Target tuning - Revision history

FloraC: + link to my (half-usable) script

2026-03-19T17:42:04Z

+ link to my (half-usable) script

← Older revision		Revision as of 17:42, 19 March 2026
Line 44:		Line 44:

	Starting from the same six intervals of the 5-odd-limit diamond and adding 2, we find after computing the normal interval lists that the three subgroups are [2, 3], [2, 5], and [2, 5/3]. Computing the projection matrix and from thence the tuning in each case, we find that the minimax tuning is [{{monzo\| 1 0 0 }}, {{monzo\| 1 0 1/4 }}, {{monzo\| 0 0 1 }}], the projection matrix for 1/4-comma meantone.		Starting from the same six intervals of the 5-odd-limit diamond and adding 2, we find after computing the normal interval lists that the three subgroups are [2, 3], [2, 5], and [2, 5/3]. Computing the projection matrix and from thence the tuning in each case, we find that the minimax tuning is [{{monzo\| 1 0 0 }}, {{monzo\| 1 0 1/4 }}, {{monzo\| 0 0 1 }}], the projection matrix for 1/4-comma meantone.

			== External links ==
			* [https://github.com/FloraCanou/temperament_evaluator/wiki/Target-tuning Github \| ''Target tuning'' · FloraCanou/temperament_evaluator Wiki] – includes Python script to compute target tunings.

	== Notes ==		== Notes ==

FloraC: + general formulation (including the requested link to D&D's guide). Mention minimean tuning (no contents yet)

2026-03-19T09:38:18Z

+ general formulation (including the requested link to D&D's guide). Mention minimean tuning (no contents yet)

@@ Line 1: / Line 1: @@
-By a '''target tuning''' is meant a tuning for a [[regular temperament]] which has been optimized with respect to a set of target [[interval]]s. Usually this set of intervals is derived from a finite set of [[pitch class|octave-equivalence classes]], and octaves are taken to be pure 2's. Moreover, normally the intervals in question are rational numbers. Below we will make all of these assumptions, and will discuss the two most important target tunings, minimax and least squares.
+A '''target tuning''' for a [[regular temperament|temperament]] is a tuning which has been optimized with respect to a set of target [[interval]]s.
-== Least squares tunings ==
+The generic formulation for any target tuning is as follows.
 By assumption we have a finite set of rational intervals defining [[interval class]]es, which we may take to be [[octave reduction|octave-reduced]] intervals, expressed logarithmically in terms of [[binary logarithm]]. The least squares tuning, ''T'', is the tuning for a particular regular temperament which minimizes the sum of the squares of the errors, ∑<sub>''i''</sub> ({{nowrap|''T''(''q<sub>i</sub>'') − log<sub>2</sub>(''q<sub>i</sub>''))<sup>2</sup>}}, where ''q<sub>i</sub>'' are the rational intervals of the target set.
@@ Line 12: / Line 28: @@
 For a rank-''r'' temperament, we can form a list of candidate sets of eigenmonzos by taking {{nowrap|''r'' − 1}} elements from the list of target intervals, and adding 2 to the set, leading to an ''r''-element set. To avoid duplications and sets with rank less than ''r'', we can take the [[normal forms|normal interval list]] defined by each of these ''r''-element sets, discarding any with less than ''r'' elements. This means we are compiling a list of rank-''r'' [[just intonation subgroup]]s, which are [[eigenmonzo subgroup]]s for the corresponding tuning. For each of these these subgroups, we may find a corresponding projection matrix as the matrix with the ''r'' generators of the subgroup as eigenmonzos and a basis for the commas as left eigenvectors with eigenvalue zero. These projection matrices define tunings, and the tuning, if unique, with the least maximum error is the minimax tuning. However, this tuning may not be unique, in which case we may break the tie by discarding the bounding intervals and repeat minimax on the rest of the intervals until a unique tuning is found. This is the same tuning achieved by minimizing the ''p''-norm error as ''p'' approaches infinity<ref group="note">See [[Dave Keenan & Douglas Blumeyer's guide to RTT/Tuning computation #Tie breaking: power limit method]] for details.</ref><ref group="note">Historically, [[Gene Ward Smith]] proposed breaking the tie by falling back to least squares tuning in the set of minimax tunings.</ref>.
 == Example ==

FloraC: Replace Gene's tie-breaking method in favor of a more reasonable one. Misc. clarification/cleanup

2026-03-18T13:21:21Z

Replace Gene's tie-breaking method in favor of a more reasonable one. Misc. clarification/cleanup

← Older revision		Revision as of 13:21, 18 March 2026
Line 2:		Line 2:

	== Least squares tunings ==		== Least squares tunings ==
	By assumption we have a finite set of rational intervals defining interval ~~classes~~, which we may take to be octave reduced ~~to {{nowrap\|0 < ''r'' < 1}}~~, ~~where the intervals are~~ expressed logarithmically in terms of ~~log base two~~. The least squares tuning, ''T'', is the tuning for a particular regular temperament which minimizes the sum of the squares of the errors, ∑<sub>''i''</sub> ({{nowrap\|''T''(''q<sub>i</sub>'') − log<sub>2</sub>(''q<sub>i</sub>''))<sup>2</sup>}}, where ''q<sub>i</sub>'' are the rational intervals of the target set. Note that most commonly, the target set is a [[tonality diamond]] (reduced to lowest terms and duplicates removed), since these are the intervals in a [[harmonic series]] up to some odd integer ''d'', and hence may be considered the [[consonance]]s of the ''d''-odd-limit.		By assumption we have a finite set of rational intervals defining [[interval class]]es, which we may take to be [[octave reduction\|octave-reduced]] intervals, expressed logarithmically in terms of [[binary logarithm]]. The least squares tuning, ''T'', is the tuning for a particular regular temperament which minimizes the sum of the squares of the errors, ∑<sub>''i''</sub> ({{nowrap\|''T''(''q<sub>i</sub>'') − log<sub>2</sub>(''q<sub>i</sub>''))<sup>2</sup>}}, where ''q<sub>i</sub>'' are the rational intervals of the target set.

