Tour of regular temperaments: Difference between revisions
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<h2>IMPORTED REVISION FROM WIKISPACES</h2> | <h2>IMPORTED REVISION FROM WIKISPACES</h2> | ||
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=Regular temperaments= | =Regular temperaments= | ||
Regular temperaments are non-Just tunings wherein the infinite number of intervals in p-limit Just intonation (or any subgroup thereof) are mapped to a smaller (though still infinite) set of tempered intervals, by "tempering" (deliberately mistuning) some of the ratios such that a comma (or set of commas) "vanishes" by becoming a unison. The utility of regular temperament is partly to produce scales that are simpler and have more consonances than strict JI, while maintaining a high level of concordance (or similarity to JI), and partly to introduce useful "puns" as commas | Regular temperaments are non-Just tunings wherein the infinite number of intervals in p-limit Just intonation (or any subgroup thereof) are mapped to a smaller (though still infinite) set of tempered intervals, by "tempering" (deliberately mistuning) some of the ratios such that a comma (or set of commas) "vanishes" by becoming a unison. The utility of regular temperament is partly to produce scales that are simpler and have more consonances than strict JI, while maintaining a high level of concordance (or similarity to JI), and partly to introduce useful "puns" as commas are tempered out. Temperaments effectively reduce the "dimensionality" of JI, thereby simplifying the pitch relationships. For instance, the pitch relationships in 7-limit JI can be thought of as 4-dimensional, with each prime up to 7 (2, 3, 5, and 7) representing an axis, and all intervals would be located by a four-dimensional set of coordinates. In a 7-limit regular temperament, however, the dimensionality is reduced in some way, depending on which commas (and how many) are tempered out, and intervals can be located with a set of one-, two-, or three-dimensional coordinates (depending on the number of commas that have been tempered out, or in other words the "rank" of the temperament). | ||
A rank r [[http://en.wikipedia.org/wiki/Regular_temperament|regular temperament]] in a particular tuning may be defined by giving r multiplicatively independent real numbers, which can be multiplied together to produce the intervals attainable in the temperament. A rank r temperament will be defined by r generators, and thus r vals. An [[abstract regular temperament]] can be defined in various ways, for instance by giving a set of [[comma|commas]] tempered out by the temperament, or a set of r independent [[Vals and Tuning Space|vals]] defining the mapping of the temperament. A characteristic feature of any temperament tempering out a comma are the [[comma pump examples|comma pumps]] of the comma, which are sequences of harmonically related notes or chords which return to their starting point when tempered, but which would not do so in just intonation. An example is the pump I-vii-IV-ii-V-I of meantone temperament. | A rank r [[http://en.wikipedia.org/wiki/Regular_temperament|regular temperament]] in a particular tuning may be defined by giving r multiplicatively independent real numbers, which can be multiplied together to produce the intervals attainable in the temperament. A rank r temperament will be defined by r generators, and thus r vals. An [[abstract regular temperament]] can be defined in various ways, for instance by giving a set of [[comma|commas]] tempered out by the temperament, or a set of r independent [[Vals and Tuning Space|vals]] defining the mapping of the temperament. A characteristic feature of any temperament tempering out a comma are the [[comma pump examples|comma pumps]] of the comma, which are sequences of harmonically related notes or chords which return to their starting point when tempered, but which would not do so in just intonation. An example is the pump I-vii-IV-ii-V-I of meantone temperament. | ||
