cents and the 13edo superfifth at 738.5 cents, is approximately 2 degrees of 65edo. Therefore, an instrument fretted to 13edo, with open strings tuned to 3-limit intervals such as 4/3, 3/2, 9/8, 16/9 etc, will approximate a subset of 65edo. For an example of this, see Rubble: a Xenuke Unfolded.

# Intervals

Degree | Cents | Ups and Downs | |
---|---|---|---|

0 | 0.0000 | P1 | D |

1 | 18.4615 | ^1 | ^D |

2 | 36.9231 | ^^1 | ^^D |

3 | 55.3846 | vvm2 | vvEb |

4 | 73.84615 | vm2 | vEb |

5 | 92.3077 | m2 | Eb |

6 | 110.7692 | A1/^m2 | D#/^Eb |

7 | 129.2308 | v~2 | ^^Eb |

8 | 147.6923 | ~2 | vvvE |

9 | 166.15385 | ^~2 | vvE |

10 | 184.6154 | vM2 | vE |

11 | 203.0769 | M2 | E |

12 | 221.5385 | ^M2 | ^E |

13 | 240 | ^^M2 | ^^E |

14 | 258.4615 | vvm3 | vvF |

15 | 276.9231 | vm3 | vF |

16 | 295.3846 | m3 | F |

17 | 313.84615 | ^m3 | ^F |

18 | 332.3077 | v~3 | ^^F |

19 | 350.7692 | ~3 | ^^^F |

20 | 369.2308 | ^~3 | vvF# |

21 | 387.6923 | vM3 | vF# |

22 | 406.15385 | M3 | F# |

23 | 424.6154 | ^M3 | ^F# |

24 | 443.0769 | ^^M3 | ^^F# |

25 | 461.5385 | vv4 | vvG |

26 | 480 | v4 | vG |

27 | 498.4615 | P4 | G |

28 | 516.9231 | ^4 | ^G |

29 | 535.3846 | v~4 | ^^G |

30 | 553.84615 | ~4 | ^^^G |

31 | 572.3077 | ^~4/vd5 | vvG#/vAb |

32 | 590.7692 | vA4/d5 | vG#/Ab |

33 | 609.2308 | A4/^d5 | G#/^Ab |

34 | 627.6923 | ^A4/v~5 | ^G#/^^Ab |

35 | 646.1538 | ~5 | vvvA |

36 | 664.6154 | ^~5 | vvA |

37 | 683.0769 | v5 | vA |

38 | 701.5385 | P5 | A |

39 | 720 | ^5 | ^A |

40 | 738.4615 | ^^5 | ^^A |

41 | 756.9231 | vvm6 | vvBb |

42 | 775.3846 | vm6 | vBb |

43 | 793.84615 | m6 | Bb |

44 | 812.3077 | ^m6 | ^Bb |

45 | 830.7692 | v~6 | ^^Bb |

46 | 849.2308 | ~6 | vvvB |

47 | 867.6923 | ^~6 | vvB |

48 | 886.15385 | vM6 | vB |

49 | 904.6154 | M6 | B |

50 | 923.0769 | ^M6 | ^B |

51 | 941.5385 | ^^M6 | ^^B |

52 | 960 | vvm7 | vvC |

53 | 978.4615 | vm7 | vC |

54 | 996.9231 | m7 | C |

55 | 1015.3846 | ^m7 | ^C |

56 | 1033.84615 | v~7 | ^^C |

57 | 1052.3077 | ~7 | ^^^C |

58 | 1070.7692 | ^~7 | vvC# |

59 | 1089.2308 | vM7 | vC# |

60 | 1107.6923 | M7 | C# |

61 | 1126.15385 | ^M7 | ^C# |

62 | 1144.6154 | ^^M7 | ^^C# |

63 | 1163.0769 | vv8 | vvD |

64 | 1181.5385 | v8 | vD |

65 | 1200 | P8 | D |