Tour of regular temperaments: 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:x31eq|x31eq]] and made on <tt>2007-08-10 08:04:41 UTC</tt>.<br>

: This revision was by author [[User:x31eq|x31eq]] and made on <tt>2007-08-10 08:26:54 UTC</tt>.<br>

: The original revision id was <tt>~~6728049~~</tt>.<br>

: The original revision id was <tt>6728403</tt>.<br>

: The revision comment was: <tt>~~Canasta isn't about equal temperaments~~</tt><br>

: The revision comment was: <tt>More details</tt><br>

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

<h4>Original Wikitext content:</h4>

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===Magic[[#magic]]===

Magic is based on a chain of major thirds.

Magic is based on a chain of major thirds. It's optimal, in a sense, for 9-limit harmony with a tuning close to 41-EDO. It's more accurate than meantone and simpler than schismatic. It works with 19 and 22 note scales. Five major thirds approximate 3/1. Twelve major thirds, less an octave, approximate 7/1. It's a little tricky to work with because 3/2 fifths are a relatively complex interval and it doesn't naturally work with scales of around 7 notes to the octave.

===Meantone[[#meantone]]===

This is the most familiar of the rank 2 temperaments. The syntonic comma, 81/80 is tempered out; any intervals that differ by 81/80 in just intonation are tempered to the same interval in meantone temperament.

This is the most familiar of the rank 2 temperaments. The syntonic comma, 81/80 is tempered out; any intervals that differ by 81/80 in just intonation are tempered to the same interval in meantone temperament. This means four fiths approximate 5/1.

===Miracle[[#miracle]]===

Miracle temperament divides the fifth into 6 equal steps. A 21-note scale called "blackjack" and a 31-note scale called "canasta" have some useful properties.

Miracle temperament divides the fifth into 6 equal steps. A 21-note scale called "blackjack" and a 31-note scale called "canasta" have some useful properties. It's the most efficient 11-limit temperament for many purposes with a tuning close to 72-EDO.

===Orwell[[#orwell]]===

So called because 19/84 (as a fraction of the octave) is a possible generator of this temperament, orwell divides the interval of a twelfth (a tempered 3/1 frequency ratio) into 7 equal steps.

So called because 19/84 (as a fraction of the octave) is a possible generator of this temperament, orwell divides the interval of a twelfth (a tempered 3/1 frequency ratio) into 7 equal steps. It's compatible with 22, 31 and 53-EDO. It's reasonable in the 7-limit and naturally extends into the 11-limit.

===Pajara[[#pajara]]===

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===Schismatic (helmholtz, garibaldi)[[#schismatic]]===

Schismatic temperament reduces the size of the perfect fifth by a fraction of a schisma (the difference between a major third and a diminished fourth, 32805/32768).

Schismatic temperament reduces the size of the perfect fifth by a fraction of a schisma (the difference between a major third and a diminished fourth, 32805/32768). It's much more accurate than meantone with manageable complexity. It also works well in the 7-limit but with lower accuracy.

==Rank 3 temperaments==

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<h3 id="toc4"><a name="x-Rank 2 (including &quot;linear&quot;) temperaments-Magic"></a>Magic<a name="magic"></a></h3>

<br />

Magic is based on a chain of major thirds.<br />

Magic is based on a chain of major thirds. It's optimal, in a sense, for 9-limit harmony with a tuning close to 41-EDO. It's more accurate than meantone and simpler than schismatic. It works with 19 and 22 note scales. Five major thirds approximate 3/1. Twelve major thirds, less an octave, approximate 7/1. It's a little tricky to work with because 3/2 fifths are a relatively complex interval and it doesn't naturally work with scales of around 7 notes to the octave.<br />

<br />

<h3 id="toc5"><a name="x-Rank 2 (including &quot;linear&quot;) temperaments-Meantone"></a>Meantone<a name="meantone"></a></h3>

<br />

This is the most familiar of the rank 2 temperaments. The syntonic comma, 81/80 is tempered out; any intervals that differ by 81/80 in just intonation are tempered to the same interval in meantone temperament.<br />

