EDT
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=Division of the tritave (3/1) into n equal parts= 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". 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. There are other uses, or conceptualizations, of tritave-based tunings. Purely intuitive use of these myriad, assuredly xenharmonic structures comes to mind (see "EDO" versus "equal temperament"). Another intent might be 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. (And given the stable nature of octave-based systems, some aesthetic overlap even in the most tritave-equivalent of music, would be forseeable.) 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 which is relevant to inharmonicity. Below is a large list of EDT's; additionally, some equal divisions of the tritave are known by alternate names or have special interest: [[5edt]] (Tritave counterpart of Magic) [[6edt]] (Tritave counterpart of Hanson) [[7edt]] (Tritave counterpart of Orwell) [[8edt]] (Tritave counterpart of Blacksmith) [[11edt]] "Euler Temperament" [[BP|"Bohlen-Pierce" or "BP"]] [[19ED3|"Bernhard Stopper"]] [[39edt]] Triple Bohlen-Pierce (Erlich) =Individual pages for EDT's= || [[edt|1edt]] || [[2edt]] || [[3edt]] || [[4edt]] || [[5edt]] || [[6edt]] || [[7edt]] || [[8edt]] || [[9edt]] || [[10edt]] || [[11edt]] || [[12edt]] || || [[13edt]] || [[14edt]] || [[15edt]] || [[16edt]] || [[17edt]] || [[18edt]] || [[19edt]] || [[20edt]] || [[21edt]] || [[22edt]] || [[23edt]] || [[24edt]] || || [[25edt]] || [[26edt]] || [[27edt]] || [[28edt]] || [[29edt]] || [[30edt]] || [[31edt]] || [[32edt]] || [[33edt]] || [[34edt]] || [[35edt]] || [[36edt]] || || [[37edt]] || [[38edt]] || [[39edt]] || [[40edt]] || [[41edt]] || [[42edt]] || [[43edt]] || [[44edt]] || [[45edt]] || [[46edt]] || [[47edt]] || [[48edt]] || Larger divisions of the tritave include [[52edt]], [[56edt]], [[71edt]], [[75edt]], [[88edt]], [[131edt]], [[245edt]] and [[316edt]]. Because the Bohlen Pierce family of equally tempered tritaves, which follow the MOS 4L+5s for integer L and s, is arguably as important to the set of EDT as 5L+2s diatonicism is to octave temperaments, here is a list of them. It's not like you can't do the math but this makes it quicker for you. L=1 s=0 4 edt L=1 s=1 9 edt (5flat40 7sharp18) L=2 s=1 13 (5flat7 7flat3) L=3 s=1 17 (5sharp10 7flat12) L=3 s=2 22 (~14edo) L=4 s=1 21 L=4 s=3 31 L=5 s=1 25 L=5 s=2 30 (~19edo) (5sharp3 7flat8) L=5 s=3 35 (~22edo) (5flat14 7sharp0) L=5 s=4 40 L=6 s=1 29 L=6 s=5 49 (~31EDO) (5sharp8 7sharp8) (Schism*) L=7 s=1 33 L=7 s=2 38 L=7 s=3 43 (~27edo) (5sharp0 7flat6) L=7 s=4 48 (5flat13 7flat0) L=7 s=5 53 L=7 s=6 58 5sharp1 7sharp10 (Schism*) *Schism, by which I mean, the most accurate value for 5/3 and-or 7/3 is found outside the 4L+5s MOS. [Also, the way I see it, as 4edt and 9edt are comparable to 5edo and 7edo, then the "counterparts" of Blackwood and Whitewood would be found in multiples therein and would be octatonic and octadecatonic, eg. 12edt and 27edt. Alas, Ryan has priority ;) ] =EDO-EDT correspondence= ||~ EDO ||~ EDT ||~ Comments || || [[5edo]] || [[8edt]] || 8edt is equivalent to 5edo with ~11 cent octave compression. Equivalently, 5edo is 8edt with ~18 cent stretched tritaves. [[Patent val|Patent vals]] match through the 13 limit. || || || [[9edt]] || || || [[6edo]] || || || || || [[10edt]] || || || [[7edo]] || [[11edt]] || 11edt is equivalent to 7edo with ~10 cent stretched octaves. Patent vals differ in the 7 limit, but neither can really be