39edo: Difference between revisions

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=<span style="color: #9900ab; font-family: 'Times New Roman',Times,serif; font-size: 113%;">39 tone equal temperament</span>= 

**39-EDO, 39-ED2** or **39-tET** divides the Octave (Ditave 2/1) in 39 equal parts of 30.76923 Cents each one. If we take 22\39 as a fifth, can be used in Mavila Temperament, and from that point of view seems to have attracted the attention of the Armodue school, an Italian group that use the scheme of [[xenharmonic/7L 2s|Superdiatonic]] LLLsLLLLs like a basical scale for notation and theory, suited in [[xenharmonic/16edo|16-ED2]], and allied systems: [[xenharmonic/25edo|25-ED2]] [1/3-tone 3;2]; [[xenharmonic/41edo|41-ED2]] [1/5-tone 5;3]; and [[xenharmonic/57edo|57]] ED2 [1/7-tone 7;4]. **Hornbostel Temperaments** is included too with: [[xenharmonic/23edo|23-ED2]] [1/3-tone 3;1]; 39-ED2 [1/5-tone 5;2] & [[xenharmonic/62edo|62-ED2]] [1/8-tone 8;3]. [[223edo|223-ED2]], the best accuracy for Hornbostel temperament fits very good with Armodue like 1/29-tone 29;10 version. Note that [[101edo|101]], [[131edo|131]], [[177edo|177]] & [[200edo|200]] ED2s are tempered systems that [[http://www.h-pi.com/eop-ogolevets.html|Alexei Ogolevets]] (Ukraine, 1891 - 1967) was proposing in his List of Temperaments, in which the Armodue system fits very well in all these.
However, its 23\39 fifth, 5.737 Cents sharp, is in much better tune than the Mavila fifth which like all Mavila fifths is very, very flat, in this case, 25 Cents flat. Together with its best third which is the familiar 400 cents of 12 equal, we get a system which tempers out the diesis, 128/125, and the amity comma, 1600000/1594323. We have two choices for a map for 7, but the sharp one works better with the 3 and 5, which adds 64/63 and 126/125 to the list. Tempering out both 128/125 and 64/63 makes 39EDO, in some few ways, allied to 12-ET in supporting augene temperament, and is in fact, an excellent choice for an augene tuning, but one difference is that 39 has a fine 11, and adding it to consideration we find that 39-EDO tempers out 99/98 and 121/120 also. This better choice for 39et is <39 62 91 110 135|.
A particular anecdote with this 39 divisions per 2/1 was made in the Teliochordon, in 1788 by Charles Clagget (Ireland, 1740? - 1820), a little extract [[http://ml.oxfordjournals.org/content/76/2/291.extract.jpg|here]].

==__**39-EDO Intervals**__== 
|| **ARMODUE NOMENCLATURE 5;2 RELATION** ||
|| * **‡** = Semisharp (1/5-tone up)
* **b** = Flat (3/5-tone down)
* **#** = Sharp (3/5-tone up)
* **v** = Semiflat (1/5-tone down) ||

||~ **Degrees** ||~ **Armodue note** ||~ **Cents size** ||~ **[[xenharmonic/Nearest just interval|Nearest Just I]]nterval** ||~ **Cents value** ||~ **Error** ||~ 11-limit Ratio Assuming
<39 62 91 110 135| [[Val]] ||
|| 0 || 1 || 0 || 1/1 || 0 || None || 1/1 ||
|| 1 || 1‡ (9#) || 30.7692 || 57/56 || 30.6421 || +0.1271 ||   ||
|| 2 || 2b || 61.5385 || 29/28 || 60.7513 || +0.7872 ||   ||
|| 3 || 1# || 92.3077 || 39/37 || 91.1386 || +1.1691 ||   ||
|| 4 || 2v || 123.0769 || 44/41 || 122.2555 || +0.8214 ||   ||
|| 5 || 2 || 153.8462 || 35/32 || 155.1396 || -1.2934 || 12/11, 11/10 ||
|| 6 || 2‡ || 184.6154 || 10/9 || 182.4037 || +2.2117 || 10/9 ||
|| 7**·** || 3b || 215.3846 || 17/15 || 216.6867 || -1.3021 || 8/7, 9/8 ||
|| 8 || 2# || 246.1538 || 15/13 || 247.7411 || -1.5873 ||   ||
|| 9 || 3v || 276.9231 || 27/23 || 277.5907 || -0.6676 || 7/6 ||
|| 10 || 3 || 307.6923 || 43/36 || 307.6077 || +0.0846 || 6/5 ||
|| 11 || 3‡ || 338.4615 || 17/14 || 336.1295 || +2.332 || 11/9 ||
|| 12**·** || 4b || 369.2308 || 26/21 || 369.7468 || -0.516 ||   ||
|| 13 || 3# || 400 || 34/27 || 399.0904 || +0.9096 || 5/4 ||
|| 14 || 4v (5b) || 430.7692 || 41/32 || 429.0624 || +1.7068 || 9/7, 14/11 ||
|| 15 || 4 || 461.5385 || 30/23 || 459.9944 || +1.5441 ||   ||
|| 16 || 4‡ (5v) || 492.3077 || 85/64 || 491.2691 || +1.0386 || 4/3 ||
|| 17**·** || 5 || 523.0769 || 23/17 || 523.3189 || -0.242 ||   ||
|| 18 || 5‡ (4#) || 553.8462 || 11/8 || 551.3179 || +2.5283 || 11/8 ||
|| 19 || 6b || 584.6154 || 7/5 || 582.5122 || +2.1032 || 7/5 ||
|| 20 || 5# || 615.3846 || 10/7 || 617.4878 || -2.1032 || 10/7 ||
|| 21 || 6v || 646.1538 || 16/11 || 648.6821 || -2.5283 || 16/11 ||
|| 22**·** || 6 || 676.9231 || 34/23 || 676.6811 || +0.242 ||   ||
|| 23 || 6‡ || 707.6923 || 128/85 || 708.7309 || -1.0386 || 3/2 ||
|| 24 || 7b || 738.4615 || 23/15 || 740.0056 || -1.5441 ||   ||
|| 25 || 6# || 769.2308 || 64/41 || 770.9376 || -1.7068 || 14/9, 11/7 ||
|| 26 || 7v || 800 || 27/17 || 800.9096 || -0.9096 || 8/5 ||
|| 27**·** || 7 || 830.7692 || 21/13 || 830.2532 || +0.516 ||   ||
|| 28 || 7‡ || 861.5385 || 28/17 || 863.8705 || -2.332 || 18/11 ||
|| 29 || 8b || 892.3077 || 72/43 || 892.3923 || -0.0846 || 5/3 ||
|| 30 || 7# || 923.0769 || 46/27 || 922.4093 || +0.6676 || 12/7 ||
|| 31 || 8v || 953.8462 || 26/15 || 952.2589 || +1.5873 ||   ||
|| 32**·** || 8 || 984.6154 || 30/17 || 983.3133 || +1.3021 || 7/4, 16/9 ||
|| 33 || 8‡ || 1015.3846 || 9/5 || 1017.5963 || -2.2117 || 9/5 ||
|| 34 || 9b || 1046.1538 || 64/35 || 1044.8604 || +1.2934 || 11/6, 20/11 ||
|| 35 || 8# || 1076.9231 || 41/22 || 1077.7445 || -0.8214 ||   ||
|| 36 || 9v (1b) || 1107.6923 || 74/39 || 1108.8614 || -1.1691 ||   ||
|| 37 || 9 || 1138.4615 || 56/29 || 1139.2487 || -0.7872 ||   ||
|| 38 || 9‡ (1v) || 1169.2308 || 112/57 || 1169.3579 || -0.1271 ||   ||
|| 39**··**(or 0) || 1 || 1200 || 2/1 || 1200 || None ||   ||

