Xenharmonic Wiki

10edo: Difference between revisions

← Older edit
VisualWikitext
Wikispaces>vaisvil
**Imported revision 265197380 - Original comment: **
MisterShafXen (talk | contribs)
1,074 edits
m Fixed typo
Tags: Visual edit Mobile edit Mobile web edit Advanced mobile edit
 
(209 intermediate revisions by 53 users not shown)
Line 1: Line 1:
<h2>IMPORTED REVISION FROM WIKISPACES</h2>
{{interwiki
This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
| de = 10-EDO
: This revision was by author [[User:vaisvil|vaisvil]] and made on <tt>2011-10-16 10:53:57 UTC</tt>.<br>
| en = 10edo
: The original revision id was <tt>265197380</tt>.<br>
| es =
: The revision comment was: <tt></tt><br>
| ja = 10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>
}}
<h4>Original Wikitext content:</h4>
{{Infobox ET}}
<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">10edo, or 10-tone equal temperament, is a tuning system which divides the [[octave]] into 10 equal parts of exactly 120 [[cent]]s. It can be thought of as two circles of [[5edo]] separated by 120 cents (or 5 circles of [[2edo]]). It adds to 5edo a small neutral second (or large minor 2nd) and its inversion a large neutral seventh (or small major 7th); an excellent approximation of [[13_8|13/8]] and its inversion [[16_13|16/13]]; and the droll 600-cent tritone that appears in every even-numbered EDO. Taking the the 360 cent large neutral third as a generator produces a heptatonic [[MOSScales|moment of symmetry scale]] of the form 1 2 1 2 1 2 1 ([[3L 4s|3L 4s - mosh]]). While not an integral or gap edo, it is a [[The Riemann Zeta Function and Tuning#Zeta%20EDO%20lists|zeta peak edo]]. One way to interpret it in terms of a temperament of Just intonation is as a 2.7.13.15 subgroup, such that 105/104, 225/224, and 16807/16384 are tempered out. It can also be treated as a full 13-limit temperament, but it is a closer match to the aforementioned subgroup.
{{ED intro}}


[[toc|flat]]
== Theory ==
----
10edo contains all the intervals of [[5edo]], but also adds another copy of it separated by 120 [[cent]]s. The new intervals have sizes of 120{{c}}, 360{{c}}, 600{{c}}, 840{{c}}, and 1080{{c}}. The 120{{c}} interval can be treated a small neutral second or large minor 2nd, and its inversion a large neutral seventh or small major 7th, with the 120{{c}} and 1080{{c}} intervals being close (about 0.6{{c}} off) to [[15/14]] and [[28/15]] respectively. The 360{{c}} interval is a large neutral third, being about 0.5{{c}} sharp of [[16/13]], with its inversion being equally close to [[13/8]]. Finally, the 600{{c}} interval is the tritone that appears in every even-numbered edo, including [[12edo]].


=[[media type="custom" key="10886138"]]=
Taking the the 360{{c}} large neutral third as a [[generator]] produces a heptatonic [[MOS scale|moment of symmetry scale]] with step sizes {{nowrap|2 1 1 2 1 2 1}} (pattern [[3L&nbsp;4s]], or "mosh"), which is the most [[Diatonic scale|diatonic]]-like scale in 10edo excluding the 5edo [[collapsed]] diatonic scale, and can be seen as a [[neutralized]] diatonic scale.  
=Intervals=
|| Degree || Cent
Values || Approximate
Ratios* || Name ||
|| 0 || 0 || 1/1 || unison ||
|| 1 || 120 || 15/14, 16/15, 13/14 || small neutral second/large minor second ||
|| 2 || 240 || 15/13, 8/7 || second/third ||
|| 3 || 360 || 16/13 || large neutral third ||
|| 4 || 480 || 64/49, 169/128 || smaller fourth ||
|| 5 || 600 || 91/64, 169/120 || tritone ||
|| 6 || 720 || 49/32, 256/169 || bigger fifth ||
|| 7 || 840 || 13/8 || neutral sixth ||
|| 8 || 960 || 7/4, 26/15, 125/72 || sixth/seventh ||
|| 9 || 1080 || 15/8, 28/15, 13/7 || small major 7th ||
|| 10 || 1200 || 2/1 || octave ||
*based on treating 10-EDO as a 2.7.13.15 subgroup temperament.


=Linear temperaments=
It shares [[5edo]]'s approximation quality in the [[2.3.7 subgroup]], though the detuned fifth could be seen as a bigger problem with the more fine division of steps. Compared to 5edo, 10edo attains more accuracy in the full [[7-limit]], by including a better approximation of [[5/4]] at 360{{c}}, resulting in the better tuning of various intervals including 5, such as [[16/15]] and [[7/5]]. However, the approximation to 5/4 is still over 25{{c}} flat, and this interval is also equated with [[6/5]] (which is even more inaccurate, at 44{{c}} sharp), tempering out [[25/24]] and resulting in the [[dicot]] exotemperament. Thus, if one wishes to represent JI with 10edo, it is best to use prime [[5/1|5]] carefully or not at all.
||~ Periods
per octave ||~ Generator ||~ Temperament(s) ||
|| 1 || 1\10 || Messed-up [[Negri|negri]] (or [[Miracle|miracle]]) ||
|| 1 || 3\10 || [[Dicot]]/[[Beatles|beatles]]/neutral thirds scale ||
|| 2 || 1\10 || Messed-up [[pajara]] ||
|| 2 || 2\10 || [[Decimal]] / messed-up [[Lemba|lemba]] ||
|| 5 || 1\10 || [[Blackwood]]/[[blacksmith]] ||
=Commas=
10 EDO tempers out the following commas. (Note: This assumes the val &lt; 10 16 23 28 35 37 |.)


