マッピング

レギュラーテンペラメントは単なるピッチの集合以上のものである。純正律サブグループからそのピッチ集合の特定の音高に対応付ける一貫したルールを持つものである。（「抽象的なレギュラーテンペラメント」は決まった音高の集合ですらない。ピッチは決まっておらず、対応付けるルールが示す構造がテンペラメントを特徴づける。）この一貫したルールはマッピングと呼ばれている。マッピングは「この純正律の音をこのテンペラメントのどの音で演奏すればいいの？」に答えるものである。答えはその純正律の音の「テンパーされたバージョン」で、それは状況によってかなり近く近似されたりかなり外れて近似されたりする。

単純に一番近い（四捨五入的な意味で）音で演奏すればいいんじゃないかと思うかもしれない、しかし、それは（通常）全くレギュラーテンペラメントにならない！ それは一貫した結果を得られない。同じ純正音程が、現れた場所によって異なる音程に対応付けされてしまうのだ。レギュラーテンペラメントのマッピングというのはそれぞれの純正音程を常に同じテンパーされた音程で表現する。結果として、複雑な純正音程はサイズ的に一番近い音程にはならないことが多い。

A note on mathematical terminology

数学の用語で "mapping"（写像）は "map" や "function"（関数）の同義語であるが、RTTにおいては「マッピング」を線形写像の意味で使う。そして行列の形で取り扱う。

Equal temperament mappings

An equal temperament, also known as a rank-1 temperament (see below for a discussion of rank), is not merely a set of equally spaced pitches. An equal temperament consists of

1. A JI subgroup that is being represented, such as "5-limit JI", and
2. A mapping that assigns every pitch of this JI subgroup to a note of the equal temperament (which can be represented as an integer).

As an example, let's consider the familiar 12edo considered as a 3-limit temperament. For concreteness, let's use A440 as the base note. In this case the JI subgroup is the 3-limit, that is, all pitches that form a JI interval with A440 whose prime factorization contains no primes other than 2 and 3. For people familiar with mathematical notation, this can be written as

[math]\displaystyle{ \left\{440\cdot 2^a\cdot 3^b\,\middle|\,a,b\in\mathbb Z\right\} }[/math]

Let's use integers to represent the 12edo notes, so that A440 is note 0, the B♭ above that is 1, the A♭ below it is −1, and so on. Then the mapping is simply expressed by saying that each factor of 2 counts for 12 steps, and each factor of 3 counts for 19 steps (because 3/1, or 1901.955… cents, is approximated as 1900 cents, or 19 steps of 12edo). (If you want a mathematical formula, that means that the above expression is mapped to 12a + 19b.) So, for example, 1/1 is mapped to note 0, which is exactly A440; 2/1 is mapped to note 12, the A one octave higher; 3/2 is mapped to note 7 (the E above A440); and 3¹²/2¹⁹ (the Pythagorean comma) is mapped to 0, the same note as 1/1.

Contrast with rounding

Now, consider the pitch 3³⁶/2⁵⁷. In JI, this pitch is 70.38… cents above A440, so the closest 12edo note to it is B♭. However, if you apply the mapping formula, you see that it is mapped to note 0 (A), not note 1 (B♭). Why is this? The pitch 3³⁶/2⁵⁷is three Pythagorean commas above A. If each Pythagorean comma is represented by 0 steps, then since 0 + 0 + 0 = 0 the pitch 3³⁶/2⁵⁷ must be represented by A, even though in JI it's closer to B♭. Mapping it to B♭ would require one of the three Pythagorean commas to be represented by 1 complete step (100 cents)! This illustrates the difference between regular mapping and rounding.

Notation

In regular temperament theory there is a special notation for this kind of JI mapping. We notate the 3-limit 12edo temperament described above as "⟨12 19]", because the first prime (2) is mapped to 12 steps, and the second prime (3) is mapped to 19 steps. This mathematical object is known as a "mapping matrix" and it summarizes all the information in the mapping in a very compact form. Since this is an equal temperament, the mapping matrix contains only one row, and since it's a 3-limit temperament, the mapping matrix contains two columns, representing the primes 2 and 3.

Many 12edo temperaments

Now, let's consider 12edo, not as a 3-limit temperament, but as a 5-limit temperament. This temperament maps all the 3-limit JI intervals in the same way as above, but in addition also maps the rest of the 5-limit JI intervals. Its mapping matrix is ⟨12 19 28]. It's important to keep in mind that this is, technically speaking, a different regular temperament than ⟨12 19], even though they would both be referred to as "12-tone equal temperament" in common parlance.

Furthermore, consider 12edo as an 11-limit temperament. What is its mapping matrix? It actually depends whether you consider 11/8 a "very sharp D" or a "very flat D♯". This choice results in two different mappings, ⟨12 19 28 34 41] and ⟨12 19 28 34 42]. The latter has a more accurate 11/8, but the former has more accurate versions of other intervals, including 12/11. In the language of regular temperament theory, these are simply two different 11-limit temperaments that both happen to have 12 steps per octave. Phrases like "11-limit 12edo" are thus ambiguous because they don't specify the mapping, and therefore don't refer to a specific temperament.

Strictly speaking, "5-limit 12edo" or even "3-limit 12edo" are also ambiguous, because ⟨12 19 27], for example, is a valid temperament even though it's much less accurate than ⟨12 19 28]. In this temperament 5/4 would be represented as 3 steps of 12edo, or 300 cents. For practical purposes, of course, the ambiguity doesn't appear until higher limits.

Linear temperament mappings

Now let's consider a temperament that does not consist of a single chain of equally spaced notes. For example, consider conventional music notation without enharmonic equivalence. Every note of this system can be expressed as some combination of octaves and perfect fourths. For example:

• E5 = A440 + 1 octave − 1 perfect fourths
• B♭4 = A440 − 3 octaves + 5 perfect fourths
• A♯ = A440 + 4 octaves − 7 perfect fourths

In other words, every note can be represented as an ordered pair of integers (x, y) where x is the number of octaves from A440 (positive is up, negative is down), and y is the number of perfect fourths.