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Wikispaces>Osmiorisbendi **Imported revision 246188973 - Original comment: ** |
Wikispaces>Kosmorsky **Imported revision 246623789 - Original comment: ** |
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<h2>IMPORTED REVISION FROM WIKISPACES</h2> | <h2>IMPORTED REVISION FROM WIKISPACES</h2> | ||
This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br> | This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br> | ||
: This revision was by author [[User: | : 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> | : The original revision id was <tt>246623789</tt>.<br> | ||
: The revision comment was: <tt></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> | 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> | <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= | <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) | [[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><!-- ws:start:WikiTextHeadingRule:0:&lt;h1&gt; --><h1 id="toc0"><a name="Division of the tritave (3/1) into n equal parts"></a><!-- ws:end:WikiTextHeadingRule:0 -->Division of the tritave (3/1) into n equal parts</h1> | <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><!-- ws:start:WikiTextHeadingRule:0:&lt;h1&gt; --><h1 id="toc0"><a name="Division of the tritave (3/1) into n equal parts"></a><!-- ws:end:WikiTextHeadingRule:0 -->Division of the tritave (3/1) into n equal parts</h1> | ||
<br /> | <br /> | ||
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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="/5edt">5edt</a> (Tritave counterpart of Magic)<br /> | ||
<a class="wiki_link" href="/6edt">6edt</a> (Tritave counterpart of Hanson)<br /> | <a class="wiki_link" href="/6edt">6edt</a> (Tritave counterpart of Hanson)<br /> | ||
Revision as of 22:32, 17 August 2011
IMPORTED REVISION FROM WIKISPACES
This is an imported revision from Wikispaces. The revision metadata is included below for reference:
- This revision was by author Kosmorsky and made on 2011-08-17 22:32:46 UTC.
- The original revision id was 246623789.
- The revision comment was:
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.
Original Wikitext content:
=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) [[6edt]] (Tritave counterpart of Hanson) [[7edt]] (Tritave counterpart of Orwell) [[8edt]] (Tritave counterpart of Blacksmith) [[9edt]] [[10edt]] [[11edt]] (Euler Temperament) [[12edt]] [[BP|13 (Bohlen-Pierce)]] [[14edt]] (Contains the Anti-Lambda MOS) [[15edt]] [[16edt]] [[17edt]] [[18edt]] [[19ED3|19 (Bernhard Stopper)]] [[20edt]] [[21edt]] [[22edt]] [[23edt]] [[24edt]] [[25edt]] [[26edt]] [[27edt]] [[28edt]] ... [[39edt]] Triple Bohlen-Pierce (Erlich)
Original HTML content:
<html><head><title>edt</title></head><body><!-- ws:start:WikiTextHeadingRule:0:<h1> --><h1 id="toc0"><a name="Division of the tritave (3/1) into n equal parts"></a><!-- ws:end:WikiTextHeadingRule:0 -->Division of the tritave (3/1) into n equal parts</h1> <br /> <br /> 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.<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 /> <a class="wiki_link" href="/7edt">7edt</a> (Tritave counterpart of Orwell)<br /> <a class="wiki_link" href="/8edt">8edt</a> (Tritave counterpart of Blacksmith)<br /> <a class="wiki_link" href="/9edt">9edt</a><br /> <a class="wiki_link" href="/10edt">10edt</a><br /> <a class="wiki_link" href="/11edt">11edt</a> (Euler Temperament)<br /> <a class="wiki_link" href="/12edt">12edt</a><br /> <a class="wiki_link" href="/BP">13 (Bohlen-Pierce)</a><br /> <a class="wiki_link" href="/14edt">14edt</a> (Contains the Anti-Lambda MOS)<br /> <a class="wiki_link" href="/15edt">15edt</a><br /> <a class="wiki_link" href="/16edt">16edt</a><br /> <a class="wiki_link" href="/17edt">17edt</a><br /> <a class="wiki_link" href="/18edt">18edt</a><br /> <a class="wiki_link" href="/19ED3">19 (Bernhard Stopper)</a><br /> <a class="wiki_link" href="/20edt">20edt</a><br /> <a class="wiki_link" href="/21edt">21edt</a><br /> <a class="wiki_link" href="/22edt">22edt</a><br /> <a class="wiki_link" href="/23edt">23edt</a><br /> <a class="wiki_link" href="/24edt">24edt</a><br /> <a class="wiki_link" href="/25edt">25edt</a><br /> <a class="wiki_link" href="/26edt">26edt</a><br /> <a class="wiki_link" href="/27edt">27edt</a><br /> <a class="wiki_link" href="/28edt">28edt</a><br /> ...<br /> <a class="wiki_link" href="/39edt">39edt</a> Triple Bohlen-Pierce (Erlich)</body></html>