# 80edo

The **80 equal temperament**, often abbreviated 80-tET, 80-EDO, or 80-ET, is the scale derived by dividing the octave into 80 equally-sized steps. Each step is exactly 15 cents.

## Theory

80et is the first equal temperament that represents the 19-limit tonality diamond consistently, though it barely manages to do so.

80et tempers out 176/175 and 540/539 in the 11-limit, 169/168, 325/324, 351/350, 352/351, 364/363 and 1001/1000 in the 13-limit, 136/135, 221/220, 256/255, 289/288, 561/560, 595/594, 715/714, 936/935, 1275/1274 in the 17-limit, 190/189, 286/285, 361/360, 400/399, 456/455, 476/475, 969/968, 1331/1330, 1445/1444, 1521/1520, 1540/1539 and 1729/1728 in the 19-limit, not to mention such important non-superparticular commas as 2048/2025, 4000/3969, 1728/1715 and 3136/3125.

80et provides the optimal patent val for 5-limit diaschismic, for 13-limit srutal, and for 7-, 11- and 13-limit bidia. It is a good tuning for various temperaments in canou family, especially in higher limits.

80et supports a profusion of 19-limit (and lower) rank two temperaments which have mostly not been explored. We might mention:

31&80 <<7 6 15 27 -24 -23 -20 ... ||

72&80 <<24 30 40 24 32 24 0 ... ||

34&80 <<2 -4 -50 22 16 2 -40 ... ||

46&80 <<2 -4 30 22 16 2 40 ... ||

29&80 <<3 34 45 33 24 -37 20 ... ||

12&80 <<4 -8 -20 -36 32 4 0 ... ||

22&80 <<6 -10 12 -14 -32 6 -40 ... ||

58&80 <<6 -10 12 -14 -32 6 40 ... ||

41&80 <<7 26 25 -3 -24 -33 20 ... ||

In each case, the numbers joined by an ampersand represent 19-limit patent vals (meaning obtained by rounding to the nearest integer) and the first and most important part of the wedgie is given.

## Intervals

Degree | Cents | Approximate Ratios* |
---|---|---|

0 | 0 | 1/1 |

1 | 15 | 64/63 |

2 | 30 | 81/80, 50/49 |

3 | 45 | 36/35, 49/48, 34/33 |

4 | 60 | 28/27, 33/32, 26/25, 35/34 |

5 | 75 | 25/24, 22/21, 27/26 |

6 | 90 | 21/20, 19/18, 20/19 |

7 | 105 | 16/15, 17/16, 18/17 |

8 | 120 | 15/14, 14/13 |

9 | 135 | 13/12 |

10 | 150 | 12/11 |

11 | 165 | 11/10 |

12 | 180 | 10/9, 21/19 |

13 | 195 | 19/17 |

14 | 210 | 9/8, 17/15 |

15 | 225 | 8/7 |

16 | 240 | |

17 | 255 | 81/70, 15/13, 22/19 |

18 | 270 | 7/6 |

19 | 285 | 13/11, 20/17 |

20 | 300 | 25/21, 19/16 |

21 | 315 | 6/5 |

22 | 330 | 17/14 |

23 | 345 | 11/9 |

24 | 360 | 16/13 |

25 | 375 | 21/17 |

26 | 390 | 5/4 |

27 | 405 | 24/19, 19/15 |

28 | 420 | 14/11 |

29 | 435 | 9/7 |

30 | 450 | 35/27, 13/10, 22/17 |

31 | 465 | 17/13 |

32 | 480 | 21/16, 25/19 |

33 | 495 | 4/3 |

34 | 510 | |

35 | 525 | 19/14 |

36 | 540 | 26/19 |

37 | 555 | 11/8 |

38 | 570 | 18/13 |

39 | 585 | 7/5 |

40 | 600 | 17/12, 24/17 |

… | … | … |

* based on treating 80edo as a 19-limit temperament; other approaches are possible.

## Just approximation

prime 2 | prime 3 | prime 5 | prime 7 | prime 11 | prime 13 | prime 17 | prime 19 | prime 23 | prime 29 | prime 31 | ||
---|---|---|---|---|---|---|---|---|---|---|---|---|

Error | absolute (¢) | 0.00 | +3.04 | +3.69 | +6.17 | +3.68 | -0.53 | +0.04 | +2.49 | +1.73 | +5.42 | -5.04 |

relative (%) | 0.0 | +20.3 | +24.6 | +41.1 | +24.5 | -3.5 | +0.3 | +16.6 | +11.5 | +36.2 | -33.6 |