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