Harmony of 23edo: Difference between revisions

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If you take a look at the intervals of [[23edo]], you'll find that this system does not contain good representations of the harmonics 3, 5, 7, 11, or 13, which appear as central in most just intonation systems. Rather than trivialize 23edo by calling it "atonal" or "inharmonic", you should consider the higher-limit harmonies that could serve as useful sonorities, perhaps even "consonances," in the context of careful composition. [[23edo]] contains intervals which approach very well the harmonics 9, 17, 21, 23, 33, 35, 55, 79 & 117. Let's compare the cents values to see how close 23edo intervals come to these harmonics (and other intervals):
|| Degrees || Armodue note || Cents sizes || [[nearest harmonic|Nearest Harmonic]] || Cents || "Error" ||
|| 0 || 1 || 0 || 1/1 || 0.000 || none ||
|| **1** || **1t (2b)** || **52.174** || **33/32** || **53.273** || **-1.099** ||
|| **2** || **2v (1#)** || **104.348** || **17/16** || **104.955** || **-0.607** ||
|| **3** || **2** || **156.522** || **35/32** || **155.140** || **-1.382** ||
|| **4·** || **2t (3b)** || **208.696** || **9/8** || **203.910** || **+4.786** ||
|| 5 || 3v (2#) || 260.869 || 50/43 || 261.110 || +0.241 ||
|| 6 || 3 || 313.043 || [[6_5|6/5]] || 315.641 || -2.598 ||
|| **7·** || **3t (4b)** || **365.217** || **79/64** || **364.537** || **-0.68** ||
|| 8 || 4v (3#) || 417.391 || [[14_11|14/11]] || 417.508 || -0.117 ||
|| **9** || **4 (5v)** || **469.565** || **21/16** || **470.781** || **-1.216** ||
|| 10**·** || 5 (4t) || 521.739 || 23/17 || 523.319 || -1.58 ||
|| 11 || 5t (6b) || 573.913 || 32/23 || 571.726 || +2.187 ||
|| **12** || **6v (5#)** || **626.087** || **23/16** || **628.274** || **-2.187** ||
|| 13**·** || 6 || 678.261 || 34/23 || 676.681 || +1.58 ||
|| 14 || 6t (7b) || 730.435 || 32/21 || 729.219 || +1.216 ||
|| 15 || 7v (6#) || 782.609 || [[11_7|11/7]] || 782.492 || +0.117 ||
|| 16**·** || 7 || 834.783 || 34/21 || 834.175 || +0.608 ||
|| 17 || 7t (8b) || 886.957 || [[5_3|5/3]] || 884.359 || +2.598 ||
|| **18** || **8v (7#)** || **939.130** || **55/32** || **937.632** || **+1.498** ||
|| 19**·** || 8 || 991.304 || 39/22 || 991.165 || +0.139 ||
|| **20** || **8t (9b)** || **1043.478** || **117/64** || **1044.438** || **-0.96** ||
|| 21 || 9v (8#) || 1095.652 || 32/17 || 1095.045 || +0.607 ||
|| 22 || 9 (1v) || 1147.826 || 64/33 || 1146.727 || +1.099 ||
|| **23·· (or 0)** || **1 (9t)** || **1200.000** || **2/1** || **1200.000** || **none** ||
You'll see that intervals of 23edo come within 5 cents of 9/8; 3 cents of 23/16; 2 cents of 33/32, 21/16, 35/32, & 55/32; & 1 cent of 17/16, 79/64, & 117/64. (<**And let's also note the excellent representations of 14/11 and its inverse, 11/7!!! In fact they might be considered good enought that a chain of 23 such intervals would be a reasonable way to acoustically tune this temperament** -- AKJ) Of course, it also has perfect unisons & octaves, by definition. This means we could potentially build a very strange (& slightly mistuned) harmonic chord which, reduced to within one octave, we could write as frequency ratios 64:66:68:70:72:79:84:92:110:117. I find this cluster a little hard to listen to, whether tuned to JI or 23edo, so I'd like to consider smaller chords, triads & tetrads, as a starting point.

I'd also like to set an arbitrary limit on how high up the harmonic series we will go. I'll set my limit at the 23rd harmonic. I'll consider harmonics 1, 9, 17, 21, & 23, excluding (at least for now) 33, 55, 79, & 117. Those sonorities could no doubt prove useful to a thoughful composer, but for this study, I'll leave them out.

Thus we produce ten triads, five tetrads, & one quintad, 16 chords, which, with their inversions (given), doubles to 32 chords. I've written then in a closed position (within one octave), & I recommend trying different voicings. Moving chord tones up & down by octaves, you can unmuddy a muddy chord.

==[[Triad]]s== 
**16:17:18, degrees 0, 2, 4 (inversion 0, 19, 21)**.
17/16 (104.955, error -0.607)
18/16 = 9/8 (203.910, error +4.786)
18/17 (98.955, error: +5.393)

===16:17:21, degrees 0, 2, 9 (inversion 0, 14, 21).=== 
17/16 (104.955, error -0.607)
21/16 (470.781, error -1.216)
21/17 (365.825, error: -0.608)

===16:17:23, degrees 0, 2, 12 (inversion 0, 11, 21).=== 
17/16 (104.955, error -0.607)
23/16 (628.274, error -2.187)
23/17 (523.319, error: -1.578)

===16:18:21, degrees 0, 4, 9 (inversion 0, 14, 19).=== 
18/16 = 9/8 (203.910, error +4.786)
21/16 (470.781, error -1.216)
21/18 = 7/6 (266.871, error: -6.001)

===16:18:23, degrees 0, 4, 12 (inversion 0, 11, 19).=== 
18/16 = 9/8 (203.910, error +4.786)
23/16 (628.274, error -2.187)
23/18 (424.364, error: -6.973)

===16:21:23, degrees 0, 9, 12 (inversion 0, 11, 14).=== 
21/16 (470.781, error -1.216)
23/16 (628.274, error -2.187)
23/21 (157.493, error: -0.971)

===17:18:21, degrees 0, 2, 7 (inversion 0, 16, 21).=== 
18/17 (98.955, error: +5.393)
21/17 (365.825, error: -0.608)
21/18 = 7/6 (266.871, error: -6.001)

===17:18:23, degrees 0, 2, 10 (inversion 0, 13, 21).=== 
18/17 (98.955, error: +5.393)
23/17 (523.319, error: -1.578)
23/18 (424.364, error: -6.973)

===17:21:23, degrees 0, 7, 10 (inversion 0, 13, 16).=== 
21/17 (365.825, error: -0.608)
23/17 (523.319, error: -1.578)
23/21 (157.493, error: -0.971)

===18:21:23, degrees 0, 5, 8 (inversion 0, 15, 18).=== 
21/18 = 7/6 (266.871, error: -6.001)
23/18 (424.364, error: -6.973)
23/21 (157.493, error: -0.971)


==[[Tetrad]]s== 
**16:17:18:21, degrees 0, 2, 4, 9 (inversion 0, 14, 19, 21)**.
17/16 (104.955, error -0.607)
18/16 = 9/8 (203.910, error +4.786)
21/16 (470.781, error -1.216)
18/17 (98.955, error: +5.393)
21/17 (365.825, error: -0.608)
21/18 = 7/6 (266.871, error: -6.001)

===16:17:18:23, degrees 0, 2, 4, 12 (inversion 0, 11 19, 21).=== 
17/16 (104.955, error -0.607)
18/16 = 9/8 (203.910, error +4.786)
23/16 (628.274, error -2.187)
18/17 (98.955, error: +5.393)
23/17 (523.319, error: -1.578)
23/18 (424.364, error: -6.973)

===16:17:21:23, degrees 0, 2, 9, 12 (inversion 0, 11, 14, 21).=== 
17/16 (104.955, error -0.607)
21/16 (470.781, error -1.216)
23/16 (628.274, error -2.187)
21/17 (365.825, error: -0.608)
23/17 (523.319, error: -1.578)
23/21 (157.493, error: -0.971)

===16:18:21:23, degrees 0, 4, 9, 12 (inversion 0, 11, 14, 19).=== 
18/16 = 9/8 (203.910, error +4.786)
21/16 (470.781, error -1.216)
23/16 (628.274, error -2.187)
21/18 = 7/6 (266.871, error: -6.001)
23/18 (424.364, error: -6.973)
23/21 (157.493, error: -0.971)

===17:18:21:23, degrees 0, 2, 7, 10 (inversion 0, 13, 16, 21).=== 
18/17 (98.955, error: +5.393)
21/17 (365.825, error: -0.608)
23/17 (523.319, error: -1.578)
21/18 = 7/6 (266.871, error: -6.001)
23/18 (424.364, error: -6.973)
23/21 (157.493, error: -0.971)


==[[Quintad]]== 
**16:17:18:21:23, degrees 0, 2, 4, 9, 12 (inversion 0, 11, 14, 19, 21)**.
17/16 (104.955, error -0.607)
18/16 = 9/8 (203.910, error +4.786)
21/16 (470.781, error -1.216)
23/16 (628.274, error -2.187)
18/17 (98.955, error: +5.393)
21/17 (365.825, error: -0.608)
23/17 (523.319, error: -1.578)
21/18 = 7/6 (266.871, error: -6.001)
23/18 (424.364, error: -6.973)
23/21 (157.493, error: -0.971)

Original HTML content:

