Patent val: Difference between revisions
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The '''patent val''' (aka nearest edomapping) for some EDO is the val that you obtain by simply finding the closest rounded-off approximation to each [[prime]] in the tuning. For example, the patent val for 17-EDO is <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 == | |||
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 <17 27 39 47|, since 16.9 * log2(7) = 47.444, which rounds down to 47. | |||
There are other vals or edomappings besides the patent or nearest one. You may prefer to use the <17 27 40| val as the 5-limit 17-equal val instead, which rather than <17 27 39| treats 424 cents as 5/4. This val has lower Tenney-Euclidean error than the 17-EDO patent val. However, while <17 27 39| may not necessarily be the "best" val for 17-equal for all purposes, it is the obvious, or "patent" val, that you get by naively rounding primes off within the EDO and taking no further considerations into account. However, <17 27 40| is the generalized patent val for 17.1, since 17.1 * log2(5) = 39.705, which rounds up to 40. | There are other vals or edomappings besides the patent or nearest one. You may prefer to use the <17 27 40| val as the 5-limit 17-equal val instead, which rather than <17 27 39| treats 424 cents as 5/4. This val has lower Tenney-Euclidean error than the 17-EDO patent val. However, while <17 27 39| may not necessarily be the "best" val for 17-equal for all purposes, it is the obvious, or "patent" val, that you get by naively rounding primes off within the EDO and taking no further considerations into account. However, <17 27 40| is the generalized patent val for 17.1, since 17.1 * log2(5) = 39.705, which rounds up to 40. | ||
=Further explanation= | == Further explanation == | ||
A [[p-limit|p-limit]] [[Vals_and_Tuning_Space|val]] contains the number of steps it takes to get to each prime number up to p, in prime number order: | A [[p-limit|p-limit]] [[Vals_and_Tuning_Space|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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Thus, the way to get the p-limit patent val for N-EDO is to multiply <1 1.585 2.322 2.807 ... log2(p) | by N. Then, since you can't take fractional steps in an EDO, you round the results to the nearest integers. | Thus, the way to get the p-limit patent val for N-EDO is to multiply <1 1.585 2.322 2.807 ... log2(p) | by N. Then, since you can't take fractional steps in an EDO, you round the results to the nearest integers. | ||
=A 12 EDO Example= | == A 12-EDO Example == | ||
Multiplying 12 times <1 1.585 2.322 2.807 3.459| | Multiplying 12 times <1 1.585 2.322 2.807 3.459| | ||
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which is the '''11-limit patent val for [[12edo|12edo]]'''. | which is the '''11-limit patent val for [[12edo|12edo]]'''. | ||
=An alternate and expanded example for 31 EDO= | == An alternate and expanded example for 31 EDO == | ||
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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Note that these are the same answers you would get if you multiplied 31 times <1 1.585 2.322 2.807 3.459 3.700 4.087 4.248 | and rounded the result. | Note that these are the same answers you would get if you multiplied 31 times <1 1.585 2.322 2.807 3.459 3.700 4.087 4.248 | and rounded the result. | ||
=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^(700/1200), 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^(700/1200), or 1.4983070769. | ||
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That doesn't make 31 EDO better or worse than 12; it just means there's more error in the 3/1 ratio in 31 EDO than in 12 EDO. If you run these calculations for 5/1 using the patent vals for 12 EDO and 31 EDO, you'll find that 5/1 has more error in 12 EDO than in 31 EDO: 5.0396842 vs. 5.002262078, respectively. 31 EDO may therefore be preferred by people who like sweeter thirds (5/4 ratios) and are willing to have flatter fifths (3/2 ratios). | That doesn't make 31 EDO better or worse than 12; it just means there's more error in the 3/1 ratio in 31 EDO than in 12 EDO. If you run these calculations for 5/1 using the patent vals for 12 EDO and 31 EDO, you'll find that 5/1 has more error in 12 EDO than in 31 EDO: 5.0396842 vs. 5.002262078, respectively. 31 EDO may therefore be preferred by people who like sweeter thirds (5/4 ratios) and are willing to have flatter fifths (3/2 ratios). | ||
=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". | ||
[[Category:Theory]] | |||
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[[Category: | [[Category:Definition]] | ||
[[Category: | [[Category:Math]] | ||
[[Category: | [[Category:Val]] | ||
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