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<h2>IMPORTED REVISION FROM WIKISPACES</h2>
{{Interwiki
This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
| en = 12edo
: This revision was by author [[User:guest|guest]] and made on <tt>2012-02-15 07:42:22 UTC</tt>.<br>
| de = 12-EDO
: The original revision id was <tt>301936130</tt>.<br>
| es = 12 EDO
: The revision comment was: <tt></tt><br>
| ja = 12平均律
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br>
| ro = 12DEO
<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">12[[EDO]], perhaps better known as 12et since it really is a temperament, is the predominating tuning system in the world today. It achieved that position because it is the smallest equal division which can seriously claim to represent [[5-limit]] harmony, and because as 1/12 Pythagorean comma meantone, it represents [[Meantone|meantone]].
{{Infobox ET}}
{{Wikipedia|12 equal temperament}}
{{ED intro}} It is the predominating tuning system in the world today.


It divides the octave into twelve equal parts, each of exactly 100 cents each unless octave shrinking or stretching is employed. Its has a fifth which is quite good at two cents flat. It has a major third which is 13 2/3 cents sharp, which works well enough for some styles of music and is not really adequate for others, and a minor third which is flat by even more, 15 2/3 cents. It is probably not an accident that as tuning in European music became increasingly close to 12et, the style of the music changed so that the defects of 12et appeared less evident, though it should be borne in mind that in actual performance these are often reduced by the tuning adaptations of the performers.
== Theory ==
12edo achieved its position as the standard Western tuning system through a combination of theoretical properties and practicality. It is the smallest number of equal divisions of the octave ([[edo]]) which can seriously claim to represent [[5-limit]] harmony, and it represents a [[meantone]] temperament, tempering out [[81/80]], equating four [[3/2|perfect fifths]] with the [[5/1|5th harmonic]].


Its seventh partial is sharp by over 31 cents, and stands out distinctly from the rest of the chord in a tetrad. Such tetrads are often being used as dominant seventh chords in functional harmony, for which the 5-limit JI version would be 1-5/4-3/2-16/9, and while 12et officially supports septimal meantone via the [[Vals and Tuning Space|val]] &lt;12 19 28 34|, its credentials in the 7-limit department are distinctly cheesy. It cannot be said to represent 11 or 13 at all, though it does a quite credible 17 and an even better 19. Nevertheless its relative tuning accuracy is quite high, and 12edo is the fourth [[The Riemann Zeta Function and Tuning#Zeta%20EDO%20lists|zeta integral edo]].
It divides the octave into twelve equal parts, each of exactly 100 [[cent]]s. It has a [[3/2|fifth]] which is quite accurate at 700 cents, two cents flat of [[just intonation]]. It has a [[5/4|major third]] which is 13.7 cents sharp of just, which, while reasonable for its size, is unsatisfactory for some. The [[6/5|minor third]] is even less accurate, being 15.6 cents flat of just.


In terms of the kernel, which is to say the commas it tempers out, it tempers out the Pythagorean comma, 3^12/2^19, the Didymas comma, 81/80, the diesis, 128/125, the diaschisma, 2048/2025, the Archytas comma, 64/63, the septimal quartertone, 36/35, the jubilisma, 50/49, the septimal semicomma, 126/125, and the septimal kleisma, 225/224. Each of these affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.
Before people used 12edo, people used a variety of [[historical temperaments]] such as [[quarter-comma meantone]], and later [[well temperament]]s. By the 20th century, 12edo became dominant primarily due to practical considerations for keyboard instruments and its ability to handle modulation across all keys with reasonable intonation. In actual performance, these deviations from just intonation are often reduced by the tuning adaptations of skilled performers. Modern music theory has increasingly treated 12edo as a system in its own right rather than as an approximation of just intonation or meantone, leading to theoretical approaches such as {{w|serialism}} and much of {{w|jazz harmony}} that derive from 12edo's structure as an equal division rather than its underlying temperament properties.{{cn}}


==Linear temperaments==
12edo is the basic example of a [[:Category:12-tone scales|dodecatonic]] scale and can be considered the simplest well temperament, where all twelve fifths are the same.
||~ Periods
per octave ||~ Generator ||~ Temperaments ||
|| 1 || 1\12 || [[Ripple]] ||
|| 1 || 5\12 || [[Meantone]]/[[dominant]] ||
|| 2 || 1\12 || [[Srutal]]/[[pajara]] ||
|| 3 || 1\12 || [[Augmented]] ||
|| 4 || 1\12 || [[Diminished]] ||
|| 6 || 1\12 || [[Hexe]] ||
==Commas==
12 EDO [[tempering out|tempers out]] the following [[comma]]s. (Note: This assumes val &lt; 12 19 28 34 42 44 |.)


||~ Comma ||~ Monzo ||~ Value (Cents) ||~ Name 1 ||~ Name 2 ||~ Name 3 ||
The 7th harmonic ([[7/4]]) is represented by the diatonic [[minor seventh]], which is sharp by 31 cents, and as such 12edo tempers out [[64/63]]. The deviation explains why minor sevenths tend to stand out distinctly from the rest of the chord in a [[tetrad]]. Such tetrads are often used as [[dominant seventh chord]]s in [[diatonic functional harmony|functional harmony]], for which the 5-limit JI version would be [[36:45:54:64|1–5/4–3/2–16/9]], and while 12et officially [[support]]s septimal meantone for tempering out [[126/125]] and [[225/224]] via its [[patent val]] of {{val| 12 19 28 34}}, its approximations of [[7-limit]] intervals are not very accurate. It cannot be said to represent [[11/1|11]] or [[13/1|13]] at all, though it does a quite credible [[17/1|17]] and an even better [[19/1|19]]. Nevertheless, its relative tuning accuracy is quite high, and 12edo is the fourth [[zeta integral edo]].
||= 531441/524288 || | -19 12 &gt; ||&gt; 23.46 ||= Pythagorean Comma ||=  ||=  ||
||= 648/625 || | 3 4 -4 &gt; ||&gt; 62.57 ||= Major Diesis ||= Diminished Comma ||=  ||
||= 128/125 || | 7 0 -3 &gt; ||&gt; 41.06 ||= Diesis ||= Augmented Comma ||=  ||
||= 81/80 || | -4 4 -1 &gt; ||&gt; 21.51 ||= Syntonic Comma ||= Didymos Comma ||= Meantone Comma ||
||= 2048/2025 || | 11 -4 -2 &gt; ||&gt; 19.55 ||= Diaschisma ||=  ||=  ||
||= 5201701/5149091 || | 26 -12 -3 &gt; ||&gt; 17.60 ||= Misty Comma ||=  ||=  ||
||= 32805/32768 || | -15 8 1 &gt; ||&gt; 1.95 ||= Schisma ||=  ||=  ||
||=  || | 161 -84 -12 &gt; ||&gt; 0.02 ||= Atom ||=  ||=  ||
||= 36/35 || | 2 2 -1 -1 &gt; ||&gt; 48.77 ||= Septimal Quarter Tone ||=  ||=  ||
||= 50/49 || | 1 0 2 -2 &gt; ||&gt; 34.98 ||= Tritonic Diesis ||= Jubilisma ||=  ||
||= 64/63 || | 6 -2 0 -1 &gt; ||&gt; 27.26 ||= Septimal Comma ||= Archytas' Comma ||= Leipziger Komma ||
||= 3125/3087 || | 0 -2 5 -3 &gt; ||&gt; 21.18 ||= Gariboh ||=  ||=  ||
||= 126/125 || | 1 2 -3 1 &gt; ||&gt; 13.79 ||= Septimal Semicomma ||= Starling Comma ||=  ||
||= 4000/3969 || | 5 -4 3 -2 &gt; ||&gt; 13.47 ||= Octagar ||=  ||=  ||
||= 321489/320000 || | -9 8 -4 2 &gt; ||&gt; 8.04 ||= Varunisma ||=  ||=  ||
||= 225/224 || | -5 2 2 -1 &gt; ||&gt; 7.71 ||= Septimal Kleisma ||= Marvel Comma ||=  ||
||= 3136/3125 || | 6 0 -5 2 &gt; ||&gt; 6.08 ||= Hemimean ||=  ||=  ||
||= 5120/5103 || | 10 -6 1 -1 &gt; ||&gt; 5.76 ||= Hemifamity ||=  ||=  ||
||= 4994735/4983772 || | 25 -14 0 -1 &gt; ||&gt; 3.80 ||= Garischisma ||=  ||=  ||
||= 703125/702464 || | -11 2 7 -3 &gt; ||&gt; 1.63 ||= Meter ||=  ||=  ||
||= 250047/250000 || | -4 6 -6 3 &gt; ||&gt; 0.33 ||= Landscape Comma ||=  ||=  ||
||= 99/98 || | -1 2 0 -2 1 &gt; ||&gt; 17.58 ||= Mothwellsma ||=  ||=  ||
||= 100/99 || | 2 -2 2 0 -1 &gt; ||&gt; 17.40 ||= Ptolemisma ||=  ||=  ||
||= 176/175 || | 4 0 -2 -1 1 &gt; ||&gt; 9.86 ||= Valinorsma ||=  ||=  ||
||= 896/891 || | 7 -4 0 1 -1 &gt; ||&gt; 9.69 ||= Pentacircle ||=  ||=  ||
||= 441/440 || | -3 2 -1 2 -1 &gt; ||&gt; 3.93 ||= Werckisma ||=  ||=  ||
||= 9801/9800 || | -3 4 -2 -2 2 &gt; ||&gt; 0.18 ||= Kalisma ||= Gauss' Comma ||=  ||
||= 91/90 || | -1 -2 -1 1 0 1 &gt; ||&gt; 19.13 ||= Superleap ||=  ||=  ||</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;12edo&lt;/title&gt;&lt;/head&gt;&lt;body&gt;12&lt;a class="wiki_link" href="/EDO"&gt;EDO&lt;/a&gt;, perhaps better known as 12et since it really is a temperament, is the predominating tuning system in the world today. It achieved that position because it is the smallest equal division which can seriously claim to represent &lt;a class="wiki_link" href="/5-limit"&gt;5-limit&lt;/a&gt; harmony, and because as 1/12 Pythagorean comma meantone, it represents &lt;a class="wiki_link" href="/Meantone"&gt;meantone&lt;/a&gt;.&lt;br /&gt;
&lt;br /&gt;
It divides the octave into twelve equal parts, each of exactly 100 cents each unless octave shrinking or stretching is employed. Its has a fifth which is quite good at two cents flat. It has a major third which is 13 2/3 cents sharp, which works well enough for some styles of music and is not really adequate for others, and a minor third which is flat by even more, 15 2/3 cents. It is probably not an accident that as tuning in European music became increasingly close to 12et, the style of the music changed so that the defects of 12et appeared less evident, though it should be borne in mind that in actual performance these are often reduced by the tuning adaptations of the performers.&lt;br /&gt;
&lt;br /&gt;
Its seventh partial is sharp by over 31 cents, and stands out distinctly from the rest of the chord in a tetrad. Such tetrads are often being used as dominant seventh chords in functional harmony, for which the 5-limit JI version would be 1-5/4-3/2-16/9, and while 12et officially supports septimal meantone via the &lt;a class="wiki_link" href="/Vals%20and%20Tuning%20Space"&gt;val&lt;/a&gt; &amp;lt;12 19 28 34|, its credentials in the 7-limit department are distinctly cheesy. It cannot be said to represent 11 or 13 at all, though it does a quite credible 17 and an even better 19. Nevertheless its relative tuning accuracy is quite high, and 12edo is the fourth &lt;a class="wiki_link" href="/The%20Riemann%20Zeta%20Function%20and%20Tuning#Zeta%20EDO%20lists"&gt;zeta integral edo&lt;/a&gt;.&lt;br /&gt;
&lt;br /&gt;
In terms of the kernel, which is to say the commas it tempers out, it tempers out the Pythagorean comma, 3^12/2^19, the Didymas comma, 81/80, the diesis, 128/125, the diaschisma, 2048/2025, the Archytas comma, 64/63, the septimal quartertone, 36/35, the jubilisma, 50/49, the septimal semicomma, 126/125, and the septimal kleisma, 225/224. Each of these affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.&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-Linear temperaments"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:0 --&gt;Linear temperaments&lt;/h2&gt;


