Sound comprises physical vibrations having frequency from around 50 to 20,000 Hz. It is defined by the sensitivity of human hearing rather than any physical property of the vibration; sound is what you can hear. A pitch is a sound with a specific fundamental frequency, often with some harmonic frequencies audible as multiples of that frequency.
In music we discretize the continuum, selecting certain specific pitches, which we call notes, to use in making music. The notes are named and organized into scales, and a piece of music can be pretty thoroughly described, for instance in sheet music, in terms of its notes and their timing.
The absolute frequencies of the notes are not important but it is theorized that the mathematical relationships of their frequencies are the basis of harmony, with some ratios of multiple frequencies sounding more pleasant that others. For instance if you double the frequency of a note you’ll get its octave, a new note that sounds the same as the first, but higher. It is believed that other consonant intervals – fifths, thirds, etc. – can be similarly derived from simple frequency ratios.
A kind of scale can be constructed by taking the fifth of the fifth (ratio 3:2 of the original pitch) and the fifth of that and so on, shifting the results into a target octave. It is believed in both music theory and science that the pitch series resulting from this recursive pattern of simple fractions has special significance within the auditory system, and this mystical marriage of mathematics and biology is the basis of the traditional harmonic system. The mathematical justification blesses and guides the aesthetic perception.
Due to some inconsistencies between the math and the biology, in the real world the pitches have to be adjusted slightly. With temper-tuning and other factors very few of the intervals align precisely with the expected fractions in reality. For instance, a major third interval – C to E, for instance – is defined theoretically as a frequency ratio of 5:4, or 1.25, but in practice we use the ratio 1.259921, which is approximately 5:3.96850278708. Not a simple ratio, but kind of close.
Still, our belief system insists that the human ear and nervous system find certain simple-fraction intervals pleasant and other, more complicated, ratios unpleasant; music composed or performed in the civilized world since Mesopotamian times is based on this assumption.
There is no known evolutionary advantage for a species to be able to analyze the ratios of multiple simultaneous audio frequencies in real time, but the human nervous system has evolved uniquely to do this – why? Whether or not simple ratios are biologically significant, the fact is that we are able to discriminate nuances of pitch intervals, for instance we know instantly if someone is playing out of tune, and we can easily tell a major chord from a minor one. There is no plausible explanation for the evolution of this sophisticated function, which amounts to a math co-processor within the nervous system. We rarely stop to appreciate how complicated that function is, and how unlikely it is to have emerged without being adaptive. There are some proposed theories about it but no consensus, and the function is not necessary for anything other than listening to music.
Cognitive functions that detect and analyze patterns do offer clear evolutionary advantages, especially for uderstanding language but also in high level planning, identification of causality, social coordination, and integration of individuals and groups with the environment. But the particuar auditory ability to decompose sounds into their frequencies and determine with zero apparent latency the ratios of the various frequencies to one another only supports the detection of musical harmony.
The musical instruments that we use are, with a few exceptions, designed to play the permitted discretized tones. Some, such as trombones and non-fretted string instruments, can play a continuum of frequencies but of course you take a lot of lessons to learn “in tune.” Most instruments have keys or frets or valves or holes to allow the musician to play proper, permissibly discretized, named pitches.
As a result of the cultural heritage of theoretical scales and physical instruments to support them, scale-based musical sound is familiar to almost all populations on the earth (where the word “almost” only serves to hedge the bet). Various scales may be locally favored, including microtonal scales, but all known music adheres to some sort of discretized framework of pitch.
INTELLECTUAL DOMINANCE
The human species is one leaf on one branch of the tree of evolved primates. We are not put on earth by God, we do not have “spirits” that other animals lack. We were not seeded here by aliens nor have any exotic supernatural provenance on earth; we are nothing more than a species of monkey. There is no evidence that contradicts or undermines this view. If it makes you feel better, you can say we’re the “best monkey.” I personally don’t need that.
The assumption that we were put here by a a higher power to dominate nature, which is a domain from which we are excluded, is false but universally believed, at least at a daily, practical level.
