Tour of regular temperaments: Difference between revisions

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=== 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 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.

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

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

These are families defined by a ya or 5-limit comma. As we go up in rank two, the various 5-limit temperaments often break up as families of related temperaments, depending on how higher primes are mapped (or equivalently, on which higher limit commas are introduced.) The same comment applies to 7-limit temperaments and rank three, etc. Members of families and their relationships can be classified by the [[Normal lists|normal comma list]] of the various temperaments. Families include weak extensions as well as strong, in other words, the pergen shown here may change.

These are families defined by a ya or 5-limit comma. As we go up in rank two, the various 5-limit temperaments often break up as families of related temperaments, depending on how higher primes are mapped (or equivalently, on which higher limit commas are introduced.) The same comment applies to 7-limit temperaments and rank three, etc. Members of families and their relationships can be classified by the [[Normal lists|normal comma list]] of the various temperaments. Families include weak extensions as well as strong, in other words, the [[pergen]] shown here may change.

; [[Meantone family|Meantone or Gu family]] (P8, P5)

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; [[Ripple family|Ripple or Quingu family]] (P8, P4/5)

: This tempers out the ripple comma, 6561/6250 = {{Monzo| -1 8 -5 }}, which equates a stack of four [[10/9]]'s with [[8/5]]. As one might expect, [[12edo|12EDO]] is about as accurate as it can be.

: This tempers out the ripple comma, 6561/6250 = {{Monzo| -1 8 -5 }}, which equates a stack of four [[10/9]]'s with [[8/5]], and five of them with [[16/9]]. The generator is [[27/25]], two of which equals 10/9, three of which equals [[6/5]], and five of which equals [[4/3]]. 5/4 is equated to an octave minus 8 generators. As one might expect, [[12edo|12EDO]] is about as accurate as it can be.

; [[Passion family|Passion or Saquingu family]] (P8, P4/5)

: This tempers out the passion comma, 262144/253125 = {{monzo| 18 -4 -5 }}, which equates a stack of ~~five~~ [[16/15]]'s with [[4/3]].

: This tempers out the passion comma, 262144/253125 = {{monzo| 18 -4 -5 }}, which equates a stack of four [[16/15]]'s with [[5/4]], and five of them with [[4/3]].

; [[Amity family|Amity or Saquinyo family]] (P8, P11/5)

: This tempers out the [[amity comma]], 1600000/1594323 = {{Monzo|9 -13 5}}. The generator is 243/200 = ~339.5¢, five of which make ~8/3. 5/4 is equated to 4 octaves minus 13 generators, or fifths minus 3 generators. An obvious 11-limit interpretation of the generator is 11/9, which makes Saquinlo. An obvious 13-limit interpretation of the generator is 39/32, which makes Lala-quintho.

: This tempers out the [[amity comma]], 1600000/1594323 = {{Monzo|9 -13 5}}. The generator is 243/200 = ~339.5¢, five of which make ~8/3. 5/4 is equated to 4 octaves minus 13 generators, or 2 fifths minus 3 generators. An obvious 11-limit interpretation of the generator is 11/9, which makes Saquinlo. An obvious 13-limit interpretation of the generator is 39/32, which makes Lala-quintho.

; [[Magic family|Magic or Laquinyo family]] (P8, P12/5)

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; Lulubi clan (P8/2, P5)

: This 2.3.11 clan tempers out the alpharabian limma, [[128/121]]. Both 11/8 and 16/11 is equated to half-octave period. This clan includes as a strong extension the pajaric temperament, which is in the diaschismic family.

: This low-accuracy 2.3.11 clan tempers out the alpharabian limma, [[128/121]]. Both 11/8 and 16/11 are equated to half-octave period. This clan includes as a strong extension the pajaric temperament, which is in the diaschismic family.

; Rastmic or Neutral or Lulu clan (P8, P5/2)

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=== Clans defined by a 3.5.7 (yaza noca) comma ===

These are defined by a yaza noca or 7-limit-no-twos comma. Any no-twos comma is a ratio of odd numbers. If no other commas are tempered out, It is a non-octave tuning, with a period of a perfect 12th or tritave, 3/1 (or some fraction of that). See also [[subgroup temperaments]]. The pergen's multigen (the 2nd term, omitting any fraction) is always a 5-limit-no-twos ratio such as 5/3, 9/5, 25/9, etc. In a noca subgroup, "compound" means increased by 3/1 not 2/1.

These are defined by a yaza noca or 7-limit-no-twos comma. Any no-twos comma is a ratio of odd numbers. If no other commas are tempered out, It is a non-octave tuning, with a period of a perfect 12th or tritave, 3/1 (or some fraction of that). See also [[subgroup temperaments]]. The pergen's multigen (the 2nd term, omitting any fraction) is always a 5-limit-no-twos ratio such as 5/3, 9/5, 25/9, etc. In any noca subgroup, "compound" means increased by 3/1 not 2/1.

