34edo: Difference between revisions
Wikispaces>PiotrGrochowski **Imported revision 591574080 - Original comment: ** |
Wikispaces>JosephRuhf **Imported revision 596940418 - 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:JosephRuhf|JosephRuhf]] and made on <tt>2016-10-25 08:35:04 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>596940418</tt>.<br> | ||
: The revision comment was: <tt></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> | 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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//Viewed in light of Western diatonic theory, the three extra steps (of 34-et compared to 31-et) in effect widen the intervals between C and D, F and G, and A and B [that is: 6 5 3 6 5 6 3], thus making a distinction between major tones, ratio 9/8 and minor tones, ratio 10/9.// ([[http://en.wikipedia.org/wiki/34_equal_temperament|Wikipedia]]) | //Viewed in light of Western diatonic theory, the three extra steps (of 34-et compared to 31-et) in effect widen the intervals between C and D, F and G, and A and B [that is: 6 5 3 6 5 6 3], thus making a distinction between major tones, ratio 9/8 and minor tones, ratio 10/9.// ([[http://en.wikipedia.org/wiki/34_equal_temperament|Wikipedia]]) | ||
* The | * The sharpening of ~13 cents of 11/8 can fit with the 9/8 and 13/8 which both are about 7 cents sharp. This the basis of a subtle trick: the guitarist tunes the high 'E' string flat by several cents, enough to be imperceptible in many contexts, but which makes chords/harmonies against those several intervals tuned more justly. | ||
Likewise the 16-cent flat 27\34 approximate 7/4 can be musically useful. It is an improvement over the yet sharper "dominant seventh" found in jazz - which some listeners are | Likewise the 16-cent flat 27\34 approximate 7/4 can be musically useful. It is an improvement over the yet sharper "dominant seventh" found in jazz - which some listeners are accustomed to. The ability to tolerate these errors may depend on subtle natural changes in mood. A few cents either way can bother the hell out of one, but on other days you might spend an hour not knowing of the strings are, or being able to, tuned. Nevertheless [[68edo]] (34 x 2) preserves the structure and has these intervals 7/8 and 11/8 in more perfect form... nearly just. | ||
=34edo and phi= | =34edo and phi= | ||
As a Fibonacci number, 34edo contains a fraction of an octave which is close approximation to the irrational interval phi -- 21 degrees of 34edo, approximately 741.2 cents. Repeated iterations of this interval generates [[xenharmonic/MOSScales|Moment of Symmetry]] scales with near-phi relationships between the step sizes. As a 2.3.5.13 temperament, the 21\34 generator is an approximate 20/13, and the temperament tempers out 512/507 and 140625/140608. From the tempering of 512/507, two 16/13 neutral thirds are an approximate 3/2, defining an essentially tempered neutral triad with a sharp rather than a flat fifth. Yes. But, to be clear the harmonic ratio of phi is ~ 833 cents, and the equal divisions of octave approximating this interval closely are 13edo and [[36edo]]. | As a Fibonacci number, 34edo contains a fraction of an octave which is close approximation to the irrational interval phi -- 21 degrees of 34edo, approximately 741.2 cents. Repeated iterations of this interval generates [[xenharmonic/MOSScales|Moment of Symmetry]] scales with near-phi relationships between the step sizes. As a 2.3.5.13 temperament, the 21\34 generator is an approximate 20/13, and the temperament tempers out 512/507 and [[tel:140625/140608|140625/140608]]. From the tempering of 512/507, two 16/13 neutral thirds are an approximate 3/2, defining an essentially tempered neutral triad with a sharp rather than a flat fifth. Yes. But, to be clear the harmonic ratio of phi is ~ 833 cents, and the equal divisions of octave approximating this interval closely are 13edo and [[36edo]]. | ||
=Rank two temperaments= | =Rank two temperaments= | ||
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||= 20000/19683 || | 5 -9 4 > ||> 27.660 ||= Minimal Diesis, Tetracot Comma || | ||= 20000/19683 || | 5 -9 4 > ||> 27.660 ||= Minimal Diesis, Tetracot Comma || | ||
||= 2048/2025 || | 11 -4 -2 > ||> 19.553 ||= Diaschisma || | ||= 2048/2025 || | 11 -4 -2 > ||> 19.553 ||= Diaschisma || | ||
||= 393216/390625 || | 17 1 -8 > ||> 11.445 ||= Würschmidt comma || | ||= [[tel:393216/390625|393216/390625]] || | 17 1 -8 > ||> 11.445 ||= Würschmidt comma || | ||
||= 15625/15552 || | -6 -5 6 > ||> 8.1073 ||= Kleisma, Semicomma Majeur || | ||= 15625/15552 || | -6 -5 6 > ||> 8.1073 ||= Kleisma, Semicomma Majeur || | ||
||= 1212717/1210381 || | 23 6 -14 > ||> 3.338 ||= Vishnuzma, Semisuper || | ||= [[tel:1212717/1210381|1212717/1210381]] || | 23 6 -14 > ||> 3.338 ||= Vishnuzma, Semisuper || | ||
||= 1029/1000 || | -3 1 -3 3 > ||> 49.492 ||= Keega || | ||= 1029/1000 || | -3 1 -3 3 > ||> 49.492 ||= Keega || | ||
||= [[50_49|50/49]] || | 1 0 2 -2 > ||> 34.976 ||= Fifty forty-nine || | ||= [[50_49|50/49]] || | 1 0 2 -2 > ||> 34.976 ||= Fifty forty-nine || | ||
