Xen concepts for beginners: Difference between revisions

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== Basic RTT ==

Assuming several things from common 12edo practice, JI has several disadvantages. To get infinite modulation and use exactly the same chord on every note, we need infinitely many notes unlike the finitely many notes of 12edo. JI also has small intervals that may be undesirable, called commas. This is the problem that regular temperament theory (RTT) exists to solve. Regular temperaments equate certain intervals by considering the difference between them as a comma and "tempering out" the difference.

Assuming several things from common 12edo practice, JI has several disadvantages. To get infinite modulation and use exactly the same chord on every note, we need infinitely many notes unlike the finitely many notes of 12edo. JI also has small intervals that may be undesirable, called commas. This is the problem that [[regular temperament theory]] (RTT) exists to solve. Regular temperaments equate certain intervals by considering the difference between them as a comma and "[[tempering out]]" the difference.

RTT views edos as regular temperaments. To simplify the infinite JI space to a finite set, we need to deform the intervals so that certain chosen intervals vanish. We can also approach simplifying JI ratios from edos themselves, namely how edos approximate each prime. This is called a val. Vals map primes to a set number of edo steps and thus tell us how many edo steps each interval in JI is mapped to. The 12edo patent val in the 5-limit is <12 19 28], as the 12edo intervals that are closest to 2/1, 3/1 and 5/1 are 12, 19 and 28 steps respectively.

RTT views edos as regular temperaments. To simplify the infinite JI space to a finite set, we need to deform the intervals so that certain chosen intervals vanish. We can also approach simplifying JI ratios from edos themselves, namely how edos approximate each prime. This is called a [[val]]. Vals map primes to a set number of edo steps and thus tell us how many edo steps each interval in JI is mapped to. The usual 12edo val (called the 12edo [[patent val]]) in the 5-limit is <12 19 28], as the 12edo intervals that are closest to 2/1, 3/1 and 5/1 are 12, 19 and 28 steps respectively.

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 [-4 4 -1>).

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 [-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 <12 19 28] and [-4 4 -1> is 0.

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 == Basic RTT ==
-Assuming several things from common 12edo practice, JI has several disadvantages. To get infinite modulation and use exactly the same chord on every note, we need infinitely many notes unlike the finitely many notes of 12edo. JI also has small intervals that may be undesirable, called commas. This is the problem that regular temperament theory (RTT) exists to solve. Regular temperaments equate certain intervals by considering the difference between them as a comma and "tempering out" the difference.
+Assuming several things from common 12edo practice, JI has several disadvantages. To get infinite modulation and use exactly the same chord on every note, we need infinitely many notes unlike the finitely many notes of 12edo. JI also has small intervals that may be undesirable, called commas. This is the problem that [[regular temperament theory]] (RTT) exists to solve. Regular temperaments equate certain intervals by considering the difference between them as a comma and "[[tempering out]]" the difference.
-RTT views edos as regular temperaments. To simplify the infinite JI space to a finite set, we need to deform the intervals so that certain chosen intervals vanish. We can also approach simplifying JI ratios from edos themselves, namely how edos approximate each prime. This is called a val. Vals map primes to a set number of edo steps and thus tell us how many edo steps each interval in JI is mapped to. The 12edo patent val in the 5-limit is <12 19 28], as the 12edo intervals that are closest to 2/1, 3/1 and 5/1 are 12, 19 and 28 steps respectively.
+RTT views edos as regular temperaments. To simplify the infinite JI space to a finite set, we need to deform the intervals so that certain chosen intervals vanish. We can also approach simplifying JI ratios from edos themselves, namely how edos approximate each prime. This is called a [[val]]. Vals map primes to a set number of edo steps and thus tell us how many edo steps each interval in JI is mapped to. The usual 12edo val (called the 12edo [[patent val]]) in the 5-limit is <12 19 28], as the 12edo intervals that are closest to 2/1, 3/1 and 5/1 are 12, 19 and 28 steps respectively.
-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 [-4 4 -1>).
+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 [-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 <12 19 28] and [-4 4 -1> is 0.