Our species’ primary adaptation is the ability to use language to represent the perceived environment and communicate information referring to it. This adaptation allows us to manipulate the environment in new ways, and inevitably produces an exaggerated sense of separateness from perceived nature in the experienced sensation of subject versus object.
Human dominance of the environment (“technology”) operates through a process of isolating parts of the environment as objects, labeling them, reducing them to linguistic descriptions, and operating in imagination on the descriptions. It’s a powerful algorithm. By the way, I don’t care if you disagree with this summary of it. At the end of this piece, see if I have made my point or not, and we can decide if it’s worth arguing about.
As part of the simplification process we also manipulate the actual environment by revising the shapes and functions of things to better fit their descriptions. “Geometrical shapes” – easily described simple forms – are combined to construct new aggregate objects that can be described in language, including the language of mathematics. This is described as “tool-building” and it has resulted in the invention of an entire artificial environment that we can live our whole lives in. Our experienced environment is primarily built of simplified surfaces, such as flat walls, ubiquitous right angles, rounded curves, and other regular shapes. Even water, the ultimate embodiment of formlessness, is contained in a well-organized system of round, linear pipes.
Music is the application of this process to sound. Points from the continuum of frequencies were selected by some ancient process of intellectual simplification and labeling – choosing intervals that fit a pleasing mathematical expectation – and the rearrangements of these discretized pitches comprise an auditory artform.
INDOMITABLE NATURE
While we strive to surround ourselves with describable, predictable, controllable geometric shapes, none of nature outside our cultural sphere conforms to our requirements. Nature (including our human environment) is not made of objects but of constantly changing relationships of fields of matter and force; objects are a by-product of the labeling function of language. Almost nothing outside the artificial human environment is made with flat surfaces, smooth curves, right angles. Even riverbeds shift over time. Everything outside our safe place is, you could say, texture.
Nature’s failure to conform to the dominance of mental simplification sometimes breaks into our artificial sphere as well, in the form of accidents and coincidences, and also in some necessary activities that defy logic and intellect.
An example is cooking. Food is not geometrical. Cooking requires such non-geometric activities as stirring, boiling, cutting, mashing, scooping, etc. that cannot be described in language or mathematics. Tell me, what shape is broccoli?
Other examples include housekeeping, where dust, spills, and other non-geometrical natural intrusions must be constantly removed or organized; also washing and folding clothing and other soft materials; and activities such as child-raising where the random impulses of nature sometimes explode spontaneously at unpredictable times and must be trained to fit geometric constraints.
This isn’t the place for it but we could build up a system of understanding with a dichotomy similar to, say, Yin and Yang, to describe activities that can be reduced to intellectual objectification and those that cannot. Perhaps such a theory would involve speculation about sex and gender, exploitation, intuition, value, and the imposition of paternalistic economic assumptions on daily life, do you think?
FAMILIARITY
Our imagination allows us to picture new objects that are slightly different from familiar ones; we can’t imagine what is truly unknown. We can only communicate with familiar vocabularies, and even our dreams offer modified forms of existing things, for instance dream monsters may have wings or claws, drawing their fantastic qualities from familiar things. Objects in dream or imagination may defy the laws of physics, doing things like flying, but only because we know what flying is. Imaginary things are mostly unfamiliar composites of familiar things, which may include shadows and sounds from unseen sources in another room. And so science, art, language, and culture proceed by baby steps from where they are to a nearby state.
New music differs incrementally from familiar music. It is usually possible to guess the year, or at least the decade, in which a piece was composed or performed, simply from audible qualities that typify the era. This is because new music sounds the same as familiar existing music. When it changes, the new music sounds like the music that preceded it, modified slightly. And as time moves on, creations retain the typical qualities of their birth date.
Innovation in any field is adopted only if it sufficiently resembles the current familiar forms in that field. This is a partly a function of limited imagination and partly a function of social pressure to conform to norms; in any case being limited to incremental innovation is where humans fall far beneath the potential offered by the world.
We are not close to perfection in anything. Problems in any field could be solved better by sampling from the entire problem space, rather than the familiar edges of it. The best solutions to any problem may – and probably do – exist in completely unknown and unfamiliar approaches. In this discussion, a beautiful piece of music is a kind of problem solution. There is some chance that the most beautiful piece of music in the world will sound almost exactly like this week’s hit record. That chance is near zero; still, that’s where we’ll look.