==__Instruments (prototypes):__== 

[[image:TECLADO 39-EDD.PNG width="800" height="467"]]
//An illustrative image of a 39-ED2 keyboard//
[[image:xenharmonic/Custom_700mm_5-str_Tricesanonaphonic_Guitar.png width="826" height="203" caption="39-EDD fretboard visualization"]]


==**__39 tone equal [[xenharmonic/modes|modes]]__:**== 

15 15 9 - [[xenharmonic/MOSScales|MOS]] of type [[xenharmonic/2L 1s|2L 1s]]
14 14 11 - [[xenharmonic/MOSScales|MOS]] of type [[xenharmonic/2L 1s|2L 1s]]
13 13 13 = [[xenharmonic/3edo|3edo]]
11 11 11 6 - [[xenharmonic/MOSScales|MOS]] of type [[xenharmonic/3L 1s|3L 1s]]
10 10 10 9 - [[xenharmonic/MOSScales|MOS]] of type [[xenharmonic/3L 1s|3L 1s]]
11 3 11 11 3 - [[xenharmonic/MOSScales|MOS]] of type [[3L 2s|3L 2s (Father pentatonic)]]
11 3 11 3 11 - <span style="cursor: pointer;">[[xenharmonic/MOSScales|MOS]]</span> of type <span style="color: #660000; cursor: pointer;">[[3L 2s|3L 2s (Father pentatonic)]]</span>
9 6 9 9 6 - [[xenharmonic/MOSScales|MOS]] of type [[3L 2s|3L 2s (Father pentatonic)]]
9 6 9 6 9 - <span style="cursor: pointer;">[[xenharmonic/MOSScales|MOS]]</span> of type <span style="color: #660000; cursor: pointer;">[[3L 2s|3L 2s (Father pentatonic)]]</span>
9 9 9 9 3 - [[xenharmonic/MOSScales|MOS]] of type [[4L 1s|4L 1s (Bug pentatonic)]]
9 3 9 9 9 - <span style="cursor: pointer;">[[xenharmonic/MOSScales|MOS]]</span> of type <span style="color: #660000; cursor: pointer;">[[4L 1s|4L 1s (Bug pentatonic)]]</span>
8 8 8 8 7 - [[xenharmonic/MOSScales|MOS]] of type [[4L 1s|4L 1s (Bug pentatonic)]]
10 3 10 3 10 3 - [[xenharmonic/MOSScales|MOS]] of type [[3L 3s|3L 3s (Augmented hexatonic)]]
9 4 9 4 9 4 - [[xenharmonic/MOSScales|MOS]] of type [[3L 3s|3L 3s (Augmented hexatonic)]]
8 5 8 5 8 5 - [[xenharmonic/MOSScales|MOS]] of type [[3L 3s|3L 3s (Augmented hexatonic)]]
7 7 7 7 7 4 - [[xenharmonic/MOSScales|MOS]] of type [[xenharmonic/5L 1s|5L 1s (Grumpy hexatonic)]]
7 4 7 7 7 7 - <span style="cursor: pointer;">[[xenharmonic/MOSScales|MOS]]</span> of type <span style="cursor: pointer;">[[xenharmonic/5L 1s|5L 1s (Grumpy hexatonic)]]</span>
3 9 3 9 3 9 3 - [[xenharmonic/MOSScales|MOS]] of type [[3L 4s|3L 4s (Mosh heptatonic)]]
5 5 7 5 5 5 7 - [[xenharmonic/MOSScales|MOS]] of type [[2L 5s|2L 5s (heptatonic Mavila Anti-Diatonic)]]
5 5 5 7 5 5 7 - [[xenharmonic/MOSScales|MOS]] of type [[2L 5s|2L 5s (heptatonic Mavila Anti-Diatonic)]]
5 7 5 5 7 5 5 - [[xenharmonic/MOSScales|MOS]] of type [[2L 5s|2L 5s (heptatonic Mavila Anti-Diatonic)]]
6 3 6 6 3 6 6 3 - [[xenharmonic/MOSScales|MOS]] of type [[5L 3s|5L 3s (unfair Father octatonic)]]
5 5 5 5 5 5 5 4 - [[xenharmonic/MOSScales|MOS]] of type [[xenharmonic/7L 1s|7L 1s (Grumpy octatonic)]]
5 4 5 5 5 5 5 5 - [[xenharmonic/MOSScales|MOS]] of type [[xenharmonic/7L 1s|7L 1s (Grumpy octatonic)]]
**5 5 5 2 5 5 5 5 2** - [[xenharmonic/MOSScales|MOS]] of type [[7L 2s|7L 2s (nonatonic Mavila Superdiatonic)]]
5 5 2 5 5 5 2 5 5 - [[xenharmonic/MOSScales|MOS]] of type [[7L 2s|7L 2s (nonatonic Mavila Superdiatonic)]]
5 5 3 5 5 3 5 5 3 - [[xenharmonic/MOSScales|MOS]] of type [[6L 3s|6L 3s (unfair Augmented nonatonic)]]
5 4 4 5 4 4 5 4 4 - [[xenharmonic/MOSScales|MOS]] of type [[3L 6s|3L 6s (fair Augmented nonatonic)]]
4 4 4 4 4 4 4 4 4 3 - [[xenharmonic/MOSScales|MOS]] of type [[xenharmonic/9L 1s|9L 1s (Grumpy decatonic)]]
4 4 3 4 4 4 4 4 4 4 - [[xenharmonic/MOSScales|MOS]] of type [[xenharmonic/9L 1s|9L 1s (Grumpy decatonic)]]
**3 3 5 3 3 3 5 3 3 3 5** - [[xenharmonic/MOSScales|MOS]] of type [[3L 8s|3L 8s (Anti-Sensi hendecatonic)]]
3 3 3 3 3 3 3 3 3 3 3 3 3 = [[xenharmonic/13edo|13edo]]
**3 3 3 2 3 3 3 3 2 3 3 3 3 2** - [[xenharmonic/MOSScales|MOS]] of type [[11L 3s|11L 3s (Ketradektriatoh tetradecatonic)]]
3 2 3 3 2 3 2 3 3 2 3 2 3 3 2 - [[xenharmonic/MOSScales|MOS]] of type [[xenharmonic/9L 6s|9L 6s]]
3 2 3 2 3 2 2 3 2 3 2 3 2 3 2 2 - [[xenharmonic/MOSScales|MOS]] of type [[xenharmonic/7L 9s|7L 9s]]
**2 2 3 2 2 2 3 2 2 3 2 2 3 2 2 2 3** - [[xenharmonic/MOSScales|MOS]] of type [[xenharmonic/5L 12s|5L 12s]]
2 2 2 2 2 3 2 2 2 2 2 3 2 2 2 2 2 3 - [[xenharmonic/MOSScales|MOS]] of type [[xenharmonic/3L 15s|3L 15s]]
**3 1 3 1 3 1 3 1 3 1 3 1 3 1 3 1 3 1 3** - <span style="cursor: pointer;">[[xenharmonic/MOSScales|MOS]]</span> of type [[xenharmonic/10L 9s|10L 9s]]
2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 1 - [[xenharmonic/MOSScales|MOS]] of type [[xenharmonic/19L 1s|19L 1s]]
2 2 2 1 2 2 2 1 2 2 2 1 2 2 2 2 1 2 2 2 2 1 - [[xenharmonic/MOSScales|MOS]] of type [[xenharmonic/17L 5s|17L 5s]]
**2 2 1 2 2 1 2 2 1 2 2 1 2 2 1 2 2 2 1 2 2 2 1** - [[xenharmonic/MOSScales|MOS]] of type [[xenharmonic/16L 7s|16L 7s]]
2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 - [[xenharmonic/MOSScales|MOS]] of type [[xenharmonic/13L 13s|13L 13s]]
**2 1 1 2 1 1 2 1 1 2 1 1 2 1 1 2 1 1 2 1 1 2 1 1 2 1 1 2 1** - [[xenharmonic/MOSScales|MOS]] of type [[xenharmonic/10L 19s|10L 19s]]
2 1 1 1 2 1 1 1 2 1 1 1 2 1 1 1 2 1 1 1 2 1 1 1 2 1 1 1 2 1 1 - [[xenharmonic/MOSScales|MOS]] of type [[xenharmonic/8L 23s|8L 23s]]

Original HTML content:

