User:VectorGraphics/Monzo notation: Difference between revisions

Line 28:

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

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:

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 [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]].

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.

(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>

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

( \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 28: / Line 28: @@
 : ''See also: [[Val]], [[Keenan's explanation of vals]], [[Vals and tuning space]] (more mathematical)''
-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:
+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 [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]].
+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.
+(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>
-[ \begin{matrix} a_1 & a_2 & \ldots & a_n \end{matrix} ][ \begin{matrix} b_1 & b_2 & \ldots & b_n \end{matrix} ] \\
+( \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 ==