Tour of regular temperaments: Difference between revisions
Wikispaces>hstraub **Imported revision 5478247 - Original comment: ** |
Wikispaces>hstraub **Imported revision 5510469 - Original comment: More anchors** |
||
| Line 1: | Line 1: | ||
<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:hstraub|hstraub]] and made on <tt>2007-06- | : This revision was by author [[User:hstraub|hstraub]] and made on <tt>2007-06-26 09:04:28 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>5510469</tt>.<br> | ||
: The revision comment was: <tt></tt><br> | : The revision comment was: <tt>More anchors</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> | ||
<h4>Original Wikitext content:</h4> | <h4>Original Wikitext content:</h4> | ||
| Line 10: | Line 10: | ||
[[Equal Temperaments| Equal temperaments]] (abbreviated ET or tET) and equal divisions of the octave (abbreviated EDO) are similar concepts, although there are distinctions in the way these terms are used. An EDO is simply a division of the octave into equal steps (specifically, steps of equal size in cents). An ET, on the other hand, is a temperament, an altered representation of some subset of the intervals of just intonation. The familiar 12-note equal temperament (12-ET) reduces the size of the perfect fifth (about 701.955 cents) by 1/12 of the Pythagorean comma, resulting in a fifth of 700.0 cents. | [[Equal Temperaments| Equal temperaments]] (abbreviated ET or tET) and equal divisions of the octave (abbreviated EDO) are similar concepts, although there are distinctions in the way these terms are used. An EDO is simply a division of the octave into equal steps (specifically, steps of equal size in cents). An ET, on the other hand, is a temperament, an altered representation of some subset of the intervals of just intonation. The familiar 12-note equal temperament (12-ET) reduces the size of the perfect fifth (about 701.955 cents) by 1/12 of the Pythagorean comma, resulting in a fifth of 700.0 cents. | ||
==Rank 2 (including "linear") temperaments== | ==Rank 2 (including "linear") temperaments[[#lineartemperaments]]== | ||
Meantone is a familar historical temperament based on a chain of fifths (or fourths), but it is only one of many possibilities for temperaments based on a chain of generating intervals. These are referred to as "rank 2" temperaments, since they are based on a set of two linearly independent intervals. One of these intervals (typically an octave or fraction of an octave) can be selected as the "period", and another interval, smaller than the period, is referred to as the "generator". | Meantone is a familar historical temperament based on a chain of fifths (or fourths), but it is only one of many possibilities for temperaments based on a chain of generating intervals. These are referred to as "rank 2" temperaments, since they are based on a set of two linearly independent intervals. One of these intervals (typically an octave or fraction of an octave) can be selected as the "period", and another interval, smaller than the period, is referred to as the "generator". | ||
| Line 70: | Line 70: | ||
<a class="wiki_link" href="/Equal%20Temperaments"> Equal temperaments</a> (abbreviated ET or tET) and equal divisions of the octave (abbreviated EDO) are similar concepts, although there are distinctions in the way these terms are used. An EDO is simply a division of the octave into equal steps (specifically, steps of equal size in cents). An ET, on the other hand, is a temperament, an altered representation of some subset of the intervals of just intonation. The familiar 12-note equal temperament (12-ET) reduces the size of the perfect fifth (about 701.955 cents) by 1/12 of the Pythagorean comma, resulting in a fifth of 700.0 cents.<br /> | <a class="wiki_link" href="/Equal%20Temperaments"> Equal temperaments</a> (abbreviated ET or tET) and equal divisions of the octave (abbreviated EDO) are similar concepts, although there are distinctions in the way these terms are used. An EDO is simply a division of the octave into equal steps (specifically, steps of equal size in cents). An ET, on the other hand, is a temperament, an altered representation of some subset of the intervals of just intonation. The familiar 12-note equal temperament (12-ET) reduces the size of the perfect fifth (about 701.955 cents) by 1/12 of the Pythagorean comma, resulting in a fifth of 700.0 cents.