パテントヴァル

いくつかの平均律における特徴的なヴァルとは、単純にチューニングの各素数に最も近いヴァルのことである。

The patent val (a.k.a. nearest edomapping) for an edo is a list of numbers you obtain by finding the closest rounded approximation to each prime harmonic in the tuning, assuming octaves are pure (or in other words, assuming the edo number is an integer). The basic application of a patent val is that you round prime harmonics to edosteps, and then deduce the number of steps of an arbitrary just interval based on its prime factorization.

例えば、17平均律の特徴的なvalは<17 27 39|であり、それは2/1のためのマッピングに最も近いのは17ステップであり、3/1のためのマッピングに最も近いのが27ステップであり、5/1のためのマッピングに最も近いのが39ステップであることを示す。このことはすなわち、もしオクターブがピュアならば、3/2は706セントであり、本来702セントであることから、大きく変化されている。そして5/4は353セントとなり、こちらも本来386セントであることから大きく変化されている。

For example, the patent val for 17edo 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

→詳細は「 Uniform map 」を参照

このヴァルはオクターブのステップの数において、本来の数から拡大されることができる。例えば、16.9のための7リミットにおける特徴的なヴァルは、<17 27 39 47|であり、16.9 * log2(7) = 47.444であることから、それは47の端数を切り捨てている。

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 × log₂7 = 47.444, which rounds down to 47.

a visualization of all possible GPVs through the 13-limit up to 99et (any vertical slice is a GPV)

5リミットの17平均律ヴァルの代わりとして、<17 27 40|を利用するならば、むしろ5/4としての424セントとして扱う<17 27 39|である。このヴァル17平均律の特徴的なヴァルと比べ、Tenney-Euclideanエラーよりも小さい。しかしながら、<17 27 39|はたぶん、必ずしも17平均律において、完全なベストではない。それは明らかに、「特徴的な」ヴァルは、EDOとなるよう単純に丸め込まれた素数であり、それ以上の理由のための深い熟考は存在しない。

There are other vals worth considering besides the patent val. Consider the case of 5-limit 17et. ⟨17 27 39] is the patent val, meaning each prime individually is as closely approximated as possible (again, assuming pure octaves). However, if that constraint is lifted, and we're allowed to choose the next-closest approximations for prime 5, the overall damage to the consonances we care about can be reduced; in other words, even though 39 steps can take you just a tiny bit closer to prime 5 than 40 steps can, this is a naïve choice which does not take into account whether the errors tend to cancel or reinforce in simple ratios that combine different primes. Considering the problem more deeply in this manner may lead to choosing ⟨17 27 40] instead. And there are other harmonic reasons to choose ⟨17 27 40] over ⟨17 27 39] as well; it tempers different commas.

<17 27 40|は17.1の特徴的なヴァルであり、17.1 * log2(5) = 39.705であるため、40に近い。

We can show that ⟨17 27 40] is a generalized patent val because it would be the patent val for 17.1et: 17.1 × log₂5 = 39.705, which rounds up to 40. Essentially this is showing that there does exist some generator size, 2^1/17.1, for which it is truly the case that 17, 27, and 40 are the respective best approximations of primes 2, 3, and 5. That is, we are not "forcing" an interpretation of a prime which is not closest to the truth. A counterexample would be ⟨17 27 41]: it is possible to find a generator that maps 2 to 17 steps and 5 to 41 steps, but it would require 3 to be 28 steps (this type of information can be read easily off the nearby visualization).

Another name for generalized patent val is uniform map (and an integer uniform map, or simple map, is another name for patent val).

Further explanation

A p-limit val contains the number of steps it takes to get to each prime number up to p, in prime number order:

⟨[2/1] [3/1] [5/1] [7/1] … [p/1]]

Given N-edo, the equal division of the octave into N parts, we may define vals that map a specific number of N-edo steps to these primes.

For any prime p we can find a corresponding p-limit val in a canonical manner by scalar multiplying ⟨1 log₂3 log₂5 … log₂p] 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, log₂3 steps to get to 3/1, log₂5 to get to 5/1, and so on. This gives us ⟨1 1.585 2.322 2.807 3.459 … log₂p].

Then ask, "How many more N-edo steps does it take to get to the same places?" One 12edo step is 1/12th of an octave, by definition; therefore, you need 12 times as many steps to reach 2/1, 3/1, 5/1, … Similarly, one 31edo step is 1/31 of an octave, so you need 31 times as many steps to reach 2/1, 3/1, 5/1, …

Thus, the way to get the p-limit patent val for N-edo is to multiply ⟨1 1.585 2.322 2.807 … log₂p] by N. Then, since you can't take fractional steps in an edo, you round the results to the nearest integers.

Examples

Example for 12edo

Multiplying 12 times ⟨1 1.585 2.322 2.807 3.459]

yields ⟨12 19.020 27.863 33.688 41.513],

rounded to ⟨12 19 28 34 42],

which is the 11-limit patent val for 12edo.

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:

⟨[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 31edo. The 2-limit patent val is ⟨31].

What's the number of steps to 3/1?

The step size for 31edo is 38.70967742 cents.

3/1 is 1901.96 in cents.

1901.96 cents / 38.70967742 cents/step = 49.13383752 steps.

This is an edo, so we can't take 0.13383752 steps. Instead, we round. This is clearly closer to 49 steps, so that's the "obvious" or "patent" choice. The 3-limit patent val is

⟨31 49].

Doing the same thing up through 17, and we get a 17-limit patent val of

⟨31 49 72 87 107 115 127]

To see how to extend from one limit to another, we may look at what to do for 19/1 and use that to go from the 17-limit to the 19-limit.

19/1 = 5097.51 cents, 5097.51 / 38.70967742 cents/step = 131.6857529 steps. Round to get 132. The 19-limit patent val is

⟨31 49 72 87 107 115 127 132]

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.