EDO vs ET: 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:genewardsmith|genewardsmith]] and made on <tt>~~2011~~-07-~~18 19~~:58:43 UTC</tt>.<br>

: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2012-09-24 10:37:12 UTC</tt>.<br>

: The original revision id was <tt>~~241855209~~</tt>.<br>

: The original revision id was <tt>367276002</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>

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There are many EDOs which, by virtue of some freak miracle of mathematics, happen to sound a lot like Just Intonation. Some say this is the case with 12-EDO, and that one of the reasons for its popularity is because it sounds enough like 5-limit (or perhaps even 7-limit) JI to please the masses, while being incredibly practical and convenient. Some also say that EDOs like 19, 22, 31, 34, 41, 46 or 53 sound even more like JI than 12-EDO, and this indeed seems to be the case. Because of this acoustic similarity between equal tunings and Just tunings, lots of people like to treat equal tunings as //approximations// to JI--in other words, temperaments. With EDOs such as these, it is possible to gain some level of insight into their harmonic structures by thinking in terms of approximate JI, and thus this approach has been very useful to many people.

However, it is not always true that those using EDOs are interested in approximating JI, nor is it true that describing all EDOs in terms of approximate JI is universally helpful or illuminating. As an example of the former, consider the atonalists, a loose school of 20th-century composers who sought to embrace the "equality" of equal temperament by treating every note as having equal musical importance, and thereby escape connotations of tonality that had previously defined Western classical music. ~~As an example of the latter, consider~~ 7-EDO: there is not a single triadic sonority within the EDO that is concordant enough to plausibly be conflated with Just Intonation, and attempts to describe its harmonic structures in terms of Just ratios is often more confusing than it is illuminating. It is not impossible to treat 7-EDO as an equal temperament, but ~~doing so begs~~ the question of what is being gained in the process, which ~~cannot be had by ignoring JI~~ all ~~together~~.

However, it is not always true that those using EDOs are interested in approximating JI, nor is it true that describing all EDOs in terms of approximate JI is universally helpful or illuminating. As an example of the former, consider the atonalists, a loose school of 20th-century composers who sought to embrace the "equality" of equal temperament by treating every note as having equal musical importance, and thereby escape connotations of tonality that had previously defined Western classical music.

Consider also 7-EDO: there is not a single triadic sonority within the EDO that is concordant enough to plausibly be conflated with Just Intonation, and attempts to describe its harmonic structures in terms of Just ratios is often more confusing than it is illuminating. It is not impossible to treat 7-EDO as an equal temperament, but the question arises of what is being gained in the process. One answer to that is that 7-EDO treated as a temperament, even though it is not actually used as one, is basic to Western musical theory. One step is a tone, two steps a third, three steps a fourth, four steps a fifth, five steps a sixth, six steps a seventh, and seven steps an octave. These can be major, minor, diminished or augmented, which are all the same to 7-EDO. It is understood that the perfect octave is a 2, and the perfect fifth must approximate 3/2; if the perfect major third approximates 5/4 then we have the <7 11 16| val of 7-EDO as a temperament lying behind the terminology of Western music.

==The EDO paradigm==

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There are many EDOs which, by virtue of some freak miracle of mathematics, happen to sound a lot like Just Intonation. Some say this is the case with 12-EDO, and that one of the reasons for its popularity is because it sounds enough like 5-limit (or perhaps even 7-limit) JI to please the masses, while being incredibly practical and convenient. Some also say that EDOs like 19, 22, 31, 34, 41, 46 or 53 sound even more like JI than 12-EDO, and this indeed seems to be the case. Because of this acoustic similarity between equal tunings and Just tunings, lots of people like to treat equal tunings as <em>approximations</em> to JI--in other words, temperaments. With EDOs such as these, it is possible to gain some level of insight into their harmonic structures by thinking in terms of approximate JI, and thus this approach has been very useful to many people.<br />