||~ Comma ||~ Monzo ||~ Value (Cents) ||~ Name 1 ||~ Name 2 ||~ Name 3 ||
Even if 10edo isn't directly used to represent JI, it could still serve as a structural archetype for the 7-limit. The fact that 25/24 is tempered out means that the 5-limit major triad [[4:5:6]] and minor triad [[10:12:15|1/(6:5:4)]] are mapped to the same number of scale steps in the 10-form, a feature shared with [[7edo]] and the [[heptatonic]] system used in western music. 10edo additionally sends [[49/48]] to the unison, meaning the 7-limit triad [[4:6:7]] and its inverse [[14:21:24|1/(12:8:7)]] are the same number of scale steps in a decatonic system as well, and therefore also the [[4:5:6:7]] major and [[70:84:105:120|1/(12:10:8:7)]] minor tetrads as well. Tempering out 25/24 and 49/48 leads to the [[decimal]] exotemperament (which is named after 10edo). A more accurate temperament based off of the 10-form that doesn't temper out 25/24 or 49/48 is [[pajara]], which shares many desireable properties with diatonic<ref>Erlich, Paul. "Tuning, Tonality and 22-Tone Temperament." Xenharmonicon 17, 1998. [http://sethares.engr.wisc.edu/paperspdf/Erlich-22.pdf http://sethares.engr.wisc.edu/paperspdf/Erlich-22.pdf]</ref>.
||= 256/243 || | 8 -5 &gt; ||&gt; 90.22 ||= Limma ||= Pythagorean Minor 2nd ||  ||
||= 16875/16384 || | -14 3 4 &gt; ||&gt; 51.12 ||= Negri Comma ||= Double Augmentation Diesis ||  ||
||= 9931568/9752117 || | -25 7 6 &gt; ||&gt; 31.57 ||= Ampersand's Comma ||=  ||  ||
||= 2048/2025 || | 11 -4 -2 &gt; ||&gt; 19.55 ||= Diaschisma ||=  ||  ||
||= 525/512 || | -9 1 2 1 &gt; ||&gt; 43.41 ||= Avicennma ||= Avicenna's Enharmonic Diesis ||  ||
||= 49/48 || | -4 -1 0 2 &gt; ||&gt; 35.70 ||= Slendro Diesis ||=  ||  ||
||= 50/49 || | 1 0 2 -2 &gt; ||&gt; 34.98 ||= Tritonic Diesis ||= Jubilisma ||  ||
||= 686/675 || | 1 -3 -2 3 &gt; ||&gt; 27.99 ||= Senga ||=  ||  ||
||= 64/63 || | 6 -2 0 -1 &gt; ||&gt; 27.26 ||= Septimal Comma ||= Archytas' Comma || Leipziger Komma ||
||= 9859966/9733137 || | -10 7 8 -7 &gt; ||&gt; 22.41 ||= Blackjackisma ||=  ||  ||
||= 1029/1024 || | -10 1 0 3 &gt; ||&gt; 8.43 ||= Gamelisma ||=  ||  ||
||= 225/224 || | -5 2 2 -1 &gt; ||&gt; 7.71 ||= Septimal Kleisma ||= Marvel Comma ||  ||
||= 16875/16807 || | 0 3 4 -5 &gt; ||&gt; 6.99 ||= Mirkwai ||=  ||  ||
||= 6772805/6751042 || | 11 -10 -10 10 &gt; ||&gt; 5.57 ||= Linus ||=  ||  ||
||= 2401/2400 || | -5 -1 -2 4 &gt; ||&gt; 0.72 ||= Breedsma ||=  ||  ||
||= 243/242 || | -1 5 0 0 -2 &gt; ||&gt; 7.14 ||= Rastma ||=  ||  ||
||= 385/384 || | -7 -1 1 1 1 &gt; ||&gt; 4.50 ||= Keenanisma ||=  ||  ||
||= 441/440 || | -3 2 -1 2 -1 &gt; ||&gt; 3.93 ||= Werckisma ||=  ||  ||
||= 540/539 || | 2 3 1 -2 -1 &gt; ||&gt; 3.21 ||= Swetisma ||=  ||  ||
||= 3025/3024 || | -4 -3 2 -1 2 &gt; ||&gt; 0.57 ||= Lehmerisma ||=  ||  ||
||= 91/90 || | -1 -2 -1 1 0 1 &gt; ||&gt; 19.13 ||= Superleap ||=  ||  ||
||= 676/675 || | 2 -3 -2 0 0 2 &gt; ||&gt; 2.56 ||= Parizeksma ||=  ||  ||
=Music=
[[http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html|Ten Fingers]] &lt;span class="ymp-btn-page-play ymp-media-0b41d890e4b2e2ea729c8a1c199295e9"&gt;[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Ten_Fingers.mp3|play]]&lt;/span&gt; by [[Bill Sethares]] (synth guitar)
[[http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html|Circle of Thirds]] &lt;span class="ymp-btn-page-play ymp-media-86d97882546354d45034d3c272cbb56c"&gt;[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Circle_of_Thirds.mp3|play]]&lt;/span&gt; by Bill Sethares (synth ens.)
[[http://www.akjmusic.com/works.html|10_fantasy]] &lt;span class="ymp-btn-page-play ymp-media-9ccb949290bc6c1cc7c1a4be8065871d"&gt;[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/AKJ/10_fantasy.mp3|play]]&lt;/span&gt; by [[Aaron Krister Johnson]] (synth monody)
&lt;span class="ymp-btn-page-play ymp-media-edd919cb4d239805e1b5cba4375a1e93"&gt;[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Hunt/Prelude%20in%2010ET.mp3|Prelude in 10ET]]&lt;/span&gt; by [[Aaron Andrew Hunt]]
[[http://www.ziaspace.com/ZIA/mp3s/Future.html|Future]] &lt;span class="ymp-btn-page-play ymp-media-a5c7622ba1020de4aa13d58769a6c357"&gt;[[http://ziaspace.com/ZIA/mp3s/Future.mp3|play]]&lt;/span&gt; and [[http://www.ziaspace.com/ZIA/mp3s/Sol.html|Sol]] &lt;span class="ymp-btn-page-play ymp-media-aa7c2b6de113578643d9a38ca7aa70e4"&gt;[[http://ziaspace.com/ZIA/mp3s/Sol.mp3|play]]&lt;/span&gt; by ZIA (synths and voice in 10)
&lt;span class="ymp-btn-page-play ymp-media-08eecd4b04b03702c59dba2eb80c1428"&gt;[[http://azuma-asobi.com/Music/RhinoPrelude-2003-02-22.mp3|Prelude]]&lt;/span&gt; by [[Rick McGowan]] (Rhino synthesizer)
&lt;span class="ymp-btn-page-play ymp-media-a9a8a238b39d19122886e0c933d01cca"&gt;[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20Ideas%20on%20the%20Waterfall%20of%20Expression.mp3|Ideas on the Waterfall of Expression]]&lt;/span&gt; by [[Iglashion Jones]] (synth)
&lt;span class="ymp-btn-page-play ymp-media-1824c99256ed13ed49cba489058a8493"&gt;[[http://micro.soonlabel.com/10edo/daily20110713_q49_10_viola_duet_and_gongs.mp3|For two violas and gongs]]&lt;/span&gt; by [[@http://www.chrisvaisvil.com|Chris Vaisvil (website)]] [[@http://chrisvaisvil.com/?p=1039|more composition information]]
[[file:10preview.ogg]] A sample of orchestral possibilities made using ZynAddSubFx under Linux (cenobyte)
[[file:decexperiment.ogg]]3 tracks made in ZynAddSubFx simply mixed in Audacity (cenobyte)
[[http://micro.soonlabel.com/10edo/10_earwigs_invasive.mp3|10 Earwigs Invasive]] by [[@http://chrisvaisvil.com/?p=1397|Chris Vaisvil]]
=**Instruments**=


