19edo: Difference between revisions
Wikispaces>genewardsmith **Imported revision 278439320 - 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:Andrew_Heathwaite|Andrew_Heathwaite]] and made on <tt>2011-12-31 01:49:49 UTC</tt>.<br> | ||
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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> | ||
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=Theory= | =Theory= | ||
In music, **19 equal temperament**, called 19-TET, 19-[[EDO]], or 19-ET, is the scale derived by dividing the [[octave]] into 19 [[equal]]ly large steps. Each step represents a frequency ratio of the 19th root of 2, or 63.16 [[cent]] | In music, **19 equal temperament**, called 19-TET, 19-[[EDO]], or 19-ET, is the scale derived by dividing the [[octave]] into 19 [[equal]]ly large steps. Each step represents a frequency ratio of the 19th root of 2, or 63.16 [[cent|cents]]. It is the 8th [[prime numbers|prime]] edo, following [[17edo]] and coming before [[23edo]]. | ||
Interest in this tuning system goes back to the sixteenth century, when composer Guillaume Costeley used it in his chanson //Seigneur Dieu ta pitié// of 1558. Costeley understood and desired the circulating aspect of this tuning; in 1577 music theorist Francisco de Salinas in effect proposed it. Salinas discussed [[1-3 Syntonic Comma Meantone|1/3-comma meantone]], in which the fifth is of size 694.786 cents; the fifth of 19-et is 694.737, which is only a twentieth of a cent flatter. Salinas suggested tuning nineteen tones to the octave to this tuning, which fails to close by less than a cent, so that his suggestion is effectively 19-et. In the nineteenth century mathematician and music theorist Wesley Woolhouse proposed it as a more practical alternative to meantone tunings he regarded as better, such as [[50edo|50 equal temperament]] ([[http://sonic-arts.org/monzo/woolhouse/essay.htm|summary of Woolhouse's essay]]). | Interest in this tuning system goes back to the sixteenth century, when composer Guillaume Costeley used it in his chanson //Seigneur Dieu ta pitié// of 1558. Costeley understood and desired the circulating aspect of this tuning; in 1577 music theorist Francisco de Salinas in effect proposed it. Salinas discussed [[1-3 Syntonic Comma Meantone|1/3-comma meantone]], in which the fifth is of size 694.786 cents; the fifth of 19-et is 694.737, which is only a twentieth of a cent flatter. Salinas suggested tuning nineteen tones to the octave to this tuning, which fails to close by less than a cent, so that his suggestion is effectively 19-et. In the nineteenth century mathematician and music theorist Wesley Woolhouse proposed it as a more practical alternative to meantone tunings he regarded as better, such as [[50edo|50 equal temperament]] ([[http://sonic-arts.org/monzo/woolhouse/essay.htm|summary of Woolhouse's essay]]). | ||
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The most salient characteristic of 19-et is that, having an almost just minor third and perfect fifths and major thirds about seven cents narrow, it serves as a good tuning for [[Meantone family|meantone]] temperament. It is also a suitable for [[Regular Temperaments#magic|magic/muggles]] temperament, because five of its major thirds are equivalent to one of its //twelfths.// For both of these there are more optimal tunings: the fifth of 19-et is flatter than the usual for meantone, and a more accurate approximation is [[31edo|31 equal temperament]]. Similarly, the generating interval of magic temperament is a major third, and again 19-et's is flatter; [[41edo|41 equal temperament]] more closely matches it. It does make for a good tuning for muggles, which in 19et is the same as magic. | The most salient characteristic of 19-et is that, having an almost just minor third and perfect fifths and major thirds about seven cents narrow, it serves as a good tuning for [[Meantone family|meantone]] temperament. It is also a suitable for [[Regular Temperaments#magic|magic/muggles]] temperament, because five of its major thirds are equivalent to one of its //twelfths.// For both of these there are more optimal tunings: the fifth of 19-et is flatter than the usual for meantone, and a more accurate approximation is [[31edo|31 equal temperament]]. Similarly, the generating interval of magic temperament is a major third, and again 19-et's is flatter; [[41edo|41 equal temperament]] more closely matches it. It does make for a good tuning for muggles, which in 19et is the same as magic. | ||
