User:VectorGraphics/Monzo notation JI initial

Monzos are a way of notating musical intervals that essentially represents a "formula" for that interval.

In tuning theory, intervals within tuning systems (whether just intonation, EDOs, or regular temperaments) are often thought of as being composed by stacking (multiplying) different types of basic intervals, called "generators" or "basis elements" (which for reference make up the "basis"), and it is useful to be able to write an interval directly in terms of the number of generators of each type it contains. This can be seen as a "formula" for the interval. An interval can be written in terms of basis intervals p and counts/exponents x as p₁^x₁ * p₂^x₂ * ... * p_n^x_n, and in monzo form as p₁.p₂.....p_n [x₁ x₂ ... x_n], where the x values are restricted to rational numbers (and often integers).

In general, if a tuning system is being represented by a given number of generators, then that number of generators is always necessary to fully represent the system, even if the intervals themselves are different, so a subgroup represented by 3 basis intervals can never be fully represented by less than three.

Monzos in just intonation

The most common use of monzos is to notate intervals in just intonation, like 3/2 and 6/5. In other words, rational numbers. We don't need to think about what the basis intervals will be here, since the fundamental theorem of arithmetic has got our back!

Any rational number can be generated by stacking some number of each of the primes (2, 3, 5, 7, etc). For example, 6/5 is 2*3*5^-1.

Usually, just intonation systems fall into a "subgroup", where a finite subset of primes (or in some cases, composite intervals) are treated as generators (for example, 2, 3, and 7), restricting the space of intervals available to a finite-dimensional system. The most common type of prime subgroup is a prime limit, where all the primes up to a given value are included. For example, 3/2, 6/5, 25/24, and 128/125 are in the 5-limit, but 7/4 isn't, because it contains a factor of 7 (which is greater than 5).

So, let's write the "formula" for 6/5. To reach it, you multiply by 2 and 3 once each, and divide by 5 once. To write the monzo, you start with the kinds of intervals you're stacking, separated by a period (so, 2.3.5), and then write the number of intervals of each type included in the target interval's "formula", separated by spaces and enclosed in square brackets (so, [1 1 -1]). The complete monzo combines these two pieces of information - 2.3.5 [1 1 -1] - but in fact, prime limits are so common that they're usually fully omitted from monzos that are written in them, and just inferred from the number of entries. So, you can simply write [1 1 -1] and people will understand that the basis is 2.3.5. (However, to write an interval as a monzo in, for example, 2.3.7, or even more unusual subgroups like 2.3.(13/5), you should keep the basis explicitly written.)

Monzos are often used in just intonation as it can be hard to tell what a number factors into at a glance, and because most musical tuning systems that approximate just intonation preserve the relationships between just intervals that a monzo exposes.

Stacking monzos

Stacking intervals written in monzo form simply involves writing the intervals in the same basis, and adding each corresponding entry together. For example, going back to our example with diatonic and chromatic semitones, a minor third plus a major third equals a perfect fifth, and that's reflected by adding the monzos for each:

m2.A1 [2 1] + [2 2] = [(2+2) (2+1)] = [4 3]

And indeed, m2.A1 [4 3] is the monzo for a perfect fifth in the m2.A1 basis.

Monzos and vals

See also: Val, Keenan's explanation of vals, Vals and tuning space (more mathematical)

Monzos in just intonation are also important because they enable us to see how any JI interval "maps" onto a val. This mapping is expressed by writing the val and the monzo together, such as (12 19 28)[-4 4 -1]. The mapping is extremely easily to calculate: simply multiply together each component in the same position on both sides of the line, and add the results together. This is perhaps best demonstrated by example:

[math]\displaystyle{ ( \begin{matrix} 12 & 19 & 28 \end{matrix} )[ \begin{matrix} -4 & 4 & -1 \end{matrix} ] \\ = 12 \cdot (-4) + 19 \cdot 4 + 28 \cdot (-1) \\ = 0 }[/math]

In this case, the val (12 19 28) is the patent val for 12-TET, which essentially tells us how many steps of 12edo, if taken as a 5-limit system, represent each of the primes of the 5-limit (2, 3, and 5), and can be seen as a very simple mapping matrix.

[-4 4 1] is the monzo notation of 81/80, or the syntonic comma separating simple 5-limit intervals from nearby simple 3-limit intervals.