10-EDO lends itself exceptionally well to guitar (and other fretted strings), on account of the fact that five of its flat 4ths (at 480 cents) exactly spans two octaves (480*5=2400), meaning the open strings can be uniformly tuned in 4ths. This allows for greater uniformity in chord and scale fingering patterns than in 12-TET, making it exceptionally easy to learn. For instance, the fingering for an "E" chord would be 0-2-2-1-0-0 (low to high), an "A" chord would be 0-0-2-2-1-0, and a "D" chord would be 1-0-0-2-2-1. This is also the case in all EDOs which are multiples of 5, but in 10-EDO it is particularly simple.
Since the neutral third is very close to 16/13, 10edo is usable as a 2.3.5.7.13 temperament, which includes 5edo's representation of 2.3.7; however, it is not without high damage. For one, all of [[9/7]], [[13/10]], [[21/16]], and [[4/3]] are equated to a flat fourth (or an extremely sharp supermajor third), tempering out [[28/27]], [[40/39]], [[49/48]], [[64/63]], [[91/90]], and [[105/104]]. Also, 5-limit major and minor thirds are equated as mentioned before (tempering out 25/24), and the third is also equated to 16/13, tempering out 40/39 and [[65/64]]. Additionally, 5-limit augmented and diminished intervals are equated with nearby septimal intervals (tempering out [[225/224]]), and since 3/2 is tuned sharp and 5/4 is tuned flat, the syntonic comma is exaggerated to a full step, or 120{{c}}. More accurately, it can be seen as a 2.7.13.15 temperament, restricting the 3.5 subgroup to powers of 15.


[[image:Decaphonic_Classic_Guitar.png caption="A Decaphonic (10-EDO) Classical Guitar"]][[image:decaphonic-uke.JPG width="526" height="406"]][[media type="custom" key="10021077"]]</pre></div>
By treating 360{{c}} as 11/9, we arrive at 11/8 = 600{{c}} (tempering out [[144/143]] and [[243/242]]), which allows 10edo to be treated as a full [[13-limit]] temperament. However, it is more accurate as a no-11 system.   
<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;10edo&lt;/title&gt;&lt;/head&gt;&lt;body&gt;10edo, or 10-tone equal temperament, is a tuning system which divides the &lt;a class="wiki_link" href="/octave"&gt;octave&lt;/a&gt; into 10 equal parts of exactly 120 &lt;a class="wiki_link" href="/cent"&gt;cent&lt;/a&gt;s. It can be thought of as two circles of &lt;a class="wiki_link" href="/5edo"&gt;5edo&lt;/a&gt; separated by 120 cents (or 5 circles of &lt;a class="wiki_link" href="/2edo"&gt;2edo&lt;/a&gt;). It adds to 5edo a small neutral second (or large minor 2nd) and its inversion a large neutral seventh (or small major 7th); an excellent approximation of &lt;a class="wiki_link" href="/13_8"&gt;13/8&lt;/a&gt; and its inversion &lt;a class="wiki_link" href="/16_13"&gt;16/13&lt;/a&gt;; and the droll 600-cent tritone that appears in every even-numbered EDO. Taking the the 360 cent large neutral third as a generator produces a heptatonic &lt;a class="wiki_link" href="/MOSScales"&gt;moment of symmetry scale&lt;/a&gt; of the form 1 2 1 2 1 2 1 (&lt;a class="wiki_link" href="/3L%204s"&gt;3L 4s - mosh&lt;/a&gt;). While not an integral or gap edo, it is a &lt;a class="wiki_link" href="/The%20Riemann%20Zeta%20Function%20and%20Tuning#Zeta%20EDO%20lists"&gt;zeta peak edo&lt;/a&gt;. One way to interpret it in terms of a temperament of Just intonation is as a 2.7.13.15 subgroup, such that 105/104, 225/224, and 16807/16384 are tempered out. It can also be treated as a full 13-limit temperament, but it is a closer match to the aforementioned subgroup.&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextTocRule:14:&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:14 --&gt;&lt;!-- ws:start:WikiTextTocRule:15: --&gt;&lt;a href="#toc0"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:15 --&gt;&lt;!-- ws:start:WikiTextTocRule:16: --&gt; | &lt;a href="#Intervals"&gt;Intervals&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:16 --&gt;&lt;!-- ws:start:WikiTextTocRule:17: --&gt; | &lt;a href="#Linear temperaments"&gt;Linear temperaments&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:17 --&gt;&lt;!-- ws:start:WikiTextTocRule:18: --&gt; | &lt;a href="#Commas"&gt;Commas&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:18 --&gt;&lt;!-- ws:start:WikiTextTocRule:19: --&gt; | &lt;a href="#Music"&gt;Music&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:19 --&gt;&lt;!-- ws:start:WikiTextTocRule:20: --&gt; | &lt;a href="#Instruments"&gt;Instruments&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:20 --&gt;&lt;!-- ws:start:WikiTextTocRule:21: --&gt;
&lt;!-- ws:end:WikiTextTocRule:21 --&gt;&lt;hr /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:2:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc0"&gt;&lt;!-- ws:end:WikiTextHeadingRule:2 --&gt;&lt;!-- ws:start:WikiTextMediaRule:0:&amp;lt;img src=&amp;quot;http://www.wikispaces.com/site/embedthumbnail/custom/10886138?h=0&amp;amp;w=0&amp;quot; class=&amp;quot;WikiMedia WikiMediaCustom&amp;quot; id=&amp;quot;wikitext@@media@@type=&amp;amp;quot;custom&amp;amp;quot; key=&amp;amp;quot;10886138&amp;amp;quot;&amp;quot; title=&amp;quot;Custom Media&amp;quot;/&amp;gt; --&gt;&lt;script type="text/javascript" src="http://mediaplayer.yahoo.com/js"&gt;
&lt;/script&gt;&lt;!-- ws:end:WikiTextMediaRule:0 --&gt;&lt;/h1&gt;
&lt;!-- ws:start:WikiTextHeadingRule:4:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc1"&gt;&lt;a name="Intervals"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:4 --&gt;Intervals&lt;/h1&gt;
   


&lt;table class="wiki_table"&gt;
10edo is a [[zeta peak edo]], due to its relatively decent tunings of the harmonics 2, 3, 5, 7, 13, and 17. 10edo is also the smallest edo that maintains [[minimal consistent EDOs|25% or lower relative error]] on all of the first eight harmonics of the [[harmonic series]].
    &lt;tr&gt;
        &lt;td&gt;Degree&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;Cent&lt;br /&gt;
Values&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;Approximate&lt;br /&gt;
Ratios*&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;Name&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;0&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;0&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;1/1&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;unison&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;120&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;15/14, 16/15, 13/14&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;small neutral second/large minor second&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;240&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;15/13, 8/7&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;second/third&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;360&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;16/13&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;large neutral third&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;480&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;64/49, 169/128&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;smaller fourth&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;600&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;91/64, 169/120&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;tritone&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;720&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;49/32, 256/169&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;bigger fifth&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;840&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;13/8&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;neutral sixth&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;960&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;7/4, 26/15, 125/72&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;sixth/seventh&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;1080&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;15/8, 28/15, 13/7&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;small major 7th&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;1200&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;2/1&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;octave&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
&lt;/table&gt;


