Val: Difference between revisions

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The 2022 version used a plus sign (+) in place of the caret and a minus sign (-) in place of the vee.

~~== Vals in JI subgroups ==~~

We can generalize the concept of monzos and vals from the ''p''-limit (for some prime ''p'') to other [[JI subgroup]]s. This can be useful when considering different edo tunings of [[subgroup temperaments]]. [[Gene Ward Smith]] called these "[[sval]]s", short for "[[subgroup val]]s", and correspondingly "[[smonzo]]s" as short for "[[subgroup monzo]]s".

To notate a subgroup val, we typically precede the "bra" (angle bracket) notation with an indicator regarding the subgroup (and choice of basis, as we don't have to use only ascending primes). 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.

Note that we could, for instance, use a different basis for the same subgroup - for instance, we could instead write "2.3.21 {{val| 12 19 53 }}", which is the 12 equal patent val in the "2.3.21" subgroup. Since the "2.3.21" subgroup is the same as the "2.3.7" subgroup, just written with a different basis, these two apparently "different" svals represent the same map from this subgroup to a rank-1 generator chain. (It is a matter of semantics if these are thought of as "different" svals or "the same sval" written using a different basis.)

~~Svals can also be written using subgroups that don't involve primes, e.g. 2.3.7.13/5 {{val| 46 73 129 63 }}.~~

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 {{monzo| -4 2 -1 }}", and it is thus easy to see that the 2.9.5 13p val above makes 81/80 [[vanish]]:~~

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

~~== Vals in regular temperaments ==~~

~~{{Main| Tmonzos and tvals }}~~

There is also a notion of a ''tempered val'' on a group of ''tempered monzos'', representing intervals in some [[regular temperament]]. These names are sometimes abbreviated as ''tval'' and ''tmonzo'', respectively. Typically, this is made explicit by writing the generators beforehand. When the tempered intervals have accepted names, such as in [[meantone]], we can use names like ''P8'' and ''P5'', so that the tval P8.P5 {{val| 12 7 }} represents the 12edo "patent tval" in meantone (given that particular basis). If the intervals do not 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 }}.

== Vals vs. mappings ==

A val is more specific than a mapping, both as in the general mathematical sense as well as [[mapping|the regular temperament sense]]:

# A val can be thought of as a mapping with one row. Put another way, the rows of mappings are vals. To be mathematically precise, a val is a specific type of [[Wikipedia:Linear_map|(linear) mapping]] called a [[Wikipedia:Linear_form|"linear form", or "linear functional"]], which means that its output is a [[Wikipedia:Scalar_(mathematics)|scalar]], or in other words, a single number. This corresponds to the fact that a val must be a 1xM array of numbers, or in other words a [[Wikipedia:Vector_(mathematics_and_physics)|vector]] (specifically a [[Wikipedia:Row_and_column_vectors|row vector]], AKA covector).

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''Main article: [[Subgroup monzos and vals]]''

A subgroup val ~~specifies~~ the ~~tunings of~~ an ~~arbitrary set~~ of ~~intervals~~, ~~which~~ can be ~~stacked~~ to ~~form all~~ the ~~intervals~~ in a subgroup. ~~For example~~, the subgroup ~~val~~ 2.3.7 ~~⟨5 8 14] tells you~~ that prime 2 is ~~mapped~~ to 5 steps, ~~prime~~ 3 ~~is mapped~~ to 8 steps, and ~~prime 7~~ is ~~mapped~~ to ~~14 steps~~, ~~without specifying a mapping for~~ 5.

We can generalize the concept of monzos and vals from the ''p''-limit (for some prime ''p'') to other [[JI subgroup]]s. This can be useful when considering different edo tunings of [[subgroup temperaments]]. [[Gene Ward Smith]] called these "[[sval]]s", short for "[[subgroup val]]s", and correspondingly "[[smonzo]]s" as short for "[[subgroup monzo]]s".

To notate a subgroup val, we typically precede the "bra" (angle bracket) notation with an indicator regarding the subgroup (and choice of basis, as we don't have to use only ascending primes). 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.

Note that we could, for instance, use a different basis for the same subgroup - for instance, we could instead write "2.3.21 {{val| 12 19 53 }}", which is the 12 equal patent val in the "2.3.21" subgroup. Since the "2.3.21" subgroup is the same as the "2.3.7" subgroup, just written with a different basis, these two apparently "different" svals represent the same map from this subgroup to a rank-1 generator chain. (It is a matter of semantics if these are thought of as "different" svals or "the same sval" written using a different basis.)

Svals can also be written using subgroups that don't involve primes, e.g. 2.3.7.13/5 {{val| 46 73 129 63 }}.

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 {{monzo| -4 2 -1 }}", and it is thus easy to see that the 2.9.5 13p val above makes 81/80 [[vanish]]:

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

=== Tempered val ===

''Main article: [[Tempered monzos and vals]]''

A tempered val ~~specifies the mappings~~ of [[~~Mapped interval|~~tempered intervals]] (that ~~is, intervals in~~ the ~~space of~~ intervals ~~generated by~~ a temperament). ~~For example~~, ~~⟨15 2] is the tval for porcupine~~ (~~with generators ~~~2 ~~and ~10~~/~~9) in 15edo. There are also tempered tuning maps~~.

