User:VectorGraphics/Monzo notation: Difference between revisions

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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).

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 ==

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 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).
+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 ==