Monzo: Difference between revisions

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 {{monzo| 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 [[Harmonic Limit|prime limit]].

Monzos can be thought of as counterparts to [[vals]]. When notating just intonation, they only permit integers as their entries.

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 [[Wikipedia: Stigler%27s law of eponymy|Stigler's law]] many times over. More descriptive but longer terms include '''prime-count vector'''<ref>Used by [[Douglas Blumeyer]] and [[Dave Keenan]] on this wiki, notably in [[Dave Keenan & Douglas Blumeyer's guide to RTT]]</ref>, '''prime-exponent vector'''<ref>[http://tonalsoft.com/enc/m/monzo.aspx Tonalsoft | ''Monzo'']</ref>, and in the context of just intonation, '''harmonic space coordinates'''<ref>[https://www.plainsound.org/HEJI/ Plainsound Music Edition | ''Plainsound Harmonic Space Calculator'']</ref>.

For example, the interval [[15/8]] can be thought of as having <math>5 \cdot 3</math> in the numerator, and <math>2 \cdot 2 \cdot 2</math> in the denominator. This can be compactly represented by the expression <math>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 {{monzo| … }} brackets, hence yielding {{monzo| -3 1 1 }}.  

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

Here are a few [[7-limit]] monzos:

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) \\

In this case, the val {{val| 12 19 28 }} is the [[patent val]] for [[12-equal]], and {{monzo| -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&mdash;in other words, it is tempered out&mdash;which tells us that 12-equal is a [[meantone]] temperament. It is noteworthy that almost the entirety of Western music composed in the [[Historical temperaments|Renaissance]] and from the sixteenth century onwards, particularly Western music composed for 12-tone circulating temperaments ([[12edo|12 equal]] and unequal [[well temperament]]s), 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.

\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