*based on treating 10-EDO as a 2.7.13.15 subgroup temperament.&lt;br /&gt;
Thanks to its sevenths, 10edo is an ideal tuning for its size for [[metallic harmony]].
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:6:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc2"&gt;&lt;a name="Linear temperaments"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:6 --&gt;Linear temperaments&lt;/h1&gt;


&lt;table class="wiki_table"&gt;
=== Prime harmonics ===
    &lt;tr&gt;
{{Harmonics in equal|10}}
        &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;Temperament(s)&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\10&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;Messed-up &lt;a class="wiki_link" href="/Negri"&gt;negri&lt;/a&gt; (or &lt;a class="wiki_link" href="/Miracle"&gt;miracle&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;3\10&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;a class="wiki_link" href="/Dicot"&gt;Dicot&lt;/a&gt;/&lt;a class="wiki_link" href="/Beatles"&gt;beatles&lt;/a&gt;/neutral thirds scale&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\10&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;Messed-up &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;2&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;2\10&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;a class="wiki_link" href="/Decimal"&gt;Decimal&lt;/a&gt; / messed-up &lt;a class="wiki_link" href="/Lemba"&gt;lemba&lt;/a&gt;&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;1\10&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;a class="wiki_link" href="/Blackwood"&gt;Blackwood&lt;/a&gt;/&lt;a class="wiki_link" href="/blacksmith"&gt;blacksmith&lt;/a&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
&lt;/table&gt;


&lt;!-- ws:start:WikiTextHeadingRule:8:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc3"&gt;&lt;a name="Commas"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:8 --&gt;Commas&lt;/h1&gt;
== Intervals ==
10 EDO tempers out the following commas. (Note: This assumes the val &amp;lt; 10 16 23 28 35 37 |.)&lt;br /&gt;
{| class="wikitable right-1 right-2 center-7 center-8"
&lt;br /&gt;
|-
! Degree
! Cents
! Approximate ratios<ref group="note">{{sg|limit=2.15.7.13-subgroup}}</ref>
! Additional ratios<br />of 3, 5, and 9<ref group="note">Adding the ratios of 3, 5, and 9 introduces greater [[error]] while giving several more harmonic identities to the 10-edo intervals</ref>
! Interval names
! colspan="3" | [[Ups and downs notation]]<br />([[Enharmonic unisons in ups and downs notation|EUs]]: vvA1 and m2)
! Audio
|-
| 0
| 0
| 1/1
| 256/243, 50/49, 25/24
| unison
| unison, min 2nd
| P1, m2
| D, Eb
| [[File:0-0 unison.mp3|frameless]]
|-
| 1
| 120
| 16/15, 15/14, 14/13
| 10/9, 13/12, 81/80
| neutral second
| mid 2nd
| ~2
| ^D, vE
| [[File:0-120 neutral second (10-EDO).mp3|frameless]]
|-
| 2
| 240
| 8/7, 15/13, 144/125, 224/195
| 9/8, 7/6
| hemifourth, major second, minor third
| maj 2nd, min 3rd
| M2, m3
| E, F
| [[File:0-240 second, third (5-EDO).mp3|frameless]]
|-
| 3
| 360
| [[16/13]]
| 5/4
| neutral third
| mid 3rd
| ~3
| ^F, vG
| [[File:0-360 neutral third (10-EDO).mp3|frameless]]
|-
| 4
| 480
| 64/49, 169/128
| 4/3, 9/7, 13/10
| perfect fourth
| maj 3rd, perf 4th
| M3, P4
| F#, G
| [[File:0-480 fourth (5-EDO).mp3|frameless]]
|-
| 5
| 600
| 91/64, 128/91, 169/120, 240/169
| 7/5, 10/7, 13/9, 18/13
| hemioctave, tritone
| up 4th, down 5th
| ^4, v5
| ^G, vA
| [[File:0-600 (12-EDO).mp3|frameless]]
|-
| 6
| 720
| 49/32, 256/169
| 3/2, 14/9, 20/13
| perfect fifth
| perf 5th, min 6th
| P5, m6
| A, Bb
| [[File:0-720 fifth (5-EDO).mp3|frameless]]
|-
| 7
| 840
| [[13/8]]
| 8/5
| neutral sixth
| mid 6th
| ~6
| ^A, vB
| [[File:0-840 neutral sixth (10-EDO).mp3|frameless]]
|-
| 8
| 960
| 7/4, 26/15, 125/72, 195/112
| 16/9, 12/7
| hemitwelfth, major sixth, minor seventh
| maj 6th, min 7th
| M6, m7
| B, C
| [[File:0-960 sixth, seventh (5-EDO).mp3|frameless]]
|-
| 9
| 1080
| 15/8, 28/15, 13/7
| 9/5, 24/13, 160/81
| neutral seventh
| mid 7th
| ~7
| ^C, vD
| [[File:0-1080 major seventh (10-EDO).mp3|frameless]]
|-
| 10
| 1200
| 2/1
| 243/128, 49/25, 48/25
| octave
| maj 7th, octave
| M7, P8
| C#, D
| [[File:0-1200 octave.mp3|frameless]]
|}
<references group="note" />


== Notation ==
=== Ups and downs notation ===
The interval table above shows the diatonic notation, generated by 5ths (6\10, representing 3/2). Alternative notations include pentatonic fifth-generated and heptatonic 3rd-generated.


&lt;table class="wiki_table"&gt;
==== Pentatonic 5th-generated ====
    &lt;tr&gt;
'''D * E * G * A * C * D''' (generator = 3/2 = 6\10 = perfect 5thoid)
        &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;256/243&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;| 8 -5 &amp;gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: right;"&gt;90.22&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;Limma&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;Pythagorean Minor 2nd&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td style="text-align: center;"&gt;16875/16384&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;| -14 3 4 &amp;gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: right;"&gt;51.12&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;Negri Comma&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;Double Augmentation Diesis&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td style="text-align: center;"&gt;9931568/9752117&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;| -25 7 6 &amp;gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: right;"&gt;31.57&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;Ampersand's Comma&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&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&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td style="text-align: center;"&gt;525/512&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;| -9 1 2 1 &amp;gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: right;"&gt;43.41&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;Avicennma&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;Avicenna's Enharmonic Diesis&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td style="text-align: center;"&gt;49/48&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;| -4 -1 0 2 &amp;gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: right;"&gt;35.70&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;Slendro Diesis&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&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&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td style="text-align: center;"&gt;686/675&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;| 1 -3 -2 3 &amp;gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: right;"&gt;27.99&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;Senga&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&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&gt;Leipziger Komma&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td style="text-align: center;"&gt;9859966/9733137&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;| -10 7 8 -7 &amp;gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: right;"&gt;22.41&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;Blackjackisma&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td style="text-align: center;"&gt;1029/1024&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;| -10 1 0 3 &amp;gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: right;"&gt;8.43&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;Gamelisma&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&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&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td style="text-align: center;"&gt;16875/16807&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;| 0 3 4 -5 &amp;gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: right;"&gt;6.99&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;Mirkwai&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td style="text-align: center;"&gt;6772805/6751042&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;| 11 -10 -10 10 &amp;gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: right;"&gt;5.57&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;Linus&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td style="text-align: center;"&gt;2401/2400&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;| -5 -1 -2 4 &amp;gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: right;"&gt;0.72&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;Breedsma&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td style="text-align: center;"&gt;243/242&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;| -1 5 0 0 -2 &amp;gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: right;"&gt;7.14&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;Rastma&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td style="text-align: center;"&gt;385/384&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;| -7 -1 1 1 1 &amp;gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: right;"&gt;4.50&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;Keenanisma&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&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&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td style="text-align: center;"&gt;540/539&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;| 2 3 1 -2 -1 &amp;gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: right;"&gt;3.21&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;Swetisma&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td style="text-align: center;"&gt;3025/3024&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;| -4 -3 2 -1 2 &amp;gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: right;"&gt;0.57&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;Lehmerisma&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&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&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td style="text-align: center;"&gt;676/675&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;| 2 -3 -2 0 0 2 &amp;gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: right;"&gt;2.56&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;Parizeksma&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
&lt;/table&gt;


