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<h2>IMPORTED REVISION FROM WIKISPACES</h2>
<h2>IMPORTED REVISION FROM WIKISPACES</h2>
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: This revision was by author [[User:keenanpepper|keenanpepper]] and made on <tt>2011-09-07 02:27:39 UTC</tt>.<br>
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: The original revision id was <tt>251460534</tt>.<br>
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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">=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 apart 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. Some put a great emphasis on timbre, claiming that a lack of even harmonics (octaves) emphasizes the property of tritave equivalence, or at least enhances the perception of [[BP]] harmonies, though the necessity of this claim is arguable. 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.
Western music generally revolves around the principle of **octave equivalence**: notes an octave apart are often perceived in western music as being the same //chroma// but differing in pitch height. As the octave corresponds to a 2/1 frequency ratio, it has been proposed that the next-simplest after the octave, the 3/1, can also be used to evoke a sense of chroma equivalence. This interval corresponds to a perfect twelfth in the diatonic scale, but when used to refer to an equivalence interval it is often called the "tritave".


The [[BP|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 [[MOSScales|MOS]] that are most naturally formed from these harmonics are of the forms 4L+1s (pentatonic) and 4L+5s (nonatonic), and through these formulae many equal divisions can be derived. 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 true nonatonic 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, formations with Large and small steps of different relative sizes are no less valuable, for their capability of representing other intervals and harmonics (eg. 17edt), for the use of extended harmonies that would be tempered together with only 13 tones, and to allow use of other non-nonatonic scale formations.
It has been argued that pitches a tritave apart can never truly be heard as equivalent in all of the ways that octaves are, with some claiming that the [[@http://www.mmk.ei.tum.de/persons/ter/top/octequiv.html|tonotopic representation of the mammalian auditory system]] is inherently biased towards octave-equivalence. With proper context, experience, and training, however, at least some people find that they can experience some degree of tritave equivalence. Either way, it is certain that musically valuable organizations of pitch can arise through the equal division of non-octave intervals, regardless of whether the period is perceived as being truly chroma-equivalent, and as such the multitude of equal divisions of the tritave are rich and ripe for exploration.
 
The [[BP|Bohlen-Pierce (BP) scale]], most commonly consisting of 13 equal divisions of the tritave (although a justly-intoned version exists as well), seems to have been the first such arrangement to be seriously studied and made into music. The BP scale is derived from the hypothesis that the tritave is best heard as equivalent if played on timbres consisting only of odd harmonics, so that the 2/1 interval and its multiples never appear, and if all factors of two are eliminated from the ratios approximated. The necessity of this claim is arguable, as some have stated that they can perceive tritave equivalence even if timbres with all harmonics are used, but some have also stated that they feel that BP harmonies are at the least enriched by using odd-only timbres.
 
If factors of two are eliminated, the simplest possible triad is (1):3:5:7:(9), with 1 and 9 in parentheses as they're equivalent to 3. Hence, 3:5:7 can be viewed as the fundamental consonant triad of BP music. The linear temperament that best approximates these chords is called the Bohlen-Pierce linear temperament, eliminating 245/243, and generally forms [[MOSScales|MOS]] of the forms 4L+1s (pentatonic) and 4L+5s (nonatonic), and through these formulae many equal divisions can be derived. This temperament is lowest in badness in the 3.5.7 subgroup and serves a function analogous to meantone in the 5-limit.
 
As far as EDTs supporting this temperament are concerned, an apt analogy can be drawn with EDOs supporting meantone: 5EDT and 9EDT are to BP what 5EDO and 7EDO are to meantone. However, in contrast to meantone, the simplest EDO supporting the BP nonatonic scale - 13edt, the traditional tempered BP scale - is the most accurate and lowest in tuning error until 56EDT. However, there are many EDTs supporting BP temperament that, despite being slightly higher in error, are no less valuable, both for their capability of representing higher-limit intervals and harmonics (eg. 17edt), for the use of extended harmonies that would be tempered together with only 13 tones, as well as to allow use of other non-nonatonic scale formations.


