Val: Difference between revisions
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<span style="display: block; text-align: right;">[[:de:Val|Deutsch]] - [[ヴァル|日本語]]</span> | <span style="display: block; text-align: right;">[[:de:Val|Deutsch]] - [[ヴァル|日本語]]</span> | ||
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'''What are vals and what are they for?''' | '''What are vals and what are they for?''' | ||
=Definition= | ==Definition== | ||
A val is a map representing how to view the intervals in a single [[periods_and_generators|chain of generators]] as the tempered versions of intervals in just intonation. They form the link between things like EDOs and JI, and by doing so form the basis for all of regular temperament theory. It's very common for vals to refer to EDOs specifically, although they also show us how to relate larger chains of generators to JI as well (such as a stack of meantone fifths). | A val is a map representing how to view the intervals in a single [[periods_and_generators|chain of generators]] as the tempered versions of intervals in just intonation. They form the link between things like EDOs and JI, and by doing so form the basis for all of regular temperament theory. It's very common for vals to refer to EDOs specifically, although they also show us how to relate larger chains of generators to JI as well (such as a stack of meantone fifths). | ||
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Vals are important because they provide a way to mathematically formalize how, specifically, the intervals in a random chain of generators 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|comma pumps]] 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. | Vals are important because they provide a way to mathematically formalize how, specifically, the intervals in a random chain of generators 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|comma pumps]] 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. | ||
= | ==Example EDO== | ||
Consider the 5-limit val <12 19 28]. This val tells us that you should view 12 generator steps as mapping to the octave 2/1. Since the temperament which maps 12 generator steps to the octave is 12-EDO, this means you're describing 12-EDO. | Consider the 5-limit val <12 19 28]. This val tells us that you should view 12 generator steps as mapping to the octave 2/1. Since the temperament which maps 12 generator steps to the octave is 12-EDO, this means you're describing 12-EDO. | ||
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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 <12 19 28 33] val, and if you'd like to say that 123000 cents is 7/4, that would be represented by the <12 19 28 1254] val. | If for some strange reason you'd instead like to say that 900 cents is 7/4, then that would be represented by the <12 19 28 33] val, and if you'd like to say that 123000 cents is 7/4, that would be represented by the <12 19 28 1254] val. | ||
=Shorthand Notation= | ==Shorthand Notation== | ||
Given an explicit or assumed limit, any [[Patent_val|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 17-EDO, <17 27 39], can be called simply, "17". | Given an explicit or assumed limit, any [[Patent_val|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 17-EDO, <17 27 39], can be called simply, "17". | ||
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<ul><li>For each wart, the letter specifies which prime approximation is being altered, so that the nth letter of the alphabet refers to the nth prime.</li><li>A letter which appears m times refers to the (m+1)th most accurate mapping for that prime.</li><li>If a number representing a val is wartless, it is taken to mean the patent val.</li></ul> | <ul><li>For each wart, the letter specifies which prime approximation is being altered, so that the nth letter of the alphabet refers to the nth prime.</li><li>A letter which appears m times refers to the (m+1)th most accurate mapping for that prime.</li><li>If a number representing a val is wartless, it is taken to mean the patent val.</li></ul> | ||
See also: [[Monzos_and_Interval_Space|Monzos and Interval Space]], [[Patent_val|Patent val]], [[Vals_and_Tuning_Space|Vals and Tuning Space]], [[Optimal_patent_val|Optimal patent val]] [[Category:definition]] | See also: [[Monzos_and_Interval_Space|Monzos and Interval Space]], [[Patent_val|Patent val]], [[Vals_and_Tuning_Space|Vals and Tuning Space]], [[Optimal_patent_val|Optimal patent val]] | ||
[[Category:definition]] | |||
[[Category:intervals]] | [[Category:intervals]] | ||
[[Category:theory]] | [[Category:theory]] | ||
[[Category:tuning]] | [[Category:tuning]] | ||
[[Category:val]] | [[Category:val]] | ||