This is the most familiar of the rank 2 temperaments. The syntonic comma, 81/80 is tempered out; any intervals that differ by 81/80 in just intonation are tempered to the same interval in meantone temperament. This means four fiths approximate 5/1.<br />

<br />

<h3 id="toc6"><a name="x-Rank 2 (including &quot;linear&quot;) temperaments-Miracle"></a>Miracle<a name="miracle"></a></h3>

<br />

Miracle temperament divides the fifth into 6 equal steps. A 21-note scale called &quot;blackjack&quot; and a 31-note scale called &quot;canasta&quot; have some useful properties.<br />

Miracle temperament divides the fifth into 6 equal steps. A 21-note scale called &quot;blackjack&quot; and a 31-note scale called &quot;canasta&quot; have some useful properties. It's the most efficient 11-limit temperament for many purposes with a tuning close to 72-EDO.<br />

<br />

<h3 id="toc7"><a name="x-Rank 2 (including &quot;linear&quot;) temperaments-Orwell"></a>Orwell<a name="orwell"></a></h3>

<br />

So called because 19/84 (as a fraction of the octave) is a possible generator of this temperament, orwell divides the interval of a twelfth (a tempered 3/1 frequency ratio) into 7 equal steps.<br />

So called because 19/84 (as a fraction of the octave) is a possible generator of this temperament, orwell divides the interval of a twelfth (a tempered 3/1 frequency ratio) into 7 equal steps. It's compatible with 22, 31 and 53-EDO. It's reasonable in the 7-limit and naturally extends into the 11-limit.<br />

<br />

<h3 id="toc8"><a name="x-Rank 2 (including &quot;linear&quot;) temperaments-Pajara"></a>Pajara<a name="pajara"></a></h3>

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<h3 id="toc10"><a name="x-Rank 2 (including &quot;linear&quot;) temperaments-Schismatic (helmholtz, garibaldi)"></a>Schismatic (helmholtz, garibaldi)<a name="schismatic"></a></h3>

<br />

Schismatic temperament reduces the size of the perfect fifth by a fraction of a schisma (the difference between a major third and a diminished fourth, 32805/32768).<br />

Schismatic temperament reduces the size of the perfect fifth by a fraction of a schisma (the difference between a major third and a diminished fourth, 32805/32768). It's much more accurate than meantone with manageable complexity. It also works well in the 7-limit but with lower accuracy.<br />

