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 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). 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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** 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.

* 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 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). 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.
@@ Line 61: / Line 61: @@
 ** 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.
+* 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>