&lt;!-- ws:start:WikiTextHeadingRule:10:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc4"&gt;&lt;a name="Music"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:10 --&gt;Music&lt;/h1&gt;
D - D^/Ev - E - E^/Gv - G - G^/Av - A - A^/Cv - C - C^/Dv - D
&lt;a class="wiki_link_ext" href="http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html" rel="nofollow"&gt;Ten Fingers&lt;/a&gt; &lt;span class="ymp-btn-page-play ymp-media-0b41d890e4b2e2ea729c8a1c199295e9"&gt;&lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Ten_Fingers.mp3" rel="nofollow"&gt;play&lt;/a&gt;&lt;/span&gt; by &lt;a class="wiki_link" href="/Bill%20Sethares"&gt;Bill Sethares&lt;/a&gt; (synth guitar)&lt;br /&gt;
 
&lt;a class="wiki_link_ext" href="http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html" rel="nofollow"&gt;Circle of Thirds&lt;/a&gt; &lt;span class="ymp-btn-page-play ymp-media-86d97882546354d45034d3c272cbb56c"&gt;&lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Circle_of_Thirds.mp3" rel="nofollow"&gt;play&lt;/a&gt;&lt;/span&gt; by Bill Sethares (synth ens.)&lt;br /&gt;
1 - ^1/vs3 - s3 - ^s3/v4d - 4d - ^4d/v5d - 5d - ^5d/vs7 - s7 - ^s7/v8d - 8d (s = sub-, d = -oid)
&lt;a class="wiki_link_ext" href="http://www.akjmusic.com/works.html" rel="nofollow"&gt;10_fantasy&lt;/a&gt; &lt;span class="ymp-btn-page-play ymp-media-9ccb949290bc6c1cc7c1a4be8065871d"&gt;&lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/AKJ/10_fantasy.mp3" rel="nofollow"&gt;play&lt;/a&gt;&lt;/span&gt; by &lt;a class="wiki_link" href="/Aaron%20Krister%20Johnson"&gt;Aaron Krister Johnson&lt;/a&gt; (synth monody)&lt;br /&gt;
 
&lt;span class="ymp-btn-page-play ymp-media-edd919cb4d239805e1b5cba4375a1e93"&gt;&lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Hunt/Prelude%20in%2010ET.mp3" rel="nofollow"&gt;Prelude in 10ET&lt;/a&gt;&lt;/span&gt; by &lt;a class="wiki_link" href="/Aaron%20Andrew%20Hunt"&gt;Aaron Andrew Hunt&lt;/a&gt;&lt;br /&gt;
pentatonic circles of fifths: ...D - A - E - C - G - D... and ...^D - ^A - ^E - ^C - ^G - ^D... (or equivalently ...vD - vA - vE - vC - vG - vD...)
&lt;a class="wiki_link_ext" href="http://www.ziaspace.com/ZIA/mp3s/Future.html" rel="nofollow"&gt;Future&lt;/a&gt; &lt;span class="ymp-btn-page-play ymp-media-a5c7622ba1020de4aa13d58769a6c357"&gt;&lt;a class="wiki_link_ext" href="http://ziaspace.com/ZIA/mp3s/Future.mp3" rel="nofollow"&gt;play&lt;/a&gt;&lt;/span&gt; and &lt;a class="wiki_link_ext" href="http://www.ziaspace.com/ZIA/mp3s/Sol.html" rel="nofollow"&gt;Sol&lt;/a&gt; &lt;span class="ymp-btn-page-play ymp-media-aa7c2b6de113578643d9a38ca7aa70e4"&gt;&lt;a class="wiki_link_ext" href="http://ziaspace.com/ZIA/mp3s/Sol.mp3" rel="nofollow"&gt;play&lt;/a&gt;&lt;/span&gt; by ZIA (synths and voice in 10)&lt;br /&gt;
 
&lt;span class="ymp-btn-page-play ymp-media-08eecd4b04b03702c59dba2eb80c1428"&gt;&lt;a class="wiki_link_ext" href="http://azuma-asobi.com/Music/RhinoPrelude-2003-02-22.mp3" rel="nofollow"&gt;Prelude&lt;/a&gt;&lt;/span&gt; by &lt;a class="wiki_link" href="/Rick%20McGowan"&gt;Rick McGowan&lt;/a&gt; (Rhino synthesizer)&lt;br /&gt;
pentatonic circles of fifths: ...1 - 5d - s3 - s7 - 4d - 1... and ...^1 - ^5d - ^s3 - ^s7 - ^4d - ^1... (or equivalently ...v1 - v5d - vs3 - vs7 - v4d - v1...)
&lt;span class="ymp-btn-page-play ymp-media-a9a8a238b39d19122886e0c933d01cca"&gt;&lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20Ideas%20on%20the%20Waterfall%20of%20Expression.mp3" rel="nofollow"&gt;Ideas on the Waterfall of Expression&lt;/a&gt;&lt;/span&gt; by &lt;a class="wiki_link" href="/Iglashion%20Jones"&gt;Iglashion Jones&lt;/a&gt; (synth)&lt;br /&gt;
 