<html><head><title>Harmony of 23edo</title></head><body>If you take a look at the intervals of <a class="wiki_link" href="/23edo">23edo</a>, you'll find that this system does not contain good representations of the harmonics 3, 5, 7, 11, or 13, which appear as central in most just intonation systems. Rather than trivialize 23edo by calling it &quot;atonal&quot; or &quot;inharmonic&quot;, you should consider the higher-limit harmonies that could serve as useful sonorities, perhaps even &quot;consonances,&quot; in the context of careful composition. <a class="wiki_link" href="/23edo">23edo</a> contains intervals which approach very well the harmonics 9, 17, 21, 23, 33, 35, 55, 79 &amp; 117. Let's compare the cents values to see how close 23edo intervals come to these harmonics (and other intervals):<br />


<table class="wiki_table">
    <tr>
        <td>Degrees<br />
</td>
        <td>Armodue note<br />
</td>
        <td>Cents sizes<br />
</td>
        <td><a class="wiki_link" href="/nearest%20harmonic">Nearest Harmonic</a><br />
</td>
        <td>Cents<br />
</td>
        <td>&quot;Error&quot;<br />
</td>
    </tr>
    <tr>
        <td>0<br />
</td>
        <td>1<br />
</td>
        <td>0<br />
</td>
        <td>1/1<br />
</td>
        <td>0.000<br />
</td>
        <td>none<br />
</td>
    </tr>
    <tr>
        <td><strong>1</strong><br />
</td>
        <td><strong>1t (2b)</strong><br />
</td>
        <td><strong>52.174</strong><br />
</td>
        <td><strong>33/32</strong><br />
</td>
        <td><strong>53.273</strong><br />
</td>
        <td><strong>-1.099</strong><br />
</td>
    </tr>
    <tr>
        <td><strong>2</strong><br />
</td>
        <td><strong>2v (1#)</strong><br />
</td>
        <td><strong>104.348</strong><br />
</td>
        <td><strong>17/16</strong><br />
</td>
        <td><strong>104.955</strong><br />
</td>
        <td><strong>-0.607</strong><br />
</td>
    </tr>
    <tr>
        <td><strong>3</strong><br />
</td>
        <td><strong>2</strong><br />
</td>
        <td><strong>156.522</strong><br />
</td>
        <td><strong>35/32</strong><br />
</td>
        <td><strong>155.140</strong><br />
</td>
        <td><strong>-1.382</strong><br />
</td>
    </tr>
    <tr>
        <td><strong>4·</strong><br />
</td>
        <td><strong>2t (3b)</strong><br />
</td>
        <td><strong>208.696</strong><br />
</td>
        <td><strong>9/8</strong><br />
</td>
        <td><strong>203.910</strong><br />
</td>
        <td><strong>+4.786</strong><br />
</td>
    </tr>
    <tr>
        <td>5<br />
</td>
        <td>3v (2#)<br />
</td>
        <td>260.869<br />
</td>
        <td>50/43<br />
</td>
        <td>261.110<br />
</td>
        <td>+0.241<br />
</td>
    </tr>
    <tr>
        <td>6<br />
</td>
        <td>3<br />
</td>
        <td>313.043<br />
</td>
        <td><a class="wiki_link" href="/6_5">6/5</a><br />
</td>
        <td>315.641<br />
</td>
        <td>-2.598<br />
</td>
    </tr>
    <tr>
        <td><strong>7·</strong><br />
</td>
        <td><strong>3t (4b)</strong><br />
</td>
        <td><strong>365.217</strong><br />
</td>
        <td><strong>79/64</strong><br />
</td>
        <td><strong>364.537</strong><br />
</td>
        <td><strong>-0.68</strong><br />
</td>
    </tr>
    <tr>
        <td>8<br />
</td>
        <td>4v (3#)<br />
</td>
        <td>417.391<br />
</td>
        <td><a class="wiki_link" href="/14_11">14/11</a><br />
</td>
        <td>417.508<br />
</td>
        <td>-0.117<br />
</td>
    </tr>
    <tr>
        <td><strong>9</strong><br />
</td>
        <td><strong>4 (5v)</strong><br />
</td>
        <td><strong>469.565</strong><br />
</td>
        <td><strong>21/16</strong><br />
</td>
        <td><strong>470.781</strong><br />
</td>
        <td><strong>-1.216</strong><br />
</td>
    </tr>
    <tr>
        <td>10<strong>·</strong><br />
</td>
        <td>5 (4t)<br />
</td>
        <td>521.739<br />
</td>
        <td>23/17<br />
</td>
        <td>523.319<br />
</td>
        <td>-1.58<br />
</td>
    </tr>
    <tr>
        <td>11<br />
</td>
        <td>5t (6b)<br />
</td>
        <td>573.913<br />
</td>
        <td>32/23<br />
</td>
        <td>571.726<br />
</td>
        <td>+2.187<br />
</td>
    </tr>
    <tr>
        <td><strong>12</strong><br />
</td>
        <td><strong>6v (5#)</strong><br />
</td>
        <td><strong>626.087</strong><br />
</td>
        <td><strong>23/16</strong><br />
</td>
        <td><strong>628.274</strong><br />
</td>
        <td><strong>-2.187</strong><br />
</td>
    </tr>
    <tr>
        <td>13<strong>·</strong><br />
</td>
        <td>6<br />
</td>
        <td>678.261<br />
</td>
        <td>34/23<br />
</td>
        <td>676.681<br />
</td>
        <td>+1.58<br />
</td>
    </tr>
    <tr>
        <td>14<br />
</td>
        <td>6t (7b)<br />
</td>
        <td>730.435<br />
</td>
        <td>32/21<br />
</td>
        <td>729.219<br />
</td>
        <td>+1.216<br />
</td>
    </tr>
    <tr>
        <td>15<br />
</td>
        <td>7v (6#)<br />
</td>
        <td>782.609<br />
</td>
        <td><a class="wiki_link" href="/11_7">11/7</a><br />
</td>
        <td>782.492<br />
</td>
        <td>+0.117<br />
</td>
    </tr>
    <tr>
        <td>16<strong>·</strong><br />
</td>
        <td>7<br />
</td>
        <td>834.783<br />
</td>
        <td>34/21<br />
</td>
        <td>834.175<br />
</td>
        <td>+0.608<br />
</td>
    </tr>
    <tr>
        <td>17<br />
</td>
        <td>7t (8b)<br />
</td>
        <td>886.957<br />
</td>
        <td><a class="wiki_link" href="/5_3">5/3</a><br />
</td>
        <td>884.359<br />
</td>
        <td>+2.598<br />
</td>
    </tr>
    <tr>
        <td><strong>18</strong><br />
</td>
        <td><strong>8v (7#)</strong><br />
</td>
        <td><strong>939.130</strong><br />
</td>
        <td><strong>55/32</strong><br />
</td>
        <td><strong>937.632</strong><br />
</td>
        <td><strong>+1.498</strong><br />
</td>
    </tr>
    <tr>
        <td>19<strong>·</strong><br />
</td>
        <td>8<br />
</td>
        <td>991.304<br />
</td>
        <td>39/22<br />
</td>
        <td>991.165<br />
</td>
        <td>+0.139<br />
</td>
    </tr>
    <tr>
        <td><strong>20</strong><br />
</td>
        <td><strong>8t (9b)</strong><br />
</td>
        <td><strong>1043.478</strong><br />
</td>
        <td><strong>117/64</strong><br />
</td>
        <td><strong>1044.438</strong><br />
</td>
        <td><strong>-0.96</strong><br />
</td>
    </tr>
    <tr>
        <td>21<br />
</td>
        <td>9v (8#)<br />
</td>
        <td>1095.652<br />
</td>
        <td>32/17<br />
</td>
        <td>1095.045<br />
</td>
        <td>+0.607<br />
</td>
    </tr>
    <tr>
        <td>22<br />
</td>
        <td>9 (1v)<br />
</td>
        <td>1147.826<br />
</td>
        <td>64/33<br />
</td>
        <td>1146.727<br />
</td>
        <td>+1.099<br />
</td>
    </tr>
    <tr>
        <td><strong>23·· (or 0)</strong><br />
</td>
        <td><strong>1 (9t)</strong><br />
</td>
        <td><strong>1200.000</strong><br />
</td>
        <td><strong>2/1</strong><br />
</td>
        <td><strong>1200.000</strong><br />
</td>
        <td><strong>none</strong><br />
</td>
    </tr>
</table>

