Patent val: Difference between revisions

The '''patent val''' (aka '''nearest edomapping''') 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.  

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 * 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 {{val| 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, {{val| 17 27 40 }} is the generalized patent val for 17.1, since 17.1 * log2(5) = 39.705, which rounds up to 40.

A [[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:

{{val| [2/1] [3/1] [5/1] [7/1] ... [p/1] }}

For any prime p we can find a corresponding p-limit val in a canonical manner by [http://en.wikipedia.org/wiki/Scalar_multiplication scalar multiplying] {{val| 1 [[log2|log2]](3) log2(5) ... log(p) }} by N and rounding to the nearest integer. In general this is not guaranteed to be the most accurate available val, but if N-edo has enough relative accuracy in the p-limit, it will be. The name ''patent'' comes from the fact that "patent" in one sense of the word is a synonym for "obvious"; the patent val may or may not be the best choice but it's the obvious choice.

One way to think of this process is to first ask, "How many 1200-cent steps (octaves) does it take to get to each prime?" It takes one full-octave step to get to 2/1, log2(3) steps to get to 3/1, log2(5) to get to 5/1, and so on. This gives us {{val| 1 1.585 2.322 2.807 3.459 ... log2(p) }}.

Thus, the way to get the p-limit patent val for N-EDO is to multiply {{val| 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.

Multiplying 12 times {{val| 1 1.585 2.322 2.807 3.459 }}

yields {{val| 12 19.020 27.863 33.688 41.513 }},

rounded to {{val| 12 19 28 34 42 }},

{{val|  [2/1] [3/1] [5/1] [7/1] [etc.] }}

By definition, for any EDO, the number of steps to 2/1 is the EDO division: 31 for 31 EDO. The 2-limit patent val is {{val| 31 }}.

{{val| 31 49 }}. Doing the same thing up through 17, and we get a 17-limit patent val of

{{val| 31 49 72 87 107 115 127 }}

{{val| 31 49 72 87 107 115 127 132 }}

Note that these are the same answers you would get if you multiplied 31 times {{val| 1 1.585 2.322 2.807 3.459 3.700 4.087 4.248 }} and rounded the result.

As stated above, the 2/1 in the patent val is perfect. The patent val for 12 EDO, {{val| 12 19 28 34 42 (etc) }}, implies that it takes 12 steps to get the octave, which is does: 12 steps * 100 cents / step = 1200 cents = 1 octave.