Infinite Divisibility of Frequencies, Notes, Pitches

Estimated read time (minus contemplative pauses): 37 min.

As a young musician I was often told by my elders that infinitely many intermediate pitches reside between any two keys on the piano. Call this the claim.

I doubt the claim. In this post, I’ll say why. Then I’ll say why I think the claim is worth thinking about, and why perhaps the good intentions behind it are at least as important as whether or not it’s true.

There are at least two interpretations of the claim.

First interpretation: The pitches produced by piano keys (or by whatever instrument) can be represented numerically. The A4 key (the 49th key on a conventional 88-key piano), in standard (by which I here mean “conventional 21st-century Western”) tuning, produces an instance of the note A at a frequency of 440Hz. (In a moment I’ll tighten up my use of the words “pitch,” “note,” and “frequency,” and will provide more detail about the processes I’m describing here.) The 50th key produces an A♯ at 446Hz*. Between 440 and 446, there are infinitely many real numbers, each of which is a frequency that corresponds to some pitch. This is the uncountably infinite interpretation.

[*This estimate comes from the following standard formula. Calculate the frequency for a note n half steps away from a given note by multiplying the starting note’s frequency value by 2(n/12) (i.e., the 12th root of 2 raised to the power of n). Since A♯/B♭ is one half step above A-440, we get

440 × 2(1/12) ≈ 466.16376151808991640720312, 

which is still rounded, as it’s an irrational number whose digits go on forever without falling into a repeating pattern.]

Second interpretation: Only some of the numbers between 440 and 446 correspond to produceable pitches. This is still an infinite set, however. I’ll call this the countably infinite interpretation. I’m designating it as countable on the weak assumption that there is some discoverable rule making it possible to order a subset of the produceable pitches in one-to-one correspondence with the natural numbers (something that cannot be done with the uncountably infinite set of real numbers, not even just the subset of real numbers between 440 and 441). This assumption is charitable to the claim.

I doubt that people who make the claim have thought it through. The extent that they have a vague idea of what they mean seems to align best with the uncountable interpretation. So, I will first consider why I think that interpretation highly implausible. Then I’ll say why the countable interpretation seems unlikely to fare any better.

To make my point, I’ll switch to an instrument whose pitches are more readily manipulated: the guitar. On a standardly tuned guitar, I can play A-440 on the fifth fret of the high E string. I can play A♯-446 on the same string, one fret up. Or I can reach A♯-446 by bending from a lower fret. I can also hit several audibly distinguishable notes in between, such as a quarter or an eighth of a step above A.

Such notes, which exist “outside” or “between” those of the official Western chromatic scale, are very common. For example, when I vibrato on guitar, I slightly vary the pitch away from and then back to A-440. This can take various forms depending on technique and equipment—e.g., with one or more fretting fingers, by bending the guitar at the joint, with a whammy bar, by depressing the strings behind the nut with the picking hand. It can be fast, slow, wide, tight. Vibrato is an essential element of a player’s distinct style. And it usually requires going outside of the official scale.

(There are techniques for achieving the effect with an unwavering pitch, some of them complicated and technologically assisted, that I won’t get into here. But I will say that they can be used to thicken a vocal recording, and are commonly used in putting together synthesizer sounds of various kinds.)

There are also explicit non-Western tuning systems that break the octave into more than the 12 notes produced by keys 49 through 60 on a standardly tuned piano. And plenty of composers brought up in Western traditions have used microtonal systems—e.g., a 24-note rather than 12-note tuning—in order to expand their baseline pitch palette.

I’ll say more about such music in the conclusion of this post. The point right now is that there clearly are plenty of audible notes between any two keys on the piano—thus the ease with which a piano can go terribly out of tune! But how many?

From A4 to A♯4—or 440Hz to 446Hz here—is a half step. (I say “here” because some folks opt, for example, for  A4 = 441Hz. Again, I’ll tighten up definitions in a moment.) Theoretically, you can cut this interval in half as many times as you like. But practically you can’t. For one thing, you can’t bend a string one-over-2-to-the-100th of the way from A to A♯. (Maybe a robot can, but I’ll get to that in a moment.)

The other problem is that, even if you can produce that bend, there are limits to what a human can perceive. To test yourself, check out this Online Tone Generator. Listen to 440. (From here on, I’ll mostly omit “Hz” to avoid clutter.) A quarter step from there is at about 453. Most of us can hear that. An eight step is at about 446.5. Still audible. A 16th is at 443.25. Still audible. Eventually the difference will be near 440.1 (I can hear this) and 440.01, which I can’t hear and/or my headphones can’t generate. Forget about 440.0000000000000000001.

As you get closer to 440, the experience you have is simply of 440. Or, more precisely, the name you put on it will be the one most conventionally given to the experience associated with that frequency: A. I could just as easily have said “the experience you have is simply of 440.0000000000000000001,” as those would sound identical to 440 to any human. But by convention we associate A with 440.

This convention carries over to measuring devices as well. A guitar tuner will register 440.0000000000000000001 as 440. I imagine that whatever detection element vibrates in the tuner in response to the air molecules set in motion by a frequency generator has its limits, as does whatever it is that turns that element’s vibrations into electrical impulses that can be measured numerically (which number is pointed to by a needle or pops up on a digital readout). Perhaps there are some extremely precise, fine-grain tuners—industrial-strength oscilloscopes—out there. They’ll all have their limits.

And so, two problems arise with translating the uncountable interpretation from theory into reality. First, there is the problem of finding materials in the world that can indeed oscillate at every real number between 440 and 466. Second is finding some conscious entity whose perceptual equipment can turn those uncountably infinite oscillations into pitch experiences. (These are actually the same problem, but let’s conceive of them as two for now.)

Before going on, I should define some terms in order to state these problems more precisely. (I won’t further define mathematical terms such as “uncountable,” as what I’ve said already provides meaningful context enough for those unfamiliar with such terms.)

