Xen concepts for beginners: Difference between revisions

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There are various temperaments in xen with varying levels of practicality. The most important one to know is probably [[Meantone]] temperament, which equates four fifths ((3/2)^4 = 81/16) with a major third plus two octaves (5/4 * 4 = 5 = 80/16), which is encoded by tempering out the syntonic comma [[81/80]] (monzo {{monzo| -4 4 -1 }}).

A val tempers out a comma if the ~~dot product of~~ the val ~~and~~ the ~~monzo of the comma~~ is 0. 12edo is a Meantone edo because the ~~dot product of the vectors~~ {{val| 12 19 28 }} ~~and~~ {{monzo| -4 4 -1 }} is 0:

A val tempers out a comma if, when you construct the comma from primes according to their tunings in the val, the result is 0 cents or the unison. For example, 12edo is a Meantone edo because:

* The patent val for 12edo in the 5-limit is {{val| 12 19 28}}.

* The comma 81/80 has monzo {{monzo| -4 4 -1 }}.

* Constructing the tuning of a comma from mappings of primes involves multiplying each entry in the val to a corresponding entry in the comma's monzo, and then adding the resulting numbers together; this operation is called a "dot product".

** 12*-4 = -48, corresponding to going down 4 octaves.

** 19*4 = 76, corresponding to going up 4 perfect twelfths (or, to going up 4 octaves and 4 fifths).

** 28*-1 = -28, corresponding to dividing by 5 (going down two octaves and a major third).

** (76 - 48) - 28 = 0

* Since the result is 0, 12edo supports Meantone.

<math>\vmp{12 & 19 & 28}{-4 & 4 & -1} = 12 * \left(-4\right) + 19 * 4 + 28 * \left(-1\right) = -48 + 76 - 28 = 0.</math>

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 There are various temperaments in xen with varying levels of practicality. The most important one to know is probably [[Meantone]] temperament, which equates four fifths ((3/2)^4 = 81/16) with a major third plus two octaves (5/4 * 4 = 5 = 80/16), which is encoded by tempering out the syntonic comma [[81/80]] (monzo {{monzo| -4 4 -1 }}).
-A val tempers out a comma if the dot product of the val and the monzo of the comma is 0. 12edo is a Meantone edo because the dot product of the vectors {{val| 12 19 28 }} and {{monzo| -4 4 -1 }} is 0:
+A val tempers out a comma if, when you construct the comma from primes according to their tunings in the val, the result is 0 cents or the unison. For example, 12edo is a Meantone edo because:
+* The patent val for 12edo in the 5-limit is {{val| 12 19 28}}.
+* The comma 81/80 has monzo {{monzo| -4 4 -1 }}.
+* Constructing the tuning of a comma from mappings of primes involves multiplying each entry in the val to a corresponding entry in the comma's monzo, and then adding the resulting numbers together; this operation is called a "dot product".
+** 12*-4 = -48, corresponding to going down 4 octaves.
+** 19*4 = 76, corresponding to going up 4 perfect twelfths (or, to going up 4 octaves and 4 fifths).
+** 28*-1 = -28, corresponding to dividing by 5 (going down two octaves and a major third).
+** (76 - 48) - 28 = 0
+* Since the result is 0, 12edo supports Meantone.
 <math>\vmp{12 & 19 & 28}{-4 & 4 & -1} = 12 * \left(-4\right) + 19 * 4 + 28 * \left(-1\right) = -48 + 76 - 28 = 0.</math>