&lt;table class="wiki_table"&gt;
Stacking the fifth twelve times returns the pitch to the starting point, so that the Pythagorean comma, [[Pythagorean comma|3<sup>12</sup>/2<sup>19</sup>]], is tempered out. Three major thirds equal an octave, so the lesser diesis, [[128/125]], is tempered out. Four minor thirds also equal an octave, so the greater diesis, [[648/625]], is tempered out. These features have been widely utilized in contemporary music. The [[duodene]] is an unequal 12-note scale in 5-limit just intonation which observes these commas, and can be considered a [[detempering|detemper]] of 12edo. Other [[comma]]s 12et [[tempering out|tempers out]] include the diaschisma, [[2048/2025]], and the schisma, [[32805/32768]], and in the 7-limit, the septimal quartertone, [[36/35]], and the jubilisma, [[50/49]]. Each affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.
    &lt;tr&gt;
        &lt;th&gt;Periods&lt;br /&gt;
per octave&lt;br /&gt;
&lt;/th&gt;
        &lt;th&gt;Generator&lt;br /&gt;
&lt;/th&gt;
        &lt;th&gt;Temperaments&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;1\12&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;a class="wiki_link" href="/Ripple"&gt;Ripple&lt;/a&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;1&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;5\12&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;a class="wiki_link" href="/Meantone"&gt;Meantone&lt;/a&gt;/&lt;a class="wiki_link" href="/dominant"&gt;dominant&lt;/a&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;1\12&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;a class="wiki_link" href="/Srutal"&gt;Srutal&lt;/a&gt;/&lt;a class="wiki_link" href="/pajara"&gt;pajara&lt;/a&gt;&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;1\12&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;a class="wiki_link" href="/Augmented"&gt;Augmented&lt;/a&gt;&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;1\12&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;a class="wiki_link" href="/Diminished"&gt;Diminished&lt;/a&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;1\12&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;a class="wiki_link" href="/Hexe"&gt;Hexe&lt;/a&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
&lt;/table&gt;


&lt;!-- ws:start:WikiTextHeadingRule:2:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc1"&gt;&lt;a name="x-Commas"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:2 --&gt;Commas&lt;/h2&gt;
12edo also offers very good approximations to intervals in the 2.3.17.19 subgroup. This indicates one way to use 12edo that deviates from common-practice harmony; for instance the cluster chord 8:17:36:76 is well represented, as well as the [[16:19:24]] otonal minor triad.
12 EDO &lt;a class="wiki_link" href="/tempering%20out"&gt;tempers out&lt;/a&gt; the following &lt;a class="wiki_link" href="/comma"&gt;comma&lt;/a&gt;s. (Note: This assumes val &amp;lt; 12 19 28 34 42 44 |.)&lt;br /&gt;
&lt;br /&gt;