“Familiar” is more important to human culture than “better.” A listener constantly compares the sound of music to existing templates, allowing it to reawaken the feelings of personal experiences associated with features of that sound; this evocation of familiarity is an important part of the power of music. But personal nostalgic listening is the lowest quality music appreciation that exists; any music, heard often enough, becomes familiar and triggers subsequent nostalgia, it doesn’t matter if the music is any good. A higher quality listening experience emerges from attending to the literal sound, in the present moment, of the instruments, the patterns of melodies and harmonies and timbre, the intention of the composer and of the performing interpreter of the piece, nuances implied by themes and motifs and phrasing – the music itself. This kind of listening transcends cultural restrictions on understanding, and centers the listener in the infinite present moment. Nostalgia is an important social force that induces coherence in a population and implies shared norms and rules to live by, but artists who target the nostalgic listener necessarily produce inferior work. So even if the “good audience” is only a fiction, it is a fiction that drives the genius of art in all modalities, not just music.
MUSICAL PROPOSAL
Given a discrete set of allowable scale-pitches, the space of allowable combinations or intervals is very large but finite. (I am speaking in terms of simultaneous vertical sound-structures but harmonic progression and melody are implicit as well.)
The space of combinations of pitches drawn from a continuum is infinite, even if you only consider pairs of simultaneous pitches – two-note chords.
There are almost certainly intervals in the continuum that are impossible in the discrete system and which are harmonious or pleasing to the ear.
It may be impossible to judge continuous harmony given the prejudice of familiarity with certain patterns of discretized sound; maybe the only thing that matters in music is familiarity. I am not being sarcastic, maybe musical sounds are nothing more than tribal calls, pack signatures, that we are attracted to because they remind us who we are. On the other hand, it may be that presently-unknown intervals can sound harmonious, and that good music can be composed of them.
Given a continuous frequency space, it will be impossible to list all consonant intervals or describe the rules that generate them, i.e., frequency ratios, named notes and intervals, chords, melodic sequences, etc.
Linguistic intellect will not be very helpful in composing music if sounds cannot be discretized with prescribed values, labeled, and manipulated in representation. Only direct perceptual prehension will be useful; you have to hear it to know if it works.
There is no qualitative definition for goodness in music. Generally, good music is music that some people (reference group TBA) say is good. On the other hand, some music that people love is shit.
In practice, timbre is just as important as pitch for determining whether people will like a piece of music. It is not at all unusual to have mediocre or even terrible music that sounds good – it can be full of cliches and plagiarism, non sequitur phrasing that leads to nowhere, intrusive notes that interrupt the flow of melody, but with a good blend of timbres and a little reverb all of this can be overlooked.
As a prediction, I’d say it is almost certain that most music sampled from unfamiliar regions of the continuous pitch matrix will sound bad to most people, including – and maybe in particular – trained and sophisticated listeners. On the other hand it is not certain, but is possible and even likely, that music carefully sampled from the continuous space can be beautiful, expressive, and likeable, and that as it becomes more familiar it can be appreciated by ordinary listeners.
Sampling randomly from the continuous space is less likely to produce good music than sampling random patterns from the discrete space (cat walking on a piano keyboard) would be. That is, it is not very fucking likely. The vast majority of random vertical or horizontal tone combinations, discretized or continuous, sound bad, and continuity of pitch offers infinitely more opportunities for painful dissonance. In discrete pitches, the worst we know is the tritone or the minor second interval, both of which are still usable in popular and classical music. Intervals from a continuous field could be uglier than those.
A computer scientist searching for a problem solution first defines a search space, then a strategy for searching for, they hope, the global optimum – the best problem solution. No composer does that. A composer learns about music that already exists, learns about ideas that other people have had, and tries to create something that is a tolerable increment different from that. Too far and the audience won’t accept it, too close and the artist’s breathtaking genius will not be apparent.
To annotate continuous music you would probably need to use an illustration. Formulas or numeric notation may be accurate but would not be useful for a performing musician. Dots sprinkled across lines and spaces will not be helpful.
