User:Dummy index/ヴァル

Template:Beginner-ja ヴァル は音程をジェネレーターの連鎖の上で（つまり平均律上で）考える方法を表した線形写像である。これにより、純正律の音程が平均律で近似される。EDOとJIの間をつなげ、レギュラーテンペラメントのすべての基礎といえるものである。通常ヴァルはEDOを参照するのに使われる。それが大きいジェネレーター（ミーントーンの5度のような）の積み重ねとJIとの関係をも示していることは気に留めておいてほしい。

定義

ヴァルはそのリミット内のすべての音程を、含まれるそれぞれの素数がそれぞれジェネレーター何ステップ分になるかを単純に示すことで平均律にマッピングする。どの正の有理数も素数の積として記述でき、素数をジェネレーターのステップ数にマッピングできることはつまりどの正の有理数もマッピングできるということを意味する。同様にしてヴァルはテンパーされた 3/2 やテンパーされた 5/4 がジェネレーターの連鎖のどこに位置するのかを示してくれる。

ヴァルは通常、⟨a b c d e f …] のように記述される。ここでスペースで区切られた項はそれぞれ 2, 3, 5, 7, 11, 13… のように素数に対応している。最後の最大の素数がリミット p となる。

Vals are important in regular temperament theory 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 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.

For a more mathematically intensive introduction to vals, see Vals and tuning space. For the characterization of higher-rank temperaments, see Mapping.

EDO の例

5リミットのヴァルとして ⟨12 19 28] を考える。このヴァルは、2/1 つまりオクターブに対するマッピングが 12 ステップであることを述べている。オクターブを 12 ステップにマップするテンペラメントは12平均律である。また、2/1 だけではなく、19 ステップがテンパーされた 3/1 を表し、28 ステップがテンパーされた 5/1 を表すことを明示的に述べている。

さて、この12平均律を7リミットに拡張することを想定する。もし12平均律の 10 ステップ (1000 セント) がテンパーされた 7/4 (本来は 968.83 セント) を表すという態度をとるなら、7/1 は、7/4 にオクターブを 2 個重ねたものだが、それは 10 ステップ＋ 12 ステップ＋ 12 ステップ＝ 34 ステップと等しくなる。この議論はしたがって、7リミットの ⟨12 19 28 34] と表現できる。

もし何か奇妙な理由で 7/4 を 900 セントだということにするなら、それは ⟨12 19 28 33] と表現されるし、7/4 を 123000 セントだとしたいなら、それは ⟨12 19 28 1254] と表現される。

短縮記法

Wart notation

wartsとも呼ばれる記法。Herman MillerとGraham Breedによって開発された。

Given an explicit or assumed limit, any 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 17edo, ⟨17 27 39], can be called simply, "17".

The patent val for any edo will use the most accurate mapping for each individual prime when pure octaves are assumed. However, it may often be the case that one wants to refer to vals other than the patent val. For example, the 5-limit val ⟨17 27 40], which maps the 5/4 to the 424-cent interval rather than the 353-cent interval, is not the patent val for 17edo but may be preferred because it is lower in overall error. Nonpatent vals are specified by adding a wart to the end of their name which specifies their deviation from the patent val. In this case, we want to specify that the 5/1 has been changed to use its second-most accurate mapping. Since 5 is the third prime number, we add the third letter of the alphabet to the end of the edo number, to form "17c".

If we wanted to use the third-most accurate mapping for 5, ⟨17 27 38], we would write "17cc". In 17edo, the approximation of the prime-5 component is raised for an odd, and lowered for an even, amount of c letters: = 39, c = 40, cc = 38, ccc = 41, cccc = 37.

The general rules:

Wart letters specify prime approximations being altered from the patent val. The n-th letter of the alphabet refers to the n-th prime: a~2, b~3, c~5, d~7, e~11 etc.
A letter which appears m times refers to the (m + 1)-th most accurate mapping for that prime.
So, if a number representing a val is wartless, it is taken to mean the patent val.
In Graham Breed's temperament finder, the wart letter "p" is used to make explicit that the number refers to the patent val (though the letter originally was intended to stand for "prime"). Note that "p" is logically the letter for prime 53.
A wart letter may prefix the number, in which case it specifies the corresponding prime as the interval of equivalence to be divided by the following number. For example, b13 refers to the patent val of 13ed3. The octave is assumed, so "a" is typically not written out.
In Graham Breed's temperament finder, the wart letter "q" and after refers each non-prime basis of composite/fractional subgroup, respectively and temporarily.

Sparse Offset Val notation

In 2022 Mike Battaglia proposed SOV notation as a way to be explicit about which primes are being affected and in which direction. In 2024 it was further refined by him and Lumi Pakkanen to be more analogous to Ups and downs notation.

Generalized patent vals are notated using the number of divisions followed by square brackets e.g. 17[] for ⟨17 27 39].