=== Prime harmonics ===
{{Harmonics in equal|12|prec=2}}


&lt;table class="wiki_table"&gt;
=== Subsets and supersets ===
    &lt;tr&gt;
12edo contains [[2edo]], [[3edo]], [[4edo]], and [[6edo]] as subsets. It is the 5th [[highly composite edo]], 12 being both a superabundant and a highly composite number. 12edo is also the only known edo aside from 2edo that is both [[the Riemann zeta function and tuning|strict zeta]] and highly composite.  
        &lt;th&gt;Comma&lt;br /&gt;
&lt;/th&gt;
        &lt;th&gt;Monzo&lt;br /&gt;
&lt;/th&gt;
        &lt;th&gt;Value (Cents)&lt;br /&gt;
&lt;/th&gt;
        &lt;th&gt;Name 1&lt;br /&gt;
&lt;/th&gt;
        &lt;th&gt;Name 2&lt;br /&gt;
&lt;/th&gt;
        &lt;th&gt;Name 3&lt;br /&gt;
&lt;/th&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td style="text-align: center;"&gt;531441/524288&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;| -19 12 &amp;gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: right;"&gt;23.46&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;Pythagorean Comma&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td style="text-align: center;"&gt;648/625&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;| 3 4 -4 &amp;gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: right;"&gt;62.57&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;Major Diesis&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;Diminished Comma&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td style="text-align: center;"&gt;128/125&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;| 7 0 -3 &amp;gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: right;"&gt;41.06&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;Diesis&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;Augmented Comma&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td style="text-align: center;"&gt;81/80&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;| -4 4 -1 &amp;gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: right;"&gt;21.51&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;Syntonic Comma&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;Didymos Comma&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;Meantone Comma&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td style="text-align: center;"&gt;2048/2025&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;| 11 -4 -2 &amp;gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: right;"&gt;19.55&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;Diaschisma&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td style="text-align: center;"&gt;5201701/5149091&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;| 26 -12 -3 &amp;gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: right;"&gt;17.60&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;Misty Comma&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td style="text-align: center;"&gt;32805/32768&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;| -15 8 1 &amp;gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: right;"&gt;1.95&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;Schisma&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;| 161 -84 -12 &amp;gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: right;"&gt;0.02&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;Atom&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td style="text-align: center;"&gt;36/35&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;| 2 2 -1 -1 &amp;gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: right;"&gt;48.77&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;Septimal Quarter Tone&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td style="text-align: center;"&gt;50/49&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;| 1 0 2 -2 &amp;gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: right;"&gt;34.98&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;Tritonic Diesis&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;Jubilisma&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td style="text-align: center;"&gt;64/63&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;| 6 -2 0 -1 &amp;gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: right;"&gt;27.26&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;Septimal Comma&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;Archytas' Comma&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;Leipziger Komma&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td style="text-align: center;"&gt;3125/3087&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;| 0 -2 5 -3 &amp;gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: right;"&gt;21.18&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;Gariboh&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td style="text-align: center;"&gt;126/125&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;| 1 2 -3 1 &amp;gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: right;"&gt;13.79&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;Septimal Semicomma&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;Starling Comma&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td style="text-align: center;"&gt;4000/3969&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;| 5 -4 3 -2 &amp;gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: right;"&gt;13.47&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;Octagar&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td style="text-align: center;"&gt;321489/320000&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;| -9 8 -4 2 &amp;gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: right;"&gt;8.04&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;Varunisma&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td style="text-align: center;"&gt;225/224&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;| -5 2 2 -1 &amp;gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: right;"&gt;7.71&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;Septimal Kleisma&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;Marvel Comma&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td style="text-align: center;"&gt;3136/3125&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;| 6 0 -5 2 &amp;gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: right;"&gt;6.08&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;Hemimean&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td style="text-align: center;"&gt;5120/5103&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;| 10 -6 1 -1 &amp;gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: right;"&gt;5.76&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;Hemifamity&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td style="text-align: center;"&gt;4994735/4983772&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;| 25 -14 0 -1 &amp;gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: right;"&gt;3.80&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;Garischisma&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td style="text-align: center;"&gt;703125/702464&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;| -11 2 7 -3 &amp;gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: right;"&gt;1.63&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;Meter&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td style="text-align: center;"&gt;250047/250000&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;| -4 6 -6 3 &amp;gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: right;"&gt;0.33&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;Landscape Comma&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td style="text-align: center;"&gt;99/98&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;| -1 2 0 -2 1 &amp;gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: right;"&gt;17.58&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;Mothwellsma&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td style="text-align: center;"&gt;100/99&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;| 2 -2 2 0 -1 &amp;gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: right;"&gt;17.40&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;Ptolemisma&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td style="text-align: center;"&gt;176/175&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;| 4 0 -2 -1 1 &amp;gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: right;"&gt;9.86&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;Valinorsma&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td style="text-align: center;"&gt;896/891&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;| 7 -4 0 1 -1 &amp;gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: right;"&gt;9.69&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;Pentacircle&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td style="text-align: center;"&gt;441/440&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;| -3 2 -1 2 -1 &amp;gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: right;"&gt;3.93&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;Werckisma&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td style="text-align: center;"&gt;9801/9800&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;| -3 4 -2 -2 2 &amp;gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: right;"&gt;0.18&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;Kalisma&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;Gauss' Comma&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td style="text-align: center;"&gt;91/90&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;| -1 -2 -1 1 0 1 &amp;gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: right;"&gt;19.13&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;Superleap&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
&lt;/table&gt;


&lt;/body&gt;&lt;/html&gt;</pre></div>
[[24edo]], which doubles it, improves significantly on approximations to 11 and 13, with 13 tuned sharp. [[36edo]], which triples it, improves on harmonics 7 and 13, but has the 13 tuned flat instead of sharp. [[72edo]] is a notable zeta-record edo, and [[60edo|60-]], [[84edo|84-]], and [[96edo]] all see utilities. Notable [[rank-2 temperament]]s that augment 12edo with extra [[generator]]s include [[compton]] and [[catler]].
 
== Intervals ==
{| class="wikitable center-all"
|+ style="font-size: 105%;" | Intervals of 12edo
|-
! [[Degree]]
! [[Cent]]s
! [[Interval region]]
! style="width: 165px;" | Approximated 5-limit<br>JI intervals (error in [[¢]])
! Audio
! style="width: 330px;" | Higher limit interpretations<ref group="note">Intervals of the 2.3.5.7.17.19 subgroup, with a few additional interpretations</ref>
|-
| 0
| 0
| Unison (prime)
| [[1/1]] (just)
| [[File:piano_0_1edo.mp3]]
|
|-
| 1
| 100
| Minor second
| [[256/243]] (+9.775)<br>[[16/15]] (−11.731)<br>[[25/24]] (+29.328)
| [[File:piano_1_12edo.mp3]]
| [[28/27]] (+37.039), [[21/20]] (+15.533), [[15/14]] (−19.443)<br>[[17/16]] (−4.955), [[18/17]] (+1.045)<br>[[19/18]] (+6.397), [[20/19]] (+11.199)
|-
| 2
| 200
| Major second
| [[9/8]] (−3.910)<br>[[10/9]] (+17.596)
| [[File:piano_1_6edo.mp3]]
| [[8/7]] (−31.174), [[28/25]] (+3.802)<br>[[17/15]] (−16.687), [[19/17]] (+7.442),<br>[[55/49]] (+0.020), [[64/57]] (−0.532)
|-
| 3
| 300
| Minor third
| [[32/27]] (+5.865)<br>[[6/5]] (−15.641)<br>[[75/64]] (+25.418)
| [[File:piano_1_4edo.mp3]]
| [[7/6]] (+33.129), [[25/21]] (−1.847)<br>[[19/16]] (+2.487)
|-
| 4
| 400
| Major third
| [[81/64]] (−7.820)<br>[[5/4]] (+13.686)<br> [[32/25]] (-27.373)
| [[File:piano_1_3edo.mp3]]
| [[63/50]] (−0.108), [[9/7]] (−35.084)<br>[[34/27]] (+0.910), [[24/19]] (−4.442)
|-
| 5
| 500
| Fourth
| [[4/3]] (+1.955)<br> [[27/20]] (-19.551)
| [[File:piano_5_12edo.mp3]]
| [[21/16]] (-29.219)
|-
| 6
| 600
| [[Tritone]]
| [[25/18]] (+31.283)<br>[[36/25]] (-31.283)<br>[[45/32]] (+9.776)<br>[[64/45]] (−9.776)
| [[File:piano_1_2edo.mp3]]
| [[7/5]] (+17.488), [[10/7]] (−17.488)<br>[[24/17]] (+3.000), [[17/12]] (−3.000)<br>[[99/70]] (−0.088), [[140/99]] (+0.088)
|-
| 7
| 700
| Fifth
| [[3/2]] (−1.955)<br>[[40/27]] (+19.551)
| [[File:piano_7_12edo.mp3]]
| [[32/21]] (+29.219)
|-
| 8
| 800
| Minor sixth
| [[128/81]] (+7.820)<br>[[8/5]] (−13.686)<br>[[25/16]] (+27.373)
| [[File:piano_2_3edo.mp3]]
| [[14/9]] (+35.084), [[100/63]] (+0.108)<br>[[19/12]] (+4.442), [[27/17]] (−0.910)
|-
| 9
| 900
| Major sixth
| [[27/16]] (−5.865)<br>[[5/3]] (+15.641)<br>[[128/75]] (-25.418)
| [[File:piano_3_4edo.mp3]]
| [[12/7]] (−33.129), [[42/25]] (+1.847)<br>[[32/19]] (−2.487)
|-
| 10
| 1000
| Minor seventh
| [[16/9]] (+3.910)<br>[[9/5]] (−17.596)
| [[File:piano_5_6edo.mp3]]
| [[7/4]] (+31.174), [[25/14]] (−3.802)<br>[[30/17]] (+16.687), [[34/19]] (−7.442)<br>[[98/55]] (-0.020), [[57/32]] (+0.532)
|-
| 11
| 1100
| Major seventh
| [[243/128]] (-9.775)<br>[[15/8]] (+11.731)<br>[[48/25]] (−29.328)
| [[File:piano_11_12edo.mp3]]
| [[28/15]] (+19.443), [[40/21]] (−15.533), [[27/14]] (−37.039)<br>[[32/17]] (+4.955), [[17/9]] (−1.045)<br>[[36/19]] (-6.397), [[19/10]] (-11.199)
|-
| 12
| 1200
| Octave
| [[2/1]] (just)
| [[File:piano_1_1edo.mp3]]
|
|}
<references group="note" />
 
== Notation ==
The intervals and notes of 12edo have standard names from classical music theory. This classical notation system, which was in use before 12edo with other tuning systems based on chains of fifths, is sometimes called the [[chain-of-fifths notation]] or extended Pythagorean notation.
 
{{Sharpness-sharp1|12}}
 
The subsets {{EDOs|1edo, 2edo, 3edo, 4edo and 6edo}} can all be written using 12edo [[subset notation]].
 
Any 12edo note or interval can be [[Enharmonic unison|respelled enharmonically]] by adding a [[pythagorean comma]] to it or subtracting one from it.
 
