# 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 | Size (Cents) |
---|---|

0 | 0.0000 |

1 | 18.4615 |

2 | 36.9231 |

3 | 55.3846 |

4 | 73.8462 |

5 | 92.3077 |

6 | 110.7692 |

7 | 129.2308 |

8 | 147.6923 |

9 | 166.1538 |

10 | 184.6154 |

11 | 203.0769 |

12 | 221.5385 |

13 | 240.0000 |

14 | 258.4615 |

15 | 276.9231 |

16 | 295.3846 |

17 | 313.8462 |

18 | 332.3077 |

19 | 350.7692 |

20 | 369.2308 |

21 | 387.6923 |

22 | 406.1538 |

23 | 424.6154 |

24 | 443.0769 |

25 | 461.5385 |

26 | 480.0000 |

27 | 498.4615 |

28 | 516.9231 |

29 | 535.3846 |

30 | 553.8462 |

31 | 572.3077 |

32 | 590.7692 |

33 | 609.2308 |

34 | 627.6923 |

35 | 646.1538 |

36 | 664.6154 |

37 | 683.0769 |

38 | 701.5385 |

39 | 720.0000 |

40 | 738.4615 |

41 | 756.9231 |

42 | 775.3846 |

43 | 793.8462 |

44 | 812.3077 |

45 | 830.7692 |

46 | 849.2308 |

47 | 867.6923 |

48 | 886.1538 |

49 | 904.6154 |

50 | 923.0769 |

51 | 941.5385 |

52 | 960.0000 |

53 | 978.4615 |

54 | 996.9231 |

55 | 1015.3846 |

56 | 1033.8462 |

57 | 1052.3077 |

58 | 1070.7692 |

59 | 1089.2308 |

60 | 1107.6923 |

61 | 1126.1538 |

62 | 1144.6154 |

63 | 1163.0769 |

64 | 1181.5385 |

65 | 1200.0000 |