Val: Difference between revisions

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This is also known as '''warts''' or '''wart notation'''. It was developed by [[Herman Miller]] and [[Graham Breed]].

Given an explicit or assumed limit, any [[patent val]] can simply be represented by stating its first coefficient - the digit representing how many generators map to 2/1. For example, the 5-limit patent val for 17edo, {{val| 17 27 39 }}, can be called simply, "17".

Given an explicit or assumed limit, any [[patent val]] can simply be represented by stating its first coefficient – the digit representing how many generators map to 2/1. For example, the 5-limit patent val for 17edo, {{val| 17 27 39 }}, can be called simply, "17".

The patent val for any ~~EDO~~ will use the most accurate mapping for each individual prime when pure octaves are assumed. However, it may often be the case that one wants to refer to vals other than the patent val. For example, the 5-limit val {{val| 17 27 40 }}, which maps the 5/4 to the 424 cent interval rather than the 353 cent interval, is not the patent val for 17edo but may be preferred because it is lower in overall error. ~~Non-patent~~ vals are specified by adding a ''wart'' to the end of their name which specifies their deviation from the patent val. In this case, we want to specify that the 5/1 has been changed to use its second-most accurate mapping. Since 5 is the third prime number, we add the third letter of the alphabet to the end of the ~~EDO~~ number, to form "17c".

The patent val for any edo will use the most accurate mapping for each individual prime when pure octaves are assumed. However, it may often be the case that one wants to refer to vals other than the patent val. For example, the 5-limit val {{val| 17 27 40 }}, which maps the 5/4 to the 424-cent interval rather than the 353-cent interval, is not the patent val for 17edo but may be preferred because it is lower in overall error. Nonpatent vals are specified by adding a ''wart'' to the end of their name which specifies their deviation from the patent val. In this case, we want to specify that the 5/1 has been changed to use its second-most accurate mapping. Since 5 is the third prime number, we add the third letter of the alphabet to the end of the edo number, to form "17c".

If we wanted to use the third-most accurate mapping for 5, {{val| 17 27 38 }}, we'd write "17cc". In [[17edo]], the approximation of the prime-5 component is raised for an odd, and lowered for an even, amount of c letters: <code></code> = 39, <code>c</code> = 40, <code>cc</code> = 38, <code>ccc</code> = 41, <code>cccc</code> = 37.

If we wanted to use the third-most accurate mapping for 5, {{val| 17 27 38 }}, we would write "17cc". In [[17edo]], the approximation of the prime-5 component is raised for an odd, and lowered for an even, amount of c letters: <code></code> = 39, <code>c</code> = 40, <code>cc</code> = 38, <code>ccc</code> = 41, <code>cccc</code> = 37.

The general rules:

* Wart letters specify prime approximations being altered from the patent val.

* Wart letters specify prime approximations being altered from the patent val. The ''n''-th letter of the alphabet refers to the ''n''-th prime: a~2, b~3, c~5, d~7, e~11 etc.

* The ''n''-th letter of the alphabet refers to the ''n''-th prime: a~2, b~3, c~5, d~7, e~11 etc.

* A letter which appears ''m'' times refers to the (''m'' + 1)-th most accurate mapping for that prime.

* A letter which appears m times refers to the (''m'' + 1)-th most accurate mapping for that prime.

* So, if a number representing a val is wartless, it is taken to mean the patent val.

* ''So, if a number representing a val is wartless, it is taken to mean the patent val''.

* In Gramham Breed's temperament finder, the wart letter "p" is used to make explicit that the number refers to the patent val (though the letter originally was intended to stand for "prime"). Note that "p" is logically the letter for prime 53.

* ~~The~~ wart letter p is used to make explicit that the number refers to the patent val (~~however,~~ the letter p was ~~not chosen~~ to stand for "~~patent val~~"; that is ~~a coincidence. The creators of wart notation do not use~~ the ~~term "patent val". The~~ letter ~~originally was intended to stand~~ for "prime".)

* A wart letter may ''prefix'' the number, in which case it specifies the corresponding prime as the [[interval of equivalence]] to be divided by the following number. For example, b13 refers to the patent val of [[13edt|13ed3]]. The octave is assumed, so "a" is typically not written out.

* ~~When a~~ wart letter ~~prefixes~~ the number, ~~this means that~~ it's the ~~first~~ prime of the ~~map~~. For example, b13 refers to a ~~map for the equal temperament corresponding to 13-ED3~~.