	We may find the least squares tuning in various ways, one of which is to start from a matrix '''R''' whose rows are the monzos of the target set, and a matrix '''U''' whose rows are [[val]]s spanning the temperament. From '''U''' we form the matrix '''V''' by taking the [[~~Normal lists~~\|normal val list]] for '''U''' and removing the first ("period") row. A list of [[~~Eigenmonzo~~\|eigenmonzos (unchanged-intervals)]] {{nowrap\|'''E'''' {{=}} '''VR'''<sup>T</sup>'''R'''}}, where the <sup>T</sup> denotes the matrix transpose, can now be obtained by matrix multiplication, and to this we add a row for the monzo of 2. The [[~~Fractional monzos\|~~projection matrix]] for the least squares tuning is then the square matrix with fractional monzo rows, the rows of '''E''', including 2, as eigenmonzos, that is left eigenvectors with eigenvalue one, and any basis for the commas of the temperament as left eigenvectors with eigenvalues zero.		Most commonly, the target set is a [[tonality diamond]], since these are the intervals in a [[harmonic series]] up to some odd integer ''d'', and hence may be considered the [[consonance]]s of the ''d''-odd-limit. By convention, intervals of the tonality diamond are reduced to lowest terms and have duplicates removed. If duplicates are not removed, the results are usually different.

			We may find the least squares tuning in various ways, one of which is to start from a matrix '''R''' whose rows are the monzos of the target set, and a matrix '''U''' whose rows are [[val]]s spanning the temperament. From '''U''' we form the matrix '''V''' by taking the [[normal forms\|normal val list]] for '''U''' and removing the first ("period") row. A list of [[eigenmonzo\|eigenmonzos (unchanged-intervals)]] {{nowrap\|'''E'''' {{=}} '''VR'''<sup>T</sup>'''R'''}}, where the <sup>T</sup> denotes the matrix transpose, can now be obtained by matrix multiplication, and to this we add a row for the monzo of 2. The [[projection matrix]] for the least squares tuning is then the square matrix with fractional monzo rows, the rows of '''E''', including 2, as eigenmonzos, that is left eigenvectors with eigenvalue one, and any basis for the commas of the temperament as left eigenvectors with eigenvalues zero.

	== Minimax tuning ==		== Minimax tuning ==
	Starting with the same assumptions as with least squares tuning, minimax tuning works similarly, except instead of minimizing the squares of the errors, we minimize {{nowrap\|max<sub>''i''</sub>{{!}}''T''(''q<sub>i</sub>'') − log<sub>2</sub>(''q<sub>i</sub>''){{!}} }}, the maximum error over all the target intervals. This can be solved by setting the minimization up as a {{w\|linear programming}} problem, but another approach leads as before to a projection matrix with fractional monzo rows.		Starting with the same assumptions as with least squares tuning, minimax tuning works similarly, except instead of minimizing the squares of the errors, we minimize {{nowrap\|max<sub>''i''</sub>{{!}}''T''(''q<sub>i</sub>'') − log<sub>2</sub>(''q<sub>i</sub>''){{!}} }}, the maximum error over all the target intervals. This can be solved by setting the minimization up as a {{w\|linear programming}} problem, but another approach leads as before to a projection matrix with fractional monzo rows.

	For a rank-''r'' temperament, we can form a list of candidate sets of eigenmonzos by taking {{nowrap\|''r'' − 1}} elements from the list of target intervals, and adding 2 to the set, leading to an ''r''-element set. To avoid duplications and sets with rank less than ''r'', we can take the [[~~Normal lists~~\|normal interval list]] defined by each of these ''r''-element sets, discarding any with less than ''r'' elements. This means we are compiling a list of rank-''r'' [[just intonation subgroup]]s, which are [[eigenmonzo subgroup]]s for the corresponding tuning. For each of these these subgroups, we may find a corresponding projection matrix as the matrix with the ''r'' generators of the subgroup as eigenmonzos and a basis for the commas as left eigenvectors with eigenvalue zero. These projection matrices define tunings, and the tuning, if unique, with the least maximum error is the minimax tuning. However, this tuning may not be unique, in which case we may break the tie by ~~choosing~~ the tuning, ~~among~~ the set of ~~least maximum error~~ tunings~~, with the smallest sum of errors squared~~.		For a rank-''r'' temperament, we can form a list of candidate sets of eigenmonzos by taking {{nowrap\|''r'' − 1}} elements from the list of target intervals, and adding 2 to the set, leading to an ''r''-element set. To avoid duplications and sets with rank less than ''r'', we can take the [[normal forms\|normal interval list]] defined by each of these ''r''-element sets, discarding any with less than ''r'' elements. This means we are compiling a list of rank-''r'' [[just intonation subgroup]]s, which are [[eigenmonzo subgroup]]s for the corresponding tuning. For each of these these subgroups, we may find a corresponding projection matrix as the matrix with the ''r'' generators of the subgroup as eigenmonzos and a basis for the commas as left eigenvectors with eigenvalue zero. These projection matrices define tunings, and the tuning, if unique, with the least maximum error is the minimax tuning. However, this tuning may not be unique, in which case we may break the tie by discarding the bounding intervals and repeat minimax on the rest of the intervals until a unique tuning is found. This is the same tuning achieved by minimizing the ''p''-norm error as ''p'' approaches infinity<ref group="note">See [[Dave Keenan & Douglas Blumeyer's guide to RTT/Tuning computation #Tie breaking: power limit method]] for details.</ref><ref group="note">Historically, [[Gene Ward Smith]] proposed breaking the tie by falling back to least squares tuning in the set of minimax tunings.</ref>.