<br />

However, it is not always true that those using EDOs are interested in approximating JI, nor is it true that describing all EDOs in terms of approximate JI is universally helpful or illuminating. As an example of the former, consider the atonalists, a loose school of 20th-century composers who sought to embrace the &quot;equality&quot; of equal temperament by treating every note as having equal musical importance, and thereby escape connotations of tonality that had previously defined Western classical music. ~~As an example of the latter, consider~~ 7-EDO: there is not a single triadic sonority within the EDO that is concordant enough to plausibly be conflated with Just Intonation, and attempts to describe its harmonic structures in terms of Just ratios is often more confusing than it is illuminating. It is not impossible to treat 7-EDO as an equal temperament, but ~~doing so begs~~ the question of what is being gained in the process, which ~~cannot be had by ignoring JI~~ all ~~together~~.<br />

However, it is not always true that those using EDOs are interested in approximating JI, nor is it true that describing all EDOs in terms of approximate JI is universally helpful or illuminating. As an example of the former, consider the atonalists, a loose school of 20th-century composers who sought to embrace the &quot;equality&quot; of equal temperament by treating every note as having equal musical importance, and thereby escape connotations of tonality that had previously defined Western classical music. <br />

<br />

Consider also 7-EDO: there is not a single triadic sonority within the EDO that is concordant enough to plausibly be conflated with Just Intonation, and attempts to describe its harmonic structures in terms of Just ratios is often more confusing than it is illuminating. It is not impossible to treat 7-EDO as an equal temperament, but the question arises of what is being gained in the process. One answer to that is that 7-EDO treated as a temperament, even though it is not actually used as one, is basic to Western musical theory. One step is a tone, two steps a third, three steps a fourth, four steps a fifth, five steps a sixth, six steps a seventh, and seven steps an octave. These can be major, minor, diminished or augmented, which are all the same to 7-EDO. It is understood that the perfect octave is a 2, and the perfect fifth must approximate 3/2; if the perfect major third approximates 5/4 then we have the &lt;7 11 16| val of 7-EDO as a temperament lying behind the terminology of Western music.<br />

<br />

<h2 id="toc2"><a name="EDOs vs. Equal Temperaments-The EDO paradigm"></a>The EDO paradigm</h2>