<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule:2:&lt;h2&gt; --><h2 id="toc1"><a name="x-Rank 2 (including &quot;linear&quot;) temperaments"></a><!-- ws:end:WikiTextHeadingRule:2 -->Rank 2 (including &quot;linear&quot;) temperaments</h2> | <!-- ws:start:WikiTextHeadingRule:2:&lt;h2&gt; --><h2 id="toc1"><a name="x-Rank 2 (including &quot;linear&quot;) temperaments"></a><!-- ws:end:WikiTextHeadingRule:2 -->Rank 2 (including &quot;linear&quot;) temperaments<!-- ws:start:WikiTextAnchorRule:30:&lt;img src=&quot;/i/anchor.gif&quot; class=&quot;WikiAnchor&quot; alt=&quot;Anchor&quot; id=&quot;wikitext@@anchor@@lineartemperaments&quot; title=&quot;Anchor: lineartemperaments&quot;/&gt; --><a name="lineartemperaments"></a><!-- ws:end:WikiTextAnchorRule:30 --></h2> | ||
<br /> | <br /> | ||
Meantone is a familar historical temperament based on a chain of fifths (or fourths), but it is only one of many possibilities for temperaments based on a chain of generating intervals. These are referred to as &quot;rank 2&quot; temperaments, since they are based on a set of two linearly independent intervals. One of these intervals (typically an octave or fraction of an octave) can be selected as the &quot;period&quot;, and another interval, smaller than the period, is referred to as the &quot;generator&quot;.<br /> | Meantone is a familar historical temperament based on a chain of fifths (or fourths), but it is only one of many possibilities for temperaments based on a chain of generating intervals. These are referred to as &quot;rank 2&quot; temperaments, since they are based on a set of two linearly independent intervals. One of these intervals (typically an octave or fraction of an octave) can be selected as the &quot;period&quot;, and another interval, smaller than the period, is referred to as the &quot;generator&quot;.<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule:4:&lt;h3&gt; --><h3 id="toc2"><a name="x-Rank 2 (including &quot;linear&quot;) temperaments-Injera"></a><!-- ws:end:WikiTextHeadingRule:4 -->Injera<!-- ws:start:WikiTextAnchorRule: | <!-- ws:start:WikiTextHeadingRule:4:&lt;h3&gt; --><h3 id="toc2"><a name="x-Rank 2 (including &quot;linear&quot;) temperaments-Injera"></a><!-- ws:end:WikiTextHeadingRule:4 -->Injera<!-- ws:start:WikiTextAnchorRule:31:&lt;img src=&quot;/i/anchor.gif&quot; class=&quot;WikiAnchor&quot; alt=&quot;Anchor&quot; id=&quot;wikitext@@anchor@@injera&quot; title=&quot;Anchor: injera&quot;/&gt; --><a name="injera"></a><!-- ws:end:WikiTextAnchorRule:31 --></h3> | ||
<br /> | <br /> | ||
Injera has a half-octave period and a small step-sized generator, which is the difference between a half-octave and a perfect fifth. It differs from pajara temperament in having a slightly smaller generator, and a different mapping of the fifth and seventh harmonics.<br /> | Injera has a half-octave period and a small step-sized generator, which is the difference between a half-octave and a perfect fifth. It differs from pajara temperament in having a slightly smaller generator, and a different mapping of the fifth and seventh harmonics.<br /> | ||
| Line 82: | Line 82: | ||
The kleismic family of temperaments is based on a chain of minor thirds.<br /> | The kleismic family of temperaments is based on a chain of minor thirds.<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule:8:&lt;h3&gt; --><h3 id="toc4"><a name="x-Rank 2 (including &quot;linear&quot;) temperaments-Magic"></a><!-- ws:end:WikiTextHeadingRule:8 -->Magic<!-- ws:start:WikiTextAnchorRule: | <!-- ws:start:WikiTextHeadingRule:8:&lt;h3&gt; --><h3 id="toc4"><a name="x-Rank 2 (including &quot;linear&quot;) temperaments-Magic"></a><!-- ws:end:WikiTextHeadingRule:8 -->Magic<!-- ws:start:WikiTextAnchorRule:32:&lt;img src=&quot;/i/anchor.gif&quot; class=&quot;WikiAnchor&quot; alt=&quot;Anchor&quot; id=&quot;wikitext@@anchor@@magic&quot; title=&quot;Anchor: magic&quot;/&gt; --><a name="magic"></a><!-- ws:end:WikiTextAnchorRule:32 --></h3> | ||
<br /> | <br /> | ||
