Harmony of 23edo: Difference between revisions

Line 1:

<h2>IMPORTED REVISION FROM WIKISPACES</h2>

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: This revision was by author [[User:Andrew_Heathwaite|Andrew_Heathwaite]] and made on <tt>2009-06-~~25 19~~:42:48 UTC</tt>.<br>

: This revision was by author [[User:Andrew_Heathwaite|Andrew_Heathwaite]] and made on <tt>2009-06-30 13:06:54 UTC</tt>.<br>

: The original revision id was <tt>~~79464781~~</tt>.<br>

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Line 18:

|| 18 || 939.130 || 55/32 || 937.632 || +1.498 ||

|| 20 || 1043.478 || 117/64 || 1044.438 || -.960 ||

|| 23 (or 0) || 1200.000 || 2/1 || 1200.000 || none ||

|| **23 (or 0)** || **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, & 55/32; & 1 cent of 17/16, 79/64, & 117/64. 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 64:66:68:72:79:84:92:110:117:132. I find this cluster a little hard to listen to on its own, whether tuned to JI or 23edo, so I'd like to consider smaller chords, triads & tetrads, as a starting point.

Line 24:

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.

~~The 9th harmonic~~, ~~being the most out-of-tune as compared with Just Intonation (off by nearly 5 cents) may sometimes add a roughness to these chords that some would consider undesirable. That said~~, ~~I'll leave it in for now, but I'll remain alert to the affect of the 9~~.

Thus we produce ten triads, five tetrads, & one quintad.

So let's look at some chords. I'll show them first reduced to within one octave, then I'll suggest some other voicings. The notation 2^ will mean the second degree of 23edo, raised one octave; 2v will mean the second degree of 23edo, lowered one octave.

==Triads==

===16:17:18, degrees 0, 2, 4.===

~~Inner interval:~~

17/16 (104.955, error -.607)

9/8 (203.910, error +4.786)

18/17 (98.955, error: +5.393)

This chord represents a tight cluster, like C, C#, D on the piano. Spreading out the voicing would help make it more palatable, but not much. Whenever degrees 0 & 2 appear together, if you want a more harmonious sound, I'd recommend raising degree 2 one or more octaves. 17/8 sounds much smoother to me than 17/16. In this case, you could put 0 in the first octave, 4 in the second octave, & 2 in the third octave, thus reflecting the overtone series a little better, if it suits you.

===16:17:21, degrees 0, 2, 9.===

~~Inner interval:~~

17/16 (104.955, error -.607)

21/16 (470.781, error -1.216)

21/17 (365.825, error: -.608)

I like how this chord sounds with the 17/16 raised one octave. It doesn't hurt to raise the 21/16 as well. The 21/17 interval is very well in tune, & unfamiliar to our ears. 367 cents sometimes sounds a bit like a flat major third, & in a context like this, I think it sounds rather pleasant.

===16:17:23, degrees 0, 2, 12.===

~~Inner interval:~~

17/16 (104.955, error -.607)

23/16 (628.274, error -2.187)

23/17 (523.319, error: -1.578)

~~Again, I recommend raising 17/16 & perhaps 23/16. That 522 cent interval, which sounds so often like a bad 4/3, fits nicely in a chord like this.~~

===16:18:21, degrees 0, 4, 9.===

~~Inner interval:~~

9/8 (203.910, error +4.786)

21/16 (470.781, error -1.216)

21/18 = 7/6 (266.871, error: -6.001)

That 9/8 introduces a mistuning, as it always will in 23edo, but not a very serious one. I find the 261 cent interval a very convincing 7/6, & one of the points of harmonic rest in this scale. You could try putting degree 4 up an octave (making it reflect 9/4) to give the chord some room to breathe.