<br />

<h2 id="toc11"><a name="x-Rank 3 temperaments"></a>Rank 3 temperaments</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:x31eq|x31eq]] and made on <tt>2007-08-10 08:04:41 UTC</tt>.<br>
+: This revision was by author [[User:x31eq|x31eq]] and made on <tt>2007-08-10 08:26:54 UTC</tt>.<br>
-: The original revision id was <tt>6728049</tt>.<br>
+: The original revision id was <tt>6728403</tt>.<br>
-: The revision comment was: <tt>Canasta isn't about equal temperaments</tt><br>
+: The revision comment was: <tt>More details</tt><br>
 The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br>
 <h4>Original Wikitext content:</h4>
@@ Line 24: / Line 24: @@
 ===Magic[[#magic]]===
-Magic is based on a chain of major thirds.
+Magic is based on a chain of major thirds. It's optimal, in a sense, for 9-limit harmony with a tuning close to 41-EDO. It's more accurate than meantone and simpler than schismatic. It works with 19 and 22 note scales. Five major thirds approximate 3/1. Twelve major thirds, less an octave, approximate 7/1. It's a little tricky to work with because 3/2 fifths are a relatively complex interval and it doesn't naturally work with scales of around 7 notes to the octave.
 ===Meantone[[#meantone]]===
-This is the most familiar of the rank 2 temperaments. The syntonic comma, 81/80 is tempered out; any intervals that differ by 81/80 in just intonation are tempered to the same interval in meantone temperament.
+This is the most familiar of the rank 2 temperaments. The syntonic comma, 81/80 is tempered out; any intervals that differ by 81/80 in just intonation are tempered to the same interval in meantone temperament. This means four fiths approximate 5/1.
 ===Miracle[[#miracle]]===
-Miracle temperament divides the fifth into 6 equal steps. A 21-note scale called "blackjack" and a  31-note scale called "canasta" have some useful properties.
+Miracle temperament divides the fifth into 6 equal steps. A 21-note scale called "blackjack" and a 31-note scale called "canasta" have some useful properties. It's the most efficient 11-limit temperament for many purposes with a tuning close to 72-EDO.
 ===Orwell[[#orwell]]===
-So called because 19/84 (as a fraction of the octave) is a possible generator of this temperament, orwell divides the interval of a twelfth (a tempered 3/1 frequency ratio) into 7 equal steps.
+So called because 19/84 (as a fraction of the octave) is a possible generator of this temperament, orwell divides the interval of a twelfth (a tempered 3/1 frequency ratio) into 7 equal steps. It's compatible with 22, 31 and 53-EDO. It's reasonable in the 7-limit and naturally extends into the 11-limit.
 ===Pajara[[#pajara]]===
@@ Line 48: / Line 48: @@
 ===Schismatic (helmholtz, garibaldi)[[#schismatic]]===
-Schismatic temperament reduces the size of the perfect fifth by a fraction of a schisma (the difference between a major third and a diminished fourth, 32805/32768).
+Schismatic temperament reduces the size of the perfect fifth by a fraction of a schisma (the difference between a major third and a diminished fourth, 32805/32768). It's much more accurate than meantone with manageable complexity. It also works well in the 7-limit but with lower accuracy.
 ==Rank 3 temperaments==
@@ Line 84: / Line 84: @@
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   &lt;br /&gt;
-Magic is based on a chain of major thirds.&lt;br /&gt;
+Magic is based on a chain of major thirds. It's optimal, in a sense, for 9-limit harmony with a tuning close to 41-EDO. It's more accurate than meantone and simpler than schismatic. It works with 19 and 22 note scales. Five major thirds approximate 3/1. Twelve major thirds, less an octave, approximate 7/1. It's a little tricky to work with because 3/2 fifths are a relatively complex interval and it doesn't naturally work with scales of around 7 notes to the octave.&lt;br /&gt;
 &lt;br /&gt;
 &lt;!-- ws:start:WikiTextHeadingRule:10:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc5"&gt;&lt;a name="x-Rank 2 (including &amp;quot;linear&amp;quot;) temperaments-Meantone"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:10 --&gt;Meantone&lt;!-- ws:start:WikiTextAnchorRule:33:&amp;lt;img src=&amp;quot;/i/anchor.gif&amp;quot; class=&amp;quot;WikiAnchor&amp;quot; alt=&amp;quot;Anchor&amp;quot; id=&amp;quot;wikitext@@anchor@@meantone&amp;quot; title=&amp;quot;Anchor: meantone&amp;quot;/&amp;gt; --&gt;&lt;a name="meantone"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextAnchorRule:33 --&gt;&lt;/h3&gt;
   &lt;br /&gt;
-This is the most familiar of the rank 2 temperaments. The syntonic comma, 81/80 is tempered out; any intervals that differ by 81/80 in just intonation are tempered to the same interval in meantone temperament.&lt;br /&gt;
