# 103edo

← 102edo | 103edo | 104edo → |

**103 equal divisions of the octave** (abbreviated **103edo** or **103ed2**), also called **103-tone equal temperament** (**103tet**) or **103 equal temperament** (**103et**) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 103 equal parts of about 11.650 ¢ each. Each step represents a frequency ratio of 2^{1/103}, or the 103rd root of 2.

## Theory

103edo is a good miracle tuning, especially for the 7-limit, and for benediction and hemisecordite, two of the 13-limit extensions of miracle. It tempers out 78732/78125 in the 5-limit; 225/224, 1029/1024 and 2401/2400 in the 7-limit; 243/242, 441/440 and 540/539 in the 11-limit; 351/350 and 847/845 in the 13-limit. In the 13-limit it provides the optimal patent val for marvel temperament as well as benediction and hemisecordite.

In 103edo, all intervals within the 17-odd-limit are consistent, with the sole exception of 9/8 and its octave complement 16/9, which barely miss (relative error 50.2%).

### Prime harmonics

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

Error | absolute (¢) | +0.00 | -2.93 | -1.85 | -1.84 | -3.75 | -1.69 | -0.10 | +5.40 | +0.85 | -4.33 | -3.29 |

relative (%) | +0 | -25 | -16 | -16 | -32 | -15 | -1 | +46 | +7 | -37 | -28 | |

Steps (reduced) |
103 (0) |
163 (60) |
239 (33) |
289 (83) |
356 (47) |
381 (72) |
421 (9) |
438 (26) |
466 (54) |
500 (88) |
510 (98) |

### Subsets and supersets

103edo is the 27th prime edo.

## Intervals

*Main article: Table of 103edo intervals*

## Regular temperament properties

Subgroup | Comma List | Mapping | Optimal 8ve Stretch (¢) |
Tuning Error | |
---|---|---|---|---|---|

Absolute (¢) | Relative (%) | ||||

2.3 | [-163 103⟩ | [⟨103 166]] | +0.923 | 0.924 | 7.92 |

2.3.5 | 78732/78125, 34171875/33554432 | [⟨103 166 239]] | +0.881 | 0.757 | 6.49 |

2.3.5.7 | 225/224, 1029/1024, 78732/78125 | [⟨103 166 239 289]] | +0.824 | 0.663 | 5.68 |

2.3.5.7.11 | 225/224, 243/242, 385/384, 43923/43750 | [⟨103 166 239 289 356]] | +0.876 | 0.602 | 5.16 |

2.3.5.7.11.13 | 225/224, 243/242, 351/350, 385/384, 847/845 | [⟨103 166 239 289 356 381]] | +0.806 | 0.571 | 4.90 |

2.3.5.7.11.13.17 | 225/224, 243/242, 273/272, 351/350, 375/374, 847/845 | [⟨103 166 239 289 356 381 421]] | +0.694 | 0.595 | 5.10 |

Its closest zeta peak index, 596zpi, stretches the octave by +0.739 cents. This expansion is both consistent and uniquely consistent within the 15-integer limit.

### Rank-2 temperaments

Periods per 8ve |
Generator (Reduced) |
Cents (Reduced) |
Associated Ratio |
Temperaments |
---|---|---|---|---|

1 | 3\103 | 34.951 | 1990656/1953125 | Gammic (5-limit) |

1 | 5\103 | 58.252 | 27/26 | Hemisecordite |

1 | 9\103 | 104.854 | 17/16 | Septendesemi |

1 | 10\103 | 116.505 | 15/14~16/15 | Miracle / benediction |

1 | 16\103 | 186.408 | 10/9 | Mintone |

1 | 20\103 | 233.010 | 8/7 | Slendric |

1 | 21\103 | 244.660 | 15/13 | Subsemifourth |

1 | 26\103 | 303.013 | 25/21 | Quinmite |

1 | 31\103 | 361.165 | 16/13 | Phicordial |

1 | 37\103 | 431.06 | 77/60 | Lockerbie |

1 | 38\103 | 442.708 | 162/125 | Sensei |

1 | 39\103 | 454.369 | 13/10 | Fibo |

1 | 40\103 | 466.019 | 55/42 | Hemiseptisix |

1 | 42\103 | 489.320 | 65/49 | Catafourth |

1 | 45\103 | 524.272 | 65/48 | Widefourth |

1 | 47\103 | 547.573 | 11/8 | Heinz |

1 | 48\103 | 559.223 | 242/175 | Tritriple |

1 | 50\103 | 582.524 | 7/5 | Neptune |