User:MasonGreen1/Naughty and nice harmonics: 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:

: This revision was by author [[User:MasonGreen1|MasonGreen1]] and made on <tt>2015-11-23 14:53:24 UTC</tt>.

: This revision was by author [[User:MasonGreen1|MasonGreen1]] and made on <tt>2015-11-23 14:54:32 UTC</tt>.

: The original revision id was <tt>~~567517025~~</tt>.

: The original revision id was <tt>567517253</tt>.

: The revision comment was: <tt></tt>

The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.

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It might be a good idea to use a modified [[The Riemann Zeta Function and Tuning|Z function]] when analyzing higher edos, removing the component corresponding to the 11th harmonic (thus making it a no-elevens Z function) while increasing the weighting of the component corresponding to 19. Using such a function even more firmly establishes 12edo's position as supreme among small edos, since 12edo matches the 19th harmonic very closely while avoiding the 11th.

Among higher edos, the one that has the most to gain is [[53edo]], which closely matches the 13 and 19, while matching the 17 less well and completely avoiding the 11.

Among higher edos, the one that has the most to gain is [[53edo]], which closely matches the 13 and 19, while matching the 17 less well and completely avoiding the 11. Even using the //un//modified Z function, 53edo is already a zeta peak, integral, //and// gap edo (and it shares that rare distinction with 12edo). Using the modified Z-function improves its standing even further.

~~Even using~~ the ~~//un//modified Z function, 53edo is already a~~ zeta ~~peak, integral, //and// gap edo~~ (and ~~it shares that rare distinction with 12edo~~)~~. Using the modified Z-function improves its standing even further.~~

Of the other two "triple zeta edos" in this range (19edo and 31edo), 19edo fares fairly well, too; while it does not match the "nice" 19th or the 17th, it does match the 13th and avoids the "naughty" 11th. 31edo, on the other hand, does not look nearly as good when using the modified Z-function, since it closely matches the 11th "naughty" harmonic while not matching the 13th, 17th, or "nice" 19th nearly as well.

19edo fares fairly well, too; while it does not match the "nice" 19th or the 17th, it does match the 13th and avoids the "naughty" 11th. 31edo, on the other hand, does not look nearly as good when using the modified Z-function, since it closely matches the 11th "naughty" harmonic while not matching the 13th, 17th, or "nice" 19th nearly as well.

Thus, if "naughtiness and niceness" of higher harmonics is taken into account, 53edo looks like one of the best candidates for extending 12edo. It sounds similar to 12edo in many respects (both tunings closely match the perfect fifth, and while 53edo is not a meantone tuning, it does closely approximate Pythagorean tuning, as does 12edo). 53edo does not contain 12edo as a subset, but [[60edo]] (which is consistent as a no-elevens, no-seventeens 27 limit system) does, making that tuning a good option as well.</pre></div>

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It might be a good idea to use a modified <a class="wiki_link" href="/The%20Riemann%20Zeta%20Function%20and%20Tuning">Z function</a> when analyzing higher edos, removing the component corresponding to the 11th harmonic (thus making it a no-elevens Z function) while increasing the weighting of the component corresponding to 19. Using such a function even more firmly establishes 12edo's position as supreme among small edos, since 12edo matches the 19th harmonic very closely while avoiding the 11th.

Among higher edos, the one that has the most to gain is <a class="wiki_link" href="/53edo">53edo</a>, which closely matches the 13 and 19, while matching the 17 less well and completely avoiding the 11.~~ ~~

Among higher edos, the one that has the most to gain is <a class="wiki_link" href="/53edo">53edo</a>, which closely matches the 13 and 19, while matching the 17 less well and completely avoiding the 11. Even using the unmodified Z function, 53edo is already a zeta peak, integral, and gap edo (and it shares that rare distinction with 12edo). Using the modified Z-function improves its standing even further.

~~ ~~

Even using the unmodified Z function, 53edo is already a zeta peak, integral, and gap edo (and it shares that rare distinction with 12edo). Using the modified Z-function improves its standing even further.

19edo fares fairly well, too; while it does not match the &quot;nice&quot; 19th or the 17th, it does match the 13th and avoids the &quot;naughty&quot; 11th. 31edo, on the other hand, does not look nearly as good when using the modified Z-function, since it closely matches the 11th &quot;naughty&quot; harmonic while not matching the 13th, 17th, or &quot;nice&quot; 19th nearly as well.

Of the other two &quot;triple zeta edos&quot; in this range (19edo and 31edo), 19edo fares fairly well, too; while it does not match the &quot;nice&quot; 19th or the 17th, it does match the 13th and avoids the &quot;naughty&quot; 11th. 31edo, on the other hand, does not look nearly as good when using the modified Z-function, since it closely matches the 11th &quot;naughty&quot; harmonic while not matching the 13th, 17th, or &quot;nice&quot; 19th nearly as well.

Thus, if &quot;naughtiness and niceness&quot; of higher harmonics is taken into account, 53edo looks like one of the best candidates for extending 12edo. It sounds similar to 12edo in many respects (both tunings closely match the perfect fifth, and while 53edo is not a meantone tuning, it does closely approximate Pythagorean tuning, as does 12edo). 53edo does not contain 12edo as a subset, but <a class="wiki_link" href="/60edo">60edo</a> (which is consistent as a no-elevens, no-seventeens 27 limit system) does, making that tuning a good option as well.</body></html></pre></div>

