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.

For a more mathematical discussion, see also Monzos and Interval Space.

Etymology

Monzos are named in honor of Joe Monzo.

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's 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

Monzos 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.

To write a JI ratio as a monzo in a JI subgroup, we choose a basis for the subgroup and factor an interval into the basis elements as we factor an interval in the p-limit into primes at most p. Then we write the monzo so as to explicitly state what basis elements we factor the intervals into and how many of each basis element the interval has in the factorization. For example, we can write 81/80 = 9²/(2⁴ 5¹) in the 2.9.5 subgroup as [2: -4, 9: 2, 5: -1⟩. (We reserve the notation [a b c ...⟩ and ⟨a b c ...] for the p-limit.)

Vals can be defined the same way in other subgroups as well; they represent how a subgroup is (viewed as being) tuned in terms of that edo's steps. For example, 13edo's "2.9.5 patent val" can be written as ⟨2: 13, 9: 41, 5: 30], since 13edo's best approximation to the 9th harmonic is 41\13 (reduces to 2\13) and its best approximation to the 5th harmonic is 30\13 (reduces to 4\13). To see that this val "tempers out 81/80", we do the same operation (of matching up and multiplying the components and summing the products) as described in the previous section:

⟨2: 13, 9: 41, 5: 30][2: -4, 9: 2, 5: -1⟩ = 13*-4 + 41*2 + 30*-1 = 0.

Monzos in temperaments

We write a tempered interval (an interval in a regular temperament) by taking a set of generators (for rank-2 temperaments, this will be the period and the generator), then writing what JI ratio each generator approximates (distinguished from pure JI intervals by putting a tilde before it), followed by the number of that specified generator that the interval has. For example, the major third in meantone temperament can be written as [~2: ‐2, ~3/2: 4⟩, meaning "4 perfect fifths minus 2 octaves".

Similarly, edo tunings of a temperament can be given in terms of vals, by specifying how many edo steps are used for each generator. For example, 31edo's tuning of meantone temperament can be written as ⟨~2: 31, ~3/2: 18].