Magic is based on a chain of major thirds.<br /> | Magic is based on a chain of major thirds.<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule:10:&lt;h3&gt; --><h3 id="toc5"><a name="x-Rank 2 (including &quot;linear&quot;) temperaments-Meantone"></a><!-- ws:end:WikiTextHeadingRule:10 -->Meantone<!-- ws:start:WikiTextAnchorRule: | <!-- ws:start:WikiTextHeadingRule:10:&lt;h3&gt; --><h3 id="toc5"><a name="x-Rank 2 (including &quot;linear&quot;) temperaments-Meantone"></a><!-- ws:end:WikiTextHeadingRule:10 -->Meantone<!-- ws:start:WikiTextAnchorRule:33:&lt;img src=&quot;/i/anchor.gif&quot; class=&quot;WikiAnchor&quot; alt=&quot;Anchor&quot; id=&quot;wikitext@@anchor@@meantone&quot; title=&quot;Anchor: meantone&quot;/&gt; --><a name="meantone"></a><!-- ws:end:WikiTextAnchorRule:33 --></h3> | ||
<br /> | <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.<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule:12:&lt;h3&gt; --><h3 id="toc6"><a name="x-Rank 2 (including &quot;linear&quot;) temperaments-Miracle"></a><!-- ws:end:WikiTextHeadingRule:12 -->Miracle<!-- ws:start:WikiTextAnchorRule: | <!-- ws:start:WikiTextHeadingRule:12:&lt;h3&gt; --><h3 id="toc6"><a name="x-Rank 2 (including &quot;linear&quot;) temperaments-Miracle"></a><!-- ws:end:WikiTextHeadingRule:12 -->Miracle<!-- ws:start:WikiTextAnchorRule:34:&lt;img src=&quot;/i/anchor.gif&quot; class=&quot;WikiAnchor&quot; alt=&quot;Anchor&quot; id=&quot;wikitext@@anchor@@miracle&quot; title=&quot;Anchor: miracle&quot;/&gt; --><a name="miracle"></a><!-- ws:end:WikiTextAnchorRule:34 --></h3> | ||
<br /> | <br /> | ||
Miracle temperament divides the fifth into 6 equal steps. A 21-note scale called &quot;blackjack&quot; and a <a class="wiki_link" href="/Equal%20Temperaments"> Equal temperaments</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 <a class="wiki_link" href="/Equal%20Temperaments"> Equal temperaments</a>31-note scale called &quot;canasta&quot; have some useful properties.<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule:14:&lt;h3&gt; --><h3 id="toc7"><a name="x-Rank 2 (including &quot;linear&quot;) temperaments-Orwell"></a><!-- ws:end:WikiTextHeadingRule:14 -->Orwell<!-- ws:start:WikiTextAnchorRule: | <!-- ws:start:WikiTextHeadingRule:14:&lt;h3&gt; --><h3 id="toc7"><a name="x-Rank 2 (including &quot;linear&quot;) temperaments-Orwell"></a><!-- ws:end:WikiTextHeadingRule:14 -->Orwell<!-- ws:start:WikiTextAnchorRule:35:&lt;img src=&quot;/i/anchor.gif&quot; class=&quot;WikiAnchor&quot; alt=&quot;Anchor&quot; id=&quot;wikitext@@anchor@@orwell&quot; title=&quot;Anchor: orwell&quot;/&gt; --><a name="orwell"></a><!-- ws:end:WikiTextAnchorRule:35 --></h3> | ||
<br /> | <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.<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule:16:&lt;h3&gt; --><h3 id="toc8"><a name="x-Rank 2 (including &quot;linear&quot;) temperaments-Pajara"></a><!-- ws:end:WikiTextHeadingRule:16 -->Pajara<!-- ws:start:WikiTextAnchorRule: | <!-- ws:start:WikiTextHeadingRule:16:&lt;h3&gt; --><h3 id="toc8"><a name="x-Rank 2 (including &quot;linear&quot;) temperaments-Pajara"></a><!-- ws:end:WikiTextHeadingRule:16 -->Pajara<!-- ws:start:WikiTextAnchorRule:36:&lt;img src=&quot;/i/anchor.gif&quot; class=&quot;WikiAnchor&quot; alt=&quot;Anchor&quot; id=&quot;wikitext@@anchor@@pajara&quot; title=&quot;Anchor: pajara&quot;/&gt; --><a name="pajara"></a><!-- ws:end:WikiTextAnchorRule:36 --></h3> | ||
<br /> | <br /> | ||
Pajara is one of the best known of the temperaments which divide the octave into two equal periods. The small intervals 50/49 and 64/63 are tempered out. The generator of pajara is the difference between a perfect fifth and a half-octave.<br /> | Pajara is one of the best known of the temperaments which divide the octave into two equal periods. The small intervals 50/49 and 64/63 are tempered out. The generator of pajara is the difference between a perfect fifth and a half-octave.<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule:18:&lt;h3&gt; --><h3 id="toc9"><a name="x-Rank 2 (including &quot;linear&quot;) temperaments-Porcupine"></a><!-- ws:end:WikiTextHeadingRule:18 -->Porcupine<!-- ws:start:WikiTextAnchorRule: | <!-- ws:start:WikiTextHeadingRule:18:&lt;h3&gt; --><h3 id="toc9"><a name="x-Rank 2 (including &quot;linear&quot;) temperaments-Porcupine"></a><!-- ws:end:WikiTextHeadingRule:18 -->Porcupine<!-- ws:start:WikiTextAnchorRule:37:&lt;img src=&quot;/i/anchor.gif&quot; class=&quot;WikiAnchor&quot; alt=&quot;Anchor&quot; id=&quot;wikitext@@anchor@@porcupine&quot; title=&quot;Anchor: porcupine&quot;/&gt; --><a name="porcupine"></a><!-- ws:end:WikiTextAnchorRule:37 --></h3> | ||
