Monzo

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 This is a beginner page. It is written to allow new readers to learn about the basics of the topic easily.The corresponding expert page for this topic is Monzos and interval space.

A monzo is a way of notating a JI interval that allows us to express directly how any "composite" interval is represented in terms of those simpler prime intervals. They are typically written using the notation [a b c d e f …, where the columns represent how the primes 2, 3, 5, 7, 11, 13, etc, in that order, contribute to the interval's prime factorization, up to some prime limit.

Monzos can be thought of as counterparts to vals. Like vals, they also only permit integers as their entries (unless otherwise specified).

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. More descriptive but longer terms include prime-count vector[1], prime-exponent vector[2], and in the context of just intonation, harmonic space coordinates[3].

Examples

For example, the interval 15/8 can be thought of as having $5 \cdot 3$ in the numerator, and $2 \cdot 2 \cdot 2$ in the denominator. This can be compactly represented by the expression $2^{-3} \cdot 3^1 \cdot 5^1$, which is exactly equal to 15/8. We construct the monzo by taking the exponent from each prime, in order, and placing them within the [ brackets, hence yielding [-3 1 1.

Practical hint: the monzo template helps you getting correct brackets (read more…).

Here are some common 5-limit monzos, for your reference:

Ratio Monzo
3/2 [-1 1 0
5/4 [-2 0 1
9/8 [-3 2 0
81/80 [-4 4 -1

Here are a few 7-limit monzos:

Ratio Monzo
7/4 [-2 0 0 1
7/6 [-1 -1 0 1
7/5 [0 0 -1 1

Relationship with vals

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

Monzos are 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:

$\left\langle \begin{matrix} 12 & 19 & 28 \end{matrix} \mid \begin{matrix} -4 & 4 & -1 \end{matrix} \right\rangle \\ = 12 \cdot (-4) + 19 \cdot 4 + 28 \cdot (-1) \\ = 0$

In this case, the val 12 19 28] is the patent val for 12-equal, and [-4 4 -1 is 81/80, or the syntonic comma. The fact that ⟨ 12 19 28 | -4 4 -1 ⟩ tells us that 81/80 is mapped to 0 steps in 12-equal—in other words, it is tempered out—which tells us that 12-equal 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:

$\left\langle \begin{matrix} a_1 & a_2 & \ldots & a_n \end{matrix} \mid \begin{matrix} b_1 & b_2 & \ldots & b_n \end{matrix} \right\rangle \\ = a_1 b_1 + a_2 b_2 + \ldots + a_n b_n$