200edo: Difference between revisions

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<h2>IMPORTED REVISION FROM WIKISPACES</h2>
{{Infobox ET}}
This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
{{ED intro}}
: This revision was by author [[User:Osmiorisbendi|Osmiorisbendi]] and made on <tt>2010-06-06 17:24:01 UTC</tt>.<br>
: The original revision id was <tt>147267985</tt>.<br>
: The revision comment was: <tt></tt><br>
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br>
<h4>Original Wikitext content:</h4>
<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html">=&lt;span style="color: #007027; font-size: 118%;"&gt;200 tone equal temperament&lt;/span&gt;=


**200 tone equal modes:**
== Theory ==
34 34 15 34 34 34 15 = Pytagorean
200edo contains a [[perfect fifth]] of exactly 702 cents and a [[perfect fourth]] of exactly 498 cents, which is accurate due to 200 being the denominator of a continued fraction convergent to log<sub>2</sub>(3/2). Only about 0.045 cents sharp, it is the next best fifth in absolute error after [[53edo]]'s. In light of having its perfect fifth precise and the step divisible by 9, it is essentially a perfect edo for [[Carlos Alpha]], even up many octaves (the difference between 13 steps of 200edo and 1 step of Carlos Alpha is only 0.03501 cents).
32 32 20 32 32 32 20 = Meantone
 
27 27 27 27 27 27 27 11 = Porcupine
It [[tempering out|tempers out]] the [[schisma]] (32805/32768) and the quartemka, {{monzo| 2 -32 21 }} in the 5-limit, and the [[gamelisma]], 1029/1024, in the [[7-limit]], so that it [[support]]s the [[guiron]] temperament.
26 26 26 9 26 26 26 26 9 = The most important Mavila (Armodue ultra-extended)</pre></div>
 
<h4>Original HTML content:</h4>
One step of 200edo is close to [[289/288]]. Unfortunately, it is not compatible with any obvious 2.3.17 subgroup mappings of 200edo.
<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html">&lt;html&gt;&lt;head&gt;&lt;title&gt;200edo&lt;/title&gt;&lt;/head&gt;&lt;body&gt;&lt;!-- ws:start:WikiTextHeadingRule:0:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc0"&gt;&lt;a name="x200 tone equal temperament"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:0 --&gt;&lt;span style="color: #007027; font-size: 118%;"&gt;200 tone equal temperament&lt;/span&gt;&lt;/h1&gt;
 
&lt;br /&gt;
=== Prime harmonics ===
&lt;strong&gt;200 tone equal modes:&lt;/strong&gt;&lt;br /&gt;
{{Harmonics in equal|200}}
34 34 15 34 34 34 15 = Pytagorean&lt;br /&gt;
 
32 32 20 32 32 32 20 = Meantone&lt;br /&gt;
=== Subsets and supersets ===
27 27 27 27 27 27 27 11 = Porcupine&lt;br /&gt;
200 factorizes as 2<sup>3</sup> × 5<sup>2</sup>, and has subset edos {{EDOs| 2, 4, 5, 8, 10, 20, 25, 40, 50, 100 }}.
26 26 26 9 26 26 26 26 9 = The most important Mavila (Armodue ultra-extended)&lt;/body&gt;&lt;/html&gt;</pre></div>
 
[[400edo]], which doubles it, provides good correction for the harmonics 5 and 7, and makes for a strong 19-limit system.
 
== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal<br />8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| 317 -200 }}
| {{mapping| 200 317 }}
| −0.0142
| 0.0142
| 0.24
|-
| 2.3.5
| 32805/32768, {{monzo| 2 -32 21 }}
| {{mapping| 200 317 464 }}
| +0.3226
| 0.4767
| 7.95
|-
| 2.3.5.7
| 1029/1024, 10976/10935, 390625/387072
| {{mapping| 200 317 464 561 }}
| +0.4937
| 0.5082
| 8.47
|}
 
=== 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
|-
| 1
| 23\200
| 138.00
| 27/25
| [[Quartemka]]
|-
| 1
| 39\200
| 234.00
| 8/7
| [[Guiron]]
|-
| 1
| 83\200
| 498.00
| 4/3
| [[Helmholtz (temperament)|Helmholtz]]
|}
<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
 
