Patent val: Difference between revisions
Cmloegcmluin (talk | contribs) →Generalized patent val: include a handy diagram and rewrite explanation for clarity and motivation |
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| ja = 特徴的なヴァル | | ja = 特徴的なヴァル | ||
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The '''patent val''' (aka '''nearest | The '''patent val''' (aka '''nearest EDO-mapping''') for some EDO is the val that you obtain by finding the closest rounded approximation to each [[prime]] in the tuning. For example, the patent val for 17-EDO is {{val| 17 27 39 }}, indicating that the closest mapping for 2/1 is 17 steps, the closest mapping for 3/1 is 27 steps, and the closest mapping for 5/1 is 39 steps. This means, if octaves are pure, that 3/2 is 706 cents, which is what you get if you round off 3/2 to the closest location in 17-equal, and that 5/4 is 353 cents, which is what you get is you round off 5/4 to the closest location in 17-equal. | ||
== Generalized patent val == | == Generalized patent val == | ||
This val can be extended to the case where the number of steps in an octave is a real number rather than an integer; this is called a '''generalized patent val''', or '''GPV'''. For instance the 7-limit generalized patent val for 16.9 is {{val| 17 27 39 47 }}, since 16.9 × log<sub>2</sub>7 = 47.444, which rounds down to 47. | This val can be extended to the case where the number of steps in an octave is a real number rather than an integer; this is called a '''generalized patent val''', or '''GPV'''. For instance the 7-limit generalized patent val for 16.9 is {{val| 17 27 39 47 }}, since 16.9 × log<sub>2</sub>7 = 47.444, which rounds down to 47. | ||
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== Further explanation == | == Further explanation == | ||
A [[harmonic limit|''p''-limit]] [[val]] contains the number of steps it takes to get to each prime number up to ''p'', in prime number order: | A [[harmonic limit|''p''-limit]] [[val]] contains the number of steps it takes to get to each prime number up to ''p'', in prime number order: | ||
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== Examples == | == Examples == | ||
=== Example for | === Example for 12EDO === | ||
Multiplying 12 times {{val| 1 1.585 2.322 2.807 3.459 }} | Multiplying 12 times {{val| 1 1.585 2.322 2.807 3.459 }} | ||
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which is the ''11-limit patent val for [[12edo]]''. | which is the ''11-limit patent val for [[12edo]]''. | ||
=== Alternate and expanded example for | === Alternate and expanded example for 31EDO === | ||
As stated above, the val contains the number of steps it takes to get to a given prime number, in prime number order: | As stated above, the val contains the number of steps it takes to get to a given prime number, in prime number order: | ||
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== How this defines a rank-1 temperament == | == How this defines a rank-1 temperament == | ||
A val defines a rank 1 temperament by defining the deliberate introduction of an error into one or more primes. In 12 EDO, for instance, the perfect fifth (ratio 3/2, or exactly 1.5) is mapped to 700 cents, which is actually just barely flat: a ratio of 2<sup>(700/1200)</sup>, or 1.4983070769. | A val defines a rank 1 temperament by defining the deliberate introduction of an error into one or more primes. In 12 EDO, for instance, the perfect fifth (ratio 3/2, or exactly 1.5) is mapped to 700 cents, which is actually just barely flat: a ratio of 2<sup>(700/1200)</sup>, or 1.4983070769. | ||
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== How this relates to commas == | == How this relates to commas == | ||
These deliberate errors ensure that certain commas get tempered out. The patent vals for both 12 EDO and 31 EDO temper out 81/80. Here are the calculations: | These deliberate errors ensure that certain commas get tempered out. The patent vals for both 12 EDO and 31 EDO temper out 81/80. Here are the calculations: | ||
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You're dividing 81 by 80, so (assuming we're starting at zero, though it works no matter where you start) you add the steps for 81 (+196) and subtract the steps for 80 (-196). 196-196 = 0. This means that it takes zero steps to reach 81/80 – in other words, 81/80 "vanishes". | You're dividing 81 by 80, so (assuming we're starting at zero, though it works no matter where you start) you add the steps for 81 (+196) and subtract the steps for 80 (-196). 196-196 = 0. This means that it takes zero steps to reach 81/80 – in other words, 81/80 "vanishes". | ||
== See also == | |||
* [[Patent val list of small EDOs]] | |||
[[Category:Regular temperament theory]] | [[Category:Regular temperament theory]] | ||