{| class="wikitable center-all"
|+ style="font-size: 105%;" | Notation of 12edo
|-
! rowspan="2" | [[Degree]]
! rowspan="2" | [[Cent]]s
! colspan="2" | [[Chain-of-fifths notation|Standard notation]]
|-
! Diatonic ([[5L&nbsp;2s]]) interval names
! Note names (on D)
|-
| 0
| 0
| '''Perfect unison (P1)'''
| '''D'''
|-
| 1
| 100
| Augmented unison (A1)<br>Minor second (m2)
| D#<br>Eb
|-
| 2
| 200
| '''Major second (M2)'''<br>Diminished third (d3)
| '''E'''<br>Fb
|-
| 3
| 300
| Augmented second (A2)<br>'''Minor third (m3)'''
| E#<br>'''F'''
|-
| 4
| 400
| Major third (M3)<br>Diminished fourth (d4)
| F#<br>Gb
|-
| 5
| 500
| '''Perfect fourth (P4)'''
| '''G'''
|-
| 6
| 600
| Augmented fourth (A4)<br>Diminished fifth (d5)
| G#<br>Ab
|-
| 7
| 700
| '''Perfect fifth (P5)'''
| A
|-
| 8
| 800
| Augmented fifth (A5)<br>Minor sixth (m6)
| A#<br>Bb
|-
| 9
| 900
| '''Major sixth (M6)'''<br>Diminished seventh (d7)
| '''B'''<br>Cb
|-
| 10
| 1000
| Augmented sixth (A6)<br>'''Minor seventh (m7)'''
| B#<br>'''C'''
|-
| 11
| 1100
| Major seventh (M7)<br>Diminished octave (d8)
| C#<br>Db
|-
| 12
| 1200
| '''Perfect octave (P8)'''
| '''D'''
|}
 
In 12edo:
* [[Ups and downs notation]] is identical to standard notation;
* Mixed [[sagittal notation]] is identical to standard notation, but pure sagittal notation exchanges sharps (&#x266F;) and flats (&#x266D;) for sagittal sharp ([[File:Sagittal sharp.png]]) and sagittal flat ([[File:Sagittal flat.png]]) respectively.
 
=== Sagittal notation ===
This notation uses the same sagittal sequence as EDOs [[5edo#Sagittal notation|5]], [[19edo#Sagittal notation|19]], and [[26edo#Sagittal notation|26]], is a subset of the notations for EDOs [[24edo#Sagittal notation|24]], [[36edo#Sagittal notation|36]], [[48edo#Sagittal notation|48]], [[60edo#Sagittal notation|60]], [[72edo#Sagittal notation|72]], and [[84edo#Sagittal notation|84]], and is a superset of the notation for [[6edo#Sagittal notation|6-EDO]].
 
==== Evo flavor ====
{{Sagittal chart|Evo}}
 
Because it includes no Sagittal symbols, this Evo Sagittal notation is also a conventional notation.
 
==== Revo flavor ====
{{Sagittal chart}}
 
== Solfege ==
{| class="wikitable center-all"
|+ style="font-size: 105%;" | Solfege of 12edo
|-
! [[Degree]]
! [[Cents]]
! Standard [[solfege]]<br>(movable do)
! [[Uniform solfege]]<br>(2-3 vowels)
|-
| 0
| 0
| Do
| Da
|-
| 1
| 100
| Di (A1)<br>Ra (m2)
| Du (A1)<br>Fra (m2)
|-
| 2
| 200
| Re
| Ra
|-
| 3
| 300
| Ri (A2)<br>Me (m3)
| Ru (A2)<br>Na (m3)
|-
| 4
| 400
| Mi
| Ma (M3)<br>Fo (d4)
|-
| 5
| 500
| Fa
| Mu (A3)<br>Fa (P4)
|-
| 6
| 600
| Fi (A4)<br>Se (d5)
| Pa (A4)<br>Sha (d5)
|-
| 7
| 700
| So
| Sa
|-
| 8
| 800
| Si (A5)<br>Le (m6)
| Su (A5)<br>Fla (m6)
|-
| 9
| 900
| La
| La (M6)<br>Tho (d7)
|-
| 10
| 1000
| Li (A6)<br>Te (m7)
| Lu (A6)<br>Tha (m7)
|-
| 11
| 1100
| Ti
| Ta (M7)<br>Do (d8)
|-
| 12
| 1200
| Do
| Da
|}
 
== Approximation to JI ==
[[File:12ed2-5Limit.svg|250px|thumb|right|alt=alt : Your browser has no SVG support.|Selected 5-limit intervals approximated in 12edo]]
 
=== 15-odd-limit interval mappings ===
{{Q-odd-limit intervals|12}}
{{Q-odd-limit intervals|12.1|apx=val|header=none|tag=none|title=15-odd-limit intervals by 12f val mapping}}
 
== 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| -19 12 }}
| {{Mapping| 12 19 }}
| +0.62
| 0.62
| 0.62
|-
| 2.3.5
| 81/80, 128/125
| {{Mapping| 12 19 28 }}
| −1.56
| 3.11
| 3.11
|-
| 2.3.5.7
| 36/35, 50/49, 64/63
| {{Mapping| 12 19 28 34 }}
| −3.95
| 4.92
| 4.94
|-
| 2.3.5.7.17
| 36/35, 50/49, 51/49, 64/63
| {{Mapping| 12 19 28 34 49 }}
| −2.92
| 4.86
| 4.87
|-
| 2.3.5.7.17.19
| 36/35, 50/49, 51/49, 57/56, 64/63
| {{Mapping| 12 19 28 34 49 51 }}
| −2.53
| 4.52
| 4.53
|- style="border-top: double;"
| 2.3.5.17
| 51/50, 81/80, 128/125
| {{Mapping| 12 19 28 49 }}
| −0.87
| 2.95
| 2.95
|-
| 2.3.5.17.19
| 51/50, 76/75, 81/80, 128/125
| {{Mapping| 12 19 28 49 51 }}
| −0.81
| 2.64
| 2.64
|}
* 12et is monotonic to the [[11-odd-limit]]. It is the first equal temperament to achieve this.
* 12et has a lower relative error than any previous equal temperaments in the [[3-limit|3-]], [[5-limit|5-]], [[7-limit|7-]], and [[11-limit]]. The next equal temperaments doing better in those subgroups are [[41edo|41]], [[19edo|19]], 19, [[22edo|22]], respectively.
* 12et is even more prominent in the 2.3.5.7.17.19 subgroup, and the next equal temperament that does this better is [[72edo|72]].
 
=== Uniform maps ===
{{Uniform map|edo=12}}
 
=== Commas ===
12edo [[tempering out|tempers out]] the following [[comma]]s. This assumes the [[val]] {{val| 12 19 28 34 42 44 49 51}}.
 
