2000edo: Difference between revisions
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{{Infobox ET}} | {{Infobox ET}} | ||
{{EDO intro|2000}} | {{EDO intro|2000}} | ||
2000 = 2<sup>4</sup> × 5<sup>3</sup> | == Theory == | ||
It 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]] [[Tenney-Euclidean temperament measures#TE simple badness|relative error]] being [[1578edo]]. The only ones superior to it in the [[23-limit]] are 1578 and [[1889edo]]. | |||
2000 = 2<sup>4</sup> × 5<sup>3</sup> , and its divisors are {{EDOs|1, 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. | |||
=== Prime harmonics === | === Prime harmonics === | ||
{{Harmonics in equal|2000|columns=13}} | {{Harmonics in equal|2000|columns=13}} | ||
[[Category:Equal divisions of the octave|####]] <!-- 4-digit number --> | [[Category:Equal divisions of the octave|####]] | ||
== Regular temperament properties == | |||
2000edo has the smallest relative error than any previous temperament in the 19-limit. It is only bettered by [[2460edo]]. | |||
=== Rank-2 temperaments === | |||
{| class="wikitable center-all left-5" | |||
|+Table of rank-2 temperaments by generator | |||
! Periods<br>per octave | |||
! Generator<br>(reduced) | |||
! Cents<br>(reduced) | |||
! Associated<br>ratio | |||
! Temperaments | |||
|- | |||
| 20 | |||
| 287\2000<br>(87\2000) | |||
| 172.2<br>(52.2) | |||
| 169/153<br>(?) | |||
| [[Calcium]] | |||
|} | |||
<!-- 4-digit number --> | |||
[[Category:29-limit]] | [[Category:29-limit]] | ||
Revision as of 21:43, 7 November 2022
| ← 1999edo | 2000edo | 2001edo → |
Theory
It 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.
2000 = 24 × 53 , and its divisors are 1, 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.
Prime harmonics
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 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 | -0.062 |
| 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 | -10.4 | |
| 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) |
10715 (715) | |
Regular temperament properties
2000edo has the smallest relative error than any previous temperament in the 19-limit. It is only bettered by 2460edo.
Rank-2 temperaments
| Periods per octave |
Generator (reduced) |
Cents (reduced) |
Associated ratio |
Temperaments |
|---|---|---|---|---|
| 20 | 287\2000 (87\2000) |
172.2 (52.2) |
169/153 (?) |
Calcium |