# 2022edo

← 2021edo | 2022edo | 2023edo → |

**2022 equal divisions of the octave** (abbreviated **2022edo** or **2022ed2**), also called **2022-tone equal temperament** (**2022tet**) or **2022 equal temperament** (**2022et**) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 2022 equal parts of about 0.593 ¢ each. Each step represents a frequency ratio of 2^{1/2022}, or the 2022nd root of 2.

## Theory

2022edo is only consistent to the 5-odd-limit since harmonic 7 is about halfway between its steps. Nonetheless, it offers good appoximations of the 2.3.5.11.17.29.41.43.53.61 subgroup. When using smaller numbers, 2.3.5.11 is a good choice, and if rougher errors are allowed, no-sevens 29-limit is a satisfactory choice.

In the 5-limit, 2022edo supports the pirate temperament, 323 & 407, and tempers out the [-90 -15 49⟩ comma.

In the 2.3.5.11 subgroup, 2022edo supports the rank-3 temperament that eliminates the [25 -17 -23 16⟩ comma. If the 11-limit is taken as a whole, 2022edo tempers out 3025/3024 and 4375/4374 when it is 7/4 is put on the 1633rd step (2022d val), and 41503/41472 with 250047/250000 when using the 1632nd step of the patent val.

In the 2.5.11.17.29.41.43.53.61 subgroup, 2022edo tempers out 17630/17629, 18491/18490, 21200/21199, and 22528/22525.

If the 29-limit is taken as a whole even including the 7-limit inconsistency, 2022edo tempers out 2002/2001, 3451/3450, 5104/5103, and 16445/16443.

### Prime harmonics

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

Error | Absolute (¢) | +0.000 | +0.122 | +0.036 | -0.280 | +0.017 | -0.172 | +0.089 | -0.184 | +0.212 | +0.096 | -0.228 |

Relative (%) | +0.0 | +20.6 | +6.1 | -47.2 | +2.9 | -28.9 | +15.0 | -30.9 | +35.8 | +16.2 | -38.5 | |

Steps (reduced) |
2022 (0) |
3205 (1183) |
4695 (651) |
5676 (1632) |
6995 (929) |
7482 (1416) |
8265 (177) |
8589 (501) |
9147 (1059) |
9823 (1735) |
10017 (1929) |

### Subsets and supersets

Since 2022 factors into 2 × 3 × 337, 2022edo has subset edos 2, 3, 6, 337, 674, and 1011.

## Regular temperament properties

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

Absolute (¢) | Relative (%) | ||||

2.3 | [3205 -2022⟩ | [⟨2022 3205]] | -0.038534 | 0.038533 | 6.493 |

2.3.5 | [25 -48 22⟩, [-90 -15 49⟩ | [⟨2022 3205 4695]] | -0.030920 | 0.033254 | 5.603 |

2.3.5.11.13.17.19.23.29 | 2431/2430, 2755/2754, 3520/3519, 142025/141984, 2582624/2581875, 9096256/9092061, 11293425/11290976, 51054848/51046875 | [⟨2022 3205 4695 6955 7482 8265 8589 9147 9283]] | -0.010752 | 0.036910 | 6.219 |

## Music

*Noble Gas*(2021)