26th-octave temperaments: Difference between revisions

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**Imported revision 300337262 - Original comment: **
- bosonic (more properly addressed in varunismic temps)
 
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<h2>IMPORTED REVISION FROM WIKISPACES</h2>
{{Technical data page}}
This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
{{Infobox fractional-octave|26}}
: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2012-02-09 19:41:35 UTC</tt>.<br>
All temperaments on this page have a period that is 1/26th of an octave. However, the {{monzo| -41 26 }} comma is not tempered out. Thus the 3/2 is not that of [[26edo]]. However, 7/4 is, as is 11/8 in the bosonic temperaments.
: The original revision id was <tt>300337262</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">[[toc|flat]]


=Bosonic=
26edo is very accurate for 7th harmonic, the [[26-7-comma]] ({{monzo| 73 0 0 -26 }}, the amount by which 26 septimal whole tones ([[8/7]]) exceed 5 octaves) is tempered out by 26-fold multiple edos up to 1456 (such as [[26edo|26-]], [[130edo|130-]], [[286edo|286-]] or [[546edo|546edo]]).
Commas: 321489/320000, 589824/588245


POTE generator: ~3/2 = 701.250
Temperaments discussed elsewhere include [[varunismic temperaments #Bosonic|bosonic]]. Considered below is iron.  


Map: [&lt;26 0 -22 73|, &lt;0 1 2 0|]
== Iron ==
Wedgie: &lt;&lt;26 52 0 22 -73 -146||
Iron may be described as the {{nowrap| 130 & 494 }} temperament. It was named by [[Eliora]] in 2023 after the 26th element.
EDOs: 26, 130, 546, 676, 806c
Badness: 0.1558


==11-limit==
[[Subgroup]]: 2.3.5.7
Commas: 441/440, 8019/8000, 65536/65219


POTE generator: ~3/2 = 701.559
[[Comma list]]: 2460375/2458624, 2147483648/2144153025


Map: [&lt;26 0 -22 73 90|, &lt;0 1 2 0 0|]
{{Mapping|legend=1| 26 0 184 73 | 0 1 -3 0 }}
EDOs: 26, 104c, 130
: mapping generators: ~17280/16807, ~3/2
Badness: 0.0652


==13-limit==
[[Optimal tuning]]s:
Commas: 351/350, 364/363, 441/440, 15379/15360
* [[WE]]: ~17280/16807 = 46.1521{{c}}, ~3/2 = 701.9475{{c}}
: [[error map]]: {{val| -0.047 -0.054 -0.039 +0.274 }}
* [[CWE]]: ~17280/16807 = 46.1538{{c}}, ~3/2 = 701.9779{{c}}
: error map: {{val| 0.000 +0.023 +0.060 +0.405 }}


POTE generator: ~3/2 = 701.546
{{Optimal ET sequence|legend=1| 104, 130, 364, 494, 1118, 2730d, 3848dd }}


Map: [&lt;26 0 -22 73 90 55|, &lt;0 1 2 0 0 1|]
[[Badness]] (Sintel): 2.61
EDOs: 26, 104c, 130
Badness: 0.0329


=Fermionic=
=== 11-limit ===
Commas: 540/539, 78408/78125, 177147/176000
Subgroup: 2.3.5.7.11


POTE generator: ~3/2 = 701.077
Comma list: 9801/9800, 131072/130977, 759375/758912


Map: [&lt;26 0 -22 73 -116|, &lt;0 1 2 0 5|]
Mapping: {{mapping| 26 0 184 73 296 | 0 1 -3 0 -5 }}
EDOs: 130, 286, 416, 546, 1508bcd
Badness: 0.0906


==13-limit==
Optimal tunings:
Commas: 351/350, 540/539, 40656/40625, 142884/142805
* WE: ~77/75 = 46.1519{{c}}, ~3/2 = 701.9621{{c}}
* CWE: ~77/75 = 46.1538{{c}}, ~3/2 = 701.9967{{c}}


POTE generator: ~3/2 = 701.038
{{Optimal ET sequence|legend=0| 104e, 130, 364, 494, 1118, 1612d, 2106d }}


