2000edo: Difference between revisions
mNo edit summary |
Cleanup; clarify the title row of the rank-2 temp table; -redundant categories |
||
| Line 4: | Line 4: | ||
== Theory == | == 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]] [[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]], and in the 19-limit, nothing smaller defeats it. | 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]] [[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]], and in the 19-limit, nothing smaller defeats it. | ||
=== Prime harmonics === | === Prime harmonics === | ||
{{Harmonics in equal|2000|columns=13}} | {{Harmonics in equal|2000|columns=13}} | ||
=== Subsets and supersets === | |||
2000 = 2<sup>4</sup> × 5<sup>3</sup>, and its divisors are {{EDOs| 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 == | == Regular temperament properties == | ||
| Line 17: | Line 18: | ||
|+Table of rank-2 temperaments by generator | |+Table of rank-2 temperaments by generator | ||
! Periods<br>per 8ve | ! Periods<br>per 8ve | ||
! Generator | ! Generator* | ||
! Cents | ! Cents* | ||
! Associated<br>Ratio | ! Associated<br>Ratio | ||
! Temperaments | ! Temperaments | ||
| Line 34: | Line 35: | ||
| [[Mercury]] | | [[Mercury]] | ||
|} | |} | ||
<nowiki>*</nowiki> [[Normal lists|octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct | |||
[[ | |||
[[ | |||
Revision as of 13:57, 15 October 2023
| ← 1999edo | 2000edo | 2001edo → |
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 | 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) | |
Subsets and supersets
2000 = 24 × 53, and its 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 |
| 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