<html><head><title>39edo</title></head><body><!-- ws:start:WikiTextHeadingRule:0:&lt;h1&gt; --><h1 id="toc0"><a name="x39 tone equal temperament"></a><!-- ws:end:WikiTextHeadingRule:0 --><span style="color: #9900ab; font-family: 'Times New Roman',Times,serif; font-size: 113%;">39 tone equal temperament</span></h1>
 <br />
<strong>39-EDO, 39-ED2</strong> or <strong>39-tET</strong> divides the Octave (Ditave 2/1) in 39 equal parts of 30.76923 Cents each one. If we take 22\39 as a fifth, can be used in Mavila Temperament, and from that point of view seems to have attracted the attention of the Armodue school, an Italian group that use the scheme of <a class="wiki_link" href="http://xenharmonic.wikispaces.com/7L%202s">Superdiatonic</a> LLLsLLLLs like a basical scale for notation and theory, suited in <a class="wiki_link" href="http://xenharmonic.wikispaces.com/16edo">16-ED2</a>, and allied systems: <a class="wiki_link" href="http://xenharmonic.wikispaces.com/25edo">25-ED2</a> [1/3-tone 3;2]; <a class="wiki_link" href="http://xenharmonic.wikispaces.com/41edo">41-ED2</a> [1/5-tone 5;3]; and <a class="wiki_link" href="http://xenharmonic.wikispaces.com/57edo">57</a> ED2 [1/7-tone 7;4]. <strong>Hornbostel Temperaments</strong> is included too with: <a class="wiki_link" href="http://xenharmonic.wikispaces.com/23edo">23-ED2</a> [1/3-tone 3;1]; 39-ED2 [1/5-tone 5;2] &amp; <a class="wiki_link" href="http://xenharmonic.wikispaces.com/62edo">62-ED2</a> [1/8-tone 8;3]. <a class="wiki_link" href="/223edo">223-ED2</a>, the best accuracy for Hornbostel temperament fits very good with Armodue like 1/29-tone 29;10 version. Note that <a class="wiki_link" href="/101edo">101</a>, <a class="wiki_link" href="/131edo">131</a>, <a class="wiki_link" href="/177edo">177</a> &amp; <a class="wiki_link" href="/200edo">200</a> ED2s are tempered systems that <a class="wiki_link_ext" href="http://www.h-pi.com/eop-ogolevets.html" rel="nofollow">Alexei Ogolevets</a> (Ukraine, 1891 - 1967) was proposing in his List of Temperaments, in which the Armodue system fits very well in all these.<br />
However, its 23\39 fifth, 5.737 Cents sharp, is in much better tune than the Mavila fifth which like all Mavila fifths is very, very flat, in this case, 25 Cents flat. Together with its best third which is the familiar 400 cents of 12 equal, we get a system which tempers out the diesis, 128/125, and the amity comma, 1600000/1594323. We have two choices for a map for 7, but the sharp one works better with the 3 and 5, which adds 64/63 and 126/125 to the list. Tempering out both 128/125 and 64/63 makes 39EDO, in some few ways, allied to 12-ET in supporting augene temperament, and is in fact, an excellent choice for an augene tuning, but one difference is that 39 has a fine 11, and adding it to consideration we find that 39-EDO tempers out 99/98 and 121/120 also. This better choice for 39et is &lt;39 62 91 110 135|.<br />
A particular anecdote with this 39 divisions per 2/1 was made in the Teliochordon, in 1788 by Charles Clagget (Ireland, 1740? - 1820), a little extract <a class="wiki_link_ext" href="http://ml.oxfordjournals.org/content/76/2/291.extract.jpg" rel="nofollow">here</a>.<br />
<br />
<!-- ws:start:WikiTextHeadingRule:2:&lt;h2&gt; --><h2 id="toc1"><a name="x39 tone equal temperament-39-EDO Intervals"></a><!-- ws:end:WikiTextHeadingRule:2 --><u><strong>39-EDO Intervals</strong></u></h2>
 

<table class="wiki_table">
    <tr>
        <td><strong>ARMODUE NOMENCLATURE 5;2 RELATION</strong><br />
</td>
    </tr>
    <tr>
        <td><ul><li><strong>‡</strong> = Semisharp (1/5-tone up)</li><li><strong>b</strong> = Flat (3/5-tone down)</li><li><strong>#</strong> = Sharp (3/5-tone up)</li><li><strong>v</strong> = Semiflat (1/5-tone down)</li></ul></td>
    </tr>
</table>