== Scales ==
* 34 34 15 34 34 34 15 = [[5L 2s|Pythagorean tuning]]
* 32 32 20 32 32 32 20 = [[5L 2s|Meantone tuning]] in the same way of [[50edo]]
* 27 27 27 27 27 27 27 11 = [[7L 1s|Porcupine tuning]]
* 26 26 26 9 26 26 26 26 9 = [[7L 2s|Superdiatonic tuning]]
* 24 24 24 16 24 24 24 24 16 = [[7L 2s|Superdiatonic tuning]] in the same way of [[25edo]]
* 22 22 8 22 22 22 8 22 22 22 8 = [[8L 3s|Sensi]]
* 16 16 16 8 16 16 16 16 8 16 16 16 16 8 = [[11L 3s|Ketradektriatoh tuning]]
 
== Music ==
; [[Francium]]
* "On Fire" from ''Mysteries'' (2023) – [https://open.spotify.com/track/6janPwh3S8FLgIzWf9S0oQ Spotify] | [https://francium223.bandcamp.com/track/on-fire Bandcamp] | [https://www.youtube.com/watch?v=S1NKb_EoYrw YouTube]
 
; [[Claudi Meneghin]]
* ''Fugue on Elgar’s Enigma Theme'' – [https://www.youtube.com/watch?v=h4rjMFAzjow YouTube] | [http://soonlabel.com/xenharmonic/archives/1324 soonlabel archive]{{dead link}} | [http://soonlabel.com/xenharmonic/wp-content/uploads/2013/10/Claudi_Meneghin_Enigma_Fugue.mp3 play]{{dead link}}
 
[[Category:3-limit record edos|###]] <!-- 3-digit number -->
[[Category:Listen]]

Latest revision as of 12:51, 26 March 2026

← 199edo 200edo 201edo →
Prime factorization 23 × 52
Step size 6 ¢ 
Fifth 117\200 (702 ¢)
(semiconvergent)
Semitones (A1:m2) 19:15 (114 ¢ : 90 ¢)
Consistency limit 9
Distinct consistency limit 9

200 equal divisions of the octave (abbreviated 200edo or 200ed2), also called 200-tone equal temperament (200tet) or 200 equal temperament (200et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 200 equal parts of exactly 6 ¢ each. Each step represents a frequency ratio of 21/200, or the 200th root of 2.

Theory

200edo contains a perfect fifth of exactly 702 cents and a perfect fourth of exactly 498 cents, which is accurate due to 200 being the denominator of a continued fraction convergent to log2(3/2). Only about 0.045 cents sharp, it is the next best fifth in absolute error after 53edo's. In light of having its perfect fifth precise and the step divisible by 9, it is essentially a perfect edo for Carlos Alpha, even up many octaves (the difference between 13 steps of 200edo and 1 step of Carlos Alpha is only 0.03501 cents).

It tempers out the schisma (32805/32768) and the quartemka, [2 -32 21⟩ in the 5-limit, and the gamelisma, 1029/1024, in the 7-limit, so that it supports the guiron temperament.

One step of 200edo is close to 289/288. Unfortunately, it is not compatible with any obvious 2.3.17 subgroup mappings of 200edo.

Prime harmonics

Approximation of prime harmonics in 200edo
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) +0.00 +0.04 -2.31 -2.83 +0.68 -0.53 -2.96 +2.49 +1.73 +2.42 +0.96
Relative (%) +0.0 +0.7 -38.6 -47.1 +11.4 -8.8 -49.3 +41.4 +28.8 +40.4 +16.1
Steps
(reduced) 		200
(0) 		317
(117) 		464
(64) 		561
(161) 		692
(92) 		740
(140) 		817
(17) 		850
(50) 		905
(105) 		972
(172) 		991
(191)

Subsets and supersets

200 factorizes as 23 × 52, and has subset edos 2, 4, 5, 8, 10, 20, 25, 40, 50, 100.

400edo, which doubles it, provides good correction for the harmonics 5 and 7, and makes for a strong 19-limit system.

Regular temperament properties

Subgroup Comma list Mapping Optimal
8ve stretch (¢) 		Tuning error
Absolute (¢) Relative (%)
2.3 [317 -200⟩ [⟨200 317]] −0.0142 0.0142 0.24
2.3.5 32805/32768, [2 -32 21⟩ [⟨200 317 464]] +0.3226 0.4767 7.95
2.3.5.7 1029/1024, 10976/10935, 390625/387072 [⟨200 317 464 561]] +0.4937 0.5082 8.47

Rank-2 temperaments

Table of rank-2 temperaments by generator
Periods
per 8ve 		Generator* Cents* Associated
ratio* 		Temperaments
1 23\200 138.00 27/25 Quartemka
1 39\200 234.00 8/7 Guiron
1 83\200 498.00 4/3 Helmholtz

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

Scales

Music

Francium
Claudi Meneghin
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