User:FloraC/Hard problems of harmony and psychoacoustically supported optimization: Difference between revisions
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[[File:Frequency spectrum of just major triad in semisine wave.png|thumb|800px|Figure 1: Frequency spectrum of just major triad in semisine wave]] | [[File:Frequency spectrum of just major triad in semisine wave.png|thumb|800px|Figure 1: Frequency spectrum of just major triad in semisine wave]] | ||
The same cannot be assumed for tuning optimization, since that is a vastly different scenario. In a just major triad, the 15th harmonic exists in three ways: as the harmonic of the root, of the 3rd harmonic, and of the 5th harmonic. Figure 1 is the frequency spectrum of the triad played in the semisine waveform, which has been proposed as the standard ear-training waveform in ''Proposed Standard Ear-Training Waveform''. | The same cannot be assumed for tuning optimization, since that is a vastly different scenario. In a just major triad, the 15th harmonic exists in three ways: as the harmonic of the root, of the 3rd harmonic, and of the 5th harmonic. Figure 1 is the frequency spectrum of the triad played in the semisine waveform, which has been proposed as the standard ear-training waveform in [[User:FloraC/Proposed standard ear-training waveform|''Proposed Standard Ear-Training Waveform'']]. | ||
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 | 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 | ||
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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, the Euclidean | 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. | ||
== 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'' – 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! 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. | |||
Try thinking of it this way: one could spend their life making music of only plain octaves and fifths without being bored at all. That was what happened in many cultures around the world and no one seemed to have a problem. It is actually our expression of harmonic feelings in intricate multidigit ratios that is the more peculiar endeavor. | |||
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 | |||
$$ | |||
W = \operatorname {diag} (1/\log_2 (Q)) | |||
$$ | |||
Thus, 9 has exactly twice the tolerance as 3, just as a temperament will tune it: if 3 has +1 cent of error, 9 must have +2 cents since 9 counts as two steps along the path of 3. In general, the Tenney weight makes the complexity curve of all integer harmonics a smooth one. The same is not true for any other weights, such as the equilateral weight we are addressing next. | |||
The equilateral weight puts all prime harmonics on a sort of equal footing regardless of their inherent difference, by which is meant, for example, prime 2 lies as far from the origin as prime 13. It is represented by an identity matrix: | |||
$$ | |||
W = I | |||
$$ | |||
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 | |||
$$ | |||
W = \operatorname {diag} (1/Q) | |||
$$ | |||
That is also very useful because the tuning will converge even if higher and higher prime harmonics join the party. Theoretically, it can be used to define a kind of "no-limit" tuning as well as other related measurements, which is not possible with Tenney or equilateral weights since they would oscillate forever. | |||
The situation here is rather like the order of norms assessed in the last chapter. Tuning by the Tenney weight is clearly a generalist's choice, making fewer assumptions and bearing the unique property of a smooth complexity curve for all integer harmonics. The equilateral and Wilson weights are defensible in their own ways. | |||
We should also note that the equilateral and Wilson weights render rational tuning maps in terms of monzo lists (i.e. projection map), whereas Tenney's is transcendental due to the nature of logarithm. By the law that algebraic matrices only have algebraic eigenvectors, both equilateral and Wilson tunings make certain intervals pure and certain other intervals equally off from pure, relating a sense of well compromise. Again in the universe of Tenney weight the appeal can be logically dismissed as follows: due to the nature of logarithm no interval should be tuned pure in the first place except when explicitly told to be (i.e. constrained). By committing to Tenney tunings one should remind themself of that whenever in doubt. Otherwise, if one is indeed after purely tuned intervals, there are many hidden options besides equilateral and Wilson. | |||
Consider an algebraic approximation to Tenney, that is, a tuning with pure intervals but not changing the bias much towards lower or higher harmonics. Notice the CTE tuning of 5-limit meantone is suspiciously close to 2/9-comma. Let us make up a Euclidean norm that results in exactly 2/9-comma. It turns out the weight must be | |||
$$ | |||
W = \operatorname {diag} (\left\langle \begin{matrix} w_1 & \sqrt{2} & 1 \end{matrix} \right]) | |||
$$ | |||
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 sqrt (2), very close to log<sub>3</sub> (5) in Tenney. In general, the weight ratio between ''q'' and 3 should be close to log<sub>3</sub> (''q'') and the exact values are left to readers to experiment with. | |||
== Chapter V. Towards an Optimization Strategy == | == Chapter V. Towards an Optimization Strategy == | ||