Monzo: Difference between revisions

Line 59:

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

Monzos 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 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:

&~~lt; 12~~ 19 28 | -4 4 -1 ~~>~~

<math>

\left\langle \begin{matrix} 12 & 19 & 28 \end{matrix} \mid \begin{matrix} -4 & 4 & -1 \end{matrix} \right\rangle \\

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

= 0

</math>

~~<math>(12⋅~~-4~~) + (19⋅4) + (28⋅~~-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 – aka it is 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.

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 – aka it is 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.

In general:

~~'''In general:~~ &~~lt; a b c | d e f~~ &~~gt;~~ = ad + be + ~~cf'''~~

<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 \\

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

</math>

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== Monzos in JI subgroups ==

@@ Line 59: / Line 59: @@
 : ''See also: [[Val]], [[Keenan's explanation of vals]], [[Vals and tuning space]] (more mathematical)''
-Monzos 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 &lt; 12 19 28 | -4 4 -1 &gt;. 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 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:
-&lt; 12 19 28 | -4 4 -1 &gt;
+<math>
+\left\langle \begin{matrix} 12 & 19 & 28 \end{matrix} \mid \begin{matrix} -4 & 4 & -1 \end{matrix} \right\rangle \\
+= 12 \times (-4) + 19 \times 4 + 28 \times (-1) \\
+= 0
+</math>
-<math>(12⋅-4) + (19⋅4) + (28⋅-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 – aka it is 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.
-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 &lt; 12 19 28 | -4 4 -1 &gt; tells us that 81/80 is mapped to 0 steps in 12-equal – aka it is 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.
+In general:
-'''In general: &lt; a b c | d e f &gt; = ad + be + cf'''
+<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 \\
+= a_1 b_1 + a_2 b_2 + \ldots + a_n b_n
+</math>
 <!--
 == Monzos in JI subgroups ==