== Vals in JI subgroups ==

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 This is also known as '''warts''' or '''wart notation'''. It was developed by [[Herman Miller]] and [[Graham Breed]].
-Given an explicit or assumed limit, any [[patent val]] can simply be represented by stating its first coefficient - the digit representing how many generators map to 2/1. For example, the 5-limit patent val for 17edo, {{val| 17 27 39 }}, can be called simply, "17".
+Given an explicit or assumed limit, any [[patent val]] can simply be represented by stating its first coefficient – the digit representing how many generators map to 2/1. For example, the 5-limit patent val for 17edo, {{val| 17 27 39 }}, can be called simply, "17".
-The patent val for any EDO will use the most accurate mapping for each individual prime when pure octaves are assumed. However, it may often be the case that one wants to refer to vals other than the patent val. For example, the 5-limit val {{val| 17 27 40 }}, which maps the 5/4 to the 424 cent interval rather than the 353 cent interval, is not the patent val for 17edo but may be preferred because it is lower in overall error. Non-patent vals are specified by adding a ''wart'' to the end of their name which specifies their deviation from the patent val. In this case, we want to specify that the 5/1 has been changed to use its second-most accurate mapping. Since 5 is the third prime number, we add the third letter of the alphabet to the end of the EDO number, to form "17c".
+The patent val for any edo will use the most accurate mapping for each individual prime when pure octaves are assumed. However, it may often be the case that one wants to refer to vals other than the patent val. For example, the 5-limit val {{val| 17 27 40 }}, which maps the 5/4 to the 424-cent interval rather than the 353-cent interval, is not the patent val for 17edo but may be preferred because it is lower in overall error. Nonpatent vals are specified by adding a ''wart'' to the end of their name which specifies their deviation from the patent val. In this case, we want to specify that the 5/1 has been changed to use its second-most accurate mapping. Since 5 is the third prime number, we add the third letter of the alphabet to the end of the edo number, to form "17c".
-If we wanted to use the third-most accurate mapping for 5, {{val| 17 27 38 }}, we'd write "17cc". In [[17edo]], the approximation of the prime-5 component is raised for an odd, and lowered for an even, amount of c letters: <code></code> = 39, <code>c</code> = 40, <code>cc</code> = 38, <code>ccc</code> = 41, <code>cccc</code> = 37.
+If we wanted to use the third-most accurate mapping for 5, {{val| 17 27 38 }}, we would write "17cc". In [[17edo]], the approximation of the prime-5 component is raised for an odd, and lowered for an even, amount of c letters: <code></code> = 39, <code>c</code> = 40, <code>cc</code> = 38, <code>ccc</code> = 41, <code>cccc</code> = 37.
 The general rules:
-* Wart letters specify prime approximations being altered from the patent val.
+* Wart letters specify prime approximations being altered from the patent val. The ''n''-th letter of the alphabet refers to the ''n''-th prime: a~2, b~3, c~5, d~7, e~11 etc.
-* The ''n''-th letter of the alphabet refers to the ''n''-th prime: a~2, b~3, c~5, d~7, e~11 etc.
+* A letter which appears ''m'' times refers to the (''m'' + 1)-th most accurate mapping for that prime.
-* A letter which appears m times refers to the (''m'' + 1)-th most accurate mapping for that prime.
+* So, if a number representing a val is wartless, it is taken to mean the patent val.
-* ''So, if a number representing a val is wartless, it is taken to mean the patent val''.
+* In Gramham Breed's temperament finder, the wart letter "p" is used to make explicit that the number refers to the patent val (though the letter originally was intended to stand for "prime"). Note that "p" is logically the letter for prime 53.
-* The wart letter p is used to make explicit that the number refers to the patent val (however, the letter p was not chosen to stand for "patent val"; that is a coincidence. The creators of wart notation do not use the term "patent val". The letter originally was intended to stand for "prime".)
+* A wart letter may ''prefix'' the number, in which case it specifies the corresponding prime as the [[interval of equivalence]] to be divided by the following number. For example, b13 refers to the patent val of [[13edt|13ed3]]. The octave is assumed, so "a" is typically not written out.
-* When a wart letter prefixes the number, this means that it's the first prime of the map. For example, b13 refers to a map for the equal temperament corresponding to 13-ED3.
 == Vals in JI subgroups ==