@@ 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:MasonGreen1|MasonGreen1]] and made on <tt>2015-11-23 14:53:24 UTC</tt>.<br>
+: This revision was by author [[User:MasonGreen1|MasonGreen1]] and made on <tt>2015-11-23 14:54:32 UTC</tt>.<br>
-: The original revision id was <tt>567517025</tt>.<br>
+: The original revision id was <tt>567517253</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 27: / Line 27: @@
 It might be a good idea to use a modified [[The Riemann Zeta Function and Tuning|Z function]] when analyzing higher edos, removing the component corresponding to the 11th harmonic (thus making it a no-elevens Z function) while increasing the weighting of the component corresponding to 19. Using such a function even more firmly establishes 12edo's position as supreme among small edos, since 12edo matches the 19th harmonic very closely while avoiding the 11th.
-Among higher edos, the one that has the most to gain is [[53edo]], which closely matches the 13 and 19, while matching the 17 less well and completely avoiding the 11.
+Among higher edos, the one that has the most to gain is [[53edo]], which closely matches the 13 and 19, while matching the 17 less well and completely avoiding the 11. Even using the //un//modified Z function, 53edo is already a zeta peak, integral, //and// gap edo (and it shares that rare distinction with 12edo). Using the modified Z-function improves its standing even further.
-Even using the //un//modified Z function, 53edo is already a zeta peak, integral, //and// gap edo (and it shares that rare distinction with 12edo). Using the modified Z-function improves its standing even further.
+Of the other two "triple zeta edos" in this range (19edo and 31edo), 19edo fares fairly well, too; while it does not match the "nice" 19th or the 17th, it does match the 13th and avoids the "naughty" 11th. 31edo, on the other hand, does not look nearly as good when using the modified Z-function, since it closely matches the 11th "naughty" harmonic while not matching the 13th, 17th, or "nice" 19th nearly as well.
-edo fares fairly well, too; while it does not match the "nice" 19th or the 17th, it does match the 13th and avoids the "naughty" 11th. 31edo, on the other hand, does not look nearly as good when using the modified Z-function, since it closely matches the 11th "naughty" harmonic while not matching the 13th, 17th, or "nice" 19th nearly as well.
 Thus, if "naughtiness and niceness" of higher harmonics is taken into account, 53edo looks like one of the best candidates for extending 12edo. It sounds similar to 12edo in many respects (both tunings closely match the perfect fifth, and while 53edo is not a meantone tuning, it does closely approximate Pythagorean tuning, as does 12edo). 53edo does not contain 12edo as a subset, but [[60edo]] (which is consistent as a no-elevens, no-seventeens 27 limit system) does, making that tuning a good option as well.</pre></div>
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 It might be a good idea to use a modified &lt;a class="wiki_link" href="/The%20Riemann%20Zeta%20Function%20and%20Tuning"&gt;Z function&lt;/a&gt; when analyzing higher edos, removing the component corresponding to the 11th harmonic (thus making it a no-elevens Z function) while increasing the weighting of the component corresponding to 19. Using such a function even more firmly establishes 12edo's position as supreme among small edos, since 12edo matches the 19th harmonic very closely while avoiding the 11th.&lt;br /&gt;
 &lt;br /&gt;
-Among higher edos, the one that has the most to gain is &lt;a class="wiki_link" href="/53edo"&gt;53edo&lt;/a&gt;, which closely matches the 13 and 19, while matching the 17 less well and completely avoiding the 11.&lt;br /&gt;
+Among higher edos, the one that has the most to gain is &lt;a class="wiki_link" href="/53edo"&gt;53edo&lt;/a&gt;, which closely matches the 13 and 19, while matching the 17 less well and completely avoiding the 11. Even using the &lt;em&gt;un&lt;/em&gt;modified Z function, 53edo is already a zeta peak, integral, &lt;em&gt;and&lt;/em&gt; gap edo (and it shares that rare distinction with 12edo). Using the modified Z-function improves its standing even further.&lt;br /&gt;
-&lt;br /&gt;
-Even using the &lt;em&gt;un&lt;/em&gt;modified Z function, 53edo is already a zeta peak, integral, &lt;em&gt;and&lt;/em&gt; gap edo (and it shares that rare distinction with 12edo). Using the modified Z-function improves its standing even further.&lt;br /&gt;
 &lt;br /&gt;
-edo fares fairly well, too; while it does not match the &amp;quot;nice&amp;quot; 19th or the 17th, it does match the 13th and avoids the &amp;quot;naughty&amp;quot; 11th. 31edo, on the other hand, does not look nearly as good when using the modified Z-function, since it closely matches the 11th &amp;quot;naughty&amp;quot; harmonic while not matching the 13th, 17th, or &amp;quot;nice&amp;quot; 19th nearly as well.&lt;br /&gt;
+Of the other two &amp;quot;triple zeta edos&amp;quot; in this range (19edo and 31edo), 19edo fares fairly well, too; while it does not match the &amp;quot;nice&amp;quot; 19th or the 17th, it does match the 13th and avoids the &amp;quot;naughty&amp;quot; 11th. 31edo, on the other hand, does not look nearly as good when using the modified Z-function, since it closely matches the 11th &amp;quot;naughty&amp;quot; harmonic while not matching the 13th, 17th, or &amp;quot;nice&amp;quot; 19th nearly as well.&lt;br /&gt;
 &lt;br /&gt;
 Thus, if &amp;quot;naughtiness and niceness&amp;quot; of higher harmonics is taken into account, 53edo looks like one of the best candidates for extending 12edo. It sounds similar to 12edo in many respects (both tunings closely match the perfect fifth, and while 53edo is not a meantone tuning, it does closely approximate Pythagorean tuning, as does 12edo). 53edo does not contain 12edo as a subset, but &lt;a class="wiki_link" href="/60edo"&gt;60edo&lt;/a&gt; (which is consistent as a no-elevens, no-seventeens 27 limit system) does, making that tuning a good option as well.&lt;/body&gt;&lt;/html&gt;</pre></div>