<br />


<table class="wiki_table">
    <tr>
        <th><strong>Degrees</strong><br />
</th>
        <th><strong>Armodue note</strong><br />
</th>
        <th><strong>Cents size</strong><br />
</th>
        <th><strong><a class="wiki_link" href="http://xenharmonic.wikispaces.com/Nearest%20just%20interval">Nearest Just I</a>nterval</strong><br />
</th>
        <th><strong>Cents value</strong><br />
</th>
        <th><strong>Error</strong><br />
</th>
        <th>11-limit Ratio Assuming<br />
&lt;39 62 91 110 135| <a class="wiki_link" href="/Val">Val</a><br />
</th>
    </tr>
    <tr>
        <td>0<br />
</td>
        <td>1<br />
</td>
        <td>0<br />
</td>
        <td>1/1<br />
</td>
        <td>0<br />
</td>
        <td>None<br />
</td>
        <td>1/1<br />
</td>
    </tr>
    <tr>
        <td>1<br />
</td>
        <td>1‡ (9#)<br />
</td>
        <td>30.7692<br />
</td>
        <td>57/56<br />
</td>
        <td>30.6421<br />
</td>
        <td>+0.1271<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>2<br />
</td>
        <td>2b<br />
</td>
        <td>61.5385<br />
</td>
        <td>29/28<br />
</td>
        <td>60.7513<br />
</td>
        <td>+0.7872<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>3<br />
</td>
        <td>1#<br />
</td>
        <td>92.3077<br />
</td>
        <td>39/37<br />
</td>
        <td>91.1386<br />
</td>
        <td>+1.1691<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>4<br />
</td>
        <td>2v<br />
</td>
        <td>123.0769<br />
</td>
        <td>44/41<br />
</td>
        <td>122.2555<br />
</td>
        <td>+0.8214<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>5<br />
</td>
        <td>2<br />
</td>
        <td>153.8462<br />
</td>
        <td>35/32<br />
</td>
        <td>155.1396<br />
</td>
        <td>-1.2934<br />
</td>
        <td>12/11, 11/10<br />
</td>
    </tr>
    <tr>
        <td>6<br />
</td>
        <td>2‡<br />
</td>
        <td>184.6154<br />
</td>
        <td>10/9<br />
</td>
        <td>182.4037<br />
</td>
        <td>+2.2117<br />
</td>
        <td>10/9<br />
</td>
    </tr>
    <tr>
        <td>7<strong>·</strong><br />
</td>
        <td>3b<br />
</td>
        <td>215.3846<br />
</td>
        <td>17/15<br />
</td>
        <td>216.6867<br />
</td>
        <td>-1.3021<br />
</td>
        <td>8/7, 9/8<br />
</td>
    </tr>
    <tr>
        <td>8<br />
</td>
        <td>2#<br />
</td>
        <td>246.1538<br />
</td>
        <td>15/13<br />
</td>
        <td>247.7411<br />
</td>
        <td>-1.5873<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>9<br />
</td>
        <td>3v<br />
</td>
        <td>276.9231<br />
</td>
        <td>27/23<br />
</td>
        <td>277.5907<br />
</td>
        <td>-0.6676<br />
</td>
        <td>7/6<br />
</td>
    </tr>
    <tr>
        <td>10<br />
</td>
        <td>3<br />
</td>
        <td>307.6923<br />
</td>
        <td>43/36<br />
</td>
        <td>307.6077<br />
</td>
        <td>+0.0846<br />
</td>
        <td>6/5<br />
</td>
    </tr>
    <tr>
        <td>11<br />
</td>
        <td>3‡<br />
</td>
        <td>338.4615<br />
</td>
        <td>17/14<br />
</td>
        <td>336.1295<br />
</td>
        <td>+2.332<br />
</td>
        <td>11/9<br />
</td>
    </tr>
    <tr>
        <td>12<strong>·</strong><br />
</td>
        <td>4b<br />
</td>
        <td>369.2308<br />
</td>
        <td>26/21<br />
</td>
        <td>369.7468<br />
</td>
        <td>-0.516<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>13<br />
</td>
        <td>3#<br />
</td>
        <td>400<br />
</td>
        <td>34/27<br />
</td>
        <td>399.0904<br />
</td>
        <td>+0.9096<br />
</td>
        <td>5/4<br />
</td>
    </tr>
    <tr>
        <td>14<br />
</td>
        <td>4v (5b)<br />
</td>
        <td>430.7692<br />
</td>
        <td>41/32<br />
</td>
        <td>429.0624<br />
</td>
        <td>+1.7068<br />
</td>
        <td>9/7, 14/11<br />
</td>
    </tr>
    <tr>
        <td>15<br />
</td>
        <td>4<br />
</td>
        <td>461.5385<br />
</td>
        <td>30/23<br />
</td>
        <td>459.9944<br />
</td>
        <td>+1.5441<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>16<br />
</td>
        <td>4‡ (5v)<br />
</td>
        <td>492.3077<br />
</td>
        <td>85/64<br />
</td>
        <td>491.2691<br />
</td>
        <td>+1.0386<br />
</td>
        <td>4/3<br />
</td>
    </tr>
    <tr>
        <td>17<strong>·</strong><br />
</td>
        <td>5<br />
</td>
        <td>523.0769<br />
</td>
        <td>23/17<br />
</td>
        <td>523.3189<br />
</td>
        <td>-0.242<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>18<br />
</td>
        <td>5‡ (4#)<br />
</td>
        <td>553.8462<br />
</td>
        <td>11/8<br />
</td>
        <td>551.3179<br />
</td>
        <td>+2.5283<br />
</td>