There is also a notion of a ''tempered val'' on a group of ''tempered monzos'', representing intervals in some [[regular temperament]]. These names are sometimes abbreviated as ''tval'' and ''tmonzo'', respectively. Typically, this is made explicit by writing the generators beforehand. When the tempered intervals have accepted names, such as in [[meantone]], we can use names like ''P8'' and ''P5'', so that the tval P8.P5 {{val| 12 7 }} represents the 12edo "patent tval" in meantone (given that particular basis). If the intervals do not 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 142: / Line 142: @@
 The 2022 version used a plus sign (+) in place of the caret and a minus sign (-) in place of the vee.
-== Vals in JI subgroups ==
-We can generalize the concept of monzos and vals from the ''p''-limit (for some prime ''p'') to other [[JI subgroup]]s. This can be useful when considering different edo tunings of [[subgroup temperaments]]. [[Gene Ward Smith]] called these "[[sval]]s", short for "[[subgroup val]]s", and correspondingly "[[smonzo]]s" as short for "[[subgroup monzo]]s".
-To notate a subgroup val, we typically precede the "bra" (angle bracket) notation with an indicator regarding the subgroup (and choice of basis, as we don't have to use only ascending primes). 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.
-Note that we could, for instance, use a different basis for the same subgroup - for instance, we could instead write "2.3.21 {{val| 12 19 53 }}", which is the 12 equal patent val in the "2.3.21" subgroup. Since the "2.3.21" subgroup is the same as the "2.3.7" subgroup, just written with a different basis, these two apparently "different" svals represent the same map from this subgroup to a rank-1 generator chain. (It is a matter of semantics if these are thought of as "different" svals or "the same sval" written using a different basis.)
-Svals can also be written using subgroups that don't involve primes, e.g. 2.3.7.13/5 {{val| 46 73 129 63 }}.
-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 {{monzo| -4 2 -1 }}", and it is thus easy to see that the 2.9.5 13p val above makes 81/80 [[vanish]]:
- &#x27E8;13, 41, 30|2^-4, 9^2, 5^-1&#x27E9; = 13*-4 + 41*2 + 30*-1 = 0.
-== Vals in regular temperaments ==
-{{Main| Tmonzos and tvals }}
-There is also a notion of a ''tempered val'' on a group of ''tempered monzos'', representing intervals in some [[regular temperament]]. These names are sometimes abbreviated as ''tval'' and ''tmonzo'', respectively. Typically, this is made explicit by writing the generators beforehand. When the tempered intervals have accepted names, such as in [[meantone]], we can use names like ''P8'' and ''P5'', so that the tval P8.P5 {{val| 12 7 }} represents the 12edo "patent tval" in meantone (given that particular basis). If the intervals do not 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 }}.
 == Vals vs. mappings ==
 A val is more specific than a mapping, both as in the general mathematical sense as well as [[mapping|the regular temperament sense]]:
 # A val can be thought of as a mapping with one row. Put another way, the rows of mappings are vals. To be mathematically precise, a val is a specific type of [[Wikipedia:Linear_map|(linear) mapping]] called a [[Wikipedia:Linear_form|"linear form", or "linear functional"]], which means that its output is a [[Wikipedia:Scalar_(mathematics)|scalar]], or in other words, a single number. This corresponds to the fact that a val must be a 1xM array of numbers, or in other words a [[Wikipedia:Vector_(mathematics_and_physics)|vector]] (specifically a [[Wikipedia:Row_and_column_vectors|row vector]], AKA covector).
@@ Line 187: / Line 166: @@
 ''Main article: [[Subgroup monzos and vals]]''
-A subgroup val specifies the tunings of an arbitrary set of intervals, which can be stacked to form all the intervals in a subgroup. For example, the subgroup val 2.3.7 ⟨5 8 14] tells you that prime 2 is mapped to 5 steps, prime 3 is mapped to 8 steps, and prime 7 is mapped to 14 steps, without specifying a mapping for 5.
+We can generalize the concept of monzos and vals from the ''p''-limit (for some prime ''p'') to other [[JI subgroup]]s. This can be useful when considering different edo tunings of [[subgroup temperaments]]. [[Gene Ward Smith]] called these "[[sval]]s", short for "[[subgroup val]]s", and correspondingly "[[smonzo]]s" as short for "[[subgroup monzo]]s".
+To notate a subgroup val, we typically precede the "bra" (angle bracket) notation with an indicator regarding the subgroup (and choice of basis, as we don't have to use only ascending primes). 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.
+Note that we could, for instance, use a different basis for the same subgroup - for instance, we could instead write "2.3.21 {{val| 12 19 53 }}", which is the 12 equal patent val in the "2.3.21" subgroup. Since the "2.3.21" subgroup is the same as the "2.3.7" subgroup, just written with a different basis, these two apparently "different" svals represent the same map from this subgroup to a rank-1 generator chain. (It is a matter of semantics if these are thought of as "different" svals or "the same sval" written using a different basis.)
+Svals can also be written using subgroups that don't involve primes, e.g. 2.3.7.13/5 {{val| 46 73 129 63 }}.
+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 {{monzo| -4 2 -1 }}", and it is thus easy to see that the 2.9.5 13p val above makes 81/80 [[vanish]]:
+ &#x27E8;13, 41, 30|2^-4, 9^2, 5^-1&#x27E9; = 13*-4 + 41*2 + 30*-1 = 0.
 === Tempered val ===
 ''Main article: [[Tempered monzos and vals]]''
-A tempered val specifies the mappings of [[Mapped interval|tempered intervals]] (that is, intervals in the space of intervals generated by a temperament). For example, ⟨15 2] is the tval for porcupine (with generators ~2 and ~10/9) in 15edo. There are also tempered tuning maps.
+There is also a notion of a ''tempered val'' on a group of ''tempered monzos'', representing intervals in some [[regular temperament]]. These names are sometimes abbreviated as ''tval'' and ''tmonzo'', respectively. Typically, this is made explicit by writing the generators beforehand. When the tempered intervals have accepted names, such as in [[meantone]], we can use names like ''P8'' and ''P5'', so that the tval P8.P5 {{val| 12 7 }} represents the 12edo "patent tval" in meantone (given that particular basis). If the intervals do not 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 ==