# 15ed4

← 13ed4 | 15ed4 | 17ed4 → |

**15 equal divisions of the 4th harmonic** (abbreviated **15ed4**) is a nonoctave tuning system that divides the interval of 4/1 into 15 equal parts of exactly 160 ¢ each. Each step represents a frequency ratio of 4^{1/15}, or the 15th root of 4. It is every second step of 15edo.

Lookalikes: 12edt

## Basics

15ED4 is a macrotonal tuning system that divides the double octave (4/1) into 15 equally spaced pitches each 160 cents apart. It can be viewed as a subset of 15 EDO that repeats at two octaves rather than one. It has a big xen appeal as it is nonoctave but at the same time contains the double octave allowing for a sort of best of both worlds approach. 15ED4 doesn't do Just Intonation well for the most part but it does represent 7/4 and 11/10 rather well, so it can be viewed as a 3.4.7.10.11 subgroup temperament tempering out 28/27, 49/48, 55/54, and 77/75 (as well as 12EDT).

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

Error | Absolute (¢) | +80.0 | +18.0 | +0.0 | -66.3 | -62.0 | -8.8 | +80.0 | +36.1 | +13.7 | +8.7 | +18.0 |

Relative (%) | +50.0 | +11.3 | +0.0 | -41.4 | -38.7 | -5.5 | +50.0 | +22.6 | +8.6 | +5.4 | +11.3 | |

Steps (reduced) |
8 (8) |
12 (12) |
15 (0) |
17 (2) |
19 (4) |
21 (6) |
23 (8) |
24 (9) |
25 (10) |
26 (11) |
27 (12) |

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

Error | Absolute (¢) | +39.5 | +71.2 | -48.3 | +0.0 | +55.0 | -43.9 | +22.5 | -66.3 | +9.2 | -71.3 | +11.7 |

Relative (%) | +24.7 | +44.5 | -30.2 | +0.0 | +34.4 | -27.4 | +14.1 | -41.4 | +5.8 | -44.6 | +7.3 | |

Steps (reduced) |
28 (13) |
29 (14) |
29 (14) |
30 (0) |
31 (1) |
31 (1) |
32 (2) |
32 (2) |
33 (3) |
33 (3) |
34 (4) |

## MOS Scales

There are a variety of Mos scales available. William Lynch considers the symmetrical MOS: LsLsLsLs to be the most consonant scale in 15ED4. There is also a LsLLsLLLs scale which is more spacey sounding than the symmetrical MOS. The symmetrical MOS can be harmonized with c.

## Intervals

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

0 | 0 | 1/1 |

1 | 160 | 11/10, 21/19, 23/21 |

2 | 320 | 17/14, 23/19 |

3 | 480 | 17/13 |

4 | 640 | 13/9, 19/13 |

5 | 800 | 11/7 |

6 | 960 | 19/11 |

7 | 1120 | 17/9, 19/10, 21/11 |

8 | 1280 | 19/9, 21/10, 23/11 |

9 | 1440 | 23/10 |

10 | 1600 | |

11 | 1760 | |

12 | 1920 | 3/1 |

13 | 2080 | 10/3, 23/7 |

14 | 2240 | 11/3 |

15 | 2400 |