When I was a kid, my father, a composer, once asked me to define music. “Sound, crafted to produce an emotional response of some kind” I said, as accurately as I can remember.
While my father accepted the answer, as would I from a 5 or 6 year old, it’s far too qualitative to satisfy the 39 year old me. What is sound? And, more importantly, what makes it, whatever it is, music?
The first question is easy. Wikipedia defines it as “vibration that typically propagates as an audible wave of pressure, through a transmission medium such as a gas, liquid or solid.” I have nothing to add to that. There is, however, no objective way to define what makes the sound music. Indeed, to mathematically analyze an audio signal that is music, we must make assumptions that do not hold in general case.
Consider the sound of a turbocharged V8, such as the engine found in the latest BMW M5, or a 65° Ferrari V12. It is not unusual to hear auto enthusiasts call them “music.” Yet, I would dare to assume, their neighbors probably disagree.
Another extreme example is John Cage’s famous composition 4’33”, which contains no sound what so ever in itself, instead encouraging the audience to concentrate on the background noises they usually tend to ignore. Obviously, there is no way to analyze the properties of the sound that forms the composition, as the sound heard by the listener isn’t even defined by the composer.
While there clearly is no objective way of even distinguishing music from what is just a sound, most music shares easily identifiable characteristics. In particular, it is usually easy to find a single frequency whose integral multiples have strong peaks in the spectrum, and another, subsonic frequency, whose integral divisions modulate the audible frequencies. The former is related to the qualitative concept of pitch, while the latter is an integral multiple of what is perceived as the tempo of the piece.
In addition to these trivially recognizable frequencies, there are others that have no rational relationship with either. Depending on their frequencies and their distances from the peaks, they may be perceived as noise, tempo variation, vibrato, tremolo, envelopes, or, indeed, anything else that isn’t trivial rhythm or harmony. Telling these concepts apart mathematically is a challenge, although on the brain of a trained listener it imposes none.
To complicate things even further, the spectrum itself isn’t unambiguous, either. The spectrum of a sound is a sum of sinusoidal frequency components, and for a frequency to exist, there must be repetition. This means that to analyze the spectrum of a fragment of a sound, we must mathematically extend the fragment to repeat infinitely.
Thus, the spectrum not only depends on the sound we are analyzing, but on the way we analyze it, as well. It is not merely that we get different approximations of some accurate, correct spectrum, although that is something many texts seem to imply. Instead, since no real world signal repeats infinitely, there is no “correct” spectrum.
The practical implication of this is that what we perceive as a static sound and what we perceive as variation of the sound is not mathematically distinguishable. Obviously, if we take the spectrum of an entire composition, there can be no spectral variation, as we only have one spectrum. The spectrum still contains all the information necessary to restore the original signal; the entire piece has been made into one, long waveform, whose spectrum contains the rhythm and structure encoded in the way frequencies are spread around the peaks.
Analyzing short fragments cannot be perfect, either, because there is no well-defined way to choose the length of the fragment, and even if there were, the audible spectrum overlaps with the modulation frequencies that are perceived as variation over time. In many real world situations, where audio signals containing music need to be analyzed in real time, the short fragments are the only option.
When the signal to analyze is a single tone, played, for example, on a monophonic instrument, it consists of a series of cycles that can be trivially identified by zero crossing detection. This is an important special case, as we have an unambiguous way to analyze the temporal spectrum of the signal by repeating an exact cycle of the waveform instead of a fragment with an arbitrary or heuristically chosen length.
The catch is that repeating a single cycle of almost any audio signal sounds cold, lifeless, and boring comparing to the original sound, no matter how static we might have subjectively perceived it. Organ sounds, invariable over time by the design of the instrument, sound somewhat similar, as do synthesizers playing basic waveforms without modulation, but such perfectly periodic sounds are rare.
In practice, at least for now and in the foreseeable future, everything in the realm of music technology, from building acoustic instruments and concert halls to authoring DAW plug-ins, requires qualitative, and thus subjective, analysis of sound and music. As the analysis itself is subjective, so are the products based on it. No piece of equipment or software can be perfect in any objective sense. This is what makes it all so challenging and satisfying at the same time.
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