===16:18:23, degrees 0, 4, 12.===

~~Inner interval:~~

9/8 (203.910, error +4.786)

23/16 (628.274, error -2.187)

23/18 (424.364, error: -6.973)

~~Here, the errors compound & create an almost 7-cent mistuning in 23/18. The chord can still work very well, though, especially with degree 4 raised an octave.~~

===16:21:23, degrees 0, 9, 12.===

~~Inner interval:~~

21/16 (470.781, error -1.216)

23/16 (628.274, error -2.187)

23/21 (157.493, error: -.971)

~~With no~~ 9/8, ~~this chord contains very little~~ error ~~from just~~. ~~That~~ 157 ~~cent interval can sound pretty harsh~~, ~~however~~, ~~even if it happens to fit~~ 23/21 ~~pretty well~~. ~~After all~~, 23/21 ~~is pretty out there~~. ~~You could soften it with octave separation~~, ~~if you like~~. ~~Of course~~, ~~it makes a very nice scale step~~. </pre></div>

===17:18:21, degrees 0, 2, 7.===

18/17 (98.955, error: +5.393)

21/17 (365.825, error: -.608)

21/18 = 7/6 (266.871, error: -6.001)

===17:18:23, degrees 0, 2, 10.===

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.===

21/17 (365.825, error: -.608)

23/17 (523.319, error: -1.578)

23/21 (157.493, error: -.971)

===18:21:23, degrees 0, 5, 8.===

21/18 = 7/6 (266.871, error: -6.001)

23/18 (424.364, error: -6.973)

23/21 (157.493, error: -.971)

==Tetrads==

===16:17:18:21, degrees 0, 2, 4, 9.===

17/16 (104.955, error -.607)

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: -.608)

21/18 = 7/6 (266.871, error: -6.001)

===16:17:18:23, degrees 0, 2, 4, 12.===

17/16 (104.955, error -.607)

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.===

17/16 (104.955, error -.607)

21/16 (470.781, error -1.216)

23/16 (628.274, error -2.187)

21/17 (365.825, error: -.608)

23/17 (523.319, error: -1.578)

23/21 (157.493, error: -.971)

===16:18:21:23, degrees 0, 4, 9, 12.===

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: -.971)

===17:18:21:23, degrees 0, 2, 7, 10.===

==Quintad==

===16:17:18:21:23, degrees 0, 2, 4, 9, 12.===

17/16 (104.955, error -.607)

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: -.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: -.971)

23/21 (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"><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 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;nonharmonic,&quot; I'd like to consider 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 harmonics 9, 17, 21, 23, 33, 55, 79, &amp; 117. Let's compare the cents values to see how close 23edo intervals come to these harmonics.<br />

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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 />

~~The 9th harmonic~~, ~~being the most out~~-of-~~tune as compared with Just Intonation~~ (~~off by nearly~~ 5 ~~cents~~) ~~may sometimes add~~ a ~~roughness to these chords that some would consider undesirable~~. ~~That said~~, ~~I'll leave it in for now~~, ~~but I'll remain alert to the affect of the~~ 9.<br />

Thus we produce ten triads, five tetrads, &amp; one quintad.<br />

<br />

<h2 id="toc0"><a name="x-Triads"></a>Triads</h2>

<br />

<h3 id="toc1"><a name="x-Triads-16:17:18, degrees 0, 2, 4."></a>16:17:18, degrees 0, 2, 4.</h3>

17/16 (104.955, error -.607)<br />

9/8 (203.910, error +4.786)<br />

18/17 (98.955, error: +5.393)<br />

<br />

<h3 id="toc2"><a name="x-Triads-16:17:21, degrees 0, 2, 9."></a>16:17:21, degrees 0, 2, 9.</h3>

17/16 (104.955, error -.607)<br />

21/16 (470.781, error -1.216)<br />

21/17 (365.825, error: -.608)<br />

<br />

<h3 id="toc3"><a name="x-Triads-16:17:23, degrees 0, 2, 12."></a>16:17:23, degrees 0, 2, 12.</h3>