You'll see that intervals of 23edo come within 5 cents of 9/8; 3 cents of 23/16; 2 cents of 33/32, 21/16, 35/32, &amp; 55/32; &amp; 1 cent of 17/16, 79/64, &amp; 117/64. (&lt;<strong>And let's also note the excellent representations of 14/11 and its inverse, 11/7!!! In fact they might be considered good enought that a chain of 23 such intervals would be a reasonable way to acoustically tune this temperament</strong> -- AKJ) Of course, it also has perfect unisons &amp; octaves, by definition. This means we could potentially build a very strange (&amp; slightly mistuned) harmonic chord which, reduced to within one octave, we could write as frequency ratios 64:66:68:70:72:79:84:92:110:117. I find this cluster a little hard to listen to, whether tuned to JI or 23edo, so I'd like to consider smaller chords, triads &amp; tetrads, as a starting point.<br />
<br />
I'd also like to set an arbitrary limit on how high up the harmonic series we will go. I'll set my limit at the 23rd harmonic. I'll consider harmonics 1, 9, 17, 21, &amp; 23, excluding (at least for now) 33, 55, 79, &amp; 117. Those sonorities could no doubt prove useful to a thoughful composer, but for this study, I'll leave them out.<br />
<br />
Thus we produce ten triads, five tetrads, &amp; one quintad, 16 chords, which, with their inversions (given), doubles to 32 chords. I've written then in a closed position (within one octave), &amp; I recommend trying different voicings. Moving chord tones up &amp; down by octaves, you can unmuddy a muddy chord.<br />
<br />
<!-- ws:start:WikiTextHeadingRule:0:&lt;h2&gt; --><h2 id="toc0"><a name="x-Triads"></a><!-- ws:end:WikiTextHeadingRule:0 --><a class="wiki_link" href="/Triad">Triad</a>s</h2>
 <strong>16:17:18, degrees 0, 2, 4 (inversion 0, 19, 21)</strong>.<br />
17/16 (104.955, error -0.607)<br />
18/16 = 9/8 (203.910, error +4.786)<br />
18/17 (98.955, error: +5.393)<br />
<br />
<!-- ws:start:WikiTextHeadingRule:2:&lt;h3&gt; --><h3 id="toc1"><a name="x-Triads-16:17:21, degrees 0, 2, 9 (inversion 0, 14, 21)."></a><!-- ws:end:WikiTextHeadingRule:2 -->16:17:21, degrees 0, 2, 9 (inversion 0, 14, 21).</h3>
 17/16 (104.955, error -0.607)<br />
21/16 (470.781, error -1.216)<br />
21/17 (365.825, error: -0.608)<br />
<br />
<!-- ws:start:WikiTextHeadingRule:4:&lt;h3&gt; --><h3 id="toc2"><a name="x-Triads-16:17:23, degrees 0, 2, 12 (inversion 0, 11, 21)."></a><!-- ws:end:WikiTextHeadingRule:4 -->16:17:23, degrees 0, 2, 12 (inversion 0, 11, 21).</h3>
 17/16 (104.955, error -0.607)<br />
23/16 (628.274, error -2.187)<br />
23/17 (523.319, error: -1.578)<br />
<br />
<!-- ws:start:WikiTextHeadingRule:6:&lt;h3&gt; --><h3 id="toc3"><a name="x-Triads-16:18:21, degrees 0, 4, 9 (inversion 0, 14, 19)."></a><!-- ws:end:WikiTextHeadingRule:6 -->16:18:21, degrees 0, 4, 9 (inversion 0, 14, 19).</h3>
 18/16 = 9/8 (203.910, error +4.786)<br />
21/16 (470.781, error -1.216)<br />
21/18 = 7/6 (266.871, error: -6.001)<br />
<br />
<!-- ws:start:WikiTextHeadingRule:8:&lt;h3&gt; --><h3 id="toc4"><a name="x-Triads-16:18:23, degrees 0, 4, 12 (inversion 0, 11, 19)."></a><!-- ws:end:WikiTextHeadingRule:8 -->16:18:23, degrees 0, 4, 12 (inversion 0, 11, 19).</h3>
 18/16 = 9/8 (203.910, error +4.786)<br />
23/16 (628.274, error -2.187)<br />
23/18 (424.364, error: -6.973)<br />
<br />
<!-- ws:start:WikiTextHeadingRule:10:&lt;h3&gt; --><h3 id="toc5"><a name="x-Triads-16:21:23, degrees 0, 9, 12 (inversion 0, 11, 14)."></a><!-- ws:end:WikiTextHeadingRule:10 -->16:21:23, degrees 0, 9, 12 (inversion 0, 11, 14).</h3>
 21/16 (470.781, error -1.216)<br />
23/16 (628.274, error -2.187)<br />
23/21 (157.493, error: -0.971)<br />
<br />
<!-- ws:start:WikiTextHeadingRule:12:&lt;h3&gt; --><h3 id="toc6"><a name="x-Triads-17:18:21, degrees 0, 2, 7 (inversion 0, 16, 21)."></a><!-- ws:end:WikiTextHeadingRule:12 -->17:18:21, degrees 0, 2, 7 (inversion 0, 16, 21).</h3>
 18/17 (98.955, error: +5.393)<br />
21/17 (365.825, error: -0.608)<br />
21/18 = 7/6 (266.871, error: -6.001)<br />
<br />
<!-- ws:start:WikiTextHeadingRule:14:&lt;h3&gt; --><h3 id="toc7"><a name="x-Triads-17:18:23, degrees 0, 2, 10 (inversion 0, 13, 21)."></a><!-- ws:end:WikiTextHeadingRule:14 -->17:18:23, degrees 0, 2, 10 (inversion 0, 13, 21).</h3>
 18/17 (98.955, error: +5.393)<br />
23/17 (523.319, error: -1.578)<br />
23/18 (424.364, error: -6.973)<br />
<br />
<!-- ws:start:WikiTextHeadingRule:16:&lt;h3&gt; --><h3 id="toc8"><a name="x-Triads-17:21:23, degrees 0, 7, 10 (inversion 0, 13, 16)."></a><!-- ws:end:WikiTextHeadingRule:16 -->17:21:23, degrees 0, 7, 10 (inversion 0, 13, 16).</h3>
 21/17 (365.825, error: -0.608)<br />
23/17 (523.319, error: -1.578)<br />
23/21 (157.493, error: -0.971)<br />
<br />
<!-- ws:start:WikiTextHeadingRule:18:&lt;h3&gt; --><h3 id="toc9"><a name="x-Triads-18:21:23, degrees 0, 5, 8 (inversion 0, 15, 18)."></a><!-- ws:end:WikiTextHeadingRule:18 -->18:21:23, degrees 0, 5, 8 (inversion 0, 15, 18).</h3>
 21/18 = 7/6 (266.871, error: -6.001)<br />
23/18 (424.364, error: -6.973)<br />
23/21 (157.493, error: -0.971)<br />
<br />
<br />
<!-- ws:start:WikiTextHeadingRule:20:&lt;h2&gt; --><h2 id="toc10"><a name="x-Tetrads"></a><!-- ws:end:WikiTextHeadingRule:20 --><a class="wiki_link" href="/Tetrad">Tetrad</a>s</h2>
 <strong>16:17:18:21, degrees 0, 2, 4, 9 (inversion 0, 14, 19, 21)</strong>.<br />
17/16 (104.955, error -0.607)<br />
18/16 = 9/8 (203.910, error +4.786)<br />
21/16 (470.781, error -1.216)<br />
18/17 (98.955, error: +5.393)<br />
21/17 (365.825, error: -0.608)<br />
21/18 = 7/6 (266.871, error: -6.001)<br />
<br />
<!-- ws:start:WikiTextHeadingRule:22:&lt;h3&gt; --><h3 id="toc11"><a name="x-Tetrads-16:17:18:23, degrees 0, 2, 4, 12 (inversion 0, 11 19, 21)."></a><!-- ws:end:WikiTextHeadingRule:22 -->16:17:18:23, degrees 0, 2, 4, 12 (inversion 0, 11 19, 21).</h3>
 17/16 (104.955, error -0.607)<br />
18/16 = 9/8 (203.910, error +4.786)<br />
23/16 (628.274, error -2.187)<br />
18/17 (98.955, error: +5.393)<br />
23/17 (523.319, error: -1.578)<br />
23/18 (424.364, error: -6.973)<br />
<br />
<!-- ws:start:WikiTextHeadingRule:24:&lt;h3&gt; --><h3 id="toc12"><a name="x-Tetrads-16:17:21:23, degrees 0, 2, 9, 12 (inversion 0, 11, 14, 21)."></a><!-- ws:end:WikiTextHeadingRule:24 -->16:17:21:23, degrees 0, 2, 9, 12 (inversion 0, 11, 14, 21).</h3>
 17/16 (104.955, error -0.607)<br />
21/16 (470.781, error -1.216)<br />
23/16 (628.274, error -2.187)<br />
21/17 (365.825, error: -0.608)<br />
23/17 (523.319, error: -1.578)<br />
23/21 (157.493, error: -0.971)<br />
<br />
<!-- ws:start:WikiTextHeadingRule:26:&lt;h3&gt; --><h3 id="toc13"><a name="x-Tetrads-16:18:21:23, degrees 0, 4, 9, 12 (inversion 0, 11, 14, 19)."></a><!-- ws:end:WikiTextHeadingRule:26 -->16:18:21:23, degrees 0, 4, 9, 12 (inversion 0, 11, 14, 19).</h3>
 18/16 = 9/8 (203.910, error +4.786)<br />
21/16 (470.781, error -1.216)<br />
23/16 (628.274, error -2.187)<br />
21/18 = 7/6 (266.871, error: -6.001)<br />
23/18 (424.364, error: -6.973)<br />
23/21 (157.493, error: -0.971)<br />
<br />
<!-- ws:start:WikiTextHeadingRule:28:&lt;h3&gt; --><h3 id="toc14"><a name="x-Tetrads-17:18:21:23, degrees 0, 2, 7, 10 (inversion 0, 13, 16, 21)."></a><!-- ws:end:WikiTextHeadingRule:28 -->17:18:21:23, degrees 0, 2, 7, 10 (inversion 0, 13, 16, 21).</h3>
 18/17 (98.955, error: +5.393)<br />
21/17 (365.825, error: -0.608)<br />
23/17 (523.319, error: -1.578)<br />
21/18 = 7/6 (266.871, error: -6.001)<br />
23/18 (424.364, error: -6.973)<br />
23/21 (157.493, error: -0.971)<br />
<br />
<br />
<!-- ws:start:WikiTextHeadingRule:30:&lt;h2&gt; --><h2 id="toc15"><a name="x-Quintad"></a><!-- ws:end:WikiTextHeadingRule:30 --><a class="wiki_link" href="/Quintad">Quintad</a></h2>
 <strong>16:17:18:21:23, degrees 0, 2, 4, 9, 12 (inversion 0, 11, 14, 19, 21)</strong>.<br />
17/16 (104.955, error -0.607)<br />
18/16 = 9/8 (203.910, error +4.786)<br />
21/16 (470.781, error -1.216)<br />
23/16 (628.274, error -2.187)<br />
18/17 (98.955, error: +5.393)<br />
21/17 (365.825, error: -0.608)<br />
23/17 (523.319, error: -1.578)<br />
21/18 = 7/6 (266.871, error: -6.001)<br />
23/18 (424.364, error: -6.973)<br />
23/21 (157.493, error: -0.971)</body></html>

