Monzo: Difference between revisions

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== Examples ==

For example, the interval 15/8 can be thought of as having 5*3 in the numerator, and 2*2*2 in the denominator. This can be compactly represented by the expression 2^-3 * 3^1 * 5^1, which is exactly equal to 15/8. We construct the monzo by taking the exponent from each prime, in order, and placing them within the {{Monzo| ... }} brackets, hence yielding {{Monzo|-3 1 1}}.

For example, the interval 15/8 can be thought of as having 5*3 in the numerator, and 2*2*2 in the denominator. This can be compactly represented by the expression <math>2^{-3} \cdot 3^1 \cdot 5^1</math>, which is exactly equal to 15/8. We construct the monzo by taking the exponent from each prime, in order, and placing them within the {{Monzo| ... }} brackets, hence yielding {{Monzo|-3 1 1}}.

:''Practical hint:'' Because the pipe symbol and the greater sign have special meaning in wiki syntax and HTML, there is a helper template ([[Template:Monzo]]) that can be used like this <code><nowiki>{{Monzo|arguments}}</nowiki></code> to get the monzo brackets (<code>{{Monzo|arguments}}</code>) from it.

:'''Practical hint:''' Because the pipe symbol and the greater sign have special meaning in wiki syntax and HTML, there is a helper template ([[Template:Monzo]]) that can be used like this <code><nowiki>{{Monzo|arguments}}</nowiki></code> to get the monzo brackets (<code>{{Monzo|arguments}}</code>) from it.

Here are some common 5-limit monzos, for your reference:

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< 12 19 28 | -4 4 -1 >

(12*-4) + (~~19*4~~) + (~~28*1~~)~~<span style="">~~ = </~~span~~>0

In this case, the val < 12 19 28 | is the [[Patent_val|patent val]] for 12-equal, and | -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's 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.

@@ Line 18: / Line 18: @@
 == Examples ==
-For example, the interval 15/8 can be thought of as having 5*3 in the numerator, and 2*2*2 in the denominator. This can be compactly represented by the expression 2^-3 * 3^1 * 5^1, which is exactly equal to 15/8. We construct the monzo by taking the exponent from each prime, in order, and placing them within the {{Monzo| ... }} brackets, hence yielding {{Monzo|-3 1 1}}.
+For example, the interval 15/8 can be thought of as having 5*3 in the numerator, and 2*2*2 in the denominator. This can be compactly represented by the expression <math>2^{-3} \cdot 3^1 \cdot 5^1</math>, which is exactly equal to 15/8. We construct the monzo by taking the exponent from each prime, in order, and placing them within the {{Monzo| ... }} brackets, hence yielding {{Monzo|-3 1 1}}.
-:''Practical hint:'' Because the pipe symbol and the greater sign have special meaning in wiki syntax and HTML, there is a helper template ([[Template:Monzo]]) that can be used like this <code><nowiki>{{Monzo|arguments}}</nowiki></code> to get the monzo brackets (<code>{{Monzo|arguments}}</code>) from it.
+:'''Practical hint:''' Because the pipe symbol and the greater sign have special meaning in wiki syntax and HTML, there is a helper template ([[Template:Monzo]]) that can be used like this <code><nowiki>{{Monzo|arguments}}</nowiki></code> to get the monzo brackets (<code>{{Monzo|arguments}}</code>) from it.
 Here are some common 5-limit monzos, for your reference:
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 &lt; 12 19 28 | -4 4 -1 &gt;
-(12*-4) + (19*4) + (28*1)<span style=""> = </span>0
+<math>(12⋅-4) + (19⋅4) + (28⋅1) = 0</math>
 In this case, the val &lt; 12 19 28 | is the [[Patent_val|patent val]] for 12-equal, and | -4 4 -1 &gt; 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's 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.