Tour of regular temperaments: Difference between revisions

(2 intermediate revisions by one other user not shown)

Line 11:

These are families defined by a 3-limit (color name: wa) comma. If only primes 2 and 3 are part of the [[subgroup]], the comma creates a rank-1 temperament, an [[edo]]. But if another prime such as 5 is present, the comma creates a rank-2 temperament. Since edos are discussed elsewhere, this section assumes the presence of at least one additional prime. The rank-2 temperament created consists of multiple "copies" of an edo. The edo copies can be thought of as being offset from one another by a small comma. This small comma is represented in the [[pergen]] by ^1.

; [[~~Limmic temperaments|~~Blackwood family]] (P8/5, ^1)

; [[Blackwood family]] (P8/5, ^1)

: This family tempers out the [[limma]], {{monzo| 8 -5 }} (256/243). It equates 5 fifths with 3 octaves, which creates multiple copies of [[5edo]]. The fifth is ~~~720¢~~, quite sharp. The only member of this family is 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 fifth. Its color name is Sawati.

: This family tempers out the [[Pythagorean limma]], {{monzo| 8 -5 }} (256/243). It equates 5 fifths with 3 octaves, which creates multiple copies of [[5edo]]. The fifth is ~720{{c}}, quite sharp. The only notable member of this family is 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{{c}}, to match the sharp fifth. Its color name is Sawati.

; [[Whitewood family]] (P8/7, ^1)

Line 171:

; [[Gammic family]] (P8, P5/20)

: The gammic family tempers out the [[gammic comma]], {{monzo| -29 -11 20 }}. Nine generators of about 35{{c}} equals ~6/5, eleven equals ~5/4 and twenty equals ~3/2. 34edo is an obvious tuning. The head of the family is 5-limit gammic, whose generator chain is [[Carlos Gamma]]. Another member is the [[neptune]] temperament. Its color name is Laquinquadyoti.

: The gammic family tempers out the [[gammic comma]], {{monzo| -29 -11 20 }}. Nine generators of about 35{{c}} equals ~6/5, eleven equals ~5/4 and twenty equals ~3/2. 34edo is an obvious tuning. The head of the family is 5-limit gammic, whose generator chain is [[Carlos Gamma]]. Another member is the [[Neptune (temperament)|neptune]] temperament. Its color name is Laquinquadyoti.

=== Clans defined by a 2.3.7 comma ===

Line 223:

; [[Buzzardsmic clan]] (P8, P12/4)

: This clan tempers out the [[buzzardsma]], {{monzo| 16 -3 0 -4 }} (65536/64827). Its generator is ~21/16. Four generators makes ~3/1. 7/4 is equated to 2 octaves minus 3 generators. Its color name is Saquadruti. This clan includes as a strong extension the [[Vulture family #Septimal vulture|vulture]] temperament, which is in the vulture family.

~~; [[Cloudy clan]] (P8/5, P5)~~

: This clan tempers out the [[cloudy comma]], {{monzo| -14 0 0 5 }} (16807/16384). It has a period of 1/5 octave, which represents ~8/7. The generator is ~3/2. Unlike the blackwood or Sawati family, ~3/2 is not equated with three-fifths of an octave, resulting in very small intervals. Its color name is Laquinzoti.

; Quinruti clan (P8, P5/5)

Line 705:

Line 702:

== Miscellaneous other temperaments ==

~~; [[Limmic temperaments]]~~

~~: Various subgroup temperaments all tempering out the limma, 256/243.~~

; [[Fractional-octave temperaments]]

: These temperaments all have a fractional-octave period.

@@ Line 11: / Line 11: @@
 These are families defined by a 3-limit (color name: wa) comma. If only primes 2 and 3 are part of the [[subgroup]], the comma creates a rank-1 temperament, an [[edo]]. But if another prime such as 5 is present, the comma creates a rank-2 temperament. Since edos are discussed elsewhere, this section assumes the presence of at least one additional prime. The rank-2 temperament created consists of multiple "copies" of an edo. The edo copies can be thought of as being offset from one another by a small comma. This small comma is represented in the [[pergen]] by ^1.
-; [[Limmic temperaments|Blackwood family]] (P8/5, ^1)
+; [[Blackwood family]] (P8/5, ^1)
-: This family tempers out the [[limma]], {{monzo| 8 -5 }} (256/243). It equates 5 fifths with 3 octaves, which creates multiple copies of [[5edo]]. The fifth is ~720¢, quite sharp. The only member of this family is 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 fifth. Its color name is Sawati.
+: This family tempers out the [[Pythagorean limma]], {{monzo| 8 -5 }} (256/243). It equates 5 fifths with 3 octaves, which creates multiple copies of [[5edo]]. The fifth is ~720{{c}}, quite sharp. The only notable member of this family is 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{{c}}, to match the sharp fifth. Its color name is Sawati.
 ; [[Whitewood family]] (P8/7, ^1)
@@ Line 171: / Line 171: @@
 ; [[Gammic family]] (P8, P5/20)
-: The gammic family tempers out the [[gammic comma]], {{monzo| -29 -11 20 }}. Nine generators of about 35{{c}} equals ~6/5, eleven equals ~5/4 and twenty equals ~3/2. 34edo is an obvious tuning. The head of the family is 5-limit gammic, whose generator chain is [[Carlos Gamma]]. Another member is the [[neptune]] temperament. Its color name is Laquinquadyoti.
+: The gammic family tempers out the [[gammic comma]], {{monzo| -29 -11 20 }}. Nine generators of about 35{{c}} equals ~6/5, eleven equals ~5/4 and twenty equals ~3/2. 34edo is an obvious tuning. The head of the family is 5-limit gammic, whose generator chain is [[Carlos Gamma]]. Another member is the [[Neptune (temperament)|neptune]] temperament. Its color name is Laquinquadyoti.
 === Clans defined by a 2.3.7 comma ===
@@ Line 223: / Line 223: @@
 ; [[Buzzardsmic clan]] (P8, P12/4)
 : This clan tempers out the [[buzzardsma]], {{monzo| 16 -3 0 -4 }} (65536/64827). Its generator is ~21/16. Four generators makes ~3/1. 7/4 is equated to 2 octaves minus 3 generators. Its color name is Saquadruti. This clan includes as a strong extension the [[Vulture family #Septimal vulture|vulture]] temperament, which is in the vulture family.
-; [[Cloudy clan]] (P8/5, P5)
-: This clan tempers out the [[cloudy comma]], {{monzo| -14 0 0 5 }} (16807/16384). It has a period of 1/5 octave, which represents ~8/7. The generator is ~3/2. Unlike the blackwood or Sawati family, ~3/2 is not equated with three-fifths of an octave, resulting in very small intervals. Its color name is Laquinzoti.
 ; Quinruti clan (P8, P5/5)
@@ Line 705: / Line 702: @@
 == Miscellaneous other temperaments ==
-; [[Limmic temperaments]]
-: Various subgroup temperaments all tempering out the limma, 256/243.
 ; [[Fractional-octave temperaments]]
 : These temperaments all have a fractional-octave period.