17/16 (104.955, error -.607)<br />

23/16 (628.274, error -2.187)<br />

23/17 (523.319, error: -1.578)<br />

<br />

<h3 id="toc4"><a name="x-Triads-16:18:21, degrees 0, 4, 9."></a>16:18:21, degrees 0, 4, 9.</h3>

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 />

~~So let's look at some chords~~. ~~I'll show them first reduced to within one octave~~, ~~then I'll suggest some other voicings~~. ~~The notation~~ 2~~^ will mean the second degree of 23edo, raised one octave~~; ~~2v will mean the second degree of 23edo~~, ~~lowered one octave~~.<br />

<h3 id="toc5"><a name="x-Triads-16:18:23, degrees 0, 4, 12."></a>16:18:23, degrees 0, 4, 12.</h3>

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 />

<h3 id="~~toc0~~"><a name="x--16:17:18, degrees 0, 2, 4."></a>16:17:18, degrees 0, 2, 4.</h3>

<h3 id="toc6"><a name="x-Triads-16:21:23, degrees 0, 9, 12."></a>16:21:23, degrees 0, 9, 12.</h3>

~~Inner interval:~~<br />

21/16 (470.781, error -1.216)<br />

18/17 (98.~~955~~, error~~: +5~~.~~393~~)<br />

23/16 (628.274, error -2.187)<br />

This chord represents a tight cluster, like C, C#, D on the piano. Spreading out the voicing would help make it more palatable, but not much. Whenever degrees 0 &amp; 2 appear together, if you want a more harmonious sound, I'd recommend raising degree 2 one or more octaves. 17/~~8 sounds much smoother to me than 17/16~~. ~~In this case~~, ~~you could put 0 in the first octave, 4 in the second octave, &amp; 2 in the third octave, thus reflecting the overtone series a little better, if it suits you~~.<br />

23/21 (157.493, error: -.971)<br />

<br />

<h3 id="~~toc1~~"><a name="x--16:17:21, degrees 0, 2, 9."></a>16:17:21, degrees 0, 2, 9.</h3>

<h3 id="toc7"><a name="x-Triads-17:18:21, degrees 0, 2, 7."></a>17:18:21, degrees 0, 2, 7.</h3>

~~Inner interval~~:<br />

18/17 (98.955, error: +5.393)<br />

21/17 (365.825, error: -.608)<br />

~~I like how this chord sounds with the 17/16 raised one octave. It doesn't hurt to raise the~~ 21/~~16 as well. The 21~~/~~17 interval is very well in tune, &amp; unfamiliar to our ears~~. ~~367 cents sometimes sounds a bit like a flat major third, &amp; in a context like this~~, ~~I think it sounds rather pleasant~~.<br />

21/18 = 7/6 (266.871, error: -6.001)<br />

<br />

<h3 id="~~toc2~~"><a name="x--16:17:23, degrees 0, 2, 12."></a>16:17:23, degrees 0, 2, 12.</h3>

<h3 id="toc8"><a name="x-Triads-17:18:23, degrees 0, 2, 10."></a>17:18:23, degrees 0, 2, 10.</h3>

~~Inner interval~~:<br />

18/17 (98.955, error: +5.393)<br />

23/17 (523.319, error: -1.578)<br />

~~Again~~, ~~I recommend raising~~ 17/16 &amp; ~~perhaps~~ 23/16. ~~That 522 cent interval~~, ~~which sounds so often like a bad 4~~/3, ~~fits nicely in a chord like this~~.<br />

23/18 (424.364, error: -6.973)<br />

<br />

<h3 id="toc9"><a name="x-Triads-17:21:23, degrees 0, 7, 10."></a>17:21:23, degrees 0, 7, 10.</h3>