However, for all of these 19-et has the practical advantage of requiring fewer pitches, which makes physical realizations of it easier to build. (Many 19-et instruments have been built.) 19-et is in fact the second equal temperament, after 12-et which is able to deal with [[Harmonic Limit|5-limit]] music in a tolerable manner, and is the fifth (after 12) [[The Riemann Zeta Function and Tuning#Zeta | However, for all of these 19-et has the practical advantage of requiring fewer pitches, which makes physical realizations of it easier to build. (Many 19-et instruments have been built.) 19-et is in fact the second equal temperament, after 12-et which is able to deal with [[Harmonic Limit|5-limit]] music in a tolerable manner, and is the fifth (after 12) [[The Riemann Zeta Function and Tuning#Zeta%20EDO%20lists|zeta integral edo]]. It is less successful with [[7-limit]] (but still better than 12-et), as it eliminates the distinction between a septimal minor third ([[7_6|7/6]]), and a septimal whole tone ([[8_7|8/7]]). 19-EDO also has the advantage of being excellent for negri, keemun, godzilla, magic/muggles and triton/liese, and fairly decent for sensi. Keemun and Negri are of particular note for being very simple 7-limit temperaments, with their MOS scales in 19-EDO offering a great abundance of septimal tetrads. The [[Graham complexity]] of a 7-limit tetrad is 6 for keemun, 7 for negri, 8 for godzilla, 10 for meantone/flattone, 11 for triton, 12 for magic/muggles and 13 for sensi. | ||
==Intervals and linear temperaments== | ==Intervals and linear temperaments== | ||
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<!-- ws:start:WikiTextHeadingRule:1:&lt;h1&gt; --><h1 id="toc0"><a name="Theory"></a><!-- ws:end:WikiTextHeadingRule:1 -->Theory</h1> | <!-- ws:start:WikiTextHeadingRule:1:&lt;h1&gt; --><h1 id="toc0"><a name="Theory"></a><!-- ws:end:WikiTextHeadingRule:1 -->Theory</h1> | ||
<br /> | <br /> | ||
In music, <strong>19 equal temperament</strong>, called 19-TET, 19-<a class="wiki_link" href="/EDO">EDO</a>, or 19-ET, is the scale derived by dividing the <a class="wiki_link" href="/octave">octave</a> into 19 <a class="wiki_link" href="/equal">equal</a>ly large steps. Each step represents a frequency ratio of the 19th root of 2, or 63.16 <a class="wiki_link" href="/cent"> | In music, <strong>19 equal temperament</strong>, called 19-TET, 19-<a class="wiki_link" href="/EDO">EDO</a>, or 19-ET, is the scale derived by dividing the <a class="wiki_link" href="/octave">octave</a> into 19 <a class="wiki_link" href="/equal">equal</a>ly large steps. Each step represents a frequency ratio of the 19th root of 2, or 63.16 <a class="wiki_link" href="/cent">cents</a>. It is the 8th <a class="wiki_link" href="/prime%20numbers">prime</a> edo, following <a class="wiki_link" href="/17edo">17edo</a> and coming before <a class="wiki_link" href="/23edo">23edo</a>.<br /> | ||
<br /> | <br /> | ||
Interest in this tuning system goes back to the sixteenth century, when composer Guillaume Costeley used it in his chanson <em>Seigneur Dieu ta pitié</em> of 1558. Costeley understood and desired the circulating aspect of this tuning; in 1577 music theorist Francisco de Salinas in effect proposed it. Salinas discussed <a class="wiki_link" href="/1-3%20Syntonic%20Comma%20Meantone">1/3-comma meantone</a>, in which the fifth is of size 694.786 cents; the fifth of 19-et is 694.737, which is only a twentieth of a cent flatter. Salinas suggested tuning nineteen tones to the octave to this tuning, which fails to close by less than a cent, so that his suggestion is effectively 19-et. In the nineteenth century mathematician and music theorist Wesley Woolhouse proposed it as a more practical alternative to meantone tunings he regarded as better, such as <a class="wiki_link" href="/50edo">50 equal temperament</a> (<a class="wiki_link_ext" href="http://sonic-arts.org/monzo/woolhouse/essay.htm" rel="nofollow">summary of Woolhouse's essay</a>).<br /> | Interest in this tuning system goes back to the sixteenth century, when composer Guillaume Costeley used it in his chanson <em>Seigneur Dieu ta pitié</em> of 1558. Costeley understood and desired the circulating aspect of this tuning; in 1577 music theorist Francisco de Salinas in effect proposed it. Salinas discussed <a class="wiki_link" href="/1-3%20Syntonic%20Comma%20Meantone">1/3-comma meantone</a>, in which the fifth is of size 694.786 cents; the fifth of 19-et is 694.737, which is only a twentieth of a cent flatter. Salinas suggested tuning nineteen tones to the octave to this tuning, which fails to close by less than a cent, so that his suggestion is effectively 19-et. In the nineteenth century mathematician and music theorist Wesley Woolhouse proposed it as a more practical alternative to meantone tunings he regarded as better, such as <a class="wiki_link" href="/50edo">50 equal temperament</a> (<a class="wiki_link_ext" href="http://sonic-arts.org/monzo/woolhouse/essay.htm" rel="nofollow">summary of Woolhouse's essay</a>).<br /> | ||