        <td>11/8<br />
</td>
    </tr>
    <tr>
        <td>19<br />
</td>
        <td>6b<br />
</td>
        <td>584.6154<br />
</td>
        <td>7/5<br />
</td>
        <td>582.5122<br />
</td>
        <td>+2.1032<br />
</td>
        <td>7/5<br />
</td>
    </tr>
    <tr>
        <td>20<br />
</td>
        <td>5#<br />
</td>
        <td>615.3846<br />
</td>
        <td>10/7<br />
</td>
        <td>617.4878<br />
</td>
        <td>-2.1032<br />
</td>
        <td>10/7<br />
</td>
    </tr>
    <tr>
        <td>21<br />
</td>
        <td>6v<br />
</td>
        <td>646.1538<br />
</td>
        <td>16/11<br />
</td>
        <td>648.6821<br />
</td>
        <td>-2.5283<br />
</td>
        <td>16/11<br />
</td>
    </tr>
    <tr>
        <td>22<strong>·</strong><br />
</td>
        <td>6<br />
</td>
        <td>676.9231<br />
</td>
        <td>34/23<br />
</td>
        <td>676.6811<br />
</td>
        <td>+0.242<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>23<br />
</td>
        <td>6‡<br />
</td>
        <td>707.6923<br />
</td>
        <td>128/85<br />
</td>
        <td>708.7309<br />
</td>
        <td>-1.0386<br />
</td>
        <td>3/2<br />
</td>
    </tr>
    <tr>
        <td>24<br />
</td>
        <td>7b<br />
</td>
        <td>738.4615<br />
</td>
        <td>23/15<br />
</td>
        <td>740.0056<br />
</td>
        <td>-1.5441<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>25<br />
</td>
        <td>6#<br />
</td>
        <td>769.2308<br />
</td>
        <td>64/41<br />
</td>
        <td>770.9376<br />
</td>
        <td>-1.7068<br />
</td>
        <td>14/9, 11/7<br />
</td>
    </tr>
    <tr>
        <td>26<br />
</td>
        <td>7v<br />
</td>
        <td>800<br />
</td>
        <td>27/17<br />
</td>
        <td>800.9096<br />
</td>
        <td>-0.9096<br />
</td>
        <td>8/5<br />
</td>
    </tr>
    <tr>
        <td>27<strong>·</strong><br />
</td>
        <td>7<br />
</td>
        <td>830.7692<br />
</td>
        <td>21/13<br />
</td>
        <td>830.2532<br />
</td>
        <td>+0.516<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>28<br />
</td>
        <td>7‡<br />
</td>
        <td>861.5385<br />
</td>
        <td>28/17<br />
</td>
        <td>863.8705<br />
</td>
        <td>-2.332<br />
</td>
        <td>18/11<br />
</td>
    </tr>
    <tr>
        <td>29<br />
</td>
        <td>8b<br />
</td>
        <td>892.3077<br />
</td>
        <td>72/43<br />
</td>
        <td>892.3923<br />
</td>
        <td>-0.0846<br />
</td>
        <td>5/3<br />
</td>
    </tr>
    <tr>
        <td>30<br />
</td>
        <td>7#<br />
</td>
        <td>923.0769<br />
</td>
        <td>46/27<br />
</td>
        <td>922.4093<br />
</td>
        <td>+0.6676<br />
</td>
        <td>12/7<br />
</td>
    </tr>
    <tr>
        <td>31<br />
</td>
        <td>8v<br />
</td>
        <td>953.8462<br />
</td>
        <td>26/15<br />
</td>
        <td>952.2589<br />
</td>
        <td>+1.5873<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>32<strong>·</strong><br />
</td>
        <td>8<br />
</td>
        <td>984.6154<br />
</td>
        <td>30/17<br />
</td>
        <td>983.3133<br />
</td>
        <td>+1.3021<br />
</td>
        <td>7/4, 16/9<br />
</td>
    </tr>
    <tr>
        <td>33<br />
</td>
        <td>8‡<br />
</td>
        <td>1015.3846<br />
</td>
        <td>9/5<br />
</td>
        <td>1017.5963<br />
</td>
        <td>-2.2117<br />
</td>
        <td>9/5<br />
</td>
    </tr>
    <tr>
        <td>34<br />
</td>
        <td>9b<br />
</td>
        <td>1046.1538<br />
</td>
        <td>64/35<br />
</td>
        <td>1044.8604<br />
</td>
        <td>+1.2934<br />
</td>
        <td>11/6, 20/11<br />
</td>
    </tr>
    <tr>
        <td>35<br />
</td>
        <td>8#<br />
</td>
        <td>1076.9231<br />
</td>
        <td>41/22<br />
</td>
        <td>1077.7445<br />
</td>
        <td>-0.8214<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>36<br />
</td>
        <td>9v (1b)<br />
</td>
        <td>1107.6923<br />
</td>
        <td>74/39<br />
</td>
        <td>1108.8614<br />
</td>
        <td>-1.1691<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>37<br />
</td>
        <td>9<br />
</td>
        <td>1138.4615<br />
</td>
        <td>56/29<br />
</td>
        <td>1139.2487<br />
</td>
        <td>-0.7872<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>38<br />
</td>
        <td>9‡ (1v)<br />
</td>
        <td>1169.2308<br />
</td>
        <td>112/57<br />
</td>
        <td>1169.3579<br />
</td>
        <td>-0.1271<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>39<strong>··</strong>(or 0)<br />
</td>
        <td>1<br />
</td>
        <td>1200<br />
</td>
        <td>2/1<br />
</td>
        <td>1200<br />
</td>
        <td>None<br />
</td>
        <td><br />
</td>
    </tr>
</table>

