# 1619edo

← 1618edo | 1619edo | 1620edo → |

**1619 equal divisions of the octave** (abbreviated **1619edo**), or **1619-tone equal temperament** (**1619tet**), **1619 equal temperament** (**1619et**) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 1619 equal parts of about 0.741 ¢ each. Each step of 1619edo represents a frequency ratio of 2^{1/1619}, or the 1619th root of 2.

## Theory

1619edo is excellent in the 13-limit, where it tempers out 4225/4224, 4375/4374, 6656/6655, 78125/78078, and 117649/117612. It also notably tempers out quartisma (117440512/117406179) and 123201/123200.

1619edo tunes keenanisma very finely, to 6 steps, and can use it as a microchroma. 1619edo has 7/6 on 360th step, a highly divisible number, 27/25 on 180th, and 33/32 on 72nd as a consequence of tempering out the commas. This means that 72ed33/32 is virtually equivalent to 1619edo. When it comes to using 33/32 as the generator, 1619edo supports the ravine temperament, which tempers out 196625/196608, 200000/199927, 2912000/2910897, and 3764768/3764475.

Since 33/32 is close to 1\45, 7\6 is close to 1\9, and 385/384 is close to 1\270, 1619edo can be thought of as 1620edo where one step was extracted and all others were moved into a more harmonically just position. It achieves this because 1620edo is contorted 270edo in the 11-limit, and its 13/8 is on the flat side coming from 324edo, and thus when it is octave stretched, steps sharpen enough to arrive at 1619edo's 13-limit excellence.

1619edo supports a very precise rank two temperament, 19 & 1619, which uses 6/5 as a generator and has a comma basis 4375/4374, 91125/91091, 196625/196608, and 54925000/54908469.

1619edo supports the keenanose temperament, which has comma basis 4225/4224, 4375/4374, 6656/6655, and 151263/151250. Keenanisma is the generator in the keenanose temperament, 270 & 1619, in which it highlights the relationship between 270 keenanismas and the octave. It also achieves this since 270 × 6 = 1620, and 1619 is 1 short of that and also excellent in the 13-limit.

Another temperament which highlights the interval relationships in 1619edo is 45 & 1619, called *decigrave*, since 10 steps make a 7/6, which is referred to as the grave minor third sometimes. It has a comma basis 4225/4224, 4375/4374, 6656/6655, [23 5 13 -23 1 0⟩ in the 13-limit. Its generator is 36 steps, which represents 65/64 and 66/65 tempered together, and 2 of them make 33/32. 5 of them make 27/25, and 10 of them make 7/6.

1619edo supports the 494 & 1619 temperament called moulin, with the comma basis of 4225/4224, 4375/4374, 6656/6655, 91125/91091. The 25-tone scale of moulin is capable of supporting the 8:11:13 triad, as it takes less than 25 notes to map the 11th and 13th harmonics.

### The Vidarines

1619edo supports vidar, which has the comma basis 4225/4224, 4375/4374, and 6656/6655. In addition, it contains a wealth of rank-two 13-limit temperaments that are produced by adding one comma on top of the vidar comma basis;. Temperaments described above such as decigrave, keenanose, moulin, are members of this collection. Eliora proposes the name *The Vidarines* for this collection of temperaments.

A quick summary is shown below.

Temperament | Generator associated ratio |
Completing comma |
---|---|---|

Keenanose (270 & 1619) | 385/384 | 151263/151250 |

Decigrave (45 & 1619) | 66/65 ~ 65/64 | [23 5 13 -23 1 0⟩ |

Moulin (494 & 1619) | 13/11 | 91125/91091 |

46 & 1619 | 3328/3087 | [-18 9 -2 8 -3 -1⟩ |

178 & 1619 | 4429568/4084101 | [-29 10 2 12 -3 -4⟩ |

224 & 1619 | 256/175 | 18753525/18743296 |

764 & 1619 | 12375/8918 | 52734375/52706752 |

901 & 1619 | 104/99 | 34875815625/34843787264 |

While abigail is a member of the vidarines, 1619edo does not support it because abigail is a period-2 temperament, and 1619 is an odd number.

### Prime harmonics

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

Error | absolute (¢) | +0.000 | -0.040 | -0.149 | -0.080 | +0.134 | -0.009 | +0.295 | -0.293 | +0.262 | -0.053 | +0.116 |

relative (%) | +0 | -5 | -20 | -11 | +18 | -1 | +40 | -39 | +35 | -7 | +16 | |

Steps (reduced) |
1619 (0) |
2566 (947) |
3759 (521) |
4545 (1307) |
5601 (744) |
5991 (1134) |
6618 (142) |
6877 (401) |
7324 (848) |
7865 (1389) |
8021 (1545) |

### Subsets and supersets

1619edo is the 256th prime edo.

## Selected intervals

Step | Cents | Ratio | Name* |
---|---|---|---|

0 | 0.000 | 1/1 | prime, unison |

6 | 4.447 | 385/384 | keenanisma |

72 | 53.366 | 33/32 | al-Farabi quarter-tone |

360 | 266.831 | 7/6 | septimal subminor third, grave minor third |

744 | 551.451 | 11/8 | 11th harmonic, undecimal superfourth |

1134 | 840.519 | 13/8 | 13th harmonic, tridecimal neutral sixth |

1619 | 1200.000 | 2/1 | perfect octave |

* named in accordance to their most just 13-limit counterpart using the names accepted on the wiki.

## Regular temperament properties

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

Absolute (¢) | Relative (%) | ||||

2.3 | [-2566 1619⟩ | [⟨1619 2566]] | +0.0127 | 0.0127 | 1.71 |

2.3.5 | [-69 45 -1⟩, [-82 -1 36⟩ | [⟨1619 2566 3759]] | +0.0299 | 0.0265 | 3.57 |

2.3.5.7 | 4375/4374, 52734375/52706752, [-67 14 6 11⟩ | [⟨1619 2566 3759 4545]] | +0.0295 | 0.0229 | 3.09 |

2.3.5.7.11 | 4375/4374, 117649/117612, 759375/758912, [24 -6 0 1 -5⟩ | [⟨1619 2566 3759 4545 5601]] | +0.0159 | 0.0341 | 4.60 |

2.3.5.7.11.13 | 4225/4224, 4375/4374, 6656/6655, 78125/78078, 117649/117612 | [⟨1619 2566 3759 4545 5601 5991]] | +0.0136 | 0.0315 | 4.26 |

### Rank-2 temperaments

Periods per Octave |
Generator* | Cents* | Associated Ratio |
Temperaments |
---|---|---|---|---|

1 | 6\1619 | 4.447 | 385/384 | Keenanose |

1 | 36\1619 | 26.683 | 65/64 | Decigrave |

1 | 72\1619 | 53.366 | 33/32 | Ravine |

1 | 112\1619 | 83.014 | 1573/1500 | Acrosextilififths |

1 | 390\1619 | 289.067 | 13/11 | Moulin |

1 | 426\1619 | 315.750 | 6/5 | Oviminor |

1 | 587\1619 | 435.083 | 9/7 | Supermajor |

1 | 672\1619 | 498.085 | 4/3 | Counterschismic |

* octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if it is distinct