EDT: Difference between revisions

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<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:~~Osmiorisbendi~~|~~Osmiorisbendi~~]] and made on <tt>2011-08-~~16 03~~:57:38 UTC</tt>.<br>

: This revision was by author [[User:Kosmorsky|Kosmorsky]] and made on <tt>2011-08-17 22:32:46 UTC</tt>.<br>

: The original revision id was <tt>~~246188973~~</tt>.<br>

: The original revision id was <tt>246623789</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">=Division of the tritave (3/1) into n equal parts=

After the octave (roughly 2:1 but it has been tuned sharp and flat for various reasons), the next simple "frame interval" available is the ratio 3:1. Among other names, the third harmonic has been called the "perfect twelfth" "triple" or "tritave". There has been argument whether pitches a tritave removed can be heard as equivalent, but with proper context and/or experience, at least some people find that they can. Arguably that is the single criterion for calling the tritave a true frame interval. But it is certain that musically valuable organizations of pitch can arise through the equal division of non-octave intervals regardless of equivalence, and either way, the multitude of equal divisions of the tritave are rich and ripe for exploration.

The Bohlen-Pierce (BP) scale seems to have been the first such arrangement to be seriously studied and made into music. As the equivalent harmonics are 1:3:9:27 etc., filling in 3-9 isoharmonically, one arrives at the fundamental consonant triad of BP music - 3:5:7:(9). The MOS that are most naturally formed from these harmonics are of the forms 4L+1s (pentatonic) and 4L+5s (nonatonic). Which brings forward one analogy with diatonic music (3rd and 5th harmonics under octaves) or diatonic function in general: that 4edt and 9edt can be directly compared to 5edo and 7edo (and indeed they sound like they can). In contrast to the state of meantone temperaments, the simplest L=2 s=1 (13edt, the traditional tempered BP scale) is the most accurate and evenly-tempered for a long way. But the formations with Large and small steps of different relative sizes are no less important, for example by being capable of representing other intervals and harmonics.

[[5edt]] (Tritave counterpart of Magic)

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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>edt</title></head><body><h1 id="toc0"><a name="Division of the tritave (3/1) into n equal parts"></a>Division of the tritave (3/1) into n equal parts</h1>

<br />

After the octave (roughly 2:1 but it has been tuned sharp and flat for various reasons), the next simple &quot;frame interval&quot; available is the ratio 3:1. Among other names, the third harmonic has been called the &quot;perfect twelfth&quot; &quot;triple&quot; or &quot;tritave&quot;. There has been argument whether pitches a tritave removed can be heard as equivalent, but with proper context and/or experience, at least some people find that they can. Arguably that is the single criterion for calling the tritave a true frame interval. But it is certain that musically valuable organizations of pitch can arise through the equal division of non-octave intervals regardless of equivalence, and either way, the multitude of equal divisions of the tritave are rich and ripe for exploration.<br />

<br />

The Bohlen-Pierce (BP) scale seems to have been the first such arrangement to be seriously studied and made into music. As the equivalent harmonics are 1:3:9:27 etc., filling in 3-9 isoharmonically, one arrives at the fundamental consonant triad of BP music - 3:5:7:(9). The MOS that are most naturally formed from these harmonics are of the forms 4L+1s (pentatonic) and 4L+5s (nonatonic). Which brings forward one analogy with diatonic music (3rd and 5th harmonics under octaves) or diatonic function in general: that 4edt and 9edt can be directly compared to 5edo and 7edo (and indeed they sound like they can). In contrast to the state of meantone temperaments, the simplest L=2 s=1 (13edt, the traditional tempered BP scale) is the most accurate and evenly-tempered for a long way. But the formations with Large and small steps of different relative sizes are no less important, for example by being capable of representing other intervals and harmonics.<br />

