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Harmony of 23edo: Difference between revisions

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<h2>IMPORTED REVISION FROM WIKISPACES</h2>
If you take a look at the intervals of [[23edo|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|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):
This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
 
: This revision was by author [[User:keenanpepper|keenanpepper]] and made on <tt>2012-01-22 14:51:51 UTC</tt>.<br>
{| class="wikitable"
: The original revision id was <tt>294270202</tt>.<br>
|-
: The revision comment was: <tt></tt><br>
| | Degrees
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br>
| | Armodue note
<h4>Original Wikitext content:</h4>
| | Cents sizes
<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):
| | [[nearest_harmonic|Nearest Harmonic]]
|| Degrees || Armodue note || Cents sizes || [[nearest harmonic|Nearest Harmonic]] || Cents || "Error" ||
| | Cents
|| 0 || 1 || 0 || 1/1 || 0.000 || none ||
| | "Error"
|| **1** || **1t (2b)** || **52.174** || **33/32** || **53.273** || **-1.099** ||
|-
|| **2** || **2v (1#)** || **104.348** || **17/16** || **104.955** || **-0.607** ||
| | 0
|| **3** || **2** || **156.522** || **35/32** || **155.140** || **+1.382** ||
| | 1
|| **4·** || **2t (3b)** || **208.696** || **9/8** || **203.910** || **+4.786** ||
| | 0
|| 5 || 3v (2#) || 260.869 || 50/43 || 261.110 || -0.241 ||
| | 1/1
|| 6 || 3 || 313.043 || [[6_5|6/5]] || 315.641 || -2.598 ||
| | 0.000
|| **7·** || **3t (4b)** || **365.217** || **79/64** || **364.537** || **+0.68** ||
| | none
|| 8 || 4v (3#) || 417.391 || [[14_11|14/11]] || 417.508 || -0.117 ||
|-
|| **9** || **4 (5v)** || **469.565** || **21/16** || **470.781** || **-1.216** ||
| | '''1'''
|| 10**·** || 5 (4t) || 521.739 || 23/17 || 523.319 || -1.58 ||
| | '''1t (2b)'''
|| 11 || 5t (6b) || 573.913 || 32/23 || 571.726 || +2.187 ||
| | '''52.174'''
|| **12** || **6v (5#)** || **626.087** || **23/16** || **628.274** || **-2.187** ||
| | '''33/32'''
|| 13**·** || 6 || 678.261 || 34/23 || 676.681 || +1.58 ||
| | '''53.273'''
|| 14 || 6t (7b) || 730.435 || 32/21 || 729.219 || +1.216 ||
| | '''-1.099'''
|| 15 || 7v (6#) || 782.609 || [[11_7|11/7]] || 782.492 || +0.117 ||
|-
|| 16**·** || 7 || 834.783 || 34/21 || 834.175 || +0.608 ||
| | '''2'''
|| 17 || 7t (8b) || 886.957 || [[5_3|5/3]] || 884.359 || +2.598 ||
| | '''2v (1#)'''
|| **18** || **8v (7#)** || **939.130** || **55/32** || **937.632** || **+1.498** ||
| | '''104.348'''
|| 19**·** || 8 || 991.304 || 39/22 || 991.165 || +0.139 ||
| | '''17/16'''
|| **20** || **8t (9b)** || **1043.478** || **117/64** || **1044.438** || **-0.96** ||
| | '''104.955'''
|| 21 || 9v (8#) || 1095.652 || 32/17 || 1095.045 || +0.607 ||
| | '''-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** ||
| | '''3'''
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.
| | '''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, &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.


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, 35, 55, 79, &amp; 117. Those sonorities could no doubt prove useful to a thoughful composer, but for this study, I'll leave them out.
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, 35, 55, 79, &amp; 117. Those sonorities could no doubt prove useful to a thoughful composer, but for this study, I'll leave them out.
Line 38: Line 184:
Thus we produce ten triads, five tetrads, &amp; one pentad, 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.
Thus we produce ten triads, five tetrads, &amp; one pentad, 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.