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P-limit Rank 2 temperaments map all intervals of p-limit JI using a set of 2-dimensional coordinates, thus rank-2 temperaments are said to have two generators (though they may have any number of step-sizes). This means that a rank-2 temperament is defined by a set of 2 vals, one val for each generator. Rank-2 temperaments can be reduced to a related rank-1 temperaments by tempering out an additional comma that is not already tempered out. For example, 5-limit meantone temperament, which is rank-2 (defined by tempering the Syntonic comma of 81/80 out of 3-dimensional 5-limit JI), can be reduced to 12-TET by tempering out the Pythagorean comma. | P-limit Rank 2 temperaments map all intervals of p-limit JI using a set of 2-dimensional coordinates, thus rank-2 temperaments are said to have two generators (though they may have any number of step-sizes). This means that a rank-2 temperament is defined by a set of 2 vals, one val for each generator. Rank-2 temperaments can be reduced to a related rank-1 temperaments by tempering out an additional comma that is not already tempered out. For example, 5-limit meantone temperament, which is rank-2 (defined by tempering the Syntonic comma of 81/80 out of 3-dimensional 5-limit JI), can be reduced to 12-TET by tempering out the Pythagorean comma. | ||
==Families== | ==Families== | ||
As we go up from rank two, the various 5-limit temperaments often break up as families of related temperaments, depending on how higher primes are mapped (or equivalently, on which higher limit commas are introduced.) Members of families and their relationships can be classified by the [[Normal lists|normal comma list]] of the various temperaments, where a **comma** is a small interval, not a square or cube or other power, which is tempered out by the temperament. | As we go up from rank two, the various 5-limit temperaments often break up as families of related temperaments, depending on how higher primes are mapped (or equivalently, on which higher limit commas are introduced.) Members of families and their relationships can be classified by the [[Normal lists|normal comma list]] of the various temperaments, where a **comma** is a small interval, not a square or cube or other power, which is tempered out by the temperament. | ||
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This tempers out 27/25, the large limma or bug comma. | This tempers out 27/25, the large limma or bug comma. | ||
===[[Father family]]=== | ===[[Father family]]=== | ||
This tempers out 16/15, the just diatonic semitone. | This tempers out 16/15, the just diatonic semitone. | ||
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This tempers out the mutt comma, |-44 -3 21>, leading to some strange properties. | This tempers out the mutt comma, |-44 -3 21>, leading to some strange properties. | ||
==Clans== | ==Clans== | ||
===[[Gamelismic clan]]=== | ===[[Gamelismic clan]]=== | ||
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This tempers out the mirkwai comma, |0 3 4 -5> = 16875/16807, a no-twos comma (ratio of odd numbers.) | This tempers out the mirkwai comma, |0 3 4 -5> = 16875/16807, a no-twos comma (ratio of odd numbers.) | ||
=Temperaments for a given comma= | =Temperaments for a given comma= | ||
===[[Septisemi temperaments]]=== | ===[[Septisemi temperaments]]=== | ||
These are very low complexity temperaments tempering out the minor septimal semitone, 21/20 and hence equating 5/4 with 7/4. | These are very low complexity temperaments tempering out the minor septimal semitone, 21/20 and hence equating 5/4 with 7/4. | ||
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These are low complexity, high error temperaments tempering out the septimal quarter-tone, 36/35. | These are low complexity, high error temperaments tempering out the septimal quarter-tone, 36/35. | ||
===[[Greenwoodmic temperaments]]=== | ===[[Greenwoodmic temperaments]]=== | ||
These temper out the greenwoodma, |-3 4 1 -2> = 405/392. | These temper out the greenwoodma, |-3 4 1 -2> = 405/392. | ||
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These temper out |-5 2 2 -1> = 225/224, the marvel comma, and include negri, sharp, mavila, wizard, tritonic, septimin, slender, triton, escapade and marvo. Considered elsewhere are meantone, miracle, magic, pajara, orwell, catakleismic, garibaldi, august and compton. | These temper out |-5 2 2 -1> = 225/224, the marvel comma, and include negri, sharp, mavila, wizard, tritonic, septimin, slender, triton, escapade and marvo. Considered elsewhere are meantone, miracle, magic, pajara, orwell, catakleismic, garibaldi, august and compton. | ||
===[[Orwellismic temperaments]]=== | ===[[Orwellismic temperaments]]=== | ||
These temper out |6 3 -1 -3> = 1728/1715, the orwellisma. | These temper out |6 3 -1 -3> = 1728/1715, the orwellisma. | ||