+This is the most familiar of the rank 2 temperaments. The syntonic comma, 81/80 is tempered out; any intervals that differ by 81/80 in just intonation are tempered to the same interval in meantone temperament. This means four fiths approximate 5/1.&lt;br /&gt;
 &lt;br /&gt;
 &lt;!-- ws:start:WikiTextHeadingRule:12:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc6"&gt;&lt;a name="x-Rank 2 (including &amp;quot;linear&amp;quot;) temperaments-Miracle"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:12 --&gt;Miracle&lt;!-- ws:start:WikiTextAnchorRule:34:&amp;lt;img src=&amp;quot;/i/anchor.gif&amp;quot; class=&amp;quot;WikiAnchor&amp;quot; alt=&amp;quot;Anchor&amp;quot; id=&amp;quot;wikitext@@anchor@@miracle&amp;quot; title=&amp;quot;Anchor: miracle&amp;quot;/&amp;gt; --&gt;&lt;a name="miracle"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextAnchorRule:34 --&gt;&lt;/h3&gt;
   &lt;br /&gt;
-Miracle temperament divides the fifth into 6 equal steps. A 21-note scale called &amp;quot;blackjack&amp;quot; and a  31-note scale called &amp;quot;canasta&amp;quot; have some useful properties.&lt;br /&gt;
+Miracle temperament divides the fifth into 6 equal steps. A 21-note scale called &amp;quot;blackjack&amp;quot; and a 31-note scale called &amp;quot;canasta&amp;quot; have some useful properties. It's the most efficient 11-limit temperament for many purposes with a tuning close to 72-EDO.&lt;br /&gt;
 &lt;br /&gt;
 &lt;!-- ws:start:WikiTextHeadingRule:14:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc7"&gt;&lt;a name="x-Rank 2 (including &amp;quot;linear&amp;quot;) temperaments-Orwell"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:14 --&gt;Orwell&lt;!-- ws:start:WikiTextAnchorRule:35:&amp;lt;img src=&amp;quot;/i/anchor.gif&amp;quot; class=&amp;quot;WikiAnchor&amp;quot; alt=&amp;quot;Anchor&amp;quot; id=&amp;quot;wikitext@@anchor@@orwell&amp;quot; title=&amp;quot;Anchor: orwell&amp;quot;/&amp;gt; --&gt;&lt;a name="orwell"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextAnchorRule:35 --&gt;&lt;/h3&gt;
   &lt;br /&gt;
-So called because 19/84 (as a fraction of the octave) is a possible generator of this temperament, orwell divides the interval of a twelfth (a tempered 3/1 frequency ratio) into 7 equal steps.&lt;br /&gt;
+So called because 19/84 (as a fraction of the octave) is a possible generator of this temperament, orwell divides the interval of a twelfth (a tempered 3/1 frequency ratio) into 7 equal steps. It's compatible with 22, 31 and 53-EDO. It's reasonable in the 7-limit and naturally extends into the 11-limit.&lt;br /&gt;
 &lt;br /&gt;
 &lt;!-- ws:start:WikiTextHeadingRule:16:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc8"&gt;&lt;a name="x-Rank 2 (including &amp;quot;linear&amp;quot;) temperaments-Pajara"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:16 --&gt;Pajara&lt;!-- ws:start:WikiTextAnchorRule:36:&amp;lt;img src=&amp;quot;/i/anchor.gif&amp;quot; class=&amp;quot;WikiAnchor&amp;quot; alt=&amp;quot;Anchor&amp;quot; id=&amp;quot;wikitext@@anchor@@pajara&amp;quot; title=&amp;quot;Anchor: pajara&amp;quot;/&amp;gt; --&gt;&lt;a name="pajara"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextAnchorRule:36 --&gt;&lt;/h3&gt;
@@ Line 108: / Line 108: @@
 &lt;!-- ws:start:WikiTextHeadingRule:20:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc10"&gt;&lt;a name="x-Rank 2 (including &amp;quot;linear&amp;quot;) temperaments-Schismatic (helmholtz, garibaldi)"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:20 --&gt;Schismatic (helmholtz, garibaldi)&lt;!-- ws:start:WikiTextAnchorRule:38:&amp;lt;img src=&amp;quot;/i/anchor.gif&amp;quot; class=&amp;quot;WikiAnchor&amp;quot; alt=&amp;quot;Anchor&amp;quot; id=&amp;quot;wikitext@@anchor@@schismatic&amp;quot; title=&amp;quot;Anchor: schismatic&amp;quot;/&amp;gt; --&gt;&lt;a name="schismatic"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextAnchorRule:38 --&gt;&lt;/h3&gt;
   &lt;br /&gt;
-Schismatic temperament reduces the size of the perfect fifth by a fraction of a schisma (the difference between a major third and a diminished fourth, 32805/32768).&lt;br /&gt;
+Schismatic temperament reduces the size of the perfect fifth by a fraction of a schisma (the difference between a major third and a diminished fourth, 32805/32768). It's much more accurate than meantone with manageable complexity. It also works well in the 7-limit but with lower accuracy.&lt;br /&gt;
 &lt;br /&gt;
 &lt;!-- ws:start:WikiTextHeadingRule:22:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc11"&gt;&lt;a name="x-Rank 3 temperaments"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:22 --&gt;Rank 3 temperaments&lt;/h2&gt;