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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:keenanpepper|keenanpepper]] and made on <tt>2012-03-07 17:10:54 UTC</tt>.<br>
: The original revision id was <tt>308834064</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">The 13 equal division of 3, the tritave, divides it into 13 equal parts of 146.304 cents each, corresponding to 8.202 edo. An alternative name for it is the [[Bohlen-Pierce]] scale. In the 7-limit, it tempers out 245/243 and 3125/3087, the same commas as [[Sensamagic clan#Bohpier|bohpier temperament]]. It is less impressive in higher p-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.


Below is a plot of the [[The Riemann Zeta Function and Tuning#Removing%20primes|no-twos Z-function]], in terms of which 13edt is the fourth no-twos zeta peak edt.
'''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.  


[[image:13edt.png]]
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.


==Intervals==
13edt can be described as approximately 8.202[[edo]]. This implies that each step of 13edt can be approximated by 5 steps of [[41edo]].
||~ Steps ||~ Cents ||~ Corresponding JI intervals ||~ Comments ||~ Generator for... ||
|| 1 || 146.3 || 27/25~49/45 ||  ||  ||
|| 2 || 292.6 || 25/21 ||  || Sirius ||
|| 3 || 438.9 || 9/7 ||  || Linear BP ||
|| 4 || 585.2 || 7/5 ||  || Canopus ||
|| 5 || 731.5 ||  || False 3/2 ||  ||
|| 6 || 877.8 || 5/3 ||  || Arcturus ||
|| 7 || 1024.1 || 9/5 ||  || Arcturus ||
|| 8 || 1170.4 ||  || False 2/1 ||  ||
|| 9 || 1316.7 || 15/7 ||  || Canopus ||
|| 10 || 1463.0 || 7/3 ||  || Linear BP ||
|| 11 || 1609.3 || 63/25 ||  || Sirius ||
|| 12 || 1755.7 || 25/9 ||  ||  ||
|| 13 || 1902.0 || 3/1 || Tritave ||  ||</pre></div>
<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;The 13 equal division of 3, the tritave, divides it into 13 equal parts of 146.304 cents each, corresponding to 8.202 edo. An alternative name for it is the &lt;a class="wiki_link" href="/Bohlen-Pierce"&gt;Bohlen-Pierce&lt;/a&gt; scale. In the 7-limit, it tempers out 245/243 and 3125/3087, the same commas as &lt;a class="wiki_link" href="/Sensamagic%20clan#Bohpier"&gt;bohpier temperament&lt;/a&gt;. It is less impressive in higher p-limits, but makes for excellent no-twos 7-limit harmony. For higher limits, the multiples of 13 &lt;a class="wiki_link" href="/26edt"&gt;26edt&lt;/a&gt;, &lt;a class="wiki_link" href="/39edt"&gt;39edt&lt;/a&gt; and &lt;a class="wiki_link" href="/52edt"&gt;52edt&lt;/a&gt; come to the fore.&lt;br /&gt;
&lt;br /&gt;
Below is a plot of the &lt;a class="wiki_link" href="/The%20Riemann%20Zeta%20Function%20and%20Tuning#Removing%20primes"&gt;no-twos Z-function&lt;/a&gt;, in terms of which 13edt is the fourth no-twos zeta peak edt.&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextLocalImageRule:172:&amp;lt;img src=&amp;quot;/file/view/13edt.png/250612880/13edt.png&amp;quot; alt=&amp;quot;&amp;quot; title=&amp;quot;&amp;quot; /&amp;gt; --&gt;&lt;img src="/file/view/13edt.png/250612880/13edt.png" alt="13edt.png" title="13edt.png" /&gt;&lt;!-- ws:end:WikiTextLocalImageRule:172 --&gt;&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:0:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc0"&gt;&lt;a name="x-Intervals"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:0 --&gt;Intervals&lt;/h2&gt;


&lt;table class="wiki_table"&gt;
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]].
    &lt;tr&gt;
        &lt;th&gt;Steps&lt;br /&gt;
&lt;/th&gt;
        &lt;th&gt;Cents&lt;br /&gt;
&lt;/th&gt;
        &lt;th&gt;Corresponding JI intervals&lt;br /&gt;
&lt;/th&gt;
        &lt;th&gt;Comments&lt;br /&gt;
&lt;/th&gt;
        &lt;th&gt;Generator for...&lt;br /&gt;
&lt;/th&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;1&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;146.3&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;27/25~49/45&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;2&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;292.6&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;25/21&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;Sirius&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;3&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;438.9&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;9/7&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;Linear BP&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;4&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;585.2&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;7/5&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;Canopus&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;5&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;731.5&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;False 3/2&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;6&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;877.8&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;5/3&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;Arcturus&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;7&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;1024.1&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;9/5&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;Arcturus&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;8&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;1170.4&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;False 2/1&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;9&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;1316.7&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;15/7&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;Canopus&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;10&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;1463.0&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;7/3&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;Linear BP&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;11&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;1609.3&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;63/25&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;Sirius&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;12&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;1755.7&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;25/9&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;13&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;1902.0&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;3/1&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;Tritave&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
&lt;/table&gt;