@@ Line 1: / Line 1: @@
 <h2>IMPORTED REVISION FROM WIKISPACES</h2>
 This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
-: This revision was by author [[User:xenwolf|xenwolf]] and made on <tt>2010-11-11 11:31:28 UTC</tt>.<br>
+: This revision was by author [[User:Osmiorisbendi|Osmiorisbendi]] and made on <tt>2011-09-20 00:44:25 UTC</tt>.<br>
-: The original revision id was <tt>178620635</tt>.<br>
+: The original revision id was <tt>255950126</tt>.<br>
-: The revision comment was: <tt>links to some interesting intervals added</tt><br>
+: The revision comment was: <tt></tt><br>
 The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br>
 <h4>Original Wikitext content:</h4>
-<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html">If you take a look at the intervals of [[23edo]], you'll find that this system does not contain good representations of harmonics 3, 5, 7, 11, or 13, which appear as central in most just intonation systems. Rather than trivialize 23edo by calling it "atonal" or "nonharmonic," I'd like to consider higher-limit harmonies that could serve as useful sonorities, perhaps even "consonances," in the context of careful composition. [[23edo]] contains intervals which approach harmonics 9, 17, 21, 23, 33, 39, 43 &amp; 117. Let's compare the cents values to see how close 23edo intervals come to these harmonics.
+<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">If you take a look at the intervals of [[23edo]], you'll find that this system does not contain good representations of the harmonics 3, 5, 7, 11, or 13, which appear as central in most just intonation systems. Rather than trivialize 23edo by calling it "atonal" or "inharmonic", you should consider the higher-limit harmonies that could serve as useful sonorities, perhaps even "consonances," in the context of careful composition. [[23edo]] contains intervals which approach very well the harmonics 9, 17, 21, 23, 33, 35, 55, 79 &amp; 117. Let's compare the cents values to see how close 23edo intervals come to these harmonics (and other intervals):
+|| Degrees || Armodue note || Cents sizes || [[nearest harmonic|Nearest Harmonic]] || Cents || "Error" ||
-|| degrees || note || cents || [[nearest harmonic]] || cents || "error" ||
+|| 0 || 1 || 0 || 1/1 || 0.000 || none ||
-|| **0** || **1** || **0** || **1/1** || **0.000** || **none** ||
+|| **1** || **1t (2b)** || **52.174** || **33/32** || **53.273** || **-1.099** ||
-|| 1 || 1# || 52.174 || 33/32 || 53.273 || -1.099 ||
+|| **2** || **2v (1#)** || **104.348** || **17/16** || **104.955** || **-0.607** ||
-|| **2** || **2b** || **104.348** || **17/16** || **104.955** || **-0.607** ||
+|| **3** || **2** || **156.522** || **35/32** || **155.140** || **-1.382** ||
-|| 3 || 2 || 156.522 || 23/21 || 157.493 || -0.971 ||
+|| **4·** || **2t (3b)** || **208.696** || **9/8** || **203.910** || **+4.786** ||
-|| **4·** || **2#** || **208.696** || **9/8** || **203.910** || **+4.786** ||
+|| 5 || 3v (2#) || 260.869 || 50/43 || 261.110 || +0.241 ||
-|| 5 || 3b || 260.869 || 43/37 || 260.174 || +0.695 ||
+|| 6 || 3 || 313.043 || [[6_5|6/5]] || 315.641 || -2.598 ||
-|| **6** || **3** || **313.043** || **[[6_5|6/5]]** || **315.641** || **-2.598** ||
+|| **7·** || **3t (4b)** || **365.217** || **79/64** || **364.537** || **-0.68** ||
-|| **7·** || **3#** || **365.217** || **21/17** || **365.825** || **-0.608** ||
+|| 8 || 4v (3#) || 417.391 || [[14_11|14/11]] || 417.508 || -0.117 ||
-|| 8 || 4b || 417.391 || [[14_11|14/11]] || 417.508 || -0.117 ||
+|| **9** || **4 (5v)** || **469.565** || **21/16** || **470.781** || **-1.216** ||
-|| **9** || **4** || **469.565** || **21/16** || **470.781** || **-1.216** ||
+|| 10**·** || 5 (4t) || 521.739 || 23/17 || 523.319 || -1.58 ||
-|| **10·** || **5** || **521.739** || **23/17** || **523.319** || **-1.58** ||
+|| 11 || 5t (6b) || 573.913 || 32/23 || 571.726 || +2.187 ||
-|| 11 || 5# || 573.913 || 32/23 || 571.726 || +2.187 ||
+|| **12** || **6v (5#)** || **626.087** || **23/16** || **628.274** || **-2.187** ||
-|| **12** || **6b** || **626.087** || **23/16** || **628.274** || **-2.187** ||
+|| 13**·** || 6 || 678.261 || 34/23 || 676.681 || +1.58 ||
-|| **13·** || **6** || **678.261** || **34/23** || **676.681** || **+1.58** ||
+|| 14 || 6t (7b) || 730.435 || 32/21 || 729.219 || +1.216 ||
-|| 14 || 6# || 730.435 || 32/21 || 729.219 || +1.216 ||
+|| 15 || 7v (6#) || 782.609 || [[11_7|11/7]] || 782.492 || +0.117 ||
-|| 15 || 7b || 782.609 || [[11_7|11/7]] || 782.492 || +0.117 ||
+|| 16**·** || 7 || 834.783 || 34/21 || 834.175 || +0.608 ||
-|| **16·** || **7** || **834.783** || **34/21** || **834.175** || **+0.608** ||
+|| 17 || 7t (8b) || 886.957 || [[5_3|5/3]] || 884.359 || +2.598 ||
-|| 17 || 7#(A) || 886.957 || [[5_3|5/3]] || 884.359 || +2.598 ||
+|| **18** || **8v (7#)** || **939.130** || **55/32** || **937.632** || **+1.498** ||
-|| 18 || 8b || 939.130 || 43/25 || 938.890 || +0.24 ||
+|| 19**·** || 8 || 991.304 || 39/22 || 991.165 || +0.139 ||
-|| **19·** || **8** || **991.304** || **39/22** || **991.165** || **+0.139** ||
+|| **20** || **8t (9b)** || **1043.478** || **117/64** || **1044.438** || **-0.96** ||
-|| **20** || **8#** || **1043.478** || **117/64** || **1044.438** || **-0.96** ||
+|| 21 || 9v (8#) || 1095.652 || 32/17 || 1095.045 || +0.607 ||
-|| 21 || 9b || 1095.652 || 32/17 || 1095.045 || +0.607 ||
+|| 22 || 9 (1v) || 1147.826 || 64/33 || 1146.727 || +1.099 ||
-|| 22 || 9 || 1147.826 || 64/33 || 1146.727 || +1.099 ||
+|| **23·· (or 0)** || **1 (9t)** || **1200.000** || **2/1** || **1200.000** || **none** ||
-|| **23·· (or 0)** || **1** || **1200.000** || **2/1** || **1200.000** || **none** ||
 You'll see that intervals of 23edo come within 5 cents of 9/8; 3 cents of 23/16; 2 cents of 33/32, 21/16, 35/32, &amp; 55/32; &amp; 1 cent of 17/16, 79/64, &amp; 117/64. (&lt;**And let's also note the excellent representations of 14/11 and its inverse, 11/7!!! In fact they might be considered good enought that a chain of 23 such intervals would be a reasonable way to acoustically tune this temperament** -- AKJ) Of course, it also has perfect unisons &amp; octaves, by definition. This means we could potentially build a very strange (&amp; slightly mistuned) harmonic chord which, reduced to within one octave, we could write as frequency ratios 64:66:68:70:72:79:84:92:110:117. I find this cluster a little hard to listen to, whether tuned to JI or 23edo, so I'd like to consider smaller chords, triads &amp; tetrads, as a starting point.
@@ Line 40: / Line 39: @@
 ==[[Triad]]s==
+**16:17:18, degrees 0, 2, 4 (inversion 0, 19, 21)**.
-===16:17:18, degrees 0, 2, 4 (inversion 0, 19, 21).===
+/16 (104.955, error -0.607)
-/16 (104.955, error -.607)
 /16 = 9/8 (203.910, error +4.786)
 /17 (98.955, error: +5.393)
 ===16:17:21, degrees 0, 2, 9 (inversion 0, 14, 21).===
-/16 (104.955, error -.607)
+/16 (104.955, error -0.607)
 /16 (470.781, error -1.216)
-/17 (365.825, error: -.608)
+/17 (365.825, error: -0.608)
 ===16:17:23, degrees 0, 2, 12 (inversion 0, 11, 21).===
-/16 (104.955, error -.607)
+/16 (104.955, error -0.607)
 /16 (628.274, error -2.187)
 /17 (523.319, error: -1.578)
@@ Line 69: / Line 67: @@
 /16 (470.781, error -1.216)
 /16 (628.274, error -2.187)
-/21 (157.493, error: -.971)
+/21 (157.493, error: -0.971)
 ===17:18:21, degrees 0, 2, 7 (inversion 0, 16, 21).===
 /17 (98.955, error: +5.393)
-/17 (365.825, error: -.608)
+/17 (365.825, error: -0.608)
 /18 = 7/6 (266.871, error: -6.001)
@@ Line 82: / Line 80: @@
 ===17:21:23, degrees 0, 7, 10 (inversion 0, 13, 16).===
-/17 (365.825, error: -.608)
+/17 (365.825, error: -0.608)
 /17 (523.319, error: -1.578)
-/21 (157.493, error: -.971)
+/21 (157.493, error: -0.971)
 ===18:21:23, degrees 0, 5, 8 (inversion 0, 15, 18).===
 /18 = 7/6 (266.871, error: -6.001)
 /18 (424.364, error: -6.973)
-/21 (157.493, error: -.971)
+/21 (157.493, error: -0.971)
 ==[[Tetrad]]s==
+**16:17:18:21, degrees 0, 2, 4, 9 (inversion 0, 14, 19, 21)**.
