User:FloraC/Hard problems of harmony and psychoacoustically supported optimization: Difference between revisions

# Are divisive ratios more important than multiplicative ratios?<ref>Prior to this material, the two problems are often said in the other order, but this essay inverts them since weight is usually considered before the skew in tuning optimization. </ref>

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.  

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 ratios; 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 above, 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.  

If we play such a triad in a tempered tuning profile, the quality of the chord is determined by how the three components said above line up. In a tuning profile characterized by the mistuning map

the ~15/1 will be a combination of harmonics with pitch errors of -''ε'', 0, and +''ε''. In addition, the harmonic itself can be played as a dyad and its pitch error is 0.  

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. Ɛ₂ comes out superior to Ɛ₁ as it perfectly hits 5/3 whereas Ɛ₁'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.  

 

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  

Let us denote the just tuning map in cents by T_J, the mistuning map Ɛ 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 log₂ (''q'') times that of prime 2. The TOP mistuning map of 5-limit meantone is

The mistuning map is

The mistuning map is

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.  

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> This essay 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.  

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 a piece. 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.  

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/log₂ (''q''), represented by the matrix

W = \operatorname {diag} (1/\log_2 (Q))

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 logically holds itself as it demands the same absolute tolerance for all primes, and highlights higher primes in a mild manner if the standard is, as they argued, a diminishing tolerance.  

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

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

with ''w''₁ 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 log₅ (3) in Tenney. In general, the weight ratio between ''q'' and 3 should be close to log_''q'' (3) and the exact values are left to readers to experiment with.  

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_''n''E tuning''' (for '''constrained Hahn[''n'']-Euclidean tuning''').

 

The only part that needs explanation is the Hahn[''n''] weight, an adaptation to the original Hahn distance.  

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.  

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

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

which indicates that the prime ''q'' in Q has the weight equal to floor (log_''q'' (''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_Q (''n'') = log₂ (''n'')/log₂ (Q) and log₂ (''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.  

That defines the H_''n''C, H_''n''E, and H_''n''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_''n''E tuning. For example, CH₁₃E, CH₁₄E, CH₁₅E, and CH₁₆E are all equivalent and one should always write CH₁₆E.  

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.  

! Temperament !! Mistuning Map (CTE) !! Mistuning Map (CH₂₄E)

| 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 }}

| 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 }}

| 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 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 }}

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