; Arcturus or Rutribiyo Noca clan (P12, M6)

@@ 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 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.
 ; [[Limma family|Limma or Sawa family]] (P8/5, ^1)
@@ Line 23: / Line 23: @@
 === Families defined by a 2.3.5 (ya) comma ===
-These are families defined by a ya or 5-limit comma. As we go up in rank two, the various 5-limit temperaments often break up as families of related temperaments, depending on how higher primes are mapped (or equivalently, on which higher limit commas are introduced.) The same comment applies to 7-limit temperaments and rank three, etc. Members of families and their relationships can be classified by the [[Normal lists|normal comma list]] of the various temperaments. Families include weak extensions as well as strong, in other words, the pergen shown here may change.
+These are families defined by a ya or 5-limit comma. As we go up in rank two, the various 5-limit temperaments often break up as families of related temperaments, depending on how higher primes are mapped (or equivalently, on which higher limit commas are introduced.) The same comment applies to 7-limit temperaments and rank three, etc. Members of families and their relationships can be classified by the [[Normal lists|normal comma list]] of the various temperaments. Families include weak extensions as well as strong, in other words, the [[pergen]] shown here may change.
 ; [[Meantone family|Meantone or Gu family]] (P8, P5)
@@ Line 74: / Line 74: @@
 ; [[Ripple family|Ripple or Quingu family]] (P8, P4/5)
-: This tempers out the ripple comma, 6561/6250 = {{Monzo| -1 8 -5 }}, which equates a stack of four [[10/9]]'s with [[8/5]]. As one might expect, [[12edo|12EDO]] is about as accurate as it can be.
+: This tempers out the ripple comma, 6561/6250 = {{Monzo| -1 8 -5 }}, which equates a stack of four [[10/9]]'s with [[8/5]], and five of them with [[16/9]]. The generator is [[27/25]], two of which equals 10/9, three of which equals [[6/5]], and five of which equals [[4/3]]. 5/4 is equated to an octave minus 8 generators. As one might expect, [[12edo|12EDO]] is about as accurate as it can be.
 ; [[Passion family|Passion or Saquingu family]] (P8, P4/5)
-: This tempers out the passion comma, 262144/253125 = {{monzo| 18 -4 -5 }}, which equates a stack of five [[16/15]]'s with [[4/3]].
+: This tempers out the passion comma, 262144/253125 = {{monzo| 18 -4 -5 }}, which equates a stack of four [[16/15]]'s with [[5/4]], and five of them with [[4/3]].
 ; [[Amity family|Amity or Saquinyo family]] (P8, P11/5)
-: This tempers out the [[amity comma]], 1600000/1594323 = {{Monzo|9 -13 5}}. The generator is 243/200 = ~339.5¢, five of which make ~8/3. 5/4 is equated to 4 octaves minus 13 generators, or  fifths minus 3 generators. An obvious 11-limit interpretation of the generator is 11/9, which makes Saquinlo. An obvious 13-limit interpretation of the generator is 39/32, which makes Lala-quintho.
+: This tempers out the [[amity comma]], 1600000/1594323 = {{Monzo|9 -13 5}}. The generator is 243/200 = ~339.5¢, five of which make ~8/3. 5/4 is equated to 4 octaves minus 13 generators, or 2 fifths minus 3 generators. An obvious 11-limit interpretation of the generator is 11/9, which makes Saquinlo. An obvious 13-limit interpretation of the generator is 39/32, which makes Lala-quintho.
 ; [[Magic family|Magic or Laquinyo family]] (P8, P12/5)
@@ Line 216: / Line 216: @@
 ; Lulubi clan (P8/2, P5)
-: This 2.3.11 clan tempers out the alpharabian limma, [[128/121]]. Both 11/8 and 16/11 is equated to half-octave period. This clan includes as a strong extension the pajaric temperament, which is in the diaschismic family.
+: This low-accuracy 2.3.11 clan tempers out the alpharabian limma, [[128/121]]. Both 11/8 and 16/11 are equated to half-octave period. This clan includes as a strong extension the pajaric temperament, which is in the diaschismic family.
 ; Rastmic or Neutral or Lulu clan (P8, P5/2)
@@ Line 261: / Line 261: @@
 === Clans defined by a 3.5.7 (yaza noca) comma ===
-These are defined by a yaza noca or 7-limit-no-twos comma. Any no-twos comma is a ratio of odd numbers. If no other commas are tempered out, It is a non-octave tuning, with a period of a perfect 12th or tritave, 3/1 (or some fraction of that). See also [[subgroup temperaments]]. The pergen's multigen (the 2nd term, omitting any fraction) is always a 5-limit-no-twos ratio such as 5/3, 9/5, 25/9, etc. In a noca subgroup, "compound" means increased by 3/1 not 2/1.
+These are defined by a yaza noca or 7-limit-no-twos comma. Any no-twos comma is a ratio of odd numbers. If no other commas are tempered out, It is a non-octave tuning, with a period of a perfect 12th or tritave, 3/1 (or some fraction of that). See also [[subgroup temperaments]]. The pergen's multigen (the 2nd term, omitting any fraction) is always a 5-limit-no-twos ratio such as 5/3, 9/5, 25/9, etc. In any noca subgroup, "compound" means increased by 3/1 not 2/1.
 ; Arcturus or Rutribiyo Noca clan (P12, M6)

Tour of regular temperaments: Difference between revisions

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