-===16:17:18:21, degrees 0, 2, 4, 9 (inversion 0, 14, 19, 21).===
+/16 (104.955, error -0.607)
-/16 (104.955, error -.607)
 /16 = 9/8 (203.910, error +4.786)
 /16 (470.781, error -1.216)
 /17 (98.955, error: +5.393)
-/17 (365.825, error: -.608)
+/17 (365.825, error: -0.608)
 /18 = 7/6 (266.871, error: -6.001)
 ===16:17:18:23, degrees 0, 2, 4, 12 (inversion 0, 11 19, 21).===
-/16 (104.955, error -.607)
+/16 (104.955, error -0.607)
 /16 = 9/8 (203.910, error +4.786)
 /16 (628.274, error -2.187)
@@ Line 110: / Line 108: @@
 ===16:17:21:23, degrees 0, 2, 9, 12 (inversion 0, 11, 14, 21).===
-/16 (104.955, error -.607)
+/16 (104.955, error -0.607)
 /16 (470.781, error -1.216)
 /16 (628.274, error -2.187)
-/17 (365.825, error: -.608)
+/17 (365.825, error: -0.608)
 /17 (523.319, error: -1.578)
-/21 (157.493, error: -.971)
+/21 (157.493, error: -0.971)
 ===16:18:21:23, degrees 0, 4, 9, 12 (inversion 0, 11, 14, 19).===
@@ Line 123: / Line 121: @@
 /18 = 7/6 (266.871, error: -6.001)
 /18 (424.364, error: -6.973)
-/21 (157.493, error: -.971)
+/21 (157.493, error: -0.971)
 ===17:18:21:23, degrees 0, 2, 7, 10 (inversion 0, 13, 16, 21).===
 /17 (98.955, error: +5.393)
-/17 (365.825, error: -.608)
+/17 (365.825, error: -0.608)
 /17 (523.319, error: -1.578)
 /18 = 7/6 (266.871, error: -6.001)
 /18 (424.364, error: -6.973)
-/21 (157.493, error: -.971)
+/21 (157.493, error: -0.971)
 ==[[Quintad]]==
+**16:17:18:21:23, degrees 0, 2, 4, 9, 12 (inversion 0, 11, 14, 19, 21)**.
-===16:17:18:21:23, degrees 0, 2, 4, 9, 12 (inversion 0, 11, 14, 19, 21).===
+/16 (104.955, error -0.607)
-/16 (104.955, error -.607)
 /16 = 9/8 (203.910, error +4.786)
 /16 (470.781, error -1.216)
 /16 (628.274, error -2.187)
 /17 (98.955, error: +5.393)
-/17 (365.825, error: -.608)
+/17 (365.825, error: -0.608)
 /17 (523.319, error: -1.578)
 /18 = 7/6 (266.871, error: -6.001)
 /18 (424.364, error: -6.973)
-/21 (157.493, error: -.971)
+/21 (157.493, error: -0.971)</pre></div>
-/21 (157.493, error: -.971)</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;Harmony of 23edo&lt;/title&gt;&lt;/head&gt;&lt;body&gt;If you take a look at the intervals of &lt;a class="wiki_link" href="/23edo"&gt;23edo&lt;/a&gt;, you'll find that this system does not contain good representations of harmonics 3, 5, 7, 11, or 13, which appear as central in most just intonation systems. Rather than trivialize 23edo by calling it &amp;quot;atonal&amp;quot; or &amp;quot;nonharmonic,&amp;quot; I'd like to consider higher-limit harmonies that could serve as useful sonorities, perhaps even &amp;quot;consonances,&amp;quot; in the context of careful composition. &lt;a class="wiki_link" href="/23edo"&gt;23edo&lt;/a&gt; contains intervals which approach harmonics 9, 17, 21, 23, 33, 39, 43 &amp;amp; 117. Let's compare the cents values to see how close 23edo intervals come to these harmonics.&lt;br /&gt;
+<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;Harmony of 23edo&lt;/title&gt;&lt;/head&gt;&lt;body&gt;If you take a look at the intervals of &lt;a class="wiki_link" href="/23edo"&gt;23edo&lt;/a&gt;, you'll find that this system does not contain good representations of the harmonics 3, 5, 7, 11, or 13, which appear as central in most just intonation systems. Rather than trivialize 23edo by calling it &amp;quot;atonal&amp;quot; or &amp;quot;inharmonic&amp;quot;, you should consider the higher-limit harmonies that could serve as useful sonorities, perhaps even &amp;quot;consonances,&amp;quot; in the context of careful composition. &lt;a class="wiki_link" href="/23edo"&gt;23edo&lt;/a&gt; contains intervals which approach very well the harmonics 9, 17, 21, 23, 33, 35, 55, 79 &amp;amp; 117. Let's compare the cents values to see how close 23edo intervals come to these harmonics (and other intervals):&lt;br /&gt;
-&lt;br /&gt;
 &lt;table class="wiki_table"&gt;
      &lt;tr&gt;
-         &lt;td&gt;degrees&lt;br /&gt;
+         &lt;td&gt;Degrees&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;note&lt;br /&gt;
+         &lt;td&gt;Armodue note&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;cents&lt;br /&gt;
+         &lt;td&gt;Cents sizes&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;&lt;a class="wiki_link" href="/nearest%20harmonic"&gt;nearest harmonic&lt;/a&gt;&lt;br /&gt;
+         &lt;td&gt;&lt;a class="wiki_link" href="/nearest%20harmonic"&gt;Nearest Harmonic&lt;/a&gt;&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;cents&lt;br /&gt;
+         &lt;td&gt;Cents&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;&amp;quot;error&amp;quot;&lt;br /&gt;
+         &lt;td&gt;&amp;quot;Error&amp;quot;&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
@@ Line 170: / Line 166: @@
          &lt;td&gt;0&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;&lt;strong&gt;1&lt;/strong&gt;&lt;br /&gt;
+         &lt;td&gt;1&lt;br /&gt;
 &lt;/td&gt;
          &lt;td&gt;0&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;&lt;strong&gt;1/1&lt;/strong&gt;&lt;br /&gt;
+         &lt;td&gt;1/1&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;&lt;strong&gt;0.000&lt;/strong&gt;&lt;br /&gt;
+         &lt;td&gt;0.000&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;&lt;strong&gt;none&lt;/strong&gt;&lt;br /&gt;
+         &lt;td&gt;none&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
      &lt;tr&gt;
-         &lt;td&gt;1&lt;br /&gt;
+         &lt;td&gt;&lt;strong&gt;1&lt;/strong&gt;&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;1#&lt;br /&gt;
+         &lt;td&gt;&lt;strong&gt;1t (2b)&lt;/strong&gt;&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;52.174&lt;br /&gt;
+         &lt;td&gt;&lt;strong&gt;52.174&lt;/strong&gt;&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;33/32&lt;br /&gt;
+         &lt;td&gt;&lt;strong&gt;33/32&lt;/strong&gt;&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;53.273&lt;br /&gt;
+         &lt;td&gt;&lt;strong&gt;53.273&lt;/strong&gt;&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;-1.099&lt;br /&gt;
+         &lt;td&gt;&lt;strong&gt;-1.099&lt;/strong&gt;&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
@@ Line 198: / Line 194: @@
          &lt;td&gt;&lt;strong&gt;2&lt;/strong&gt;&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;&lt;strong&gt;2b&lt;/strong&gt;&lt;br /&gt;
+         &lt;td&gt;&lt;strong&gt;2v (1#)&lt;/strong&gt;&lt;br /&gt;
 &lt;/td&gt;
          &lt;td&gt;&lt;strong&gt;104.348&lt;/strong&gt;&lt;br /&gt;
@@ Line 210: / Line 206: @@
      &lt;/tr&gt;
      &lt;tr&gt;
-         &lt;td&gt;3&lt;br /&gt;
+         &lt;td&gt;&lt;strong&gt;3&lt;/strong&gt;&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;2&lt;br /&gt;
+         &lt;td&gt;&lt;strong&gt;2&lt;/strong&gt;&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;156.522&lt;br /&gt;
+         &lt;td&gt;&lt;strong&gt;156.522&lt;/strong&gt;&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;23/21&lt;br /&gt;
+         &lt;td&gt;&lt;strong&gt;35/32&lt;/strong&gt;&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;157.493&lt;br /&gt;
+         &lt;td&gt;&lt;strong&gt;155.140&lt;/strong&gt;&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;-0.971&lt;br /&gt;
+         &lt;td&gt;&lt;strong&gt;-1.382&lt;/strong&gt;&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
@@ Line 226: / Line 222: @@
          &lt;td&gt;&lt;strong&gt;4·&lt;/strong&gt;&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;&lt;strong&gt;2#&lt;/strong&gt;&lt;br /&gt;
+         &lt;td&gt;&lt;strong&gt;2t (3b)&lt;/strong&gt;&lt;br /&gt;
 &lt;/td&gt;
          &lt;td&gt;&lt;strong&gt;208.696&lt;/strong&gt;&lt;br /&gt;
@@ Line 240: / Line 236: @@
          &lt;td&gt;5&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;3b&lt;br /&gt;
+         &lt;td&gt;3v (2#)&lt;br /&gt;
 &lt;/td&gt;
          &lt;td&gt;260.869&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;43/37&lt;br /&gt;