By “frequency” I mean roughly the usual. The vibrations or oscillations or cycles of some appropriately elastic medium (e.g., a string, vocal cord, air molecule), measured in units of Herz per second.

By “note” I mean the label we put, typically in a musical context, on a class of frequencies whose relations hold a certain interest for us. For example, in the 12-tone equal temperament (thus the 12th-root in the formula above) tuning system we in the West are accustomed to,“the note A” is the label we give to the set of frequency multiples that include 27.5, 55, 110, 220, 440, 880, and so on. These are conventionally numbered so that 440 can be specified as A4, 220 as A3, and so on. My distinction between frequency and note is inexact, providing a fuzzy bridge-language between the realms of frequency and pitch, the formal height of which is notation. I’ll try not to fall into babbling incoherence.

By “pitch” I mean something idiosyncratic. The phenomenological representation or experience of a specific frequency. This is a mental event, or what cognitive scientists call a “percept.” It’s what a particular instance of a note class sounds like.

Let me put some of these terms to use.

When I strike my standardly tuned guitar’s open A string, the string vibrates at 110Hz (while also vibrating at other rates along its length, generating overtones and, in accumulation, timbre; a tiny bit more about this in the closing section). This results in the guitar body and air molecules in contact with the string and guitar vibrating at the same rate, and so on, as described earlier.

In conventional A-440 tuning, these oscillatory goings-on are referred to as the note A—or specifically A2. When and if all of this results—without any unusual impediments (e.g., the frequency being changed as it is propagated through various media on its way to a brain)—in the experience of a hearer, it becomes the pitch that, for those who use such terms, is associated with the label “note A (or A2).” Pitch can exist without those goings-on (as in a dream, hallucination, or in a strong imagination), which emphasizes that pitch is a mental event distinct from vibrations that happen in the world. (Not to mention that some people, for whatever reason, will not experience a pitch when confronted with those goings-on.)

Pitch is a subset of sound. So we can now easily answer a common puzzle. A tree that falls in the woods without anyone there to hear it results in vibrating air molecules, but does not result in sound, as there is no perceptual equipment around to convert that energy into experience, into sound. I can, however, produce in my mind—I can imagine—something like the sound of a falling tree when there is no tree around. I’ll return to this idea in a moment.

For now, let’s look again at the twin problems I introduced. The first is the question of whether material can be found that can vibrate at a rate of every real number between 440 and 466. Next is the question of finding some conscious entity whose perceptual equipment can turn that stimulus—that frequency or note—into a pitch.

I don’t know if people who claim that there are infinitely many notes between any two notes really mean that there must be some entity that can actually hear the note. The tree in the woods example suggests that if the frequency occurs, then the note—e.g., A-1024th-sharp—occurs whether or not the pitch is possible. I’ll grant this. There are loads more notes than there are pitches, then. Which certainly rules out there being infinitely many pitches (see again the above Online Tone Generator demonstration): for each pitch, there are a huge number—theoretically, infinitely many—frequencies that could produce it. Which is to say that, qualitatively, pitches must be discrete, though nailing down where one actually hears a change in pitch from one to the next is difficulty and would have to be felt out, probably with limited accuracy.

An interesting consequence of this, and indicator of how difficult to make nail down, is the auditory version of the triangulation phenomenon in color experience. Suppose you perceive no difference between 440 and 440.1. Come back the next day, and 440.1 and 440.2 also sound the same. And on and on until you’re at 465.9 and 466. (This would be true, though perhaps even more disorienting and perplexing, for someone with absolute pitch. If you have absolute pitch, please run that experiment!)

At any rate, it’s uncontroversial to say that there are theoretically infinitely many frequencies that will yield the same pitch experience. But are those frequencies actually producible? Return to the guitar.

The system composed of guitar string plus human perception fails in both of the relevant respects. In the first case, I am confident that, when you bend a string that is oscillating at 440 until it oscillates at 466, you pass through a sequence of other frequencies. I don’t know what they are, but I’m confident they include 440.5, 441, 441.5, and so on until you hit 466. As well as 440.25, 440.75, and so on. I’m also confident that these frequencies result in distinct pitches.

But I doubt that the string oscillates at every real number along the way. I doubt you’ll find 440.0000000000000000001 or (insert more zeros). Which is to say that I take the sequence to be discrete, or granular. And if the string does manage this, the human will experience the pitch usually associated with 440. It’s likely that the source of that pitch is rarely an object oscillating at exactly 440, but I don’t know what those oscillations would be; maybe one of them on a low gauge high-E electric guitar string is 440.0000000000000000001. But whatever those actual numbers, I doubt they are infinitely many, and the majority will result in no distinctions in pitch.

There are likely many reasons why we can’t perceive such distinctions. One can imagine some evolutionary account, for example. I’ll save such deep digging for another day. Let’s just say it would probably be maddening to hear trillions of distinct pitches every time someone sings “Happy Birthday,” so I’m glad my brain rounds the world in this way. (The same thanks can be given for, say, not experiencing quintillions of distinct colors when attempting to differentiate a frog in a patch of grass.)

I’ll focus here, rather, on the physical limitations in play. Suppose there’s a guitar string vibrating at 440.0000000000000000001. The following must happen to convert this into a pitch. The vibrating string must cause the air (or water, etc.) molecules around it to vibrate at the same frequency. Those air molecules must make it into your ear, where the tympanic membrane, or eardrum, must also vibrate at that frequency. This results in signals being sent to the brain where neurons fire, and so on, resulting ultimately in the conversion of all this business into a pitch experience.

Throughout this process, all the physical mechanisms involved—from string, air, ear, neuron, etc.—must be delicate or fine-tuned enough to faithfully vibrate at 440.0000000000000000001. If that frequency hits the eardrum, but its constitution is such that it vibrates at some other rate, then 440.0000000000000000001 won’t make it into the brain for conversion into sound.