<br />

<a class="wiki_link" href="/5edt">5edt</a> (Tritave counterpart of Magic)<br />

<a class="wiki_link" href="/6edt">6edt</a> (Tritave counterpart of Hanson)<br />

@@ 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:Osmiorisbendi|Osmiorisbendi]] and made on <tt>2011-08-16 03:57:38 UTC</tt>.<br>
+: This revision was by author [[User:Kosmorsky|Kosmorsky]] and made on <tt>2011-08-17 22:32:46 UTC</tt>.<br>
-: The original revision id was <tt>246188973</tt>.<br>
+: The original revision id was <tt>246623789</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">=Division of the tritave (3/1) into n equal parts=
+After the octave (roughly 2:1 but it has been tuned sharp and flat for various reasons), the next simple "frame interval" available is the ratio 3:1. Among other names, the third harmonic has been called the "perfect twelfth" "triple" or "tritave". There has been argument whether pitches a tritave removed can be heard as equivalent, but with proper context and/or experience, at least some people find that they can. Arguably that is the single criterion for calling the tritave a true frame interval. But it is certain that musically valuable organizations of pitch can arise through the equal division of non-octave intervals regardless of equivalence, and either way, the multitude of equal divisions of the tritave are rich and ripe for exploration.
+The Bohlen-Pierce (BP) scale seems to have been the first such arrangement to be seriously studied and made into music. As the equivalent harmonics are 1:3:9:27 etc., filling in 3-9 isoharmonically, one arrives at the fundamental consonant triad of BP music - 3:5:7:(9). The MOS that are most naturally formed from these harmonics are of the forms 4L+1s (pentatonic) and 4L+5s (nonatonic). Which brings forward one analogy with diatonic music (3rd and 5th harmonics under octaves) or diatonic function in general: that 4edt and 9edt can be directly compared to 5edo and 7edo (and indeed they sound like they can). In contrast to the state of meantone temperaments, the simplest L=2 s=1 (13edt, the traditional tempered BP scale) is the most accurate and evenly-tempered for a long way. But the formations with Large and small steps of different relative sizes are no less important, for example by being capable of representing other intervals and harmonics.
 [[5edt]] (Tritave counterpart of Magic)
@@ Line 37: / Line 42: @@
 <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;edt&lt;/title&gt;&lt;/head&gt;&lt;body&gt;&lt;!-- ws:start:WikiTextHeadingRule:0:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc0"&gt;&lt;a name="Division of the tritave (3/1) into n equal parts"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:0 --&gt;Division of the tritave (3/1) into n equal parts&lt;/h1&gt;
   &lt;br /&gt;
+&lt;br /&gt;
+After the octave (roughly 2:1 but it has been tuned sharp and flat for various reasons), the next simple &amp;quot;frame interval&amp;quot; available is the ratio 3:1. Among other names, the third harmonic has been called the &amp;quot;perfect twelfth&amp;quot; &amp;quot;triple&amp;quot; or &amp;quot;tritave&amp;quot;. There has been argument whether pitches a tritave removed can be heard as equivalent, but with proper context and/or experience, at least some people find that they can. Arguably that is the single criterion for calling the tritave a true frame interval. But it is certain that musically valuable organizations of pitch can arise through the equal division of non-octave intervals regardless of equivalence, and either way, the multitude of equal divisions of the tritave are rich and ripe for exploration.&lt;br /&gt;
+&lt;br /&gt;
+The Bohlen-Pierce (BP) scale seems to have been the first such arrangement to be seriously studied and made into music. As the equivalent harmonics are 1:3:9:27 etc., filling in 3-9 isoharmonically, one arrives at the fundamental consonant triad of BP music - 3:5:7:(9). The MOS that are most naturally formed from these harmonics are of the forms 4L+1s (pentatonic) and 4L+5s (nonatonic). Which brings forward one analogy with diatonic music (3rd and 5th harmonics under octaves) or diatonic function in general: that 4edt and 9edt can be directly compared to 5edo and 7edo (and indeed they sound like they can). In contrast to the state of meantone temperaments, the simplest L=2 s=1 (13edt, the traditional tempered BP scale) is the most accurate and evenly-tempered for a long way. But the formations with Large and small steps of different relative sizes are no less important, for example by being capable of representing other intervals and harmonics.&lt;br /&gt;
+&lt;br /&gt;
 &lt;a class="wiki_link" href="/5edt"&gt;5edt&lt;/a&gt; (Tritave counterpart of Magic)&lt;br /&gt;
 &lt;a class="wiki_link" href="/6edt"&gt;6edt&lt;/a&gt; (Tritave counterpart of Hanson)&lt;br /&gt;