21/17 (365.825, error: -.608)<br />

23/17 (523.319, error: -1.578)<br />

23/21 (157.493, error: -.971)<br />

<br />

<h3 id="toc10"><a name="x-Triads-18:21:23, degrees 0, 5, 8."></a>18:21:23, degrees 0, 5, 8.</h3>

21/18 = 7/6 (266.871, error: -6.001)<br />

23/18 (424.364, error: -6.973)<br />

23/21 (157.493, error: -.971)<br />

<br />

<h3 id="~~toc3~~"><a name="x--16:18:21, degrees 0, 4, 9."></a>16:18:21, degrees 0, 4, 9.</h3>

<h2 id="toc11"><a name="x-Tetrads"></a>Tetrads</h2>

~~Inner interval~~:<br />

<br />

<h3 id="toc12"><a name="x-Tetrads-16:17:18:21, degrees 0, 2, 4, 9."></a>16:17:18:21, degrees 0, 2, 4, 9.</h3>

17/16 (104.955, error -.607)<br />

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: -.608)<br />

21/18 = 7/6 (266.871, error: -6.001)<br />

That 9/8 introduces a mistuning, as it always will in 23edo, but not a very serious one. I find the 261 cent interval a very convincing 7/6, &amp; one of the points of harmonic rest in this scale. You could try putting degree 4 up an octave (making it reflect 9/4) to give the chord some room to breathe.<br />

<br />

<h3 id="~~toc4~~"><a name="x--16:18:23, degrees 0, 4, 12."></a>16:18:23, degrees 0, 4, 12.</h3>

<h3 id="toc13"><a name="x-Tetrads-16:17:18:23, degrees 0, 2, 4, 12."></a>16:17:18:23, degrees 0, 2, 4, 12.</h3>

~~Inner interval~~:<br />

17/16 (104.955, error -.607)<br />

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 />

<h3 id="toc14"><a name="x-Tetrads-16:17:21:23, degrees 0, 2, 9, 12."></a>16:17:21:23, degrees 0, 2, 9, 12.</h3>

17/16 (104.955, error -.607)<br />

21/16 (470.781, error -1.216)<br />

23/16 (628.274, error -2.187)<br />

21/17 (365.825, error: -.608)<br />

23/17 (523.319, error: -1.578)<br />

23/21 (157.493, error: -.971)<br />

<br />

<h3 id="toc15"><a name="x-Tetrads-16:18:21:23, degrees 0, 4, 9, 12."></a>16:18:21:23, degrees 0, 4, 9, 12.</h3>

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 />

~~Here~~, ~~the errors compound~~ &amp; ~~create an almost 7~~-~~cent mistuning in~~ 23/18~~. The chord can still work very well~~, ~~though~~, ~~especially with degree 4 raised an octave~~.<br />

23/21 (157.493, error: -.971)<br />

<br />

<h3 id="toc16"><a name="x-Tetrads-17:18:21:23, degrees 0, 2, 7, 10."></a>17:18:21:23, degrees 0, 2, 7, 10.</h3>

<br />

<h3 id="~~toc5~~"><a name="x--16:21:23, degrees 0, 9, 12."></a>16:21:23, degrees 0, 9, 12.</h3>

<h2 id="toc17"><a name="x-Quintad"></a>Quintad</h2>

~~Inner interval~~:<br />

<br />

<h3 id="toc18"><a name="x-Quintad-16:17:18:21:23, degrees 0, 2, 4, 9, 12."></a>16:17:18:21:23, degrees 0, 2, 4, 9, 12.</h3>

17/16 (104.955, error -.607)<br />

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: -.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: -.971)<br />

~~With no 9/8, this chord contains very little error from just. That 157 cent interval can sound pretty harsh, however, even if it happens to fit~~ 23/21 ~~pretty well~~. ~~After all~~, ~~23/21 is pretty out there. You could soften it with octave separation, if you like. Of course, it makes a very nice scale step~~.</body></html></pre></div>