Clearly the process is more complicated than this. I’m not an expert in the physics, biology, or cognitive science involved, and, though I have read a fair amount about the latter two in particular, I don’t remember all of it and even what I do remember, I’m oversimplifying in order to make the simple point that it’s a lot to ask of the world for there to exist in any mind the experience of a pitch that can only be accurately characterized as a mental representation of 440.0000000000000000001 (or insert as many zeros as you like).

That is a rational number. By induction, my worries can be expressed appealing only to rational numbers, which constitute a countable set, one that is “smaller” than the uncountable set of real numbers, though it is also true that between any two rational numbers there is another rational number. Enough said, then, about why I don’t think making the set countably infinite saves the claim, though I haven’t addressed the first problem, involving material that can vibrate at any given frequency. I’ll get to that in a moment, when I propose a way to test for this.

First, I should say one final thing about the second problem, involving pitch production. I should address the fact that I’ve been working from assumptions about human perception. Maybe there’s some being out there whose perceptual equipment can convert each of an infinite number of frequencies into a distinct pitch. I doubt this for the same reason I doubt the number of possible frequencies is infinite: something in the being’s perceptual equipment itself has to physically vibrate at a given frequency in order to convert that frequency into a pitch experience.

To be clear (or not so clear), pitch just is a physical process that we talk about in vague mental terms. This involves far more than I can get into here, but I’ll risk saying a little about it. The language I choose for talking about it will depend on where I stand in the mind-body debate. I might say that the brain produces or causes pitch; or that pitch, like all manner of experience, is an emergent phenomenon, which is to say that it supervenes on the activity of brain parts and/or processes; or I might say that pitches are mental states associated with certain brain states. I might say a lot else. I don’t want to come down decisively here, except as follows.

I believe you need physical machinery—e.g., neural firings synchronized in some proportion corresponding to that of the frequencies that have entered the ear canal—for pitch to happen. (For an in-depth, clear explanation of such processes, see musician-turned-neuroscientist Daniel Levitin’s fascinating 2006 book This Is Your Brain on Music).

I hope the way I’ve worded this says only enough about dualism to imply a rejection of substance dualism of any sort (e.g., that of Descartes). That is, I assume here that minds, or mental content, are not tantamount to extension-less souls that exist distinct from any physical substrate. Maybe you disagree. Maybe you think that you’ll be able to hear all the theoretical pitches once bodily deceased, or can somehow tap into them now (meditation? psychedelics?)—can somehow free yourself of tympanic membranes and neural firings. (Which again invokes the aforementioned ability to imagine the sound of a tree falling: a non-substance-dualist might think a well-trained and vivid imagination could do it, but you still need the physically limited perceptual equipment to do its part.)

I of course don’t buy any of this. What materials vibrate in heaven? “It doesn’t work that way,” you say. Then how does it work? Even weirder: How does it work for ghosts and the incorporeal on Earth? A certain vampire on a certain vampire show spinoff comes to mind, who walks through walls and can’t pick up a pencil but somehow can stand on a floor and can convert sound and light waves into auditory and visual experiences—can in turn excite air molecules by sending them through vocal cords in order to talk and… yeah, it’s all TV magic. And that’s my point.

At any rate, my account here applies to what physical bodies can do. This doesn’t rule out some sort of dualism. Maybe Spinoza’s dual-aspect monism (which I enjoy, despite its problems) is more your cup of tea, or property dualism (which, I presume will grow in fashion as panpsychism does). As far as I’m willing to go here is to endorse a kind of conceptual or semantic dualism, wherein we can talk as though there are two problems here, one physical the other phenomenological. I think the distinction easily justified at the very least by our utter lack of an explanation for how brains turn physical stuff into experience (conceptual–semantic dualism isn’t even as strong as predicate dualism).

The upshot is that the first and second problems are really two ways of wording a single problem. This in mind, I’ll restate one last time what I’m claiming here about pitch, with an eye towards rejecting the actual, perhaps even the potential, existence of infinitely many frequencies.

It is intuitive that any given frequency struck between 440 and 466 will result in a pitch experience for a human with typical auditory faculties. It is also intuitive that the vast majority of such frequencies, even among those that a simple tone generator can produce, will result in a small number of pitches in said human (e.g., 440.001 and 440.01 will both produce in me what I call “A4”).

We humans could have been calibrated to experience any number of distinct pitch experiences in response to a given frequency (perhaps many of us are in this respect differently calibrated). Whatever the case, we will find some anchor for describing these experiences (with a common language)—e.g., 440 as A4.

I find it impossible to imagine any singular entity physically capable of perceiving every such oscillation (and what madness if it could!). Which is to say that I doubt such a creature is physically possible. Which is to say that I doubt the possibility of a non-experiencing machine that can measure every such possible oscillation.**

[**Even for the human, sophisticated frequency response need not result in a pitch. Frequencies can be felt and have influence in various ways, include those that for whatever reason don’t result in pitch. A mundane example of this is when I tune a guitar without hearing it: I need only smoothen out the “beats” as they transfer from the guitar’s ribcage to my own.]

I’ll narrow this doubt as follows. Intuitively, I don’t believe that there exists the possibility for producing—where there is no relevant physical difference between producing and detecting—just any frequency between 440 and, for simplicity, let’s say 441. I cannot prove this. But I’ll try to motivate the intuition in at two ways: with a thought experiment and with an appeal to the notion of granularity.

Bear with me with the thought experiment, as it starts out sounding like I’m proposing a real experiment, but an impossible one.

The thought experiment asks you to imagine an extremely sophisticated frequency generator and a hypersensitive frequency-detecting oscilloscope. These can be run by one or two supercomputers, but the point is that you have some elastic medium (more about which shortly) through which to propagate sound waves—one broadcasts via some vibrating membrane or fibre etc., the other receives via some other vibrating membrane or fibre etc. (the rate of which is turned into electrical impulses that can be counted, or some such).  But I’ll just call it the computer.