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


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


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


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


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


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


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


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


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


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


==Tetrads==
'''16:17:18:21, degrees 0, 2, 4, 9 (inversion 0, 14, 19, 21)'''.


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


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


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


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


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


==Pentad==
'''16:17:18:21:23, degrees 0, 2, 4, 9, 12 (inversion 0, 11, 14, 19, 21)'''.


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


 
23/21 (157.493, error: -0.971)
&lt;table class="wiki_table"&gt;
[[Category:23edo]]
    &lt;tr&gt;
[[Category:edo]]
        &lt;td&gt;Degrees&lt;br /&gt;
[[Category:harmony]]
&lt;/td&gt;
[[Category:interval]]
        &lt;td&gt;Armodue note&lt;br /&gt;
[[Category:pentad]]
&lt;/td&gt;
[[Category:tetrad]]
        &lt;td&gt;Cents sizes&lt;br /&gt;
[[Category:theory]]
&lt;/td&gt;
[[Category:triad]]
        &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;
        &lt;td&gt;&amp;quot;Error&amp;quot;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;0&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;1&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;0&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;1/1&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;0.000&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;none&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;&lt;strong&gt;1&lt;/strong&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;strong&gt;1t (2b)&lt;/strong&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;strong&gt;52.174&lt;/strong&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;strong&gt;33/32&lt;/strong&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;strong&gt;53.273&lt;/strong&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;strong&gt;-1.099&lt;/strong&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;&lt;strong&gt;2&lt;/strong&gt;&lt;br /&gt;
&lt;/td&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;
&lt;/td&gt;
        &lt;td&gt;&lt;strong&gt;17/16&lt;/strong&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;strong&gt;104.955&lt;/strong&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;strong&gt;-0.607&lt;/strong&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;&lt;strong&gt;3&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;td&gt;&lt;strong&gt;156.522&lt;/strong&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;strong&gt;35/32&lt;/strong&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;strong&gt;155.140&lt;/strong&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;strong&gt;+1.382&lt;/strong&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;&lt;strong&gt;4·&lt;/strong&gt;&lt;br /&gt;
&lt;/td&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;
&lt;/td&gt;
        &lt;td&gt;&lt;strong&gt;9/8&lt;/strong&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;strong&gt;203.910&lt;/strong&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;strong&gt;+4.786&lt;/strong&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;5&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;3v (2#)&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;260.869&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;50/43&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;261.110&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;-0.241&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;6&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;3&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;313.043&lt;br /&gt;
&lt;/td&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;315.641&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;-2.598&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;&lt;strong&gt;7·&lt;/strong&gt;&lt;br /&gt;
&lt;/td&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;79/64&lt;/strong&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;strong&gt;364.537&lt;/strong&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;strong&gt;+0.68&lt;/strong&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;8&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;4v (3#)&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;417.391&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;a class="wiki_link" href="/14_11"&gt;14/11&lt;/a&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;417.508&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;-0.117&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;&lt;strong&gt;9&lt;/strong&gt;&lt;br /&gt;
&lt;/td&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;
&lt;/td&gt;
        &lt;td&gt;&lt;strong&gt;21/16&lt;/strong&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;strong&gt;470.781&lt;/strong&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;strong&gt;-1.216&lt;/strong&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;10&lt;strong&gt;·&lt;/strong&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;5 (4t)&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;521.739&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;23/17&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;523.319&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;-1.58&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;11&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;5t (6b)&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;573.913&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;32/23&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;571.726&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;+2.187&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;&lt;strong&gt;12&lt;/strong&gt;&lt;br /&gt;
&lt;/td&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;
&lt;/td&gt;
        &lt;td&gt;&lt;strong&gt;23/16&lt;/strong&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;strong&gt;628.274&lt;/strong&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;strong&gt;-2.187&lt;/strong&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;13&lt;strong&gt;·&lt;/strong&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;6&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;678.261&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;34/23&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;676.681&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;+1.58&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;14&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;6t (7b)&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;730.435&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;32/21&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;729.219&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;+1.216&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;15&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;7v (6#)&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;782.609&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;a class="wiki_link" href="/11_7"&gt;11/7&lt;/a&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;782.492&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;+0.117&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;16&lt;strong&gt;·&lt;/strong&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;7&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;834.783&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;34/21&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;834.175&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;+0.608&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;17&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;7t (8b)&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;886.957&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;a class="wiki_link" href="/5_3"&gt;5/3&lt;/a&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;884.359&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;+2.598&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;&lt;strong&gt;18&lt;/strong&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;strong&gt;8v (7#)&lt;/strong&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;strong&gt;939.130&lt;/strong&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;strong&gt;55/32&lt;/strong&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;strong&gt;937.632&lt;/strong&gt;&lt;br /&gt;
&lt;/td&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;19&lt;strong&gt;·&lt;/strong&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;8&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;991.304&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;39/22&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;991.165&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;+0.139&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;&lt;strong&gt;20&lt;/strong&gt;&lt;br /&gt;
&lt;/td&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;
&lt;/td&gt;
        &lt;td&gt;&lt;strong&gt;117/64&lt;/strong&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;strong&gt;1044.438&lt;/strong&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;strong&gt;-0.96&lt;/strong&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;21&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;9v (8#)&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;1095.652&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;32/17&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;1095.045&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;+0.607&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;22&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;9 (1v)&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;1147.826&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;64/33&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;1146.727&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;+1.099&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &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 (9t)&lt;/strong&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;strong&gt;1200.000&lt;/strong&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;strong&gt;2/1&lt;/strong&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;strong&gt;1200.000&lt;/strong&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;strong&gt;none&lt;/strong&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
&lt;/table&gt;
 
