User:FloraC/Hard problems of harmony and psychoacoustically supported optimization - Revision history

FloraC: In this version I make it explicit that Euclidean norms are better use of optimizational resource

2025-06-22T11:02:33Z

In this version I make it explicit that Euclidean norms are better use of optimizational resource

FloraC: According to Mike, "Hahn" is skewed in the first place, so I'm claiming the unskewed variant myself

2024-04-18T12:14:33Z

According to Mike, "Hahn" is skewed in the first place, so I'm claiming the unskewed variant myself

← Older revision		Revision as of 12:14, 18 April 2024
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	== Chapter V. Towards an Optimization Strategy ==		== Chapter V. Towards an Optimization Strategy ==
	Incorporating all that have been discussed above, I recommend CTE tuning as the best general-purpose reference solution to everyone, whereas my hemi-idiosyncratic answer to tuning optimization is the '''CH<sub>''n''</sub>E tuning''' (for '''constrained ~~Hahn~~[''n'']-Euclidean tuning''').		Incorporating all that have been discussed above, I recommend CTE tuning as the best general-purpose reference solution to everyone, whereas my hemi-idiosyncratic answer to tuning optimization is based on a meticulously engineered weight function, which happens to be an unskewed version of the Hahn distance. Let us dub this the Canou[''n''] weight, and the tuning using this weight the '''CC<sub>''n''</sub>E tuning''' (for '''constrained Canou[''n'']-Euclidean tuning''').

	~~The only part that needs explanation is the Hahn[''n'']~~ weight, ~~an adaptation to~~ the ~~original Hahn distance.~~		In this weight, the ''n'' is a positive integer determining the highest relevant harmonic. The weight of any prime harmonic equals its maximum number of stacks without exceeding the ''n''-integer-limit. Different values of ''n'' can alter the relative weights of the primes.

	~~The~~ ''n'' is a positive integer determining the highest relevant harmonic. ~~Specifically, the~~ weight of any prime harmonic equals its maximum number of stacks without exceeding the ''n''-integer-limit~~, so different~~ values of ''n'' can alter the relative weights of the primes.

	To illustrate, let us set ''n'' = 9, or 9-integer-limit. Harmonic 2 can be stacked thrice, giving 8. Stacking it four times would give 16, exceeding 9. Its weight is thus 3. Harmonic 3 can be stacked twice, giving 9. Stacking it three times would give 27, exceeding 9. Its weight is thus 2. Both 5 and 7 have unit weight since they can only be stacked once in the integer limit. 11 and beyond have zero weight because they cannot be stacked at all. If optimization is to be carried out for a 13-limit temperament then we have the weights 3, 2, 1, 1, 0, 0 for primes 2 to 13. The weights are different if ''n'' = 7, or 7-integer-limit, for example. The weight of 2 is 2, of 3, 5 and 7 is unity, and of 11 and 13 zero, giving 2, 1, 1, 1, 0, 0 for primes 2 to 13.		To illustrate, let us set ''n'' = 9, or 9-integer-limit. Harmonic 2 can be stacked thrice, giving 8. Stacking it four times would give 16, exceeding 9. Its weight is thus 3. Harmonic 3 can be stacked twice, giving 9. Stacking it three times would give 27, exceeding 9. Its weight is thus 2. Both 5 and 7 have unit weight since they can only be stacked once in the integer limit. 11 and beyond have zero weight because they cannot be stacked at all. If optimization is to be carried out for a 13-limit temperament then we have the weights 3, 2, 1, 1, 0, 0 for primes 2 to 13. The weights are different if ''n'' = 7, or 7-integer-limit, for example. The weight of 2 is 2, of 3, 5 and 7 is unity, and of 11 and 13 zero, giving 2, 1, 1, 1, 0, 0 for primes 2 to 13.

	The ~~Hahn~~[''n''] weight matrix is given as		The Canou[''n''] weight matrix is given as

	$$		$$
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	which indicates that the prime ''q'' in ''Q'' has the weight equal to floor (log<sub>''q''</sub> (''n'')).		which indicates that the prime ''q'' in ''Q'' has the weight equal to floor (log<sub>''q''</sub> (''n'')).

	The Tenney weight is a special case of the ~~Hahn~~[''n''] weight, where ''n'' → infinity. The only thing that sets ~~Hahn~~[''n''] apart from Tenney is the floor function (since log<sub>''Q''</sub> (''n'') = log<sub>2</sub> (''n'')/log<sub>2</sub> (''Q'') and log<sub>2</sub> (''n'') is a constant), and its effect converges to zero as ''n'' gets sufficiently large. Conceptualizing the Tenney weight in this way is not recommended, though, because Tenney's is characteristically transcendental whereas all the other ~~Hahn~~[''n''] weights are algebraic.		The Tenney weight is a special case of the Canou[''n''] weight, where ''n'' → infinity. The only thing that sets Canou[''n''] apart from Tenney is the floor function (since log<sub>''Q''</sub> (''n'') = log<sub>2</sub> (''n'')/log<sub>2</sub> (''Q'') and log<sub>2</sub> (''n'') is a constant), and its effect converges to zero as ''n'' gets sufficiently large. Conceptualizing the Tenney weight in this way is not recommended, though, because Tenney's is characteristically transcendental whereas all the other Canou[''n''] weights are algebraic.

	That defines the H<sub>''n''</sub>C, H<sub>''n''</sub>E, and H<sub>''n''</sub>OP tunings, but if we contrain the octave to pure, it does not matter how many times the octave is stacked, making the integer limit equivalent to the smaller closest odd limit. The proposed convention is to always use the largest number ''n'' if multiple consecutive choices of ''n'' will give the same CH<sub>''n''</sub>E tuning. For example, CH<sub>13</sub>E, CH<sub>14</sub>E, CH<sub>15</sub>E, and CH<sub>16</sub>E are all equivalent and one should always write CH<sub>16</sub>E.		That defines the C<sub>''n''</sub>C, C<sub>''n''</sub>E, and C<sub>''n''</sub>OP tunings, but if we contrain the octave to pure, it does not matter how many times the octave is stacked, making the integer limit equivalent to the smaller closest odd limit. The proposed convention is to always use the largest number ''n'' if multiple consecutive choices of ''n'' will give the same CC<sub>''n''</sub>E tuning. For example, CC<sub>13</sub>E, CC<sub>14</sub>E, CC<sub>15</sub>E, and CC<sub>16</sub>E are all equivalent and one should always write CC<sub>16</sub>E.