Map: [&lt;26 0 -22 73 -116 55|, &lt;0 1 2 0 5 1|]
Badness (Sintel): 1.14
EDOs: 130, 286, 416, 546, 962bf
Badness: 0.0436


</pre></div>
=== 13-limit ===
<h4>Original HTML content:</h4>
Subgroup: 2.3.5.7.11.13
<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;11-limit comma temperaments&lt;/title&gt;&lt;/head&gt;&lt;body&gt;&lt;!-- ws:start:WikiTextTocRule:10:&amp;lt;img id=&amp;quot;wikitext@@toc@@flat&amp;quot; class=&amp;quot;WikiMedia WikiMediaTocFlat&amp;quot; title=&amp;quot;Table of Contents&amp;quot; src=&amp;quot;/site/embedthumbnail/toc/flat?w=100&amp;amp;h=16&amp;quot;/&amp;gt; --&gt;&lt;!-- ws:end:WikiTextTocRule:10 --&gt;&lt;!-- ws:start:WikiTextTocRule:11: --&gt;&lt;a href="#Bosonic"&gt;Bosonic&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:11 --&gt;&lt;!-- ws:start:WikiTextTocRule:12: --&gt;&lt;!-- ws:end:WikiTextTocRule:12 --&gt;&lt;!-- ws:start:WikiTextTocRule:13: --&gt;&lt;!-- ws:end:WikiTextTocRule:13 --&gt;&lt;!-- ws:start:WikiTextTocRule:14: --&gt; | &lt;a href="#Fermionic"&gt;Fermionic&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:14 --&gt;&lt;!-- ws:start:WikiTextTocRule:15: --&gt;&lt;!-- ws:end:WikiTextTocRule:15 --&gt;&lt;!-- ws:start:WikiTextTocRule:16: --&gt;
 
&lt;!-- ws:end:WikiTextTocRule:16 --&gt;&lt;br /&gt;
Comma list: 1716/1715, 2080/2079, 4096/4095, 91125/91091
&lt;!-- ws:start:WikiTextHeadingRule:0:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc0"&gt;&lt;a name="Bosonic"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:0 --&gt;Bosonic&lt;/h1&gt;
 
Commas: 321489/320000, 589824/588245&lt;br /&gt;
Mapping: {{mapping| 26 0 184 73 296 55 | 0 1 -3 0 -5 1 }}
&lt;br /&gt;
 
POTE generator: ~3/2 = 701.250&lt;br /&gt;
Optimal tunings:  
&lt;br /&gt;
* WE: ~77/75 = 46.1525{{c}}, ~3/2 = 701.9820{{c}}
Map: [&amp;lt;26 0 -22 73|, &amp;lt;0 1 2 0|]&lt;br /&gt;
* CWE: ~77/75 = 46.1538{{c}}, ~3/2 = 702.0051{{c}}
Wedgie: &amp;lt;&amp;lt;26 52 0 22 -73 -146||&lt;br /&gt;
 
EDOs: 26, 130, 546, 676, 806c&lt;br /&gt;
{{Optimal ET sequence|legend=0| 104e, 130, 364, 494, 1612d, 2106d }}
Badness: 0.1558&lt;br /&gt;
 
&lt;br /&gt;
Badness (Sintel): 0.711
&lt;!-- ws:start:WikiTextHeadingRule:2:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc1"&gt;&lt;a name="Bosonic-11-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:2 --&gt;11-limit&lt;/h2&gt;
 
Commas: 441/440, 8019/8000, 65536/65219&lt;br /&gt;
{{Navbox fractional-octave}}
&lt;br /&gt;
 