{| class="commatable wikitable center-all left-3 right-4 left-6"
|-
! [[Harmonic limit|Prime<br>limit]]
! [[Ratio]]<ref group="note">{{rd}}</ref>
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name
|-
| 3
| <abbr title="531441/524288">(12 digits)</abbr>
| {{monzo| -19 12 }}
| 23.46
| Lalawama / Poma
| [[Pythagorean comma]]
|-
| 5
| [[648/625]]
| {{monzo| 3 4 -4 }}
| 62.57
| Quadguma
| Diminished comma, greater diesis
|-
| 5
| <abbr title="262144/253125">(12 digits)</abbr>
| {{monzo| 18 -4 -5 }}
| 60.61
| Saquinguma
| [[Passion comma]]
|-
| 5
| [[128/125]]
| {{monzo| 7 0 -3 }}
| 41.06
| Triguma
| Augmented comma, lesser diesis
|-
| 5
| [[81/80]]
| {{monzo| -4 4 -1 }}
| 21.51
| Guma
| Syntonic comma, Didymus' comma, meantone comma
|-
| 5
| [[2048/2025]]
| {{monzo| 11 -4 -2 }}
| 19.55
| Saguguma
| Diaschisma
|-
| 5
| <abbr title="67108864/66430125">(16 digits)</abbr>
| {{monzo| 26 -12 -3 }}
| 17.60
| Sasa-triguma
| [[Misty comma]]
|-
| 5
| [[32805/32768]]
| {{monzo| -15 8 1 }}
| 1.95
| Layoma
| Schisma
|-
| 5
| <abbr title="2923003274661805836407369665432566039311865085952/2922977339492680612451840826835216578535400390625">(98 digits)</abbr>
| {{monzo| 161 -84 -12 }}
| 0.02
| Sepbisa-quadtriguma
| [[Kirnberger's atom]]
|-
| 7
| [[256/245]]
| {{monzo| 8 0 -1 -2 }}
| 76.03
| Ruruguma
| Bapbo comma
|-
| 7
| [[59049/57344]]
| {{monzo| -13 10 0 -1 }}
| 50.72
| Laruma
| Harrison's comma
|-
| 7
| [[36/35]]
| {{monzo| 2 2 -1 -1 }}
| 48.77
| Ruguma
| Mint comma, septimal quarter tone
|-
| 7
| [[50/49]]
| {{monzo| 1 0 2 -2 }}
| 34.98
| Biruyoma
| Jubilisma
|-
| 7
| [[3645/3584]]
| {{monzo| -9 6 1 -1 }}
| 29.22
| Laruyoma
| Schismean comma
|-
| 7
| [[64/63]]
| {{monzo| 6 -2 0 -1 }}
| 27.26
| Ruma
| Septimal comma
|-
| 7
| [[3125/3087]]
| {{monzo| 0 -2 5 -3 }}
| 21.18
| Triru-aquinyoma
| Gariboh comma
|-
| 7
| [[126/125]]
| {{monzo| 1 2 -3 1 }}
| 13.79
| Zotriguma
| Starling comma
|-
| 7
| [[4000/3969]]
| {{monzo| 5 -4 3 -2 }}
| 13.47
| Rurutriyoma
| Octagar comma
|-
| 7
| <abbr title="321489/320000">(12 digits)</abbr>
| {{monzo| -9 8 -4 2 }}
| 8.04
| Labizoguguma
| [[Varunisma]]
|-
| 7
| [[225/224]]
| {{monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyoma
| Marvel comma
|-
| 7
| [[3136/3125]]
| {{monzo| 6 0 -5 2 }}
| 6.08
| Zozoquinguma
| Hemimean comma
|-
| 7
| [[5120/5103]]
| {{monzo| 10 -6 1 -1 }}
| 5.76
| Saruyoma
| Hemifamity comma
|-
| 7
| [[33554432/33480783|(16 digits)]]
| {{monzo| 25 -14 0 -1 }}
| 3.80
| Sasaruma
| [[Garischisma]]
|-
| 7
| [[703125/702464|(12 digits)]]
| {{monzo| -11 2 7 -3 }}
| 1.63
| Latriru-asepyoma
| [[Metric comma]]
|-
| 7
| <abbr title="250047/250000">(12 digits)</abbr>
| {{monzo| -4 6 -6 3 }}
| 0.33
| Trizoguguma
| [[Landscape comma]]
|-
| 11
| [[128/121]]
| {{monzo| 7 0 0 0 -2 }}
| 97.36
| Lulubima
| Axirabian limma
|-
| 11
| [[45/44]]
| {{monzo| -2 2 1 0 -1 }}
| 38.91
| Luyoma
| Undecimal fifth tone
|-
| 11
| [[56/55]]
| {{monzo| 3 0 -1 1 -1 }}
| 31.19
| Luzoguma
| Undecimal tritonic comma
|-
| 11
| [[245/242]]
| {{monzo| -1 0 1 2 -2 }}
| 21.33
| Luluzozoyoma
| Frostma
|-
| 11
| [[99/98]]
| {{monzo| -1 2 0 -2 1 }}
| 17.58
| Loruruma
| Mothwellsma
|-
| 11
| [[100/99]]
| {{monzo| 2 -2 2 0 -1 }}
| 17.40
| Luyoyoma
| Ptolemisma
|-
| 11
| [[176/175]]
| {{monzo| 4 0 -2 -1 1 }}
| 9.86
| Loruguguma
| Valinorsma
|-
| 11
| [[896/891]]
| {{monzo| 7 -4 0 1 -1 }}
| 9.69
| Saluzoma
| Pentacircle comma
|-
| 11
| [[441/440]]
| {{monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozoguma
| Werckisma
|-
| 11
| [[9801/9800]]
| {{monzo| -3 4 -2 -2 2 }}
| 0.18
| Biloruguma
| Kalisma
|-
| 13
| [[65/64]]
| {{monzo| -6 0 1 0 0 1 }}
| 26.84
| Thoyoma
| Wilsorma
|-
| 13
| [[91/90]]
| {{monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozoguma
| Superleap comma, biome comma
|-
| 13
| [[144/143]]
| {{monzo| 4 2 0 0 -1 -1 }}
| 12.06
| Thuluma
| Grossma
|-
| 13
| [[1001/1000]]
| {{monzo| -3 0 -3 1 1 1 }}
| 1.73
| Tholozotriguma
| Fairytale comma, sinbadma
|-
| 13
| [[4096/4095]]
| {{monzo| 12 -2 -1 -1 0 -1 }}
| 0.42
| Sathuruguma
| Minisma
|-
| 17
| [[51/50]]
| {{monzo| -1 1 -2 0 0 0 1 }}
| 34.28
| Soguguma
| Large septendecimal sixth tone
|-
| 17
| [[52/51]]
| {{monzo| 2 -1 0 0 0 1 -1 }}
| 33.62
| Suthoma
| Small septendecimal sixth tone
|-
| 17
| [[136/135]]
| {{monzo| 3 -3 -1 0 0 0 1 }}
| 12.78
| Soguma
| Diatisma, fiventeen comma
|-
| 17
| [[256/255]]
| {{monzo| 8 -1 -1 0 0 0 -1 }}
| 6.78
| Suguma
| Charisma, septendecimal kleisma
|-
| 17
| [[289/288]]
| {{monzo| -5 -2 0 0 0 0 2 }}
| 6.00
| Sosoma
| Semitonisma
|-
| 17
| [[2601/2600]]
| {{monzo| -3 2 -2 0 0 -1 2 }}
| 0.67
| Sosothuguguma
| Sextantonisma
|-
| 19
| [[39/38]]
| {{monzo| -1 1 0 0 0 1 0 -1 }}
| 44.97
| Nuthoma
| Undevicesimal two-ninth tone
|-
| 19
| [[96/95]]
| {{monzo| 5 1 -1 0 0 0 0 -1 }}
| 18.13
| Nuguma
| 19th-partial chroma
|-
| 19
| [[153/152]]
| {{monzo| -3 2 0 0 0 0 1 -1}}
| 11.35
| Nusoma
| Ganassisma
|-
| 19
| [[171/170]]
| {{monzo| -1 2 -1 0 0 0 -1 1 }}
| 10.15
| Nosuguma
| Malcolmisma
|-
| 19
| [[324/323]]
| {{monzo| 2 4 0 0 0 0 -1 -1 }}
| 5.35
| Nusuma
| Photisma
|-
| 19
| [[361/360]]
| {{monzo| -3 -2 -1 0 0 0 0 2 }}
| 4.80
| Nonoguma
| Go comma
|-
|19
|[[513/512]]
|{{Monzo|9 3 0 0 0 0 0 -1}}
|3.37
|Lanoma
|Boethius' comma
|}
<references group="note" />
 
=== Rank-2 temperaments ===
{| class="wikitable center-1 center-2"
|-
! Periods<br>per 8ve
! Generator*
! Pergen
! Temperaments
|-
| 1
| 1\12
| (P8, P4/5)
| [[Ripple]], [[passion]]
|-
| 1
| 5\12
| (P8, P5)
| [[Meantone]] / [[dominant (temperament)|dominant]]
|-
| 2
| 5\12 (1\12)
| (P8/2, P5)
| [[Pajara]], [[injera]]
|-
| 3
| 5\12 (1\12)
| (P8/3, P5)
| [[Augmented (temperament)|Augmented]] / [[august]]
|-
| 4
| 5\12 (1\12)
| (P8/4, P5)
| [[Diminished (temperament)|Diminished]]
|-
| 6
| 5\12 (1\12)
| (P8/6, P5)
| [[Hexe]]
|}
<nowiki>*</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
 
Rank-2 temperaments to which 12et can be [[detempering|detempered]] include [[compton]] (12 & 72), [[schismic]]/[[garibaldi]] (41 & 53), and [[diaschismic]] (46 & 58). Rank-3 temperaments to which 12et can be detempered include [[marvel]], [[starling]]/[[thrush]], [[aberschismic]], [[orthoschismic]], and [[cassaschismic]]. For more comprehensive lists, see:
* [[List of 12et rank two temperaments by badness]]
* [[List of 12et rank two temperaments by complexity]]
* [[List of edo-distinct 12f rank two temperaments]]
* [[Schismic–commatic equivalence continuum]]
 
== Octave stretch or compression ==
Whether there is intonational improvement from [[stretched and compressed tuning|octave stretch and compression]] for 12edo varies by context. A slight compression such as what is given by [[40ed10]] and [[zpi|34zpi]] shows improved intonation of harmonics [[5/1|5]] and [[7/1|7]] at the cost of worse [[2/1|2]] and [[3/1|3]]; while stretching the octave for a purer 3 and for a better match of the inharmonicity on string instruments, like those in [[7edf]], [[19edt]], or [[31ed6]], also makes sense.
 
