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13edt: 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>
[[File:13edt.png|thumb|alt=13edt.png|A plot of the [[the Riemann zeta function and tuning#Removing primes|no-twos Z-function]], in terms of which 13edt is the fourth no-twos zeta peak [[EDT]].]]
: This revision was by author [[User:xenwolf|xenwolf]] and made on <tt>2011-08-16 03:41:10 UTC</tt>.<br>
 
: The original revision id was <tt>246187601</tt>.<br>
'''13 equal divisions of the tritave''' ('''13edt''') is the [[nonoctave]] [[tuning system]] derived by dividing the [[tritave]] (3/1) into 13 equal steps of 146.3 [[cent]]s each, or the thirteenth root of 3. It is best known as the equal-tempered version of the [[Bohlen–Pierce]] scale, and therefore has received by far the most attention among equal divisions of the tritave.
: 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>
It provides an excellent approximation to the [[3.5.7 subgroup]], especially for its size, being comparable to [[34edo]]'s accuracy in the 5-limit. In this subgroup, it tempers out [[245/243]] and [[3125/3087]], the same commas as [[Sensamagic_clan#Bohpier|bohpier temperament]]. It is less impressive in higher prime limits, but makes for excellent no-twos 7-limit harmony. For higher limits, the multiples of 13 ([[26edt]], [[39edt]], and [[52edt]]) come to the fore.
<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">see [[BP]] (Bohlen-Pierce)</pre></div>
13edt can be described as approximately 8.202[[edo]]. This implies that each step of 13edt can be approximated by 5 steps of [[41edo]].
<h4>Original HTML 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;width:200%;white-space: pre-wrap ! important" class="old-revision-html">&lt;html&gt;&lt;head&gt;&lt;title&gt;13edt&lt;/title&gt;&lt;/head&gt;&lt;body&gt;see &lt;a class="wiki_link" href="/BP"&gt;BP&lt;/a&gt; (Bohlen-Pierce)&lt;/body&gt;&lt;/html&gt;</pre></div>
In the [[no-2]] [[3/1]]-[[equave]]-[[7-limit]], [[13edt]] maintains the smallest relative error of any EDT until [[258edt]] and [[271edt]], and the smallest absolute error until [[56edt]].
 
== Theory ==
{{Harmonics in equal|13|3|1|prec=2|intervals=odd}}
{{Harmonics in equal|13|3|1|prec=2|intervals=odd|start=12}}
 
* [[Relationship between Bohlen-Pierce and octave-ful temperaments]]
 
== Intervals ==
{{Main|Intervals of BP}}
 
{| class="wikitable center-all right-2 right-3"
|-
! Steps
! [[Cent]]s
! [[Hekt]]s
! [[4L 5s (3/1-equivalent)|Enneatonic]]<br />degree
! Corresponding<br />3.5.7 subgroup<br />intervals
! [[Lambda ups and downs notation|Lambda]]<br />(sLsLsLsLs, {{nowrap|E {{=}} 1/1}})
|-
| 0
| 0
| 0
| P1
| 1/1
| E
|-
| 1
| 146.3
| 100
| A1/m2
| [[49/45]] (−1.1{{c}}); [[27/25]] (+13.1{{c}})
| F
|-
| 2
| 292.6
| 200
| M2/d3
| [[25/21]] (−9.2{{c}})
| F#, Gb
|-
| 3
| 438.9
| 300
| A2/P3/d4
| [[9/7]] (+3.8{{c}})
| G
|-
| 4
| 585.2
| 400
| A3/m4/d5
| [[7/5]] (+2.7{{c}})
| H
|-
| 5
| 731.5
| 500
| M4/m5
| [[75/49]] (−5.4{{c}})
| H#, Jb
|-
| 6
| 877.8
| 600
| A4/M5
| [[5/3]] (−6.5{{c}})
| J
|-
| 7
| 1024.1
| 700
| A5/m6/d7
| [[9/5]] (+6.5{{c}})
| A
|-
| 8
| 1170.4
| 800
| M6/m7
| [[49/25]] (+5.4{{c}})
| A#, Bb
|-
| 9
| 1316.7
| 900
| A6/M7/d8
| [[15/7]] (−2.7{{c}})
| B
|-
| 10
| 1463.0
| 1000
| P8/d9
| [[7/3]] (−3.8{{c}})
| C
|-
| 11
| 1609.3
| 1100
| A8/m9
| [[63/25]] (+9.2{{c}})
| C#, Db
|-
| 12
| 1755.7
| 1200
| M9/d10
| [[135/49]] (+1.1{{c}}); [[25/9]] (−13.1{{c}})
| D
|-
| 13
| 1902.0
| 1300
| A9/P10
| [[3/1]]
| E
|}
 
== JI approximation ==
[[File:13ed3-17-001.svg|alt=alt : Your browser has no SVG support.]]
 
== 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>Equave stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 3.5.7
| 245/243, 3125/3087
| [{{val| 13 19 23 }}] (b13)
| +1.393
| 1.150
| 0.79
|}
 
=== Rank-2 temperaments ===
{| class="wikitable center-all right-3 left-5"
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
|-
! Periods<br />per tritave
! Generator<br />(reduced)
! Cents<br />(reduced)
! Associated<br />ratio
! Temperament
|-
| 1
| 1\13
| 146.30
| 49/45
| [[Procyon]]
|-
| 1
| 2\13
| 292.61
| 25/21
| [[Sirius]]
|-
| 1
| 3\13
| 438.91
| 9/7
| [[BPS]]
|-
| 1
| 4\13
| 585.22
| 7/5
| [[Canopus]]
|-
| 1
| 5\13
| 731.63
| 75/49
|
|-
| 1
| 6\13
| 877.83
| 5/3
| [[Arcturus]]
|}
 
== See also ==
* [[Bohlen-p_et]]
* [[Catalog of 3.5.7 subgroup rank two temperaments]]
* [[No-twos subgroup temperaments#3.5.7 subgroup temperaments]]
* [[19ed5|19ED5]]: relative ED5
* [[23ed7|23ED7]]: relative ED7
 
[[Category:Tritave]]
[[Category:Macrotonal]]
[[Category:Nonoctave]]
[[Category:Bohlen–Pierce]]
Retrieved from "https://en.xen.wiki/w/13edt"