Tour of regular temperaments
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==Equal temperaments== [[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[[#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". ===Injera[[#injera]]=== 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. ===Kleismic (hanson, keemun)=== The kleismic family of temperaments is based on a chain of minor thirds. ===Magic[[#magic]]=== 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 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. 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. 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]]=== 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. ===Porcupine[[#porcupine]]=== Porcupine temperament divides the perfect fourth into three equal parts. ===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). 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== 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. ===Marvel[[#marvel]]=== Tempers out 225/224 ===Starling[[#starling]]=== Tempers out 126/125 ===Wonder[[#wonder]]=== Tempers out 243/242 and 441/440
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<html><head><title>Tour of Regular Temperaments</title></head><body><!-- ws:start:WikiTextHeadingRule:0:<h2> --><h2 id="toc0"><a name="x-Equal temperaments"></a><!-- ws:end:WikiTextHeadingRule:0 -->Equal temperaments</h2> <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 /> <!-- ws:start:WikiTextHeadingRule:2:<h2> --><h2 id="toc1"><a name="x-Rank 2 (including "linear") temperaments"></a><!-- ws:end:WikiTextHeadingRule:2 -->Rank 2 (including "linear") temperaments<!-- ws:start:WikiTextAnchorRule:30:<img src="/i/anchor.gif" class="WikiAnchor" alt="Anchor" id="wikitext@@anchor@@lineartemperaments" title="Anchor: lineartemperaments"/> --><a name="lineartemperaments"></a><!-- ws:end:WikiTextAnchorRule:30 --></h2> <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 "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".<br /> <br /> <!-- ws:start:WikiTextHeadingRule:4:<h3> --><h3 id="toc2"><a name="x-Rank 2 (including "linear") temperaments-Injera"></a><!-- ws:end:WikiTextHeadingRule:4 -->Injera<!-- ws:start:WikiTextAnchorRule:31:<img src="/i/anchor.gif" class="WikiAnchor" alt="Anchor" id="wikitext@@anchor@@injera" title="Anchor: injera"/> --><a name="injera"></a><!-- ws:end:WikiTextAnchorRule:31 --></h3> <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 /> <br /> <!-- ws:start:WikiTextHeadingRule:6:<h3> --><h3 id="toc3"><a name="x-Rank 2 (including "linear") temperaments-Kleismic (hanson, keemun)"></a><!-- ws:end:WikiTextHeadingRule:6 -->Kleismic (hanson, keemun)</h3> <br /> The kleismic family of temperaments is based on a chain of minor thirds.<br /> <br /> <!-- ws:start:WikiTextHeadingRule:8:<h3> --><h3 id="toc4"><a name="x-Rank 2 (including "linear") temperaments-Magic"></a><!-- ws:end:WikiTextHeadingRule:8 -->Magic<!-- ws:start:WikiTextAnchorRule:32:<img src="/i/anchor.gif" class="WikiAnchor" alt="Anchor" id="wikitext@@anchor@@magic" title="Anchor: magic"/> --><a name="magic"></a><!-- ws:end:WikiTextAnchorRule:32 --></h3> <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 /> <!-- ws:start:WikiTextHeadingRule:10:<h3> --><h3 id="toc5"><a name="x-Rank 2 (including "linear") temperaments-Meantone"></a><!-- ws:end:WikiTextHeadingRule:10 -->Meantone<!-- ws:start:WikiTextAnchorRule:33:<img src="/i/anchor.gif" class="WikiAnchor" alt="Anchor" id="wikitext@@anchor@@meantone" title="Anchor: meantone"/> --><a name="meantone"></a><!-- ws:end:WikiTextAnchorRule:33 --></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. This means four fiths approximate 5/1.<br /> <br /> <!-- ws:start:WikiTextHeadingRule:12:<h3> --><h3 id="toc6"><a name="x-Rank 2 (including "linear") temperaments-Miracle"></a><!