# 13ed5

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Prime factorization
13 (prime)
Step size
214.332¢
Octave
6\13ed5 (1285.99¢)
Twelfth
9\13ed5 (1928.99¢)

(semiconvergent)
Consistency limit
3
Distinct consistency limit
3

← 12ed5 | 13ed5 | 14ed5 → |

(semiconvergent)

**13 equal divisions of the 5th harmonic** (abbreviated **13ed5**) is a nonoctave tuning system that divides the interval of 5/1 into 13 equal parts of about 214 ¢ each. Each step represents a frequency ratio of 5^{1/13}, or the 13th root of 5.

What is known as "father" has a large 3:2 which warps the heptatonic scale to an octatonic.

Analogously, in hyperpyth, 13ed5 has an analogously large "13/5” which warps the scale similarly. It is not readily apparent what consonance(s) it approximates, but some listeners have noted the sound to be enjoyable.

18ed5 also does this; the "13/5" is actually 50 cents sharp, while 8/3 is much closer.

## Harmonics

Harmonic | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | |
---|---|---|---|---|---|---|---|---|---|---|---|---|

Error | Absolute (¢) | +86.0 | +27.0 | -42.3 | +0.0 | -101.3 | +60.5 | +43.6 | +54.1 | +86.0 | -79.0 | -15.3 |

Relative (%) | +40.1 | +12.6 | -19.8 | +0.0 | -47.3 | +28.2 | +20.4 | +25.2 | +40.1 | -36.9 | -7.1 | |

Steps (reduced) |
6 (6) |
9 (9) |
11 (11) |
13 (0) |
14 (1) |
16 (3) |
17 (4) |
18 (5) |
19 (6) |
19 (6) |
20 (7) |

Harmonic | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | |
---|---|---|---|---|---|---|---|---|---|---|---|---|

Error | Absolute (¢) | +60.4 | -67.9 | +27.0 | -84.7 | +24.7 | -74.3 | +46.5 | -42.3 | +87.5 | +7.0 | -70.0 |

Relative (%) | +28.2 | -31.7 | +12.6 | -39.5 | +11.5 | -34.7 | +21.7 | -19.8 | +40.8 | +3.3 | -32.6 | |

Steps (reduced) |
21 (8) |
21 (8) |
22 (9) |
22 (9) |
23 (10) |
23 (10) |
24 (11) |
24 (11) |
25 (12) |
25 (12) |
25 (12) |

## Intervals

Steps | Cents | Approximate Ratios |
---|---|---|

0 | 0 | 1/1 |

1 | 214.332 | 17/15, 19/17 |

2 | 428.664 | 9/7, 13/10, 19/15, 22/17 |

3 | 642.995 | 10/7, 13/9, 19/13, 22/15 |

4 | 857.327 | |

5 | 1071.659 | 13/7 |

6 | 1285.991 | 19/9, 21/10, 23/11 |

7 | 1500.323 | |

8 | 1714.655 | 19/7 |

9 | 1928.986 | |

10 | 2143.318 | 7/2, 17/5 |

11 | 2357.65 | |

12 | 2571.982 | 22/5 |

13 | 2786.314 | 5/1 |