&lt;span class="ymp-btn-page-play ymp-media-1824c99256ed13ed49cba489058a8493"&gt;&lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/10edo/daily20110713_q49_10_viola_duet_and_gongs.mp3" rel="nofollow"&gt;For two violas and gongs&lt;/a&gt;&lt;/span&gt; by &lt;a class="wiki_link_ext" href="http://www.chrisvaisvil.com" rel="nofollow" target="_blank"&gt;Chris Vaisvil (website)&lt;/a&gt; &lt;a class="wiki_link_ext" href="http://chrisvaisvil.com/?p=1039" rel="nofollow" target="_blank"&gt;more composition information&lt;/a&gt;&lt;br /&gt;
(s- = sub-, -d = -oid, see [[5edo#Alternative%20notations|5edo notation]])
&lt;!-- ws:start:WikiTextFileRule:521:&amp;lt;img src=&amp;quot;http://www.wikispaces.com/site/embedthumbnail/file/10preview.ogg?h=52&amp;amp;w=320&amp;quot; class=&amp;quot;WikiFile&amp;quot; id=&amp;quot;wikitext@@file@@10preview.ogg&amp;quot; title=&amp;quot;File: 10preview.ogg&amp;quot; width=&amp;quot;320&amp;quot; height=&amp;quot;52&amp;quot; /&amp;gt; --&gt;&lt;div class="objectEmbed"&gt;&lt;a href="/file/view/10preview.ogg/246873297/10preview.ogg" onclick="ws.common.trackFileLink('/file/view/10preview.ogg/246873297/10preview.ogg');"&gt;&lt;img src="http://www.wikispaces.com/i/mime/32/empty.png" height="32" width="32" alt="10preview.ogg" /&gt;&lt;/a&gt;&lt;div&gt;&lt;a href="/file/view/10preview.ogg/246873297/10preview.ogg" onclick="ws.common.trackFileLink('/file/view/10preview.ogg/246873297/10preview.ogg');" class="filename" title="10preview.ogg"&gt;10preview.ogg&lt;/a&gt;&lt;br /&gt;&lt;ul&gt;&lt;li&gt;&lt;a href="/file/detail/10preview.ogg"&gt;Details&lt;/a&gt;&lt;/li&gt;&lt;li&gt;&lt;a href="/file/view/10preview.ogg/246873297/10preview.ogg"&gt;Download&lt;/a&gt;&lt;/li&gt;&lt;li style="color: #666"&gt;3 MB&lt;/li&gt;&lt;/ul&gt;&lt;/div&gt;&lt;/div&gt;&lt;!-- ws:end:WikiTextFileRule:521 --&gt; A sample of orchestral possibilities made using ZynAddSubFx under Linux (cenobyte)&lt;br /&gt;
 
&lt;!-- ws:start:WikiTextFileRule:522:&amp;lt;img src=&amp;quot;http://www.wikispaces.com/site/embedthumbnail/file/decexperiment.ogg?h=52&amp;amp;w=320&amp;quot; class=&amp;quot;WikiFile&amp;quot; id=&amp;quot;wikitext@@file@@decexperiment.ogg&amp;quot; title=&amp;quot;File: decexperiment.ogg&amp;quot; width=&amp;quot;320&amp;quot; height=&amp;quot;52&amp;quot; /&amp;gt; --&gt;&lt;div class="objectEmbed"&gt;&lt;a href="/file/view/decexperiment.ogg/246971619/decexperiment.ogg" onclick="ws.common.trackFileLink('/file/view/decexperiment.ogg/246971619/decexperiment.ogg');"&gt;&lt;img src="http://www.wikispaces.com/i/mime/32/empty.png" height="32" width="32" alt="decexperiment.ogg" /&gt;&lt;/a&gt;&lt;div&gt;&lt;a href="/file/view/decexperiment.ogg/246971619/decexperiment.ogg" onclick="ws.common.trackFileLink('/file/view/decexperiment.ogg/246971619/decexperiment.ogg');" class="filename" title="decexperiment.ogg"&gt;decexperiment.ogg&lt;/a&gt;&lt;br /&gt;&lt;ul&gt;&lt;li&gt;&lt;a href="/file/detail/decexperiment.ogg"&gt;Details&lt;/a&gt;&lt;/li&gt;&lt;li&gt;&lt;a href="/file/view/decexperiment.ogg/246971619/decexperiment.ogg"&gt;Download&lt;/a&gt;&lt;/li&gt;&lt;li style="color: #666"&gt;2 MB&lt;/li&gt;&lt;/ul&gt;&lt;/div&gt;&lt;/div&gt;&lt;!-- ws:end:WikiTextFileRule:522 --&gt;3 tracks made in ZynAddSubFx simply mixed in Audacity (cenobyte)&lt;br /&gt;
[[Enharmonic unison]]: vvs3
&lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/10edo/10_earwigs_invasive.mp3" rel="nofollow"&gt;10 Earwigs Invasive&lt;/a&gt; by &lt;a class="wiki_link_ext" href="http://chrisvaisvil.com/?p=1397" rel="nofollow" target="_blank"&gt;Chris Vaisvil&lt;/a&gt;&lt;br /&gt;
 
&lt;!-- ws:start:WikiTextHeadingRule:12:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc5"&gt;&lt;a name="Instruments"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:12 --&gt;&lt;strong&gt;Instruments&lt;/strong&gt;&lt;/h1&gt;
==== Heptatonic 3rd-generated ====
&lt;br /&gt;
'''D E * F G * A B * C D''' (generator = 3\10 = perfect 3rd)
10-EDO lends itself exceptionally well to guitar (and other fretted strings), on account of the fact that five of its flat 4ths (at 480 cents) exactly spans two octaves (480*5=2400), meaning the open strings can be uniformly tuned in 4ths. This allows for greater uniformity in chord and scale fingering patterns than in 12-TET, making it exceptionally easy to learn. For instance, the fingering for an &amp;quot;E&amp;quot; chord would be 0-2-2-1-0-0 (low to high), an &amp;quot;A&amp;quot; chord would be 0-0-2-2-1-0, and a &amp;quot;D&amp;quot; chord would be 1-0-0-2-2-1. This is also the case in all EDOs which are multiples of 5, but in 10-EDO it is particularly simple.&lt;br /&gt;
 
&lt;br /&gt;
D - E - E#/Fb - F - G - G#/Ab - A - B - B#/Cb - C - D
&lt;!-- ws:start:WikiTextLocalImageRule:519:&amp;lt;img src=&amp;quot;/file/view/Decaphonic_Classic_Guitar.png/234752078/Decaphonic_Classic_Guitar.png&amp;quot; alt=&amp;quot;A Decaphonic (10-EDO) Classical Guitar&amp;quot; title=&amp;quot;A Decaphonic (10-EDO) Classical Guitar&amp;quot; /&amp;gt; --&gt;&lt;table class="captionBox"&gt;&lt;tr&gt;&lt;td class="captionedImage"&gt;&lt;img src="/file/view/Decaphonic_Classic_Guitar.png/234752078/Decaphonic_Classic_Guitar.png" alt="Decaphonic_Classic_Guitar.png" title="Decaphonic_Classic_Guitar.png" /&gt;&lt;/td&gt;&lt;/tr&gt;&lt;tr&gt;&lt;td class="imageCaption"&gt;A Decaphonic (10-EDO) Classical Guitar&lt;/td&gt;&lt;/tr&gt;&lt;/table&gt;&lt;!-- ws:end:WikiTextLocalImageRule:519 --&gt;&lt;!-- ws:start:WikiTextLocalImageRule:520:&amp;lt;img src=&amp;quot;/file/view/decaphonic-uke.JPG/251070060/526x406/decaphonic-uke.JPG&amp;quot; alt=&amp;quot;&amp;quot; title=&amp;quot;&amp;quot; style=&amp;quot;height: 406px; width: 526px;&amp;quot; /&amp;gt; --&gt;&lt;img src="/file/view/decaphonic-uke.JPG/251070060/526x406/decaphonic-uke.JPG" alt="decaphonic-uke.JPG" title="decaphonic-uke.JPG" style="height: 406px; width: 526px;" /&gt;&lt;!-- ws:end:WikiTextLocalImageRule:520 --&gt;&lt;!-- ws:start:WikiTextMediaRule:1:&amp;lt;img src=&amp;quot;http://www.wikispaces.com/site/embedthumbnail/custom/10021077?h=0&amp;amp;w=0&amp;quot; class=&amp;quot;WikiMedia WikiMediaCustom&amp;quot; id=&amp;quot;wikitext@@media@@type=&amp;amp;quot;custom&amp;amp;quot; key=&amp;amp;quot;10021077&amp;amp;quot;&amp;quot; title=&amp;quot;Custom Media&amp;quot;/&amp;gt; --&gt;&lt;script type="text/javascript" src="http://webplayer.yahooapis.com/player.js"&gt;
 
