User:VectorGraphics/Monzo notation: Difference between revisions

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~~This page serves as an introduction to~~ '''~~monzos~~''', a way of notating musical intervals.

M'''onzos''' are a way of notating musical intervals that essentially represents a "formula" for that interval.

In tuning theory, intervals within tuning systems (whether just intonation, EDOs, or regular temperaments) are often thought of as being composed by stacking different types of basic intervals, called "generators" or "basis elements" (which for reference make up the "basis"), and it is useful to be able to write an interval directly in terms of the number of generators of each type it contains. This can be seen as a "formula" for the interval.

In tuning theory, intervals within tuning systems (whether just intonation, EDOs, or regular temperaments) are often thought of as being composed by [[stacking]] (multiplying) different types of basic intervals, called "generators" or "basis elements" (which for reference make up the "basis"), and it is useful to be able to write an interval directly in terms of the number of generators of each type it contains. This can be seen as a "formula" for the interval. An interval can be written in terms of basis intervals ''p'' and counts/exponents ''x'' as ''p1''^''x1'' * ''p2''^''x2'' * ... * ''pn^xn'', and in monzo form as ''p1'''.'''p2'''.'''...'''.'''pn [x1 x2 ... xn],'' where the ''x'' values are restricted to rational numbers (and often integers).

== Monzos in the diatonic scale ==

In general, if a tuning system is being represented by a given number of generators, then that number of generators is always necessary to fully represent the system, even if the intervals themselves are different, so a subgroup represented by 3 basis intervals can never be fully represented by less than three.

== Example: Monzos in the diatonic scale ==

For example, to reach a minor third in diatonic, you can go up two diatonic semitones (m2) and one chromatic semitone (A1). So, to write this information, you start with the kinds of intervals you're stacking, separated by a period (so ''m2.A1''), and then write the number of intervals of each type included in the target interval's "formula", separated by spaces and enclosed in square brackets (so ''[2 1]''). The completed formula, ''m2.A1 [2 1]'', is called a "monzo" (more specifically, this one is a tmonzo), and it essentially tells you how to get to the interval you want by only stepping up or down by the intervals on the left.

@@ Line 1: / Line 1: @@
-This page serves as an introduction to '''monzos''', a way of notating musical intervals.
+M'''onzos''' are a way of notating musical intervals that essentially represents a "formula" for that interval.
-In tuning theory, intervals within tuning systems (whether just intonation, EDOs, or regular temperaments) are often thought of as being composed by stacking different types of basic intervals, called "generators" or "basis elements" (which for reference make up the "basis"), and it is useful to be able to write an interval directly in terms of the number of generators of each type it contains. This can be seen as a "formula" for the interval.
+In tuning theory, intervals within tuning systems (whether just intonation, EDOs, or regular temperaments) are often thought of as being composed by [[stacking]] (multiplying) different types of basic intervals, called "generators" or "basis elements" (which for reference make up the "basis"), and it is useful to be able to write an interval directly in terms of the number of generators of each type it contains. This can be seen as a "formula" for the interval. An interval can be written in terms of basis intervals ''p'' and counts/exponents ''x'' as ''p<sub>1</sub>''^''x<sub>1</sub>'' * ''p<sub>2</sub>''^''x<sub>2</sub>'' * ... * ''p<sub>n</sub>^x<sub>n</sub>'', and in monzo form as ''p<sub>1</sub>'''.'''p<sub>2</sub>'''.'''...'''.'''p<sub>n</sub> [x<sub>1</sub> x<sub>2</sub> ... x<sub>n</sub>],'' where the ''x'' values are restricted to rational numbers (and often integers).
-== Monzos in the diatonic scale ==
+In general, if a tuning system is being represented by a given number of generators, then that number of generators is always necessary to fully represent the system, even if the intervals themselves are different, so a subgroup represented by 3 basis intervals can never be fully represented by less than three.
+== Example: Monzos in the diatonic scale ==
 For example, to reach a minor third in diatonic, you can go up two diatonic semitones (m2) and one chromatic semitone (A1). So, to write this information, you start with the kinds of intervals you're stacking, separated by a period (so ''m2.A1''), and then write the number of intervals of each type included in the target interval's "formula", separated by spaces and enclosed in square brackets (so ''[2 1]''). The completed formula, ''m2.A1 [2 1]'', is called a "monzo" (more specifically, this one is a tmonzo), and it essentially tells you how to get to the interval you want by only stepping up or down by the intervals on the left.