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Porwell temperaments temper out the porwell comma, |11 1 -3 -2> = 6144/6125. | Porwell temperaments temper out the porwell comma, |11 1 -3 -2> = 6144/6125. | ||
===[[Stearnsmic temperaments]]=== | ===[[Stearnsmic temperaments]]=== | ||
Stearnsmic temperaments temper out the stearnsma, |1 10 0 -6> = 118098/117649. | Stearnsmic temperaments temper out the stearnsma, |1 10 0 -6> = 118098/117649. | ||
===[[Hemimage temperaments]]=== | ===[[Hemimage temperaments]]=== | ||
Hemimage temperaments temper out the hemimage comma, |5 -7 -1 3> = 10976/10935. | Hemimage temperaments temper out the hemimage comma, |5 -7 -1 3> = 10976/10935. | ||
===[[Horwell temperaments]]=== | ===[[Horwell temperaments]]=== | ||
Horwell temperaments temper out the horwell comma, |-16 1 5 1> = 65625/65536. | Horwell temperaments temper out the horwell comma, |-16 1 5 1> = 65625/65536. | ||
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A ragismic temperament is one which tempers out |-1 -7 4 1> = 4375/4374. These are not by any means all microtemperaments, but those which are not highly accurate are probably best discussed under another heading. Accurate ones include ennealimmal, supermajor, enneadecal, amity, mitonic, parakleismic, gamera and vishnu. Pontiac belongs on the list but falls under the schismatic family rubric. | A ragismic temperament is one which tempers out |-1 -7 4 1> = 4375/4374. These are not by any means all microtemperaments, but those which are not highly accurate are probably best discussed under another heading. Accurate ones include ennealimmal, supermajor, enneadecal, amity, mitonic, parakleismic, gamera and vishnu. Pontiac belongs on the list but falls under the schismatic family rubric. | ||
===[[Landscape microtemperaments]]=== | ===[[Landscape microtemperaments]]=== | ||
A landscape temperament is one which tempers out |-4 6 -6 3> = 250047/250000. | A landscape temperament is one which tempers out |-4 6 -6 3> = 250047/250000. | ||
===[[31 comma temperaments]]=== | ===[[31 comma temperaments]]=== | ||
These all have period 1/31 of an octave. | These all have period 1/31 of an octave. | ||
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Vsrious theoretical solutions have been put forward for the vexing problem of how to indicate and define the tuning of Turkish [[Arabic, Turkish, Persian|maqam music]] in a systematic way. This includes, in effect, certain linear temperaments. | Vsrious theoretical solutions have been put forward for the vexing problem of how to indicate and define the tuning of Turkish [[Arabic, Turkish, Persian|maqam music]] in a systematic way. This includes, in effect, certain linear temperaments. | ||
===[[Very low accuracy temperaments]]=== | ===[[Very low accuracy temperaments]]=== | ||
All hope abandon ye who enter here. | All hope abandon ye who enter here. | ||
===[[Very high accuracy temperaments]]=== | ===[[Very high accuracy temperaments]]=== | ||
Microtemperaments which don't fit in elsewhere. | Microtemperaments which don't fit in elsewhere. | ||
===[[11-limit comma temperaments]]=== | ===[[11-limit comma temperaments]]=== | ||
=Rank 3 temperaments= | =Rank 3 temperaments= | ||
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The hemimean family of rank three temperaments tempers out the hemimean comma, 3136/3125. | The hemimean family of rank three temperaments tempers out the hemimean comma, 3136/3125. | ||
===[[Mirwomo family]]=== | ===[[Mirwomo family]]=== | ||
The mirwomo family of rank three temperaments tempers out the mirwomo comma, 33075/32768. | The mirwomo family of rank three temperaments tempers out the mirwomo comma, 33075/32768. | ||
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<!-- ws:start:WikiTextHeadingRule:0:&lt;h1&gt; --><h1 id="toc0"><a name="Regular temperaments"></a><!-- ws:end:WikiTextHeadingRule:0 -->Regular temperaments</h1> | <!-- ws:start:WikiTextHeadingRule:0:&lt;h1&gt; --><h1 id="toc0"><a name="Regular temperaments"></a><!-- ws:end:WikiTextHeadingRule:0 -->Regular temperaments</h1> | ||