== Scales ==
{{Main| List of MOS scales in 12edo }}
 
The two most common 12edo mos scales are meantone[5] and meantone[7].
* Diatonic (meantone) 5L2s 2221221 (generator = 7\12)
* Pentatonic (meantone) 2L3s 22323 (generator = 7\12)
* Diminished 4L4s 12121212 (generator = 1\12, period = 3\12)
 
=== Non-mos scales ===
Due to 12edo's dominance, some non-mos scales are also widely used in many musical practices around the world.
 
* Harmonic major – 2212132
* Melodic major – 2212122
* Hungarian minor – 2131131
* Maqam hijaz / double harmonic major – 1312131
* 5-odd-limit tonality diamond – 3112113
[[File:12edo modes.pdf|thumb]]
 
== Well temperaments ==
:''For a list of historical well temperaments, see [[Well temperament]].''
 
* [[Cauldron]]
* [[Bifrost]]
* [[Grail]]
* [[Secor5 23TX]]
* [[Secor wt10]]
* [[Sabat1]]
* [[Sabat2]]
 
== Music ==
{{Catrel|12edo tracks}}
 
The vast majority of music today is in 12edo or 12edo with slight modifications. As such, one can easily find 12edo music by going on any music streaming service.
 
== See also ==
* [[Lumatone mapping for 12edo]]
* [[:purdal:12-EDD]]{{dead link}}
* [[Near12]] – a just intonation scale where every interval is within 12.5 cents of a 12edo step
 
== External links ==
* [http://tonalsoft.com/enc/number/12edo.aspx 12-tone equal-temperament] on [[Tonalsoft Encyclopedia]]
 
[[Category:3-limit record edos|##]] <!-- 2-digit number -->
[[Category:Historical]]
[[Category:Meantone]]

Latest revision as of 05:27, 6 June 2026

← 11edo 12edo 13edo →
Prime factorization 22 × 3 (highly composite)
Step size 100 ¢ (by definition) 
Fifth 7\12 (700 ¢)
(convergent)
Semitones (A1:m2) 1:1 (100 ¢ : 100 ¢)
Consistency limit 9
Distinct consistency limit 5
English Wikipedia has an article on:

12 equal divisions of the octave (abbreviated 12edo or 12ed2), also called 12-tone equal temperament (12tet) or 12 equal temperament (12et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 12 equal parts of exactly 100 ¢ each. Each step represents a frequency ratio of 21/12, or the 12th root of 2. It is the predominating tuning system in the world today.

Theory

12edo achieved its position as the standard Western tuning system through a combination of theoretical properties and practicality. It is the smallest number of equal divisions of the octave (edo) which can seriously claim to represent 5-limit harmony, and it represents a meantone temperament, tempering out 81/80, equating four perfect fifths with the 5th harmonic.

It divides the octave into twelve equal parts, each of exactly 100 cents. It has a fifth which is quite accurate at 700 cents, two cents flat of just intonation. It has a major third which is 13.7 cents sharp of just, which, while reasonable for its size, is unsatisfactory for some. The minor third is even less accurate, being 15.6 cents flat of just.

Before people used 12edo, people used a variety of historical temperaments such as quarter-comma meantone, and later well temperaments. By the 20th century, 12edo became dominant primarily due to practical considerations for keyboard instruments and its ability to handle modulation across all keys with reasonable intonation. In actual performance, these deviations from just intonation are often reduced by the tuning adaptations of skilled performers. Modern music theory has increasingly treated 12edo as a system in its own right rather than as an approximation of just intonation or meantone, leading to theoretical approaches such as serialism and much of jazz harmony that derive from 12edo's structure as an equal division rather than its underlying temperament properties.[citation needed]

12edo is the basic example of a dodecatonic scale and can be considered the simplest well temperament, where all twelve fifths are the same.

The 7th harmonic (7/4) is represented by the diatonic minor seventh, which is sharp by 31 cents, and as such 12edo tempers out 64/63. The deviation explains why minor sevenths tend to stand out distinctly from the rest of the chord in a tetrad. Such tetrads are often used as dominant seventh chords in functional harmony, for which the 5-limit JI version would be 1–5/4–3/2–16/9, and while 12et officially supports septimal meantone for tempering out 126/125 and 225/224 via its patent val of ⟨12 19 28 34], its approximations of 7-limit intervals are not very accurate. It cannot be said to represent 11 or 13 at all, though it does a quite credible 17 and an even better 19. Nevertheless, its relative tuning accuracy is quite high, and 12edo is the fourth zeta integral edo.

Stacking the fifth twelve times returns the pitch to the starting point, so that the Pythagorean comma, 312/219, is tempered out. Three major thirds equal an octave, so the lesser diesis, 128/125, is tempered out. Four minor thirds also equal an octave, so the greater diesis, 648/625, is tempered out. These features have been widely utilized in contemporary music. The duodene is an unequal 12-note scale in 5-limit just intonation which observes these commas, and can be considered a detemper of 12edo. Other commas 12et tempers out include the diaschisma, 2048/2025, and the schisma, 32805/32768, and in the 7-limit, the septimal quartertone, 36/35, and the jubilisma, 50/49. Each affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.

12edo also offers very good approximations to intervals in the 2.3.17.19 subgroup. This indicates one way to use 12edo that deviates from common-practice harmony; for instance the cluster chord 8:17:36:76 is well represented, as well as the 16:19:24 otonal minor triad.

Prime harmonics

Approximation of prime harmonics in 12edo
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) +0.00 -1.96 +13.69 +31.17 +48.68 -40.53 -4.96 +2.49 -28.27 -29.58 -45.04
Relative (%) +0.0 -2.0 +13.7 +31.2 +48.7 -40.5 -5.0 +2.5 -28.3 -29.6 -45.0
Steps
(reduced) 		12
(0) 		19
(7) 		28
(4) 		34
(10) 		42
(6) 		44
(8) 		49
(1) 		51
(3) 		54
(6) 		58
(10) 		59
(11)

Subsets and supersets

12edo contains 2edo, 3edo, 4edo, and 6edo as subsets. It is the 5th highly composite edo, 12 being both a superabundant and a highly composite number. 12edo is also the only known edo aside from 2edo that is both strict zeta and highly composite.

24edo, which doubles it, improves significantly on approximations to 11 and 13, with 13 tuned sharp. 36edo, which triples it, improves on harmonics 7 and 13, but has the 13 tuned flat instead of sharp. 72edo is a notable zeta-record edo, and 60-, 84-, and 96edo all see utilities. Notable rank-2 temperaments that augment 12edo with extra generators include compton and catler.

Intervals

Intervals of 12edo
Degree Cents Interval region Approximated 5-limit
JI intervals (error in ¢) 		Audio Higher limit interpretations[note 1]
0 0 Unison (prime) 1/1 (just)
1 100 Minor second 256/243 (+9.775)
16/15 (−11.731)
25/24 (+29.328) 		28/27 (+37.039), 21/20 (+15.533), 15/14 (−19.443)
17/16 (−4.955), 18/17 (+1.045)
19/18 (+6.397), 20/19 (+11.199)
2 200 Major second 9/8 (−3.910)
10/9 (+17.596) 		8/7 (−31.174), 28/25 (+3.802)
17/15 (−16.687), 19/17 (+7.442),
55/49 (+0.020), 64/57 (−0.532)
3 300 Minor third 32/27 (+5.865)
6/5 (−15.641)
75/64 (+25.418) 		7/6 (+33.129), 25/21 (−1.847)
19/16 (+2.487)
4 400 Major third 81/64 (−7.820)
5/4 (+13.686)
32/25 (-27.373) 		63/50 (−0.108), 9/7 (−35.084)
34/27 (+0.910), 24/19 (−4.442)
5 500 Fourth 4/3 (+1.955)
27/20 (-19.551) 		21/16 (-29.219)
6 600 Tritone 25/18 (+31.283)
36/25 (-31.283)
45/32 (+9.776)
64/45 (−9.776) 		7/5 (+17.488), 10/7 (−17.488)
24/17 (+3.000), 17/12 (−3.000)
99/70 (−0.088), 140/99 (+0.088)
7 700 Fifth 3/2 (−1.955)
40/27 (+19.551) 		32/21 (+29.219)
8 800 Minor sixth 128/81 (+7.820)
8/5 (−13.686)
25/16 (+27.373) 		14/9 (+35.084), 100/63 (+0.108)
19/12 (+4.442), 27/17 (−0.910)
9 900 Major sixth 27/16 (−5.865)
5/3 (+15.641)
128/75 (-25.418) 		12/7 (−33.129), 42/25 (+1.847)
32/19 (−2.487)
10 1000 Minor seventh 16/9 (+3.910)
9/5 (−17.596) 		7/4 (+31.174), 25/14 (−3.802)
30/17 (+16.687), 34/19 (−7.442)
98/55 (-0.020), 57/32 (+0.532)
11 1100 Major seventh 243/128 (-9.775)
15/8 (+11.731)
48/25 (−29.328) 		28/15 (+19.443), 40/21 (−15.533), 27/14 (−37.039)
32/17 (+4.955), 17/9 (−1.045)
36/19 (-6.397), 19/10 (-11.199)
12 1200 Octave 2/1 (just)
  1. ↑ Intervals of the 2.3.5.7.17.19 subgroup, with a few additional interpretations

Notation

The intervals and notes of 12edo have standard names from classical music theory. This classical notation system, which was in use before 12edo with other tuning systems based on chains of fifths, is sometimes called the chain-of-fifths notation or extended Pythagorean notation.