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<em>Viewed in light of Western diatonic theory, the three extra steps (of 34-et compared to 31-et) in effect widen the intervals between C and D, F and G, and A and B [that is: 6 5 3 6 5 6 3], thus making a distinction between major tones, ratio 9/8 and minor tones, ratio 10/9.</em> (<a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/34_equal_temperament" rel="nofollow">Wikipedia</a>)<br /> | <em>Viewed in light of Western diatonic theory, the three extra steps (of 34-et compared to 31-et) in effect widen the intervals between C and D, F and G, and A and B [that is: 6 5 3 6 5 6 3], thus making a distinction between major tones, ratio 9/8 and minor tones, ratio 10/9.</em> (<a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/34_equal_temperament" rel="nofollow">Wikipedia</a>)<br /> | ||
<br /> | <br /> | ||
<ul><li>The | <ul><li>The sharpening of ~13 cents of 11/8 can fit with the 9/8 and 13/8 which both are about 7 cents sharp. This the basis of a subtle trick: the guitarist tunes the high 'E' string flat by several cents, enough to be imperceptible in many contexts, but which makes chords/harmonies against those several intervals tuned more justly.</li></ul><br /> | ||
Likewise the 16-cent flat 27\34 approximate 7/4 can be musically useful. It is an improvement over the yet sharper &quot;dominant seventh&quot; found in jazz - which some listeners are | Likewise the 16-cent flat 27\34 approximate 7/4 can be musically useful. It is an improvement over the yet sharper &quot;dominant seventh&quot; found in jazz - which some listeners are accustomed to. The ability to tolerate these errors may depend on subtle natural changes in mood. A few cents either way can bother the hell out of one, but on other days you might spend an hour not knowing of the strings are, or being able to, tuned. Nevertheless <a class="wiki_link" href="/68edo">68edo</a> (34 x 2) preserves the structure and has these intervals 7/8 and 11/8 in more perfect form... nearly just.<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule:2:&lt;h1&gt; --><h1 id="toc1"><a name="x34edo and phi"></a><!-- ws:end:WikiTextHeadingRule:2 -->34edo and phi</h1> | <!-- ws:start:WikiTextHeadingRule:2:&lt;h1&gt; --><h1 id="toc1"><a name="x34edo and phi"></a><!-- ws:end:WikiTextHeadingRule:2 -->34edo and phi</h1> | ||
As a Fibonacci number, 34edo contains a fraction of an octave which is close approximation to the irrational interval phi -- 21 degrees of 34edo, approximately 741.2 cents. Repeated iterations of this interval generates <a class="wiki_link" href="http://xenharmonic.wikispaces.com/MOSScales">Moment of Symmetry</a> scales with near-phi relationships between the step sizes. As a 2.3.5.13 temperament, the 21\34 generator is an approximate 20/13, and the temperament tempers out 512/507 and 140625/140608. From the tempering of 512/507, two 16/13 neutral thirds are an approximate 3/2, defining an essentially tempered neutral triad with a sharp rather than a flat fifth. Yes. But, to be clear the harmonic ratio of phi is ~ 833 cents, and the equal divisions of octave approximating this interval closely are 13edo and <a class="wiki_link" href="/36edo">36edo</a>.<br /> | As a Fibonacci number, 34edo contains a fraction of an octave which is close approximation to the irrational interval phi -- 21 degrees of 34edo, approximately 741.2 cents. Repeated iterations of this interval generates <a class="wiki_link" href="http://xenharmonic.wikispaces.com/MOSScales">Moment of Symmetry</a> scales with near-phi relationships between the step sizes. As a 2.3.5.13 temperament, the 21\34 generator is an approximate 20/13, and the temperament tempers out 512/507 and [[tel:140625/140608|140625/140608]]. From the tempering of 512/507, two 16/13 neutral thirds are an approximate 3/2, defining an essentially tempered neutral triad with a sharp rather than a flat fifth. Yes. But, to be clear the harmonic ratio of phi is ~ 833 cents, and the equal divisions of octave approximating this interval closely are 13edo and <a class="wiki_link" href="/36edo">36edo</a>.<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule:4:&lt;h1&gt; --><h1 id="toc2"><a name="Rank two temperaments"></a><!-- ws:end:WikiTextHeadingRule:4 -->Rank two temperaments</h1> | <!-- ws:start:WikiTextHeadingRule:4:&lt;h1&gt; --><h1 id="toc2"><a name="Rank two temperaments"></a><!-- ws:end:WikiTextHeadingRule:4 -->Rank two temperaments</h1> | ||
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</tr> | </tr> | ||
<tr> | <tr> | ||
<td style="text-align: center;">393216/390625<br /> | <td style="text-align: center;">[[tel:393216/390625|393216/390625]]<br /> | ||
</td> | </td> | ||
<td>| 17 1 -8 &gt;<br /> | <td>| 17 1 -8 &gt;<br /> | ||
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</tr> | </tr> | ||
<tr> | <tr> | ||
<td style="text-align: center;">1212717/1210381<br /> | <td style="text-align: center;">[[tel:1212717/1210381|1212717/1210381]]<br /> | ||
</td> | </td> | ||
<td>| 23 6 -14 &gt;<br /> | <td>| 23 6 -14 &gt;<br /> | ||