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Regular temperaments are non-Just tunings wherein the infinite number of intervals in p-limit Just intonation (or any subgroup thereof) are mapped to a smaller (though still infinite) set of tempered intervals, by &quot;tempering&quot; (deliberately mistuning) some of the ratios such that a comma (or set of commas) &quot;vanishes&quot; by becoming a unison. The utility of regular temperament is partly to produce scales that are simpler and have more consonances than strict JI, while maintaining a high level of concordance (or similarity to JI), and partly to introduce useful &quot;puns&quot; as commas | Regular temperaments are non-Just tunings wherein the infinite number of intervals in p-limit Just intonation (or any subgroup thereof) are mapped to a smaller (though still infinite) set of tempered intervals, by &quot;tempering&quot; (deliberately mistuning) some of the ratios such that a comma (or set of commas) &quot;vanishes&quot; by becoming a unison. The utility of regular temperament is partly to produce scales that are simpler and have more consonances than strict JI, while maintaining a high level of concordance (or similarity to JI), and partly to introduce useful &quot;puns&quot; as commas are tempered out. Temperaments effectively reduce the &quot;dimensionality&quot; of JI, thereby simplifying the pitch relationships. For instance, the pitch relationships in 7-limit JI can be thought of as 4-dimensional, with each prime up to 7 (2, 3, 5, and 7) representing an axis, and all intervals would be located by a four-dimensional set of coordinates. In a 7-limit regular temperament, however, the dimensionality is reduced in some way, depending on which commas (and how many) are tempered out, and intervals can be located with a set of one-, two-, or three-dimensional coordinates (depending on the number of commas that have been tempered out, or in other words the &quot;rank&quot; of the temperament).<br /> | ||
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A rank r <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Regular_temperament" rel="nofollow">regular temperament</a> in a particular tuning may be defined by giving r multiplicatively independent real numbers, which can be multiplied together to produce the intervals attainable in the temperament. A rank r temperament will be defined by r generators, and thus r vals. An <a class="wiki_link" href="/abstract%20regular%20temperament">abstract regular temperament</a> can be defined in various ways, for instance by giving a set of <a class="wiki_link" href="/comma">commas</a> tempered out by the temperament, or a set of r independent <a class="wiki_link" href="/Vals%20and%20Tuning%20Space">vals</a> defining the mapping of the temperament. A characteristic feature of any temperament tempering out a comma are the <a class="wiki_link" href="/comma%20pump%20examples">comma pumps</a> of the comma, which are sequences of harmonically related notes or chords which return to their starting point when tempered, but which would not do so in just intonation. An example is the pump I-vii-IV-ii-V-I of meantone temperament.<br /> | A rank r <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Regular_temperament" rel="nofollow">regular temperament</a> in a particular tuning may be defined by giving r multiplicatively independent real numbers, which can be multiplied together to produce the intervals attainable in the temperament. A rank r temperament will be defined by r generators, and thus r vals. An <a class="wiki_link" href="/abstract%20regular%20temperament">abstract regular temperament</a> can be defined in various ways, for instance by giving a set of <a class="wiki_link" href="/comma">commas</a> tempered out by the temperament, or a set of r independent <a class="wiki_link" href="/Vals%20and%20Tuning%20Space">vals</a> defining the mapping of the temperament. A characteristic feature of any temperament tempering out a comma are the <a class="wiki_link" href="/comma%20pump%20examples">comma pumps</a> of the comma, which are sequences of harmonically related notes or chords which return to their starting point when tempered, but which would not do so in just intonation. An example is the pump I-vii-IV-ii-V-I of meantone temperament.<br /> | ||
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<!-- ws:start:WikiTextHeadingRule:10:&lt;h2&gt; --><h2 id="toc5"><a name="Rank 2 (including &quot;linear&quot;) temperaments-Families"></a><!-- ws:end:WikiTextHeadingRule:10 -->Families</h2> | <!-- ws:start:WikiTextHeadingRule:10:&lt;h2&gt; --><h2 id="toc5"><a name="Rank 2 (including &quot;linear&quot;) temperaments-Families"></a><!-- ws:end:WikiTextHeadingRule:10 -->Families</h2> | ||
