2000edo: Difference between revisions
simplified links, expanded links to original lemma where appropriate, replaced the word beat |
m Text replacement - "Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct" to "Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct" |
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{{Infobox ET}} | |||
{{ED intro}} | |||
== Theory == | |||
2000edo is [[consistency|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 [[1578edo|1578-]] and [[1889edo]], and in the 19-limit, nothing smaller defeats it. | |||
[[Category: | === Prime harmonics === | ||
{{Harmonics in equal|2000|columns=12}} | |||
{{Harmonics in equal|2000|start=13|columns=12|collapsed=1|title=Approximation of prime harmonics in 2000edo (continued)}} | |||
=== Subsets and supersets === | |||
2000 = {{factorization|2000}}, and its nontrivial 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 == | |||
2000edo has the smallest relative error than any previous equal temperaments in the 19-limit. It is only bettered by [[2460edo]]. | |||
=== Rank-2 temperaments === | |||
{| class="wikitable center-all left-5" | |||
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator | |||
|- | |||
! Periods<br />per 8ve | |||
! Generator* | |||
! Cents* | |||
! Associated<br />ratio* | |||
! Temperaments | |||
|- | |||
| 20 | |||
| 287\2000<br />(87\2000) | |||
| 172.2<br />(52.2) | |||
| 169/153<br />(?) | |||
| [[Calcium]] | |||
|- | |||
|25 | |||
|301\2000<br />(1\2000) | |||
|180.6<br />(0.6) | |||
|272/245<br />(?) | |||
|[[Hemimanganese]] | |||
|- | |||
| 80 | |||
| 619\2000<br />(19\2000) | |||
| 371.4<br />(11.4) | |||
| 2275/1836<br />(?) | |||
| [[Mercury]] | |||
|} | |||
<nowiki />* [[Normal forms #Equave-reduced-generator form|Octave-reduced form]], reduced to the first half-octave, and [[normal forms #Minimal-generator form|minimal form]] in parentheses if distinct | |||
== Music == | |||
; [[Eliora]] | |||
* ''[https://www.youtube.com/watch?v=gM4dfrF5wPg Fugue, but Not (in A Mercury & Bidia)]'' (2024) | |||
[[Category:Listen]] | |||
Latest revision as of 13:31, 13 March 2026
| ← 1999edo | 2000edo | 2001edo → |
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 21/2000, 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 = 24 × 53, 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 distinct