23/21 (157.493, error: -.971)</body></html></pre></div>

@@ 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:Andrew_Heathwaite|Andrew_Heathwaite]] and made on <tt>2009-06-25 19:42:48 UTC</tt>.<br>
+: This revision was by author [[User:Andrew_Heathwaite|Andrew_Heathwaite]] and made on <tt>2009-06-30 13:06:54 UTC</tt>.<br>
-: The original revision id was <tt>79464781</tt>.<br>
+: The original revision id was <tt>79928089</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>
@@ Line 18: / Line 18: @@
 || 18 || 939.130 || 55/32 || 937.632 || +1.498 ||
 || 20 || 1043.478 || 117/64 || 1044.438 || -.960 ||
-|| 23 (or 0) || 1200.000 || 2/1 || 1200.000 || none ||
+|| **23 (or 0)** || **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, &amp; 55/32; &amp; 1 cent of 17/16, 79/64, &amp; 117/64. 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 64:66:68:72:79:84:92:110:117:132. I find this cluster a little hard to listen to on its own, whether tuned to JI or 23edo, so I'd like to consider smaller chords, triads &amp; tetrads, as a starting point.
@@ Line 24: / Line 24: @@
 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.
-The 9th harmonic, being the most out-of-tune as compared with Just Intonation (off by nearly 5 cents) may sometimes add a roughness to these chords that some would consider undesirable. That said, I'll leave it in for now, but I'll remain alert to the affect of the 9.
+Thus we produce ten triads, five tetrads, &amp; one quintad.
-So let's look at some chords. I'll show them first reduced to within one octave, then I'll suggest some other voicings. The notation 2^ will mean the second degree of 23edo, raised one octave; 2v will mean the second degree of 23edo, lowered one octave.
+==Triads==
 ===16:17:18, degrees 0, 2, 4.===
-Inner interval:
+/16 (104.955, error -.607)
+/8 (203.910, error +4.786)
 /17 (98.955, error: +5.393)
-This chord represents a tight cluster, like C, C#, D on the piano. Spreading out the voicing would help make it more palatable, but not much. Whenever degrees 0 &amp; 2 appear together, if you want a more harmonious sound, I'd recommend raising degree 2 one or more octaves. 17/8 sounds much smoother to me than 17/16. In this case, you could put 0 in the first octave, 4 in the second octave, &amp; 2 in the third octave, thus reflecting the overtone series a little better, if it suits you.
 ===16:17:21, degrees 0, 2, 9.===
-Inner interval:
+/16 (104.955, error -.607)
+/16 (470.781, error -1.216)
 /17 (365.825, error: -.608)
-I like how this chord sounds with the 17/16 raised one octave. It doesn't hurt to raise the 21/16 as well. The 21/17 interval is very well in tune, &amp; unfamiliar to our ears. 367 cents sometimes sounds a bit like a flat major third, &amp; in a context like this, I think it sounds rather pleasant.
 ===16:17:23, degrees 0, 2, 12.===
-Inner interval:
+/16 (104.955, error -.607)
+/16 (628.274, error -2.187)
 /17 (523.319, error: -1.578)
-Again, I recommend raising 17/16 &amp; perhaps 23/16. That 522 cent interval, which sounds so often like a bad 4/3, fits nicely in a chord like this.
 ===16:18:21, degrees 0, 4, 9.===
-Inner interval:
+/8 (203.910, error +4.786)
+/16 (470.781, error -1.216)
 /18 = 7/6 (266.871, error: -6.001)
-That 9/8 introduces a mistuning, as it always will in 23edo, but not a very serious one. I find the 261 cent interval a very convincing 7/6, &amp; one of the points of harmonic rest in this scale. You could try putting degree 4 up an octave (making it reflect 9/4) to give the chord some room to breathe.
 ===16:18:23, degrees 0, 4, 12.===
-Inner interval:
+/8 (203.910, error +4.786)
+/16 (628.274, error -2.187)
 /18 (424.364, error: -6.973)
-Here, the errors compound &amp; create an almost 7-cent mistuning in 23/18. The chord can still work very well, though, especially with degree 4 raised an octave.
 ===16:21:23, degrees 0, 9, 12.===
