Tour of regular temperaments: Difference between revisions
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The following is a tour of many of the [[regular temperament]]s that contributors to this wiki have found notable. It is of course not meant to be comprehensive, and is not the result of a systematic search. | The following is a tour of many of the [[regular temperament]]s that contributors to this wiki have found notable. It is of course not meant to be comprehensive, and is not the result of a systematic search. | ||
= Rank-2 temperaments = | |||
A p-limit rank-2 temperament maps all intervals of p-limit JI using a set of 2-dimensional coordinates, thus a rank-2 temperament is said to have two generators, though it may have any number of step-sizes. This means that a rank-2 temperament is defined by a period-generator mapping, a set of 2 vals, one val for each generator. The larger generator is called the period, as the temperament will repeat at that interval, and is often a fraction of an octave; if it is exactly an octave, the temperament is said to be a '''linear temperament'''. Rank-2 temperaments can be reduced to a related rank-1 temperament by tempering out an additional comma. 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-ET by tempering out the Pythagorean comma. | |||
== Families defined by a 2.3 (wa) comma == | == Families defined by a 2.3 (wa) comma == | ||
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===[[Blackwood|Blackwood or Sawa family]] (P8/5, ^1)=== | ===[[Blackwood|Blackwood or Sawa family]] (P8/5, ^1)=== | ||
This family tempers out the limma, {{Monzo|8 -5 0}} = 256/243, which implies [[ | This family tempers out the limma, {{Monzo|8 -5 0}} = 256/243, which implies [[5edo|5EDO]]. | ||
===[[Apotome family|Apotome or Lawa family]] (P8/7, ^1)=== | ===[[Apotome family|Apotome or Lawa family]] (P8/7, ^1)=== | ||
This family tempers out the apotome, {{Monzo|-11 7 0}} = 2187/2048, which implies [[ | This family tempers out the apotome, {{Monzo|-11 7 0}} = 2187/2048, which implies [[7edo|7EDO]]. | ||
===[[Pythagorean family|Pythagorean or Lalawa family]] (P8/12, ^1)=== | ===[[Pythagorean family|Pythagorean or Lalawa family]] (P8/12, ^1)=== | ||
The Pythagorean family tempers out the [[Pythagorean comma]], 531441/524288 = {{Monzo|-19 12 0}}, which implies [[ | The Pythagorean family tempers out the [[Pythagorean comma]], 531441/524288 = {{Monzo|-19 12 0}}, which implies [[12edo|12EDO]]. This family includes the compton and catler temperaments. Temperaments in this family tend to have a period of 1/12th octave The 5-limit compton temperament can be thought of as multiple rings of 12-edo, offset from one another by a justly tuned 5/4, or alternatively by a tempered 81/80. Several 12-edo instruments slightly detuned from each other provide an easy way to make music with these temperaments. | ||
===[[Counterpyth family|Counterpyth or Tribisawa family]] (P8/41, ^1)=== | |||
The Counterpyth family tempers out the [[41-comma|counterpyth comma]], {{Monzo| 65 -41}}, which implies [[41edo|41EDO]]. | |||
===[[Mercator family|Mercator family]] (P8/53, ^1)=== | ===[[Mercator family|Mercator or Quadbilawa family]] (P8/53, ^1)=== | ||
The Mercator family tempers out the [[Mercator's comma]], {{Monzo| -84 53 }}, which implies [[53edo]]. | The Mercator family tempers out the [[Mercator's comma]], {{Monzo| -84 53 }}, which implies [[53edo|53EDO]]. | ||
== Families defined by a 2.3.5 (ya) comma == | == Families defined by a 2.3.5 (ya) comma == | ||
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This tempers out the vishnuzma, {{Monzo|23 6 -14}}, or the amount by which seven chromatic semitones (25/24) fall short of a perfect fourth (4/3), or (4/3)/(25/24)^7. The period is ~{{Monzo|-11 -3 7}} and the generator is ~25/24. 5/4 is equated to 1 period minus 3 generators. | This tempers out the vishnuzma, {{Monzo|23 6 -14}}, or the amount by which seven chromatic semitones (25/24) fall short of a perfect fourth (4/3), or (4/3)/(25/24)^7. The period is ~{{Monzo|-11 -3 7}} and the generator is ~25/24. 5/4 is equated to 1 period minus 3 generators. | ||
===[[Mutt | ===[[Mutt temperament|Mutt or Trila-septriyo family]] (P8/3, ccP4/7)=== | ||
This tempers out the [[mutt_comma|mutt comma]], {{Monzo|-44 -3 21}, leading to some strange properties. Seven ~5/4 generators equals a double-compound 4th = ~16/3. The third-octave period is <u>not</u> 5/4, thus the generator is equivalently a period minus ~5/4, only about 14¢. The L/s ratio is very lopsided, and scales resemble a "fuzzy" augmented chord. | This tempers out the [[mutt_comma|mutt comma]], {{Monzo|-44 -3 21}, leading to some strange properties. Seven ~5/4 generators equals a double-compound 4th = ~16/3. The third-octave period is <u>not</u> 5/4, thus the generator is equivalently a period minus ~5/4, only about 14¢. The L/s ratio is very lopsided, and scales resemble a "fuzzy" augmented chord. | ||