	Specifically designed to ''my'' taste, another special case of note is setting ''n'' = 24, or ~~Hahn~~[24]. The entries are 4, 2, 1, 1, 1, 1, 1, 1, 1 for primes 2 to 23, and primes beyond 23 are never optimized for. The octave matters not, so you can see its only difference from the equilateral weight is that not 3 but 9 is treated as a prime, meaning every two steps along the path of 3 counts as one.		Specifically designed to ''my'' taste, another special case of note is setting ''n'' = 24, or Canou[24]. This shall be the default ''n''. The entries are 4, 2, 1, 1, 1, 1, 1, 1, 1 for primes 2 to 23, and primes beyond 23 are never optimized for. The octave matters not, so you can see its only difference from the equilateral weight is that not 3 but 9 is treated as a prime, meaning every two steps along the path of 3 counts as one.

	Let us tune some temperaments!		Let us tune some temperaments!
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	{\| class="wikitable"		{\| class="wikitable"
	\|-		\|-
	! Temperament !! Error Map (CTE) !! Error Map (CH<sub>24</sub>E)		! Temperament !! Error Map (CTE) !! Error Map (CC<sub>24</sub>E)
	\|-		\|-
	\| Meantone, 5-limit \|\| {{val\| 0 −4.7407 +2.5436 }} \|\| {{val\| 0 −4.3013 +4.3013 }}		\| Meantone, 5-limit \|\| {{val\| 0 −4.7407 +2.5436 }} \|\| {{val\| 0 −4.3013 +4.3013 }}
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	== Release Notes ==		== Release Notes ==
	© ~~2023~~ Flora Canou		© 2023–2024 Flora Canou

	Version Stable 3		Version Stable 4

	This work is licensed under the [https://creativecommons.org/licenses/by-sa/4.0/ Creative Commons Attribution-ShareAlike 4.0 International License].		This work is licensed under the [https://creativecommons.org/licenses/by-sa/4.0/ Creative Commons Attribution-ShareAlike 4.0 International License].

FloraC: Style and misc., bump the version number for all the small changes

2023-12-26T07:15:47Z

Style and misc., bump the version number for all the small changes

← Older revision		Revision as of 07:15, 26 December 2023
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	$$		$$
	W = \operatorname {diag} (1/\log_2 (Q))		W = \operatorname {diag} \left( 1/\log_2 (Q) \right)
	$$		$$

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	$$		$$
	W = \operatorname {diag} (\left\langle \begin{matrix} w_1 & \sqrt{2} & 1 \end{matrix} \right])		W = \operatorname {diag} \left( \left\langle \begin{matrix} w_1 & \sqrt{2} & 1 \end{matrix} \right] \right)
	$$		$$

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	$$		$$
	W = \operatorname {diag} (\operatorname {floor} (\log_Q (n)))		W = \operatorname {diag} \left( \operatorname {floor} \left( \log_Q (n) \right) \right)
	$$		$$

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	! Temperament !! Error Map (CTE) !! Error Map (CH<sub>24</sub>E)		! Temperament !! Error Map (CTE) !! Error Map (CH<sub>24</sub>E)
	\|-		\|-
	\| 5-limit ~~meantone~~ \|\| {{val\| 0 −4.7407 +2.5436 }} \|\| {{val\| 0 −4.3013 +4.3013 }}		\| Meantone, 5-limit \|\| {{val\| 0 −4.7407 +2.5436 }} \|\| {{val\| 0 −4.3013 +4.3013 }}
	\|-		\|-
	\| 7-limit ~~meantone~~ \|\| {{val\| 0 −5.0029 +1.4948 +0.6955 }} \|\| {{val\| 0 −4.9439 +1.7308 +1.2853 }}		\| Meantone, 7-limit \|\| {{val\| 0 −5.0029 +1.4948 +0.6955 }} \|\| {{val\| 0 −4.9439 +1.7308 +1.2853 }}
	\|-		\|-
	\| 2.3.7 ~~superpyth~~ \|\| {{val\| 0 +7.6398 +11.9845 }} \|\| {{val\| 0 +6.8160 +13.6320 }}		\| Superpyth, 2.3.7 subgroup \|\| {{val\| 0 +7.6398 +11.9845 }} \|\| {{val\| 0 +6.8160 +13.6320 }}
	\|-		\|-
	\| 7-limit ~~superpyth~~ \|\| {{val\| 0 +7.6357 +0.0023 +11.9928 }} \|\| {{val\| 0 +7.5618 −0.6629 +12.1406 }}		\| Superpyth, 7-limit \|\| {{val\| 0 +7.6357 +0.0023 +11.9928 }} \|\| {{val\| 0 +7.5618 −0.6629 +12.1406 }}
	\|-		\|-
	\| 11-limit ~~sensamagic~~ \|\| {{val\| 0 +1.8187 −0.7280 −2.1970 +0.2160 }} \|\| {{val\| 0 +1.6927 −0.9366 −2.3952 +0.5219 }}		\| Sensamagic, 11-limit \|\| {{val\| 0 +1.8187 −0.7280 −2.1970 +0.2160 }} \|\| {{val\| 0 +1.6927 −0.9366 −2.3952 +0.5219 }}
	\|-		\|-
	\| 13-limit ~~marvel (hecate)~~ \|\| {{val\| 0 −0.3917 −3.2984 +0.3314 −1.9273 −0.3053 }} \|\| {{val\| 0 −0.3922 −3.5071 −0.0870 −1.3006 +0.1105 }}		\| Marvel (hecate), 13-limit \|\| {{val\| 0 −0.3917 −3.2984 +0.3314 −1.9273 −0.3053 }} \|\| {{val\| 0 −0.3922 −3.5071 −0.0870 −1.3006 +0.1105 }}
	\|-		\|-
	\| 13-limit ~~pele~~ \|\| {{val\| 0 +1.4848 +0.5796 −2.5714 +1.1774 +1.6483 }} \|\| {{val\| 0 +1.5434 +0.7961 −2.7066 +0.8077 +1.1028 }}		\| Pele, 13-limit \|\| {{val\| 0 +1.4848 +0.5796 −2.5714 +1.1774 +1.6483 }} \|\| {{val\| 0 +1.5434 +0.7961 −2.7066 +0.8077 +1.1028 }}
	\|}		\|}
	</center>		</center>
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	© 2023 Flora Canou		© 2023 Flora Canou

	Version Stable 2		Version Stable 3

	This work is licensed under the [https://creativecommons.org/licenses/by-sa/4.0/ Creative Commons Attribution-ShareAlike 4.0 International License].		This work is licensed under the [https://creativecommons.org/licenses/by-sa/4.0/ Creative Commons Attribution-ShareAlike 4.0 International License].