	== Example ==		== Example ==
Line 23:		Line 25:

	Starting from the same six intervals of the 5-odd-limit diamond and adding 2, we find after computing the normal interval lists that the three subgroups are [2, 3], [2, 5], and [2, 5/3]. Computing the projection matrix and from thence the tuning in each case, we find that the minimax tuning is [{{monzo\| 1 0 0 }}, {{monzo\| 1 0 1/4 }}, {{monzo\| 0 0 1 }}], the projection matrix for 1/4-comma meantone.		Starting from the same six intervals of the 5-odd-limit diamond and adding 2, we find after computing the normal interval lists that the three subgroups are [2, 3], [2, 5], and [2, 5/3]. Computing the projection matrix and from thence the tuning in each case, we find that the minimax tuning is [{{monzo\| 1 0 0 }}, {{monzo\| 1 0 1/4 }}, {{monzo\| 0 0 1 }}], the projection matrix for 1/4-comma meantone.

			== Notes ==
			<references group="note"/>

	[[Category:Math]]		[[Category:Math]]
	[[Category:Regular temperament tuning]]		[[Category:Regular temperament tuning]]

ArrowHead294: Formatting

2025-06-27T19:21:02Z

Formatting

Sintel: -legacy

2025-03-29T18:11:36Z

-legacy

← Older revision		Revision as of 18:11, 29 March 2025
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	~~{{Legacy}}~~
	By a '''target tuning''' is meant a tuning for a [[regular temperament]] which has been optimized with respect to a set of target [[interval]]s. Usually this set of intervals is derived from a finite set of [[pitch class\|octave-equivalence classes]], and octaves are taken to be pure 2's. Moreover, normally the intervals in question are rational numbers. Below we will make all of these assumptions, and will discuss the two most important target tunings, minimax and least squares.		By a '''target tuning''' is meant a tuning for a [[regular temperament]] which has been optimized with respect to a set of target [[interval]]s. Usually this set of intervals is derived from a finite set of [[pitch class\|octave-equivalence classes]], and octaves are taken to be pure 2's. Moreover, normally the intervals in question are rational numbers. Below we will make all of these assumptions, and will discuss the two most important target tunings, minimax and least squares.

Lériendil at 16:52, 17 February 2025

2025-02-17T16:52:08Z

← Older revision		Revision as of 16:52, 17 February 2025
Line 1:		Line 1:
	{{Legacy~~\|Target_tuning~~}}		{{Legacy}}
	By a '''target tuning''' is meant a tuning for a [[regular temperament]] which has been optimized with respect to a set of target [[interval]]s. Usually this set of intervals is derived from a finite set of [[pitch class\|octave-equivalence classes]], and octaves are taken to be pure 2's. Moreover, normally the intervals in question are rational numbers. Below we will make all of these assumptions, and will discuss the two most important target tunings, minimax and least squares.		By a '''target tuning''' is meant a tuning for a [[regular temperament]] which has been optimized with respect to a set of target [[interval]]s. Usually this set of intervals is derived from a finite set of [[pitch class\|octave-equivalence classes]], and octaves are taken to be pure 2's. Moreover, normally the intervals in question are rational numbers. Below we will make all of these assumptions, and will discuss the two most important target tunings, minimax and least squares.

Lériendil: deploying new cat

2025-02-17T16:15:13Z

deploying new cat

← Older revision		Revision as of 16:15, 17 February 2025
Line 1:		Line 1:
			{{Legacy\|Target_tuning}}
	By a '''target tuning''' is meant a tuning for a [[regular temperament]] which has been optimized with respect to a set of target [[interval]]s. Usually this set of intervals is derived from a finite set of [[pitch class\|octave-equivalence classes]], and octaves are taken to be pure 2's. Moreover, normally the intervals in question are rational numbers. Below we will make all of these assumptions, and will discuss the two most important target tunings, minimax and least squares.		By a '''target tuning''' is meant a tuning for a [[regular temperament]] which has been optimized with respect to a set of target [[interval]]s. Usually this set of intervals is derived from a finite set of [[pitch class\|octave-equivalence classes]], and octaves are taken to be pure 2's. Moreover, normally the intervals in question are rational numbers. Below we will make all of these assumptions, and will discuss the two most important target tunings, minimax and least squares.

