Eigenmonzo: Difference between revisions

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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 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.

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

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

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

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

== With respect to the projection matrix ==

The "eigen" part of the term "eigenmonzo" comes from the fact that these intervals are [~~https~~:~~//en.wikipedia.org/wiki/Eigenvalues_and_eigenvectors~~ eigenvectors] of the tuning's [[projection matrix]] (not the [[Temperament_mapping_matrices|temperament's mapping matrix]]). Only eigenvectors of the projection matrix with [~~https~~:~~//en.wikipedia.org/wiki/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 "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 [[Temperament_mapping_matrices|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.

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.

== See also ==

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

* [[Eigenmonzo subgroup]]

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

[[Category:Regular temperament theory]]

* [[~~eigenmonzo subgroup~~]]

[[Category:Terms]]

@@ Line 3: / Line 3: @@
 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 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.
-For any pure-octave temperament tuning, {{monzo|1}}, aka 1:2, is an eigenmonzo.
+For any pure-octave temperament tuning, {{monzo| 1 }}, aka 1:2, is an eigenmonzo.
-A [[rank]]-n temperament can have up to n different eigenmonzos — one for each [[generator]].
+A [[rank]]-''n'' temperament can have up to ''n'' different eigenmonzos — one for each [[generator]].
 == With respect to the projection matrix ==
-The "eigen" part of the term "eigenmonzo" comes from the fact that these intervals are [https://en.wikipedia.org/wiki/Eigenvalues_and_eigenvectors eigenvectors] of the tuning's [[projection matrix]] (not the [[Temperament_mapping_matrices|temperament's mapping matrix]]). Only eigenvectors of the projection matrix with [https://en.wikipedia.org/wiki/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 "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 [[Temperament_mapping_matrices|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.
+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.
 == See also ==
+* [[Fractional monzo]]: for more mathematical information
+* [[Eigenmonzo subgroup]]
-* [[fractional monzo]]: for more mathematical information
+[[Category:Regular temperament theory]]
-* [[eigenmonzo subgroup]]
+[[Category:Terms]]