An important reason to discretize music is to make it intellectually manageable. We can say “a perfect fifth” and understand one another; such linguistic shorthand does not exist in a continuous sound space.
An important reason to use the continuous range is that it almost surely contains regions of unimaginably beautiful harmony. It can be a way to experience the beauty of actual nature, unboxed from language and the simplification imposed by human intellect.
An effect of using it may be to awaken people from the lazy habit of expecting their art to be served to them without effort on their part, carrying them back in fantasy to sentimental high points in their lives. It is not the artist’s job to make people “feel good.” They may have to slough off some preconceptions to appreciate the unfamiliar.
In attempting group improvisation in a continuous pitch environment, the infinity of available sound would allow incredibly dissonant cacophony. We can picture a room full of chimps at typewriters, randomly writing Shakespeare. Now imagine a room full of chimps with trombones. On the other hand, ignoring such things as scales and chords might make the improvisation more facile and interesting.
Exploring the improvisation question is made more difficult because there is not much in the way of continuous-pitch instruments. Some continuous microtonal synth keyboards have been developed, and a continuous woodwind instrument has been invented but is rarely seen. Unfretted string instruments can be used as well as instruments where the tension on the strings can be varied by pressure or bending, and the trombone offers one vision of a possibility for brass instruments, a slider that changes the length of the resonating tube – the slide whistle is similar in controlling pitch. The theremin offers a monophonic approach that can be adapted to function as a synthesizer interface, and there are probably other continuous instruments, but they are not the norm. At this time the best approach will be a synthesizer that is not controlled by a keyboard, perhaps programmed separately from performance, probably graphically.
All comments here about discretization of pitch apply to tempo, as well. We restrict the initiation and duration of tones to carefully defined fractions of a regular beat, imagining notes on a discrete “piano roll” grid or staff. Time is continuous and can be treated as such, with or without the liberation of pitch. Disregarding weird time signatures, which fit inside the borders of normal tradition, experimentation with time has mostly consisted of variations on rubato, swing or grooves where musicians play ahead of or behind the beat to some extent, or polyrhythmic juxtaposition of regularized temporal pulses. But it is possible to free the timing of notes from any rules or restraints, to place notes at the aesthetically appropriate moments of continuous time regardless of rules or traditions.
As a test, I composed and recorded a dozen or so pieces with neither pitch or temporal domain discretization, and a couple of pieces with one or the other. Granted, my compositional skills are limited, but as a proof of concept I feel comfortable saying that the ear can get used to sounds without a rules-based framework. Some of my pieces, or some pieces of some of the pieces, sound fine but “different.” As a composer, my experience was that I could not write blindly from imagination, I had to listen to each phrase in its context and change it until it fit. It’s tedious but also I am working without any precedent or habits to fall back on. The point being, music without discretized pitch or time can sound like music, and even like good music, though habits in listening and creating will need to evolve.
CURIOSITY
In conclusion, an infinity of musical possibilities remain in the great domain of Nothingness because mortal musicians are unable to imagine music without the familiar rules. We have the technology to choose tones from the continuous regions between the usual notes. Such experimentation is sure to be a commercial failure initially, with zero probability of becoming popular with ordinary audiences, at least at first. It is possible they will come to accept it, though, as audiences have accepted atonality as well as various modes and unfamiliar scales in classical music, “outside” chord extensions and melodies in jazz, microtones in blues, and other exceptions.
The science of economics is a Borgesian one where music and art are treated as products categorized by their “market” and their value is literally their sales value, but of course artists don’t see it that way; they make things but can’t explain why. (Music is a notoriously terrible way to make a living.)
The best reason to sample from the unknown is curiosity, to see what happens, and what can be made with new combinations of sounds. Our entire body of human knowledge is an infinitesimal island of beliefs that we huddle on in the midst of a wild and unknown and limitless universe. We don’t know what we don’t know but we know it’s a lot. We have divided the perceptible world into labeled units, and we can deal with those discrete units, but omnipresent curiosity whispers to us that there is more, always more, in the infinity between known facts.