(12 19 28)[-4 4 -1] tells us that 81/80 is mapped to 0 steps in 12-TET—in other words, it is tempered out—which tells us that 12-TET is a meantone temperament. It is noteworthy that almost the entirety of Western music composed in the Renaissance and from the sixteenth century onwards, particularly Western music composed for 12-tone circulating temperaments (12 equal and unequal well temperaments), is made possible by the tempering out of 81/80, and that almost all aspects of modern common practice Western music theory (chords and scales) in both classical and non-classical music genres are based exclusively on meantone.

In general:

[math]\displaystyle{ ( \begin{matrix} a_1 & a_2 & \ldots & a_n \end{matrix} )[ \begin{matrix} b_1 & b_2 & \ldots & b_n \end{matrix} ] \\ = a_1 b_1 + a_2 b_2 + \ldots + a_n b_n }[/math]

History and terminology

Monzos are named in honor of Joseph Monzo, given by Gene Ward Smith in July 2003. These were also previously called factorads by John Chalmers in Xenharmonikôn 1, although the basic idea goes back at least as far as Adriaan Fokker and probably further back, so that the entire naming situation can be viewed as an example of Stigler's law many times over.

Monzos outside just intonation

The concept of monzos can be generalized beyond just intonation, however, to refer to intervals in other systems, like diatonic harmony (which will be the example for this section). To reach a minor third in diatonic, you can go up two diatonic semitones (m2) and one chromatic semitone (A1). As such, the completed monzo is m2.A1 [2 1].

However, you don't have to use semitones to notate diatonic intervals: in fact, that example, while intuitive, is kind of weird! It is more common to denote diatonic intervals by their location on the circle of fifths rather than by their number of semitones of each type. However, you also need to account for octave-equivalence. Stacking fifths can never get you to an octave, so it's necessary to add the octave as another basis interval.

To reach a minor third on the circle of fifths, you go down 3 fifths, but now you're at the minor third two octaves below your starting point, so to correct that, you go up two octaves. So, a more conventional monzo for a minor third is P8.P5 [2 -3].

Monzos in regular temperaments

A regular temperament's basis can be specified using a mapping matrix, which essentially "flattens" an input basis down to the basis of a regular temperament. For example, meantone's mapping matrix, [[1 0 -4], [0 1 4]], transforms the normal 3-dimensional 2.3.5 just intonation basis into the 2-dimensional ~2.~3/2 meantone basis in a specific way where the just intonation intervals 9/8 and 10/9 get mapped to the same place in the tempered system. The tempered interval space follows a very similar structure to our diatonic interval space from before, but now gives each diatonic interval an interpretation in just intonation. As in, ~2.~3/2 [2 -3] in meantone temperament corresponds to our earlier P8.P5 [2 -3] for a minor third, and what "meantone" is doing is saying that this minor third specifically refers to the tempered interval corresponding to 6/5 and 32/27. Usually, context makes it clear which temperament is being referred to, but if it doesn't, a mapping matrix on a JI basis not only specifies the temperament, but what the basis you're using is. For example, to get our basis based on chromatic and diatonic semitones, but for meantone, you can use a different meantone mapping, specifically [[5 8 12], [7 11 16]].

Just intonation monzos are technically in temperaments as well, simply the trivial temperament of that just intonation subgroup (i.e. pythagorean for 3-limit, classical for 5-limit, etc), which just maps every interval to itself.

Monzos in equal temperaments

A monzo can also have rational entries, indicating multiplying by a root of one of the basis intervals. Take 12edo. It can be seen as having only one basis interval, the octave 2/1, identically to the 2-limit, but the monzos for 12edo intervals are increments of 1/12, rather than 1, denoting multiplications by 2^(1/12) (or 100 cents). The 7-step interval, for example, is 2 [7/12], but this is more commonly written as 7\12<2> or simply 7\12, with a backslash instead of a normal slash.

Note that this is the pure 2-limit interpretation of 12edo, with no tempering being applied, which is unusual in tuning theory (usually, the 12edo-step will be seen as a tempered version of a higher-limit interval, like 16/15, and so the rules from the previous section apply). A more familiar example might be 2.3 [-1/2 1/2], which denotes a neutral third that perfectly divides the perfect fifth 3/2 in two.

Additional notes

• In the linear algebra formalism, monzos are represented by vectors in a vector space corresponding to the system being used. In this context, the right square bracket is replaced with an angle bracket ⟩ to denote that a vector is being represented.
• Due to factors such as dissatisfaction with the term "monzo" and overreliance on the linear algebra formalism, they have many different names, such as "ket", "prime-count vector", "prime-exponent vector", "interval vector", simply "vector", or "interval coordinates".