Val: Difference between revisions

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Also note that in practice vals are ''very far'' from just any list of positive integers; rather, they are generally equal to or one off from the lists of integers that correspond to a ''patent val''.

=== ~~Warts and generalized~~ patent vals ===

=== Generalized patent vals ===

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This works by instead of doing log2(''p'') (where ''p'' is prime) we use log2.01…(''p'') or something to that effect, where 2.01/1 is our altered version of 2/1. The ''val'' produced by a slight alteration is usually the same, so there are actually continuous ranges where the val produced is the same.

For example, let us say we want to interpret [[104edo]] (104-tone equal temperament) as a [[19-limit]] temperament; there is two possible mappings to use for 5; all primes up to and including 19 are sharp except for 5 which is quite flat, which causes a lot of inconsistencies; therefore a more natural val to use than the patent val is using the second-best mapping for 5, as log2(5) × 104 = 241.4805 is very close to exactly off anyways, and given the precision of 104edo, using the second-best mapping is very reasonable, as usually the sharpness of prime 5 cancels out with the sharpness of other primes when constructing ratios from them. But if we basically always want to use the patent val except for a slight modification to a second-best mapping for a handful of primes, do we really need to specify the full val every time? The answer is of course no.

For example, let us say we want to interpret [[104edo]] (104-tone equal temperament) as a [[19-limit]] temperament; there is two possible mappings to use for 5; all primes up to and including 19 are sharp except for 5 which is quite flat, which causes a lot of inconsistencies; therefore a more natural val to use than the patent val is using the second-best mapping for 5, as log2(5) × 104 = 241.4805 is very close to exactly off anyways, and given the precision of 104edo, using the second-best mapping is very reasonable, as usually the sharpness of prime 5 cancels out with the sharpness of other primes when constructing ratios from them.

~~We can specify the patent val of ''N''-edo as just ''N'', then we can specify each prime we want to map to the second best approximation by a cryptic letter shorthand:~~

* adding a means you make the mapping of 2 worse

* adding b means you make the mapping of 3 worse

* adding c means you make the mapping of 5 worse

* adding d means you make the mapping of 7 worse

* etc.

So we can refer to "104c" (not to be confused with 104{{cent}} (cents)), where we mnemonically think "a, b, '''c'''; 3rd letter; 3rd prime is 2, 3, '''5'''; there is one c so we make the mapping of prime 5 worse (further from just) ''once'' compared to patent".

Perhaps a much clearer notation is to specify explicitly which primes are altered how many times in which direction from the patent val; the notation used by [https://sintel.pythonanywhere.com sintel's temperament finder] would be: 104[+5] and if we also "''warted''" prime 23 (which is similarly flat to prime 5), it'd be 104[+5, +23], which corresponds in the other notation to 104ci where i is the 9th letter of the alphabet and 23 is the 9th prime. They differ on how to notate more than one wart though; with [+5, +23] the direction is ''always'' sharpwards, and [-5, -23] would be flatwards, while with "ci" it's based on "second-best, third-best, etc."; so repeated warts with letters have alternating error while repeated +'s and -'s always have the same kind of error. Slight variations of the notation that [[User:sintel|sintel]] uses have been reinvented and suggested by multiple people, so the one in use by a temperament finder is prioritized here.

== Applications ==

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If for some strange reason you'd instead like to say that 900 cents is 7/4, then that would be represented by the {{val| 12 19 28 33 }} val (notated 12d), and if you'd like to say that 1100 cents is 7/4, that would be represented by the {{val| 12 19 28 35 }} (12dd) val.

== ~~Warts explanation~~ ==

== Shorthand notations ==

If we basically always want to use the patent val except for a slight modification to a second-best mapping for a handful of primes, it will be tedious to specify the full val every time, so the following shorthand notations are developed to address that.

