Can music time signatures really be irrational?

Question

In this video around the 5 minute mark, the presenter mentions that "irrational units of time cannot exist because of Planck's constant", and therefore music time signatures like 2:√2 cannot be perfectly performed in the real world (he used Conlan Nancarrow's Study 32 as an example).

Does this mean that the "units of time" are in a way discrete, i.e. there is a smallest unit of time and all others will be multiples of it, similar to the natural numbers and the unit 1? And does this imply that, whenever a music piece with an irrational time signature is performed by electronic means or a player piano, what is really happening is just a very good rational approximation of the signature? i.e. if the signature was π:1 (Pi:1), then it would really be something like 3927:1250? Am I understanding this all wrong?

Answer 1

I'll give this another spin:

Can music time signatures really be rational?

Which I'd answer: no, not really. Rationality is a mathematical concept, depending on an exact, axiomatic notion of numbers. Now, sure, any ordinary piece of music will use a rational signature given by integer numbers on paper, or in the DAW you use. Time is conceptually divided in a rational way in these compositions.

But is the conceptual level really what the music is? I wouldn't be sure. To the listener, music is first just fluctuations in the air pressure field. Our brains do a remarkable job of disassembling that mess again... they will notice that there are certain individual voices. They will perceive that these voices are somehow synchronised, that there are some scales of repetition etc.. If they know something about music they'll usually also be able to guess from that again what the numbers on the sheet were that the composer wrote.
But this is not quite an objective, reproducable process. Upon close investigation, you will find out that any human performance has slight tempo etc. fluctuations intrinsic in it. And, which may be more surprising, even a purely electronic, “exact” sequenced piece, when analysed from the audio mix, will seem to have such fluctuations, on a much smaller scale. The reason is actually mathematically akin to the quantum-mechanical Heisenberg uncertainty relation^†: whenever you pack information into any sort of waves (be it electromagnetic radio waves or acoustic sound waves), you must make a tradeoff of frequency accuracy vs time accuracy. In exact terminology, transients in signal that has a frequency-bandwidth f_U can have a time-accuracy of at most ≈ ¹⁄_fU.

Human hearing has a bandwidth of <20 kHz. Signals in that range have in principle a time certainty of at most 50 μs. In fact, humans can determine signal time only to about 3 ms... but whatever the exact numbers, there is in principle a limit to the precision. Therefore, it can never be possible to objectively distinguish a piece in ⁴⁄₄ time from one in ^4.00000001⁄₄, or in fact from the irrational ^{cosh(2.0634371)}⁄₄.

So why can we still be sure a given piece is in ⁴⁄₄ time, and not in ^{cosh(2.0634371)}⁄₄? Occam's razor. The simplest possible model that fits – within the available accuracy – the observed data, is the one that brings us closest to understanding what's going on.

Now there's an interesting detail: what do I mean by “the simplest possible model”? Simplicity can't really be defined, it is different for every human what seems simple and what seems complicated. In information theory, there exists a thing called Kolmogorov complexity. It's a tricky quantity, in fact it can't be computed – but it's still well-defined and can be decently approximated by the shortest program someone will send in to a given challenge on CodeGolf.StackExchange.

For example, ⁴⁄₄ has a complexity of at most three characters or 24 bits, whereas the number 2.0634371 alone has a complexity of at least 30 bits.

But there are examples that are less clear-cut. In particular, there are some irrational numbers whose complexity we should consider rather low. The number π has a complexity of, like, one character, whereas the rational 3.1416 has already more than 16 bits complexity.

Therefore I would argue that a piece in ^π⁄₄ should be considered to be in ^π⁄₄ time, not in ³⁹²⁷⁄₅₀₀₀.

Can music time signatures really be irrational?

Yes they can, in every sense that the concept of time signatures can make sense at all.