-- ws:end:WikiTextHeadingRule:12 -->Miracle<!-- ws:start:WikiTextAnchorRule:34:<img src="/i/anchor.gif" class="WikiAnchor" alt="Anchor" id="wikitext@@anchor@@miracle" title="Anchor: miracle"/> --><a name="miracle"></a><!-- ws:end:WikiTextAnchorRule:34 --></h3> <br /> 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.<br /> <br /> <!-- ws:start:WikiTextHeadingRule:14:<h3> --><h3 id="toc7"><a name="x-Rank 2 (including "linear") temperaments-Orwell"></a><!-- ws:end:WikiTextHeadingRule:14 -->Orwell<!-- ws:start:WikiTextAnchorRule:35:<img src="/i/anchor.gif" class="WikiAnchor" alt="Anchor" id="wikitext@@anchor@@orwell" title="Anchor: orwell"/> --><a name="orwell"></a><!-- ws:end:WikiTextAnchorRule:35 --></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. 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 /> <!-- ws:start:WikiTextHeadingRule:16:<h3> --><h3 id="toc8"><a name="x-Rank 2 (including "linear") temperaments-Pajara"></a><!-- ws:end:WikiTextHeadingRule:16 -->Pajara<!-- ws:start:WikiTextAnchorRule:36:<img src="/i/anchor.gif" class="WikiAnchor" alt="Anchor" id="wikitext@@anchor@@pajara" title="Anchor: pajara"/> --><a name="pajara"></a><!-- ws:end:WikiTextAnchorRule:36 --></h3> <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 /> <!-- ws:start:WikiTextHeadingRule:18:<h3> --><h3 id="toc9"><a name="x-Rank 2 (including "linear") temperaments-Porcupine"></a><!-- ws:end:WikiTextHeadingRule:18 -->Porcupine<!-- ws:start:WikiTextAnchorRule:37:<img src="/i/anchor.gif" class="WikiAnchor" alt="Anchor" id="wikitext@@anchor@@porcupine" title="Anchor: porcupine"/> --><a name="porcupine"></a><!-- ws:end:WikiTextAnchorRule:37 --></h3> <br /> Porcupine temperament divides the perfect fourth into three equal parts.<br /> <br /> <!-- ws:start:WikiTextHeadingRule:20:<h3> --><h3 id="toc10"><a name="x-Rank 2 (including "linear") temperaments-Schismatic (helmholtz, garibaldi)"></a><!-- ws:end:WikiTextHeadingRule:20 -->Schismatic (helmholtz, garibaldi)<!-- ws:start:WikiTextAnchorRule:38:<img src="/i/anchor.gif" class="WikiAnchor" alt="Anchor" id="wikitext@@anchor@@schismatic" title="Anchor: schismatic"/> --><a name="schismatic"></a><!-- ws:end:WikiTextAnchorRule:38 --></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). It's much more accurate than meantone with manageable complexity. It also works well in the 7-limit but with lower accuracy.<br /> <br /> <!-- ws:start:WikiTextHeadingRule:22:<h2> --><h2 id="toc11"><a name="x-Rank 3 temperaments"></a><!-- ws:end:WikiTextHeadingRule:22 -->Rank 3 temperaments</h2> <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 /> <!-- ws:start:WikiTextHeadingRule:24:<h3> --><h3 id="toc12"><a name="x-Rank 3 temperaments-Marvel"></a><!-- ws:end:WikiTextHeadingRule:24 -->Marvel<!-- ws:start:WikiTextAnchorRule:39:<img src="/i/anchor.gif" class="WikiAnchor" alt="Anchor" id="wikitext@@anchor@@marvel" title="Anchor: marvel"/> --><a name="marvel"></a><!-- ws:end:WikiTextAnchorRule:39 --></h3> <br /> Tempers out 225/224<br /> <br /> <!-- ws:start:WikiTextHeadingRule:26:<h3> --><h3 id="toc13"><a name="x-Rank 3 temperaments-Starling"></a><!-- ws:end:WikiTextHeadingRule:26 -->Starling<!-- ws:start:WikiTextAnchorRule:40:<img src="/i/anchor.gif" class="WikiAnchor" alt="Anchor" id="wikitext@@anchor@@starling" title="Anchor: starling"/> --><a name="starling"></a><!-- ws:end:WikiTextAnchorRule:40 --></h3> <br /> Tempers out 126/125<br /> <br /> <!-- ws:start:WikiTextHeadingRule:28:<h3> --><h3 id="toc14"><a name="x-Rank 3 temperaments-Wonder"></a><!-- ws:end:WikiTextHeadingRule:28 -->Wonder<!-- ws:start:WikiTextAnchorRule:41:<img src="/i/anchor.gif" class="WikiAnchor" alt="Anchor" id="wikitext@@anchor@@wonder" title="Anchor: wonder"/> --><a name="wonder"></a><!-- ws:end:WikiTextAnchorRule:41 --></h3> <br /> Tempers out 243/242 and 441/440</body></html>