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;amp; 55/32; &amp;amp; 1 cent of 17/16, 79/64, &amp;amp; 117/64. (&amp;lt;&lt;strong&gt;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&lt;/strong&gt; -- AKJ) Of course, it also has perfect unisons &amp;amp; octaves, by definition. This means we could potentially build a very strange (&amp;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;amp; tetrads, as a starting point.&lt;br /&gt;
&lt;br /&gt;
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;amp; 23, excluding (at least for now) 33, 35, 55, 79, &amp;amp; 117. Those sonorities could no doubt prove useful to a thoughful composer, but for this study, I'll leave them out.&lt;br /&gt;
&lt;br /&gt;
Thus we produce ten triads, five tetrads, &amp;amp; one pentad, 16 chords, which, with their inversions (given), doubles to 32 chords. I've written then in a closed position (within one octave), &amp;amp; I recommend trying different voicings. Moving chord tones up &amp;amp; down by octaves, you can unmuddy a muddy chord.&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:0:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc0"&gt;&lt;a name="x-Triads"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:0 --&gt;Triads&lt;/h2&gt;
&lt;strong&gt;16:17:18, degrees 0, 2, 4 (inversion 0, 19, 21)&lt;/strong&gt;.&lt;br /&gt;
17/16 (104.955, error -0.607)&lt;br /&gt;
18/16 = 9/8 (203.910, error +4.786)&lt;br /&gt;
18/17 (98.955, error: +5.393)&lt;br /&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-16:17:21, 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;
17/16 (104.955, error -0.607)&lt;br /&gt;
21/16 (470.781, error -1.216)&lt;br /&gt;
21/17 (365.825, error: -0.608)&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-16:17:23, 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;
17/16 (104.955, error -0.607)&lt;br /&gt;
23/16 (628.274, error -2.187)&lt;br /&gt;
23/17 (523.319, error: -1.578)&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-16:18:21, 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;
18/16 = 9/8 (203.910, error +4.786)&lt;br /&gt;
21/16 (470.781, error -1.216)&lt;br /&gt;
21/18 = 7/6 (266.871, error: -6.001)&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-16:18:23, 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;
18/16 = 9/8 (203.910, error +4.786)&lt;br /&gt;
23/16 (628.274, error -2.187)&lt;br /&gt;
23/18 (424.364, error: -6.973)&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-16:21:23, 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;
21/16 (470.781, error -1.216)&lt;br /&gt;
23/16 (628.274, error -2.187)&lt;br /&gt;
23/21 (157.493, error: -0.971)&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-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;
18/17 (98.955, error: +5.393)&lt;br /&gt;
21/17 (365.825, error: -0.608)&lt;br /&gt;
21/18 = 7/6 (266.871, error: -6.001)&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-17:18:23, 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;
18/17 (98.955, error: +5.393)&lt;br /&gt;
23/17 (523.319, error: -1.578)&lt;br /&gt;
