Eigenmonzo: Difference between revisions

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An '''eigenmonzo''' is a ~~just intonation~~ interval that, when ~~tempered in~~ a ~~given~~ tuning of a given temperament, ~~remains justly tuned~~.

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.

~~For example~~, ~~{{monzo|0 0 1}} is an eigenmonzo~~ in ~~quarter-comma meantone~~. This ~~is a fancy way~~ of ~~saying that 5:4 is justly tuned in quarter-comma meantone~~.

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.

For any pure-octave temperament tuning, {{monzo|~~1}}~~, ~~aka~~ 2:1, is an ~~eigenmonzo~~.

A popular example of an unchanged interval is the JI interval 5/1, when it is mapped by [[quarter-comma meantone]]; because this temperament tuning's generator is defined as exactly one-quarter the size of the interval 5/1, it remains justly tuned.

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

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.

However, the definition of eigenmonzo is more precise; along with corresponding to an eigenvector, it must be scaled by a factor of 1 (i.e. left unchanged). In other words, there are many intervals that correspond to eigenvectors of a projection but that are not unchanged intervals in the corresponding tuning (most notably, any comma tempered out by a temperament is scaled to 0, but that is obviously not unchanged).

== See also ==

* [[Eigenmonzo subgroup]]

* [[~~fractional monzo~~]]: ~~for more mathematical information~~

[[Category:Regular temperament theory]]

* [[~~eigenmonzo subgroup~~]]

[[Category:Terms]]

[[Category:Math]]

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-An '''eigenmonzo''' is a just intonation interval that, when tempered in a given tuning of a given temperament, remains justly tuned.
+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.
-For example, {{monzo|0 0 1}} is an eigenmonzo in quarter-comma meantone. This is a fancy way of saying that 5:4 is justly tuned in quarter-comma meantone.
+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.
-For any pure-octave temperament tuning, {{monzo|1}}, aka 2:1, is an eigenmonzo.
+A popular example of an unchanged interval is the JI interval 5/1, when it is mapped by [[quarter-comma meantone]]; because this temperament tuning's generator is defined as exactly one-quarter the size of the interval 5/1, it remains justly tuned.
+For any pure-octave temperament tuning, 2/1 is an unchanged interval.
+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.
+However, the definition of eigenmonzo is more precise; along with corresponding to an eigenvector, it must be scaled by a factor of 1 (i.e. left unchanged). In other words, there are many intervals that correspond to eigenvectors of a projection but that are not unchanged intervals in the corresponding tuning (most notably, any comma tempered out by a temperament is scaled to 0, but that is obviously not unchanged).
 == See also ==
+* [[Eigenmonzo subgroup]]
-* [[fractional monzo]]: for more mathematical information
+[[Category:Regular temperament theory]]
-* [[eigenmonzo subgroup]]
+[[Category:Terms]]
+[[Category:Math]]