said to represent the 7th harmonic with a straight face. || || || [[12edt]] || || || [[8edo]] || || || || || [[13edt]] || The equal-tempered BP scale cannot be considered equivalent to 8edo. || || [[9edo]] || [[14edt]] || There is a lot of mismatch between the pure-octave and pure-tritave tunings, but the patent vals match through the 13 limit. || || || [[15edt]] || || || [[10edo]] || [[16edt]] || Similar situation to 5edo~8edt. Patent vals match through the 17 limit. || || || [[17edt]] || || || [[11edo]] || || || || || [[18edt]] || || || [[12edo]] || [[19edt]] || 19edt is 12edo with ~1.2 cent octave stretch. Patent vals match only through the 7 limit, but neither can be said to include 11 at all. || || || [[20edt]] || || || [[13edo]] || || || || || [[21edt]] || || || || [[22edt]] || || || [[14edo]] || || || || || [[23edt]] || || || [[15edo]] || [[24edt]] || This is only a rough correspondence, as the (5n)edo ~ (8n)edt sequence begins to break down. The patent vals match only through the 5 limit. || || [[16edo]] || [[25edt]] || Also only a rough correspondence; 25edt corresponds to 16edo with ~17 cent octave stretch. Patent vals match through the 5 limit. || || || [[26edt]] || Double BP || || [[17edo]] || [[27edt]] || 27edt is 17edo with ~2.5 cent compressed octaves. With the exception of 5 (which neither represents well), patent vals match through the 13 limit. || || || [[28edt]] || || || [[18edo]] || || || || || [[29edt]] || || || [[19edo]] || [[30edt]] || 30edt is 19edo with ~5 cent stretched octaves. Patent vals match through the 7 limit. || || || [[31edt]] || || || [[20edo]] || || || || || [[32edt]] || || || || [[33edt]] || || || [[21edo]] || || || || || [[34edt]] || || || [[22edo]] || [[35edt]] || 35edt is 22edo with ~4 cent compressed octaves. Patent vals match through the 11 limit. || || || [[36edt]] || || || [[23edo]] || || || || || [[37edt]] || || || [[24edo]] || [[38edt]] || Same ~1.2 cent octave stretch as 12edo~19edt. Patent vals match through the 19 limit. || || || [[39edt]] || [[Triple BP]] || || [[25edo]] || || || || || [[40edt]] || || || [[26edo]] || [[41edt]] || 41edt is 26edo with ~6 cent stretched octaves. Patent vals match through the 7 limit. || || || [[42edt]] || || || [[27edo]] || [[43edt]] || 43edt is 27edo with ~6 cent compressed octaves. Patent vals match through the 7 limit. || || || [[44edt]] || || || [[28edo]] || || || || || [[45edt]] || || || [[29edo]] || [[46edt]] || 46edt is 29edo with ~0.9 cent compressed octaves. Patent vals match through the 89 limit. (Really! I checked!) ||
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 />
Western music generally revolves around the principle of <strong>octave equivalence</strong>: notes an octave apart are often perceived in western music as being the same <em>chroma</em> 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".<br />
<br />
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 <a class="wiki_link_ext" href="http://www.mmk.ei.tum.de/persons/ter/top/octequiv.html" rel="nofollow" target="_blank">tonotopic representation of the mammalian auditory system</a> 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.<br />
<br />
The <a class="wiki_link" href="/BP">Bohlen-Pierce (BP) scale</a>, 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.<br />
<br />
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 <a class="wiki_link" href="/MOSScales">MOS</a> 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.<br />
<br />
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.<br />