=== Wart notation ===

In this notation, we can specify the patent val of ''n''-edo as just ''n'', then we can specify each prime we want to map to the second best approximation by appending a letter, called a ''wart'', after the number:

* adding a means you make the mapping of 2 worse

* adding b means you make the mapping of 3 worse

* adding c means you make the mapping of 5 worse

* adding d means you make the mapping of 7 worse

* etc.

So we can refer to {{val| 17 27 40 }} by "17c" (not to be confused with 17{{cent}} (cents)), where we mnemonically think "a, b, '''c'''; 3rd letter; 3rd prime is 2, 3, '''5'''; there is one 'c' so we make the mapping of prime 5 worse (further from just) once compared to patent".

The general rules:

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In 2022 [[User:Mike Battaglia|Mike Battaglia]] proposed '''SOV notation''' as a way to be explicit about which primes are being affected and in which direction. In 2024 it was further refined by him and [[User:Frostburn|Lumi Pakkanen]] to be more analogous to [[Ups and downs notation]].

~~Generalized~~ patent vals are notated using the number of divisions followed by square brackets e.g. 17[] for {{val| 17 27 39 }}.

In this notation, patent vals are notated using the number of divisions followed by square brackets e.g. 17[] for {{val| 17 27 39 }}. To indicate a wider mapping for a prime, it is prefixed with a carret (^) e.g. 17[^5] for {{val| 17 27 40 }}. To indicate a narrower mapping for a prime it is prefixed with a vee (v) e.g. 17[v5] for {{val| 17 27 38 }}. The prefixes stack e.g. 17[^^5] corresponds to {{val| 17 27 41 }}. Multiple modifications are separated by commas (and optionally spaces) e.g. 17[v3, ^5] for {{val| 17 26 40 }}.

To indicate a wider mapping for a prime it is prefixed with a carret (^) e.g. 17[^5] for {{val| 17 27 40 }}.

To indicate a narrower mapping for a prime it is prefixed with a vee (v) e.g. 17[v5] for {{val| 17 27 38 }}.

The prefixes stack e.g. 17[^^5] corresponds to {{val| 17 27 41 }}.

Multiple modifications are separated by commas (and optionally spaces) e.g. 17[v3, ^5] for {{val| 17 26 40 }}.

The interval of equivalence may be prefixed in square brackets e.g. [3]13[] for {{val| 8 13 19 }} (subgroup 2.3.5).

The subgroup may be made explicit separated by an "at" sign (@) at the end e.g. 46[]@2.3.7.13/5 for {{val| 46 73 129 63 }} (subgroup 2.3.7.13/5).

The subgroup may be made explicit separated by an "at" sign (@) at the end e.g. 46[]@2.3.7.13/5 for {{val| 46 73 129 63 }} (subgroup 2.3.7.13/5). Formal primes are treated the same way as actual primes e.g. 46[^13/5]@2.3.7.13/5 for {{val| 46 73 129 64 }} (subgroup 2.3.7.13/5)

Formal primes are treated the same way as actual primes e.g. 46[^13/5]@2.3.7.13/5 for {{val| 46 73 129 64 }} (subgroup 2.3.7.13/5)

For patent vals the empty square brackets are optional when using an "at" sign. The subgroup itself is optional if its obvious from context e.g. 12@ for {{val|12 19 28 }} (subgroup 2.3.5).

For patent vals the empty square brackets are optional when using an "at" sign. The subgroup itself is optional if its obvious from context e.g. 12@ for {{val| 12 19 28 }} (subgroup 2.3.5).

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

The 2022 version used a plus sign (+) in place of the caret and a minus sign (-) in place of the vee. [https://sintel.pythonanywhere.com Sintel's temperament calculator] is a notable implementation of this notation.