Lériendil: Changed protection level for "Target tuning" ([Edit=Allow only administrators] (expires 17:02, 17 February 2025 (UTC)) [Move=Allow only administrators] (expires 17:02, 17 February 2025 (UTC)))

2025-02-17T16:02:26Z

Changed protection level for "Target tuning" ([Edit=Allow only administrators] (expires 17:02, 17 February 2025 (UTC)) [Move=Allow only administrators] (expires 17:02, 17 February 2025 (UTC)))

← Older revision	Revision as of 16:02, 17 February 2025
(No difference)

FloraC: Clarify about the "tonality diamond"; re-link "eigenmonzo"; -typo

2023-09-25T14:41:12Z

Clarify about the "tonality diamond"; re-link "eigenmonzo"; -typo

← Older revision		Revision as of 14:41, 25 September 2023
Line 2:		Line 2:

	== Least squares tunings ==		== Least squares tunings ==
	By assumption we have a finite set of rational intervals defining interval classes, which we may take to be octave reduced to 0 < ''r'' < 1, where the intervals are expressed logarithmically in terms of log base two. The least squares tuning T is the tuning for a particular regular temperament which minimizes the sum of the squares of the errors, ∑<sub>''i''</sub> (T (''q<sub>i</sub>'') - log<sub>2</sub> (''q<sub>i</sub>''))<sup>2</sup>, where ''q<sub>i</sub>'' are the rational intervals of the target set. Note that most commonly, the target set is a [[tonality diamond]], since these are the intervals in a [[harmonic series]] up to some odd integer ''d'', and hence may be considered the [[consonance]]s of the ''d''-odd-limit.		By assumption we have a finite set of rational intervals defining interval classes, which we may take to be octave reduced to 0 < ''r'' < 1, where the intervals are expressed logarithmically in terms of log base two. The least squares tuning T is the tuning for a particular regular temperament which minimizes the sum of the squares of the errors, ∑<sub>''i''</sub> (T (''q<sub>i</sub>'') - log<sub>2</sub> (''q<sub>i</sub>''))<sup>2</sup>, where ''q<sub>i</sub>'' are the rational intervals of the target set. Note that most commonly, the target set is a [[tonality diamond]] (reduced to lowest terms and duplicates removed), since these are the intervals in a [[harmonic series]] up to some odd integer ''d'', and hence may be considered the [[consonance]]s of the ''d''-odd-limit.

	We may find the least squares tuning in various ways, one of which is to start from a matrix R whose rows are the monzos of the target set, and a matrix U whose rows are [[val]]s spanning the temperament. From U we form the matrix V by taking the [[Normal lists\|normal val list]] for U and removing the first ("period") row. A list of [[~~eigenmonzo~~]]s E = VR<sup>T</sup>R, where the <sup>T</sup> denotes the matrix transpose, can now be obtained by matrix multiplication, and to this we add a row for the monzo of 2. The [[Fractional monzos\|projection matrix]] for the least squares tuning is then the square matrix with fractional monzo rows, the rows of E, including 2, as eigenmonzos, that is left eigenvectors with eigenvalue one, and any basis for the commas of the temperament as left eigenvectors with eigenvalues zero.		We may find the least squares tuning in various ways, one of which is to start from a matrix R whose rows are the monzos of the target set, and a matrix U whose rows are [[val]]s spanning the temperament. From U we form the matrix V by taking the [[Normal lists\|normal val list]] for U and removing the first ("period") row. A list of [[Eigenmonzo\|eigenmonzos (unchanged-intervals)]] E = VR<sup>T</sup>R, where the <sup>T</sup> denotes the matrix transpose, can now be obtained by matrix multiplication, and to this we add a row for the monzo of 2. The [[Fractional monzos\|projection matrix]] for the least squares tuning is then the square matrix with fractional monzo rows, the rows of E, including 2, as eigenmonzos, that is left eigenvectors with eigenvalue one, and any basis for the commas of the temperament as left eigenvectors with eigenvalues zero.

	== Minimax tuning ==		== Minimax tuning ==
	Starting with the same assumptions as with least squares tuning, minimax tuning works similarly, except instead of minimizing the squares of the errors, we minimize max<sub>''i''</sub> \|T (''q<sub>i</sub>'') - log<sub>2</sub> (''q<sub>i</sub>'')\|, the maximum error over all the target intervals. This can be solved by setting the minimization up as a [[Wikipedia: Linear programming\|linear programming]] problem, but another approach leads as before to a projection matrix with fractional monzo rows.		Starting with the same assumptions as with least squares tuning, minimax tuning works similarly, except instead of minimizing the squares of the errors, we minimize max<sub>''i''</sub> \|T (''q<sub>i</sub>'') - log<sub>2</sub> (''q<sub>i</sub>'')\|, the maximum error over all the target intervals. This can be solved by setting the minimization up as a [[Wikipedia: Linear programming\|linear programming]] problem, but another approach leads as before to a projection matrix with fractional monzo rows.

	For a rank-''r'' temperament, we can form a list of candidate sets of ~~[[Eigenmonzo\|~~eigenmonzos ~~(unchanged-intervals)]]~~ by taking ''r'' - 1 elements from the list of target intervals, and adding 2 to the set, leading to an ''r''-element set. To avoid duplications and sets with rank less than ''r'', we can take the [[Normal lists\|normal interval list]] defined by each of these ''r''-element sets, discarding any with less than ''r'' elements. This means we are compiling a list of rank-''r'' [[just intonation subgroup]]s, which are [[eigenmonzo subgroup]]s for the corresponding tuning. For each of these these subgroups, we may find a corresponding projection matrix as the matrix with the ''r'' generators of the subgroup as eigenmonzos and a basis for the commas as left eigenvectors with eigenvalue zero. These projection matrices define tunings, and the tuning, if unique, with the least maximum error is the minimax tuning. However, this tuning may not be unique, in which case we may break the tie by choosing the tuning, among the set of least maximum error tunings, with the smallest sum of errors squared.		For a rank-''r'' temperament, we can form a list of candidate sets of eigenmonzos by taking ''r'' - 1 elements from the list of target intervals, and adding 2 to the set, leading to an ''r''-element set. To avoid duplications and sets with rank less than ''r'', we can take the [[Normal lists\|normal interval list]] defined by each of these ''r''-element sets, discarding any with less than ''r'' elements. This means we are compiling a list of rank-''r'' [[just intonation subgroup]]s, which are [[eigenmonzo subgroup]]s for the corresponding tuning. For each of these these subgroups, we may find a corresponding projection matrix as the matrix with the ''r'' generators of the subgroup as eigenmonzos and a basis for the commas as left eigenvectors with eigenvalue zero. These projection matrices define tunings, and the tuning, if unique, with the least maximum error is the minimax tuning. However, this tuning may not be unique, in which case we may break the tie by choosing the tuning, among the set of least maximum error tunings, with the smallest sum of errors squared.

