Xen concepts for beginners: Difference between revisions

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== Interval math ==

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* ''Logarithmic units'' such as [[cents]] and [[edo]] steps that treat intervals we hear as equal as the same additive unit

To stack two intervals, we use different types of operations for the two kinds of units. To stack two intervals written as ratios, we *multiply*, whereas to stack two intervals written as cents or edo steps, we *add* the intuitive way. To "unstack" an interval from another interval, we *divide* the respective ratios and *subtract* logarithmic units. To convert between cents and ratios we use the following formulas:

<math>

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</math>

The unison has frequency ratio 1/1 and is 0 cents. The octave has frequency ratio 2/1 and is exactly 1200 cents. A 12edo~~/12tet semitone~~ has frequency ratio 2~~^(1~~/~~12)~~ and is exactly 100 cents.

The unison has frequency ratio 1/1 and is 0 cents. The octave has frequency ratio 2/1 and is exactly 1200 cents. A standard semitone (in 12edo) has frequency ratio <math>\sqrt[12]{2}</math> and is exactly 100 cents (by definition).

The notation m\n means m steps of n-edo. 7\12 is 7 steps ~~out~~ of ~~12edo, the 12edo perfect fifth~~.

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.

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== Basic JI ==

[[Just intonation]] (JI) is the set of intervals that are tuned to rational frequency ratios, ones can be written as fractions of whole numbers.

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No edo interval except for the octave (2/1) and stacks of it is exact JI. A JI ratio might be far from a 12edo interval; for example 7/4 is 969 cents. This is another reason why JI is a common approach to xen.

As stacking JI ratios involves multiplying, primes are important as the simplest building blocks of arbitrary JI ratios. So we can write every ratio as a vector called a ''monzo'', a list of powers for primes~~. 81/80's monzo is [-4 4 -1>~~. We can visualize each ratio as living in some JI lattice (the set of all intervals built by stacking a finite set of basic intervals).

As stacking JI ratios involves multiplying, primes are important as the simplest building blocks of arbitrary JI ratios. So we can write every ratio as a vector called a ''monzo'', a list of powers for primes. We can visualize each ratio as living in some JI lattice (the set of all intervals built by stacking a finite set of basic intervals).

There are many approaches to JI music: lattice-based JI, constant structure scales, free JI, primodality, tonality diamonds, combination product sets...

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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 12*(-4) + 19*4 + 28*(-1) = -48 + 76 - 28 = 0.

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:

<math>\vmproduct{12 & 19 & 28}{-4 & 4 & -1} = 12 * -4 + 19 * 4 + 28 * -1 = -48 + 76 - 28 = 0.</math>

== MOS scales ==

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+{{Texmap}}
 {{Beginner}}
 == Interval math ==
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 * ''Logarithmic units'' such as [[cents]] and [[edo]] steps that treat intervals we hear as equal as the same additive unit
-To stack two intervals, we use different types of operations for the two kinds of units.  To stack two intervals written as ratios, we *multiply*, whereas to stack two intervals written as cents or edo steps, we *add* the intuitive way. To "unstack" an interval from another interval, we *divide* the respective ratios and *subtract* logarithmic units. To convert between cents and ratios we use the following formulas:
+To stack two intervals, we use different types of operations for the two kinds of units. To stack two intervals written as ratios, we *multiply*, whereas to stack two intervals written as cents or edo steps, we *add* the intuitive way. To "unstack" an interval from another interval, we *divide* the respective ratios and *subtract* logarithmic units. To convert between cents and ratios we use the following formulas:
 <math>
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 </math>
-The unison has frequency ratio 1/1 and is 0 cents. The octave has frequency ratio 2/1 and is exactly 1200 cents. A 12edo/12tet semitone has frequency ratio 2^(1/12) and is exactly 100 cents.
+The unison has frequency ratio 1/1 and is 0 cents. The octave has frequency ratio 2/1 and is exactly 1200 cents. A standard semitone (in 12edo) has frequency ratio <math>\sqrt[12]{2}</math> and is exactly 100 cents (by definition).
-The notation m\n means m steps of n-edo. 7\12 is 7 steps out of 12edo, the 12edo perfect fifth.
+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.
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 == Basic JI ==
 [[Just intonation]] (JI) is the set of intervals that are tuned to rational frequency ratios, ones can be written as fractions of whole numbers.
@@ Line 31: / Line 31: @@
 No edo interval except for the octave (2/1) and stacks of it is exact JI. A JI ratio might be far from a 12edo interval; for example 7/4 is 969 cents. This is another reason why JI is a common approach to xen.
-As stacking JI ratios involves multiplying, primes are important as the simplest building blocks of arbitrary JI ratios. So we can write every ratio as a vector called a ''monzo'', a list of powers for primes. 81/80's monzo is [-4 4 -1>. We can visualize each ratio as living in some JI lattice (the set of all intervals built by stacking a finite set of basic intervals).
+As stacking JI ratios involves multiplying, primes are important as the simplest building blocks of arbitrary JI ratios. So we can write every ratio as a vector called a ''monzo'', a list of powers for primes. We can visualize each ratio as living in some JI lattice (the set of all intervals built by stacking a finite set of basic intervals).
 There are many approaches to JI music: lattice-based JI, constant structure scales, free JI, primodality, tonality diamonds, combination product sets...
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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 12*(-4) + 19*4 + 28*(-1) = -48 + 76 - 28 = 0.
+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:
+<math>\vmproduct{12 & 19 & 28}{-4 & 4 & -1} = 12 * -4 + 19 * 4 + 28 * -1 = -48 + 76 - 28 = 0.</math>
 == MOS scales ==