Eigenmonzo - Revision history

Overthink: more correct

2026-01-01T23:59:41Z

more correct

← Older revision		Revision as of 23:59, 1 January 2026
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	For any pure-octave temperament tuning, 2/1 is an unchanged interval.		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.		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.

MisterShafXen at 14:43, 2 August 2025

2025-08-02T14:43:03Z

← Older revision		Revision as of 14:43, 2 August 2025
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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. 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.		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.

VectorGraphics at 20:45, 17 June 2025

2025-06-17T20:45:04Z

← Older revision		Revision as of 20:45, 17 June 2025
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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 [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 [[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'' different unchanged intervals—one for each [[generator]].

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

	~~The "monzo" part~~ of "eigenmonzo~~" should not be taken to imply that the interval~~ is ~~notated in monzo form. For example, if {{monzo\| 2 -1 }} is an eigenmonzo, then we may also refer~~ to ~~this same interval expressed in quotient form, 4/3, as~~ an ~~eigenmonzo.~~		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]] (vectors that are not rotated~~, ~~only~~ scaled) of the tuning's [[projection matrix]] (not the [[mapping\|temperament's mapping 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. However, only eigenvectors~~ of ~~the projection matrix with [[wikipedia: Eigenvalues and eigenvectors\|eigenvalues]] (scale factors) equal to~~ 1 (~~scaled by 1,~~ i.e. unchanged ~~intervals~~) ~~are considered to be eigenmonzos; eigenvectors with eigenvalues equal to anything else are ''not'' considered to be eigenmonzos~~. In other words, many ~~things~~ that ~~are both monzos and~~ eigenvectors are not ~~eigenmonzos. Most~~ notably, ~~eigenvectors with eigenvalues equal to 0 are the vanishing commas of the~~ temperament~~, being~~ scaled to 0, but ~~these are~~ not ~~eigenmonzos~~.

	== See also ==		== See also ==
	* [[Fractional monzo]]: for more mathematical information
	* [[Eigenmonzo subgroup]]		* [[Eigenmonzo subgroup]]

VectorGraphics at 22:11, 8 April 2025

2025-04-08T22:11:19Z

← Older revision		Revision as of 22:11, 8 April 2025
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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.

	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 [[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.

ArrowHead294 at 21:42, 23 May 2024

2024-05-23T21:42:13Z

← Older revision		Revision as of 21:42, 23 May 2024
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	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{{mdash}}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 ==		== With respect to the projection matrix ==

ArrowHead294 at 13:45, 23 May 2024

2024-05-23T13:45:22Z

← Older revision		Revision as of 13:45, 23 May 2024
Line 7:		Line 7:
	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{{mdash}}one for each [[generator]].

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

FloraC: They say they changed it to hyphenated

2023-03-08T07:10:26Z

They say they changed it to hyphenated

← Older revision		Revision as of 07:10, 8 March 2023
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.		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.

	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 [[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.

Cmloegcmluin: /* With respect to the projection matrix */ recognize other possible scaling factor sizes besides 0 and 1

2022-09-16T17:52:41Z

With respect to the projection matrix: recognize other possible scaling factor sizes besides 0 and 1

← Older revision		Revision as of 17:52, 16 September 2022
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	The "monzo" part of "eigenmonzo" should not be taken to imply that the interval is notated in monzo form. For example, if {{monzo\| 2 -1 }} is an eigenmonzo, then we may also refer to this same interval expressed in quotient form, 4/3, as an eigenmonzo.		The "monzo" part of "eigenmonzo" should not be taken to imply that the interval is notated in monzo form. For example, if {{monzo\| 2 -1 }} is an eigenmonzo, then we may also refer to this same interval expressed in quotient form, 4/3, as an eigenmonzo.

	The "eigen" part of the term "eigenmonzo" comes from the fact that these intervals are [[wikipedia: Eigenvalues and eigenvectors\|eigenvectors]] (vectors that are not rotated, only scaled) of the tuning's [[projection matrix]] (not the [[mapping\|temperament's mapping 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. However, only eigenvectors of the projection matrix with [[wikipedia: Eigenvalues and eigenvectors\|eigenvalues]] (scale factors) equal to 1 (scaled by 1, i.e. unchanged) are considered to be eigenmonzos~~, while those~~ with ~~eigenvalue~~ equal to ~~0 — which are the vanishing commas of the temperament, being that they are scaled by 0 —~~ are ''not'' considered to be eigenmonzos. In other words, ~~some~~ things that are both monzos and eigenvectors are not eigenmonzos, ~~most notably any~~ vanishing ~~comma~~ of ~~any~~ temperament.		The "eigen" part of the term "eigenmonzo" comes from the fact that these intervals are [[wikipedia: Eigenvalues and eigenvectors\|eigenvectors]] (vectors that are not rotated, only scaled) of the tuning's [[projection matrix]] (not the [[mapping\|temperament's mapping 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. However, only eigenvectors of the projection matrix with [[wikipedia: Eigenvalues and eigenvectors\|eigenvalues]] (scale factors) equal to 1 (scaled by 1, i.e. unchanged intervals) are considered to be eigenmonzos; eigenvectors with eigenvalues equal to anything else are ''not'' considered to be eigenmonzos. In other words, many things that are both monzos and eigenvectors are not eigenmonzos. Most notably, eigenvectors with eigenvalues equal to 0 are the vanishing commas of the temperament, being scaled to 0, but these are not eigenmonzos.

