Tour of regular temperaments: Difference between revisions

Line 5:

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

These are families defined by ~~a comma that uses~~ a wa or 3-limit comma. If ~~prime 5 is assumed to be~~ part of the subgroup, ~~and no other~~ comma is ~~tempered out~~, the comma creates a rank-2 temperament. ~~The comma can also stand as parent to a 7-limit or higher family when other commas~~ are ~~tempered out as well~~, ~~or when~~ prime ~~7 is assumed to be part of the subgroup~~. ~~Any~~ temperament ~~in these families~~ can be thought of as consisting of ~~mutiple~~ "copies" of an edo ~~separated~~ by a small comma. This small comma is represented in the [[pergen]] by ^1.

These are families defined by a wa or 3-limit comma. If only primes 2 and 3 are part of the [[Just intonation subgroup|subgroup]], the comma creates a rank-1 temperament, an [[EDO|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 can be thought of as consisting 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.

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

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

: This family tempers out the limma, {{nowrap|{{Monzo|8 -5 0}} {{=}} 256/243}}, which ~~implies~~ [[5edo]].

: 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.

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

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

: This family tempers out the apotome, {{nowrap|{{Monzo|-11 7 0}} {{=}} 2187/2048}}~~, which implies [[7edo]]~~.

: This family equates 7 fifths with 4 octaves, which implies [[7edo]]. It tempers out the apotome, {{nowrap|{{Monzo|-11 7}} {{=}} 2187/2048}}.

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

: This tempers out the [[Pythagorean comma]], {{nowrap|{{Monzo|-19 12 0}} {{=}} 531441/524288}}, which implies [[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 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}}, which implies [[12edo]]. This family includes the compton and catler temperaments. Temperaments in this family have a period of 1/12th octave 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.

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

Line 139:

: This tempers out the shibboleth comma, {{nowrap|{{Monzo|-5 -10 9}} {{=}} 1953125/1889568}}. Nine generators of ~6/5 equal a double compound 4th of ~16/3. 5/4 is equated to 3 octaves minus 10 generators.

; {{monzo|[Mabila family|Mabila or Sasa-quinbigu family]] (P8, c<sup}}~~4</sup>P4~~/10)

; {{monzo|[Mabila family|Mabila or Sasa-quinbigu family]] (P8, c<sup}}4P4/10)

: The mabila family tempers out the mabila comma, {{nowrap|{{Monzo|28 -3 -10}} {{=}} 268435456/263671875}}. The generator is {{nowrap|~512/375 {{=}} ~530¢}}, three generators equals ~5/2 and ten of them equals a quadruple-compound 4th of ~64/3. An obvious 11-limit interpretation of the generator is ~15/11.

Line 151:

: This tempers out the lafa comma, {{Monzo|77 -31 -12}}. The generator is {{nowrap|~4982259375/4294967296 {{=}} ~258.6¢}}. Twelve generators equals a twelfth (~3/1). 5/4 is equated to 7 octaves minus 31 generators.

; {{monzo|[Ditonmic family|Ditonmic or Lala-theyo family]] (P8, c<sup}}~~4</sup>P4~~/13)

; {{monzo|[Ditonmic family|Ditonmic or Lala-theyo family]] (P8, c<sup}}4P4/13)

: This tempers out the ditonma, {{nowrap|{{Monzo|-27 -2 13}} {{=}} 1220703125/1207959552}}. Thirteen ~{{Monzo|-12 -1 6}} generators of about 407¢ equals a quadruple-compound 4th. 5/4 is equated to 1 octave minus 2 generators. An obvious 3-limit interpretation of the generator is 81/64, which implies 53EDO, which is a good tuning for this high-accuracy family of temperaments.

Line 163:

: This tempers out the minortone comma, {{Monzo|-16 35 -17}}. The head of the family is minortonic temperament, with a generator of a minor tone (~10/9). Seventeen generators equals a double-compound 5th (~6/1). 5/4 is equated to 35 generators minus 5 octaves.

; {{monzo|[Maja family|Maja or Saseyo family]] (P8, c<sup}}~~6</sup>P4~~/17)

; {{monzo|[Maja family|Maja or Saseyo family]] (P8, c<sup}}6P4/17)

: This tempers out the maja comma, {{nowrap|{{Monzo|-3 -23 17}} {{=}} 762939453125/753145430616}}. The generator is {{nowrap|~162/125 {{=}} ~453¢}}. Seventeen generators equals a sextuple-compound 4th. 5/4 is equated to 9 octaves minus 23 generators.

; {{monzo|[Maquila family|Maquila or Trisa-segu family]] (P8, c<sup}}~~7</sup>P5~~/17)

; {{monzo|[Maquila family|Maquila or Trisa-segu family]] (P8, c<sup}}7P5/17)

: This tempers out the maquila comma, {{nowrap|562949953421312/556182861328125 {{=}} 49 -6 -17}}. The generator is {{nowrap|~512/375 {{=}} ~535¢}}. Seventeen generators equals a septuple-compound 5th. 5/4 is equated to 3 octaves minus 6 generators. An obvious 11-limit interpretation of the generator is 11/8, leading to the Lala-selo temperament. However, Lala-selo isn't nearly as accurate as Trisa-segu.