&lt;/script&gt;&lt;!-- ws:end:WikiTextMediaRule:1 --&gt;&lt;/body&gt;&lt;/html&gt;</pre></div>
P1 - m2 - M2 - P3 - m4 - M4/m5 - M5 - P6 - m7 - M7 - P8
 
genchain of 3rds: ...Db - Fb - Ab - Cb - E - G - B - D - F - A - C - E# - G# - B# - D#... ("Every good boy deserves fudge and candy")
 
genchain of 3rds: ...d8 - d3 - m5 - m7 - m2 - m4 - P6 - P1 - P3 - M5 - M7 - M2 - M4 - A6 - A1...
 
[[Enharmonic unison]]: d2
 
See below: 3L&nbsp;4s mosh notation
 
=== 3L&nbsp;4s (mosh) notation ===
See above: Heptatonic 3rd-generated notation.
 
The notation of Neutral[7]. Notes are denoted as LsLssLs = CDEFGABC, and raising and lowering by a chroma (L − s), 1 step in this instance, is denoted by ♯ and ♭.
 
{| class="wikitable center-1 right-2 center-3 mw-collapsible mw-collapsed"
! #
! Cents
! Note
! Name
! Associated ratios
|-
| 0
| 0
| C
| Perf 1sn
| 1/1
|-
| 1
| 120
| Db
| Min 2nd
| 12/11
|-
| 2
| 240
| D, Eb
| Maj 2nd, dim 3rd
| 9/8, 32/27
|-
| 3
| 360
| E
| Perf 3rd
| 11/9, 27/22
|-
| 4
| 480
| Fb
| Min 4th
| 4/3
|-
| 5
| 600
| F, Gb
| Maj 4th, min 5th
| 11/8, 16/11
|-
| 6
| 720
| G
| Maj 5th
| 3/2
|-
| 7
| 840
| A
| Perf 6th
| 18/11, 44/27
|-
| 8
| 960
| A#, Bb
| Aug 6th, min 7th
| 16/9, 27/16
|-
| 9
| 1080
| B
| Maj 7th
| 11/6
|-
| 10
| 1200
| C
| Perf 8ve
| 2/1
|}
 
=== Sagittal notation ===
This notation is a subset of the notations for edos [[20edo #Sagittal notation|20]] and [[30edo #Sagittal notation|30]] and a superset of the notation for [[5edo #Sagittal notation|5edo]].
 
==== Evo and Revo flavors ====
<imagemap>
File:10-EDO_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 319 0 479 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
rect 20 80 319 106 [[Fractional_3-limit_notation#Bad-fifths_apotome-fraction_notation | apotome-fraction notation]]
default [[File:10-EDO_Sagittal.svg]]
</imagemap>
 
==== Evo-SZ flavor ====
<imagemap>
File:10-EDO_Evo-SZ_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 315 0 475 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
rect 20 80 315 106 [[Fractional_3-limit_notation#Bad-fifths_apotome-fraction_notation | apotome-fraction notation]]
default [[File:10-EDO_Evo-SZ_Sagittal.svg]]
</imagemap>
 
Because it contains no Sagittal symbols, this Evo-SZ Sagittal notation is identical to Stein–Zimmerman notation.
 
== Approximation to JI ==
=== Selected just intervals by error ===
==== Selected 13-limit intervals ====
[[File:10ed2-001.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>8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3.5
| 25/24, 256/243
| {{Mapping| 10 16 23 }}
| -0.089
| 9.27
| 7.73
|-
| 2.3.5.7
| 25/24, 28/27, 49/48
| {{Mapping| 10 16 23 28 }}
| +0.718
| 8.15
| 6.79
|-
| 2.3.5.7.13
| 25/24, 28/27, 40/39, 49/48
| {{Mapping| 10 16 23 28 37 }}
| +0.603
| 7.30
| 6.08
|}
* 10et is lower in relative error than any previous equal temperaments in the 7- and 17-limit. The next equal temperaments doing better in those subgroups are [[12edo|12]] and [[19edo|19eg]], respectively.  
* 10et is prominent in the 2.3.7.13, 2.3.5.7.13, 2.3.7.13.17, and 2.3.5.7.13.17 subgroup. The next equal temperaments doing better in those subgroups are [[17edo|17]], 19, [[36edo|36]] and [[31edo|31]], respectively.
 
=== Uniform maps ===
{{Uniform map|edo=10}}
 
=== Commas ===
10et tempers out the following commas. This assumes the val {{val| 10 16 23 28 35 37 }}.  
 
