Tour of regular temperaments: Difference between revisions

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== 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.

A rank-2 temperament maps all JI intervals within its [[JI subgroup]] to a 2-dimensional lattice. This lattice has two generators. The larger generator is called the period, as the temperament will repeat periodically at that interval. The period is usually the octave, or some unit fraction of the octave. If the period is a full octave, the temperament is said to be a '''linear temperament'''. The smaller generator, referred to as "the" generator, is stacked repeatedly to generate a scale within that period.

A rank-2 temperament is defined by a period-generator mapping, a set of 2 vals, one val for the period and one for the generator. Each member of the JI subgroup is mapped to a period-generator pair.

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 81/80 out of 3-dimensional 5-limit JI), can be reduced to 1-dimensional 12-ET by tempering out the Pythagorean comma.

=== Families defined by a 2.3 (wa) comma ===

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; [[Limma family|Limma or Sawati family]] (P8/5, ^1)

: This family tempers out the [[limma]], {{nowrap|{{Monzo| 8 -5 }} {{=}} 256/243}}. It equates 5 fifths with 3 octaves, which creates multiple copies of [[5edo]]. The fifth is ~720¢, quite sharp. This family includes the [[Blackwood]] temperament, which is 5-limit. Blackwood's edo copies are offset from one another by 5/4, or alternatively by 81/80. 5/4 is usually tempered sharp, to match the sharp 5th.

: This family tempers out the [[limma]], {{nowrap|{{Monzo| 8 -5 }} {{=}} 256/243}}. It equates 5 fifths with 3 octaves, which creates multiple copies of [[5edo]]. The fifth is ~720¢, quite sharp. This family includes the [[Blackwood]] temperament, which is 5-limit. Blackwood's edo copies are offset from one another by 5/4, or alternatively by 81/80. 5/4 is usually tempered sharp, perhaps ~400¢, to match the sharp 5th.

; [[Apotome family|Apotome or Lawati family]] (P8/7, ^1)

: This family ~~equates 7 fifths with 4 octaves, which implies [[7edo]]. It~~ tempers out the apotome, {{nowrap|{{Monzo| -11 7 }} {{=}} 2187/2048}}. ~~Here~~, ~~the~~ fifth is ~~~685.714¢~~, which is very flat~~, leading to~~ the [[Whitewood]] temperament.

: This family tempers out the apotome, {{nowrap|{{Monzo| -11 7 }} {{=}} 2187/2048}}. It equates 7 fifths with 4 octaves, which creates multiple copies of [[7edo]]. The fifth is ~685¢, which is very flat. This family includes the [[Whitewood]] temperament.

; [[Compton family|Compton or ~~Lalawa~~ family]] (P8/12, ^1)

; [[Compton family|Compton or Lalawati family]] (P8/12, ^1)

: This tempers out the [[Pythagorean comma]], {{nowrap|{{Monzo| -19 12 0 }} {{=}} 531441/524288}}, which ~~implies~~ [[12edo]]. Temperaments in this family ~~have a period of one-twelfth octave, and~~ include ~~compton~~ and ~~catler~~. ~~The~~ 5-limit compton temperament ~~can be thought of as multiple rings of 12edo~~, offset from one another by a justly tuned 5/4, or alternatively by a tempered 81/80. Several 12edo instruments slightly detuned from each other provide an easy way to make music with these temperaments.

: This tempers out the [[Pythagorean comma]], {{nowrap|{{Monzo| -19 12 0 }} {{=}} 531441/524288}}. It equates 12 fifths with 7 octaves, which creates multiple copies of [[12edo]]. Temperaments in this family include [[Compton family|Compton]] and [[Catler]]. In the 5-limit compton temperament, the edo copies are offset from one another by a justly tuned 5/4, or alternatively by a tempered 81/80. Several 12edo instruments slightly detuned from each other provide an easy way to make music with these temperaments.

; [[Countercomp family|Countercomp or Wa-41 family]] (P8/41, ^1)

: This family tempers out the [[41-comma|Pythagorean countercomma]], {{Monzo| 65 -41 }}, which ~~implies~~ [[41edo]].

: This family tempers out the [[41-comma|Pythagorean countercomma]], {{Monzo| 65 -41 }}, which creates multiple copies of [[41edo]].

; [[Mercator family|Mercator or Wa-53 family]] (P8/53, ^1)

: This family tempers out the [[Mercator's comma]], {{Monzo| -84 53 }}, which ~~implies~~ [[53edo]].

: This family tempers out the [[Mercator's comma]], {{Monzo| -84 53 }}, which creates multiple copies of [[53edo]].