POTE generator: ~3/2 = 701.559&lt;br /&gt;
[[Category:26edo]]
&lt;br /&gt;
Map: [&amp;lt;26 0 -22 73 90|, &amp;lt;0 1 2 0 0|]&lt;br /&gt;
EDOs: 26, 104c, 130&lt;br /&gt;
Badness: 0.0652&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:4:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc2"&gt;&lt;a name="Bosonic-13-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:4 --&gt;13-limit&lt;/h2&gt;
Commas: 351/350, 364/363, 441/440, 15379/15360&lt;br /&gt;
&lt;br /&gt;
POTE generator: ~3/2 = 701.546&lt;br /&gt;
&lt;br /&gt;
Map: [&amp;lt;26 0 -22 73 90 55|, &amp;lt;0 1 2 0 0 1|]&lt;br /&gt;
EDOs: 26, 104c, 130&lt;br /&gt;
Badness: 0.0329&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:6:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc3"&gt;&lt;a name="Fermionic"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:6 --&gt;Fermionic&lt;/h1&gt;
Commas: 540/539, 78408/78125, 177147/176000&lt;br /&gt;
&lt;br /&gt;
POTE generator: ~3/2 = 701.077&lt;br /&gt;
&lt;br /&gt;
Map: [&amp;lt;26 0 -22 73 -116|, &amp;lt;0 1 2 0 5|]&lt;br /&gt;
EDOs: 130, 286, 416, 546, 1508bcd&lt;br /&gt;
Badness: 0.0906&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:8:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc4"&gt;&lt;a name="Fermionic-13-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:8 --&gt;13-limit&lt;/h2&gt;
Commas: 351/350, 540/539, 40656/40625, 142884/142805&lt;br /&gt;
&lt;br /&gt;
POTE generator: ~3/2 = 701.038&lt;br /&gt;
&lt;br /&gt;
Map: [&amp;lt;26 0 -22 73 -116 55|, &amp;lt;0 1 2 0 5 1|]&lt;br /&gt;
EDOs: 130, 286, 416, 546, 962bf&lt;br /&gt;
Badness: 0.0436&lt;/body&gt;&lt;/html&gt;</pre></div>

Latest revision as of 04:18, 21 July 2026

This is a list showing technical temperament data. For an explanation of what information is shown here, you may look at the technical data guide for regular temperaments.

← 25th-octave • 26th-octave • 27th-octave →

All temperaments on this page have a period that is 1/26th of an octave. However, the [-41 26⟩ comma is not tempered out. Thus the 3/2 is not that of 26edo. However, 7/4 is, as is 11/8 in the bosonic temperaments.

26edo is very accurate for 7th harmonic, the 26-7-comma ([73 0 0 -26⟩, the amount by which 26 septimal whole tones (8/7) exceed 5 octaves) is tempered out by 26-fold multiple edos up to 1456 (such as 26-, 130-, 286- or 546edo).

Temperaments discussed elsewhere include bosonic. Considered below is iron.

Iron

Iron may be described as the 130 & 494 temperament. It was named by Eliora in 2023 after the 26th element.

Subgroup: 2.3.5.7

Comma list: 2460375/2458624, 2147483648/2144153025

Mapping: [⟨26 0 184 73], ⟨0 1 -3 0]]

mapping generators: ~17280/16807, ~3/2

Optimal tunings:

  • WE: ~17280/16807 = 46.1521 ¢, ~3/2 = 701.9475 ¢
error map: ⟨-0.047 -0.054 -0.039 +0.274]
  • CWE: ~17280/16807 = 46.1538 ¢, ~3/2 = 701.9779 ¢
error map: ⟨0.000 +0.023 +0.060 +0.405]

Optimal ET sequence: 104, 130, 364, 494, 1118, 2730d, 3848dd

Badness (Sintel): 2.61

11-limit

Subgroup: 2.3.5.7.11

Comma list: 9801/9800, 131072/130977, 759375/758912

Mapping: [⟨26 0 184 73 296], ⟨0 1 -3 0 -5]]

Optimal tunings:

  • WE: ~77/75 = 46.1519 ¢, ~3/2 = 701.9621 ¢
  • CWE: ~77/75 = 46.1538 ¢, ~3/2 = 701.9967 ¢

Optimal ET sequence: 104e, 130, 364, 494, 1118, 1612d, 2106d

Badness (Sintel): 1.14

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 1716/1715, 2080/2079, 4096/4095, 91125/91091

Mapping: [⟨26 0 184 73 296 55], ⟨0 1 -3 0 -5 1]]

Optimal tunings:

  • WE: ~77/75 = 46.1525 ¢, ~3/2 = 701.9820 ¢
  • CWE: ~77/75 = 46.1538 ¢, ~3/2 = 702.0051 ¢

Optimal ET sequence: 104e, 130, 364, 494, 1612d, 2106d

Badness (Sintel): 0.711

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