FloraC: Style

2023-12-24T15:45:23Z

Style

← Older revision		Revision as of 15:45, 24 December 2023
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	! Temperament !! Error Map (CTE) !! Error Map (CH<sub>24</sub>E)		! Temperament !! Error Map (CTE) !! Error Map (CH<sub>24</sub>E)
	\|-		\|-
	\| 5-limit meantone \|\| {{val\| 0 -4.7407 +2.5436 }} \|\| {{val\| 0 -4.3013 +4.3013 }}		\| 5-limit meantone \|\| {{val\| 0 −4.7407 +2.5436 }} \|\| {{val\| 0 −4.3013 +4.3013 }}
	\|-		\|-
	\| 7-limit meantone \|\| {{val\| 0 -5.0029 +1.4948 +0.6955 }} \|\| {{val\| 0 -4.9439 +1.7308 +1.2853 }}		\| 7-limit meantone \|\| {{val\| 0 −5.0029 +1.4948 +0.6955 }} \|\| {{val\| 0 −4.9439 +1.7308 +1.2853 }}
	\|-		\|-
	\| 2.3.7 superpyth \|\| {{val\| 0 +7.6398 +11.9845 }} \|\| {{val\| 0 +6.8160 +13.6320 }}		\| 2.3.7 superpyth \|\| {{val\| 0 +7.6398 +11.9845 }} \|\| {{val\| 0 +6.8160 +13.6320 }}
	\|-		\|-
	\| 7-limit superpyth \|\| {{val\| 0 +7.6357 +0.0023 +11.9928 }} \|\| {{val\| 0 +7.5618 -0.6629 +12.1406 }}		\| 7-limit superpyth \|\| {{val\| 0 +7.6357 +0.0023 +11.9928 }} \|\| {{val\| 0 +7.5618 −0.6629 +12.1406 }}
	\|-		\|-
	\| 11-limit sensamagic \|\| {{val\| 0 +1.8187 -0.7280 -2.1970 +0.2160 }} \|\| {{val\| 0 +1.6927 -0.9366 -2.3952 +0.5219 }}		\| 11-limit sensamagic \|\| {{val\| 0 +1.8187 −0.7280 −2.1970 +0.2160 }} \|\| {{val\| 0 +1.6927 −0.9366 −2.3952 +0.5219 }}
	\|-		\|-
	\| 13-limit marvel (hecate) \|\| {{val\| 0 -0.3917 -3.2984 +0.3314 -1.9273 -0.3053 }} \|\| {{val\| 0 -0.3922 -3.5071 -0.0870 -1.3006 +0.1105 }}		\| 13-limit marvel (hecate) \|\| {{val\| 0 −0.3917 −3.2984 +0.3314 −1.9273 −0.3053 }} \|\| {{val\| 0 −0.3922 −3.5071 −0.0870 −1.3006 +0.1105 }}
	\|-		\|-
	\| 13-limit pele \|\| {{val\| 0 +1.4848 +0.5796 -2.5714 +1.1774 +1.6483 }} \|\| {{val\| 0 +1.5434 +0.7961 -2.7066 +0.8077 +1.1028 }}		\| 13-limit pele \|\| {{val\| 0 +1.4848 +0.5796 −2.5714 +1.1774 +1.6483 }} \|\| {{val\| 0 +1.5434 +0.7961 −2.7066 +0.8077 +1.1028 }}
	\|}		\|}
	</center>		</center>

FloraC: Another typo and style fix

2023-12-24T15:41:22Z

Another typo and style fix

← Older revision		Revision as of 15:41, 24 December 2023
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	== Chapter III. Power in Proportion ==		== Chapter III. Power in Proportion ==
	The first ever attempt at a systematic tuning solution was Paul Erlich's TOP tuning<ref>"All-Interval Tuning Schemes", ''Dave Keenan & Douglas Blumeyer's Guide to RTT''. Dave Keenan and Douglas Blumeyer. Xenharmonic Wiki. </ref>. This tuning was elegantly explained in his ''Middle Path'' paper in the case of nullity-1 (i.e. single-comma temperaments)<ref>"A Middle Path between Just Intonation and the Equal Temperaments – Part 1", ''Xenharmonikôn, An Informal Journal of Experimental Music''. Paul Erlich. </ref>. In this tuning, every prime makes an effort in the right direction to close out the comma. To illustrate, consider 5-limit meantone, and to simplify it even more, let us start with the constrained equilateral-optimal tuning (CEOP tuning) instead since its effect is the easiest to observe. The CEOP tuning of 5-limit meantone is given in terms of the projection map P as		The first ever attempt at a systematic tuning solution was Paul Erlich's TOP tuning<ref>"All-Interval Tuning Schemes", ''Dave Keenan & Douglas Blumeyer's Guide to RTT''. Dave Keenan and Douglas Blumeyer. Xenharmonic Wiki. </ref>. This tuning was elegantly explained in his ''Middle Path'' paper in the case of nullity-1 (i.e. single-comma temperaments)<ref>"A Middle Path between Just Intonation and the Equal Temperaments – Part 1", ''Xenharmonikôn, An Informal Journal of Experimental Music''. Paul Erlich. </ref>. In this tuning, every prime makes an effort in the right direction to close out the comma. To illustrate, consider 5-limit meantone, and to simplify it even more, let us start with the constrained equilateral-optimal tuning (CEOP tuning) instead since its effect is the easiest to observe. The CEOP tuning of 5-limit meantone is given in terms of the projection map ''P'' as

	$$		$$
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	The ''n'' is a positive integer determining the highest relevant harmonic. Specifically, the weight of any prime harmonic equals its maximum number of stacks without exceeding the ''n''-integer-limit, so different values of ''n'' can alter the relative weights of the primes.		The ''n'' is a positive integer determining the highest relevant harmonic. Specifically, the weight of any prime harmonic equals its maximum number of stacks without exceeding the ''n''-integer-limit, so different values of ''n'' can alter the relative weights of the primes.