To indicate a wider mapping for a prime it is prefixed with a carret (^) e.g. 17[^5] for ⟨17 27 40].

To indicate a narrower mapping for a prime it is prefixed with a vee (v) e.g. 17[v5] for ⟨17 27 38].

The prefixes stack e.g. 17[^^5] corresponds to ⟨17 27 41].

Multiple modifications are separated by commas (and optionally spaces) e.g. 17[v3, ^5] for ⟨17 26 40].

The interval of equivalence may be prefixed in square brackets e.g. [3]13[] for ⟨8 13 19] (subgroup 2.3.5).

The subgroup may be made explicit separated by an "at" sign (@) at the end e.g. 46[]@2.3.7.13/5 for ⟨46 73 129 63] (subgroup 2.3.7.13/5).

Formal primes are treated the same way as actual primes e.g. 46[^13/5]@2.3.7.13/5 for ⟨46 73 129 64] (subgroup 2.3.7.13/5)

For patent vals the empty square brackets are optional when using an "at" sign. The subgroup itself is optional if its obvious from context e.g. 12@ for ⟨12 19 28] (subgroup 2.3.5).

The 2022 version used a plus sign (+) in place of the caret and a minus sign (-) in place of the vee.

Vals in JI subgroups

We can generalize the concept of monzos and vals from the p-limit to other JI subgroups. This can be useful when considering different edo tunings of subgroup temperaments. Gene Ward Smith called these "svals" for short.

To notate a subgroup val, we typically precede the "bra" notation with an indicator regarding the subgroup (and choice of basis). For instance, the patent val for 12 equal on the 2.3.7 subgroup is often notated "2.3.7 ⟨12 19 34]". If the subgroup indicator isn't present, the subgroup can be inferred from context. It is very typical for a val with no explicit subgroup indicator to be interpreted as representing some prime limit, e.g. ⟨a b c] would represent a 5-limit val.

Note that we could, for instance, use a different basis for the same subgroup - for instance, we could instead write "2.3.21 ⟨12 19 43]", which is the 12 equal patent val in the "2.3.21" subgroup. Since the "2.3.21" subgroup is the same as the "2.3.7" subgroup, just written with a different basis, these two apparently "different" svals represent the same map from this subgroup to a rank-1 generator chain. (It is a matter of semantics if these are thought of as "different" svals or "the same sval" written using a different basis.)

Svals can also be written using subgroups that don't involve primes, e.g. 2.3.7.13/5 ⟨46 73 129 63].

Note that the notion of a "patent val" for a subgroup val may not agree with the patent val on a prime limit. For instance, 13edo's "2.9.5 patent val" can be written as "2.9.5 ⟨13 41 30], because the best approximation to 2 is 13 steps, the best approximation to 9 is 41 steps, and the best approximation to 5 is 30 steps. Note that, however, the patent val on the 2.3.5 subgroup instead maps 3/1 to 21 steps, so that the "induced 9" from the 2.3.5 patent val is not the same as the "direct 9" from the 2.9.5 patent val.

This notation is also used for subgroup monzos; e.g. 81/80 on the 2.9.5 subgroup is "2.9.5 [-4 2 -1⟩", and it is thus easy to see that the 2.9.5 13p val above makes 81/80 vanish:

⟨13, 41, 30|2^-4, 9^2, 5^-1⟩ = 13*-4 + 41*2 + 30*-1 = 0.

Vals in regular temperaments

Main article: Tmonzos and tvals

There is also a notion of a tempered val on a group of tempered monzos, representing intervals in some regular temperament. These names are sometimes abbreviated as tval and tmonzo, respectively. Typically, this is made explicit by writing the generators beforehand. When the tempered intervals have accepted names, such as in meantone, we can use names like P8 and P5, so that the tval P8.P5 ⟨12 7] represents the 12edo "patent tval" in meantone (given that particular basis). If the intervals do not have names, a transversal can be given instead, preceded with the temperament name, so that we have (meantone) 2.3/2 ⟨12 7], or (meantone) 2.3/2 ⟨31 18].

Vals vs. mappings

A val is more specific than a mapping, both as in the general mathematical sense as well as the regular temperament sense:

A val can be thought of as a mapping with one row. Put another way, the rows of mappings are vals. To be mathematically precise, a val is a specific type of (linear) mapping called a "linear form", or "linear functional", which means that its output is a scalar, or in other words, a single number. This corresponds to the fact that a val must be a 1xM array of numbers, or in other words a vector (specifically a row vector, AKA covector).
Vals must have only integer entries (when expressed in the standard, non-weighted coordinate basis).
Being short for "valuation", a val is a formal linear sum of p-adic valuations.

In practice, most single-row mappings in RTT are vals, because we usually deal with integer entries, and the other specifications only mean anything to advanced mathematicians.