== Vals vs. mappings ==

@@ Line 83: / Line 83: @@
 Also note that in practice vals are ''very far'' from just any list of positive integers; rather, they are generally equal to or one off from the lists of integers that correspond to a ''patent val''.
-=== Warts and generalized patent vals ===
+=== Generalized patent vals ===
 {{Main| Generalized patent val }}
 {{See also| patent val }}
@@ Line 91: / Line 91: @@
 This works by instead of doing log<sub>2</sub>(''p'') (where ''p'' is prime) we use log<sub>2.01…</sub>(''p'') or something to that effect, where 2.01/1 is our altered version of 2/1. The ''val'' produced by a slight alteration is usually the same, so there are actually continuous ranges where the val produced is the same.
-For example, let us say we want to interpret [[104edo]] (104-tone equal temperament) as a [[19-limit]] temperament; there is two possible mappings to use for 5; all primes up to and including 19 are sharp except for 5 which is quite flat, which causes a lot of inconsistencies; therefore a more natural val to use than the patent val is using the second-best mapping for 5, as log<sub>2</sub>(5) × 104 = 241.4805 is very close to exactly off anyways, and given the precision of 104edo, using the second-best mapping is very reasonable, as usually the sharpness of prime 5 cancels out with the sharpness of other primes when constructing ratios from them. But if we basically always want to use the patent val except for a slight modification to a second-best mapping for a handful of primes, do we really need to specify the full val every time? The answer is of course no.
+For example, let us say we want to interpret [[104edo]] (104-tone equal temperament) as a [[19-limit]] temperament; there is two possible mappings to use for 5; all primes up to and including 19 are sharp except for 5 which is quite flat, which causes a lot of inconsistencies; therefore a more natural val to use than the patent val is using the second-best mapping for 5, as log<sub>2</sub>(5) × 104 = 241.4805 is very close to exactly off anyways, and given the precision of 104edo, using the second-best mapping is very reasonable, as usually the sharpness of prime 5 cancels out with the sharpness of other primes when constructing ratios from them.
-We can specify the patent val of ''N''-edo as just ''N'', then we can specify each prime we want to map to the second best approximation by a cryptic letter shorthand:
-* adding a means you make the mapping of 2 worse
-* adding b means you make the mapping of 3 worse
-* adding c means you make the mapping of 5 worse
-* adding d means you make the mapping of 7 worse
-* etc.
-So we can refer to "104c" (not to be confused with 104{{cent}} (cents)), where we mnemonically think "a, b, '''c'''; 3rd letter; 3rd prime is 2, 3, '''5'''; there is one c so we make the mapping of prime 5 worse (further from just) ''once'' compared to patent".
-Perhaps a much clearer notation is to specify explicitly which primes are altered how many times in which direction from the patent val; the notation used by [https://sintel.pythonanywhere.com sintel's temperament finder] would be: 104[+5] and if we also "''warted''" prime 23 (which is similarly flat to prime 5), it'd be 104[+5, +23], which corresponds in the other notation to 104ci where i is the 9th letter of the alphabet and 23 is the 9th prime. They differ on how to notate more than one wart though; with [+5, +23] the direction is ''always'' sharpwards, and [-5, -23] would be flatwards, while with "ci" it's based on "second-best, third-best, etc."; so repeated warts with letters have alternating error while repeated +'s and -'s always have the same kind of error. Slight variations of the notation that [[User:sintel|sintel]] uses have been reinvented and suggested by multiple people, so the one in use by a temperament finder is prioritized here.
 == Applications ==
@@ Line 117: / Line 107: @@