	To illustrate, let us set ''n'' = 9, or 9-integer-limit. Harmonic 2 can be stacked thrice, giving 8. Stacking it four times would give 16, exceeding 9. Its weight is thus 3. Harmonic 3 can be stacked twice, giving 9. Stacking it three times would give 27, exceeding 9. Its weight is thus 2. Both 5 and 7 have ~~unity~~ weight since they can only be stacked once in the integer limit. 11 and beyond have zero weight because they cannot be stacked at all. If optimization is to be carried out for a 13-limit temperament then we have the weights 3, 2, 1, 1, 0, 0 for primes 2 to 13. The weights are different if ''n'' = 7, or 7-integer-limit, for example. The weight of 2 is 2, of 3, 5 and 7 is unity, and of 11 and 13 zero, giving 2, 1, 1, 1, 0, 0 for primes 2 to 13.		To illustrate, let us set ''n'' = 9, or 9-integer-limit. Harmonic 2 can be stacked thrice, giving 8. Stacking it four times would give 16, exceeding 9. Its weight is thus 3. Harmonic 3 can be stacked twice, giving 9. Stacking it three times would give 27, exceeding 9. Its weight is thus 2. Both 5 and 7 have unit weight since they can only be stacked once in the integer limit. 11 and beyond have zero weight because they cannot be stacked at all. If optimization is to be carried out for a 13-limit temperament then we have the weights 3, 2, 1, 1, 0, 0 for primes 2 to 13. The weights are different if ''n'' = 7, or 7-integer-limit, for example. The weight of 2 is 2, of 3, 5 and 7 is unity, and of 11 and 13 zero, giving 2, 1, 1, 1, 0, 0 for primes 2 to 13.

	The Hahn[''n''] weight matrix is given as		The Hahn[''n''] weight matrix is given as

FloraC: Fix a typo

2023-12-24T11:11:45Z

Fix a typo

← Older revision		Revision as of 11:11, 24 December 2023
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	Chordal rootedness is said with respect to an individual chord. A chord with chordal rootedness is dubbed a rooted chord. Such a chord feels distinctly resolved and self-contained: it speaks of itself as a center of reference, a result from the alignment of the chord's formal root with its virtual fundamental. Let us clarify these terms before we proceed.		Chordal rootedness is said with respect to an individual chord. A chord with chordal rootedness is dubbed a rooted chord. Such a chord feels distinctly resolved and self-contained: it speaks of itself as a center of reference, a result from the alignment of the chord's formal root with its virtual fundamental. Let us clarify these terms before we proceed.

	A formal root is simply the pitch that is denoted in terms of ratios as unity. Determination of a chord's formal root is part of the traditional pedagogy of harmony, but it is really hard to speak of as it depends on many factors: the specific chord structure, the texture and perhaps the genre of the piece. So it is often found heuristically. The bass supported by the perfect fifth is typically the best candidate of a formal root in an positive, octave-equivalent context, but such a structure is lacking in many chords especially in nontertian harmony. Another reasonable strategy in the same context is to always place the formal root on the bass. Anyway, this root is considered to be the starting point of a chord on top of which other notes are built.		A formal root is simply the pitch that is denoted in terms of ratios as unity. Determination of a chord's formal root is part of the traditional pedagogy of harmony, but it is really hard to speak of as it depends on many factors: the specific chord structure, the texture and perhaps the genre of the piece. So it is often found heuristically. The bass supported by the perfect fifth is typically the best candidate of a formal root in a positive, octave-equivalent context, but such a structure is lacking in many chords especially in nontertian harmony. Another reasonable strategy in the same context is to always place the formal root on the bass. Anyway, this root is considered to be the starting point of a chord on top of which other notes are built.

	By contrast, the chord's actual root is what we know as the virtual fundamental. This is the particular pitch on which a chord appears to be a single harmonic note after the phenomenon of timbral fusion, so analytically it emerges at the GCD of all the ratios of the chord. This root, being simultaneously "virtual" and "actual", presents an interesting case on how we think about it. Its virtuality is reflected by the fact that the pitch need not be present – neither as a note nor even as an energy stream in the spectrogram. Its actuality is evidenced by the fact that we literally hear it due to the gestalt of harmonic series.		By contrast, the chord's actual root is what we know as the virtual fundamental. This is the particular pitch on which a chord appears to be a single harmonic note after the phenomenon of timbral fusion, so analytically it emerges at the GCD of all the ratios of the chord. This root, being simultaneously "virtual" and "actual", presents an interesting case on how we think about it. Its virtuality is reflected by the fact that the pitch need not be present – neither as a note nor even as an energy stream in the spectrogram. Its actuality is evidenced by the fact that we literally hear it due to the gestalt of harmonic series.

FloraC: Final polish for this version

2023-12-24T10:14:50Z

Final polish for this version

← Older revision		Revision as of 10:14, 24 December 2023
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	That is our familiar 1/4-comma tuning. It is surprising that no interest has yet developed in tunings by the Chebyshevian norm. Compared to the 4/17-comma tuning by the Euclidean norm, The 1/4-comma tuning by the Chebyshevian norm removes all errors in prime 5 at the cost of just a little bit more in prime 3.		That is our familiar 1/4-comma tuning. It is surprising that no interest has yet developed in tunings by the Chebyshevian norm. Compared to the 4/17-comma tuning by the Euclidean norm, The 1/4-comma tuning by the Chebyshevian norm removes all errors in prime 5 at the cost of just a little bit more in prime 3.