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As we go up from rank two, the various 5-limit temperaments often break up as families of related temperaments, depending on how higher primes are mapped (or equivalently, on which higher limit commas are introduced.) Members of families and their relationships can be classified by the <a class="wiki_link" href="/Normal%20lists">normal comma list</a> of the various temperaments, where a <strong>comma</strong> is a small interval, not a square or cube or other power, which is tempered out by the temperament.<br /> | As we go up from rank two, the various 5-limit temperaments often break up as families of related temperaments, depending on how higher primes are mapped (or equivalently, on which higher limit commas are introduced.) Members of families and their relationships can be classified by the <a class="wiki_link" href="/Normal%20lists">normal comma list</a> of the various temperaments, where a <strong>comma</strong> is a small interval, not a square or cube or other power, which is tempered out by the temperament.<br /> | ||
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<!-- ws:start:WikiTextHeadingRule:48:&lt;h3&gt; --><h3 id="toc24"><a name="Rank 2 (including &quot;linear&quot;) temperaments-Families-Father family"></a><!-- ws:end:WikiTextHeadingRule:48 --><a class="wiki_link" href="/Father%20family">Father family</a></h3> | <!-- ws:start:WikiTextHeadingRule:48:&lt;h3&gt; --><h3 id="toc24"><a name="Rank 2 (including &quot;linear&quot;) temperaments-Families-Father family"></a><!-- ws:end:WikiTextHeadingRule:48 --><a class="wiki_link" href="/Father%20family">Father family</a></h3> | ||
This tempers out 16/15, the just diatonic semitone.<br /> | This tempers out 16/15, the just diatonic semitone.<br /> | ||
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<!-- ws:start:WikiTextHeadingRule:50:&lt;h3&gt; --><h3 id="toc25"><a name="Rank 2 (including &quot;linear&quot;) temperaments-Families-Sycamore family"></a><!-- ws:end:WikiTextHeadingRule:50 --><a class="wiki_link" href="/Sycamore%20family">Sycamore family</a></h3> | <!-- ws:start:WikiTextHeadingRule:50:&lt;h3&gt; --><h3 id="toc25"><a name="Rank 2 (including &quot;linear&quot;) temperaments-Families-Sycamore family"></a><!-- ws:end:WikiTextHeadingRule:50 --><a class="wiki_link" href="/Sycamore%20family">Sycamore family</a></h3> | ||
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<!-- ws:start:WikiTextHeadingRule:64:&lt;h2&gt; --><h2 id="toc32"><a name="Rank 2 (including &quot;linear&quot;) temperaments-Clans"></a><!-- ws:end:WikiTextHeadingRule:64 -->Clans</h2> | <!-- ws:start:WikiTextHeadingRule:64:&lt;h2&gt; --><h2 id="toc32"><a name="Rank 2 (including &quot;linear&quot;) temperaments-Clans"></a><!-- ws:end:WikiTextHeadingRule:64 -->Clans</h2> | ||
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<!-- ws:start:WikiTextHeadingRule:66:&lt;h3&gt; --><h3 id="toc33"><a name="Rank 2 (including &quot;linear&quot;) temperaments-Clans-Gamelismic clan"></a><!-- ws:end:WikiTextHeadingRule:66 --><a class="wiki_link" href="/Gamelismic%20clan">Gamelismic clan</a></h3> | <!-- ws:start:WikiTextHeadingRule:66:&lt;h3&gt; --><h3 id="toc33"><a name="Rank 2 (including &quot;linear&quot;) temperaments-Clans-Gamelismic clan"></a><!-- ws:end:WikiTextHeadingRule:66 --><a class="wiki_link" href="/Gamelismic%20clan">Gamelismic clan</a></h3> | ||
If a 5-limit comma defines a family of rank two temperaments, then we might say a comma belonging to another <a class="wiki_link" href="/Just%20intonation%20subgroups">subgroup</a> of the 7-limit can define a clan. In particular we might say a triprime comma (one with exactly three primes in the factorization) can define a clan. Notable among such clans are the temperaments which temper out the gamelisma, 1029/1024. We can modify the definition of <a class="wiki_link" href="/Normal%20lists">normal comma list</a> for clans by changing the ordering of prime numbers, and using this to sort out clan relationships.<br /> | If a 5-limit comma defines a family of rank two temperaments, then we might say a comma belonging to another <a class="wiki_link" href="/Just%20intonation%20subgroups">subgroup</a> of the 7-limit can define a clan. In particular we might say a triprime comma (one with exactly three primes in the factorization) can define a clan. Notable among such clans are the temperaments which temper out the gamelisma, 1029/1024. We can modify the definition of <a class="wiki_link" href="/Normal%20lists">normal comma list</a> for clans by changing the ordering of prime numbers, and using this to sort out clan relationships.<br /> | ||