+         &lt;td&gt;50/43&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;260.174&lt;br /&gt;
+         &lt;td&gt;261.110&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;+0.695&lt;br /&gt;
+         &lt;td&gt;+0.241&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
      &lt;tr&gt;
-         &lt;td&gt;&lt;strong&gt;6&lt;/strong&gt;&lt;br /&gt;
+         &lt;td&gt;6&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;&lt;strong&gt;3&lt;/strong&gt;&lt;br /&gt;
+         &lt;td&gt;3&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;&lt;strong&gt;313.043&lt;/strong&gt;&lt;br /&gt;
+         &lt;td&gt;313.043&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;&lt;strong&gt;&lt;a class="wiki_link" href="/6_5"&gt;6/5&lt;/a&gt;&lt;/strong&gt;&lt;br /&gt;
+         &lt;td&gt;&lt;a class="wiki_link" href="/6_5"&gt;6/5&lt;/a&gt;&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;&lt;strong&gt;315.641&lt;/strong&gt;&lt;br /&gt;
+         &lt;td&gt;315.641&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;&lt;strong&gt;-2.598&lt;/strong&gt;&lt;br /&gt;
+         &lt;td&gt;-2.598&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
@@ Line 268: / Line 264: @@
          &lt;td&gt;&lt;strong&gt;7·&lt;/strong&gt;&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;&lt;strong&gt;3#&lt;/strong&gt;&lt;br /&gt;
+         &lt;td&gt;&lt;strong&gt;3t (4b)&lt;/strong&gt;&lt;br /&gt;
 &lt;/td&gt;
          &lt;td&gt;&lt;strong&gt;365.217&lt;/strong&gt;&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;&lt;strong&gt;21/17&lt;/strong&gt;&lt;br /&gt;
+         &lt;td&gt;&lt;strong&gt;79/64&lt;/strong&gt;&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;&lt;strong&gt;365.825&lt;/strong&gt;&lt;br /&gt;
+         &lt;td&gt;&lt;strong&gt;364.537&lt;/strong&gt;&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;&lt;strong&gt;-0.608&lt;/strong&gt;&lt;br /&gt;
+         &lt;td&gt;&lt;strong&gt;-0.68&lt;/strong&gt;&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
@@ Line 282: / Line 278: @@
          &lt;td&gt;8&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;4b&lt;br /&gt;
+         &lt;td&gt;4v (3#)&lt;br /&gt;
 &lt;/td&gt;
          &lt;td&gt;417.391&lt;br /&gt;
@@ Line 296: / Line 292: @@
          &lt;td&gt;&lt;strong&gt;9&lt;/strong&gt;&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;&lt;strong&gt;4&lt;/strong&gt;&lt;br /&gt;
+         &lt;td&gt;&lt;strong&gt;4 (5v)&lt;/strong&gt;&lt;br /&gt;
 &lt;/td&gt;
          &lt;td&gt;&lt;strong&gt;469.565&lt;/strong&gt;&lt;br /&gt;
@@ Line 308: / Line 304: @@
      &lt;/tr&gt;
      &lt;tr&gt;
-         &lt;td&gt;&lt;strong&gt;10·&lt;/strong&gt;&lt;br /&gt;
+         &lt;td&gt;10&lt;strong&gt;·&lt;/strong&gt;&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;&lt;strong&gt;5&lt;/strong&gt;&lt;br /&gt;
+         &lt;td&gt;5 (4t)&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;&lt;strong&gt;521.739&lt;/strong&gt;&lt;br /&gt;
+         &lt;td&gt;521.739&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;&lt;strong&gt;23/17&lt;/strong&gt;&lt;br /&gt;
+         &lt;td&gt;23/17&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;&lt;strong&gt;523.319&lt;/strong&gt;&lt;br /&gt;
+         &lt;td&gt;523.319&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;&lt;strong&gt;-1.58&lt;/strong&gt;&lt;br /&gt;
+         &lt;td&gt;-1.58&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
@@ Line 324: / Line 320: @@
          &lt;td&gt;11&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;5#&lt;br /&gt;
+         &lt;td&gt;5t (6b)&lt;br /&gt;
 &lt;/td&gt;
          &lt;td&gt;573.913&lt;br /&gt;
@@ Line 338: / Line 334: @@
          &lt;td&gt;&lt;strong&gt;12&lt;/strong&gt;&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;&lt;strong&gt;6b&lt;/strong&gt;&lt;br /&gt;
+         &lt;td&gt;&lt;strong&gt;6v (5#)&lt;/strong&gt;&lt;br /&gt;
 &lt;/td&gt;
          &lt;td&gt;&lt;strong&gt;626.087&lt;/strong&gt;&lt;br /&gt;
@@ Line 350: / Line 346: @@
      &lt;/tr&gt;
      &lt;tr&gt;
-         &lt;td&gt;&lt;strong&gt;13·&lt;/strong&gt;&lt;br /&gt;
+         &lt;td&gt;13&lt;strong&gt;·&lt;/strong&gt;&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;&lt;strong&gt;6&lt;/strong&gt;&lt;br /&gt;
+         &lt;td&gt;6&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;&lt;strong&gt;678.261&lt;/strong&gt;&lt;br /&gt;
+         &lt;td&gt;678.261&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;&lt;strong&gt;34/23&lt;/strong&gt;&lt;br /&gt;
+         &lt;td&gt;34/23&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;&lt;strong&gt;676.681&lt;/strong&gt;&lt;br /&gt;
+         &lt;td&gt;676.681&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;&lt;strong&gt;+1.58&lt;/strong&gt;&lt;br /&gt;
+         &lt;td&gt;+1.58&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
@@ Line 366: / Line 362: @@
          &lt;td&gt;14&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;6#&lt;br /&gt;
+         &lt;td&gt;6t (7b)&lt;br /&gt;
 &lt;/td&gt;
          &lt;td&gt;730.435&lt;br /&gt;
@@ Line 380: / Line 376: @@
          &lt;td&gt;15&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;7b&lt;br /&gt;
+         &lt;td&gt;7v (6#)&lt;br /&gt;
 &lt;/td&gt;
          &lt;td&gt;782.609&lt;br /&gt;
@@ Line 392: / Line 388: @@
      &lt;/tr&gt;
      &lt;tr&gt;
-         &lt;td&gt;&lt;strong&gt;16·&lt;/strong&gt;&lt;br /&gt;
+         &lt;td&gt;16&lt;strong&gt;·&lt;/strong&gt;&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;&lt;strong&gt;7&lt;/strong&gt;&lt;br /&gt;
+         &lt;td&gt;7&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;&lt;strong&gt;834.783&lt;/strong&gt;&lt;br /&gt;
+         &lt;td&gt;834.783&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;&lt;strong&gt;34/21&lt;/strong&gt;&lt;br /&gt;
+         &lt;td&gt;34/21&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;&lt;strong&gt;834.175&lt;/strong&gt;&lt;br /&gt;
+         &lt;td&gt;834.175&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;&lt;strong&gt;+0.608&lt;/strong&gt;&lt;br /&gt;
+         &lt;td&gt;+0.608&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
@@ Line 408: / Line 404: @@
          &lt;td&gt;17&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;7#(A)&lt;br /&gt;
+         &lt;td&gt;7t (8b)&lt;br /&gt;
 &lt;/td&gt;
          &lt;td&gt;886.957&lt;br /&gt;
@@ Line 420: / Line 416: @@
      &lt;/tr&gt;
      &lt;tr&gt;
-         &lt;td&gt;18&lt;br /&gt;
+         &lt;td&gt;&lt;strong&gt;18&lt;/strong&gt;&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;8b&lt;br /&gt;
+         &lt;td&gt;&lt;strong&gt;8v (7#)&lt;/strong&gt;&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;939.130&lt;br /&gt;
+         &lt;td&gt;&lt;strong&gt;939.130&lt;/strong&gt;&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;43/25&lt;br /&gt;
+         &lt;td&gt;&lt;strong&gt;55/32&lt;/strong&gt;&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;938.890&lt;br /&gt;
+         &lt;td&gt;&lt;strong&gt;937.632&lt;/strong&gt;&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;+0.24&lt;br /&gt;
+         &lt;td&gt;&lt;strong&gt;+1.498&lt;/strong&gt;&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
      &lt;tr&gt;
-         &lt;td&gt;&lt;strong&gt;19·&lt;/strong&gt;&lt;br /&gt;
+         &lt;td&gt;19&lt;strong&gt;·&lt;/strong&gt;&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;&lt;strong&gt;8&lt;/strong&gt;&lt;br /&gt;
+         &lt;td&gt;8&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;&lt;strong&gt;991.304&lt;/strong&gt;&lt;br /&gt;
+         &lt;td&gt;991.304&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;&lt;strong&gt;39/22&lt;/strong&gt;&lt;br /&gt;
+         &lt;td&gt;39/22&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;&lt;strong&gt;991.165&lt;/strong&gt;&lt;br /&gt;
+         &lt;td&gt;991.165&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;&lt;strong&gt;+0.139&lt;/strong&gt;&lt;br /&gt;
+         &lt;td&gt;+0.139&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
