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| <h2>IMPORTED REVISION FROM WIKISPACES</h2>
| | A '''saddle chord''' is a chord that represents a ''saddle point'' in the [[harmonic entropy]] surface, rather than a local minimum or maximum. Because saddle points only occur in two-dimensional or higher surfaces, a saddle chord cannot be a dyad (since the harmonic entropy graph for dyads is a one-dimensional curve). It must be a triad, tetrad or higher. |
| This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
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| : This revision was by author [[User:MasonGreen1|MasonGreen1]] and made on <tt>2015-11-29 13:34:45 UTC</tt>.<br>
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| : The original revision id was <tt>568156139</tt>.<br>
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| : The revision comment was: <tt></tt><br>
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| The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br>
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| <h4>Original Wikitext content:</h4>
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| <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">A **saddle chord** is a chord that represents a //saddle point// in the [[harmonic entropy]] surface, rather than a local minimum or maximum. Because saddle points only occur in two-dimensional or higher surfaces, a saddle chord cannot be a dyad (since the harmonic entropy graph for dyads is a one-dimensional curve). It must be a triad, tetrad or higher.
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| Chords <span style="line-height: 1.5;">at or near local minima sound "clean" and have a single primary approximation just intonation approximation. For example, the justly intoned major chord 4:5:6 is a local minimum, and its approximation in 12edo is close by.</span> | | Chords <span style="line-height: 1.5;">at or near local minima sound "clean" and have a single primary approximation just intonation approximation. For example, the justly intoned major chord 4:5:6 is a local minimum, and its approximation in 12edo is close by.</span> |
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| In contrast, a chord at or near a local maximum sounds especially "dirty" and discordant. "Dirty" dyads include many [[quarter tone]]-based intervals. There are also dirty triads, for example, the quarter tone triad {0,1,2} with each note a quarter tone apart. | | In contrast, a chord at or near a local maximum sounds especially "dirty" and discordant. "Dirty" dyads include many [[quarter tone]]-based intervals. There are also dirty triads, for example, the quarter tone triad {0,1,2} in [[24edo]] with each note a quarter tone apart. |
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| For triads and higher, though, there are other possibilities, corresponding to [[https://en.wikipedia.org/wiki/Monkey_saddle|monkey saddle]] and horse saddle-type points. These chords do not have a single just approximation but rather are a compromise between multiple ones, and as such their harmonic entropy is not a local minimum (at least not in all directions). They are intermediate between consonant and "dirty" in terms of sounds.</pre></div> | | For triads and higher, though, there are other possibilities, corresponding to [https://en.wikipedia.org/wiki/Monkey_saddle monkey saddle] and horse saddle-type points. These chords do not have a single just approximation but rather are a compromise between multiple ones, and as such their harmonic entropy is not a local minimum (at least not in all directions). They are intermediate between consonant and "dirty" in terms of sounds. |
| <h4>Original HTML content:</h4>
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| <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>Saddle chord</title></head><body>A <strong>saddle chord</strong> is a chord that represents a <em>saddle point</em> in the <a class="wiki_link" href="/harmonic%20entropy">harmonic entropy</a> surface, rather than a local minimum or maximum. Because saddle points only occur in two-dimensional or higher surfaces, a saddle chord cannot be a dyad (since the harmonic entropy graph for dyads is a one-dimensional curve). It must be a triad, tetrad or higher.<br />
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| | {{todo|add examples|inline=1}} |
| Chords <span style="line-height: 1.5;">at or near local minima sound &quot;clean&quot; and have a single primary approximation just intonation approximation. For example, the justly intoned major chord 4:5:6 is a local minimum, and its approximation in 12edo is close by.</span><br />
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| | [[Category:Chord]] |
| In contrast, a chord at or near a local maximum sounds especially &quot;dirty&quot; and discordant. &quot;Dirty&quot; dyads include many <a class="wiki_link" href="/quarter%20tone">quarter tone</a>-based intervals. There are also dirty triads, for example, the quarter tone triad {0,1,2} with each note a quarter tone apart.<br />
| | [[Category:Consonance and dissonance]] |
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| For triads and higher, though, there are other possibilities, corresponding to <a class="wiki_link_ext" href="https://en.wikipedia.org/wiki/Monkey_saddle" rel="nofollow">monkey saddle</a> and horse saddle-type points. These chords do not have a single just approximation but rather are a compromise between multiple ones, and as such their harmonic entropy is not a local minimum (at least not in all directions). They are intermediate between consonant and &quot;dirty&quot; in terms of sounds.</body></html></pre></div>
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A saddle chord is a chord that represents a saddle point in the harmonic entropy surface, rather than a local minimum or maximum. Because saddle points only occur in two-dimensional or higher surfaces, a saddle chord cannot be a dyad (since the harmonic entropy graph for dyads is a one-dimensional curve). It must be a triad, tetrad or higher.
Chords at or near local minima sound "clean" and have a single primary approximation just intonation approximation. For example, the justly intoned major chord 4:5:6 is a local minimum, and its approximation in 12edo is close by.
In contrast, a chord at or near a local maximum sounds especially "dirty" and discordant. "Dirty" dyads include many quarter tone-based intervals. There are also dirty triads, for example, the quarter tone triad {0,1,2} in 24edo with each note a quarter tone apart.
For triads and higher, though, there are other possibilities, corresponding to monkey saddle and horse saddle-type points. These chords do not have a single just approximation but rather are a compromise between multiple ones, and as such their harmonic entropy is not a local minimum (at least not in all directions). They are intermediate between consonant and "dirty" in terms of sounds.
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Todo: add examples
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