Monzo: Difference between revisions

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</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 composed from the sixteenth century onwards, particularly ~~western~~ music composed for 12-equal ~~or 12-tone~~ well ~~temperaments~~, is made possible by tempering out of 81/80, ~~as illustrated 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 composed in the Renaissance and from the sixteenth century onwards, particularly Western music composed for 12-tone circulating temperaments ([[12edo|12 equal]] and unequal [[well temperament]]s), is made possible by the tempering out of 81/80, and common practice Western music theory is heavily based on meantone.

In general:

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Similarly, edo tunings of a temperament can be given in terms of (a generalized version of) vals, by specifying how many edo steps are used for each generator of the temperament. For example, [[31edo]]'s tuning of meantone temperament can be written as {{val|"2"~31, "3/2"~18}}.

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== See also ==

* [[Extended bra-ket notation]]

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 </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 composed from the sixteenth century onwards, particularly western music composed for 12-equal or 12-tone well temperaments, is made possible by tempering out of 81/80, as illustrated 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 composed in the Renaissance and from the sixteenth century onwards, particularly Western music composed for 12-tone circulating temperaments ([[12edo|12 equal]] and unequal [[well temperament]]s), is made possible by the tempering out of 81/80, and common practice Western music theory is heavily based on meantone.
 In general:
@@ Line 89: / Line 89: @@
 Similarly, edo tunings of a temperament can be given in terms of (a generalized version of) vals, by specifying how many edo steps are used for each generator of the temperament. For example, [[31edo]]'s tuning of meantone temperament can be written as {{val|"2"~31, "3/2"~18}}.
 -->
 == See also ==
 * [[Extended bra-ket notation]]