Semitones −2 −1 0 +1 +2
Symbol  
 		 
 		 
 		 
 		 

The subsets 1edo, 2edo, 3edo, 4edo and 6edo can all be written using 12edo subset notation.

Any 12edo note or interval can be respelled enharmonically by adding a pythagorean comma to it or subtracting one from it.

Notation of 12edo
Degree Cents Standard notation
Diatonic (5L 2s) interval names Note names (on D)
0 0 Perfect unison (P1) D
1 100 Augmented unison (A1)
Minor second (m2) 		D#
Eb
2 200 Major second (M2)
Diminished third (d3) 		E
Fb
3 300 Augmented second (A2)
Minor third (m3) 		E#
F
4 400 Major third (M3)
Diminished fourth (d4) 		F#
Gb
5 500 Perfect fourth (P4) G
6 600 Augmented fourth (A4)
Diminished fifth (d5) 		G#
Ab
7 700 Perfect fifth (P5) A
8 800 Augmented fifth (A5)
Minor sixth (m6) 		A#
Bb
9 900 Major sixth (M6)
Diminished seventh (d7) 		B
Cb
10 1000 Augmented sixth (A6)
Minor seventh (m7) 		B#
C
11 1100 Major seventh (M7)
Diminished octave (d8) 		C#
Db
12 1200 Perfect octave (P8) D

In 12edo:

  • Ups and downs notation is identical to standard notation;
  • Mixed sagittal notation is identical to standard notation, but pure sagittal notation exchanges sharps (♯) and flats (♭) for sagittal sharp () and sagittal flat () respectively.

Sagittal notation

This notation uses the same sagittal sequence as EDOs 5, 19, and 26, is a subset of the notations for EDOs 24, 36, 48, 60, 72, and 84, and is a superset of the notation for 6-EDO.

Evo flavor

12-EDO_Evo_Sagittal.svg

Because it includes no Sagittal symbols, this Evo Sagittal notation is also a conventional notation.

Revo flavor

12-EDO_Revo_Sagittal.svg

Solfege

Solfege of 12edo
Degree Cents Standard solfege
(movable do) 		Uniform solfege
(2-3 vowels)
0 0 Do Da
1 100 Di (A1)
Ra (m2) 		Du (A1)
Fra (m2)
2 200 Re Ra
3 300 Ri (A2)
Me (m3) 		Ru (A2)
Na (m3)
4 400 Mi Ma (M3)
Fo (d4)
5 500 Fa Mu (A3)
Fa (P4)
6 600 Fi (A4)
Se (d5) 		Pa (A4)
Sha (d5)
7 700 So Sa
8 800 Si (A5)
Le (m6) 		Su (A5)
Fla (m6)
9 900 La La (M6)
Tho (d7)
10 1000 Li (A6)
Te (m7) 		Lu (A6)
Tha (m7)
11 1100 Ti Ta (M7)
Do (d8)
12 1200 Do Da

Approximation to JI

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Selected 5-limit intervals approximated in 12edo

15-odd-limit interval mappings

The following tables show how 15-odd-limit intervals are represented in 12edo. Prime harmonics are in bold; inconsistent intervals are in italics.

Note that, since the cent was defined in terms of 12edo, the absolute and relative errors for 12edo are identical.

15-odd-limit intervals in 12edo (direct approximation, even if inconsistent)
Interval and complement Error (abs, ¢) Error (rel, %)
1/1, 2/1 0.000 0.0
3/2, 4/3 1.955 2.0
9/8, 16/9 3.910 3.9
13/11, 22/13 10.790 10.8
15/8, 16/15 11.731 11.7
5/4, 8/5 13.686 13.7
5/3, 6/5 15.641 15.6
7/5, 10/7 17.488 17.5
11/7, 14/11 17.508 17.5
9/5, 10/9 17.596 17.6
15/14, 28/15 19.443 19.4
13/7, 14/13 28.298 28.3
7/4, 8/7 31.174 31.2
7/6, 12/7 33.129 33.1
11/10, 20/11 34.996 35.0
9/7, 14/9 35.084 35.1
13/9, 18/13 36.618 36.6
15/11, 22/15 36.951 37.0
13/12, 24/13 38.573 38.6
13/8, 16/13 40.528 40.5
13/10, 20/13 45.786 45.8
11/9, 18/11 47.408 47.4
15/13, 26/15 47.741 47.7
11/8, 16/11 48.682 48.7
11/6, 12/11 49.363 49.4
15-odd-limit intervals in 12edo (patent val mapping)
Interval and complement Error (abs, ¢) Error (rel, %)
1/1, 2/1 0.000 0.0
3/2, 4/3 1.955 2.0
9/8, 16/9 3.910 3.9
15/8, 16/15 11.731 11.7
5/4, 8/5 13.686 13.7
5/3, 6/5 15.641 15.6
7/5, 10/7 17.488 17.5
11/7, 14/11 17.508 17.5
9/5, 10/9 17.596 17.6
15/14, 28/15 19.443 19.4
7/4, 8/7 31.174 31.2
7/6, 12/7 33.129 33.1
11/10, 20/11 34.996 35.0
9/7, 14/9 35.084 35.1
13/9, 18/13 36.618 36.6
15/11, 22/15 36.951 37.0
13/12, 24/13 38.573 38.6
13/8, 16/13 40.528 40.5
11/8, 16/11 48.682 48.7
11/6, 12/11 50.637 50.6
15/13, 26/15 52.259 52.3
11/9, 18/11 52.592 52.6
13/10, 20/13 54.214 54.2
13/7, 14/13 71.702 71.7
13/11, 22/13 89.210 89.2
15-odd-limit intervals by 12f val mapping
Interval and complement Error (abs, ¢) Error (rel, %)
1/1, 2/1 0.000 0.0
3/2, 4/3 1.955 2.0
9/8, 16/9 3.910 3.9
13/11, 22/13 10.790 10.8
15/8, 16/15 11.731 11.7
5/4, 8/5 13.686 13.7
5/3, 6/5 15.641 15.6
7/5, 10/7 17.488 17.5
11/7, 14/11 17.508 17.5
9/5, 10/9 17.596 17.6
15/14, 28/15 19.443 19.4
13/7, 14/13 28.298 28.3
7/4, 8/7 31.174 31.2
7/6, 12/7 33.129 33.1
11/10, 20/11 34.996 35.0
9/7, 14/9 35.084 35.1
15/11, 22/15 36.951 37.0
13/10, 20/13 45.786 45.8
15/13, 26/15 47.741 47.7
11/8, 16/11 48.682 48.7
11/6, 12/11 50.637 50.6
11/9, 18/11 52.592 52.6
13/8, 16/13 59.472 59.5
13/12, 24/13 61.427 61.4
13/9, 18/13 63.382 63.4

Regular temperament properties

Subgroup Comma list Mapping Optimal
8ve stretch (¢) 		Tuning error
Absolute (¢) Relative (%)
2.3 [-19 12⟩ [⟨12 19]] +0.62 0.62 0.62
2.3.5 81/80, 128/125 [⟨12 19 28]] −1.56 3.11 3.11
2.3.5.7 36/35, 50/49, 64/63 [⟨12 19 28 34]] −3.95 4.92 4.94
2.3.5.7.17 36/35, 50/49, 51/49, 64/63 [⟨12 19 28 34 49]] −2.92 4.86 4.87
2.3.5.7.17.19 36/35, 50/49, 51/49, 57/56, 64/63 [⟨12 19 28 34 49 51]] −2.53 4.52 4.53
2.3.5.17 51/50, 81/80, 128/125 [⟨12 19 28 49]] −0.87 2.95 2.95
2.3.5.17.19 51/50, 76/75, 81/80, 128/125 [⟨12 19 28 49 51]] −0.81 2.64 2.64
  • 12et is monotonic to the 11-odd-limit. It is the first equal temperament to achieve this.
  • 12et has a lower relative error than any previous equal temperaments in the 3-, 5-, 7-, and 11-limit. The next equal temperaments doing better in those subgroups are 41, 19, 19, 22, respectively.
  • 12et is even more prominent in the 2.3.5.7.17.19 subgroup, and the next equal temperament that does this better is 72.

