# 2000edo

← 1999edo | 2000edo | 2001edo → |

^{4}× 5^{3}**2000 equal divisions of the octave** (abbreviated **2000edo** or **2000ed2**), also called **2000-tone equal temperament** (**2000tet**) or **2000 equal temperament** (**2000et**) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 2000 equal parts of exactly 0.6 ¢ each. Each step represents a frequency ratio of 2^{1/2000}, or the 2000th root of 2.

## Theory

2000edo is distinctly consistent through the 29-odd-limit and a strong no-31's 41-limit system; the only smaller edo with a smaller 29-limit relative error being 1578edo. The only ones superior to it in the 23-limit are 1578- and 1889edo, and in the 19-limit, nothing smaller defeats it.

### Prime harmonics

Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Error | Absolute (¢) | +0.000 | +0.045 | +0.086 | +0.174 | +0.082 | +0.072 | +0.045 | +0.087 | -0.074 | +0.023 | -0.236 | +0.056 |

Relative (%) | +0.0 | +7.5 | +14.4 | +29.0 | +13.7 | +12.1 | +7.4 | +14.5 | -12.4 | +3.8 | -39.3 | +9.3 | |

Steps (reduced) |
2000 (0) |
3170 (1170) |
4644 (644) |
5615 (1615) |
6919 (919) |
7401 (1401) |
8175 (175) |
8496 (496) |
9047 (1047) |
9716 (1716) |
9908 (1908) |
10419 (419) |

Harmonic | 41 | 43 | 47 | 53 | 59 | 61 | 67 | 71 | 73 | 79 | 83 | 89 | |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Error | Absolute (¢) | -0.062 | +0.282 | -0.107 | +0.095 | -0.172 | -0.285 | -0.107 | -0.297 | +0.211 | +0.263 | -0.047 | -0.280 |

Relative (%) | -10.4 | +47.0 | -17.8 | +15.9 | -28.6 | -47.5 | -17.8 | -49.4 | +35.1 | +43.9 | -7.9 | -46.7 | |

Steps (reduced) |
10715 (715) |
10853 (853) |
11109 (1109) |
11456 (1456) |
11765 (1765) |
11861 (1861) |
12132 (132) |
12299 (299) |
12380 (380) |
12608 (608) |
12750 (750) |
12951 (951) |

### Subsets and supersets

2000 = 2^{4} × 5^{3}, and its nontrivial divisors are 2, 4, 5, 8, 10, 16, 20, 25, 40, 50, 80, 100, 125, 200, 250, 400, 500, 1000. From these, 1000edo is notable because it carries the interval size measure millioctave. It is argued that cutting millioctaves in half makes for a better interval measuring system, in light of 2000edo's high consistency limit, which introduces just interval approximations not present in 1000edo. In addition, 2000edo inherits its fifth from 200edo, where it is semiconvergent.

## Regular temperament properties

2000edo has the smallest relative error than any previous equal temperaments in the 19-limit. It is only bettered by 2460edo.

### Rank-2 temperaments

Periods per 8ve |
Generator* | Cents* | Associated Ratio* |
Temperaments |
---|---|---|---|---|

20 | 287\2000 (87\2000) |
172.2 (52.2) |
169/153 (?) |
Calcium |

25 | 301\2000 (1\2000) |
180.6 (0.6) |
272/245 (?) |
Hemimanganese |

80 | 619\2000 (19\2000) |
371.4 (11.4) |
2275/1836 (?) |
Mercury |

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