I don’t care how this is done so long as there is involved something that sufficiently merits the description “elastic medium” through which each frequency is propagated. In other words, it cannot simply be a transfer of bits from one machine to another. A crude and doomed example would be to simply play frequencies out loud via the above-linked Online Tone Generator, and try to pick them up with a standard guitar tuner. You won’t get far.

But use whatever method you like. Shoot photons or construct a tiny robot to play an ultra-thin string made of titanium or whatever you like. Change materials and methods along the way. So long as it makes sense to say, in an intuitively relevant way, “there was a hundredth of a cycle over such and such a period of time.” But don’t worry about sending it through the air, as this may not be the best transducer. Actually, you can send it via electrical impulses through a wire or fiber from broadcaster to receiver. My naive, layperson’s expectation is that there will be information loss/distortion sooner rather than later.

This presupposes, by the way, that the “there are infinite notes between keys 45 and 46 on piano” need not refer to piano. That’s no problem.

That is, I assume there are physical constraints on the rates at which stuff can vibrate, and if no stuff can vibrate at every possible numerical value, then frequencies for that numerical value do not exist, thus the frequencies—the actual existing—or possible frequencies between 440Hz and 441Hz, are finite in number.

And no matter how fine-grain you make your machinery—your vibrating medium and tuning mechanism (which could take many forms but with the same results that tighten/loosen, shorten/lengthen, etc), as you make your way to a distance of zero from some fixed point, you will eventually encounter a situation in which a change in your mechanism registers no change in your measuring device’s readout.

I’ll reinforce this claim in a moment. First, a bit about the significance of the readout. Physicist Lawrence Krauss writes in his misleadingly subtitled, but still enjoyable and (for me) informative 2017 book The Greatest Story Ever Told—So Far: Why Are We Here?, when explaining and riffing on the clock-on-train illustration popular in introductions to special relativity:

This may make it seem like a slowing of clocks is merely an illusion, but once again, measurement equals reality (p 60) …

When I am carrying an object such as a ruler, and moving fast compared to you, my ruler will be measured by you to be smaller than it is for me. … Surely, this is an illusion, you might say, because how could the same object have two different lengths? The atoms can’t be compressed together for you, but not for me. Once again, we return to the question of what is “real.” If every measurement you can perform on my ruler tells you it is 6 cm long, then it is 6 cm long. “Length” is not an abstract quantity but requires a measurement. Since measurement is observer dependent, so is length. (p. 62) …

“Reality” for each of us is simply based on what we can measure. (p. 64)

A possible, and welcome, counterpoint—about measurement being reality, not about special relativity—comes from someone who earned Krauss’s ire with a devastating review of Krauss’s previous book (which claims to explain how the universe came from “nothing” of a sort that in fact has stuff in it; more on this another day, link to come):  physicist and philosopher David Albert who, in a 2014 New York Academy of Sciences panel discussion, responds cordially and thoughtfully to a question about certain contemporary physicists being uncordially and unthoughtfully dismissive of philosophers:

There was a very long period during the 20th century where, if you grew up in a physics education and if you become puzzled about issues of the foundations of quantum mechanics, you were told in very stern language—language that could have consequences for your future career if you didn’t pay heed to it—that nowadays we know that physics is not about what the world is like, physics is about the business of predicting the positions of pointers on measuring instruments and so and and so forth.

And if you said something like, “Ok, so which department should I go to if I want to find out what the world is like? Because I thought that’s what physicists were doing”—this would be diagnosed as a worrisome sign of intellectual immaturity and so on and so forth.

And there was a very long period of people on various edges of physics and philosophers saying, “Gee, we thought the project here was to tell us in some fairly straightforward, realistically understandable sense what the world is like.”

That quote starts at 50:40 of this YouTube video called “The Origins of the Universe: Why Is There Something Rather Than Nothing?” (posted 10/20/14):

A few moments later (54:38), another panelist—philosopher and writer Jim Holt, still fresh from his wonderful 2012 book Why Does the World Exist? (in which he approaches blindingly difficult questions by alternating between two styles: one journalistically investigative, the other introspectively meditative)—points to a philosophical disagreement between Stephen Hawking and Roger Penrose that amounts to the former caring only about measurements and the latter caring about what the world is really like.

I operate on the side that wants to know what the world is really like. Which is to say, not just whatever Krauss means by “reality” (a word he puts in quotes, which leads me to wonder what he really means; clearly what the world is really and truly like doesn’t change when you get a better measurement, but then it’s not clear to me what Krauss means by “every measurement you can perform.”

This may make the broadcast–receiver test at best inconclusive, but if you can get it fine grain enough, it will tilt my intuition. That is, suppose you successfully cut the distance from 440 to 441 in half a trillion times (an approach that keeps the numbers rational and thus easily verifiable) before a small change yields the same readout. I’d start to feel that, in something more than a merely theoretical sense, this could go on indefinitely given improved machinery.

What will slow the tilt, however, is that I know this can’t go on forever. You can’t cut a portion of string in half forever, for example. Or think of it this way.

Suppose you have a vibraphone bar made of graphene or some similarly manipulable material that can be atom-thin. Start with it tuned to precisely 440Hz. I used to work at a mallet instrument factory where I tuned bars by carefully removing material while monitoring an oscilloscope. Imagine an oscilloscope that will accurately detect a rise in pitch after removing just one atom. So the bar is now at 440.000000000001 or some such. Now remove another atom. You get the idea. You’ll eventually pass 441Hz.

But along the way, you will not have made infinitely many measurements. Even if your oscilloscope isn’t up to the task, you will not have even tried to make infinitely many measurements. What more can you do? Perhaps you can start with a bar tuned slightly lower, then you’ll hit a new set of frequencies as you tun the bar. But this seems to face the same problem, though perhaps using a new material will help, or even going to a different part of the universe or traveling at different speeds with the broadcaster and receiver. I don’t know.

