Xen concepts for beginners: Difference between revisions

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The notation m\n means m steps of n-edo. For example, 12edo's perfect fifth can be denoted as 7\12, meaning "7 steps of 12-tone equal temperament".

A ~~very important~~ operation in xen math is the [[mediant]]. The mediant of two fractions, a/b and c/d, is the "freshman sum" (a+b)/(c+d). For example, the mediant of 4/3, the just perfect fourth, and 5/4, the just major third, is 9/7, the supermajor third. If two fractions are in lowest terms, their mediant is the simplest fraction that is strictly between both. The mediant is commonly used for both JI ratios and edo intervals.

A common operation in xen math is the [[mediant]]. The mediant of two fractions, a/b and c/d, is the "freshman sum" (a+b)/(c+d), which is always between a/b and c/d. For example, the mediant of 4/3, the just perfect fourth, and 5/4, the just major third, is 9/7, the supermajor third. If two fractions are in lowest terms, their mediant is the simplest fraction that is strictly between both. The mediant is commonly used for both JI ratios and edo intervals.

Another important operation is [[octave reduction|reduction]]. To reduce an interval a by an interval b means to stack or "unstack" b from a until a is at least the unison and less than b. For example, 3/1 reduced by 2/1 is 3/2.

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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. In RTT math, this can be written as:

<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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 The notation m\n means m steps of n-edo. For example, 12edo's perfect fifth can be denoted as 7\12, meaning "7 steps of 12-tone equal temperament".
-A very important operation in xen math is the [[mediant]]. The mediant of two fractions, a/b and c/d, is the "freshman sum" (a+b)/(c+d). For example, the mediant of 4/3, the just perfect fourth, and 5/4, the just major third, is 9/7, the supermajor third. If two fractions are in lowest terms, their mediant is the simplest fraction that is strictly between both. The mediant is commonly used for both JI ratios and edo intervals.
+A common operation in xen math is the [[mediant]]. The mediant of two fractions, a/b and c/d, is the "freshman sum" (a+b)/(c+d), which is always between a/b and c/d. For example, the mediant of 4/3, the just perfect fourth, and 5/4, the just major third, is 9/7, the supermajor third. If two fractions are in lowest terms, their mediant is the simplest fraction that is strictly between both. The mediant is commonly used for both JI ratios and edo intervals.
 Another important operation is [[octave reduction|reduction]]. To reduce an interval a by an interval b means to stack or "unstack" b from a until a is at least the unison and less than b. For example, 3/1 reduced by 2/1 is 3/2.
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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. In RTT math, this can be written as:
 <math>\vmp{12 & 19 & 28}{-4 & 4 & -1} = 12 * \left(-4\right) + 19 * 4 + 28 * \left(-1\right) = -48 + 76 - 28 = 0.</math>