	== Example ==		== Example ==
Line 16:		Line 16:
	R = [{{monzo\| 1 1 -1 }}, {{monzo\| -2 0 1 }}, {{monzo\| 2 -1 0 }}, {{monzo\| -1 1 0 }}, {{monzo\| 3 0 -1 }}, {{monzo\| 0 -1 1 }}]		R = [{{monzo\| 1 1 -1 }}, {{monzo\| -2 0 1 }}, {{monzo\| 2 -1 0 }}, {{monzo\| -1 1 0 }}, {{monzo\| 3 0 -1 }}, {{monzo\| 0 -1 1 }}]

	Then R<sup>T</sup>R = [[19 -2 -6], [-2 4 -2], [-6 -2 4]], a positive-definite symmetric matrix. If U = [{{val\| 1 0 -4 }}, {{val\| 0 1 4 }}], we may remove the top row and obtain V = [{{val\| 0 1 4 }}]; then VR<sup>T</sup>R is [{{monzo\| -26 -4 14 }}]; using this along with {{monzo\| 1 0 0 }} as eigenmonzos, and using {{monzo\| -4 4 -1 }} an a commatic eigenvector with eigenvalue zero, we obtain		Then R<sup>T</sup>R = [[19 -2 -6], [-2 4 -2], [-6 -2 4]], a positive-definite symmetric matrix. If U = [{{val\| 1 0 -4 }}, {{val\| 0 1 4 }}], we may remove the top row and obtain V = [{{val\| 0 1 4 }}]; then VR<sup>T</sup>R is [{{monzo\| -26 -4 14 }}]; using this along with {{monzo\| 1 0 0 }} as eigenmonzos, and using {{monzo\| -4 4 -1 }} as a commatic eigenvector with eigenvalue zero, we obtain

	[{{monzo\| 1 0 0 }}, {{monzo\| 14/13 -1/13 7/26 }}, {{monzo\| 4/13 -4/13 14/13 }}]		[{{monzo\| 1 0 0 }}, {{monzo\| 14/13 -1/13 7/26 }}, {{monzo\| 4/13 -4/13 14/13 }}]

FloraC: Fix some "bugs"; linking; recategorize

2023-09-25T14:24:21Z

Fix some "bugs"; linking; recategorize

← Older revision		Revision as of 14:24, 25 September 2023
Line 1:		Line 1:
	By a '''target tuning''' is meant a tuning for a regular temperament which has been optimized with respect to a set of target ~~intervals~~. Usually this set of intervals is derived from a finite set of octave equivalence classes, and octaves are taken to be pure 2s. Moreover, normally the intervals in question are rational numbers. Below we will make all of these assumptions, and will discuss the two most important target tunings, minimax and least squares.		By a '''target tuning''' is meant a tuning for a [[regular temperament]] which has been optimized with respect to a set of target [[interval]]s. Usually this set of intervals is derived from a finite set of [[pitch class\|octave-equivalence classes]], and octaves are taken to be pure 2's. Moreover, normally the intervals in question are rational numbers. Below we will make all of these assumptions, and will discuss the two most important target tunings, minimax and least squares.

	== Least squares tunings ==		== Least squares tunings ==
	By assumption we have a finite set of rational intervals defining interval classes, which we may take to be octave reduced to 0 < ''r'' < 1, where the intervals are expressed logarithmically in terms of log base two. The least squares tuning T is the tuning for a particular regular temperament which minimizes the sum of the squares of the errors, ∑<sub>''i''</sub> (T (''q<sub>i</sub>'') - log<sub>2</sub> (''q<sub>i</sub>''))<sup>2</sup>, where ''q<sub>i</sub>'' are the rational intervals of the target set. Note that most commonly, the target set is a tonality diamond, since these are the intervals in ~~an overtone~~ series up to some odd integer ''d'', and hence may be considered the ~~consonances~~ of the ''d''-odd-limit.		By assumption we have a finite set of rational intervals defining interval classes, which we may take to be octave reduced to 0 < ''r'' < 1, where the intervals are expressed logarithmically in terms of log base two. The least squares tuning T is the tuning for a particular regular temperament which minimizes the sum of the squares of the errors, ∑<sub>''i''</sub> (T (''q<sub>i</sub>'') - log<sub>2</sub> (''q<sub>i</sub>''))<sup>2</sup>, where ''q<sub>i</sub>'' are the rational intervals of the target set. Note that most commonly, the target set is a [[tonality diamond]], since these are the intervals in a [[harmonic series]] up to some odd integer ''d'', and hence may be considered the [[consonance]]s of the ''d''-odd-limit.

