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 those 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]].

For example, the interval 15/8 can be thought of as having <math>5⋅3</math> in the numerator, and <math>2⋅2⋅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}}.  

| {{Monzo| -1 1 0 }}

| {{Monzo| -2 0 1 }}

| {{Monzo| -3 2 0 }}

| {{Monzo| -4 4 -1 }}

! | Ratio

! | Monzo

| {{Monzo| -2 0 0 1 }}

| {{Monzo| -1 -1 0 1 }}

| {{Monzo| 0 0 -1 1 }}

In this case, the val &lt; 12 19 28 | is the [[Patent_val|patent val]] for 12-equal, and | -4 4 -1 &gt; is 81/80, or the syntonic comma. The fact that &lt; 12 19 28 | -4 4 -1 &gt; 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.

[[Category:definition]]

[[Category:intervals]]

[[Category:prime limit]]

[[Category:theory]]