<br />
<!-- ws:start:WikiTextHeadingRule:4:&lt;h2&gt; --><h2 id="toc2"><a name="x39 tone equal temperament-Instruments (prototypes):"></a><!-- ws:end:WikiTextHeadingRule:4 --><u>Instruments (prototypes):</u></h2>
 <br />
<!-- ws:start:WikiTextLocalImageRule:686:&lt;img src=&quot;/file/view/TECLADO%2039-EDD.PNG/390052498/800x467/TECLADO%2039-EDD.PNG&quot; alt=&quot;&quot; title=&quot;&quot; style=&quot;height: 467px; width: 800px;&quot; /&gt; --><img src="/file/view/TECLADO%2039-EDD.PNG/390052498/800x467/TECLADO%2039-EDD.PNG" alt="TECLADO 39-EDD.PNG" title="TECLADO 39-EDD.PNG" style="height: 467px; width: 800px;" /><!-- ws:end:WikiTextLocalImageRule:686 --><br />
<em>An illustrative image of a 39-ED2 keyboard</em><br />
<!-- ws:start:WikiTextLocalImageRule:687:&lt;img src=&quot;http://xenharmonic.wikispaces.com/file/view/Custom_700mm_5-str_Tricesanonaphonic_Guitar.png/258445130/826x203/Custom_700mm_5-str_Tricesanonaphonic_Guitar.png&quot; alt=&quot;39-EDD fretboard visualization&quot; title=&quot;39-EDD fretboard visualization&quot; style=&quot;height: 203px; width: 826px;&quot; /&gt; --><table class="captionBox"><tr><td class="captionedImage"><img src="http://xenharmonic.wikispaces.com/file/view/Custom_700mm_5-str_Tricesanonaphonic_Guitar.png/258445130/826x203/Custom_700mm_5-str_Tricesanonaphonic_Guitar.png" alt="Custom_700mm_5-str_Tricesanonaphonic_Guitar.png" title="Custom_700mm_5-str_Tricesanonaphonic_Guitar.png" style="height: 203px; width: 826px;" /></td></tr><tr><td class="imageCaption">39-EDD fretboard visualization</td></tr></table><!-- ws:end:WikiTextLocalImageRule:687 --><br />
<br />
<br />
<!-- ws:start:WikiTextHeadingRule:6:&lt;h2&gt; --><h2 id="toc3"><a name="x39 tone equal temperament-39 tone equal modes:"></a><!-- ws:end:WikiTextHeadingRule:6 --><strong><u>39 tone equal <a class="wiki_link" href="http://xenharmonic.wikispaces.com/modes">modes</a></u>:</strong></h2>
 <br />
15 15 9 - <a class="wiki_link" href="http://xenharmonic.wikispaces.com/MOSScales">MOS</a> of type <a class="wiki_link" href="http://xenharmonic.wikispaces.com/2L%201s">2L 1s</a><br />
14 14 11 - <a class="wiki_link" href="http://xenharmonic.wikispaces.com/MOSScales">MOS</a> of type <a class="wiki_link" href="http://xenharmonic.wikispaces.com/2L%201s">2L 1s</a><br />
13 13 13 = <a class="wiki_link" href="http://xenharmonic.wikispaces.com/3edo">3edo</a><br />
11 11 11 6 - <a class="wiki_link" href="http://xenharmonic.wikispaces.com/MOSScales">MOS</a> of type <a class="wiki_link" href="http://xenharmonic.wikispaces.com/3L%201s">3L 1s</a><br />
10 10 10 9 - <a class="wiki_link" href="http://xenharmonic.wikispaces.com/MOSScales">MOS</a> of type <a class="wiki_link" href="http://xenharmonic.wikispaces.com/3L%201s">3L 1s</a><br />
11 3 11 11 3 - <a class="wiki_link" href="http://xenharmonic.wikispaces.com/MOSScales">MOS</a> of type <a class="wiki_link" href="/3L%202s">3L 2s (Father pentatonic)</a><br />
11 3 11 3 11 - <span style="cursor: pointer;"><a class="wiki_link" href="http://xenharmonic.wikispaces.com/MOSScales">MOS</a></span> of type <span style="color: #660000; cursor: pointer;"><a class="wiki_link" href="/3L%202s">3L 2s (Father pentatonic)</a></span><br />
9 6 9 9 6 - <a class="wiki_link" href="http://xenharmonic.wikispaces.com/MOSScales">MOS</a> of type <a class="wiki_link" href="/3L%202s">3L 2s (Father pentatonic)</a><br />
9 6 9 6 9 - <span style="cursor: pointer;"><a class="wiki_link" href="http://xenharmonic.wikispaces.com/MOSScales">MOS</a></span> of type <span style="color: #660000; cursor: pointer;"><a class="wiki_link" href="/3L%202s">3L 2s (Father pentatonic)</a></span><br />
