Val: Difference between revisions
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{{Beginner|Vals and tuning space}} | {{Beginner|Vals and tuning space}} | ||
A [[val]] | A [[val]] – short for ''valuation'' – is like an algorithm or procedure for finding out how to represent intervals of [[just intonation|just intonation (JI)]] with the pitches of an [[equal tuning]] such as an [[edo]]. The basic principle of using a val is to assign [[prime harmonic]]s to edosteps, and then deduce the number of edosteps of an arbitrary interval based on its [[prime factorization]]. 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. | ||
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. | |||
== Motivation == | == Motivation == | ||
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== Applications == | == Applications == | ||
As discussed, vals are important in regular temperament theory because they provide a way to mathematically formalize how, specifically, the intervals in some set of equally spaced pitches are viewed as the tempered versions of more fundamental just intonation intervals. They can also be viewed as a way to map JI "onto" the chain, imbuing it with a harmonic context. Vals will enable you to figure out what commas your temperament eliminates, what [[comma pump]]s are available in the temperament, what the most consonant chords in the temperament are, how to optimize the octave stretch of the temperament to minimize tuning error, what EDOs support your temperament, and other operations as of yet undiscovered. | As discussed, vals are important in regular temperament theory because they provide a way to mathematically formalize how, specifically, the intervals in some set of equally spaced pitches are viewed as the tempered versions of more fundamental just intonation intervals. They can also be viewed as a way to map JI "onto" the chain, imbuing it with a harmonic context. Vals will enable you to figure out what commas your temperament eliminates, what [[comma pump]]s are available in the temperament, what the most consonant chords in the temperament are, how to optimize the octave stretch of the temperament to minimize tuning error, what EDOs support your temperament, and other operations as of yet undiscovered. | ||
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== Warts explanation == | == Warts explanation == | ||
{{ | {{Todo| rework }} | ||
The general rules: | The general rules: | ||
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=== In JI subgroups === | === In JI subgroups === | ||
{{Main| Subgroup monzos and vals }} | |||
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". | 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". | ||
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=== In regular temperaments === | === In regular temperaments === | ||
{{Main| 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 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 }}. | ||
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=== Mapping matrix === | === Mapping matrix === | ||
{{Main| 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. | 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 === | === Tuning map === | ||
{{Main| 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). | 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). | ||
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[[Category:Math]] | [[Category:Math]] | ||
[[Category:Notation]] | [[Category:Notation]] | ||
{{Todo| cleanup }} | |||