Vals and tuning space: Difference between revisions

Line 8:

__FORCETOC__

== Definition ==

A val "maps" just intonation to a certain number of steps in a chain of generators; by putting vals together we can define the mapping of a [[Regular_Temperaments|regular temperament]] and thereby define the temperament. A val is written in the form {{val|a1 a2 a3 ... ak}}, where the numbers a1 a2 a3 ... are the number of steps along the chain that the first k primes are mapped to. This can be generalized so that a1 a2 a3 ... represent the number of steps any JI basis is mapped to, whereas a JI basis for a [[Just_intonation_subgroups|just intonation subgroup]] is an independent collection of just intonation intervals, meaning that no one of them is a product of the rest.

A [[val|val]] "maps" just intonation to a certain number of steps in a chain of generators; by putting vals together we can define the mapping of a [[Regular_Temperaments|regular temperament]] and thereby define the temperament. A val is written in the form {{val|a1 a2 a3 ... ak}}, where the numbers a1 a2 a3 ... are the number of steps along the chain that the first k primes are mapped to. This can be generalized so that a1 a2 a3 ... represent the number of steps any JI basis is mapped to, whereas a JI basis for a [[Just_intonation_subgroups|just intonation subgroup]] is an independent collection of just intonation intervals, meaning that no one of them is a product of the rest.

A ''rank r'' temperament has r generators, and thus is defined by r vals. In the usual coordinates for the [[Harmonic_Limit|p-limit]], the set of generators are the first k prime numbers and the set of vals for a p-limit temperament gives you the coordinates for each prime harmonic in the p-limit. For example, all 5-limit rank-1 temperaments, or [[Equal|equal temperaments]], will be defined by a val {{val|a b c}}, where a is the number of generators it takes to reach the 2nd harmonic (2/1), b is the number of generators to reach the 3rd harmonic (3/1), and c is the number of generators it takes to reach the 5th harmonic (5/1). All 5-limit rank-2 temperaments are defined by two vals: {{monzo|{{val|a1 b1 c1}}, {{val|a2 b2 c2}}}}. Now, we locate the 2nd harmonic (2/1) with the 2-dimensional coordinates (a1, a2), sometimes written as {{monzo|a1 a2}}, meaning go up a1 of the first generator, and up a2 of the 2nd generator, to reach 2/1. Similarly, the 3rd harmonic and 5th harmonic will be reached by {{monzo|b1 b2}} and {{monzo|c1 c2}} respectively.

Line 41:

== Example ==

The rank-1 [[7-limit|7-limit]] ~~patent~~ [[~~val|~~val]] corresponding to [[31edo|31edo]] is {{val|31 49 72 87}}. This tells us that 31 steps reaches the 2, approximately 49 the 3, 72 the 5, and 87 the 7. In weighted coordinates, it becomes

The rank-1 [[7-limit|7-limit]] [[patent val]] corresponding to [[31edo|31edo]] is {{val|31 49 72 87}}. This tells us that 31 steps reaches the 2, approximately 49 the 3, 72 the 5, and 87 the 7. In weighted coordinates, it becomes

<math>\displaystyle

@@ Line 8: / Line 8: @@
 __FORCETOC__
 == Definition ==
-A val "maps" just intonation to a certain number of steps in a chain of generators; by putting vals together we can define the mapping of a [[Regular_Temperaments|regular temperament]] and thereby define the temperament. A val is written in the form {{val|a1 a2 a3 ... ak}}, where the numbers a1 a2 a3 ... are the number of steps along the chain that the first k primes are mapped to. This can be generalized so that a1 a2 a3 ... represent the number of steps any JI basis is mapped to, whereas a JI basis for a [[Just_intonation_subgroups|just intonation subgroup]] is an independent collection of just intonation intervals, meaning that no one of them is a product of the rest.
+A [[val|val]] "maps" just intonation to a certain number of steps in a chain of generators; by putting vals together we can define the mapping of a [[Regular_Temperaments|regular temperament]] and thereby define the temperament. A val is written in the form {{val|a1 a2 a3 ... ak}}, where the numbers a1 a2 a3 ... are the number of steps along the chain that the first k primes are mapped to. This can be generalized so that a1 a2 a3 ... represent the number of steps any JI basis is mapped to, whereas a JI basis for a [[Just_intonation_subgroups|just intonation subgroup]] is an independent collection of just intonation intervals, meaning that no one of them is a product of the rest.
 A ''rank r'' temperament has r generators, and thus is defined by r vals. In the usual coordinates for the [[Harmonic_Limit|p-limit]], the set of generators are the first k prime numbers and the set of vals for a p-limit temperament gives you the coordinates for each prime harmonic in the p-limit. For example, all 5-limit rank-1 temperaments, or [[Equal|equal temperaments]], will be defined by a val {{val|a b c}}, where a is the number of generators it takes to reach the 2nd harmonic (2/1), b is the number of generators to reach the 3rd harmonic (3/1), and c is the number of generators it takes to reach the 5th harmonic (5/1). All 5-limit rank-2 temperaments are defined by two vals: {{monzo|{{val|a1 b1 c1}}, {{val|a2 b2 c2}}}}. Now, we locate the 2nd harmonic (2/1) with the 2-dimensional coordinates (a1, a2), sometimes written as {{monzo|a1 a2}}, meaning go up a1 of the first generator, and up a2 of the 2nd generator, to reach 2/1. Similarly, the 3rd harmonic and 5th harmonic will be reached by {{monzo|b1 b2}} and {{monzo|c1 c2}} respectively.
@@ Line 41: / Line 41: @@
 == Example ==
-The rank-1 [[7-limit|7-limit]] patent [[val|val]] corresponding to [[31edo|31edo]] is {{val|31 49 72 87}}. This tells us that 31 steps reaches the 2, approximately 49 the 3, 72 the 5, and 87 the 7. In weighted coordinates, it becomes
+The rank-1 [[7-limit|7-limit]] [[patent val]] corresponding to [[31edo|31edo]] is {{val|31 49 72 87}}. This tells us that 31 steps reaches the 2, approximately 49 the 3, 72 the 5, and 87 the 7. In weighted coordinates, it becomes
 <math>\displaystyle