Monzo

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 simpler prime intervals. They are typically written using the notation [a₁ a₂ a₃ a₄ a₅ a₆ …⟩, where a_i are numbers that 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. When notating just intonation, they only permit integers as their entries.

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, prime-exponent vector, and in the context of just intonation, harmonic space coordinates.

Examples

To find the monzo of an interval in ratio form, factor the entire ratio as a product of primes, each raised to an exponent. For primes appearing in the denominator, these exponents will be negative. (A prime never appears in both the numerator and the denominator.) Arrange the primes in ascending order. If any primes smaller than the largest prime do not appear, include them using a zero exponent. Enter the exponents into the monzo.

For example, the interval 15/8 can be thought of as having 5 × 3 in the numerator, and 2 × 2 × 2 in the denominator. This can be compactly represented by the expression 2^-3 × 3¹ × 5¹, 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⟩.

Here are some common 5-limit monzos, along with their factorizations to show how to derive them:

Ratio	Factors	Monzo
3/2	[math]\displaystyle{ 2^{-1} \cdot 3 }[/math]	[-1 1 0⟩
5/4	[math]\displaystyle{ 2^{-2} \cdot 5 }[/math]	[-2 0 1⟩
9/8	[math]\displaystyle{ 2^{-3} \cdot 3^2 }[/math]	[-3 2 0⟩
81/80	[math]\displaystyle{ 2^{-4} \cdot 3^4 \cdot 5^{-1} }[/math]	[-4 4 -1⟩

Here are a few 7-limit monzos:

Ratio	Factors	Monzo
7/4	[math]\displaystyle{ 2^{-2} \cdot 7 }[/math]	[-2 0 0 1⟩
7/6	[math]\displaystyle{ 2^{-1} \cdot 3^{-1} \cdot 7 }[/math]	[-1 -1 0 1⟩
7/5	[math]\displaystyle{ 5^{-1} \cdot 7 }[/math]	[0 0 -1 1⟩

💡Tip:

On the wiki, the monzo template helps you getting correct brackets (read more…).

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:

[math]\displaystyle{ \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 }[/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—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:

[math]\displaystyle{ \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 }[/math]

Generalizations

Subgroup monzos

Main article: Subgroup monzos and vals

A subgroup monzo is like a standard monzo, except that it is in a just intonation subgroup that is not a prime limit. It is usually provided along with the subgroup it is in, such as 2.3.7 [1 -2 1⟩ for 14/9 or 2.3.13/5 [1 -1 1⟩ for 26/15. For subgroups with primes as basis elements (such as 2.3.7), a subgroup monzo can be seen as a standard monzo abbreviated to hide zeroes. To make this more clear, the abbreviated sections may be replaced with ellipses, such as 2.3.7 [1 -2 … 1⟩ for 14/9.

Tempered monzos

Main article: Tempered monzos and vals

A tempered monzo represents a tempered interval, or an interval in a regular temperament. It functions similarly to a subgroup monzo, except each entry represents not a JI interval but a generator for that regular temperament. For example, the tempered monzo for the major third in meantone (or more generally, the diatonic major third in a fifth-generated temperament) is ~2.~3 [-6 4⟩. Note that we write the generators with tildes to indicate that they are tempered intervals. This means that to find the major third, you go up four perfect twelfths (~3/1) and down six octaves (~2/1).

More generally, tempered monzos are applicable to any regular tuning, regardless of JI mapping, so corresponding intervals in two different regular temperaments that are tuned the same way have the same tempered monzo.

A tempered monzo may be called a generator-count vector, and conversely a standard monzo can be called a prime-count vector.

Fractional monzos

Main article: Fractional monzos

Any of the previous categories of monzo can also be a "fractional monzo", allowing entries to be fractions or non-integer rational numbers as opposed to just integers. This allows monzos to express equal divisions of just intervals and stacks thereof. For example, [-1/2 1/2⟩ is a monzo representing a neutral third equal to half of a perfect fifth, and [1/12⟩ is a monzo representing a 12edo semitone. [1/12 1/13⟩ is a monzo representing 1\12edo stacked with 1\13edt. (Numerically, this is the 156th root of 2¹³ × 3¹².) Note that we write the fractional monzo entries with forward slashes, as they represent fractions, despite writing edosteps with backslashes.

See also

• Extended bra–ket notation

External links

• Tonalsoft | Monzo