<br />
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. <strong>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.</strong> 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.<br />
<br />
There are other uses, or conceptualizations, of tritave-based tunings. Purely intuitive use of these myriad, assuredly xenharmonic structures comes to mind (see "EDO" versus "equal temperament"). Another intent might be 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. (And given the stable nature of octave-based systems, some aesthetic overlap even in the most tritave-equivalent of music, would be forseeable.) 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 which is relevant to inharmonicity.<br />
<br />
Below is a large list of EDT's; additionally, some equal divisions of the tritave are known by alternate names or have special interest:<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="/11edt">11edt</a> "Euler Temperament"<br />
<a class="wiki_link" href="/BP">"Bohlen-Pierce" or "BP"</a><br />
<a class="wiki_link" href="/19ED3">"Bernhard Stopper"</a><br />
<a class="wiki_link" href="/39edt">39edt</a> Triple Bohlen-Pierce (Erlich)<br />
<br />
<!-- ws:start:WikiTextHeadingRule:2:<h1> --><h1 id="toc1"><a name="Individual pages for EDT's"></a><!-- ws:end:WikiTextHeadingRule:2 -->Individual pages for EDT's</h1>
<br />
<table class="wiki_table">
<tr>
<td><a class="wiki_link" href="/edt">1edt</a><br />
</td>
<td><a class="wiki_link" href="/2edt">2edt</a><br />
</td>
<td><a class="wiki_link" href="/3edt">3edt</a><br />
</td>
<td><a class="wiki_link" href="/4edt">4edt</a><br />
</td>
<td><a class="wiki_link" href="/5edt">5edt</a><br />
</td>
<td><a class="wiki_link" href="/6edt">6edt</a><br />
</td>
<td><a class="wiki_link" href="/7edt">7edt</a><br />
</td>
<td><a class="wiki_link" href="/8edt">8edt</a><br />
</td>
<td><a class="wiki_link" href="/9edt">9edt</a><br />
</td>
<td><a class="wiki_link" href="/10edt">10edt</a><br />
</td>
<td><a class="wiki_link" href="/11edt">11edt</a><br />
</td>
<td><a class="wiki_link" href="/12edt">12edt</a><br />
</td>
</tr>
<tr>
<td><a class="wiki_link" href="/13edt">13edt</a><br />
</td>
<td><a class="wiki_link" href="/14edt">14edt</a><br />
</td>
<td><a class="wiki_link" href="/15edt">15edt</a><br />
</td>
<td><a class="wiki_link" href="/16edt">16edt</a><br />
</td>
<td><a class="wiki_link" href="/17edt">17edt</a><br />
</td>
<td><a class="wiki_link" href="/18edt">18edt</a><br />
</td>
<td><a class="wiki_link" href="/19edt">19edt</a><br />
</td>
<td><a class="wiki_link" href="/20edt">20edt</a><br />
</td>
<td><a class="wiki_link" href="/21edt">21edt</a><br />
</td>
<td><a class="wiki_link" href="/22edt">22edt</a><br />
</td>
<td><a class="wiki_link" href="/23edt">23edt</a><br />
</td>
<td><a class="wiki_link" href="/24edt">24edt</a><br />
</td>
</tr>
<tr>
<td><a class="wiki_link" href="/25edt">25edt</a><br />
</td>
<td><a class="wiki_link" href="/26edt">26edt</a><br />
</td>
<td><a class="wiki_link" href="/27edt">27edt</a><br />
</td>
<td><a class="wiki_link" href="/28edt">28edt</a><br />
</td>
<td><a class="wiki_link" href="/29edt">29edt</a><br />
</td>
<td><a class="wiki_link" href="/30edt">30edt</a><br />
</td>
<td><a class="wiki_link" href="/31edt">31edt</a><br />
</td>
<td><a class="wiki_link" href="/32edt">32edt</a><br />
</td>
<td><a class="wiki_link" href="/33edt">33edt</a><br />
</td>
<td><a class="wiki_link" href="/34edt">34edt</a><br />
</td>
<td><a class="wiki_link" href="/35edt">35edt</a><br />