Uniform maps

13-limit uniform maps between 11.8 and 12.2
Min. size Max. size Wart notation Map
11.7554 11.8436 12cde ⟨12 19 27 33 41 44]
11.8436 11.9329 12de ⟨12 19 28 33 41 44]
11.9329 11.9962 12e ⟨12 19 28 34 41 44]
11.9962 12.0256 12 ⟨12 19 28 34 42 44]
12.0256 12.2743 12f ⟨12 19 28 34 42 45]

Commas

12edo tempers out the following commas. This assumes the val ⟨12 19 28 34 42 44 49 51].

Prime
limit 		Ratio[note 1] Monzo Cents Color name Name
3 (12 digits) [-19 12⟩ 23.46 Lalawama / Poma Pythagorean comma
5 648/625 [3 4 -4⟩ 62.57 Quadguma Diminished comma, greater diesis
5 (12 digits) [18 -4 -5⟩ 60.61 Saquinguma Passion comma
5 128/125 [7 0 -3⟩ 41.06 Triguma Augmented comma, lesser diesis
5 81/80 [-4 4 -1⟩ 21.51 Guma Syntonic comma, Didymus' comma, meantone comma
5 2048/2025 [11 -4 -2⟩ 19.55 Saguguma Diaschisma
5 (16 digits) [26 -12 -3⟩ 17.60 Sasa-triguma Misty comma
5 32805/32768 [-15 8 1⟩ 1.95 Layoma Schisma
5 (98 digits) [161 -84 -12⟩ 0.02 Sepbisa-quadtriguma Kirnberger's atom
7 256/245 [8 0 -1 -2⟩ 76.03 Ruruguma Bapbo comma
7 59049/57344 [-13 10 0 -1⟩ 50.72 Laruma Harrison's comma
7 36/35 [2 2 -1 -1⟩ 48.77 Ruguma Mint comma, septimal quarter tone
7 50/49 [1 0 2 -2⟩ 34.98 Biruyoma Jubilisma
7 3645/3584 [-9 6 1 -1⟩ 29.22 Laruyoma Schismean comma
7 64/63 [6 -2 0 -1⟩ 27.26 Ruma Septimal comma
7 3125/3087 [0 -2 5 -3⟩ 21.18 Triru-aquinyoma Gariboh comma
7 126/125 [1 2 -3 1⟩ 13.79 Zotriguma Starling comma
7 4000/3969 [5 -4 3 -2⟩ 13.47 Rurutriyoma Octagar comma
7 (12 digits) [-9 8 -4 2⟩ 8.04 Labizoguguma Varunisma
7 225/224 [-5 2 2 -1⟩ 7.71 Ruyoyoma Marvel comma
7 3136/3125 [6 0 -5 2⟩ 6.08 Zozoquinguma Hemimean comma
7 5120/5103 [10 -6 1 -1⟩ 5.76 Saruyoma Hemifamity comma
7 (16 digits) [25 -14 0 -1⟩ 3.80 Sasaruma Garischisma
7 (12 digits) [-11 2 7 -3⟩ 1.63 Latriru-asepyoma Metric comma
7 (12 digits) [-4 6 -6 3⟩ 0.33 Trizoguguma Landscape comma
11 128/121 [7 0 0 0 -2⟩ 97.36 Lulubima Axirabian limma
11 45/44 [-2 2 1 0 -1⟩ 38.91 Luyoma Undecimal fifth tone
11 56/55 [3 0 -1 1 -1⟩ 31.19 Luzoguma Undecimal tritonic comma
11 245/242 [-1 0 1 2 -2⟩ 21.33 Luluzozoyoma Frostma
11 99/98 [-1 2 0 -2 1⟩ 17.58 Loruruma Mothwellsma
11 100/99 [2 -2 2 0 -1⟩ 17.40 Luyoyoma Ptolemisma
11 176/175 [4 0 -2 -1 1⟩ 9.86 Loruguguma Valinorsma
11 896/891 [7 -4 0 1 -1⟩ 9.69 Saluzoma Pentacircle comma
11 441/440 [-3 2 -1 2 -1⟩ 3.93 Luzozoguma Werckisma
11 9801/9800 [-3 4 -2 -2 2⟩ 0.18 Biloruguma Kalisma
13 65/64 [-6 0 1 0 0 1⟩ 26.84 Thoyoma Wilsorma
13 91/90 [-1 -2 -1 1 0 1⟩ 19.13 Thozoguma Superleap comma, biome comma
13 144/143 [4 2 0 0 -1 -1⟩ 12.06 Thuluma Grossma
13 1001/1000 [-3 0 -3 1 1 1⟩ 1.73 Tholozotriguma Fairytale comma, sinbadma
13 4096/4095 [12 -2 -1 -1 0 -1⟩ 0.42 Sathuruguma Minisma
17 51/50 [-1 1 -2 0 0 0 1⟩ 34.28 Soguguma Large septendecimal sixth tone
17 52/51 [2 -1 0 0 0 1 -1⟩ 33.62 Suthoma Small septendecimal sixth tone
17 136/135 [3 -3 -1 0 0 0 1⟩ 12.78 Soguma Diatisma, fiventeen comma
17 256/255 [8 -1 -1 0 0 0 -1⟩ 6.78 Suguma Charisma, septendecimal kleisma
17 289/288 [-5 -2 0 0 0 0 2⟩ 6.00 Sosoma Semitonisma
17 2601/2600 [-3 2 -2 0 0 -1 2⟩ 0.67 Sosothuguguma Sextantonisma
19 39/38 [-1 1 0 0 0 1 0 -1⟩ 44.97 Nuthoma Undevicesimal two-ninth tone
19 96/95 [5 1 -1 0 0 0 0 -1⟩ 18.13 Nuguma 19th-partial chroma
19 153/152 [-3 2 0 0 0 0 1 -1⟩ 11.35 Nusoma Ganassisma
19 171/170 [-1 2 -1 0 0 0 -1 1⟩ 10.15 Nosuguma Malcolmisma
19 324/323 [2 4 0 0 0 0 -1 -1⟩ 5.35 Nusuma Photisma
19 361/360 [-3 -2 -1 0 0 0 0 2⟩ 4.80 Nonoguma Go comma
19 513/512 [9 3 0 0 0 0 0 -1⟩ 3.37 Lanoma Boethius' comma
  1. ↑ Ratios longer than 10 digits are presented by placeholders with informative hints.

Rank-2 temperaments

Periods
per 8ve 		Generator* Pergen Temperaments
1 1\12 (P8, P4/5) Ripple, passion
1 5\12 (P8, P5) Meantone / dominant
2 5\12 (1\12) (P8/2, P5) Pajara, injera
3 5\12 (1\12) (P8/3, P5) Augmented / august
4 5\12 (1\12) (P8/4, P5) Diminished
6 5\12 (1\12) (P8/6, P5) Hexe

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

Rank-2 temperaments to which 12et can be detempered include compton (12 & 72), schismic/garibaldi (41 & 53), and diaschismic (46 & 58). Rank-3 temperaments to which 12et can be detempered include marvel, starling/thrush, aberschismic, orthoschismic, and cassaschismic. For more comprehensive lists, see:

Octave stretch or compression

Whether there is intonational improvement from octave stretch and compression for 12edo varies by context. A slight compression such as what is given by 40ed10 and 34zpi shows improved intonation of harmonics 5 and 7 at the cost of worse 2 and 3; while stretching the octave for a purer 3 and for a better match of the inharmonicity on string instruments, like those in 7edf, 19edt, or 31ed6, also makes sense.

Scales

Main article: List of MOS scales in 12edo

The two most common 12edo mos scales are meantone[5] and meantone[7].

  • Diatonic (meantone) 5L2s 2221221 (generator = 7\12)
  • Pentatonic (meantone) 2L3s 22323 (generator = 7\12)
  • Diminished 4L4s 12121212 (generator = 1\12, period = 3\12)

Non-mos scales

Due to 12edo's dominance, some non-mos scales are also widely used in many musical practices around the world.

  • Harmonic major – 2212132
  • Melodic major – 2212122
  • Hungarian minor – 2131131
  • Maqam hijaz / double harmonic major – 1312131
  • 5-odd-limit tonality diamond – 3112113

File:12edo modes.pdf

Well temperaments

For a list of historical well temperaments, see Well temperament.

Music

See also: Category:12edo tracks

The vast majority of music today is in 12edo or 12edo with slight modifications. As such, one can easily find 12edo music by going on any music streaming service.

See also

External links

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