{| class="commatable wikitable center-1 center-2 right-4 center-5"
|-
! [[Harmonic limit|Prime<br>limit]]
! [[Ratio]]<ref group="note">{{rd}}</ref>
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name(s)
|-
| 3
| [[256/243]]
| {{Monzo| 8 -5 }}
| 90.22
| Sawa
| Blackwood comma, Pythagorean limma
|-
| 5
| [[25/24]]
| {{Monzo| -3 -1 2 }}
| 70.67
| Yoyo
| Dicot comma, classic chroma
|-
| 5
| [[16875/16384]]
| {{Monzo| -14 3 4 }}
| 51.12
| Laquadyo
| Negri comma, double augmentation diesis
|-
| 5
| [[34171875/33554432|(16 digits)]]
| {{Monzo| -25 7 6 }}
| 31.57
| Lala-tribiyo
| [[Ampersand comma]]
|-
| 5
| [[2048/2025]]
| {{Monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 7
| [[525/512]]
| {{Monzo| -9 1 2 1 }}
| 43.41
| Lazoyoyo
| Avicennma, Avicenna's enharmonic diesis
|-
| 7
| [[49/48]]
| {{Monzo| -4 -1 0 2 }}
| 35.70
| Zozo
| Semaphoresma, slendro diesis
|-
| 7
| [[50/49]]
| {{Monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Jubilisma, tritonic diesis
|-
| 7
| [[686/675]]
| {{Monzo| 1 -3 -2 3 }}
| 27.99
| Trizo-agugu
| Senga
|-
| 7
| [[64/63]]
| {{Monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma, Archytas' comma, Leipziger Komma
|-
| 7
| <abbr title="854296875/843308032">(18 digits)</abbr>
| {{Monzo| -10 7 8 -7 }}
| 22.41
| Lasepru-aquadbiyo
| [[Blackjackisma]]
|-
| 7
| [[1029/1024]]
| {{Monzo| -10 1 0 3 }}
| 8.43
| Latrizo
| Gamelisma
|-
| 7
| [[225/224]]
| {{Monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Marvel comma, septimal kleisma
|-
| 7
| [[16875/16807]]
| {{Monzo| 0 3 4 -5 }}
| 6.99
| Quinru-aquadyo
| Mirkwai comma
|-
| 7
| <abbr title="578509309952/576650390625">(24 digits)</abbr>
| {{Monzo| 11 -10 -10 10 }}
| 5.57
| Saquinbizogu
| [[Linus comma]]
|-
| 7
| [[2401/2400]]
| {{Monzo| -5 -1 -2 4 }}
| 0.72
| Bizozogu
| Breedsma
|-
| 11
| [[243/242]]
| {{Monzo| -1 5 0 0 -2 }}
| 7.14
| Lulu
| Rastma
|-
| 11
| [[385/384]]
| {{Monzo| -7 -1 1 1 1 }}
| 4.50
| Lozoyo
| Keenanisma
|-
| 11
| [[441/440]]
| {{Monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[540/539]]
| {{Monzo| 2 3 1 -2 -1 }}
| 3.21
| Lururuyo
| Swetisma
|-
| 11
| [[3025/3024]]
| {{Monzo| -4 -3 2 -1 2 }}
| 0.57
| Loloruyoyo
| Lehmerisma
|-
| 13
| [[91/90]]
| {{Monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap comma, biome comma
|-
| 13
| [[196/195]]
| {{Monzo| 2 -1 -1 2 0 -1 }}
| 8.86
| Thuzozogu
| Mynucuma
|-
| 13
| [[676/675]]
| {{Monzo| 2 -3 -2 0 0 2 }}
| 2.56
| Bithogu
| Island comma, parizeksma
|}
<references group="note"/>
 
=== Rank-2 temperaments ===
{| class="wikitable center-1 center-2"
|-
! Periods<br>per 8ve
! Generator
! Temperament(s)
|-
| 1
| 1\10
| [[Negri]], [[miracle]] (out-of-tune)
|-
| 1
| 3\10
| [[Dicot]] / [[beatles]] (out-of-tune) / [[neutral]] (out-of-tune)
|-
| 2
| 1\10
| [[Pajara]] (out-of-tune)
|-
| 2
| 2\10
| [[Decimal]], [[lemba]] (out-of-tune)
|-
| 5
| 1\10
| [[Blackwood]]
|}
 
== Octave stretch or compression ==
If one wishes to use 10edo as a no-5s, 19-or-lower-limit tuning, then it benefits from [[octave shrinking]]. [[zpi|26zpi]] and [[36ed12]] are compressed-octave versions of 10edo.
 
If one wishes to use 10edo as a no-3s, 13-or-lower-limit tuning, then it benefits from [[octave stretching]]. [[ed7|28ed7]] is a stretched version of 10edo.
 
== Scales ==
=== MOS scales ===
* Decimal/Lemba[6] [[4L&nbsp;2s]] (period = 5\10, gen = 2\10): 2 2 1 2 2 1
* Dicot[7] [[3L&nbsp;4s]] (gen = 3\10): 1 2 1 2 1 2 1
* Negri[9] [[1L&nbsp;8s]] (gen = 1\10): 1 1 1 1 2 1 1 1 1
 
=== Other scales ===
* [[Pinetone #Pinetone pentatonic|Pinetone major pentatonic]] (subset of Dicot[7]): 2 1 3 1 3
* [[Pinetone #Pinetone pentatonic|Pinetone minor pentatonic]] (subset of Dicot[7]): 3 1 2 3 1
* [[Marvel hexatonic|Marvel augmented hexatonic]] (subset of Dicot[7]): 2 1 3 1 2 1
* Marvel double harmonic hexatonic (subset of Dicot[7]): 1 2 1 3 2 1, 1 2 3 1 2 1
* Decimal/Lemba[6] [[4M]]: 2 1 2 2 2 1
* Dicot[7] [[4M]]: 2 1 1 2 2 1 1
 
=== Horagrams ===
[[File:Screen Shot 2020-04-23 at 11.13.09 PM.png|alt=1\10 MOS|none|thumb|697x697px|1\10 mos with 1L&nbsp;1s, 1L&nbsp;2s, 1L&nbsp;3s, 1L&nbsp;4s, 1L&nbsp;5s, 1L&nbsp;6s, 1L&nbsp;7s, and 1L&nbsp;8s]]
[[File:Screen Shot 2020-04-23 at 11.13.35 PM.png|none|thumb|697x697px|3\10 mos with 1L&nbsp;1s, 1L&nbsp;2s, 3L&nbsp;1s, 3L&nbsp;4s]]
 
== Diagrams ==
[[File:10edo_wheel.png|alt=10edo wheel.png|280x285px|10edo wheel.png]]
 
== Instruments ==
10edo lends itself exceptionally well to guitar (and other fretted strings), on account of the fact that five of its flat 4ths (at 480{{c}}) exactly spans two octaves ({{nowrap|480 × 5 {{=}} 2400}}), meaning the open strings can be uniformly tuned in 4ths. This allows for greater uniformity in chord and scale fingering patterns than in 12edo, making it exceptionally easy to learn. For instance, the fingering for an "E" chord would be {{dash|0, 2, 2, 1, 0, 0}} (low to high), an "A" chord would be {{dash|0, 0, 2, 2, 1, 0}}, and a "D" chord would be {{nowrap|1, 0, 0, 2, 2, 1}}. This is also the case in all edos which are multiples of 5, but in 10-edo it is particularly simple.
 
Retuning a conventional keyboard to 10edo may be done in many ways, but neglecting or making redundant the Eb and Ab keys preserves the sLsLsLs scale on the white keys. Redundancy may make modulation easier, but another option is tuning the superfluous keys to selections from [[20edo|20edo]] which approximates the 11th harmonic with relative accuracy, among other features.
 
{| class="wikitable"
|-
| [[File:Decaphonic_Classic_Guitar.png|alt=Decaphonic_Classic_Guitar.png|Decaphonic_Classic_Guitar.png]]
|-
| A Decaphonic (10edo) Classical Guitar
|}
[[File:decaphonic-uke.JPG|alt=decaphonic-uke.JPG|526x406px|decaphonic-uke.JPG]]
 
=== Lumatone ===
''See [[Lumatone mapping for 10edo]]''.
 
== Music ==
{{Main| 10edo/Music }}
{{Catrel|10edo tracks}}
 
== References ==
<references/>
 
[[Category:10-tone scales]]
Retrieved from "https://en.xen.wiki/w/10edo"