	== See also ==		== See also ==

Cmloegcmluin: /* With respect to the projection matrix */ better if this point comes first, so the points in the next paragraph are less confusing

2022-09-16T16:56:00Z

With respect to the projection matrix: better if this point comes first, so the points in the next paragraph are less confusing

← Older revision		Revision as of 16:56, 16 September 2022
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	== With respect to the projection matrix ==		== With respect to the projection matrix ==

			The "monzo" part of "eigenmonzo" should not be taken to imply that the interval is notated in monzo form. For example, if {{monzo\| 2 -1 }} is an eigenmonzo, then we may also refer to this same interval expressed in quotient form, 4/3, as an eigenmonzo.

	The "eigen" part of the term "eigenmonzo" comes from the fact that these intervals are [[wikipedia: Eigenvalues and eigenvectors\|eigenvectors]] (vectors that are not rotated, only scaled) of the tuning's [[projection matrix]] (not the [[mapping\|temperament's mapping 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. However, only eigenvectors of the projection matrix with [[wikipedia: Eigenvalues and eigenvectors\|eigenvalues]] (scale factors) equal to 1 (scaled by 1, i.e. unchanged) are considered to be eigenmonzos, while those with eigenvalue equal to 0 — which are the vanishing commas of the temperament, being that they are scaled by 0 — are ''not'' considered to be eigenmonzos. In other words, some things that are both monzos and eigenvectors are not eigenmonzos, most notably any vanishing comma of any temperament.		The "eigen" part of the term "eigenmonzo" comes from the fact that these intervals are [[wikipedia: Eigenvalues and eigenvectors\|eigenvectors]] (vectors that are not rotated, only scaled) of the tuning's [[projection matrix]] (not the [[mapping\|temperament's mapping 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. However, only eigenvectors of the projection matrix with [[wikipedia: Eigenvalues and eigenvectors\|eigenvalues]] (scale factors) equal to 1 (scaled by 1, i.e. unchanged) are considered to be eigenmonzos, while those with eigenvalue equal to 0 — which are the vanishing commas of the temperament, being that they are scaled by 0 — are ''not'' considered to be eigenmonzos. In other words, some things that are both monzos and eigenvectors are not eigenmonzos, most notably any vanishing comma of any temperament.

	The "monzo" part of "eigenmonzo" should not be taken to imply that the interval is notated in monzo form. For example, if {{monzo\| 2 -1 }} is an eigenmonzo, then we may also refer to this same interval expressed in quotient form, 4/3, as an eigenmonzo.

	== See also ==		== See also ==

Cmloegcmluin: /* With respect to the projection matrix */ improve clarity

2022-09-16T16:55:31Z

With respect to the projection matrix: improve clarity

← Older revision		Revision as of 16:55, 16 September 2022
Line 11:		Line 11:
	== With respect to the projection matrix ==		== With respect to the projection matrix ==

	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 "eigen" part of the term "eigenmonzo" comes from the fact that these intervals are [[wikipedia: Eigenvalues and eigenvectors\|eigenvectors]] (vectors that are not rotated, only scaled) of the tuning's [[projection matrix]] (not the [[mapping\|temperament's mapping 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. However, only eigenvectors of the projection matrix with [[wikipedia: Eigenvalues and eigenvectors\|eigenvalues]] (scale factors) equal to 1 (scaled by 1, i.e. unchanged) are considered to be eigenmonzos, while those with eigenvalue equal to 0 — which are the vanishing commas of the temperament, being that they are scaled by 0 — are ''not'' considered to be eigenmonzos. In other words, some things that are both monzos and eigenvectors are not eigenmonzos, most notably any vanishing comma of any temperament.

	The "monzo" part of "eigenmonzo" should not be taken to imply that the interval is notated in monzo form. For example, if {{monzo\| 2 -1 }} is an eigenmonzo, then we may also refer to this same interval expressed in quotient form, 4/3, as an eigenmonzo.		The "monzo" part of "eigenmonzo" should not be taken to imply that the interval is notated in monzo form. For example, if {{monzo\| 2 -1 }} is an eigenmonzo, then we may also refer to this same interval expressed in quotient form, 4/3, as an eigenmonzo.