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13edt: Difference between revisions

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**Imported revision 247160127 - Original comment: **
m Text replacement - "[[the Riemann zeta function and tuning" to "[[Riemann zeta function"
 
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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 [[Riemann zeta function#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>2011-08-20 07:10:57 UTC</tt>.<br>
[[File:pts-3-5-7-e3-twtop.jpg|thumb|Projective tuning space of the [[3.5.7 subgroup]], showing 13edt near the center.]]
: The original revision id was <tt>247160127</tt>.<br>
 
: The revision comment was: <tt></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 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>
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.
<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">**13edt** is an alternative name for [[Bohlen-Pierce]].</pre></div>
 
<h4>Original HTML content:</h4>
13edt can be described as approximately 8.202[[edo]]. This implies that each step of 13edt can be approximated by 5 steps of [[41edo]].
<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;&lt;strong&gt;13edt&lt;/strong&gt; is an alternative name for &lt;a class="wiki_link" href="/Bohlen-Pierce"&gt;Bohlen-Pierce&lt;/a&gt;.&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"