Eigenmonzo: Difference between revisions

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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.

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.

A popular example of an ~~eigenmonzo~~ is the JI interval ~~{{monzo| 0 0~~ 1 ~~}}, or 1:5~~, 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 1:5, it remains justly tuned.

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, ~~{{monzo|~~ 1 ~~}}, aka 1:2,~~ is an ~~eigenmonzo~~.

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

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

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

~~From the scale's or~~ the ~~instrument's point of view~~, ~~eigenmonzos have two sides~~ of ~~information~~.

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.

* Specify the ~~interval in~~ the scale. ~~(1/4-comma meantone: specifying "M3"~~, which ~~tempers together ~~~5/4~~~81/64~6561/5120~~~.~~..) Then,~~

* Tune it to the ~~specified ratio. (1/4-comma meantone:~~ tuning ~~M3 to just 5/4~~.)

~~== With respect to~~ the ~~projection matrix ==~~

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).

~~The "eigen" part~~ of ~~the term "~~eigenmonzo~~" comes from the fact that these intervals are [[wikipedia: Eigenvalues and eigenvectors|eigenvectors]]~~ of ~~the tuning's [[projection matrix]]~~ (~~not the [[mapping|temperament's mapping matrix]]~~). ~~Only~~ eigenvectors of ~~the~~ projection ~~matrix with [[wikipedia: Eigenvalues and eigenvectors|eigenvalue]] equal to 1~~ are ~~considered eigenmonzos~~, ~~while those with eigenvalue equal~~ to 0 ~~are the vanishing commas of the temperament; in other words~~, ~~a vector~~ that is ~~a monzo and an eigenvector is~~ not ~~necessarily an eigenmonzo.~~

The "monzo" part of "eigenmonzo" should not be taken to imply that the interval is notated in monzo form, e.g. {{monzo| 2 -1 }}; for example, 4/3 may be called an eigenmonzo. But of course it must be able to specify the position in the JI lattice; can't be as 498¢.

== See also ==

* [[Fractional monzo]]: for more mathematical information

* [[Eigenmonzo subgroup]]

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-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.
+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.
-A popular example of an eigenmonzo is the JI interval {{monzo| 0 0 1 }}, or 1:5, 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 1:5, it remains justly tuned.
+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, {{monzo| 1 }}, aka 1:2, is an eigenmonzo.
+For any pure-octave temperament tuning, 2/1 is an unchanged interval.
-A [[rank]]-''n'' temperament can have up to ''n'' different eigenmonzos — one for each [[generator]].
+A [[rank]]-''n'' temperament can have up to ''n'' linearly independent unchanged intervals&mdash;one for each [[generator]].
-From the scale's or the instrument's point of view, eigenmonzos have two sides of information.
+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.
-* Specify the interval in the scale. (1/4-comma meantone: specifying "M3", which tempers together ~5/4~81/64~6561/5120~...) Then,
-* Tune it to the specified ratio. (1/4-comma meantone: tuning M3 to just 5/4.)
-== With respect to the projection matrix ==
+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).
-The "eigen" part of the term "eigenmonzo" comes from the fact that these intervals are [[wikipedia: Eigenvalues and eigenvectors|eigenvectors]] of the tuning's [[projection matrix]] (not the [[mapping|temperament's mapping matrix]]). Only eigenvectors of the projection matrix with [[wikipedia: Eigenvalues and eigenvectors|eigenvalue]] equal to 1 are considered eigenmonzos, while those with eigenvalue equal to 0 are the vanishing commas of the temperament; in other words, a vector that is a monzo and an eigenvector is not necessarily an eigenmonzo.
-The "monzo" part of "eigenmonzo" should not be taken to imply that the interval is notated in monzo form, e.g. {{monzo| 2 -1 }}; for example, 4/3 may be called an eigenmonzo. But of course it must be able to specify the position in the JI lattice; can't be as 498¢.
 == See also ==
-* [[Fractional monzo]]: for more mathematical information
 * [[Eigenmonzo subgroup]]