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~~.

A [[regular temperament]] transforms a set of untempered intervals into a set of tempered intervals, which changes the size of all of these intervals except for a few exceptions; which of these intervals that are the exceptions and do not change depends on the choice of tuning (of the temperament's generators), and thus each of these is an unchanged interval, or '''eigenmonzo''', of the tuning.

~~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 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 2:1, 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]].

== 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 "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 ==

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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.
+A [[regular temperament]] transforms a set of untempered intervals into a set of tempered intervals, which changes the size of all of these intervals except for a few exceptions; which of these intervals that are the exceptions and do not change depends on the choice of tuning (of the temperament's generators), and thus each of these is an unchanged interval, or '''eigenmonzo''', of the tuning.
-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 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 2:1, 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]].
+== 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 "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 ==