@@ 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:genewardsmith|genewardsmith]] and made on <tt>2011-07-18 19:58:43 UTC</tt>.<br>
+: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2012-09-24 10:37:12 UTC</tt>.<br>
-: The original revision id was <tt>241855209</tt>.<br>
+: The original revision id was <tt>367276002</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>
@@ Line 14: / Line 14: @@
 There are many EDOs which, by virtue of some freak miracle of mathematics, happen to sound a lot like Just Intonation. Some say this is the case with 12-EDO, and that one of the reasons for its popularity is because it sounds enough like 5-limit (or perhaps even 7-limit) JI to please the masses, while being incredibly practical and convenient. Some also say that EDOs like 19, 22, 31, 34, 41, 46 or 53 sound even more like JI than 12-EDO, and this indeed seems to be the case. Because of this acoustic similarity between equal tunings and Just tunings, lots of people like to treat equal tunings as //approximations// to JI--in other words, temperaments. With EDOs such as these, it is possible to gain some level of insight into their harmonic structures by thinking in terms of approximate JI, and thus this approach has been very useful to many people.
-However, it is not always true that those using EDOs are interested in approximating JI, nor is it true that describing all EDOs in terms of approximate JI is universally helpful or illuminating. As an example of the former, consider the atonalists, a loose school of 20th-century composers who sought to embrace the "equality" of equal temperament by treating every note as having equal musical importance, and thereby escape connotations of tonality that had previously defined Western classical music. As an example of the latter, consider 7-EDO: there is not a single triadic sonority within the EDO that is concordant enough to plausibly be conflated with Just Intonation, and attempts to describe its harmonic structures in terms of Just ratios is often more confusing than it is illuminating. It is not impossible to treat 7-EDO as an equal temperament, but doing so begs the question of what is being gained in the process, which cannot be had by ignoring JI all together.
+However, it is not always true that those using EDOs are interested in approximating JI, nor is it true that describing all EDOs in terms of approximate JI is universally helpful or illuminating. As an example of the former, consider the atonalists, a loose school of 20th-century composers who sought to embrace the "equality" of equal temperament by treating every note as having equal musical importance, and thereby escape connotations of tonality that had previously defined Western classical music.
+Consider also 7-EDO: there is not a single triadic sonority within the EDO that is concordant enough to plausibly be conflated with Just Intonation, and attempts to describe its harmonic structures in terms of Just ratios is often more confusing than it is illuminating. It is not impossible to treat 7-EDO as an equal temperament, but the question  arises of what is being gained in the process. One answer to that is that 7-EDO treated as a temperament, even though it is not actually used as one, is basic to Western musical theory. One step is a tone, two steps a third, three steps a fourth, four steps a fifth, five steps a sixth, six steps a seventh, and seven steps an octave. These can be major, minor, diminished or augmented, which are all the same to 7-EDO. It is understood that the perfect octave is a 2, and the perfect fifth must approximate 3/2; if the perfect major third approximates 5/4 then we have the &lt;7 11 16| val of 7-EDO as a temperament lying behind the terminology of Western music.
 ==The EDO paradigm==
@@ Line 32: / Line 34: @@
 There are many EDOs which, by virtue of some freak miracle of mathematics, happen to sound a lot like Just Intonation. Some say this is the case with 12-EDO, and that one of the reasons for its popularity is because it sounds enough like 5-limit (or perhaps even 7-limit) JI to please the masses, while being incredibly practical and convenient. Some also say that EDOs like 19, 22, 31, 34, 41, 46 or 53 sound even more like JI than 12-EDO, and this indeed seems to be the case. Because of this acoustic similarity between equal tunings and Just tunings, lots of people like to treat equal tunings as &lt;em&gt;approximations&lt;/em&gt; to JI--in other words, temperaments. With EDOs such as these, it is possible to gain some level of insight into their harmonic structures by thinking in terms of approximate JI, and thus this approach has been very useful to many people.&lt;br /&gt;
 &lt;br /&gt;
-However, it is not always true that those using EDOs are interested in approximating JI, nor is it true that describing all EDOs in terms of approximate JI is universally helpful or illuminating. As an example of the former, consider the atonalists, a loose school of 20th-century composers who sought to embrace the &amp;quot;equality&amp;quot; of equal temperament by treating every note as having equal musical importance, and thereby escape connotations of tonality that had previously defined Western classical music. As an example of the latter, consider 7-EDO: there is not a single triadic sonority within the EDO that is concordant enough to plausibly be conflated with Just Intonation, and attempts to describe its harmonic structures in terms of Just ratios is often more confusing than it is illuminating. It is not impossible to treat 7-EDO as an equal temperament, but doing so begs the question of what is being gained in the process, which cannot be had by ignoring JI all together.&lt;br /&gt;
+However, it is not always true that those using EDOs are interested in approximating JI, nor is it true that describing all EDOs in terms of approximate JI is universally helpful or illuminating. As an example of the former, consider the atonalists, a loose school of 20th-century composers who sought to embrace the &amp;quot;equality&amp;quot; of equal temperament by treating every note as having equal musical importance, and thereby escape connotations of tonality that had previously defined Western classical music. &lt;br /&gt;
+&lt;br /&gt;
+Consider also 7-EDO: there is not a single triadic sonority within the EDO that is concordant enough to plausibly be conflated with Just Intonation, and attempts to describe its harmonic structures in terms of Just ratios is often more confusing than it is illuminating. It is not impossible to treat 7-EDO as an equal temperament, but the question  arises of what is being gained in the process. One answer to that is that 7-EDO treated as a temperament, even though it is not actually used as one, is basic to Western musical theory. One step is a tone, two steps a third, three steps a fourth, four steps a fifth, five steps a sixth, six steps a seventh, and seven steps an octave. These can be major, minor, diminished or augmented, which are all the same to 7-EDO. It is understood that the perfect octave is a 2, and the perfect fifth must approximate 3/2; if the perfect major third approximates 5/4 then we have the &amp;lt;7 11 16| val of 7-EDO as a temperament lying behind the terminology of Western music.&lt;br /&gt;
 &lt;br /&gt;
 &lt;!-- ws:start:WikiTextHeadingRule:4:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc2"&gt;&lt;a name="EDOs vs. Equal Temperaments-The EDO paradigm"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:4 --&gt;The EDO paradigm&lt;/h2&gt;