(This also suggests the possibility of attaining higher and higher frequencies, but eventually you’ll have an atom-thin bar, or wire, that can be made no thinner. That said, there’s nothing in the claim that precludes infinitely many pitches lower than, say, 20Hz or above 20,000Hz [the generic range of human hearing].

I won’t hold the claim accountable for the idea, however, that there must be material that can vibrate at 0.0000000000001Hz; I imagine the lower limit is less constrained than the higher, as the slowest speed possible is zero.

This raises another issue with going outside of the range of human auditory perception: what counts as a cycle starts to seem arbitrary. In fact, this danger exists within the range in question as well. That is, it may be possible to appear to achieve infinitely many frequencies by arbitrarily drawing boarders around events in the world. Clearly this won’t do. I’m looking for periodicity of a sort that could at least at a stretch be conceived of as a note—as something that in some circumstance could result in a pitch. I don’t know what the constraints are in that respect. But a cycle in which “something happened and then something else happened,” where the first of those is the Big Bang and the send is the singular, composite event of “Socrates putting on his left sandal and Donald Trump sneezing” clearly won’t do.

But, yeah, for all we know: the life of our universe is but a single cycle in an audible waveform whose periods are universes that together sound, in the mind of some unfathomable god-like creature, like so much meaningless static or perhaps something sweeter.)

I’ve invoked here the notion of indivisibility of matter. But we can go even further than that, to the indivisibility of space. Which is to say the granularity of space, an idea I occasionally hear physicists murmuring about (a quick Google search on “space is granular” bears this out).

The most popular physicist I’ve seen talking about this is Max Tegmark, in his status-quo challenging 2014 book Our Mathematical University: My Quest for the Ultimate Nature of Reality. The book’s core premise is that “our physical world not only is described by mathematics, but that it is mathematics: a mathematical structure, to be precise” (p 6), a strong claim I’ve so far not heard any other professional physics type agree with.

My favorite ideas in this book full of ideas, and the reason I bought it, are those related to his skepticism about infinity (from pages 316–317):

We have no direct observational evidence for either the infinitely big or the infinitely small. We speak of infinite volumes with infinitely many planets, but our observable Universe contains only about 1089 objects (mostly photons). If space is a true continuum, then to describe even something as simple as the distance between two points requires an infinite amount of information, specified by a number with infinitely many decimal places. In practice, we physicists have never managed to measure anything to more than about sixteen decimal places.

I remember distrusting infinity already as a teenager, and the more I’ve learned, the more suspicious I’ve become. …

Among physicists, my skepticism toward infinity places me in a very small minority. Among mathematicians, infinity and the continuum used to be viewed with considerable suspicion. …

In the past century, however, infinity has become mathematically mainstream, with only a few vocal critics remaining…

So why are today’s physicists and mathematicians so enamored with infinity that it’s almost never questioned? Basically, because infinity is an extremely convenient approximation, and we haven’t discovered good alternatives. For example, consider the air in front of you. Keeping track of the positions and speeds of octillions of atoms would be hopelessly complicated. But if you ignore the fact that air is made of atoms and instead approximate it as a continuum, a smooth substance that has a density, pressure and velocity at each point, you find that this idealized air obeys a beautifully simple equation that explains almost everything we care about from how sound waves propagate through air to how winds work. Yet despite all that convenience, air isn’t truly continuous. Could it be the same way for space, time and all the other building blocks of our physical word?

Then on page 369:

A rubber band looks nice and continuous, just like space, but if you stretch it too much, it snaps. Why? Because it’s made of atoms, and with enough stretching, this granular atomic nature of the rubber becomes important. Could it be that space too has some sort of granularity on a scale that’s simply too small for us to have noticed? … We use this continuous space model in most of the physics classes we teach at MIT, but do we really know that it’s correct? Certainly not! In fact, there’s mounting evidence against it…

Tegmark asks here about space and time. But the claim is not this controversial, only goes as deep as asking for some material—see again Tegmark’s uncontroversial rubber band—that can vibrate at infinitely many frequencies between those produced by striking keys 49 and 50 on a standardly tuned piano, such that the frequencies are then propagated through some elastic medium, making its way to some perceptual equipment that turns it into pitch.

I think I’ve given good reasons to be very skeptical of this claim, even if we play loose with where those frequencies exist—e.g., we need not assume that live, in any literal sense, between keys 49 and 50 on that piano, or even that they’ve ever actually been produced: only that they do seem to have the potential to be produced. Likewise for pitch, which I hold to be conceptually (if not exactly physically) different than asking about frequency, given several factors that I think not only justify this conceptual difference, but demand it, most notably: we have no idea how stimuli are converted into experience; there is an undeniable first-person, subjective dimension to pitch; a great many measurable frequencies result in the same pitch experience (though pitch may be seen here as merely the subset of frequencies that a given brain can instantiate and thus can accommodate as as external stimuli, much like the limitations of a coin sorter of, say, American coins).

In the end, while I accept that there are theoretically—or somehow potentially, even—infinitely many frequencies, notes, and pitches, I am unable to accept that there are in fact infinitely many of these objects in the world we actually live in.

Now for some closing thoughts.

Why do I think this question is worth a few moments’ contemplation? I can think of a few answers. One is that I’m in general interested in claims that are meant to illuminate some counterintuitive feature of the world, particularly when meant to be in some way mind-blowing or mind-expanding. This claim is also about the relationship between the world as experienced—i.e., as constructed in our mind, as navigated—and the world as it is. It also goes an extra step: it’s about the world we could be constructing in our mind if only we had access to it, if only we had infinitely many keys between numbers 49 and 50 on our piano. A mind-blowing thought, except that we can only actually hear a very small subset of those keys.