&lt;/body&gt;&lt;/html&gt;</pre></div>
== 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]]

Latest revision as of 16:47, 22 May 2026

← 12edt 13edt 14edt →
Prime factorization 13 (prime)
Step size 146.304 ¢ 
Octave 8\13edt (1170.43 ¢)
Consistency limit 7
Distinct consistency limit 4
13edt.png
A plot of the no-twos Z-function, in terms of which 13edt is the fourth no-twos zeta peak EDT.

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 cents 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.

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 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.

13edt can be described as approximately 8.202edo. This implies that each step of 13edt can be approximated by 5 steps of 41edo.

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

Approximation of odd harmonics in 13edt
Harmonic 3 5 7 9 11 13 15 17 19 21 23
Error Absolute (¢) +0.00 -6.53 -3.83 +0.00 -54.80 -51.40 -6.53 +69.39 +23.14 -3.83 -15.02
Relative (%) +0.0 -4.5 -2.6 +0.0 -37.5 -35.1 -4.5 +47.4 +15.8 -2.6 -10.3
Steps
(reduced) 		13
(0) 		19
(6) 		23
(10) 		26
(0) 		28
(2) 		30
(4) 		32
(6) 		34
(8) 		35
(9) 		36
(10) 		37
(11)
Approximation of odd harmonics in 13edt
Harmonic 25 27 29 31 33 35 37 39 41 43 45
Error Absolute (¢) -13.07 +0.00 +22.59 +53.44 -54.80 -10.36 +39.74 -51.40 +8.32 +72.17 -6.53
Relative (%) -8.9 +0.0 +15.4 +36.5 -37.5 -7.1 +27.2 -35.1 +5.7 +49.3 -4.5
Steps
(reduced) 		38
(12) 		39
(0) 		40
(1) 		41
(2) 		41
(2) 		42
(3) 		43
(4) 		43
(4) 		44
(5) 		45
(6) 		45
(6)

Intervals

Main article: Intervals of BP
Steps Cents Hekts Enneatonic
degree 		Corresponding
3.5.7 subgroup
intervals 		Lambda
(sLsLsLsLs, E = 1/1)
0 0 0 P1 1/1 E
1 146.3 100 A1/m2 49/45 (−1.1 ¢); 27/25 (+13.1 ¢) F
2 292.6 200 M2/d3 25/21 (−9.2 ¢) F#, Gb
3 438.9 300 A2/P3/d4 9/7 (+3.8 ¢) G
4 585.2 400 A3/m4/d5 7/5 (+2.7 ¢) H
5 731.5 500 M4/m5 75/49 (−5.4 ¢) H#, Jb
6 877.8 600 A4/M5 5/3 (−6.5 ¢) J
7 1024.1 700 A5/m6/d7 9/5 (+6.5 ¢) A
8 1170.4 800 M6/m7 49/25 (+5.4 ¢) A#, Bb
9 1316.7 900 A6/M7/d8 15/7 (−2.7 ¢) B
10 1463.0 1000 P8/d9 7/3 (−3.8 ¢) C
11 1609.3 1100 A8/m9 63/25 (+9.2 ¢) C#, Db
12 1755.7 1200 M9/d10 135/49 (+1.1 ¢); 25/9 (−13.1 ¢) D
13 1902.0 1300 A9/P10 3/1 E

JI approximation

alt : Your browser has no SVG support.

Regular temperament properties

Subgroup Comma list Mapping Optimal
Equave stretch (¢) 		Tuning error
Absolute (¢) Relative (%)
3.5.7 245/243, 3125/3087 [⟨13 19 23]] (b13) +1.393 1.150 0.79

Rank-2 temperaments

Table of rank-2 temperaments by generator
Periods
per tritave 		Generator
(reduced) 		Cents
(reduced) 		Associated
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

Retrieved from "https://en.xen.wiki/index.php?title=13edt&oldid=230767"