-Inner interval:
+/16 (470.781, error -1.216)
+/16 (628.274, error -2.187)
 /21 (157.493, error: -.971)
-With no 9/8, this chord contains very little error from just. That 157 cent interval can sound pretty harsh, however, even if it happens to fit 23/21 pretty well. After all, 23/21 is pretty out there. You could soften it with octave separation, if you like. Of course, it makes a very nice scale step. </pre></div>
+===17:18:21, degrees 0, 2, 7.===
+/17 (98.955, error: +5.393)
+/17 (365.825, error: -.608)
+/18 = 7/6 (266.871, error: -6.001)
+===17:18:23, degrees 0, 2, 10.===
+/17 (98.955, error: +5.393)
+/17 (523.319, error: -1.578)
+/18 (424.364, error: -6.973)
+===17:21:23, degrees 0, 7, 10.===
+/17 (365.825, error: -.608)
+/17 (523.319, error: -1.578)
+/21 (157.493, error: -.971)
+===18:21:23, degrees 0, 5, 8.===
+/18 = 7/6 (266.871, error: -6.001)
+/18 (424.364, error: -6.973)
+/21 (157.493, error: -.971)
+==Tetrads==
+===16:17:18:21, degrees 0, 2, 4, 9.===
+/16 (104.955, error -.607)
+/8 (203.910, error +4.786)
+/16 (470.781, error -1.216)
+/17 (98.955, error: +5.393)
+/17 (365.825, error: -.608)
+/18 = 7/6 (266.871, error: -6.001)
+===16:17:18:23, degrees 0, 2, 4, 12.===
+/16 (104.955, error -.607)
+/8 (203.910, error +4.786)
+/16 (628.274, error -2.187)
+/17 (98.955, error: +5.393)
+/17 (523.319, error: -1.578)
+/18 (424.364, error: -6.973)
+===16:17:21:23, degrees 0, 2, 9, 12.===
+/16 (104.955, error -.607)
+/16 (470.781, error -1.216)
+/16 (628.274, error -2.187)
+/17 (365.825, error: -.608)
+/17 (523.319, error: -1.578)
+/21 (157.493, error: -.971)
+===16:18:21:23, degrees 0, 4, 9, 12.===
+/8 (203.910, error +4.786)
+/16 (470.781, error -1.216)
+/16 (628.274, error -2.187)
+/18 = 7/6 (266.871, error: -6.001)
+/18 (424.364, error: -6.973)
+/21 (157.493, error: -.971)
+===17:18:21:23, degrees 0, 2, 7, 10.===
+==Quintad==
+===16:17:18:21:23, degrees 0, 2, 4, 9, 12.===
+/16 (104.955, error -.607)
+/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 (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: -.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, 55, 79, &amp;amp; 117. Let's compare the cents values to see how close 23edo intervals come to these harmonics.&lt;br /&gt;
@@ Line 184: / Line 256: @@
      &lt;/tr&gt;
      &lt;tr&gt;
-         &lt;td&gt;23 (or 0)&lt;br /&gt;
+         &lt;td&gt;&lt;strong&gt;23 (or 0)&lt;/strong&gt;&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;1200.000&lt;br /&gt;
+         &lt;td&gt;&lt;strong&gt;1200.000&lt;/strong&gt;&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;2/1&lt;br /&gt;
+         &lt;td&gt;&lt;strong&gt;2/1&lt;/strong&gt;&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;1200.000&lt;br /&gt;
+         &lt;td&gt;&lt;strong&gt;1200.000&lt;/strong&gt;&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;none&lt;br /&gt;
+         &lt;td&gt;&lt;strong&gt;none&lt;/strong&gt;&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
@@ Line 202: / Line 274: @@
 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, 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;
-The 9th harmonic, being the most out-of-tune as compared with Just Intonation (off by nearly 5 cents) may sometimes add a roughness to these chords that some would consider undesirable. That said, I'll leave it in for now, but I'll remain alert to the affect of the 9.&lt;br /&gt;
+Thus we produce ten triads, five tetrads, &amp;amp; one quintad.&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;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."&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:2 --&gt;16:17:18, degrees 0, 2, 4.&lt;/h3&gt;
+/16 (104.955, error -.607)&lt;br /&gt;
+/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."&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:4 --&gt;16:17:21, degrees 0, 2, 9.&lt;/h3&gt;