	To evaluate, tuning by the Euclidean norm turns out advantageous not only because it is easy to compute (Euclidean being the only order of norms with analytical solutions) but because it is theoretically nice as the more capable are tasked to do proportionately more. Both Manhattan and Chebyshevian tunings show discontinuities when the complexities of the primes are at certain extreme points, and things start to break down as we approach them. Manhattan tunings ~~feel~~ strange when some primes are orders-of-magnitude more complex than the rest. Chebyshevian tunings ~~feel~~ as strange when all primes have near-equal complexities.		To evaluate, tuning by the Euclidean norm turns out advantageous not only because it is easy to compute (Euclidean being the only order of norms with analytical solutions) but because it is theoretically nice as the more capable are tasked to do proportionately more. Both Manhattan and Chebyshevian tunings show discontinuities when the complexities of the primes are at certain extreme points, and things start to break down as we approach them. Manhattan tunings show strange behaviors when some primes are orders-of-magnitude more complex than the rest. Chebyshevian tunings are as strange when all primes have near-equal complexities.

	== Chapter IV. Art of Compromise ==		== Chapter IV. Art of Compromise ==
	Tempering is the ultimate art of compromise, a global, millenium-old puzzle, for a coarse tuning of the 12 equal temperament was actually given in the ancient Chinese book ''Huai Nan Zi'' (''c''. 122 BC) – not that the concept of equal temperament was laid out in any way, but they wanted twelve Pythagorean fifths to close off at the octave!<ref>"Prince Chu Tsai-Yü's Life and Work: A Re-Evaluation of His Contribution to Equal Temperament Theory", ''Ethnomusicology''. Fritz A. Kuttner. </ref> So what about this essay? Most likely, it will be no end of a debate, but inviting more. It is high time we confront the last hard problem: compositeness of the harmonics.		Tempering is the ultimate art of compromise, a global, millenium-old puzzle, for a coarse tuning of the 12 equal temperament was actually given in the ancient Chinese book ''Huai Nan Zi'' (''c''. 122 BC) – not that the concept of equal temperament was laid out in any way, but they wanted twelve Pythagorean fifths to close off at the octave!<ref>"Prince Chu Tsai-Yü's Life and Work: A Re-Evaluation of His Contribution to Equal Temperament Theory", ''Ethnomusicology''. Fritz A. Kuttner. </ref> So what about this essay? Most likely, it will be no end of a debate, but inviting more. It is high time we confront the last hard problem: compositeness of the harmonics.

	If we play the ~~interval of 15/1~~, does it somehow suggest 5~~/1 and 3/1~~? It seems even if we do not hear 15/1 as composite, we may perceive the compositeness in some other ways, making them conceptually reducible, thus simpler, than its neighboring prime harmonics. Yet the problem definitely does not end there. Sensing compositeness sounds like a reasonable assertion, but does it make composite intervals more important, or less? Does it make composite intervals deserve more care, or less? That is essentially equivalent to asking if complexity needs more care, or less.		If we play the 15th harmonic, does it somehow suggest 3 and 5? It seems even if we do not hear 15 as composite, we may perceive the compositeness in some other ways, making them conceptually reducible, thus simpler, than its neighboring prime harmonics. Yet the problem definitely does not end there. Sensing compositeness sounds like a reasonable assertion, but does it make composite intervals more important, or less? Does it make composite intervals deserve more care, or less? That is essentially equivalent to asking if complexity needs more care, or less.

	On one hand, we want the majority of chords to be in tune, so obviously the most common intervals should get the best care. The question is then what probability distribution is followed without knowing what kind of harmony will be used in advance. A chi distribution would certainly make sense if we were to talk about randomly generated "tonal" music with no regards of psychoacoustics – since each voice's number of generator steps from the tonic was supposed to follow a normal distribution. In a world with human beings and with harmonic clarity rather than the abstract number of generator steps playing the predominant role of forming tonality, the right assumption for commonness is definitely not that but to be inversely related to complexity. The metric can be taken as the inner product of a uniform distribution and the inverse complexity, and if the uniform distribution is replaced with something that favors structurally tonal music such as a chi distribution, we obtain a commonness curve that biases heavily towards simplicity more than many would expect.		On one hand, we want the majority of chords to be in tune, so obviously the most common intervals should get the best care. The question is then what probability distribution is followed without knowing what kind of harmony will be used in advance. A chi distribution would certainly make sense if we were to talk about randomly generated "tonal" music with no regards of psychoacoustics – since each voice's number of generator steps from the tonic was supposed to follow a normal distribution. In a world with human beings and with harmonic clarity rather than the abstract number of generator steps playing the predominant role of forming tonality, the right assumption for commonness is definitely not that but to be inversely related to complexity. The metric can be taken as the inner product of a uniform distribution and the inverse complexity, and if the uniform distribution is replaced with something that favors structurally tonal music such as a chi distribution, we obtain a commonness curve that biases heavily towards simplicity more than many would expect.

FloraC: Update (clarifications and style)

2023-12-24T09:48:40Z

Update (clarifications and style)

Show changes

FloraC: Update to unify the symbols

2023-09-19T09:17:35Z

Update to unify the symbols

← Older revision		Revision as of 09:17, 19 September 2023
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	$$		$$
	~~E_1~~ = \left\langle \begin{matrix} 0 & -\~~epsilon~~ & +\~~epsilon~~ \end{matrix} \right]		\mathcal{E}_1 = \left\langle \begin{matrix} 0 & -\varepsilon & +\varepsilon \end{matrix} \right]
	$$		$$

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	$$		$$
	~~E_2~~ = \left\langle \begin{matrix} 0 & -\~~epsilon~~ & -\~~epsilon~~ \end{matrix} \right]		\mathcal{E}_2 = \left\langle \begin{matrix} 0 & -\varepsilon & -\varepsilon \end{matrix} \right]
	$$		$$

	the ~15/1 will be a combination of harmonics with pitch errors of -''ε'', -''ε'', and 0, but the played harmonic is at -2''ε''. So we see this harmonic will get pretty off the track whenever played.		the ~15/1 will be a combination of harmonics with pitch errors of -''ε'', -''ε'', and 0, but the played harmonic is at -2''ε''. So we see this harmonic will get pretty off the track whenever played.