	We may find the least squares tuning in various ways, one of which is to start from a matrix R whose rows are the monzos of the target set, and a matrix U whose rows are ~~vals~~ spanning the temperament. From U we form the matrix V by taking the [[Normal lists\|normal val list]] for U and removing the first ("period") row. A list of [[eigenmonzo]]s E = VR<sup>T</sup>R, where the T denotes the matrix transpose, can now be obtained by matrix multiplication, and to this we add a row for the monzo of 2. The [[Fractional monzos\|projection matrix]] for the least squares tuning is then the square matrix with fractional monzo rows, the rows of E, including 2, as eigenmonzos, that is left eigenvectors with eigenvalue one, and any basis for the commas of the temperament as left eigenvectors with eigenvalues zero.		We may find the least squares tuning in various ways, one of which is to start from a matrix R whose rows are the monzos of the target set, and a matrix U whose rows are [[val]]s spanning the temperament. From U we form the matrix V by taking the [[Normal lists\|normal val list]] for U and removing the first ("period") row. A list of [[eigenmonzo]]s E = VR<sup>T</sup>R, where the <sup>T</sup> denotes the matrix transpose, can now be obtained by matrix multiplication, and to this we add a row for the monzo of 2. The [[Fractional monzos\|projection matrix]] for the least squares tuning is then the square matrix with fractional monzo rows, the rows of E, including 2, as eigenmonzos, that is left eigenvectors with eigenvalue one, and any basis for the commas of the temperament as left eigenvectors with eigenvalues zero.

	== Minimax tuning ==		== Minimax tuning ==
	Starting with the same assumptions as with least squares tuning, minimax tuning works similarly, except instead of minimizing the squares of the errors, we minimize max<sub>''i''</sub> \|T (''q<sub>i</sub>'') - log<sub>2</sub> (''q<sub>i</sub>'')\|, the maximum error over all the target intervals. This can be solved by setting the minimization up as a [[Wikipedia: Linear programming\|linear programming]] problem, but another approach leads as before to a projection matrix with fractional monzo rows.		Starting with the same assumptions as with least squares tuning, minimax tuning works similarly, except instead of minimizing the squares of the errors, we minimize max<sub>''i''</sub> \|T (''q<sub>i</sub>'') - log<sub>2</sub> (''q<sub>i</sub>'')\|, the maximum error over all the target intervals. This can be solved by setting the minimization up as a [[Wikipedia: Linear programming\|linear programming]] problem, but another approach leads as before to a projection matrix with fractional monzo rows.

	For a rank ''r'' temperament, we can form a list of candidate sets of eigenmonzos by taking ''r'' - 1 elements from the list of target intervals, and adding 2 to the set, leading to an ''r'' element set. To avoid duplications and sets with rank less than ''r'', we can take the [[Normal lists\|normal interval list]] defined by each of these ''r'' element sets, discarding any with less than ''r'' elements. This means we are compiling a list of rank ''r'' [[just intonation ~~subgroups~~]], which are [[eigenmonzo subgroup]]s for the corresponding tuning. For each of these these subgroups, we may find a corresponding projection matrix as the matrix with the ''r'' generators of the subgroup as eigenmonzos and a basis for the commas as left eigenvectors with eigenvalue zero. These projection matrices define tunings, and the tuning, if unique, with the least maximum error is the minimax tuning. However, this tuning may not be unique, in which case we may break the tie by choosing the tuning, among the set of least maximum error tunings, with the smallest sum of errors squared.		For a rank-''r'' temperament, we can form a list of candidate sets of [[Eigenmonzo\|eigenmonzos (unchanged-intervals)]] by taking ''r'' - 1 elements from the list of target intervals, and adding 2 to the set, leading to an ''r''-element set. To avoid duplications and sets with rank less than ''r'', we can take the [[Normal lists\|normal interval list]] defined by each of these ''r''-element sets, discarding any with less than ''r'' elements. This means we are compiling a list of rank-''r'' [[just intonation subgroup]]s, which are [[eigenmonzo subgroup]]s for the corresponding tuning. For each of these these subgroups, we may find a corresponding projection matrix as the matrix with the ''r'' generators of the subgroup as eigenmonzos and a basis for the commas as left eigenvectors with eigenvalue zero. These projection matrices define tunings, and the tuning, if unique, with the least maximum error is the minimax tuning. However, this tuning may not be unique, in which case we may break the tie by choosing the tuning, among the set of least maximum error tunings, with the smallest sum of errors squared.

	== Example ==		== Example ==
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	R = [{{monzo\| 1 1 -1 }}, {{monzo\| -2 0 1 }}, {{monzo\| 2 -1 0 }}, {{monzo\| -1 1 0 }}, {{monzo\| 3 0 -1 }}, {{monzo\| 0 -1 1 }}]		R = [{{monzo\| 1 1 -1 }}, {{monzo\| -2 0 1 }}, {{monzo\| 2 -1 0 }}, {{monzo\| -1 1 0 }}, {{monzo\| 3 0 -1 }}, {{monzo\| 0 -1 1 }}]

	Then R<sup>T</sup>R = [[19 -2 -6], [-2 4 -2], [-6 -2 4]], a positive-definite symmetric matrix. If U = [{{val\| 1 0 -9 }}, {{val\| 0 1 4 }}], we may remove the top row and obtain V = [{{val\| 0 1 4 }}]; then VR<sup>T</sup>R is [{{monzo\| -26 -4 14 }}]; using this along with {{monzo\| 1 0 0 }} as eigenmonzos, and using {{monzo\| -4 4 -1 }} an a commatic eigenvector with eigenvalue zero, we obtain		Then R<sup>T</sup>R = [[19 -2 -6], [-2 4 -2], [-6 -2 4]], a positive-definite symmetric matrix. If U = [{{val\| 1 0 -4 }}, {{val\| 0 1 4 }}], we may remove the top row and obtain V = [{{val\| 0 1 4 }}]; then VR<sup>T</sup>R is [{{monzo\| -26 -4 14 }}]; using this along with {{monzo\| 1 0 0 }} as eigenmonzos, and using {{monzo\| -4 4 -1 }} an a commatic eigenvector with eigenvalue zero, we obtain