I also think the claim’s contemplation—along with the bigger question of why people feel moved to make it—is a kind of gateway to similar questions to do with the relationship between the world in itself and what we carry around of that world in our minds, constructed through perception and language and logics and models and so on.

(While refraining from over-excitedly indulging such notions by holding in mind John McWhorter’s sobering and excellent 2014 book The Language Hoax: Why the World Looks the Same in Any Language, whose subtitle is an obvious response to that of a 2010 book by Guy Deutscher I admittedly haven’t read: Through the Language Glass: Why the World Looks Different in Other Languages, which McWhorter claims the media have characteristically overblown despite Deutscher’s care to “argue that language’s effects on thought is modest, hedging the issue as responsibly as we would expect of academics” (McWhorter, p XV).)

My contemplation of all this leaves me ambivalent—annoyed and sympathetic. I’ll say something about each. If I seem to venture from the point, I’m not: I have my reasons.

Annoyed because the claim is false (or at least suspect), seems to be stated without much contemplation by those who say it (I’m sure I said it plenty of times to students when I was teaching music, long ago), and on top of this I get the sense that to reject the claim will get one accused of a kind of stuffy, cold-hearted, closed-minded hubris.

These adjectives don’t apply at all. In fact, the claim often strikes me as a kind of feeble excuse: “I don’t have enough colors in order to make the music I could otherwise be making.” The truth is note only that there are endless possibilities with even the 12 notes of Western tuning, but that it is rare to only rely on those 12 notes. As I noted earlier, most musicians spend much time in-between those notes (see again vibrato). But it goes deeper than this. Even on a piano, where you can’t bend (unless you reach inside to engage the strings more directly), the in-between notes are there, in the slightly off tuning between the many strings and, if that’s not enough, in each individual string itself!

The harmonic series—a series of frequencies that occur simultaneously when, for example, you strike the piano’s 49th key or you sing the note A4—is naturally out of tune by the standard of equal temperament. Even by the third partial, we are about two cents (i.e., a hundredth of a half step) above “perfect,” and at the fifth partial, we’re 14 cents below. The in-between notes—the “microtones” as they’re sometimes called—are right there even on a single, open guitar string.

How arbitrary or subjective, then, our notion—we, I’ll remind you, natural creatures that we are—of this or that tuning system being the correct one. How this interplays with our biology and meaning processing surely contributes to the complex account of, for example, our emotional response to music (a good attempt at which would acknowledge, for instance, the difference between merely sounding sad and actually inducing  sadness: the former requires only facile observations such as the, say,  E♭ in a minor C chord being in tension with the second overtone of the root note C’s, harmonic series which is E; while the latter my include this as only relatively small entry in a much richer account).

But the point here now is that the microtones—the in-between note—are there, even on the piano, an instrument without vibrating notes, and can be heard to varying degrees throughout the keyboard’s length: notice the difference in richness between playing keys number 1 and number 88.

There is far more complexity, then, in the frequencies (which, again, can apparently affect us even when we can’t hear them) and pitches than there are in the notes we put down on paper. It’s well understood that pianists make up for this with dynamics, pedals, sometimes reaching inside the keyboard, manipulation of harmonic nuance, and perhaps even digital manipulation. And, of course, writing to the instruments timbrel strengths, As gorgeous as Chopin’s music is, some of my favorite every written, I never love it as much when as performed on piano—those arrangements pull every bit of richness out of the instrument, of a sort that no other instrument as. Likewise, I never love Wagner so much as when performed by an orchestra. And Beethoven’s Grosse Fugue, as I’ve noted before, one of my all-time favorite pieces—I’ve never heard it work on any instruments other than a string quartet (guitars, forget it; piano, so-so).

At any rate, all of this has to do with timbre, which is itself largely a matter of how loudly various partials ring relative to one another, which is to say how many “off” partials are making it through. The point again: the microtonal complexity is there, even with just 12 notes.

Sometimes this is experienced as a nuisance. And not just as a phenomenon we arbitrarily call “going out of tune.” I’m reminded of a Tape Op interview I read in 2006 (Issue #55, Sept/Oct 2006) with the band Deerhoof, where it was said that, while recording, they’d retune their guitars from chord-to-chord in order to get each chord as in-tune as possible. Sometimes when you play guitar, by the way, you might slightly bend notes while playing a given chord in order to improve intonation; one of the many things an experienced guitarist does unconsciously in order to shape their sound.

That said, I’m sympathetic to the claim for several reasons. A simple one I’ve already commented on. There are non-Western tuning systems that break the octave into more than 12 notes, and this is obviously just as valid as what most of us in the U.S. are raised to think sounds “correct.” I’m hyper aware of this. As a very young child, I heard a lot of Middle Eastern music at home, which generally uses more than 12 notes. Perhaps this was why when asked to sing a half step in a music class, but in the terms, “sing the smallest interval you can sing,” I sang something more like an 8th step, and whole steps sounded huge to me (I recall in ear training mistaking a whole step for a fifth because I’d be told a fifth was a wide interval).

Plenty of folks, by the way, brought up on the Western 12-note octave also explicitly work with intervals of smaller than a half step. My favorite example from the pop-rock world are the slow single-note guitar melodies of Marty Friedman when he was in the band Cacophony back in the 1980s, beautiful stuff and hugely influential on me.

In the classical (or whatever you’d like to call it) world, there are composers who use, or have used, alternative tuning systems. One such whom I like quite a lot is composer Henk Badings, who worked extensively with a 31-note temperament, generally otherwise in what me make call a fairly traditional Western context; I happen to enjoy that context and often find Badings quite inspired. I have no need to be bathed in a wash of microtones for its own sake, as some aficionados sometimes strike me as being in search of.