</td>
<td><a class="wiki_link" href="/36edt">36edt</a><br />
</td>
</tr>
<tr>
<td><a class="wiki_link" href="/37edt">37edt</a><br />
</td>
<td><a class="wiki_link" href="/38edt">38edt</a><br />
</td>
<td><a class="wiki_link" href="/39edt">39edt</a><br />
</td>
<td><a class="wiki_link" href="/40edt">40edt</a><br />
</td>
<td><a class="wiki_link" href="/41edt">41edt</a><br />
</td>
<td><a class="wiki_link" href="/42edt">42edt</a><br />
</td>
<td><a class="wiki_link" href="/43edt">43edt</a><br />
</td>
<td><a class="wiki_link" href="/44edt">44edt</a><br />
</td>
<td><a class="wiki_link" href="/45edt">45edt</a><br />
</td>
<td><a class="wiki_link" href="/46edt">46edt</a><br />
</td>
<td><a class="wiki_link" href="/47edt">47edt</a><br />
</td>
<td><a class="wiki_link" href="/48edt">48edt</a><br />
</td>
</tr>
</table>
<br />
Larger divisions of the tritave include <a class="wiki_link" href="/52edt">52edt</a>, <a class="wiki_link" href="/56edt">56edt</a>, <a class="wiki_link" href="/71edt">71edt</a>, <a class="wiki_link" href="/75edt">75edt</a>, <a class="wiki_link" href="/88edt">88edt</a>, <a class="wiki_link" href="/131edt">131edt</a>, <a class="wiki_link" href="/245edt">245edt</a> and <a class="wiki_link" href="/316edt">316edt</a>.<br />
<br />
Because the Bohlen Pierce family of equally tempered tritaves, which follow the MOS 4L+5s for integer L and s, is arguably as important to the set of EDT as 5L+2s diatonicism is to octave temperaments, here is a list of them. It's not like you can't do the math but this makes it quicker for you.<br />
<br />
L=1 s=0 4 edt<br />
L=1 s=1 9 edt (5flat40 7sharp18)<br />
L=2 s=1 13 (5flat7 7flat3)<br />
L=3 s=1 17 (5sharp10 7flat12)<br />
L=3 s=2 22 (~14edo)<br />
L=4 s=1 21<br />
L=4 s=3 31<br />
L=5 s=1 25<br />
L=5 s=2 30 (~19edo) (5sharp3 7flat8)<br />
L=5 s=3 35 (~22edo) (5flat14 7sharp0)<br />
L=5 s=4 40<br />
L=6 s=1 29<br />
L=6 s=5 49 (~31EDO) (5sharp8 7sharp8) (Schism*)<br />
L=7 s=1 33<br />
L=7 s=2 38<br />
L=7 s=3 43 (~27edo) (5sharp0 7flat6)<br />
L=7 s=4 48 (5flat13 7flat0)<br />
L=7 s=5 53<br />
L=7 s=6 58 5sharp1 7sharp10 (Schism*)<br />
*Schism, by which I mean, the most accurate value for 5/3 and-or 7/3 is found outside the 4L+5s MOS.<br />
[Also, the way I see it, as 4edt and 9edt are comparable to 5edo and 7edo, then the "counterparts" of Blackwood and Whitewood would be found in multiples therein and would be octatonic and octadecatonic, eg. 12edt and 27edt. Alas, Ryan has priority ;) ]<br />
<br />
<!-- ws:start:WikiTextHeadingRule:4:<h1> --><h1 id="toc2"><a name="EDO-EDT correspondence"></a><!-- ws:end:WikiTextHeadingRule:4 -->EDO-EDT correspondence</h1>
<table class="wiki_table">
<tr>
<th>EDO<br />
</th>
<th>EDT<br />
</th>
<th>Comments<br />
</th>
</tr>
<tr>
<td><a class="wiki_link" href="/5edo">5edo</a><br />
</td>
<td><a class="wiki_link" href="/8edt">8edt</a><br />
</td>
<td>8edt is equivalent to 5edo with ~11 cent octave compression.<br />
Equivalently, 5edo is 8edt with ~18 cent stretched tritaves.<br />
<a class="wiki_link" href="/Patent%20val">Patent vals</a> match through the 13 limit.<br />
</td>
</tr>
<tr>
<td><br />
</td>
<td><a class="wiki_link" href="/9edt">9edt</a><br />
</td>
<td><br />
</td>
</tr>
<tr>
<td><a class="wiki_link" href="/6edo">6edo</a><br />
</td>
<td><br />
</td>
<td><br />
</td>
</tr>
<tr>
<td><br />
</td>
<td><a class="wiki_link" href="/10edt">10edt</a><br />
</td>
<td><br />
</td>
</tr>
<tr>
<td><a class="wiki_link" href="/7edo">7edo</a><br />
</td>