	Regarding 5/3, it is the opposite situation. E<sub>2</sub> comes out superior to E<sub>1</sub> as it perfectly hits 5/3 whereas E<sub>1</sub>'s ~5/3 is off by +2''ε''.		Regarding 5/3, it is the opposite situation. Ɛ<sub>2</sub> comes out superior to Ɛ<sub>1</sub> as it perfectly hits 5/3 whereas Ɛ<sub>1</sub>'s ~5/3 is off by +2''ε''.

	However, the beating occurs at ~15/1 and multiples thereof, not at ~5/3. The ~5/3, played as a nonrooted dyad, is free from a real reference point (e.g. harmonic series) for it to beat against, so it lacks relevance in tuning optimization. The only scenario to account for its accuracy is where it is played on the chord's formal root, in which case its 3rd harmonic beats against the formal root's 5th harmonic, for example. That is still not a good argument for its relative importance since we would have manipulated the chord structure just in order to obtain this result. A chord with ~15/1 played on the formal root would call for an accurate ~15/1 and then neutralize the demand for an accurate ~5/3 as previously posed. For example, the just major sixth chord 1–5/4–3/2–5/3 and the just major seventh chord 1–5/4–3/2–15/8 cancel each other out up to octave equivalence. More generally, for any chord featuring a divisive ratio on the formal root, there is a counterpart featuring a multiplicative ratio alike.		However, the beating occurs at ~15/1 and multiples thereof, not at ~5/3. The ~5/3, played as a nonrooted dyad, is free from a real reference point (e.g. harmonic series) for it to beat against, so it lacks relevance in tuning optimization. The only scenario to account for its accuracy is where it is played on the chord's formal root, in which case its 3rd harmonic beats against the formal root's 5th harmonic, for example. That is still not a good argument for its relative importance since we would have manipulated the chord structure just in order to obtain this result. A chord with ~15/1 played on the formal root would call for an accurate ~15/1 and then neutralize the demand for an accurate ~5/3 as previously posed. For example, the just major sixth chord 1–5/4–3/2–5/3 and the just major seventh chord 1–5/4–3/2–15/8 cancel each other out up to octave equivalence. More generally, for any chord featuring a divisive ratio on the formal root, there is a counterpart featuring a multiplicative ratio alike.
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	$$		$$

	Let us denote the just tuning map in cents by J, the mistuning map E is		Let us denote the just tuning map in cents by T<sub>J</sub>, the mistuning map Ɛ is

	$$		$$
	\begin{align}		\begin{align}
	E &= J(P - I) \\		\mathcal{E} &= T_J(P - I) \\
	&= \left\langle \begin{matrix} 0 & -4.3013 & +4.3013 \end{matrix} \right]		&= \left\langle \begin{matrix} 0 & -4.3013 & +4.3013 \end{matrix} \right]
	\end{align}		\end{align}
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	$$		$$
	E = \left\langle \begin{matrix} +1.6985 & -2.6921 & +3.9439 \end{matrix} \right]		\mathcal{E} = \left\langle \begin{matrix} +1.6985 & -2.6921 & +3.9439 \end{matrix} \right]
	$$		$$

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	$$		$$
	\begin{align}		\begin{align}
	E &= J(P - I) \\		\mathcal{E} &= T_J(P - I) \\
	&= \left\langle \begin{matrix} 0 & -5.0603 & +1.2651 \end{matrix} \right]		&= \left\langle \begin{matrix} 0 & -5.0603 & +1.2651 \end{matrix} \right]
	\end{align}		\end{align}
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	$$		$$
	\begin{align}		\begin{align}
	E &= J(P - I) \\		\mathcal{E} &= T_J(P - I) \\
	&= \left\langle \begin{matrix} 0 & -5.3766 & 0 \end{matrix} \right]		&= \left\langle \begin{matrix} 0 & -5.3766 & 0 \end{matrix} \right]
	\end{align}		\end{align}
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	© 2023 Flora Canou		© 2023 Flora Canou

	Version Stable 0		Version Stable 1

	This work is licensed under the [https://creativecommons.org/licenses/by-sa/4.0/ Creative Commons Attribution-ShareAlike 4.0 International License].		This work is licensed under the [https://creativecommons.org/licenses/by-sa/4.0/ Creative Commons Attribution-ShareAlike 4.0 International License].

FloraC: Done

2023-05-28T11:14:38Z

Done

@@ Line 1: / Line 1: @@
 In the study of tuning optimization, we find two blocker issues that deserve the title "hard problems of harmony". Versed in a catchy way, they are:
@@ Line 199: / Line 193: @@
 == Chapter V. Towards an Optimization Strategy ==
 == Notes ==
@@ Line 204: / Line 242: @@
 == Release Notes ==

← Older revision		Revision as of 11:02, 22 June 2025
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	Divisive ratios and multiplicative ratios are always said relative to each other. If a divisive ratio is of the form ''n''/''d'', where ''n'' and ''d'' are integers, then a multiplicative ratio is of the form ''nd''. For example, 5/3 is a divisive ratio; 15/1 is a multiplicative ratio. The question is, thus, if ratios of the form ''n''/''d'' are more important than those of the form ''nd''.		Divisive ratios and multiplicative ratios are always said relative to each other. If a divisive ratio is of the form ''n''/''d'', where ''n'' and ''d'' are integers, then a multiplicative ratio is of the form ''nd''. For example, 5/3 is a divisive ratio; 15/1 is a multiplicative ratio. The question is, thus, if ratios of the form ''n''/''d'' are more important than those of the form ''nd''.