9 9 9 9 3 - <a class="wiki_link" href="http://xenharmonic.wikispaces.com/MOSScales">MOS</a> of type <a class="wiki_link" href="/4L%201s">4L 1s (Bug pentatonic)</a><br />
9 3 9 9 9 - <span style="cursor: pointer;"><a class="wiki_link" href="http://xenharmonic.wikispaces.com/MOSScales">MOS</a></span> of type <span style="color: #660000; cursor: pointer;"><a class="wiki_link" href="/4L%201s">4L 1s (Bug pentatonic)</a></span><br />
8 8 8 8 7 - <a class="wiki_link" href="http://xenharmonic.wikispaces.com/MOSScales">MOS</a> of type <a class="wiki_link" href="/4L%201s">4L 1s (Bug pentatonic)</a><br />
10 3 10 3 10 3 - <a class="wiki_link" href="http://xenharmonic.wikispaces.com/MOSScales">MOS</a> of type <a class="wiki_link" href="/3L%203s">3L 3s (Augmented hexatonic)</a><br />
9 4 9 4 9 4 - <a class="wiki_link" href="http://xenharmonic.wikispaces.com/MOSScales">MOS</a> of type <a class="wiki_link" href="/3L%203s">3L 3s (Augmented hexatonic)</a><br />
8 5 8 5 8 5 - <a class="wiki_link" href="http://xenharmonic.wikispaces.com/MOSScales">MOS</a> of type <a class="wiki_link" href="/3L%203s">3L 3s (Augmented hexatonic)</a><br />
7 7 7 7 7 4 - <a class="wiki_link" href="http://xenharmonic.wikispaces.com/MOSScales">MOS</a> of type <a class="wiki_link" href="http://xenharmonic.wikispaces.com/5L%201s">5L 1s (Grumpy hexatonic)</a><br />
7 4 7 7 7 7 - <span style="cursor: pointer;"><a class="wiki_link" href="http://xenharmonic.wikispaces.com/MOSScales">MOS</a></span> of type <span style="cursor: pointer;"><a class="wiki_link" href="http://xenharmonic.wikispaces.com/5L%201s">5L 1s (Grumpy hexatonic)</a></span><br />
3 9 3 9 3 9 3 - <a class="wiki_link" href="http://xenharmonic.wikispaces.com/MOSScales">MOS</a> of type <a class="wiki_link" href="/3L%204s">3L 4s (Mosh heptatonic)</a><br />
5 5 7 5 5 5 7 - <a class="wiki_link" href="http://xenharmonic.wikispaces.com/MOSScales">MOS</a> of type <a class="wiki_link" href="/2L%205s">2L 5s (heptatonic Mavila Anti-Diatonic)</a><br />
5 5 5 7 5 5 7 - <a class="wiki_link" href="http://xenharmonic.wikispaces.com/MOSScales">MOS</a> of type <a class="wiki_link" href="/2L%205s">2L 5s (heptatonic Mavila Anti-Diatonic)</a><br />
5 7 5 5 7 5 5 - <a class="wiki_link" href="http://xenharmonic.wikispaces.com/MOSScales">MOS</a> of type <a class="wiki_link" href="/2L%205s">2L 5s (heptatonic Mavila Anti-Diatonic)</a><br />
6 3 6 6 3 6 6 3 - <a class="wiki_link" href="http://xenharmonic.wikispaces.com/MOSScales">MOS</a> of type <a class="wiki_link" href="/5L%203s">5L 3s (unfair Father octatonic)</a><br />
5 5 5 5 5 5 5 4 - <a class="wiki_link" href="http://xenharmonic.wikispaces.com/MOSScales">MOS</a> of type <a class="wiki_link" href="http://xenharmonic.wikispaces.com/7L%201s">7L 1s (Grumpy octatonic)</a><br />
5 4 5 5 5 5 5 5 - <a class="wiki_link" href="http://xenharmonic.wikispaces.com/MOSScales">MOS</a> of type <a class="wiki_link" href="http://xenharmonic.wikispaces.com/7L%201s">7L 1s (Grumpy octatonic)</a><br />
<strong>5 5 5 2 5 5 5 5 2</strong> - <a class="wiki_link" href="http://xenharmonic.wikispaces.com/MOSScales">MOS</a> of type <a class="wiki_link" href="/7L%202s">7L 2s (nonatonic Mavila Superdiatonic)</a><br />
5 5 2 5 5 5 2 5 5 - <a class="wiki_link" href="http://xenharmonic.wikispaces.com/MOSScales">MOS</a> of type <a class="wiki_link" href="/7L%202s">7L 2s (nonatonic Mavila Superdiatonic)</a><br />
5 5 3 5 5 3 5 5 3 - <a class="wiki_link" href="http://xenharmonic.wikispaces.com/MOSScales">MOS</a> of type <a class="wiki_link" href="/6L%203s">6L 3s (unfair Augmented nonatonic)</a><br />
5 4 4 5 4 4 5 4 4 - <a class="wiki_link" href="http://xenharmonic.wikispaces.com/MOSScales">MOS</a> of type <a class="wiki_link" href="/3L%206s">3L 6s (fair Augmented nonatonic)</a><br />