+/16 (104.955, error -.607)&lt;br /&gt;
+/16 (470.781, error -1.216)&lt;br /&gt;
+/17 (365.825, error: -.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."&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:6 --&gt;16:17:23, degrees 0, 2, 12.&lt;/h3&gt;
+/16 (104.955, error -.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."&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:8 --&gt;16:18:21, degrees 0, 4, 9.&lt;/h3&gt;
+/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;
-So let's look at some chords. I'll show them first reduced to within one octave, then I'll suggest some other voicings. The notation 2^ will mean the second degree of 23edo, raised one octave; 2v will mean the second degree of 23edo, lowered one octave.&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."&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:10 --&gt;16:18:23, degrees 0, 4, 12.&lt;/h3&gt;
+/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:0:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc0"&gt;&lt;a name="x--16:17:18, degrees 0, 2, 4."&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:0 --&gt;16:17:18, degrees 0, 2, 4.&lt;/h3&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."&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:12 --&gt;16:21:23, degrees 0, 9, 12.&lt;/h3&gt;
-  Inner interval:&lt;br /&gt;
+/16 (470.781, error -1.216)&lt;br /&gt;
-/17 (98.955, error: +5.393)&lt;br /&gt;
+/16 (628.274, error -2.187)&lt;br /&gt;
-This chord represents a tight cluster, like C, C#, D on the piano. Spreading out the voicing would help make it more palatable, but not much. Whenever degrees 0 &amp;amp; 2 appear together, if you want a more harmonious sound, I'd recommend raising degree 2 one or more octaves. 17/8 sounds much smoother to me than 17/16. In this case, you could put 0 in the first octave, 4 in the second octave, &amp;amp; 2 in the third octave, thus reflecting the overtone series a little better, if it suits you.&lt;br /&gt;
+/21 (157.493, error: -.971)&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--16:17:21, degrees 0, 2, 9."&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:2 --&gt;16:17:21, degrees 0, 2, 9.&lt;/h3&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:14:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc7"&gt;&lt;a name="x-Triads-17:18:21, degrees 0, 2, 7."&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:14 --&gt;17:18:21, degrees 0, 2, 7.&lt;/h3&gt;
-  Inner interval:&lt;br /&gt;
+/17 (98.955, error: +5.393)&lt;br /&gt;
 /17 (365.825, error: -.608)&lt;br /&gt;
-I like how this chord sounds with the 17/16 raised one octave. It doesn't hurt to raise the 21/16 as well. The 21/17 interval is very well in tune, &amp;amp; unfamiliar to our ears. 367 cents sometimes sounds a bit like a flat major third, &amp;amp; in a context like this, I think it sounds rather pleasant.&lt;br /&gt;
+/18 = 7/6 (266.871, error: -6.001)&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--16:17:23, degrees 0, 2, 12."&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:4 --&gt;16:17:23, degrees 0, 2, 12.&lt;/h3&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:16:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc8"&gt;&lt;a name="x-Triads-17:18:23, degrees 0, 2, 10."&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:16 --&gt;17:18:23, degrees 0, 2, 10.&lt;/h3&gt;
-  Inner interval:&lt;br /&gt;
+/17 (98.955, error: +5.393)&lt;br /&gt;
 /17 (523.319, error: -1.578)&lt;br /&gt;
-Again, I recommend raising 17/16 &amp;amp; perhaps 23/16. That 522 cent interval, which sounds so often like a bad 4/3, fits nicely in a chord like this.&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."&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:18 --&gt;17:21:23, degrees 0, 7, 10.&lt;/h3&gt;
+/17 (365.825, error: -.608)&lt;br /&gt;
+/17 (523.319, error: -1.578)&lt;br /&gt;
