Val: Difference between revisions

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In practice, most single-row mappings in RTT are vals, because we usually deal with integer entries, and the other specifications only mean anything to advanced mathematicians.

== ~~Vals~~ in non-~~prime~~-~~limit spaces~~ ==

== Generalizations ==

The entries of a val measure equal-tempered steps, which can be thought of either as a generator for a rank-1 temperament (and thus the structure can be generalized to account for multiple generators, resulting in a mapping matrix) or as a logarithmic interval size measure (and thus the entries can be generalized to non-integer values to create a tuning map).

=== Mapping matrix ===

A mapping matrix is the most common generalization of a val, for a rank-2 or higher temperament. As a result, it has more than one row, To be precise, there is one row for each generator of the temperament.

=== Tuning map ===

=== ~~In JI subgroups~~ ===

A tuning map generalizes a val in a different way. Instead of treating the entries of a val as equal temperament steps, it treats them as a logarithmic interval size measure (usually cents). Thus, the entries of a tuning map may be any real number. ⟨1200 1901.955] is the tuning map for the justly-tuned 3-limit, and ⟨1200 1896.8 2787.1] is the tuning map for the 5-limit tuned to meantone (specifically, 31edo).

=== Vals in non-prime-limit spaces ===

==== Subgroup vals ====

It is rather intuitive to 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".

It is rather intuitive to 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 ~~(in~~ fact, the ~~normal~~ vals introduced in this ~~page~~ can be seen as entirely contained within this special case).

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 do not have to use only ascending primes). For instance, the patent val for 12et on the 2.3.7 subgroup is often notated 2.3.7 {{val| 12 19 34 }}. If the subgroup indicator is not 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. In fact, the ordinary vals introduced in this article can be seen as entirely contained within this special case.

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.)~~

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 12et 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 subgroup vals represent the same map from this subgroup to a rank-1 generator chain.

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

Subgroup vals can also be written using subgroups that do not 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.

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, the [[patent val]] for [[13edo]] in the 2.9.5 subgroup 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 9 induced from the 5-limit patent val is not the same as the 9 directly derived from the 2.9.5-subgroup 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]]:

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 2.9.5 {{val| 13 41 30 }} above makes 81/80 [[vanish]]: {{vmprod| 13 41 30 | -4 2 -1 }} = 13 × (-4) + 41 × 2 + 30 × (-1) = 0.

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

==== Tempered vals ====

=== ~~In regular temperaments~~ ===

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There are also tempered tuning maps, covered on their respective page.

~~== Generalizations ==~~

The entries of a val measure equal-tempered steps, which can be thought of either as a generator for a rank-1 temperament (and thus the structure can be generalized to account for multiple generators, resulting in a mapping matrix) or as a logarithmic interval size measure (and thus the entries can be generalized to non-integer values to create a tuning map).

~~=== Mapping matrix ===~~

~~{{Main| Mapping }}~~

A mapping matrix is the most common generalization of a val, for a rank-2 or higher temperament. As a result, it has more than one row, To be precise, there is one row for each generator of the temperament.

~~=== Tuning map ===~~

~~{{Main| Tuning map }}~~

A tuning map generalizes a val in a different way. Instead of treating the entries of a val as equal temperament steps, it treats them as a logarithmic interval size measure (usually cents). Thus, the entries of a tuning map may be any real number. ⟨1200 1901.955] is the tuning map for the justly-tuned 3-limit, and ⟨1200 1896.8 2787.1] is the tuning map for the 5-limit tuned to meantone (specifically, 31edo).

