Monzo

This page gives a pragmatic introduction to monzos. For the formal mathematical definition, visit the page Monzos and interval space.

Definition

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).

Etymology

Monzos are named in honor of Joe Monzo, although the name was given by Gene Ward Smith. 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.

Examples

For example, the interval 15/8 can be thought of as having [math]\displaystyle{ 5⋅3 }[/math] in the numerator, and [math]\displaystyle{ 2⋅2⋅2 }[/math] in the denominator. This can be compactly represented by the expression [math]\displaystyle{ 2^{-3} \cdot 3^1 \cdot 5^1 }[/math], 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:

< 12 19 28 | -4 4 -1 >

[math]\displaystyle{ (12⋅-4) + (19⋅4) + (28⋅-1) = 0 }[/math]

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 – aka it is tempered out – which tells us that 12-equal is a meantone temperament. It is noteworthy that almost the entirety of western music, particularly western music composed for 12-equal or 12-tone well temperaments, is made possible by the above equation.

In general: < a b c | d e f > = ad + be + cf

See also

• Smonzos and svals — subgroup monzos and vals
• Tonalsoft Encyclopedia | Monzo