@@ Line 5: / Line 5: @@
 === Families defined by a 2.3 (wa) comma ===
-These are families defined by a comma that uses a wa or 3-limit comma. If prime 5 is assumed to be part of the subgroup, and no other comma is tempered out, the comma creates a rank-2 temperament. The comma can also stand as parent to a 7-limit or higher family when other commas are tempered out as well, or when prime 7 is assumed to be part of the subgroup. Any temperament in these families can be thought of as consisting of mutiple "copies" of an edo separated by a small comma. This small comma is represented in the [[pergen]] by ^1.
+These are families defined by a wa or 3-limit comma. If only primes 2 and 3 are part of the [[Just intonation subgroup|subgroup]], the comma creates a rank-1 temperament, an [[EDO|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 can be thought of as consisting 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.
-; [[Limma family|Limma or Sawa family]] (P8/5, ^1)
+; [[Limma family|Limma or Sawati family]] (P8/5, ^1)
-: This family tempers out the limma, {{nowrap|{{Monzo|8 -5 0}} {{=}} 256/243}}, which implies [[5edo]].
+: 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.
-; [[Apotome family|Apotome or Lawa family]] (P8/7, ^1)
+; [[Apotome family|Apotome or Lawati family]] (P8/7, ^1)
-: This family tempers out the apotome, {{nowrap|{{Monzo|-11 7 0}} {{=}} 2187/2048}}, which implies [[7edo]].
+: This family equates 7 fifths with 4 octaves, which implies [[7edo]]. It tempers out the apotome, {{nowrap|{{Monzo|-11 7}} {{=}} 2187/2048}}.
 ; [[Compton family|Compton or Lalawa family]] (P8/12, ^1)
-: This tempers out the [[Pythagorean comma]], {{nowrap|{{Monzo|-19 12 0}} {{=}} 531441/524288}}, which implies [[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 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}}, which implies [[12edo]]. This family includes the compton and catler temperaments. Temperaments in this family have a period of 1/12th octave 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.
 ; [[Countercomp family|Countercomp or Wa-41 family]] (P8/41, ^1)
@@ Line 139: / Line 139: @@
 : This tempers out the shibboleth comma, {{nowrap|{{Monzo|-5 -10 9}} {{=}} 1953125/1889568}}. Nine generators of ~6/5 equal a double compound 4th of ~16/3.  5/4 is equated to 3 octaves minus 10 generators.
-; {{monzo|[Mabila family|Mabila or Sasa-quinbigu family]] (P8, c<sup}}4</sup>P4/10)
+; {{monzo|[Mabila family|Mabila or Sasa-quinbigu family]] (P8, c<sup}}4P4/10)
 : The mabila family tempers out the mabila comma, {{nowrap|{{Monzo|28 -3 -10}} {{=}} 268435456/263671875}}. The generator is {{nowrap|~512/375 {{=}} ~530¢}}, three generators equals ~5/2 and ten of them equals a quadruple-compound 4th of ~64/3. An obvious 11-limit interpretation of the generator is ~15/11.
@@ Line 151: / Line 151: @@
 : This tempers out the lafa comma, {{Monzo|77 -31 -12}}. The generator is {{nowrap|~4982259375/4294967296 {{=}} ~258.6¢}}. Twelve generators equals a twelfth (~3/1). 5/4 is equated to 7 octaves minus 31 generators.
-; {{monzo|[Ditonmic family|Ditonmic or Lala-theyo family]] (P8, c<sup}}4</sup>P4/13)
+; {{monzo|[Ditonmic family|Ditonmic or Lala-theyo family]] (P8, c<sup}}4P4/13)
 : This tempers out the ditonma, {{nowrap|{{Monzo|-27 -2 13}} {{=}} 1220703125/1207959552}}. Thirteen ~{{Monzo|-12 -1 6}} generators of about 407¢ equals a quadruple-compound 4th.  5/4 is equated to 1 octave minus 2 generators. An obvious 3-limit interpretation of the generator is 81/64, which implies 53EDO, which is a good tuning for this high-accuracy family of temperaments.
@@ Line 163: / Line 163: @@
 : This tempers out the minortone comma, {{Monzo|-16 35 -17}}. The head of the family is minortonic temperament, with a generator of a minor tone (~10/9). Seventeen generators equals a double-compound 5th (~6/1). 5/4 is equated to 35 generators minus 5 octaves.
-; {{monzo|[Maja family|Maja or Saseyo family]] (P8, c<sup}}6</sup>P4/17)
+; {{monzo|[Maja family|Maja or Saseyo family]] (P8, c<sup}}6P4/17)
 : This tempers out the maja comma, {{nowrap|{{Monzo|-3 -23 17}} {{=}} 762939453125/753145430616}}. The generator is {{nowrap|~162/125 {{=}} ~453¢}}. Seventeen generators equals a sextuple-compound 4th. 5/4 is equated to 9 octaves minus 23 generators.
-; {{monzo|[Maquila family|Maquila or Trisa-segu family]] (P8, c<sup}}7</sup>P5/17)
+; {{monzo|[Maquila family|Maquila or Trisa-segu family]] (P8, c<sup}}7P5/17)
 : This tempers out the maquila comma, {{nowrap|562949953421312/556182861328125 {{=}} 49 -6 -17}}. The generator is {{nowrap|~512/375 {{=}} ~535¢}}. Seventeen generators equals a septuple-compound 5th.  5/4 is equated to 3 octaves minus 6 generators. An obvious 11-limit interpretation of the generator is 11/8, leading to the Lala-selo temperament. However, Lala-selo isn't nearly as accurate as Trisa-segu.