	[{{monzo\| 1 0 0 }}, {{monzo\| 14/13 -1/13 7/26 }}, {{monzo\| 4/13 -4/13 14/13 }}]		[{{monzo\| 1 0 0 }}, {{monzo\| 14/13 -1/13 7/26 }}, {{monzo\| 4/13 -4/13 14/13 }}]
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	[[Category:Math]]		[[Category:Math]]
	[[Category:~~Tuning~~]]		[[Category:Regular temperament tuning]]

← Older revision		Revision as of 19:21, 27 June 2025
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	== Least squares tunings ==		== Least squares tunings ==
	By assumption we have a finite set of rational intervals defining interval classes, which we may take to be octave reduced to 0 < ''r'' < 1, where the intervals are expressed logarithmically in terms of log base two. The least squares tuning T is the tuning for a particular regular temperament which minimizes the sum of the squares of the errors, ∑<sub>''i''</sub> (T (''q<sub>i</sub>'') - log<sub>2</sub> (''q<sub>i</sub>''))<sup>2</sup>, where ''q<sub>i</sub>'' are the rational intervals of the target set. Note that most commonly, the target set is a [[tonality diamond]] (reduced to lowest terms and duplicates removed), since these are the intervals in a [[harmonic series]] up to some odd integer ''d'', and hence may be considered the [[consonance]]s of the ''d''-odd-limit.		By assumption we have a finite set of rational intervals defining interval classes, which we may take to be octave reduced to {{nowrap\|0 < ''r'' < 1}}, where the intervals are expressed logarithmically in terms of log base two. The least squares tuning, ''T'', is the tuning for a particular regular temperament which minimizes the sum of the squares of the errors, ∑<sub>''i''</sub> ({{nowrap\|''T''(''q<sub>i</sub>'') − log<sub>2</sub>(''q<sub>i</sub>''))<sup>2</sup>}}, where ''q<sub>i</sub>'' are the rational intervals of the target set. Note that most commonly, the target set is a [[tonality diamond]] (reduced to lowest terms and duplicates removed), since these are the intervals in a [[harmonic series]] up to some odd integer ''d'', and hence may be considered the [[consonance]]s of the ''d''-odd-limit.

	We may find the least squares tuning in various ways, one of which is to start from a matrix R whose rows are the monzos of the target set, and a matrix U whose rows are [[val]]s spanning the temperament. From U we form the matrix V by taking the [[Normal lists\|normal val list]] for U and removing the first ("period") row. A list of [[Eigenmonzo\|eigenmonzos (unchanged-intervals)]] E = VR<sup>T</sup>R, where the <sup>T</sup> denotes the matrix transpose, can now be obtained by matrix multiplication, and to this we add a row for the monzo of 2. The [[Fractional monzos\|projection matrix]] for the least squares tuning is then the square matrix with fractional monzo rows, the rows of E, including 2, as eigenmonzos, that is left eigenvectors with eigenvalue one, and any basis for the commas of the temperament as left eigenvectors with eigenvalues zero.		We may find the least squares tuning in various ways, one of which is to start from a matrix '''R''' whose rows are the monzos of the target set, and a matrix '''U''' whose rows are [[val]]s spanning the temperament. From '''U''' we form the matrix '''V''' by taking the [[Normal lists\|normal val list]] for '''U''' and removing the first ("period") row. A list of [[Eigenmonzo\|eigenmonzos (unchanged-intervals)]] {{nowrap\|'''E'''' {{=}} '''VR'''<sup>T</sup>'''R'''}}, where the <sup>T</sup> denotes the matrix transpose, can now be obtained by matrix multiplication, and to this we add a row for the monzo of 2. The [[Fractional monzos\|projection matrix]] for the least squares tuning is then the square matrix with fractional monzo rows, the rows of '''E''', including 2, as eigenmonzos, that is left eigenvectors with eigenvalue one, and any basis for the commas of the temperament as left eigenvectors with eigenvalues zero.

	== Minimax tuning ==		== Minimax tuning ==
	Starting with the same assumptions as with least squares tuning, minimax tuning works similarly, except instead of minimizing the squares of the errors, we minimize max<sub>''i''</sub> \|T (''q<sub>i</sub>'') - log<sub>2</sub> (''q<sub>i</sub>'')\|, the maximum error over all the target intervals. This can be solved by setting the minimization up as a ~~[[Wikipedia: Linear programming~~\|linear programming]] problem, but another approach leads as before to a projection matrix with fractional monzo rows.		Starting with the same assumptions as with least squares tuning, minimax tuning works similarly, except instead of minimizing the squares of the errors, we minimize {{nowrap\|max<sub>''i''</sub>{{!}}''T''(''q<sub>i</sub>'') − log<sub>2</sub>(''q<sub>i</sub>''){{!}} }}, the maximum error over all the target intervals. This can be solved by setting the minimization up as a {{w\|linear programming}} problem, but another approach leads as before to a projection matrix with fractional monzo rows.

