Val: Difference between revisions

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A [[val]] — short for ''valuation'' — is like an algorithm or procedure for finding out how to ~~approximate~~ [[frequency ratio]]s ([[interval]]s of [[just intonation|just intonation (JI)]]) with the pitches of an [[equal tuning]] such as an [[edo]]. The basic principle of using a ''patent'' val (the most common usecase) is that you round prime harmonics to edosteps, and then deduce the "mapping" of an arbitrary interval based on its prime factorization. (The "patent" refers to the method of choosing the approximations to the prime harmonics.)

A [[val]] — short for ''valuation'' — is like an algorithm or procedure for finding out how to represent [[frequency ratio]]s ([[interval]]s of [[just intonation|just intonation (JI)]]) with the pitches of an [[equal tuning]] such as an [[edo]]. The basic principle of using a ''patent'' val (the most common usecase) is that you round prime harmonics to edosteps, and then deduce the "mapping" of an arbitrary interval based on its prime factorization. (The "patent" refers to the method of choosing the approximations to the prime harmonics.)

This therefore assumes either that you want to use an equal tuning to approximate specific harmonies or that you have some other more indirect use in mind.

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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.

== ~~Generalizations ==~~

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

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

~~''Main article: [[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 article: [[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).~~

=== ~~Subgroup val~~ ===

=== In JI subgroups ===

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

~~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".

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.

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).

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.)

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

=== ~~Tempered val~~ ===

=== In regular temperaments ===

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

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 }}.

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 article: [[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 article: [[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 7: / Line 7: @@
 {{Beginner|Vals and tuning space}}
-A [[val]] — short for ''valuation'' — is like an algorithm or procedure for finding out how to approximate [[frequency ratio]]s ([[interval]]s of [[just intonation|just intonation (JI)]]) with the pitches of an [[equal tuning]] such as an [[edo]]. The basic principle of using a ''patent'' val (the most common usecase) is that you round prime harmonics to edosteps, and then deduce the "mapping" of an arbitrary interval based on its prime factorization. (The "patent" refers to the method of choosing the approximations to the prime harmonics.)
+A [[val]] — short for ''valuation'' — is like an algorithm or procedure for finding out how to represent [[frequency ratio]]s ([[interval]]s of [[just intonation|just intonation (JI)]]) with the pitches of an [[equal tuning]] such as an [[edo]]. The basic principle of using a ''patent'' val (the most common usecase) is that you round prime harmonics to edosteps, and then deduce the "mapping" of an arbitrary interval based on its prime factorization. (The "patent" refers to the method of choosing the approximations to the prime harmonics.)
 This therefore assumes either that you want to use an equal tuning to approximate specific harmonies or that you have some other more indirect use in mind.
@@ Line 151: / Line 151: @@
 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.
-== Generalizations ==
+== Vals in non-prime-limit spaces ==
-=== Mapping matrix ===
-''Main article: [[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 article: [[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).
-=== Subgroup val ===
+=== In JI subgroups ===
 ''Main article: [[Subgroup monzos and vals]]''
-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".
+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.
+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).
 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.)
@@ Line 180: / Line 170: @@
   &#x27E8;13, 41, 30|2^-4, 9^2, 5^-1&#x27E9; = 13*-4 + 41*2 + 30*-1 = 0.
-=== Tempered val ===
+=== In regular temperaments ===
 ''Main article: [[Tempered monzos and vals]]''
 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 }}.
+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 article: [[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 article: [[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 ==