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<!-- ws:start:WikiTextHeadingRule:80:&lt;h1&gt; --><h1 id="toc40"><a name="Temperaments for a given comma"></a><!-- ws:end:WikiTextHeadingRule:80 -->Temperaments for a given comma</h1> | <!-- ws:start:WikiTextHeadingRule:80:&lt;h1&gt; --><h1 id="toc40"><a name="Temperaments for a given comma"></a><!-- ws:end:WikiTextHeadingRule:80 -->Temperaments for a given comma</h1> | ||
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<!-- ws:start:WikiTextHeadingRule:82:&lt;h3&gt; --><h3 id="toc41"><a name="Temperaments for a given comma--Septisemi temperaments"></a><!-- ws:end:WikiTextHeadingRule:82 --><a class="wiki_link" href="/Septisemi%20temperaments">Septisemi temperaments</a></h3> | <!-- ws:start:WikiTextHeadingRule:82:&lt;h3&gt; --><h3 id="toc41"><a name="Temperaments for a given comma--Septisemi temperaments"></a><!-- ws:end:WikiTextHeadingRule:82 --><a class="wiki_link" href="/Septisemi%20temperaments">Septisemi temperaments</a></h3> | ||
These are very low complexity temperaments tempering out the minor septimal semitone, 21/20 and hence equating 5/4 with 7/4.<br /> | These are very low complexity temperaments tempering out the minor septimal semitone, 21/20 and hence equating 5/4 with 7/4.<br /> | ||
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<!-- ws:start:WikiTextHeadingRule:84:&lt;h3&gt; --><h3 id="toc42"><a name="Temperaments for a given comma--Mint temperaments"></a><!-- ws:end:WikiTextHeadingRule:84 --><a class="wiki_link" href="/Mint%20temperaments">Mint temperaments</a></h3> | <!-- ws:start:WikiTextHeadingRule:84:&lt;h3&gt; --><h3 id="toc42"><a name="Temperaments for a given comma--Mint temperaments"></a><!-- ws:end:WikiTextHeadingRule:84 --><a class="wiki_link" href="/Mint%20temperaments">Mint temperaments</a></h3> | ||
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<!-- ws:start:WikiTextHeadingRule:86:&lt;h3&gt; --><h3 id="toc43"><a name="Temperaments for a given comma--Greenwoodmic temperaments"></a><!-- ws:end:WikiTextHeadingRule:86 --><a class="wiki_link" href="/Greenwoodmic%20temperaments">Greenwoodmic temperaments</a></h3> | <!-- ws:start:WikiTextHeadingRule:86:&lt;h3&gt; --><h3 id="toc43"><a name="Temperaments for a given comma--Greenwoodmic temperaments"></a><!-- ws:end:WikiTextHeadingRule:86 --><a class="wiki_link" href="/Greenwoodmic%20temperaments">Greenwoodmic temperaments</a></h3> | ||
These temper out the greenwoodma, |-3 4 1 -2&gt; = 405/392.<br /> | These temper out the greenwoodma, |-3 4 1 -2&gt; = 405/392.<br /> | ||
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<!-- ws:start:WikiTextHeadingRule:88:&lt;h3&gt; --><h3 id="toc44"><a name="Temperaments for a given comma--Avicennmic temperaments"></a><!-- ws:end:WikiTextHeadingRule:88 --><a class="wiki_link" href="/Avicennmic%20temperaments">Avicennmic temperaments</a></h3> | <!-- ws:start:WikiTextHeadingRule:88:&lt;h3&gt; --><h3 id="toc44"><a name="Temperaments for a given comma--Avicennmic temperaments"></a><!-- ws:end:WikiTextHeadingRule:88 --><a class="wiki_link" href="/Avicennmic%20temperaments">Avicennmic temperaments</a></h3> | ||
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<!-- ws:start:WikiTextHeadingRule:94:&lt;h3&gt; --><h3 id="toc47"><a name="Temperaments for a given comma--Orwellismic temperaments"></a><!-- ws:end:WikiTextHeadingRule:94 --><a class="wiki_link" href="/Orwellismic%20temperaments">Orwellismic temperaments</a></h3> | <!-- ws:start:WikiTextHeadingRule:94:&lt;h3&gt; --><h3 id="toc47"><a name="Temperaments for a given comma--Orwellismic temperaments"></a><!-- ws:end:WikiTextHeadingRule:94 --><a class="wiki_link" href="/Orwellismic%20temperaments">Orwellismic temperaments</a></h3> | ||
These temper out |6 3 -1 -3&gt; = 1728/1715, the orwellisma.<br /> | These temper out |6 3 -1 -3&gt; = 1728/1715, the orwellisma.<br /> | ||
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<!-- ws:start:WikiTextHeadingRule:96:&lt;h3&gt; --><h3 id="toc48"><a name="Temperaments for a given comma--Hemifamity temperaments"></a><!-- ws:end:WikiTextHeadingRule:96 --><a class="wiki_link" href="/Hemifamity%20temperaments">Hemifamity temperaments</a></h3> | <!-- ws:start:WikiTextHeadingRule:96:&lt;h3&gt; --><h3 id="toc48"><a name="Temperaments for a given comma--Hemifamity temperaments"></a><!-- ws:end:WikiTextHeadingRule:96 --><a class="wiki_link" href="/Hemifamity%20temperaments">Hemifamity temperaments</a></h3> | ||
