Tour of regular temperaments: Difference between revisions

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== 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.

Even less familiar than rank-2 temperaments are the [[rank-3 temperament]], 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 ===

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; [[Semicanousmic clan|Semicanousmic or Quadlo-agu clan]] (P8, P5, ^1)

: These temper out the semicanousma, {{monzo| -2 -6 -1 0 4 }} = 14641/14580. 5/4 is equated to an ila (11~~-limit no-fives no-sevens~~) interval, thus every 2.3.5.11 interval is equated to an ila interval, and both the pergen and the lattice are identical to that of ila JI. In the pergen, ^1 is either ~33/32 or ~729/704.

: These temper out the semicanousma, {{monzo| -2 -6 -1 0 4 }} = 14641/14580. 5/4 is equated to an ila (2.3.11) interval, thus every 2.3.5.11 interval is equated to an ila interval, and both the pergen and the lattice are identical to that of ila JI. In the pergen, ^1 is either ~33/32 or ~729/704.

; [[Semiporwellismic clan|Semiporwellismic or Salulugu clan]] (P8, P5, ^1)

: These temper out the semiporwellisma, {{monzo| 14 -3 -1 0 -2 }} = 16384/16335. 5/4 is equated to an ila (11~~-limit no-fives no-sevens~~) interval, thus every 2.3.5.11 interval is equated to an ila interval, and both the pergen and the lattice are identical to that of ila JI. In the pergen, ^1 is either ~33/32 or ~729/704.

: These temper out the semiporwellisma, {{monzo| 14 -3 -1 0 -2 }} = 16384/16335. 5/4 is equated to an ila (2.3.11) interval, thus every 2.3.5.11 interval is equated to an ila interval, and both the pergen and the lattice are identical to that of ila JI. In the pergen, ^1 is either ~33/32 or ~729/704.

; [[Olympic clan|Olympic or Salururu clan]] (P8, P5, ^1)

: These temper out the olympia, {{monzo| 17 -5 0 -2 -1 }} = 131072/130977. 11/8 is equated with a 2.3.7 interval, and thus every 2.3.7.11 interval is equated with a 2.3.7 interval. In the pergen, ^1 = ~64/63.

; [[Alpharabian rank three clan|Alpharabian or Laquadlo rank-3 clan]]

@@ Line 461: / Line 461: @@
 == 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.
+Even less familiar than rank-2 temperaments are the [[rank-3 temperament]], 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 ===
@@ Line 597: / Line 597: @@
 ; [[Semicanousmic clan|Semicanousmic or Quadlo-agu clan]] (P8, P5, ^1)
-: These temper out the semicanousma, {{monzo| -2 -6 -1 0 4 }} = 14641/14580. 5/4 is equated to an ila (11-limit no-fives no-sevens) interval, thus every 2.3.5.11 interval is equated to an ila interval, and both the pergen and the lattice are identical to that of ila JI.  In the pergen, ^1 is either ~33/32 or ~729/704.
+: These temper out the semicanousma, {{monzo| -2 -6 -1 0 4 }} = 14641/14580. 5/4 is equated to an ila (2.3.11) interval, thus every 2.3.5.11 interval is equated to an ila interval, and both the pergen and the lattice are identical to that of ila JI.  In the pergen, ^1 is either ~33/32 or ~729/704.
 ; [[Semiporwellismic clan|Semiporwellismic or Salulugu clan]] (P8, P5, ^1)
-: These temper out the semiporwellisma, {{monzo| 14 -3 -1 0 -2 }} = 16384/16335. 5/4 is equated to an ila (11-limit no-fives no-sevens) interval, thus every 2.3.5.11 interval is equated to an ila interval, and both the pergen and the lattice are identical to that of ila JI.  In the pergen, ^1 is either ~33/32 or ~729/704.
+: These temper out the semiporwellisma, {{monzo| 14 -3 -1 0 -2 }} = 16384/16335. 5/4 is equated to an ila (2.3.11) interval, thus every 2.3.5.11 interval is equated to an ila interval, and both the pergen and the lattice are identical to that of ila JI.  In the pergen, ^1 is either ~33/32 or ~729/704.
+; [[Olympic clan|Olympic or Salururu clan]] (P8, P5, ^1)
+: These temper out the olympia, {{monzo| 17 -5 0 -2 -1 }} = 131072/130977. 11/8 is equated with a 2.3.7 interval, and thus every 2.3.7.11 interval is equated with a 2.3.7 interval. In the pergen, ^1 = ~64/63.
 ; [[Alpharabian rank three clan|Alpharabian or Laquadlo rank-3 clan]]