<td><a class="wiki_link" href="/11edt">11edt</a><br />
</td>
<td>11edt is equivalent to 7edo with ~10 cent stretched octaves.<br />
Patent vals differ in the 7 limit, but neither can really be said<br />
to represent the 7th harmonic with a straight face.<br />
</td>
</tr>
<tr>
<td><br />
</td>
<td><a class="wiki_link" href="/12edt">12edt</a><br />
</td>
<td><br />
</td>
</tr>
<tr>
<td><a class="wiki_link" href="/8edo">8edo</a><br />
</td>
<td><br />
</td>
<td><br />
</td>
</tr>
<tr>
<td><br />
</td>
<td><a class="wiki_link" href="/13edt">13edt</a><br />
</td>
<td>The equal-tempered BP scale cannot be considered equivalent to 8edo.<br />
</td>
</tr>
<tr>
<td><a class="wiki_link" href="/9edo">9edo</a><br />
</td>
<td><a class="wiki_link" href="/14edt">14edt</a><br />
</td>
<td>There is a lot of mismatch between the pure-octave and pure-tritave tunings,<br />
but the patent vals match through the 13 limit.<br />
</td>
</tr>
<tr>
<td><br />
</td>
<td><a class="wiki_link" href="/15edt">15edt</a><br />
</td>
<td><br />
</td>
</tr>
<tr>
<td><a class="wiki_link" href="/10edo">10edo</a><br />
</td>
<td><a class="wiki_link" href="/16edt">16edt</a><br />
</td>
<td>Similar situation to 5edo~8edt. Patent vals match through the 17 limit.<br />
</td>
</tr>
<tr>
<td><br />
</td>
<td><a class="wiki_link" href="/17edt">17edt</a><br />
</td>
<td><br />
</td>
</tr>
<tr>
<td><a class="wiki_link" href="/11edo">11edo</a><br />
</td>
<td><br />
</td>
<td><br />
</td>
</tr>
<tr>
<td><br />
</td>
<td><a class="wiki_link" href="/18edt">18edt</a><br />
</td>
<td><br />
</td>
</tr>
<tr>
<td><a class="wiki_link" href="/12edo">12edo</a><br />
</td>
<td><a class="wiki_link" href="/19edt">19edt</a><br />
</td>
<td>19edt is 12edo with ~1.2 cent octave stretch. Patent vals match<br />
only through the 7 limit, but neither can be said to include 11 at all.<br />
</td>
</tr>
<tr>
<td><br />
</td>
<td><a class="wiki_link" href="/20edt">20edt</a><br />
</td>
<td><br />
</td>
</tr>
<tr>
<td><a class="wiki_link" href="/13edo">13edo</a><br />
</td>
<td><br />
</td>
<td><br />
</td>
</tr>
<tr>
<td><br />
</td>
<td><a class="wiki_link" href="/21edt">21edt</a><br />
</td>
<td><br />
</td>
</tr>
<tr>
<td><br />
</td>
<td><a class="wiki_link" href="/22edt">22edt</a><br />
</td>
<td><br />
</td>
</tr>
<tr>
<td><a class="wiki_link" href="/14edo">14edo</a><br />
</td>
<td><br />
</td>
<td><br />
</td>
</tr>
<tr>
<td><br />
</td>
<td><a class="wiki_link" href="/23edt">23edt</a><br />
</td>
<td><br />
</td>
</tr>
<tr>
<td><a class="wiki_link" href="/15edo">15edo</a><br />
</td>
<td><a class="wiki_link" href="/24edt">24edt</a><br />
</td>
<td>This is only a rough correspondence, as the (5n)edo ~ (8n)edt sequence<br />
begins to break down. The patent vals match only through the 5 limit.<br />
</td>
</tr>
<tr>
<td><a class="wiki_link" href="/16edo">16edo</a><br />
</td>
<td><a class="wiki_link" href="/25edt">25edt</a><br />
</td>
<td>Also only a rough correspondence; 25edt corresponds to 16edo<br />
with ~17 cent octave stretch. Patent vals match through the 5 limit.<br />
</td>
</tr>
<tr>
<td><br />
</td>
<td><a class="wiki_link" href="/26edt">26edt</a><br />
</td>
<td>Double BP<br />
</td>
</tr>
<tr>
<td><a class="wiki_link" href="/17edo">17edo</a><br />
</td>
<td><a class="wiki_link" href="/27edt">27edt</a><br />
</td>
<td>27edt is 17edo with ~2.5 cent compressed octaves. With the exception of<br />
5 (which neither represents well), patent vals match through the 13 limit.<br />
</td>
</tr>
<tr>
<td><br />
</td>