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<!-- ws:start:WikiTextHeadingRule:100:&lt;h3&gt; --><h3 id="toc50"><a name="Temperaments for a given comma--Stearnsmic temperaments"></a><!-- ws:end:WikiTextHeadingRule:100 --><a class="wiki_link" href="/Stearnsmic%20temperaments">Stearnsmic temperaments</a></h3> | <!-- ws:start:WikiTextHeadingRule:100:&lt;h3&gt; --><h3 id="toc50"><a name="Temperaments for a given comma--Stearnsmic temperaments"></a><!-- ws:end:WikiTextHeadingRule:100 --><a class="wiki_link" href="/Stearnsmic%20temperaments">Stearnsmic temperaments</a></h3> | ||
Stearnsmic temperaments temper out the stearnsma, |1 10 0 -6&gt; = 118098/117649.<br /> | Stearnsmic temperaments temper out the stearnsma, |1 10 0 -6&gt; = 118098/117649.<br /> | ||
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<!-- ws:start:WikiTextHeadingRule:102:&lt;h3&gt; --><h3 id="toc51"><a name="Temperaments for a given comma--Hemimage temperaments"></a><!-- ws:end:WikiTextHeadingRule:102 --><a class="wiki_link" href="/Hemimage%20temperaments">Hemimage temperaments</a></h3> | <!-- ws:start:WikiTextHeadingRule:102:&lt;h3&gt; --><h3 id="toc51"><a name="Temperaments for a given comma--Hemimage temperaments"></a><!-- ws:end:WikiTextHeadingRule:102 --><a class="wiki_link" href="/Hemimage%20temperaments">Hemimage temperaments</a></h3> | ||
Hemimage temperaments temper out the hemimage comma, |5 -7 -1 3&gt; = 10976/10935.<br /> | Hemimage temperaments temper out the hemimage comma, |5 -7 -1 3&gt; = 10976/10935.<br /> | ||
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<!-- ws:start:WikiTextHeadingRule:104:&lt;h3&gt; --><h3 id="toc52"><a name="Temperaments for a given comma--Horwell temperaments"></a><!-- ws:end:WikiTextHeadingRule:104 --><a class="wiki_link" href="/Horwell%20temperaments">Horwell temperaments</a></h3> | <!-- ws:start:WikiTextHeadingRule:104:&lt;h3&gt; --><h3 id="toc52"><a name="Temperaments for a given comma--Horwell temperaments"></a><!-- ws:end:WikiTextHeadingRule:104 --><a class="wiki_link" href="/Horwell%20temperaments">Horwell temperaments</a></h3> | ||
Horwell temperaments temper out the horwell comma, |-16 1 5 1&gt; = 65625/65536.<br /> | Horwell temperaments temper out the horwell comma, |-16 1 5 1&gt; = 65625/65536.<br /> | ||
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<!-- ws:start:WikiTextHeadingRule:106:&lt;h3&gt; --><h3 id="toc53"><a name="Temperaments for a given comma--Breedsmic temperaments"></a><!-- ws:end:WikiTextHeadingRule:106 --><a class="wiki_link" href="/Breedsmic%20temperaments">Breedsmic temperaments</a></h3> | <!-- ws:start:WikiTextHeadingRule:106:&lt;h3&gt; --><h3 id="toc53"><a name="Temperaments for a given comma--Breedsmic temperaments"></a><!-- ws:end:WikiTextHeadingRule:106 --><a class="wiki_link" href="/Breedsmic%20temperaments">Breedsmic temperaments</a></h3> | ||
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<!-- ws:start:WikiTextHeadingRule:110:&lt;h3&gt; --><h3 id="toc55"><a name="Temperaments for a given comma--Landscape microtemperaments"></a><!-- ws:end:WikiTextHeadingRule:110 --><a class="wiki_link" href="/Landscape%20microtemperaments">Landscape microtemperaments</a></h3> | <!-- ws:start:WikiTextHeadingRule:110:&lt;h3&gt; --><h3 id="toc55"><a name="Temperaments for a given comma--Landscape microtemperaments"></a><!-- ws:end:WikiTextHeadingRule:110 --><a class="wiki_link" href="/Landscape%20microtemperaments">Landscape microtemperaments</a></h3> | ||
A landscape temperament is one which tempers out |-4 6 -6 3&gt; = 250047/250000.<br /> | A landscape temperament is one which tempers out |-4 6 -6 3&gt; = 250047/250000.<br /> | ||
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<!-- ws:start:WikiTextHeadingRule:112:&lt;h3&gt; --><h3 id="toc56"><a name="Temperaments for a given comma--31 comma temperaments"></a><!-- ws:end:WikiTextHeadingRule:112 --><a class="wiki_link" href="/31%20comma%20temperaments">31 comma temperaments</a></h3> | <!-- ws:start:WikiTextHeadingRule:112:&lt;h3&gt; --><h3 id="toc56"><a name="Temperaments for a given comma--31 comma temperaments"></a><!-- ws:end:WikiTextHeadingRule:112 --><a class="wiki_link" href="/31%20comma%20temperaments">31 comma temperaments</a></h3> | ||