4 4 4 4 4 4 4 4 4 3 - <a class="wiki_link" href="http://xenharmonic.wikispaces.com/MOSScales">MOS</a> of type <a class="wiki_link" href="http://xenharmonic.wikispaces.com/9L%201s">9L 1s (Grumpy decatonic)</a><br />
4 4 3 4 4 4 4 4 4 4 - <a class="wiki_link" href="http://xenharmonic.wikispaces.com/MOSScales">MOS</a> of type <a class="wiki_link" href="http://xenharmonic.wikispaces.com/9L%201s">9L 1s (Grumpy decatonic)</a><br />
<strong>3 3 5 3 3 3 5 3 3 3 5</strong> - <a class="wiki_link" href="http://xenharmonic.wikispaces.com/MOSScales">MOS</a> of type <a class="wiki_link" href="/3L%208s">3L 8s (Anti-Sensi hendecatonic)</a><br />
3 3 3 3 3 3 3 3 3 3 3 3 3 = <a class="wiki_link" href="http://xenharmonic.wikispaces.com/13edo">13edo</a><br />
<strong>3 3 3 2 3 3 3 3 2 3 3 3 3 2</strong> - <a class="wiki_link" href="http://xenharmonic.wikispaces.com/MOSScales">MOS</a> of type <a class="wiki_link" href="/11L%203s">11L 3s (Ketradektriatoh tetradecatonic)</a><br />
3 2 3 3 2 3 2 3 3 2 3 2 3 3 2 - <a class="wiki_link" href="http://xenharmonic.wikispaces.com/MOSScales">MOS</a> of type <a class="wiki_link" href="http://xenharmonic.wikispaces.com/9L%206s">9L 6s</a><br />
3 2 3 2 3 2 2 3 2 3 2 3 2 3 2 2 - <a class="wiki_link" href="http://xenharmonic.wikispaces.com/MOSScales">MOS</a> of type <a class="wiki_link" href="http://xenharmonic.wikispaces.com/7L%209s">7L 9s</a><br />
<strong>2 2 3 2 2 2 3 2 2 3 2 2 3 2 2 2 3</strong> - <a class="wiki_link" href="http://xenharmonic.wikispaces.com/MOSScales">MOS</a> of type <a class="wiki_link" href="http://xenharmonic.wikispaces.com/5L%2012s">5L 12s</a><br />
2 2 2 2 2 3 2 2 2 2 2 3 2 2 2 2 2 3 - <a class="wiki_link" href="http://xenharmonic.wikispaces.com/MOSScales">MOS</a> of type <a class="wiki_link" href="http://xenharmonic.wikispaces.com/3L%2015s">3L 15s</a><br />
<strong>3 1 3 1 3 1 3 1 3 1 3 1 3 1 3 1 3 1 3</strong> - <span style="cursor: pointer;"><a class="wiki_link" href="http://xenharmonic.wikispaces.com/MOSScales">MOS</a></span> of type <a class="wiki_link" href="http://xenharmonic.wikispaces.com/10L%209s">10L 9s</a><br />
2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 1 - <a class="wiki_link" href="http://xenharmonic.wikispaces.com/MOSScales">MOS</a> of type <a class="wiki_link" href="http://xenharmonic.wikispaces.com/19L%201s">19L 1s</a><br />
2 2 2 1 2 2 2 1 2 2 2 1 2 2 2 2 1 2 2 2 2 1 - <a class="wiki_link" href="http://xenharmonic.wikispaces.com/MOSScales">MOS</a> of type <a class="wiki_link" href="http://xenharmonic.wikispaces.com/17L%205s">17L 5s</a><br />
<strong>2 2 1 2 2 1 2 2 1 2 2 1 2 2 1 2 2 2 1 2 2 2 1</strong> - <a class="wiki_link" href="http://xenharmonic.wikispaces.com/MOSScales">MOS</a> of type <a class="wiki_link" href="http://xenharmonic.wikispaces.com/16L%207s">16L 7s</a><br />
2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 - <a class="wiki_link" href="http://xenharmonic.wikispaces.com/MOSScales">MOS</a> of type <a class="wiki_link" href="http://xenharmonic.wikispaces.com/13L%2013s">13L 13s</a><br />
<strong>2 1 1 2 1 1 2 1 1 2 1 1 2 1 1 2 1 1 2 1 1 2 1 1 2 1 1 2 1</strong> - <a class="wiki_link" href="http://xenharmonic.wikispaces.com/MOSScales">MOS</a> of type <a class="wiki_link" href="http://xenharmonic.wikispaces.com/10L%2019s">10L 19s</a><br />
2 1 1 1 2 1 1 1 2 1 1 1 2 1 1 1 2 1 1 1 2 1 1 1 2 1 1 1 2 1 1 - <a class="wiki_link" href="http://xenharmonic.wikispaces.com/MOSScales">MOS</a> of type <a class="wiki_link" href="http://xenharmonic.wikispaces.com/8L%2023s">8L 23s</a></body></html>

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