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.

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: ''See also: [[Val]], [[Keenan's explanation of vals]], [[Vals and tuning space]] (more mathematical)''

Monzos in just intonation 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:

Monzos in just intonation are also 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>

[ \begin{matrix} 12 & 19 & 28 \end{matrix} ][ \begin{matrix} -4 & 4 & -1 \end{matrix} ] \\

( \begin{matrix} 12 & 19 & 28 \end{matrix} )[ \begin{matrix} -4 & 4 & -1 \end{matrix} ] \\

= 12 \cdot (-4) + 19 \cdot 4 + 28 \cdot (-1) \\

= 0

</math>

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~~—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 [[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|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 this case, the val (12 19 28) is the [[patent val]] for [[12-equal|12-]]TET, which essentially tells us how many steps of 12edo, if taken as a 5-limit system, represent each of the primes of the 5-limit (2, 3, and 5), and can be seen as a very simple [[Mapping|mapping matrix]].

[-4 4 1] is the monzo notation of 81/80, or the [[syntonic comma]] separating simple 5-limit intervals from nearby simple 3-limit intervals.

(12 19 28)[-4 4 -1] tells us that 81/80 is mapped to 0 steps in 12-TET—in other words, it is tempered out—which tells us that 12-TET 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|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>

~~\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~~ \\

( \begin{matrix} a_1 & a_2 & \ldots & a_n \end{matrix} )[ \begin{matrix} b_1 & b_2 & \ldots & b_n \end{matrix} ] \\

= a_1 b_1 + a_2 b_2 + \ldots + a_n b_n

</math><!--== Monzos in JI subgroups ==

@@ 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.
@@ Line 28: / Line 30: @@
 : ''See also: [[Val]], [[Keenan's explanation of vals]], [[Vals and tuning space]] (more mathematical)''
-Monzos in just intonation 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:
+Monzos in just intonation are also 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>
-[ \begin{matrix} 12 & 19 & 28 \end{matrix} ][ \begin{matrix} -4 & 4 & -1 \end{matrix} ] \\
+( \begin{matrix} 12 & 19 & 28 \end{matrix} )[ \begin{matrix} -4 & 4 & -1 \end{matrix} ] \\
 = 12 \cdot (-4) + 19 \cdot 4 + 28 \cdot (-1) \\
 = 0
 </math>
-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&#x2014;in other words, it is tempered out&#x2014;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|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 this case, the val (12 19 28) is the [[patent val]] for [[12-equal|12-]]TET, which essentially tells us how many steps of 12edo, if taken as a 5-limit system, represent each of the primes of the 5-limit (2, 3, and 5), and can be seen as a very simple [[Mapping|mapping matrix]].
+[-4 4 1] is the monzo notation of 81/80, or the [[syntonic comma]] separating simple 5-limit intervals from nearby simple 3-limit intervals.
+(12 19 28)[-4 4 -1] tells us that 81/80 is mapped to 0 steps in 12-TET&#x2014;in other words, it is tempered out&#x2014;which tells us that 12-TET 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|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>
-\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 \\
+( \begin{matrix} a_1 & a_2 & \ldots & a_n \end{matrix} )[ \begin{matrix} b_1 & b_2 & \ldots & b_n \end{matrix} ] \\
 = a_1 b_1 + a_2 b_2 + \ldots + a_n b_n
 </math><!--== Monzos in JI subgroups ==