# 7edt

← 6edt | 7edt | 8edt → |

**7 equal divisions of the tritave**, **perfect twelfth**, or **3rd harmonic** (abbreviated **7edt** or **7ed3**), is a nonoctave tuning system that divides the interval of 3/1 into 7 equal parts of about 272 ¢ each. Each step represents a frequency ratio of 3^{1/7}, or the 7th root of 3.

## Theory

Since one step of 7edt approximates a 7/6 subminor third (4.84 cents sharp) quite nicely, three steps are almost exactly 8/5 (tempering out 1728/1715, the orwellisma), and four steps are very nearly 15/8 (tempering out 2430/2401, the nuwell comma). 7edt is the lowest equal division of the tritave to accurately approximate some 7-limit harmony, along with some elements of the 11-limit, such as the 11/8 major fourth. Seven steps make up a tritave, meaning that 7edt tempers out 839808/823543, the eric comma.

Due to the proximity of the step size with 7/6, 7edt supports orwell temperament. One step of 7edt is almost identical to 12\53, the 53edo orwell generator, at about 271.698 cents. 7edt is also a good tuning for Electra temperament, with two steps of 7edt being a close approximation to 15/11.

### Harmonics

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

Error | Absolute (¢) | -113 | +0 | +45 | -69 | -113 | -108 | -68 | +0 | +89 | -76 | +45 |

Relative (%) | -41.7 | +0.0 | +16.7 | -25.5 | -41.7 | -39.9 | -25.0 | +0.0 | +32.9 | -27.9 | +16.7 | |

Steps (reduced) |
4 (4) |
7 (0) |
9 (2) |
10 (3) |
11 (4) |
12 (5) |
13 (6) |
14 (0) |
15 (1) |
15 (1) |
16 (2) |

### Prime harmonics

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

Error | Absolute (¢) | -113 | +0 | -69 | -108 | -76 | -93 | -14 | +65 | +6 | -124 | +33 |

Relative (%) | -41.7 | +0.0 | -25.5 | -39.9 | -27.9 | -34.3 | -5.2 | +23.9 | +2.2 | -45.5 | +12.0 | |

Steps (reduced) |
4 (4) |
7 (0) |
10 (3) |
12 (5) |
15 (1) |
16 (2) |
18 (4) |
19 (5) |
20 (6) |
21 (0) |
22 (1) |

## Scale degrees of 7edt

Degrees | Cents | Hekts | Approximate Ratio | Electra notation (J = 1/1) |
---|---|---|---|---|

0 | 1/1 | J | ||

1 | 271.708 | 185.714 | 7/6 | K |

2 | 543.416 | 371.429 | 15/11, 11/8 | L |

3 | 815.124 | 557.143 | 8/5 | M |

4 | 1086.831 | 742.857 | 15/8 | N |

5 | 1358.539 | 928.571 | 11/5 (11/10 plus an octave) | O |

6 | 1630.247 | 1114.286 | 18/7 (9/7 plus an octave) | P |

7 | 1901.955 | 1300 | 3/1 | J |