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Orwellismic rank-two temperaments temper out orwellisma, {{Monzo|6 3 -1 -3}} = 1728/1715. | Orwellismic rank-two temperaments temper out orwellisma, {{Monzo|6 3 -1 -3}} = 1728/1715. | ||
===[[Nuwell temperaments|Nuwell or Quadru-ayo temperaments]] | ===[[Nuwell temperaments|Nuwell or Quadru-ayo temperaments]]=== | ||
Nuwell rank-two temperaments temper out the nuwell comma, {{Monzo|1 5 1 -4}} = 2430/2401. | Nuwell rank-two temperaments temper out the nuwell comma, {{Monzo|1 5 1 -4}} = 2430/2401. | ||
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Ragismic rank-two microtemperaments temper out the ragisma, {{Monzo|-1 -7 4 1}} = 4375/4374. | Ragismic rank-two microtemperaments temper out the ragisma, {{Monzo|-1 -7 4 1}} = 4375/4374. | ||
===[[Hemifamity temperaments|Hemifamity or Saruyo temperaments]] | ===[[Hemifamity temperaments|Hemifamity or Saruyo temperaments]]=== | ||
Hemifamity rank-two temperaments temper out the hemifamity comma, {{Monzo|10 -6 1 -1}} = 5120/5103. | Hemifamity rank-two temperaments temper out the hemifamity comma, {{Monzo|10 -6 1 -1}} = 5120/5103. | ||
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These are very low complexity temperaments tempering out the minor septimal semitone, [[21/20]] and hence equating 5/3 with 7/4. | These are very low complexity temperaments tempering out the minor septimal semitone, [[21/20]] and hence equating 5/3 with 7/4. | ||
===[[Cataharry temperaments|Cataharry or Labirugu temperaments]] | ===[[Cataharry temperaments|Cataharry or Labirugu temperaments]]=== | ||
Cataharry rank-two temperaments temper out the cataharry comma, {{Monzo|-4 9 -2 -2}} = 19683/19600. | Cataharry rank-two temperaments temper out the cataharry comma, {{Monzo|-4 9 -2 -2}} = 19683/19600. | ||
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Mirwomo rank-two temperaments temper out the mirwomo comma, {{Monzo|-15 3 2 2}} = 33075/32768. | Mirwomo rank-two temperaments temper out the mirwomo comma, {{Monzo|-15 3 2 2}} = 33075/32768. | ||
===[[Landscape microtemperaments|Landscape or Trizogugu temperaments]] | ===[[Landscape microtemperaments|Landscape or Trizogugu temperaments]]=== | ||
Lanscape rank-two temperaments temper out the lanscape comma, {{Monzo|-4 6 -6 3}} = 250047/250000. | Lanscape rank-two temperaments temper out the lanscape comma, {{Monzo|-4 6 -6 3}} = 250047/250000. | ||
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Canousmic rank-two temperaments temper out the canousma, {{Monzo|4 -14 3 4}} = 4802000/4782969. | Canousmic rank-two temperaments temper out the canousma, {{Monzo|4 -14 3 4}} = 4802000/4782969. | ||
= | ===[[Triwellsmic temperaments|Triwellsmic or Tribizo-asepgu temperaments]]=== | ||
Triwellsmic rank-two temperaments temper out the ''triwellsma'' (named by [[User:Xenllium|Xenllium]]), {{Monzo|1 -1 -7 6}} = 235298/234375. | |||
===[[Hewuermera temperaments|Hewuermera or Satribiru-agu temperaments]]=== | |||
Hewuermera rank-two temperaments temper out the ''hewuermera'' comma (named by [[User:Xenllium|Xenllium]]), {{Monzo|16 2 -1 -6}} = 589824/588245. | |||
Even less familiar than rank-2 temperaments are the [[ | = Rank-3 temperaments = | ||
Even less familiar than rank-2 temperaments are the [[Planar temperament|rank-3 temperaments]], generated by 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. They can also be identified by pergen. The 3rd generator in a rank-3 pergen is usually a comma, but sometimes it's some fraction of a 5-limit or 7-limit interval. | |||
== Families defined by a 2.3.5 (ya) comma == | == Families defined by a 2.3.5 (ya) comma == | ||
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=[[Rank_four_temperaments|Rank-4 temperaments]]= | =[[Rank_four_temperaments|Rank-4 temperaments]]= | ||
Even less explored than rank three temperaments are rank four temperaments. In fact, unless one counts 7-limit JI they don't seem to have been explored at all. However, they could be used; for example [[Hobbits|hobbit scales]] can be constructed for them. | Even less explored than rank three temperaments are rank four temperaments. In fact, unless one counts 7-limit JI they don't seem to have been explored at all. However, they could be used; for example [[Hobbits|hobbit scales]] can be constructed for them. | ||
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The Archipelago is a name which has been given to the commatic realm of the [[13-limit]] comma {{Monzo|2 -3 -2 0 0 2}} = 676/675. | The Archipelago is a name which has been given to the commatic realm of the [[13-limit]] comma {{Monzo|2 -3 -2 0 0 2}} = 676/675. | ||
== [[Marveltwin]] == | ==[[Marveltwin|Marveltwin or Thoyoyo]] == | ||
This is the commatic realm of the 13-limit comma 325/324. | This is the commatic realm of the 13-limit comma 325/324. | ||