<br /> | <br /> | ||
Porcupine temperament divides the perfect fourth into three equal parts.<br /> | Porcupine temperament divides the perfect fourth into three equal parts.<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule:20:&lt;h3&gt; --><h3 id="toc10"><a name="x-Rank 2 (including &quot;linear&quot;) temperaments-Schismatic (helmholtz, garibaldi)"></a><!-- ws:end:WikiTextHeadingRule:20 -->Schismatic (helmholtz, garibaldi)<!-- ws:start:WikiTextAnchorRule: | <!-- ws:start:WikiTextHeadingRule:20:&lt;h3&gt; --><h3 id="toc10"><a name="x-Rank 2 (including &quot;linear&quot;) temperaments-Schismatic (helmholtz, garibaldi)"></a><!-- ws:end:WikiTextHeadingRule:20 -->Schismatic (helmholtz, garibaldi)<!-- ws:start:WikiTextAnchorRule:38:&lt;img src=&quot;/i/anchor.gif&quot; class=&quot;WikiAnchor&quot; alt=&quot;Anchor&quot; id=&quot;wikitext@@anchor@@schismatic&quot; title=&quot;Anchor: schismatic&quot;/&gt; --><a name="schismatic"></a><!-- ws:end:WikiTextAnchorRule:38 --></h3> | ||
<br /> | <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).<br /> | ||
| Line 114: | Line 114: | ||
Even less familiar than rank 2 temperaments are the rank 3 temperaments, based on a set of three intervals. Since these temperaments may be mapped in many different ways, it is more common to identify rank 3 temperaments by the commas they temper out.<br /> | Even less familiar than rank 2 temperaments are the rank 3 temperaments, based on a set of three intervals. Since these temperaments may be mapped in many different ways, it is more common to identify rank 3 temperaments by the commas they temper out.<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule:24:&lt;h3&gt; --><h3 id="toc12"><a name="x-Rank 3 temperaments-Marvel"></a><!-- ws:end:WikiTextHeadingRule:24 -->Marvel<!-- ws:start:WikiTextAnchorRule: | <!-- ws:start:WikiTextHeadingRule:24:&lt;h3&gt; --><h3 id="toc12"><a name="x-Rank 3 temperaments-Marvel"></a><!-- ws:end:WikiTextHeadingRule:24 -->Marvel<!-- ws:start:WikiTextAnchorRule:39:&lt;img src=&quot;/i/anchor.gif&quot; class=&quot;WikiAnchor&quot; alt=&quot;Anchor&quot; id=&quot;wikitext@@anchor@@marvel&quot; title=&quot;Anchor: marvel&quot;/&gt; --><a name="marvel"></a><!-- ws:end:WikiTextAnchorRule:39 --></h3> | ||
<br /> | <br /> | ||
Tempers out 225/224<br /> | Tempers out 225/224<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule:26:&lt;h3&gt; --><h3 id="toc13"><a name="x-Rank 3 temperaments-Starling"></a><!-- ws:end:WikiTextHeadingRule:26 -->Starling<!-- ws:start:WikiTextAnchorRule: | <!-- ws:start:WikiTextHeadingRule:26:&lt;h3&gt; --><h3 id="toc13"><a name="x-Rank 3 temperaments-Starling"></a><!-- ws:end:WikiTextHeadingRule:26 -->Starling<!-- ws:start:WikiTextAnchorRule:40:&lt;img src=&quot;/i/anchor.gif&quot; class=&quot;WikiAnchor&quot; alt=&quot;Anchor&quot; id=&quot;wikitext@@anchor@@starling&quot; title=&quot;Anchor: starling&quot;/&gt; --><a name="starling"></a><!-- ws:end:WikiTextAnchorRule:40 --></h3> | ||
<br /> | <br /> | ||
Tempers out 126/125<br /> | Tempers out 126/125<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule:28:&lt;h3&gt; --><h3 id="toc14"><a name="x-Rank 3 temperaments-Wonder"></a><!-- ws:end:WikiTextHeadingRule:28 -->Wonder<!-- ws:start:WikiTextAnchorRule: | <!-- ws:start:WikiTextHeadingRule:28:&lt;h3&gt; --><h3 id="toc14"><a name="x-Rank 3 temperaments-Wonder"></a><!-- ws:end:WikiTextHeadingRule:28 -->Wonder<!-- ws:start:WikiTextAnchorRule:41:&lt;img src=&quot;/i/anchor.gif&quot; class=&quot;WikiAnchor&quot; alt=&quot;Anchor&quot; id=&quot;wikitext@@anchor@@wonder&quot; title=&quot;Anchor: wonder&quot;/&gt; --><a name="wonder"></a><!-- ws:end:WikiTextAnchorRule:41 --></h3> | ||
<br /> | <br /> | ||
Tempers out 243/242 and 441/440</body></html></pre></div> | Tempers out 243/242 and 441/440</body></html></pre></div> | ||