== See also ==

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* [[Monzos and interval space]]

* [[Patent val]]

* ~~Definition on Tonalsoft's encyclopedia of microtonal music theory:~~ http://tonalsoft.com/enc/v/val.aspx

== External links ==

* [http://tonalsoft.com/enc/v/val.aspx Tonalsoft Encyclopedia | ''Val'']

[[Category:Regular temperament theory]]

@@ Line 159: / Line 159: @@
 In practice, most single-row mappings in RTT are vals, because we usually deal with integer entries, and the other specifications only mean anything to advanced mathematicians.
-== Vals in non-prime-limit spaces ==
+== Generalizations ==
+The entries of a val measure equal-tempered steps, which can be thought of either as a generator for a rank-1 temperament (and thus the structure can be generalized to account for multiple generators, resulting in a mapping matrix) or as a logarithmic interval size measure (and thus the entries can be generalized to non-integer values to create a tuning map).
+=== Mapping matrix ===
+{{Main| Mapping }}
+A mapping matrix is the most common generalization of a val, for a rank-2 or higher temperament. As a result, it has more than one row, To be precise, there is one row for each generator of the temperament.
+=== Tuning map ===
+{{Main| Tuning map }}
-=== In JI subgroups ===
+A tuning map generalizes a val in a different way. Instead of treating the entries of a val as equal temperament steps, it treats them as a logarithmic interval size measure (usually cents). Thus, the entries of a tuning map may be any real number. ⟨1200 1901.955] is the tuning map for the justly-tuned 3-limit, and ⟨1200 1896.8 2787.1] is the tuning map for the 5-limit tuned to meantone (specifically, 31edo).
+=== Vals in non-prime-limit spaces ===
+==== Subgroup vals ====
 {{Main| Subgroup monzos and vals }}
-It is rather intuitive to 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".
+It is rather intuitive to 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 (in fact, the normal vals introduced in this page can be seen as entirely contained within this special case).
+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 do not have to use only ascending primes). For instance, the patent val for 12et on the 2.3.7 subgroup is often notated 2.3.7 {{val| 12 19 34 }}. If the subgroup indicator is not 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. In fact, the ordinary vals introduced in this article can be seen as entirely contained within this special case.
-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.)
+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 12et 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 subgroup vals represent the same map from this subgroup to a rank-1 generator chain.
-Svals can also be written using subgroups that don't involve primes, e.g. 2.3.7.13/5 {{val| 46 73 129 63 }}.
+Subgroup vals can also be written using subgroups that do not 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.
+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, the [[patent val]] for [[13edo]] in the 2.9.5 subgroup 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 9 induced from the 5-limit patent val is not the same as the 9 directly derived from the 2.9.5-subgroup 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]]:
+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 2.9.5 {{val| 13 41 30 }} above makes 81/80 [[vanish]]: {{vmprod| 13 41 30 | -4 2 -1 }} = 13 × (-4) + 41 × 2 + 30 × (-1) = 0.
- &#x27E8;13, 41, 30|2^-4, 9^2, 5^-1&#x27E9; = 13*-4 + 41*2 + 30*-1 = 0.
+==== Tempered vals ====
-=== In regular temperaments ===
 {{Main| Tempered monzos and vals }}
@@ Line 184: / Line 194: @@
 There are also tempered tuning maps, covered on their respective page.
-== Generalizations ==
-The entries of a val measure equal-tempered steps, which can be thought of either as a generator for a rank-1 temperament (and thus the structure can be generalized to account for multiple generators, resulting in a mapping matrix) or as a logarithmic interval size measure (and thus the entries can be generalized to non-integer values to create a tuning map).
-=== Mapping matrix ===
-{{Main| Mapping }}
-A mapping matrix is the most common generalization of a val, for a rank-2 or higher temperament. As a result, it has more than one row, To be precise, there is one row for each generator of the temperament.
-=== Tuning map ===
-{{Main| Tuning map }}
-A tuning map generalizes a val in a different way. Instead of treating the entries of a val as equal temperament steps, it treats them as a logarithmic interval size measure (usually cents). Thus, the entries of a tuning map may be any real number. ⟨1200 1901.955] is the tuning map for the justly-tuned 3-limit, and ⟨1200 1896.8 2787.1] is the tuning map for the 5-limit tuned to meantone (specifically, 31edo).
 == See also ==
@@ Line 204: / Line 201: @@
 * [[Monzos and interval space]]
 * [[Patent val]]
-* Definition on Tonalsoft's encyclopedia of microtonal music theory: http://tonalsoft.com/enc/v/val.aspx
+== External links ==
+* [http://tonalsoft.com/enc/v/val.aspx Tonalsoft Encyclopedia | ''Val'']
 [[Category:Regular temperament theory]]