Not that there isn’t striking music of that sort—the gorgeous microtonal swirls that open the first prelude of Ivan Wyschnegradsky’s  “24 Péludes” (opus 22) come to mind:

Wyschnegradsky was an interesting guy who was dealing in ideas as much as in sound, in which he sought what he called “pansonority,” which is described at the Huygens-Fokker Foundation (a “centre for microtonal music”) entry on the composer: “the sound of an infinite number of tones, sounding simultaneously in an infinite sound continuum.” On a website dedicated to Wyschnegradsky, it is described as something that “includes ‘an infinity of sounds arranged at an infinitely small distance.'”

The latter quote is from a book by the composer called La Loi de la pansonorité, Contrechamps, 1996, p. 68.—which is available at Amazon in French at a very reasonable price and I’m resisting the temptation to go down its rabbit hole, but do see that it lists the three properties of “pansonorous space” (“l’espace pansonore”) as “infinity, continuity, uniformity.”)

Intriguing stuff, for sure, from a guy who did “chromatic drawings” like this one:

Ivan Wyschnegradsky
From the Association Ivan Wyschnegradsky website.

I’ll save that book—the above-linked edition of which includes contributions from others (such as microtonal composer/musicologist Pascale Criton, who dedicates her contribution to “the memory of Gilles Deleuze”)—for another day (if I translate any, I’ll share). I’d be especially curious to know what relation and distinction, if any, he explores between frequency and pitch (as I define them). At any rate, I do find it fascinating that his music extends, it seems, from a deeply considered, even well-developed, philosophical account of mind–world engagement.

My favorite composer to deal in microtones also happens to be one of my favorite composers, period: Gloria Coates. She’s been known to use microtonal tunings (as discussed in this 2010 interview with Nolan Gasser), but most of her work involving microtones, I believe, does not require special tuning, is accomplished rather with glissandi, especially on bowed string instruments (which, of course, are fretless).

From the first interview I heard with her—which I might still have on a hard drive somewhere, from a podcast series dedicated to the Naxos American Classics catalog—I have a vivid memory of her describing her notation as often indicating, for a given musician, a start and end note, along with a duration of time. The idea was to slide, at a steady rate, for that duration from the first to second note, no breaks in sound. This can take a while. And it makes the music relatively easy to play. When several musicians do this with differing instructions, the result is complex but coherent (from the human’s perspective). Or, rather, is as if Coates has an order–chaos dial over which she commands a soft touch.

The result of all this is jarringly and sometimes ominously, always transcendently, beautiful music. It boggles my mind how she manages it. But I also know she puts a lot of effort into getting those sounds from her mind into ours.

In the above-linked interview, she mentions that she in part began working with glissandi due to being a visual artist: “I was building structures visually that could also exist in sound—I think that’s how it all began.” More recently, she has explained that the visual work comes more quickly than the auditory:

As for Coates’s music, I could spend all day curating a playlist, but will restrict myself to two.

First, the second movement (“Puzzle Cannon”) of her 15th symphony:

Notice the long glissando that is the work’s centerpiece. Apropos to today’s theme, if you listen all the way through, you will experience every pitch there is for you to experience between the first and last pitch of the glissando. It’s not literally infinite, but maybe it is acceptable in a figurative sense—perhaps in the sense Wyschnegradsky means the word, perhaps in the sense my music elders meant it, or would have if they’d given it some contemplation (more on this in a moment).

Something similar occurs in the next Coates work I’ll share, but you have to be on the lookout for it. The stunningly brilliant Holographic Universe:

Whatever pitches can be produced, you will hear them in her works. There’s a temptation to call them “otherworldly,” but I reject that label: they are very much of this world (projections? abstractions? de-abstractions that reverse the abstractions we live with? saturations? lattice-illuminators? I don’t know), and that in large part accounts for their special and sometimes unsettling beauty.

For more on Coates, here’s an interview with her at Sequenza 21, and here’s her Wikipedia entry, where you’ll find a listing of compositions.

That said, I still do not think of myself as a microtone aficionado. I don’t seek out microtonal music for its own sake: I seek out music that moves me. Most of that has as its framework the current standard tuning system because that’s the overwhelming norm of the culture I was raised in. And the overwhelming majority of the performances of that sort of music that I (and you) love has as integral to its appeal a deft navigation of the “in-between” notes, in some form or another, of which there are many (e.g., vibrato). And some of the music I love involves other kinds of tunings. All tunings are perfect and none of them is: that’s nature.

(As an entryway to learning more, see the Wikipedia entries “Quarter Tone” and “Microtonal Music.”)

That said, I’ll get to my point with going on about these composers. I noted that, in the end, I reject the claim as being literally true, but also that it could, or should, be somehow figuratively true. Which is to say that I sympathize with the spirit in which it is often said. There’s a world of experience we don’t want to cut ourselves off from, particularly at an instrument as rigidly tuned as the piano—for string players, for example, playing in tune can be as difficult as trying to sing in tune, even for guitarists despite having frets (people tend to push and pull too hard on the strings, making them go sharp).

Beyond this, as composers, interpreters, and listeners, we want to realize that there is much more to the world than what notation shows us, and even than what we tend to encounter as casual participants going about our daily lives.

It’s an attempt to get us to reach further, to notice more, and so on. While I don’t like this as an excuse for not making the most of the vast possibilities afforded by even just five rigidly stable pitches, I see the point: push harder, look harder and further, bigger and smaller. I don’t believe this will generate or tap into new emotions or new perspectives that couldn’t be inspired by a brilliantly arranged set of five notes***, but such a search might find more ways to get there than originally considered, might expand one’s inner and outer vistas, and who knows what else.

[***Or just three notes, as in the second movement of György Ligeti’s Musica Ricercata, which was used to great effect in Stanley Kubrick’s 1999 film Eyes Wide Shut:

There are, of course, more than three notes there if you count the overtones. So even here, if you listen closely, you can hear notes between those strictly assigned as names to the keys.]