These all have period 1/31 of an octave.<br /> | These all have period 1/31 of an octave.<br /> | ||
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<!-- ws:start:WikiTextHeadingRule:114:&lt;h3&gt; --><h3 id="toc57"><a name="Temperaments for a given comma--Turkish maqam music temperaments"></a><!-- ws:end:WikiTextHeadingRule:114 --><a class="wiki_link" href="/Turkish%20maqam%20music%20temperaments">Turkish maqam music temperaments</a></h3> | <!-- ws:start:WikiTextHeadingRule:114:&lt;h3&gt; --><h3 id="toc57"><a name="Temperaments for a given comma--Turkish maqam music temperaments"></a><!-- ws:end:WikiTextHeadingRule:114 --><a class="wiki_link" href="/Turkish%20maqam%20music%20temperaments">Turkish maqam music temperaments</a></h3> | ||
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<!-- ws:start:WikiTextHeadingRule:116:&lt;h3&gt; --><h3 id="toc58"><a name="Temperaments for a given comma--Very low accuracy temperaments"></a><!-- ws:end:WikiTextHeadingRule:116 --><a class="wiki_link" href="/Very%20low%20accuracy%20temperaments">Very low accuracy temperaments</a></h3> | <!-- ws:start:WikiTextHeadingRule:116:&lt;h3&gt; --><h3 id="toc58"><a name="Temperaments for a given comma--Very low accuracy temperaments"></a><!-- ws:end:WikiTextHeadingRule:116 --><a class="wiki_link" href="/Very%20low%20accuracy%20temperaments">Very low accuracy temperaments</a></h3> | ||
All hope abandon ye who enter here.<br /> | All hope abandon ye who enter here.<br /> | ||
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<!-- ws:start:WikiTextHeadingRule:118:&lt;h3&gt; --><h3 id="toc59"><a name="Temperaments for a given comma--Very high accuracy temperaments"></a><!-- ws:end:WikiTextHeadingRule:118 --><a class="wiki_link" href="/Very%20high%20accuracy%20temperaments">Very high accuracy temperaments</a></h3> | <!-- ws:start:WikiTextHeadingRule:118:&lt;h3&gt; --><h3 id="toc59"><a name="Temperaments for a given comma--Very high accuracy temperaments"></a><!-- ws:end:WikiTextHeadingRule:118 --><a class="wiki_link" href="/Very%20high%20accuracy%20temperaments">Very high accuracy temperaments</a></h3> | ||
Microtemperaments which don't fit in elsewhere.<br /> | Microtemperaments which don't fit in elsewhere.<br /> | ||
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<!-- ws:start:WikiTextHeadingRule:120:&lt;h3&gt; --><h3 id="toc60"><a name="Temperaments for a given comma--11-limit comma temperaments"></a><!-- ws:end:WikiTextHeadingRule:120 --><a class="wiki_link" href="/11-limit%20comma%20temperaments">11-limit comma temperaments</a></h3> | <!-- ws:start:WikiTextHeadingRule:120:&lt;h3&gt; --><h3 id="toc60"><a name="Temperaments for a given comma--11-limit comma temperaments"></a><!-- ws:end:WikiTextHeadingRule:120 --><a class="wiki_link" href="/11-limit%20comma%20temperaments">11-limit comma temperaments</a></h3> | ||
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<!-- ws:start:WikiTextHeadingRule:122:&lt;h1&gt; --><h1 id="toc61"><a name="Rank 3 temperaments"></a><!-- ws:end:WikiTextHeadingRule:122 -->Rank 3 temperaments</h1> | <!-- ws:start:WikiTextHeadingRule:122:&lt;h1&gt; --><h1 id="toc61"><a name="Rank 3 temperaments"></a><!-- ws:end:WikiTextHeadingRule:122 -->Rank 3 temperaments</h1> | ||
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<!-- ws:start:WikiTextHeadingRule:158:&lt;h3&gt; --><h3 id="toc79"><a name="Rank 3 temperaments--Mirwomo family"></a><!-- ws:end:WikiTextHeadingRule:158 --><a class="wiki_link" href="/Mirwomo%20family">Mirwomo family</a></h3> | <!-- ws:start:WikiTextHeadingRule:158:&lt;h3&gt; --><h3 id="toc79"><a name="Rank 3 temperaments--Mirwomo family"></a><!-- ws:end:WikiTextHeadingRule:158 --><a class="wiki_link" href="/Mirwomo%20family">Mirwomo family</a></h3> | ||
The mirwomo family of rank three temperaments tempers out the mirwomo comma, 33075/32768.<br /> | The mirwomo family of rank three temperaments tempers out the mirwomo comma, 33075/32768.<br /> | ||
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<!-- ws:start:WikiTextHeadingRule:160:&lt;h3&gt; --><h3 id="toc80"><a name="Rank 3 temperaments--Dimcomp family"></a><!-- ws:end:WikiTextHeadingRule:160 --><a class="wiki_link" href="/Dimcomp%20family">Dimcomp family</a></h3> | <!-- ws:start:WikiTextHeadingRule:160:&lt;h3&gt; --><h3 id="toc80"><a name="Rank 3 temperaments--Dimcomp family"></a><!-- ws:end:WikiTextHeadingRule:160 --><a class="wiki_link" href="/Dimcomp%20family">Dimcomp family</a></h3> | ||