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In this version I make it explicit that Euclidean norms are better use of optimizational resource
 
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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 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
© 2023–2025 Flora Canou


Version Stable 3
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].
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