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The most salient characteristic of 19-et is that, having an almost just minor third and perfect fifths and major thirds about seven cents narrow, it serves as a good tuning for <a class="wiki_link" href="/Meantone%20family">meantone</a> temperament. It is also a suitable for <a class="wiki_link" href="/Regular%20Temperaments#magic">magic/muggles</a> temperament, because five of its major thirds are equivalent to one of its <em>twelfths.</em> For both of these there are more optimal tunings: the fifth of 19-et is flatter than the usual for meantone, and a more accurate approximation is <a class="wiki_link" href="/31edo">31 equal temperament</a>. Similarly, the generating interval of magic temperament is a major third, and again 19-et's is flatter; <a class="wiki_link" href="/41edo">41 equal temperament</a> more closely matches it. It does make for a good tuning for muggles, which in 19et is the same as magic.<br /> | The most salient characteristic of 19-et is that, having an almost just minor third and perfect fifths and major thirds about seven cents narrow, it serves as a good tuning for <a class="wiki_link" href="/Meantone%20family">meantone</a> temperament. It is also a suitable for <a class="wiki_link" href="/Regular%20Temperaments#magic">magic/muggles</a> temperament, because five of its major thirds are equivalent to one of its <em>twelfths.</em> For both of these there are more optimal tunings: the fifth of 19-et is flatter than the usual for meantone, and a more accurate approximation is <a class="wiki_link" href="/31edo">31 equal temperament</a>. Similarly, the generating interval of magic temperament is a major third, and again 19-et's is flatter; <a class="wiki_link" href="/41edo">41 equal temperament</a> more closely matches it. It does make for a good tuning for muggles, which in 19et is the same as magic.<br /> | ||
<br /> | <br /> | ||
However, for all of these 19-et has the practical advantage of requiring fewer pitches, which makes physical realizations of it easier to build. (Many 19-et instruments have been built.) 19-et is in fact the second equal temperament, after 12-et which is able to deal with <a class="wiki_link" href="/Harmonic%20Limit">5-limit</a> music in a tolerable manner, and is the fifth (after 12) <a class="wiki_link" href="/The%20Riemann%20Zeta%20Function%20and%20Tuning#Zeta | However, for all of these 19-et has the practical advantage of requiring fewer pitches, which makes physical realizations of it easier to build. (Many 19-et instruments have been built.) 19-et is in fact the second equal temperament, after 12-et which is able to deal with <a class="wiki_link" href="/Harmonic%20Limit">5-limit</a> music in a tolerable manner, and is the fifth (after 12) <a class="wiki_link" href="/The%20Riemann%20Zeta%20Function%20and%20Tuning#Zeta%20EDO%20lists">zeta integral edo</a>. It is less successful with <a class="wiki_link" href="/7-limit">7-limit</a> (but still better than 12-et), as it eliminates the distinction between a septimal minor third (<a class="wiki_link" href="/7_6">7/6</a>), and a septimal whole tone (<a class="wiki_link" href="/8_7">8/7</a>). 19-EDO also has the advantage of being excellent for negri, keemun, godzilla, magic/muggles and triton/liese, and fairly decent for sensi. Keemun and Negri are of particular note for being very simple 7-limit temperaments, with their MOS scales in 19-EDO offering a great abundance of septimal tetrads. The <a class="wiki_link" href="/Graham%20complexity">Graham complexity</a> of a 7-limit tetrad is 6 for keemun, 7 for negri, 8 for godzilla, 10 for meantone/flattone, 11 for triton, 12 for magic/muggles and 13 for sensi.<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule:5:&lt;h2&gt; --><h2 id="toc2"><a name="Theory-Intervals and linear temperaments"></a><!-- ws:end:WikiTextHeadingRule:5 -->Intervals and linear temperaments</h2> | <!-- ws:start:WikiTextHeadingRule:5:&lt;h2&gt; --><h2 id="toc2"><a name="Theory-Intervals and linear temperaments"></a><!-- ws:end:WikiTextHeadingRule:5 -->Intervals and linear temperaments</h2> | ||