Xen concepts for beginners: Difference between revisions

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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.

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

The easiest way to get concordance is to use low-numbered JI ratios in your interval or chord, for example 3/2 the just perfect fifth, 5/4 a just major third, and 7/5 a just tritone. When pure JI ratios are used, the acoustic effect called JI buzz occurs. When the overall chord is low number JI, such as 8:9:10:11:12:13:14, the result is very concordant.

The easiest way to get concordance (blending, buzzing chords) is to use low-numbered JI ratios in your interval or chord, for example 3/2 the just perfect fifth, 5/4 a just major third, and 7/5 a just tritone. When pure JI ratios are used, the acoustic effect called JI buzz occurs. When the overall chord is low number JI, such as 8:9:10:11:12:13:14, the result is very concordant.

No edo interval except for the octave (2/1) 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.

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The approach that tempering cares about the most is lattice-based JI. A JI lattice, or a subgroup, is built by stacking a finite set of JI intervals, usually primes such as 2, 3, 5, and 7.

There are two ways the term **limit** is used. The **p-prime-limit** is the lattice built by multiplying the primes at most p, possibly multiple times. We write a JI lattice by writing the basic intervals separated by periods. For example, 6/5 = 2 \* 3 / 5 is in the 5-limit, or the 2.3.5 subgroup, and so is 45/32 = (3 * 3 * 5) / (2 * 2 * 2 * 2 * 2). The **n-odd-limit** is the set of all JI ratios where the larger of the numerator and denominator after removing factors of 2 from the JI ratios is at most the odd number n. For example, 7/6, 13/5, 11/10, and 16/15 are all in the 15-odd limit.

There are two ways the term ''[[limit]]'' is used.

* The ''p-[[prime limit]]'' is the lattice built by multiplying the primes at most p, possibly multiple times. We write a JI lattice by writing the basic intervals separated by periods. For example, 6/5 = 2 \* 3 / 5 is in the 5-limit, or the 2.3.5 subgroup, and so is 45/32 = (3 * 3 * 5) / (2 * 2 * 2 * 2 * 2).

* The ''n-[[odd-limit]]'' is the set of all JI ratios where the larger of the numerator and denominator after removing factors of 2 from the JI ratios is at most the odd number n. For example, 7/6, 13/5, 11/10, and 16/15 are all in the 15-odd limit.

== Basic RTT ==

@@ Line 23: / Line 23: @@
 == 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.
+[[Just intonation]] (JI) is the set of intervals that are tuned to rational frequency ratios, ones can be written as fractions of whole numbers.
-The easiest way to get concordance is to use low-numbered JI ratios in your interval or chord, for example 3/2 the just perfect fifth, 5/4 a just major third, and 7/5 a just tritone. When pure JI ratios are used, the acoustic effect called JI buzz occurs. When the overall chord is low number JI, such as 8:9:10:11:12:13:14, the result is very concordant.
+The easiest way to get concordance (blending, buzzing chords) is to use low-numbered JI ratios in your interval or chord, for example 3/2 the just perfect fifth, 5/4 a just major third, and 7/5 a just tritone. When pure JI ratios are used, the acoustic effect called JI buzz occurs. When the overall chord is low number JI, such as 8:9:10:11:12:13:14, the result is very concordant.
 No edo interval except for the octave (2/1) 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.
@@ Line 35: / Line 35: @@
 The approach that tempering cares about the most is lattice-based JI. A JI lattice, or a subgroup, is built by stacking a finite set of JI intervals, usually primes such as 2, 3, 5, and 7.
-There are two ways the term **limit** is used. The **p-prime-limit** is the lattice built by multiplying the primes at most p, possibly multiple times. We write a JI lattice by writing the basic intervals separated by periods. For example, 6/5 = 2 \* 3 / 5 is in the 5-limit, or the 2.3.5 subgroup, and so is 45/32 = (3 * 3 * 5) / (2 * 2 * 2 * 2 * 2). The **n-odd-limit** is the set of all JI ratios where the larger of the numerator and denominator after removing factors of 2 from the JI ratios is at most the odd number n. For example, 7/6, 13/5, 11/10, and 16/15 are all in the 15-odd limit.
+There are two ways the term ''[[limit]]'' is used.
+* The ''p-[[prime limit]]'' is the lattice built by multiplying the primes at most p, possibly multiple times. We write a JI lattice by writing the basic intervals separated by periods. For example, 6/5 = 2 \* 3 / 5 is in the 5-limit, or the 2.3.5 subgroup, and so is 45/32 = (3 * 3 * 5) / (2 * 2 * 2 * 2 * 2).
+* The ''n-[[odd-limit]]'' is the set of all JI ratios where the larger of the numerator and denominator after removing factors of 2 from the JI ratios is at most the odd number n. For example, 7/6, 13/5, 11/10, and 16/15 are all in the 15-odd limit.
 == Basic RTT ==