And of course, diatonicism isn't the whole of octave based temperaments, and other MOSes and the equal divisions based on them may approximate other systems of harmonics altogether! For example, 15edt very well approximates the 5th and 13th harmonics, and 12edt, the 13th and 17th. **One should bear in mind that, assuming tritave equivalence, when determining which harmonics are represented, the ratios of 3 in the denominator are fungible instead of those of 2.** For example making the fifth harmonic 5:3 a "major sixth" by conventional (and arbitrarily silly for the purposes of xenharmony, even with octaves) pitch class terminology.
And of course, diatonicism isn't the whole of octave based temperaments, and other MOSes and the equal divisions based on them may approximate other systems of harmonics altogether! For example, 15edt very well approximates the 5th and 13th harmonics, and 12edt, the 13th and 17th. **One should bear in mind that, assuming tritave equivalence, when determining which harmonics are represented, the ratios of 3 in the denominator are fungible instead of those of 2.** For example making the fifth harmonic 5:3 a "major sixth" by conventional (and arbitrarily silly for the purposes of xenharmony, even with octaves) pitch class terminology.
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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">&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;
<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 apart 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. Some put a great emphasis on timbre, claiming that a lack of even harmonics (octaves) emphasizes the property of tritave equivalence, or at least enhances the perception of &lt;a class="wiki_link" href="/BP"&gt;BP&lt;/a&gt; harmonies, though the necessity of this claim is arguable. 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;
Western music generally revolves around the principle of &lt;strong&gt;octave equivalence&lt;/strong&gt;: notes an octave apart are often perceived in western music as being the same &lt;em&gt;chroma&lt;/em&gt; but differing in pitch height. As the octave corresponds to a 2/1 frequency ratio, it has been proposed that the next-simplest after the octave, the 3/1, can also be used to evoke a sense of chroma equivalence. This interval corresponds to a perfect twelfth in the diatonic scale, but when used to refer to an equivalence interval it is often called the &amp;quot;tritave&amp;quot;.&lt;br /&gt;
&lt;br /&gt;
It has been argued that pitches a tritave apart can never truly be heard as equivalent in all of the ways that octaves are, with some claiming that the &lt;a class="wiki_link_ext" href="http://www.mmk.ei.tum.de/persons/ter/top/octequiv.html" rel="nofollow" target="_blank"&gt;tonotopic representation of the mammalian auditory system&lt;/a&gt; is inherently biased towards octave-equivalence. With proper context, experience, and training, however, at least some people find that they can experience some degree of tritave equivalence. Either way, it is certain that musically valuable organizations of pitch can arise through the equal division of non-octave intervals, regardless of whether the period is perceived as being truly chroma-equivalent, and as such the multitude of equal divisions of the tritave are rich and ripe for exploration.&lt;br /&gt;
&lt;br /&gt;
The &lt;a class="wiki_link" href="/BP"&gt;Bohlen-Pierce (BP) scale&lt;/a&gt;, most commonly consisting of 13 equal divisions of the tritave (although a justly-intoned version exists as well), seems to have been the first such arrangement to be seriously studied and made into music. The BP scale is derived from the hypothesis that the tritave is best heard as equivalent if played on timbres consisting only of odd harmonics, so that the 2/1 interval and its multiples never appear, and if all factors of two are eliminated from the ratios approximated. The necessity of this claim is arguable, as some have stated that they can perceive tritave equivalence even if timbres with all harmonics are used, but some have also stated that they feel that BP harmonies are at the least enriched by using odd-only timbres.&lt;br /&gt;
&lt;br /&gt;
If factors of two are eliminated, the simplest possible triad is (1):3:5:7:(9), with 1 and 9 in parentheses as they're equivalent to 3. Hence, 3:5:7 can be viewed as the fundamental consonant triad of BP music. The linear temperament that best approximates these chords is called the Bohlen-Pierce linear temperament, eliminating 245/243, and generally forms &lt;a class="wiki_link" href="/MOSScales"&gt;MOS&lt;/a&gt; of the forms 4L+1s (pentatonic) and 4L+5s (nonatonic), and through these formulae many equal divisions can be derived. This temperament is lowest in badness in the 3.5.7 subgroup and serves a function analogous to meantone in the 5-limit.&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
The &lt;a class="wiki_link" href="/BP"&gt;Bohlen-Pierce (BP) scale&lt;/a&gt; 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 &lt;a class="wiki_link" href="/MOSScales"&gt;MOS&lt;/a&gt; that are most naturally formed from these harmonics are of the forms 4L+1s (pentatonic) and 4L+5s (nonatonic), and through these formulae many equal divisions can be derived. 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 true nonatonic 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, formations with Large and small steps of different relative sizes are no less valuable, for their capability of representing other intervals and harmonics (eg. 17edt), for the use of extended harmonies that would be tempered together with only 13 tones, and to allow use of other non-nonatonic scale formations.&lt;br /&gt;
As far as EDTs supporting this temperament are concerned, an apt analogy can be drawn with EDOs supporting meantone: 5EDT and 9EDT are to BP what 5EDO and 7EDO are to meantone. However, in contrast to meantone, the simplest EDO supporting the BP nonatonic scale - 13edt, the traditional tempered BP scale - is the most accurate and lowest in tuning error until 56EDT. However, there are many EDTs supporting BP temperament that, despite being slightly higher in error, are no less valuable, both for their capability of representing higher-limit intervals and harmonics (eg. 17edt), for the use of extended harmonies that would be tempered together with only 13 tones, as well as to allow use of other non-nonatonic scale formations.&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
And of course, diatonicism isn't the whole of octave based temperaments, and other MOSes and the equal divisions based on them may approximate other systems of harmonics altogether! For example, 15edt very well approximates the 5th and 13th harmonics, and 12edt, the 13th and 17th. &lt;strong&gt;One should bear in mind that, assuming tritave equivalence, when determining which harmonics are represented, the ratios of 3 in the denominator are fungible instead of those of 2.&lt;/strong&gt; For example making the fifth harmonic 5:3 a &amp;quot;major sixth&amp;quot; by conventional (and arbitrarily silly for the purposes of xenharmony, even with octaves) pitch class terminology.&lt;br /&gt;
And of course, diatonicism isn't the whole of octave based temperaments, and other MOSes and the equal divisions based on them may approximate other systems of harmonics altogether! For example, 15edt very well approximates the 5th and 13th harmonics, and 12edt, the 13th and 17th. &lt;strong&gt;One should bear in mind that, assuming tritave equivalence, when determining which harmonics are represented, the ratios of 3 in the denominator are fungible instead of those of 2.&lt;/strong&gt; For example making the fifth harmonic 5:3 a &amp;quot;major sixth&amp;quot; by conventional (and arbitrarily silly for the purposes of xenharmony, even with octaves) pitch class terminology.&lt;br /&gt;
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