23/18 (424.364, error: -6.973)&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-17:21:23, 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;
21/17 (365.825, error: -0.608)&lt;br /&gt;
23/17 (523.319, error: -1.578)&lt;br /&gt;
23/21 (157.493, error: -0.971)&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-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;
21/18 = 7/6 (266.871, error: -6.001)&lt;br /&gt;
23/18 (424.364, error: -6.973)&lt;br /&gt;
23/21 (157.493, error: -0.971)&lt;br /&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;Tetrads&lt;/h2&gt;
&lt;strong&gt;16:17:18:21, degrees 0, 2, 4, 9 (inversion 0, 14, 19, 21)&lt;/strong&gt;.&lt;br /&gt;
17/16 (104.955, error -0.607)&lt;br /&gt;
18/16 = 9/8 (203.910, error +4.786)&lt;br /&gt;
21/16 (470.781, error -1.216)&lt;br /&gt;
18/17 (98.955, error: +5.393)&lt;br /&gt;
21/17 (365.825, error: -0.608)&lt;br /&gt;
21/18 = 7/6 (266.871, error: -6.001)&lt;br /&gt;
&lt;br /&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;
17/16 (104.955, error -0.607)&lt;br /&gt;
18/16 = 9/8 (203.910, error +4.786)&lt;br /&gt;
23/16 (628.274, error -2.187)&lt;br /&gt;
18/17 (98.955, error: +5.393)&lt;br /&gt;
23/17 (523.319, error: -1.578)&lt;br /&gt;
23/18 (424.364, error: -6.973)&lt;br /&gt;
&lt;br /&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;
17/16 (104.955, error -0.607)&lt;br /&gt;
21/16 (470.781, error -1.216)&lt;br /&gt;
23/16 (628.274, error -2.187)&lt;br /&gt;
21/17 (365.825, error: -0.608)&lt;br /&gt;
23/17 (523.319, error: -1.578)&lt;br /&gt;
23/21 (157.493, error: -0.971)&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-16:18:21:23, 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;
18/16 = 9/8 (203.910, error +4.786)&lt;br /&gt;
21/16 (470.781, error -1.216)&lt;br /&gt;
23/16 (628.274, error -2.187)&lt;br /&gt;
21/18 = 7/6 (266.871, error: -6.001)&lt;br /&gt;
23/18 (424.364, error: -6.973)&lt;br /&gt;
23/21 (157.493, error: -0.971)&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-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;
18/17 (98.955, error: +5.393)&lt;br /&gt;
21/17 (365.825, error: -0.608)&lt;br /&gt;
23/17 (523.319, error: -1.578)&lt;br /&gt;
21/18 = 7/6 (266.871, error: -6.001)&lt;br /&gt;
23/18 (424.364, error: -6.973)&lt;br /&gt;
23/21 (157.493, error: -0.971)&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:30:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc15"&gt;&lt;a name="x-Pentad"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:30 --&gt;Pentad&lt;/h2&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;
17/16 (104.955, error -0.607)&lt;br /&gt;
18/16 = 9/8 (203.910, error +4.786)&lt;br /&gt;
21/16 (470.781, error -1.216)&lt;br /&gt;
23/16 (628.274, error -2.187)&lt;br /&gt;
18/17 (98.955, error: +5.393)&lt;br /&gt;
21/17 (365.825, error: -0.608)&lt;br /&gt;
23/17 (523.319, error: -1.578)&lt;br /&gt;
21/18 = 7/6 (266.871, error: -6.001)&lt;br /&gt;
23/18 (424.364, error: -6.973)&lt;br /&gt;
23/21 (157.493, error: -0.971)&lt;/body&gt;&lt;/html&gt;</pre></div>
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