	The problem is hard because it is not clear what is implied by importance and what context it can be applied to. Of course, importance means simplicity. But simplicity of ratios is used in two major contexts: chord construction and tuning optimization, and they correspond to distinct psychoacoustic effects. Chord construction has to do with the revelation of harmonic identities due to timbral fusion to a virtual fundamental as discussed in the last chapter, whereas tuning optimization has to do with percept formation and excitation, and to the better end, minimization of mistuned beating. These are fundamentally different effects – this essay takes the liberty of being the first to treat them separately.		The problem is hard because it is not clear what is implied by importance and what context it can be applied to. Of course, importance means simplicity, as that is what makes some intervals stand out from the rest. But simplicity of ratios is used in two major contexts: chord construction and tuning optimization, and they correspond to distinct psychoacoustic effects. Chord construction has to do with the revelation of harmonic identities due to timbral fusion to a virtual fundamental as discussed in the last chapter, whereas tuning optimization has to do with percept formation and excitation, and to the better end, minimization of mistuned beating. These are fundamentally different effects – this essay takes the liberty of being the first to treat them separately.

	The odd-limit tonality diamond fully favors divisive ratios to multiplicative ones, as the odd limit of a ratio is equal to the exponentiation of the Kees height, a norm in a lattice skewed towards divisive ratios by 1/12 turn. It is useful in just chord construction. Consider the just major triad again. While 5/1 and 3/1 are the only ratios used to build the chord, the interval between them – 5/3 – is a real, played interval, unlike the multiplicative ratio 15/1, which is not played, only present in the harmonics. Likewise, using any harmonics as components of a just chord causes all the ratios between them to be played, and thus to be emergent. Unless we stick to bare dyads, it could not be more appropriate than adopting a metric that favors divisive ratios, especially the tonality diamonds.		The odd-limit tonality diamond fully favors divisive ratios to multiplicative ones, as the odd limit of a ratio is equal to the exponentiation of the Kees height, a norm in a lattice skewed towards divisive ratios by 1/12 turn. It is useful in just chord construction. Consider the just major triad again. While 5/1 and 3/1 are the only ratios used to build the chord, the interval between them – 5/3 – is a real, played interval, unlike the multiplicative ratio 15/1, which is not played, only present in the harmonics. Likewise, using any harmonics as components of a just chord causes all the ratios between them to be played, and thus to be emergent. Unless we stick to bare dyads, it could not be more appropriate than adopting a metric that favors divisive ratios, especially the tonality diamonds.
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	That is the 1/5-comma tuning, in which harmonics 3 and 5 have an equal magnitude and an opposite sign of error.		That is the 1/5-comma tuning, in which harmonics 3 and 5 have an equal magnitude and an opposite sign of error.

	TOP tuning works principally the same, except that harmonic 2 is no longer constrained to pure and that the allowed error of ''q'' is log<sub>2</sub> (''q'') times that of prime 2. The TOP error map of 5-limit meantone is		TOP tuning works principally the same, except that harmonic 2 is no longer constrained to pure and that the allowed error of ''q'' is log<sub>2</sub>(''q'') times that of prime 2. The TOP error map of 5-limit meantone is

	$$		$$
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	That is our familiar 1/4-comma tuning. It is surprising that no interest has yet developed in tunings by the Chebyshevian norm. Compared to the 4/17-comma tuning by the Euclidean norm, The 1/4-comma tuning by the Chebyshevian norm removes all errors in prime 5 at the cost of just a little bit more in prime 3.		That is our familiar 1/4-comma tuning. It is surprising that no interest has yet developed in tunings by the Chebyshevian norm. Compared to the 4/17-comma tuning by the Euclidean norm, The 1/4-comma tuning by the Chebyshevian norm removes all errors in prime 5 at the cost of just a little bit more in prime 3.

	~~To evaluate, tuning by the~~ Euclidean ~~norm turns out advantageous not only~~ because ~~it is~~ easy to compute ~~(Euclidean being~~ the only order of norms with analytical solutions) but ~~because it is~~ theoretically nice as the ~~more capable are tasked to~~ do ~~proportionately more~~. ~~Both~~ Manhattan ~~and Chebyshevian tunings~~ show discontinuities when the complexities of the primes are at certain extreme points, and things start to break down as we approach them. Manhattan tunings show strange behaviors when some primes are orders-of-magnitude more complex than the rest. Chebyshevian tunings are as strange when all primes have near-equal complexities.		There is a belief that Euclidean norms are common simply because they are easy to compute, as that is the only order of norms with analytical solutions, but here we see they are theoretically nice as they take account of how each prime is reached in a temperament, which implies better use of optimizational resource. Taking that to the extreme, Chebyshevian norms would do even better. However, like Manhattan they show discontinuities when the complexities of the primes are at certain extreme points, and things start to break down as we approach them. Manhattan tunings show strange behaviors when some primes are orders-of-magnitude more complex than the rest, whereas Chebyshevian tunings are as strange when all primes have near-equal complexities.

	== Chapter IV. Art of Compromise ==		== Chapter IV. Art of Compromise ==
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	On the other hand, it is argued that complex intervals need relatively more care since it is harder to capture their identities. It is believed that complex intervals have a smaller range of tolerance in which their identities will be revealed, which is fairly easy to understand.		On the other hand, it is argued that complex intervals need relatively more care since it is harder to capture their identities. It is believed that complex intervals have a smaller range of tolerance in which their identities will be revealed, which is fairly easy to understand.

	The Tenney weight is the weight that <s>turns a deaf ear to</s> strikes a perfect balance on those considerations. In fact, it is the only weight in which tunings on composite subgroups coincide with tunings on prime subgroups, meaning that optimizing a temperament on 2.3.5 or 2.9.5 will render the same result for all the intervals they share. The reason is each prime ''q'' in the prime list ''Q'' has an importance rating of 1/log<sub>2</sub> (''q''), represented by the matrix		The Tenney weight is the weight that <s>turns a deaf ear to</s> strikes a perfect balance on those considerations. In fact, it is the only weight in which tunings on composite subgroups coincide with tunings on prime subgroups, meaning that optimizing a temperament on 2.3.5 or 2.9.5 will render the same result for all the intervals they share. The reason is each prime ''q'' in the prime list ''Q'' has an importance rating of 1/log<sub>2</sub>(''q''), represented by the matrix

	$$		$$
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	$$		$$

	That is pretty wrong ~~if gazed~~ from the ~~universe~~ of Tenney weight, as it makes harmonic 8 three times distant with three times the error of 13. One can immediately see the bumps in the complexity curve of integer harmonics. Nonetheless, it reasonably holds itself as it demands the same absolute tolerance for all primes. It only highlights higher primes in a mild manner if the standard is, as they argued, a diminishing tolerance.		That is pretty wrong from the perspectives of Tenney weight, as it makes harmonic 8 three times distant with three times the error of 13. One can immediately see the bumps in the complexity curve of integer harmonics. Nonetheless, it reasonably holds itself as it demands the same absolute tolerance for all primes. It only highlights higher primes in a mild manner if the standard is, as they argued, a diminishing tolerance.

