Domain basis: Difference between revisions

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A '''domain basis''' is a [[basis]] for a ''domain'', or space of inputs to a function, such as a [[regular temperament]] [[mapping]]. Since the basis for the domain is more fundamental than other types of bases, such as [[comma bases]] or [[unchanged-interval bases]], we can typically shorten this to simply "basis", and we will be largely adopting that shorthand for the remainder of the article.

A '''domain basis''' is a [[basis]] for a ''domain'', or space of inputs to a function, such as a [[regular temperament]] [[mapping]]. Since the basis for the domain is more fundamental than other types of bases, such as [[comma basis|comma bases]] or [[unchanged-interval basis|unchanged-interval bases]], we can typically shorten this to simply "basis", and we will be largely adopting that shorthand for the remainder of the article.

= Basis elements and nonstandard domain bases =

Bases are expressed as lists of '''basis elements''', the building blocks of the corresponding space, and domain bases are no different. They are notated by separating these basis elements with periods. The most common bases consist of prime numbers, which are the building blocks of the [[Wikipedia:Rational_number|rational numbers]] we use for [[just intonation|justly intoned]] interval ratios. Examples of such bases are 2.3.5, or 2.3.7.11.

Bases are expressed as lists of '''[[basis elements]]''', the building blocks of the corresponding space, and domain bases are no different. They are notated by separating these basis elements with periods. The most common bases consist of prime numbers, which are the building blocks of the [[Wikipedia:Rational_number|rational numbers]] we use for [[just intonation|justly intoned]] interval ratios. Examples of such bases are 2.3.5, or 2.3.7.11.

When left unspecified, the basis is the <math>d</math> [[prime limit]], or in other words, the sequence of the first <math>d</math> primes where <math>d</math> is the [[dimensionality]]. So, for example, given the [[mapping|regular temperament mapping]] {{rket|{{map|1 0 -4 -13 -25}} {{map|0 1 4 10 18}}}}, we should assume that its domain basis is 2.3.5.7.11. This is called a '''standard basis''', and any other basis would therefore be a '''nonstandard basis'''.

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= Bases for temperaments =

In the context of a regular temperament, a domain basis serves as a minimal representation of all the intervals this temperament can map (some of which it completely [[vanishes]]). The full set of these mappable intervals is called the domain; it, in turn, is a ''sub''space with respect to a theoretically ''full'' domain which would include all conceivable intervals able to be built from the infinitude of greater and greater primes.

In the context of a regular temperament, a domain basis serves as a minimal representation of all the intervals this temperament can map (some of which it completely [[temper out|vanishes]]). The full set of these mappable intervals is called the domain; it, in turn, is a ''sub''space with respect to a theoretically ''full'' domain which would include all conceivable intervals able to be built from the infinitude of greater and greater primes.

So, for instance, a temperament in the 2.3.5 domain cannot map the intervals 7/6 or 11/8, because there is no way to represent either of those intervals using only the primes 2, 3, and 5. It could, however, temper 6/5, 5/4, 10/9, or 9/8, etc., because those intervals ''can'' be represented using only those three primes.

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-A '''domain basis''' is a [[basis]] for a ''domain'', or space of inputs to a function, such as a [[regular temperament]] [[mapping]]. Since the basis for the domain is more fundamental than other types of bases, such as [[comma bases]] or [[unchanged-interval bases]], we can typically shorten this to simply "basis", and we will be largely adopting that shorthand for the remainder of the article.
+A '''domain basis''' is a [[basis]] for a ''domain'', or space of inputs to a function, such as a [[regular temperament]] [[mapping]]. Since the basis for the domain is more fundamental than other types of bases, such as [[comma basis|comma bases]] or [[unchanged-interval basis|unchanged-interval bases]], we can typically shorten this to simply "basis", and we will be largely adopting that shorthand for the remainder of the article.
 = Basis elements and nonstandard domain bases =
-Bases are expressed as lists of '''basis elements''', the building blocks of the corresponding space, and domain bases are no different. They are notated by separating these basis elements with periods. The most common bases consist of prime numbers, which are the building blocks of the [[Wikipedia:Rational_number|rational numbers]] we use for [[just intonation|justly intoned]] interval ratios. Examples of such bases are 2.3.5, or 2.3.7.11.
+Bases are expressed as lists of '''[[basis elements]]''', the building blocks of the corresponding space, and domain bases are no different. They are notated by separating these basis elements with periods. The most common bases consist of prime numbers, which are the building blocks of the [[Wikipedia:Rational_number|rational numbers]] we use for [[just intonation|justly intoned]] interval ratios. Examples of such bases are 2.3.5, or 2.3.7.11.
 When left unspecified, the basis is the <math>d</math> [[prime limit]], or in other words, the sequence of the first <math>d</math> primes where <math>d</math> is the [[dimensionality]]. So, for example, given the [[mapping|regular temperament mapping]] {{rket|{{map|1 0 -4 -13 -25}} {{map|0 1 4 10 18}}}}, we should assume that its domain basis is 2.3.5.7.11. This is called a '''standard basis''', and any other basis would therefore be a '''nonstandard basis'''.
@@ Line 57: / Line 57: @@
 = Bases for temperaments =
-In the context of a regular temperament, a domain basis serves as a minimal representation of all the intervals this temperament can map (some of which it completely [[vanishes]]). The full set of these mappable intervals is called the domain; it, in turn, is a ''sub''space with respect to a theoretically ''full'' domain which would include all conceivable intervals able to be built from the infinitude of greater and greater primes.
+In the context of a regular temperament, a domain basis serves as a minimal representation of all the intervals this temperament can map (some of which it completely [[temper out|vanishes]]). The full set of these mappable intervals is called the domain; it, in turn, is a ''sub''space with respect to a theoretically ''full'' domain which would include all conceivable intervals able to be built from the infinitude of greater and greater primes.
 So, for instance, a temperament in the 2.3.5 domain cannot map the intervals 7/6 or 11/8, because there is no way to represent either of those intervals using only the primes 2, 3, and 5. It could, however, temper 6/5, 5/4, 10/9, or 9/8, etc., because those intervals ''can'' be represented using only those three primes.