@@ Line 450: / Line 446: @@
          &lt;td&gt;&lt;strong&gt;20&lt;/strong&gt;&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;&lt;strong&gt;8#&lt;/strong&gt;&lt;br /&gt;
+         &lt;td&gt;&lt;strong&gt;8t (9b)&lt;/strong&gt;&lt;br /&gt;
 &lt;/td&gt;
          &lt;td&gt;&lt;strong&gt;1043.478&lt;/strong&gt;&lt;br /&gt;
@@ Line 464: / Line 460: @@
          &lt;td&gt;21&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;9b&lt;br /&gt;
+         &lt;td&gt;9v (8#)&lt;br /&gt;
 &lt;/td&gt;
          &lt;td&gt;1095.652&lt;br /&gt;
@@ Line 478: / Line 474: @@
          &lt;td&gt;22&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;9&lt;br /&gt;
+         &lt;td&gt;9 (1v)&lt;br /&gt;
 &lt;/td&gt;
          &lt;td&gt;1147.826&lt;br /&gt;
@@ Line 492: / Line 488: @@
          &lt;td&gt;&lt;strong&gt;23·· (or 0)&lt;/strong&gt;&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;&lt;strong&gt;1&lt;/strong&gt;&lt;br /&gt;
+         &lt;td&gt;&lt;strong&gt;1 (9t)&lt;/strong&gt;&lt;br /&gt;
 &lt;/td&gt;
          &lt;td&gt;&lt;strong&gt;1200.000&lt;/strong&gt;&lt;br /&gt;
@@ Line 512: / Line 508: @@
 &lt;br /&gt;
 &lt;!-- ws:start:WikiTextHeadingRule:0:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc0"&gt;&lt;a name="x-Triads"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:0 --&gt;&lt;a class="wiki_link" href="/Triad"&gt;Triad&lt;/a&gt;s&lt;/h2&gt;
-  &lt;br /&gt;
+  &lt;strong&gt;16:17:18, degrees 0, 2, 4 (inversion 0, 19, 21)&lt;/strong&gt;.&lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:2:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc1"&gt;&lt;a name="x-Triads-16:17:18, degrees 0, 2, 4 (inversion 0, 19, 21)."&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:2 --&gt;16:17:18, degrees 0, 2, 4 (inversion 0, 19, 21).&lt;/h3&gt;
+/16 (104.955, error -0.607)&lt;br /&gt;
-/16 (104.955, error -.607)&lt;br /&gt;
 /16 = 9/8 (203.910, error +4.786)&lt;br /&gt;
 /17 (98.955, error: +5.393)&lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:4:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc2"&gt;&lt;a name="x-Triads-16:17:21, degrees 0, 2, 9 (inversion 0, 14, 21)."&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:4 --&gt;16:17:21, degrees 0, 2, 9 (inversion 0, 14, 21).&lt;/h3&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:2:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc1"&gt;&lt;a name="x-Triads-16:17:21, degrees 0, 2, 9 (inversion 0, 14, 21)."&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:2 --&gt;16:17:21, degrees 0, 2, 9 (inversion 0, 14, 21).&lt;/h3&gt;
-/16 (104.955, error -.607)&lt;br /&gt;
+/16 (104.955, error -0.607)&lt;br /&gt;
 /16 (470.781, error -1.216)&lt;br /&gt;
-/17 (365.825, error: -.608)&lt;br /&gt;
+/17 (365.825, error: -0.608)&lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:6:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc3"&gt;&lt;a name="x-Triads-16:17:23, degrees 0, 2, 12 (inversion 0, 11, 21)."&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:6 --&gt;16:17:23, degrees 0, 2, 12 (inversion 0, 11, 21).&lt;/h3&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:4:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc2"&gt;&lt;a name="x-Triads-16:17:23, degrees 0, 2, 12 (inversion 0, 11, 21)."&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:4 --&gt;16:17:23, degrees 0, 2, 12 (inversion 0, 11, 21).&lt;/h3&gt;
-/16 (104.955, error -.607)&lt;br /&gt;
+/16 (104.955, error -0.607)&lt;br /&gt;
 /16 (628.274, error -2.187)&lt;br /&gt;
 /17 (523.319, error: -1.578)&lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:8:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc4"&gt;&lt;a name="x-Triads-16:18:21, degrees 0, 4, 9 (inversion 0, 14, 19)."&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:8 --&gt;16:18:21, degrees 0, 4, 9 (inversion 0, 14, 19).&lt;/h3&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:6:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc3"&gt;&lt;a name="x-Triads-16:18:21, degrees 0, 4, 9 (inversion 0, 14, 19)."&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:6 --&gt;16:18:21, degrees 0, 4, 9 (inversion 0, 14, 19).&lt;/h3&gt;
 /16 = 9/8 (203.910, error +4.786)&lt;br /&gt;
 /16 (470.781, error -1.216)&lt;br /&gt;
 /18 = 7/6 (266.871, error: -6.001)&lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:10:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc5"&gt;&lt;a name="x-Triads-16:18:23, degrees 0, 4, 12 (inversion 0, 11, 19)."&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:10 --&gt;16:18:23, degrees 0, 4, 12 (inversion 0, 11, 19).&lt;/h3&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:8:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc4"&gt;&lt;a name="x-Triads-16:18:23, degrees 0, 4, 12 (inversion 0, 11, 19)."&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:8 --&gt;16:18:23, degrees 0, 4, 12 (inversion 0, 11, 19).&lt;/h3&gt;
 /16 = 9/8 (203.910, error +4.786)&lt;br /&gt;
 /16 (628.274, error -2.187)&lt;br /&gt;
 /18 (424.364, error: -6.973)&lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:12:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc6"&gt;&lt;a name="x-Triads-16:21:23, degrees 0, 9, 12 (inversion 0, 11, 14)."&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:12 --&gt;16:21:23, degrees 0, 9, 12 (inversion 0, 11, 14).&lt;/h3&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:10:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc5"&gt;&lt;a name="x-Triads-16:21:23, degrees 0, 9, 12 (inversion 0, 11, 14)."&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:10 --&gt;16:21:23, degrees 0, 9, 12 (inversion 0, 11, 14).&lt;/h3&gt;
 /16 (470.781, error -1.216)&lt;br /&gt;
 /16 (628.274, error -2.187)&lt;br /&gt;
-/21 (157.493, error: -.971)&lt;br /&gt;
+/21 (157.493, error: -0.971)&lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:14:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc7"&gt;&lt;a name="x-Triads-17:18:21, degrees 0, 2, 7 (inversion 0, 16, 21)."&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:14 --&gt;17:18:21, degrees 0, 2, 7 (inversion 0, 16, 21).&lt;/h3&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:12:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc6"&gt;&lt;a name="x-Triads-17:18:21, degrees 0, 2, 7 (inversion 0, 16, 21)."&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:12 --&gt;17:18:21, degrees 0, 2, 7 (inversion 0, 16, 21).&lt;/h3&gt;
 /17 (98.955, error: +5.393)&lt;br /&gt;
-/17 (365.825, error: -.608)&lt;br /&gt;
+/17 (365.825, error: -0.608)&lt;br /&gt;
 /18 = 7/6 (266.871, error: -6.001)&lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:16:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc8"&gt;&lt;a name="x-Triads-17:18:23, degrees 0, 2, 10 (inversion 0, 13, 21)."&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:16 --&gt;17:18:23, degrees 0, 2, 10 (inversion 0, 13, 21).&lt;/h3&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:14:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc7"&gt;&lt;a name="x-Triads-17:18:23, degrees 0, 2, 10 (inversion 0, 13, 21)."&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:14 --&gt;17:18:23, degrees 0, 2, 10 (inversion 0, 13, 21).&lt;/h3&gt;
 /17 (98.955, error: +5.393)&lt;br /&gt;
 /17 (523.319, error: -1.578)&lt;br /&gt;
 /18 (424.364, error: -6.973)&lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:18:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc9"&gt;&lt;a name="x-Triads-17:21:23, degrees 0, 7, 10 (inversion 0, 13, 16)."&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:18 --&gt;17:21:23, degrees 0, 7, 10 (inversion 0, 13, 16).&lt;/h3&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:16:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc8"&gt;&lt;a name="x-Triads-17:21:23, degrees 0, 7, 10 (inversion 0, 13, 16)."&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:16 --&gt;17:21:23, degrees 0, 7, 10 (inversion 0, 13, 16).&lt;/h3&gt;