 If for some strange reason you'd instead like to say that 900 cents is 7/4, then that would be represented by the {{val| 12 19 28 33 }} val (notated 12d), and if you'd like to say that 1100 cents is 7/4, that would be represented by the {{val| 12 19 28 35 }} (12dd) val.
-== Warts explanation ==
+== Shorthand notations ==
-{{Todo| rework }}
+If we basically always want to use the patent val except for a slight modification to a second-best mapping for a handful of primes, it will be tedious to specify the full val every time, so the following shorthand notations are developed to address that.
+=== Wart notation ===
+In this notation, we can specify the patent val of ''n''-edo as just ''n'', then we can specify each prime we want to map to the second best approximation by appending a letter, called a ''wart'', after the number:
+* adding a means you make the mapping of 2 worse
+* adding b means you make the mapping of 3 worse
+* adding c means you make the mapping of 5 worse
+* adding d means you make the mapping of 7 worse
+* etc.
+So we can refer to {{val| 17 27 40 }} by "17c" (not to be confused with 17{{cent}} (cents)), where we mnemonically think "a, b, '''c'''; 3rd letter; 3rd prime is 2, 3, '''5'''; there is one 'c' so we make the mapping of prime 5 worse (further from just) once compared to patent".
 The general rules:
@@ Line 131: / Line 131: @@
 In 2022 [[User:Mike Battaglia|Mike Battaglia]] proposed '''SOV notation''' as a way to be explicit about which primes are being affected and in which direction. In 2024 it was further refined by him and [[User:Frostburn|Lumi Pakkanen]] to be more analogous to [[Ups and downs notation]].
-Generalized patent vals are notated using the number of divisions followed by square brackets e.g. 17[] for {{val| 17 27 39 }}.
+In this notation, patent vals are notated using the number of divisions followed by square brackets e.g. 17[] for {{val| 17 27 39 }}. To indicate a wider mapping for a prime, it is prefixed with a carret (^) e.g. 17[^5] for {{val| 17 27 40 }}. To indicate a narrower mapping for a prime it is prefixed with a vee (v) e.g. 17[v5] for {{val| 17 27 38 }}. The prefixes stack e.g. 17[^^5] corresponds to {{val| 17 27 41 }}. Multiple modifications are separated by commas (and optionally spaces) e.g. 17[v3, ^5] for {{val| 17 26 40 }}.
-To indicate a wider mapping for a prime it is prefixed with a carret (^) e.g. 17[^5] for {{val| 17 27 40 }}.
-To indicate a narrower mapping for a prime it is prefixed with a vee (v) e.g. 17[v5] for {{val| 17 27 38 }}.
-The prefixes stack e.g. 17[^^5] corresponds to {{val| 17 27 41 }}.
-Multiple modifications are separated by commas (and optionally spaces) e.g. 17[v3, ^5] for {{val| 17 26 40 }}.
 The interval of equivalence may be prefixed in square brackets e.g. [3]13[] for {{val| 8 13 19 }} (subgroup 2.3.5).
-The subgroup may be made explicit separated by an "at" sign (@) at the end e.g. 46[]@2.3.7.13/5 for {{val| 46 73 129 63 }} (subgroup 2.3.7.13/5).
+The subgroup may be made explicit separated by an "at" sign (@) at the end e.g. 46[]@2.3.7.13/5 for {{val| 46 73 129 63 }} (subgroup 2.3.7.13/5). Formal primes are treated the same way as actual primes e.g. 46[^13/5]@2.3.7.13/5 for {{val| 46 73 129 64 }} (subgroup 2.3.7.13/5)
-Formal primes are treated the same way as actual primes e.g. 46[^13/5]@2.3.7.13/5 for {{val| 46 73 129 64 }} (subgroup 2.3.7.13/5)
-For patent vals the empty square brackets are optional when using an "at" sign. The subgroup itself is optional if its obvious from context e.g. 12@ for {{val|12 19 28 }} (subgroup 2.3.5).
+For patent vals the empty square brackets are optional when using an "at" sign. The subgroup itself is optional if its obvious from context e.g. 12@ for {{val| 12 19 28 }} (subgroup 2.3.5).
-The 2022 version used a plus sign (+) in place of the caret and a minus sign (-) in place of the vee.
+The 2022 version used a plus sign (+) in place of the caret and a minus sign (-) in place of the vee. [https://sintel.pythonanywhere.com Sintel's temperament calculator] is a notable implementation of this notation.
 == Vals vs. mappings ==