=== Families defined by a 2.3.5 (ya) comma ===

@@ Line 2: / Line 2: @@
 == 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.
+A rank-2 temperament maps all JI intervals within its [[JI subgroup]] to a 2-dimensional lattice. This lattice has two generators. The larger generator is called the period, as the temperament will repeat periodically at that interval. The period is usually the octave, or some unit fraction of the octave. If the period is a full octave, the temperament is said to be a '''linear temperament'''. The smaller generator, referred to as "the" generator, is stacked repeatedly to generate a scale within that period.
+A rank-2 temperament is defined by a period-generator mapping, a set of 2 vals, one val for the period and one for the generator. Each member of the JI subgroup is mapped to a period-generator pair.
+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 81/80 out of 3-dimensional 5-limit JI), can be reduced to 1-dimensional 12-ET by tempering out the Pythagorean comma.
 === Families defined by a 2.3 (wa) comma ===
@@ Line 8: / Line 12: @@
 ; [[Limma family|Limma or Sawati family]] (P8/5, ^1)
-: This family tempers out the [[limma]], {{nowrap|{{Monzo| 8 -5 }} {{=}} 256/243}}. It equates 5 fifths with 3 octaves, which creates multiple copies of [[5edo]]. The fifth is ~720¢, quite sharp. This family includes the [[Blackwood]] temperament, which is 5-limit. Blackwood's edo copies are offset from one another by 5/4, or alternatively by 81/80. 5/4 is usually tempered sharp, to match the sharp 5th.
+: This family tempers out the [[limma]], {{nowrap|{{Monzo| 8 -5 }} {{=}} 256/243}}. It equates 5 fifths with 3 octaves, which creates multiple copies of [[5edo]]. The fifth is ~720¢, quite sharp. This family includes the [[Blackwood]] temperament, which is 5-limit. Blackwood's edo copies are offset from one another by 5/4, or alternatively by 81/80. 5/4 is usually tempered sharp, perhaps ~400¢, to match the sharp 5th.
 ; [[Apotome family|Apotome or Lawati family]] (P8/7, ^1)
-: This family equates 7 fifths with 4 octaves, which implies [[7edo]]. It tempers out the apotome, {{nowrap|{{Monzo| -11 7 }} {{=}} 2187/2048}}. Here, the fifth is ~685.714¢, which is very flat, leading to the [[Whitewood]] temperament.
+: This family tempers out the apotome, {{nowrap|{{Monzo| -11 7 }} {{=}} 2187/2048}}. It equates 7 fifths with 4 octaves, which creates multiple copies of [[7edo]]. The fifth is ~685¢, which is very flat. This family includes the [[Whitewood]] temperament.
-; [[Compton family|Compton or Lalawa family]] (P8/12, ^1)
+; [[Compton family|Compton or Lalawati family]] (P8/12, ^1)
-: This tempers out the [[Pythagorean comma]], {{nowrap|{{Monzo| -19 12 0 }} {{=}} 531441/524288}}, which implies [[12edo]]. Temperaments in this family have a period of one-twelfth octave, and include compton and catler. The 5-limit compton temperament can be thought of as multiple rings of 12edo, offset from one another by a justly tuned 5/4, or alternatively by a tempered 81/80. Several 12edo instruments slightly detuned from each other provide an easy way to make music with these temperaments.
+: This tempers out the [[Pythagorean comma]], {{nowrap|{{Monzo| -19 12 0 }} {{=}} 531441/524288}}. It equates 12 fifths with 7 octaves, which creates multiple copies of [[12edo]]. Temperaments in this family include [[Compton family|Compton]] and [[Catler]]. In the 5-limit compton temperament, the edo copies are offset from one another by a justly tuned 5/4, or alternatively by a tempered 81/80. Several 12edo instruments slightly detuned from each other provide an easy way to make music with these temperaments.
 ; [[Countercomp family|Countercomp or Wa-41 family]] (P8/41, ^1)
-: This family tempers out the [[41-comma|Pythagorean countercomma]], {{Monzo| 65 -41 }}, which implies [[41edo]].
+: This family tempers out the [[41-comma|Pythagorean countercomma]], {{Monzo| 65 -41 }}, which creates multiple copies of [[41edo]].
 ; [[Mercator family|Mercator or Wa-53 family]] (P8/53, ^1)
-: This family tempers out the [[Mercator's comma]], {{Monzo| -84 53 }}, which implies [[53edo]].
+: This family tempers out the [[Mercator's comma]], {{Monzo| -84 53 }}, which creates multiple copies of [[53edo]].
 === Families defined by a 2.3.5 (ya) comma ===