-/17 (365.825, error: -.608)&lt;br /&gt;
+/17 (365.825, error: -0.608)&lt;br /&gt;
 /17 (523.319, error: -1.578)&lt;br /&gt;
-/21 (157.493, error: -.971)&lt;br /&gt;
+/21 (157.493, error: -0.971)&lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:20:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc10"&gt;&lt;a name="x-Triads-18:21:23, degrees 0, 5, 8 (inversion 0, 15, 18)."&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:20 --&gt;18:21:23, degrees 0, 5, 8 (inversion 0, 15, 18).&lt;/h3&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:18:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc9"&gt;&lt;a name="x-Triads-18:21:23, degrees 0, 5, 8 (inversion 0, 15, 18)."&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:18 --&gt;18:21:23, degrees 0, 5, 8 (inversion 0, 15, 18).&lt;/h3&gt;
 /18 = 7/6 (266.871, error: -6.001)&lt;br /&gt;
 /18 (424.364, error: -6.973)&lt;br /&gt;
-/21 (157.493, error: -.971)&lt;br /&gt;
+/21 (157.493, error: -0.971)&lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:22:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc11"&gt;&lt;a name="x-Tetrads"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:22 --&gt;&lt;a class="wiki_link" href="/Tetrad"&gt;Tetrad&lt;/a&gt;s&lt;/h2&gt;
+&lt;br /&gt;
-  &lt;br /&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:20:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc10"&gt;&lt;a name="x-Tetrads"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:20 --&gt;&lt;a class="wiki_link" href="/Tetrad"&gt;Tetrad&lt;/a&gt;s&lt;/h2&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:24:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc12"&gt;&lt;a name="x-Tetrads-16:17:18:21, degrees 0, 2, 4, 9 (inversion 0, 14, 19, 21)."&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:24 --&gt;16:17:18:21, degrees 0, 2, 4, 9 (inversion 0, 14, 19, 21).&lt;/h3&gt;
+  &lt;strong&gt;16:17:18:21, degrees 0, 2, 4, 9 (inversion 0, 14, 19, 21)&lt;/strong&gt;.&lt;br /&gt;
-/16 (104.955, error -.607)&lt;br /&gt;
+/16 (104.955, error -0.607)&lt;br /&gt;
 /16 = 9/8 (203.910, error +4.786)&lt;br /&gt;
 /16 (470.781, error -1.216)&lt;br /&gt;
 /17 (98.955, error: +5.393)&lt;br /&gt;
-/17 (365.825, error: -.608)&lt;br /&gt;
+/17 (365.825, error: -0.608)&lt;br /&gt;
 /18 = 7/6 (266.871, error: -6.001)&lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:26:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc13"&gt;&lt;a name="x-Tetrads-16:17:18:23, degrees 0, 2, 4, 12 (inversion 0, 11 19, 21)."&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:26 --&gt;16:17:18:23, degrees 0, 2, 4, 12 (inversion 0, 11 19, 21).&lt;/h3&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:22:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc11"&gt;&lt;a name="x-Tetrads-16:17:18:23, degrees 0, 2, 4, 12 (inversion 0, 11 19, 21)."&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:22 --&gt;16:17:18:23, degrees 0, 2, 4, 12 (inversion 0, 11 19, 21).&lt;/h3&gt;
-/16 (104.955, error -.607)&lt;br /&gt;
+/16 (104.955, error -0.607)&lt;br /&gt;
 /16 = 9/8 (203.910, error +4.786)&lt;br /&gt;
 /16 (628.274, error -2.187)&lt;br /&gt;
@@ Line 581: / Line 576: @@
 /18 (424.364, error: -6.973)&lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:28:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc14"&gt;&lt;a name="x-Tetrads-16:17:21:23, degrees 0, 2, 9, 12 (inversion 0, 11, 14, 21)."&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:28 --&gt;16:17:21:23, degrees 0, 2, 9, 12 (inversion 0, 11, 14, 21).&lt;/h3&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:24:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc12"&gt;&lt;a name="x-Tetrads-16:17:21:23, degrees 0, 2, 9, 12 (inversion 0, 11, 14, 21)."&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:24 --&gt;16:17:21:23, degrees 0, 2, 9, 12 (inversion 0, 11, 14, 21).&lt;/h3&gt;
-/16 (104.955, error -.607)&lt;br /&gt;
+/16 (104.955, error -0.607)&lt;br /&gt;
 /16 (470.781, error -1.216)&lt;br /&gt;
 /16 (628.274, error -2.187)&lt;br /&gt;
-/17 (365.825, error: -.608)&lt;br /&gt;
+/17 (365.825, error: -0.608)&lt;br /&gt;
 /17 (523.319, error: -1.578)&lt;br /&gt;
-/21 (157.493, error: -.971)&lt;br /&gt;
+/21 (157.493, error: -0.971)&lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:30:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc15"&gt;&lt;a name="x-Tetrads-16:18:21:23, degrees 0, 4, 9, 12 (inversion 0, 11, 14, 19)."&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:30 --&gt;16:18:21:23, degrees 0, 4, 9, 12 (inversion 0, 11, 14, 19).&lt;/h3&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:26:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc13"&gt;&lt;a name="x-Tetrads-16:18:21:23, degrees 0, 4, 9, 12 (inversion 0, 11, 14, 19)."&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:26 --&gt;16:18:21:23, degrees 0, 4, 9, 12 (inversion 0, 11, 14, 19).&lt;/h3&gt;
 /16 = 9/8 (203.910, error +4.786)&lt;br /&gt;
 /16 (470.781, error -1.216)&lt;br /&gt;
@@ Line 595: / Line 590: @@
 /18 = 7/6 (266.871, error: -6.001)&lt;br /&gt;
 /18 (424.364, error: -6.973)&lt;br /&gt;
-/21 (157.493, error: -.971)&lt;br /&gt;
+/21 (157.493, error: -0.971)&lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:32:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc16"&gt;&lt;a name="x-Tetrads-17:18:21:23, degrees 0, 2, 7, 10 (inversion 0, 13, 16, 21)."&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:32 --&gt;17:18:21:23, degrees 0, 2, 7, 10 (inversion 0, 13, 16, 21).&lt;/h3&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:28:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc14"&gt;&lt;a name="x-Tetrads-17:18:21:23, degrees 0, 2, 7, 10 (inversion 0, 13, 16, 21)."&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:28 --&gt;17:18:21:23, degrees 0, 2, 7, 10 (inversion 0, 13, 16, 21).&lt;/h3&gt;
 /17 (98.955, error: +5.393)&lt;br /&gt;
-/17 (365.825, error: -.608)&lt;br /&gt;
+/17 (365.825, error: -0.608)&lt;br /&gt;
 /17 (523.319, error: -1.578)&lt;br /&gt;
 /18 = 7/6 (266.871, error: -6.001)&lt;br /&gt;
 /18 (424.364, error: -6.973)&lt;br /&gt;
-/21 (157.493, error: -.971)&lt;br /&gt;
+/21 (157.493, error: -0.971)&lt;br /&gt;
+&lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:34:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc17"&gt;&lt;a name="x-Quintad"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:34 --&gt;&lt;a class="wiki_link" href="/Quintad"&gt;Quintad&lt;/a&gt;&lt;/h2&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:30:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc15"&gt;&lt;a name="x-Quintad"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:30 --&gt;&lt;a class="wiki_link" href="/Quintad"&gt;Quintad&lt;/a&gt;&lt;/h2&gt;
-  &lt;br /&gt;
+  &lt;strong&gt;16:17:18:21:23, degrees 0, 2, 4, 9, 12 (inversion 0, 11, 14, 19, 21)&lt;/strong&gt;.&lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:36:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc18"&gt;&lt;a name="x-Quintad-16:17:18:21:23, degrees 0, 2, 4, 9, 12 (inversion 0, 11, 14, 19, 21)."&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:36 --&gt;16:17:18:21:23, degrees 0, 2, 4, 9, 12 (inversion 0, 11, 14, 19, 21).&lt;/h3&gt;
+/16 (104.955, error -0.607)&lt;br /&gt;
-/16 (104.955, error -.607)&lt;br /&gt;
 /16 = 9/8 (203.910, error +4.786)&lt;br /&gt;
 /16 (470.781, error -1.216)&lt;br /&gt;
 /16 (628.274, error -2.187)&lt;br /&gt;
 /17 (98.955, error: +5.393)&lt;br /&gt;
-/17 (365.825, error: -.608)&lt;br /&gt;
+/17 (365.825, error: -0.608)&lt;br /&gt;
 /17 (523.319, error: -1.578)&lt;br /&gt;
 /18 = 7/6 (266.871, error: -6.001)&lt;br /&gt;
 /18 (424.364, error: -6.973)&lt;br /&gt;
-/21 (157.493, error: -.971)&lt;br /&gt;
+/21 (157.493, error: -0.971)&lt;/body&gt;&lt;/html&gt;</pre></div>
-/21 (157.493, error: -.971)&lt;/body&gt;&lt;/html&gt;</pre></div>

Harmony of 23edo: Difference between revisions

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