EDT
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- This revision was by author Kosmorsky and made on 2011-08-17 22:46:01 UTC.
- The original revision id was 246626327.
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=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 in terms of just 5 and 7 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. And of course, other MOSes and the equal division based on them may approximate other systems of harmonics altogether. There are other uses, or conceptualizations, of tritave-based scales. Purely intuitive use of these myriad, assuredly xenharmonic scales comes to mind (see "EDO" versus "equal temperament"). Another may use them to find or define temperaments (such as Magic, Hanson, etc.), or to provide exact formulae for stretching/compressing what would musically be used as an "ordinary" octave of ~2:1. For instance, the Bernhard-Stopper (19edt) temperament, might for instance be found useful in tuning pianoforti, being equivalent to 12edo except for a 2c sharp octave, relevant to inharmonicity. [[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 in terms of just 5 and 7 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. And of course, other MOSes and the equal division based on them may approximate other systems of harmonics altogether.<br /> <br /> There are other uses, or conceptualizations, of tritave-based scales. Purely intuitive use of these myriad, assuredly xenharmonic scales comes to mind (see "EDO" versus "equal temperament"). Another may use them to find or define temperaments (such as Magic, Hanson, etc.), or to provide exact formulae for stretching/compressing what would musically be used as an "ordinary" octave of ~2:1. For instance, the Bernhard-Stopper (19edt) temperament, might for instance be found useful in tuning pianoforti, being equivalent to 12edo except for a 2c sharp octave, relevant to inharmonicity.<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>