Val: Difference between revisions

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== Vals in JI subgroups ==

We can generalize the concept of monzos and vals from the ''p''-limit to other [[JI subgroup]]s. This can be useful when considering different edo tunings of [[subgroup temperaments]]. Gene Smith called these "[Sval|svals]" for short.

We can generalize the concept of monzos and vals from the ''p''-limit to other [[JI subgroup]]s. This can be useful when considering different edo tunings of [[subgroup temperaments]]. [[Gene Ward Smith]] called these "[[Sval|svals]]" for short.

To notate a subgroup val, we typically precede the "bra" notation with an indicator regarding the subgroup (and choice of basis). For instance, the patent val for 12 equal on the 2.3.7 subgroup is often notated "2.3.7 {{val|12 19 34}}". If the subgroup indicator isn't present, the subgroup can be inferred from context. It is very typical for a val with no explicit subgroup indicator to be interpreted as representing some prime limit, e.g. {{val|a b c}} would represent a 5-limit val.

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Note that the notion of a "patent val" for a subgroup val may not agree with the patent val on a prime limit. For instance, [[13edo]]'s "2.9.5 [[patent val]]" can be written as "2.9.5 {{val|13 41 30}}, because the best approximation to 2 is 13 steps, the best approximation to 9 is 41 steps, and the best approximation to 5 is 30 steps. Note that, however, the patent val on the 2.3.5 subgroup instead maps 3/1 to 21 steps, so that the "induced 9" from the 2.3.5 patent val is not the same as the "direct 9" from the 2.9.5 patent val.

This notation is also used for subgroup monzos; e.g. [[81/80]] on the 2.9.5 subgroup is "2.9.5 {{~~val~~|-4 2 -1}}", and it is thus easy to see that the 2.9.5 13p val above tempers out 81/80:

This notation is also used for subgroup monzos; e.g. [[81/80]] on the 2.9.5 subgroup is "2.9.5 {{monzo|-4 2 -1}}", and it is thus easy to see that the 2.9.5 13p val above tempers out 81/80:

⟨13, 41, 30|2^-4, 9^2, 5^-1⟩ = 13*-4 + 41*2 + 30*-1 = 0.

== Vals in Regular Temperaments ==

There is also a notion of a "tempered val" on a group of "tempered monzos," representing intervals in some [[regular temperament]]. This name is sometimes abbreviated as [[~~Tmonzos_and_Tvals~~ "tmonzos" and "tvals"]]. Typically, this is made explicit by writing the generators beforehand. When the tempered intervals have accepted names, such as in [[meantone ~~temperament~~]], we can use names like "P8" and "P5," so that the tval "P8.P5 {{val|12 7}}" represents the 12-edo "patent tval" in meantone temperament (given that particular basis). If the intervals don't have names, a [[Transversal]] can be given instead, preceded with the temperament name, so that we have "(meantone) 2.3/2 {{val|12 7}}, or "(meantone) 2.3/2 {{val|31 18}}".

There is also a notion of a "tempered val" on a group of "tempered monzos," representing intervals in some [[regular temperament]]. This name is sometimes abbreviated as [[Tmonzos and Tvals|"tmonzos" and "tvals"]]. Typically, this is made explicit by writing the generators beforehand. When the tempered intervals have accepted names, such as in [[meantone]] temperament, we can use names like "P8" and "P5," so that the tval "P8.P5 {{val|12 7}}" represents the 12-edo "patent tval" in meantone temperament (given that particular basis). If the intervals don't have names, a [[Transversal]] can be given instead, preceded with the temperament name, so that we have "(meantone) 2.3/2 {{val|12 7}}, or "(meantone) 2.3/2 {{val|31 18}}".

== See also ==

@@ Line 47: / Line 47: @@
 == Vals in JI subgroups ==
-We can generalize the concept of monzos and vals from the ''p''-limit to other [[JI subgroup]]s. This can be useful when considering different edo tunings of [[subgroup temperaments]]. Gene Smith called these "[Sval|svals]" for short.
+We can generalize the concept of monzos and vals from the ''p''-limit to other [[JI subgroup]]s. This can be useful when considering different edo tunings of [[subgroup temperaments]]. [[Gene Ward Smith]] called these "[[Sval|svals]]" for short.
 To notate a subgroup val, we typically precede the "bra" notation with an indicator regarding the subgroup (and choice of basis). For instance, the patent val for 12 equal on the 2.3.7 subgroup is often notated "2.3.7 {{val|12 19 34}}". If the subgroup indicator isn't present, the subgroup can be inferred from context. It is very typical for a val with no explicit subgroup indicator to be interpreted as representing some prime limit, e.g. {{val|a b c}} would represent a 5-limit val.
@@ Line 57: / Line 57: @@
 Note that the notion of a "patent val" for a subgroup val may not agree with the patent val on a prime limit. For instance, [[13edo]]'s "2.9.5 [[patent val]]" can be written as "2.9.5 {{val|13 41 30}}, because the best approximation to 2 is 13 steps, the best approximation to 9 is 41 steps, and the best approximation to 5 is 30 steps. Note that, however, the patent val on the 2.3.5 subgroup instead maps 3/1 to 21 steps, so that the "induced 9" from the 2.3.5 patent val is not the same as the "direct 9" from the 2.9.5 patent val.
-This notation is also used for subgroup monzos; e.g. [[81/80]] on the 2.9.5 subgroup is "2.9.5 {{val|-4 2 -1}}", and it is thus easy to see that the 2.9.5 13p val above tempers out 81/80:
+This notation is also used for subgroup monzos; e.g. [[81/80]] on the 2.9.5 subgroup is "2.9.5 {{monzo|-4 2 -1}}", and it is thus easy to see that the 2.9.5 13p val above tempers out 81/80:
   &#x27E8;13, 41, 30|2^-4, 9^2, 5^-1&#x27E9; = 13*-4 + 41*2 + 30*-1 = 0.
 == Vals in Regular Temperaments ==
-There is also a notion of a "tempered val" on a group of "tempered monzos," representing intervals in some [[regular temperament]]. This name is sometimes abbreviated as [[Tmonzos_and_Tvals "tmonzos" and "tvals"]]. Typically, this is made explicit by writing the generators beforehand. When the tempered intervals have accepted names, such as in [[meantone temperament]], we can use names like "P8" and "P5," so that the tval "P8.P5 {{val|12 7}}" represents the 12-edo "patent tval" in meantone temperament (given that particular basis). If the intervals don't have names, a [[Transversal]] can be given instead, preceded with the temperament name, so that we have "(meantone) 2.3/2 {{val|12 7}}, or  "(meantone) 2.3/2 {{val|31 18}}".
+There is also a notion of a "tempered val" on a group of "tempered monzos," representing intervals in some [[regular temperament]]. This name is sometimes abbreviated as [[Tmonzos and Tvals|"tmonzos" and "tvals"]]. Typically, this is made explicit by writing the generators beforehand. When the tempered intervals have accepted names, such as in [[meantone]] temperament, we can use names like "P8" and "P5," so that the tval "P8.P5 {{val|12 7}}" represents the 12-edo "patent tval" in meantone temperament (given that particular basis). If the intervals don't have names, a [[Transversal]] can be given instead, preceded with the temperament name, so that we have "(meantone) 2.3/2 {{val|12 7}}, or  "(meantone) 2.3/2 {{val|31 18}}".
 == See also ==