Eigenmonzo: Difference between revisions

(One intermediate revision by one other user not shown)

Line 1:

An '''eigenmonzo''' (also known as an '''unchanged interval''') is a (usually justly intoned) interval that remains unchanged when a specific tuning of a given temperament is applied to it. The term "eigenmonzo" does not imply that the interval is notated in monzo form, so that if 2.3 [2 -1] is an eigenmonzo, then we may also refer to the same interval written as a ratio, 4/3, as an eigenmonzo.

An '''eigenmonzo''' (also known as an '''unchanged interval''') is a (usually justly intoned) interval that remains unchanged when a specific tuning of a given temperament is applied to it. The term "eigenmonzo" does not imply that the interval is notated in monzo form, so that if 2.3 {{Monzo|2 -1}} is an eigenmonzo, then we may also refer to the same interval written as a ratio, 4/3, as an eigenmonzo.

A [[regular temperament]] transforms untempered intervals into tempered intervals, which changes most of their sizes. Only a small set of exceptional intervals do not change in size. This set of unchanged intervals depends on the choice of tuning.

Line 7:

For any pure-octave temperament tuning, 2/1 is an unchanged interval.

A [[rank]]-''n'' temperament can have up to ''n'' ~~different~~ unchanged intervals—one for each [[generator]].

A [[rank]]-''n'' temperament can have up to ''n'' linearly independent unchanged intervals—one for each [[generator]].

The term "eigenmonzo" here comes from the [[linear algebra formalism]], where intervals are often represented as vectors corresponding to their [[monzos]] (and thus instances of "vector" are often replaced with "monzo"). An [[wikipedia: Eigenvalues and eigenvectors|eigenvector]] is a vector that is not rotated (only scaled) by a matrix. The etymology of "eigen" is "own" in the sense of "characteristic"; the set of unrotated vectors and their scale factors are considered to characterize the transformation represented by the matrix. In this case, the transformation matrix is the [[projection]] corresponding to the tuning of the regular temperament, which gives the conflations of the just bases with [[Radical interval|radical intervals]], such as 3/2 to 5^(1/4). Note that this is ''not'' the matrix corresponding to the [[mapping]], which cannot specify a precise tuning.

@@ Line 1: / Line 1: @@
-An '''eigenmonzo''' (also known as an '''unchanged interval''') is a (usually justly intoned) interval that remains unchanged when a specific tuning of a given temperament is applied to it. The term "eigenmonzo" does not imply that the interval is notated in monzo form, so that if 2.3 [2 -1] is an eigenmonzo, then we may also refer to the same interval written as a ratio, 4/3, as an eigenmonzo.
+An '''eigenmonzo''' (also known as an '''unchanged interval''') is a (usually justly intoned) interval that remains unchanged when a specific tuning of a given temperament is applied to it. The term "eigenmonzo" does not imply that the interval is notated in monzo form, so that if 2.3 {{Monzo|2 -1}} is an eigenmonzo, then we may also refer to the same interval written as a ratio, 4/3, as an eigenmonzo.
 A [[regular temperament]] transforms untempered intervals into tempered intervals, which changes most of their sizes. Only a small set of exceptional intervals do not change in size. This set of unchanged intervals depends on the choice of tuning.
@@ Line 7: / Line 7: @@
 For any pure-octave temperament tuning, 2/1 is an unchanged interval.
-A [[rank]]-''n'' temperament can have up to ''n'' different unchanged intervals&mdash;one for each [[generator]].
+A [[rank]]-''n'' temperament can have up to ''n'' linearly independent unchanged intervals&mdash;one for each [[generator]].
 The term "eigenmonzo" here comes from the [[linear algebra formalism]], where intervals are often represented as vectors corresponding to their [[monzos]] (and thus instances of "vector" are often replaced with "monzo"). An [[wikipedia: Eigenvalues and eigenvectors|eigenvector]] is a vector that is not rotated (only scaled) by a matrix.  The etymology of "eigen" is "own" in the sense of "characteristic"; the set of unrotated vectors and their scale factors are considered to characterize the transformation represented by the matrix. In this case, the transformation matrix is the [[projection]] corresponding to the tuning of the regular temperament, which gives the conflations of the just bases with [[Radical interval|radical intervals]], such as 3/2 to 5^(1/4). Note that this is ''not'' the matrix corresponding to the [[mapping]], which cannot specify a precise tuning.