This spirit in mind, I did a quick Google search and found several instances of some form of the claim. The vey first hit leads to a 2005 book unknown to me, by Angie Choi, called My Dreams: A Simple Guide to Dream Interpretation. The relevant passage:

Consciousness is a spectrum of different state of awareness and we constantly slide along the spectrum. Where one state begins and ends is hard to determine because there is a ceaseless spectrum between any two points of consciousness. The following example of piano notes illustrates the consciousness spectrum. Even though there are eighty-eight keys on the piano, there are many more than eighty-eight tones that the human ear can hear. In fact, between any two tones or notes on the piano such as middle C and D, there are an infinite number of tones that can be produced. Each note may be slightly different, and it may be hard to hear the difference between the preceding and succeeding notes, but they are distinct. (p 2)

A great point, even if I disagree with the factual details or find the language fuzzy or am not a devotee of systematic dream interpretation. There are experiences we’re missing out on. Musical and otherwise. (I happen to think that making the most of our experiential capacity, aesthetic and otherwise, is the closest thing there is to a “meaning of life,” but I’ll save that thought for another day.) We need to look between the cracks of what we’re generally given. Persistent musicians get there sooner or later, know it or not—bending and vibrato and so on. But this is just the beginning. For me, this is about one more tool, and a powerful one, in a musician’s (and music listener’s) toolbox; for others, the toolbox seems to be the point. Good for them.

Maybe I’m missing out on more than I realize by not digging deeper into microtonal music (or doing magic mushrooms or month-long meditation retreats, for that matter). Maybe some of those daydreaming about the notes in between the piano keys are missing the vistas available to them in the harmonic world of a single key strike.

The search continues…

Addendum: What I’ve said here is based on my observations, experiences, reading, and intuitions about a fairly broad subject matter. I welcome corrections and refining and further fleshing out from those with the appropriate expertise. If that’s you, I’d even more love to know your thoughts about a far more speculative article of mine: “Memory and Consciousness (via Audition).”

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Further Reading

4 Replies to “Infinite Divisibility of Frequencies, Notes, Pitches”

  1. Some interesting ideas. It seems to me more of a problem of measurement and the limits of human knowledge. “Can it be observed?” isn’t the same as “Does it exist?” I suppose there are some philosophies that claim that, but I wouldn’t say that microscopic distances don’t exist because my ruler only measures to 1/32 of an inch.

    There’s also a sampling limitation. 110hz means 110 compression cycles within the space of a second. Unless your measuring device (and I’m not sure how computing and/or human apperception work here) can detect a fraction of a sine wave, you’re limited to whole numbers unless you can sample/hear more than a second of a note. And I doubt that human hearing is precise enough to continually evaluate pitch after a note has been sustained for more than a second.

    I’ve run into this problem before when tuning instruments using the “graph spectrum” function on Audacity. The function only displays whole numbers, which means I have to settle for “close enough” in the low register, where there might be a difference of only 10hz between two pitches in 12TET.

    If you want to go a bit deeper, you could interrogate whether pitch itself exists. The harmonic series is a theoretical ideal that is based on a one-dimensional shape, like a string or an air column. But once you consider things like the mass or stiffness of the string or the dimensions of the air column the upper partials might not be exactly perfect ratios to the fundamental. And two- or three-dimensional vibrating objects follow their own irregular harmonic series entirely; sometimes it is close enough to the expected harmonic series that we hear it as such, but sometimes it completely loses the appearance of pitch.

    1. Hi Kyle! Thanks for the thought-provoking comment.

      I certainly agree that the readouts on measuring instruments shouldn’t be taken as the final word for what’s real (in my post, hopefully I seem more sympathetic to David Albert than to Lawrence Krauss on this point). That said, a couple of points of clarification.

      I’m defining “frequency” in the usual sense. I’d say if there’s no medium that can be structured so that it will oscillate at a given frequency, then the frequency doesn’t exist. It seems likely that such limitations do in fact obtain, given that the infinite divisibility of matter—or at least of any such medium—seems unlikely. But who knows.

      I define “pitch” as a mental event (i.e., as experience, or what is often called a “percept”). I’d say if there’s a pitch experience that cannot be had by any potentially existing mind, then the pitch doesn’t exist. In broader terms, if something is impossible to observe, such that no entity—human, nonhuman, god, or machine—can observe it, then, even though that thing exists, the observation of it doesn’t.

      In light of the above, it strikes me that, when someone says “there are infinitely many frequencies/notes/pitches between the A4 and Bb4 keys on this piano,” it’s an open question whether that claim is true. Though my intuition is that it’s false. My sense, however, is that people don’t usually mean the claim in such broad or well-defined terms. In which case, whether or not it’s clear that something false is being said will depend on what the person actually means.

      Great point about the two-/three-dimensional vibrating objects.

      I definitely take your points regarding “whether pitch itself exists,” at least, as I understand you: i.e., our tidy models of how we experience the sounds made by musical instruments (for example) are just that: tidy models. More precisely, a family of models that includes our concepts of those things, so that we can talk about them and notate them; as well as our actual mental representations/perceptions of them (perhaps independent of whether we have concepts, words, or symbols for them). I see this as similar to the question of whether triangles exist as anything other than theoretical ideals.

  2. This here is the truly good stuff on the internet. I really like your article, Mr. Wallace, and how you reference/recommend so many other things. The difference between the theoretically existent and the practically percievable is fascinating. I love the longer-and-more-intellectually-strenuous vocabulary you so unapologetically utilize, and just the quality of the content. I appreciate your thoughts and efforts to write all that out.
    I am a high school student and this will help me with my research paper, on a related topic.
    So just a message of approval and encouragement from a random reader who appreciates your efforts. I also really like math, and music, and the intersection between the two. This is the first page of yours I have read but I will likely read some of your others, on various topics.

    1. Hi – Thanks for checking it out and for your thoughtful comment! Glad to be of help with what sounds like a fascinating research topic.

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