	The Wilson weight does the opposite to the equilateral weight, as it puts 1/''q'' importance rating to the prime ''q'', represented by the matrix		The Wilson weight does the opposite to the equilateral weight, as it puts 1/''q'' importance rating to the prime ''q'', represented by the matrix
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	$$		$$

	with ''w''<sub>1</sub> being free since the octave is constrained. This weight always yields rational projection maps for some reasons. It can be used to tune all 5-limit temperaments alike, and the weight ratio between 5 and 3 is 1/sqrt (2), very close to log<sub>5</sub> (3) in Tenney. In general, the weight ratio between ''q'' and 3 should be close to log<sub>''q''</sub> (3) and the exact values are left to readers to experiment with.		with ''w''<sub>1</sub> being free since the octave is constrained. This weight always yields rational projection maps for some reasons. It can be used to tune all 5-limit temperaments alike, and the weight ratio between 5 and 3 is 1/sqrt (2), very close to log<sub>5</sub>(3) in Tenney. In general, the weight ratio between ''q'' and 3 should be close to log<sub>''q''</sub>(3) and the exact values are left to readers to experiment with.

	== Chapter V. Towards an Optimization Strategy ==		== Chapter V. Towards an Optimization Strategy ==
	Incorporating all that have been discussed above, I recommend CTE tuning as the best general-purpose reference solution to everyone, whereas my hemi-idiosyncratic answer to tuning optimization is based on a meticulously engineered weight function, which happens to be an unskewed version of the Hahn distance. Let us dub this the Canou[''n''] weight, and the tuning using this weight the '''CC<sub>''n''</sub>E tuning''' (for '''constrained Canou[''n'']~~-Euclidean~~ tuning''').		Incorporating all that have been discussed above, I recommend CTE tuning as the best general-purpose reference solution to everyone, whereas my hemi-idiosyncratic answer to tuning optimization is based on a meticulously engineered weight function, which happens to be an unskewed version of the Hahn distance. Let us dub this the Canou[''n''] weight, and the tuning using this weight the '''CC<sub>''n''</sub>E tuning''' (for '''constrained Canou[''n'']–Euclidean tuning''').

	In this weight, the ''n'' is a positive integer determining the highest relevant harmonic. The weight of any prime harmonic equals its maximum number of stacks without exceeding the ''n''-integer-limit. Different values of ''n'' can alter the relative weights of the primes.		In this weight, the ''n'' is a positive integer determining the highest relevant harmonic. The weight of any prime harmonic equals its maximum number of stacks without exceeding the ''n''-integer-limit. Different values of ''n'' can alter the relative weights of the primes.
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	$$		$$

	which indicates that the prime ''q'' in ''Q'' has the weight equal to floor (log<sub>''q''</sub> (''n'')).		which indicates that the prime ''q'' in ''Q'' has the weight equal to floor (log<sub>''q''</sub>(''n'')).

	The Tenney weight is a special case of the Canou[''n''] weight, where ''n'' → infinity. The only thing that sets Canou[''n''] apart from Tenney is the floor function (since log<sub>''Q''</sub> (''n'') = log<sub>2</sub> (''n'')/log<sub>2</sub> (''Q'') and log<sub>2</sub> (''n'') is a constant), and its effect converges to zero as ''n'' gets sufficiently large. Conceptualizing the Tenney weight in this way is not recommended, though, because Tenney's is characteristically transcendental whereas all the other Canou[''n''] weights are algebraic.		The Tenney weight is a special case of the Canou[''n''] weight, where ''n'' → infinity. The only thing that sets Canou[''n''] apart from Tenney is the floor function (since log<sub>''Q''</sub>(''n'') = log<sub>2</sub>(''n'')/log<sub>2</sub>(''Q'') and log<sub>2</sub>(''n'') is a constant), and its effect converges to zero as ''n'' gets sufficiently large. Conceptualizing the Tenney weight in this way is not recommended, though, because Tenney's is characteristically transcendental whereas all the other Canou[''n''] weights are algebraic.

	That defines the C<sub>''n''</sub>C, C<sub>''n''</sub>E, and C<sub>''n''</sub>OP tunings, but if we contrain the octave to pure, it does not matter how many times the octave is stacked, making the integer limit equivalent to the smaller closest odd limit. The proposed convention is to always use the largest number ''n'' if multiple consecutive choices of ''n'' will give the same CC<sub>''n''</sub>E tuning. For example, CC<sub>13</sub>E, CC<sub>14</sub>E, CC<sub>15</sub>E, and CC<sub>16</sub>E are all equivalent and one should always write CC<sub>16</sub>E.		That defines the C<sub>''n''</sub>C, C<sub>''n''</sub>E, and C<sub>''n''</sub>OP tunings, but if we contrain the octave to pure, it does not matter how many times the octave is stacked, making the integer limit equivalent to the smaller closest odd limit. The proposed convention is to always use the largest number ''n'' if multiple consecutive choices of ''n'' will give the same CC<sub>''n''</sub>E tuning. For example, CC<sub>13</sub>E, CC<sub>14</sub>E, CC<sub>15</sub>E, and CC<sub>16</sub>E are all equivalent and one should always write CC<sub>16</sub>E.
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	== Release Notes ==		== Release Notes ==
	© ~~2023–2024~~ Flora Canou		© 2023–2025 Flora Canou

	Version Stable 4		Version Stable 5

	This work is licensed under the [https://creativecommons.org/licenses/by-sa/4.0/ Creative Commons Attribution-ShareAlike 4.0 International License].		This work is licensed under the [https://creativecommons.org/licenses/by-sa/4.0/ Creative Commons Attribution-ShareAlike 4.0 International License].