<td><a class="wiki_link" href="/28edt">28edt</a><br />
</td>
<td><br />
</td>
</tr>
<tr>
<td><a class="wiki_link" href="/18edo">18edo</a><br />
</td>
<td><br />
</td>
<td><br />
</td>
</tr>
<tr>
<td><br />
</td>
<td><a class="wiki_link" href="/29edt">29edt</a><br />
</td>
<td><br />
</td>
</tr>
<tr>
<td><a class="wiki_link" href="/19edo">19edo</a><br />
</td>
<td><a class="wiki_link" href="/30edt">30edt</a><br />
</td>
<td>30edt is 19edo with ~5 cent stretched octaves.<br />
Patent vals match through the 7 limit.<br />
</td>
</tr>
<tr>
<td><br />
</td>
<td><a class="wiki_link" href="/31edt">31edt</a><br />
</td>
<td><br />
</td>
</tr>
<tr>
<td><a class="wiki_link" href="/20edo">20edo</a><br />
</td>
<td><br />
</td>
<td><br />
</td>
</tr>
<tr>
<td><br />
</td>
<td><a class="wiki_link" href="/32edt">32edt</a><br />
</td>
<td><br />
</td>
</tr>
<tr>
<td><br />
</td>
<td><a class="wiki_link" href="/33edt">33edt</a><br />
</td>
<td><br />
</td>
</tr>
<tr>
<td><a class="wiki_link" href="/21edo">21edo</a><br />
</td>
<td><br />
</td>
<td><br />
</td>
</tr>
<tr>
<td><br />
</td>
<td><a class="wiki_link" href="/34edt">34edt</a><br />
</td>
<td><br />
</td>
</tr>
<tr>
<td><a class="wiki_link" href="/22edo">22edo</a><br />
</td>
<td><a class="wiki_link" href="/35edt">35edt</a><br />
</td>
<td>35edt is 22edo with ~4 cent compressed octaves.<br />
Patent vals match through the 11 limit.<br />
</td>
</tr>
<tr>
<td><br />
</td>
<td><a class="wiki_link" href="/36edt">36edt</a><br />
</td>
<td><br />
</td>
</tr>
<tr>
<td><a class="wiki_link" href="/23edo">23edo</a><br />
</td>
<td><br />
</td>
<td><br />
</td>
</tr>
<tr>
<td><br />
</td>
<td><a class="wiki_link" href="/37edt">37edt</a><br />
</td>
<td><br />
</td>
</tr>
<tr>
<td><a class="wiki_link" href="/24edo">24edo</a><br />
</td>
<td><a class="wiki_link" href="/38edt">38edt</a><br />
</td>
<td>Same ~1.2 cent octave stretch as 12edo~19edt.<br />
Patent vals match through the 19 limit.<br />
</td>
</tr>
<tr>
<td><br />
</td>
<td><a class="wiki_link" href="/39edt">39edt</a><br />
</td>
<td><a class="wiki_link" href="/Triple%20BP">Triple BP</a><br />
</td>
</tr>
<tr>
<td><a class="wiki_link" href="/25edo">25edo</a><br />
</td>
<td><br />
</td>
<td><br />
</td>
</tr>
<tr>
<td><br />
</td>
<td><a class="wiki_link" href="/40edt">40edt</a><br />
</td>
<td><br />
</td>
</tr>
<tr>
<td><a class="wiki_link" href="/26edo">26edo</a><br />
</td>
<td><a class="wiki_link" href="/41edt">41edt</a><br />
</td>
<td>41edt is 26edo with ~6 cent stretched octaves.<br />
Patent vals match through the 7 limit.<br />
</td>
</tr>
<tr>
<td><br />
</td>
<td><a class="wiki_link" href="/42edt">42edt</a><br />
</td>
<td><br />
</td>
</tr>
<tr>
<td><a class="wiki_link" href="/27edo">27edo</a><br />
</td>
<td><a class="wiki_link" href="/43edt">43edt</a><br />
</td>
<td>43edt is 27edo with ~6 cent compressed octaves.<br />
Patent vals match through the 7 limit.<br />
</td>
</tr>
<tr>
<td><br />
</td>
<td><a class="wiki_link" href="/44edt">44edt</a><br />
</td>
<td><br />
</td>
</tr>
<tr>
<td><a class="wiki_link" href="/28edo">28edo</a><br />
</td>
<td><br />
</td>
<td><br />
</td>
</tr>
<tr>
<td><br />
</td>
<td><a class="wiki_link" href="/45edt">45edt</a><br />
</td>
<td><br />
</td>
</tr>
<tr>
<td><a class="wiki_link" href="/29edo">29edo</a><br />
</td>
<td><a class="wiki_link" href="/46edt">46edt</a><br />
</td>
<td>46edt is 29edo with ~0.9 cent compressed octaves.<br />
Patent vals match through the 89 limit. (Really! I checked!)<br />
</td>
</tr>
</table>
</body></html>