	For a rank-''r'' temperament, we can form a list of candidate sets of eigenmonzos by taking ''r'' - 1 elements from the list of target intervals, and adding 2 to the set, leading to an ''r''-element set. To avoid duplications and sets with rank less than ''r'', we can take the [[Normal lists\|normal interval list]] defined by each of these ''r''-element sets, discarding any with less than ''r'' elements. This means we are compiling a list of rank-''r'' [[just intonation subgroup]]s, which are [[eigenmonzo subgroup]]s for the corresponding tuning. For each of these these subgroups, we may find a corresponding projection matrix as the matrix with the ''r'' generators of the subgroup as eigenmonzos and a basis for the commas as left eigenvectors with eigenvalue zero. These projection matrices define tunings, and the tuning, if unique, with the least maximum error is the minimax tuning. However, this tuning may not be unique, in which case we may break the tie by choosing the tuning, among the set of least maximum error tunings, with the smallest sum of errors squared.		For a rank-''r'' temperament, we can form a list of candidate sets of eigenmonzos by taking {{nowrap\|''r'' − 1}} elements from the list of target intervals, and adding 2 to the set, leading to an ''r''-element set. To avoid duplications and sets with rank less than ''r'', we can take the [[Normal lists\|normal interval list]] defined by each of these ''r''-element sets, discarding any with less than ''r'' elements. This means we are compiling a list of rank-''r'' [[just intonation subgroup]]s, which are [[eigenmonzo subgroup]]s for the corresponding tuning. For each of these these subgroups, we may find a corresponding projection matrix as the matrix with the ''r'' generators of the subgroup as eigenmonzos and a basis for the commas as left eigenvectors with eigenvalue zero. These projection matrices define tunings, and the tuning, if unique, with the least maximum error is the minimax tuning. However, this tuning may not be unique, in which case we may break the tie by choosing the tuning, among the set of least maximum error tunings, with the smallest sum of errors squared.

	== Example ==		== Example ==
	Suppose our set of target intervals is the 5-odd-limit diamond {6/5, 5/4, 4/3, 3/2, 8/5, 5/3}. The corresponding matrix is		Suppose our set of target intervals is the 5-odd-limit diamond {{nowrap\|{6/5, 5/4, 4/3, 3/2, 8/5, 5/3}<nowiki />}}. The corresponding matrix is

	R = [{{monzo\| 1 1 -1 }}, {{monzo\| -2 0 1 }}, {{monzo\| 2 -1 0 }}, {{monzo\| -1 1 0 }}, {{monzo\| 3 0 -1 }}, {{monzo\| 0 -1 1 }}]		'''R''' = [{{monzo\| 1 1 -1 }}, {{monzo\| -2 0 1 }}, {{monzo\| 2 -1 0 }}, {{monzo\| -1 1 0 }}, {{monzo\| 3 0 -1 }}, {{monzo\| 0 -1 1 }}]

	Then R<sup>T</sup>R = [[19 -2 -6], [-2 4 -2], [-6 -2 4]], a positive-definite symmetric matrix. If U = [{{val\| 1 0 -4 }}, {{val\| 0 1 4 }}], we may remove the top row and obtain V = [{{val\| 0 1 4 }}]; then VR<sup>T</sup>R is [{{monzo\| -26 -4 14 }}]; using this along with {{monzo\| 1 0 0 }} as eigenmonzos, and using {{monzo\| -4 4 -1 }} as a commatic eigenvector with eigenvalue zero, we obtain		Then {{nowrap\|'''R'''<sup>T</sup>'''R''' {{=}} [[19 -2 -6], [-2 4 -2], [-6 -2 4]]}}, a positive-definite symmetric matrix. If {{nowrap\|'''U''' {{=}} [{{val\| 1 0 -4 }}, {{val\| 0 1 4 }}]<nowiki />}}, we may remove the top row and obtain {{nowrap\|'''V''' {{=}} [{{val\| 0 1 4 }}]<nowiki />}}; then {{nowrap\|'''VR'''<sup>T</sup>'''R''' {{=}} [{{monzo\| -26 -4 14 }}]<nowiki />}}; using this along with {{monzo\| 1 0 0 }} as eigenmonzos, and using {{monzo\| -4 4 -1 }} as a commatic eigenvector with eigenvalue zero, we obtain

	[{{monzo\| 1 0 0 }}, {{monzo\| 14/13 -1/13 7/26 }}, {{monzo\| 4/13 -4/13 14/13 }}]		[{{monzo\| 1 0 0 }}, {{monzo\| 14/13 -1/13 7/26 }}, {{monzo\| 4/13 -4/13 14/13 }}]
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	which is the projection matrix for the Woolhouse 7/26-comma meantone, the 5-limit least squares tuning.		which is the projection matrix for the Woolhouse 7/26-comma meantone, the 5-limit least squares tuning.

	Starting from the same six intervals of the 5-odd-limit diamond and adding 2, we find after computing the normal interval lists that the three subgroups are [2, 3], [2, 5], and [2, 5/3]. Computing the projection matrix and from thence the tuning in each case, we find that the minimax tuning is [{{monzo\| 1 0 0 }}, {{monzo\| 1 0 1/4 }}, {{monzo\| 0 0 1 }}], the projection matrix for 1/4-comma meantone.		Starting from the same six intervals of the 5-odd-limit diamond and adding 2, we find after computing the normal interval lists that the three subgroups are [2, 3], [2, 5], and [2, 5/3]. Computing the projection matrix and from thence the tuning in each case, we find that the minimax tuning is [{{monzo\| 1 0 0 }}, {{monzo\| 1 0 1/4 }}, {{monzo\| 0 0 1 }}], the projection matrix for 1/4-comma meantone.

	[[Category:Math]]		[[Category:Math]]
	[[Category:Regular temperament tuning]]		[[Category:Regular temperament tuning]]