+/21 (157.493, error: -.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."&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:20 --&gt;18:21:23, degrees 0, 5, 8.&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;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:6:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc3"&gt;&lt;a name="x--16:18:21, degrees 0, 4, 9."&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:6 --&gt;16:18:21, degrees 0, 4, 9.&lt;/h3&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;Tetrads&lt;/h2&gt;
-  Inner interval:&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-16:17:18:21, degrees 0, 2, 4, 9."&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:24 --&gt;16:17:18:21, degrees 0, 2, 4, 9.&lt;/h3&gt;
+/16 (104.955, error -.607)&lt;br /&gt;
+/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;
 /18 = 7/6 (266.871, error: -6.001)&lt;br /&gt;
-That 9/8 introduces a mistuning, as it always will in 23edo, but not a very serious one. I find the 261 cent interval a very convincing 7/6, &amp;amp; one of the points of harmonic rest in this scale. You could try putting degree 4 up an octave (making it reflect 9/4) to give the chord some room to breathe.&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--16:18:23, degrees 0, 4, 12."&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:8 --&gt;16:18:23, degrees 0, 4, 12.&lt;/h3&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:26:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc13"&gt;&lt;a name="x-Tetrads-16:17:18:23, degrees 0, 2, 4, 12."&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:26 --&gt;16:17:18:23, degrees 0, 2, 4, 12.&lt;/h3&gt;
-  Inner interval:&lt;br /&gt;
+/16 (104.955, error -.607)&lt;br /&gt;
+/8 (203.910, error +4.786)&lt;br /&gt;
+/16 (628.274, error -2.187)&lt;br /&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: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."&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:28 --&gt;16:17:21:23, degrees 0, 2, 9, 12.&lt;/h3&gt;
+/16 (104.955, error -.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 (523.319, error: -1.578)&lt;br /&gt;
+/21 (157.493, error: -.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."&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:30 --&gt;16:18:21:23, degrees 0, 4, 9, 12.&lt;/h3&gt;
+/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;
+/18 = 7/6 (266.871, error: -6.001)&lt;br /&gt;
 /18 (424.364, error: -6.973)&lt;br /&gt;
-Here, the errors compound &amp;amp; create an almost 7-cent mistuning in 23/18. The chord can still work very well, though, especially with degree 4 raised an octave.&lt;br /&gt;
+/21 (157.493, error: -.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."&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:32 --&gt;17:18:21:23, degrees 0, 2, 7, 10.&lt;/h3&gt;
+ &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--16:21:23, degrees 0, 9, 12."&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:10 --&gt;16:21:23, degrees 0, 9, 12.&lt;/h3&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;Quintad&lt;/h2&gt;
-  Inner interval:&lt;br /&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."&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:36 --&gt;16:17:18:21:23, degrees 0, 2, 4, 9, 12.&lt;/h3&gt;
+/16 (104.955, error -.607)&lt;br /&gt;
+/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 (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;
-With no 9/8, this chord contains very little error from just. That 157 cent interval can sound pretty harsh, however, even if it happens to fit 23/21 pretty well. After all, 23/21 is pretty out there. You could soften it with octave separation, if you like. Of course, it makes a very nice scale step.&lt;/body&gt;&lt;/html&gt;</pre></div>
+/21 (157.493, error: -.971)&lt;/body&gt;&lt;/html&gt;</pre></div>