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^†_{These uncertainty-effects actually have little to do with quantum fluctuations, physically – they're just mathematically analogous, but the physics is only classical mechanics.
Theoretically speaking, quantum mechanics does put an even more fundamental bound on how exactly we can measure time: time-uncertainty times energy-uncertainly must be larger than the Planck constant. (This can be seen as the reason why particle-physics experiments need to put in such enormous energies: some of the particles they're studying there have extremely short life-times, which can only be resolved by allowing for enormous energy-fluctuations.)
Applied to music, there is an energy-uncertainty of at least the average thermal energy that particles have at body temperature: Δ E ≥ k · Δ T = k · 37°C = k · 310 K ≈ 4.3×10^-21 J. If you calculate the time-uncertainty from that, you get 1.5×10^-13 s. Now, that's a time scale we can actually resolve with high-tech equipment (e.g. femtosecond lasers), way longer than the Planck time. But it's still many orders of magnitude smaller than the time-uncertainty due to classical effects, which I discussed above.}

Answer 2

This is so over-hyped as to be irrelevant for any practical purposes. The wavelength of the highest pitch sound that humans can hear is about 16mm. That is about 1,000,000,000,000,000,000,000,000,000,000,000,000 times longer than the Planck wavelength, i.e. the distance light travels travels in one Planck unit of time.

Even taking into account the approximations made by current computer systems (for example, representing numbers approximately to "only" about 16 digits of precision) the accumulating timing errors would be barely perceptible in a piece of music with a duration of a million years.

In fact the commonest use of "irrational time signatures" in music theory are not mathematically irrational numbers at all - they are simply fractions with denominators which are not powers of two, for example 4/3 (which means the length of a bar is a quarter note - 3/3 - plus one note from a triplet of 8th-notes.)

Answer 3

He's just showing off.

There's a few major reasons why what he describes doesn't matter. First and foremost, sheet music is a guide. It's not actually the music. You are always expected to put your own experience into the notes before they get called music. Thus, you would never want to play an exact transcription of the black toner on the page. It wouldn't be music.

Beyond that, let's pretend you want to play such a piece. You in theory cannot, because you cannot properly play the irrational ratios. But can you play any song? It turns out that, by that standard, you can't. Let's say you want to play a quarter note at 100BPM. Let's say you want to play an A. 100BPM is 0.6 seconds per quarter note. The frequency of A is 440Hz, so you'd have 264 oscillations in that quarter note. Now let's play a B after that. B is 493.88...Hz. Uh o. Now I need 296.328... oscillations. But if I stop right there, I'm not actually at the end of a cycle. I have to use a "windowing function" to stop the note when it's amplitude is not 0. This causes an infinite series of harmonics to be unleashed. It's no longer a pure B.

So if I try to hold myself to the standards put forth in this video, I can play a quarter note A at 100BPM, but I cannot play a quarter note of B. The standards set are simply too high to permit a B to be played.

So go out there and play real music, using the sheet music as a guide. Plank's Constant, Avagadro's Number, and all the other useful numbers in science will be there to back you up.

Answer 4

Can music time signatures really be irrational?

One thing that is worth noting is that the term irrational time signature uses 'irrational' in a different sense to the normal mathematical meaning. https://en.wikipedia.org/wiki/Time_signature#Irrational_meters.

Another thing is that I don't think there would be a commonly-understood way of notating music using standard notation in a time signature that was irrational in the normal, mathematical sense. I could be wrong there, but for the rest of my answer I'll talk about irrational ratios of time 'in general'.

"irrational units of time cannot exist because of Planck's constant"

They can exist 'conceptually', and be represented in numbers. It may not be possible to make something actually happen for that length of time - I'll let the physicists argue about that one!

.... therefore music time signatures like 2:sqrt2 cannot be perfectly performed in the real world

I think this statement is misleading for two reasons.

Firstly, Even if our timing resolution were limited by the Planck time, then we could still play pieces perfectly whose event lengths were exact multiples of the Planck time.

Secondly, in our actual real world, timing resolution is limited in all sorts of ways that are not related to the Planck time - so the Planck time wouldn't actually be the limiting factor in many (or any?) real-world situations.

Does this imply that , whenever a music piece with an irrational time signature is performed by electronic means or a player piano, what is really happening is just a very good rational approximation of the signature? i.e. if the signature was pi:1, then it would really be something like 3927:1250 ?

Well, it is true that a digital computer will need to represent an irrational number with a rational approximation. However, this approximation could be as precise as necessary - it could even be much more precise than the resolution of Planck's time!

Answer 5

It’s wrong.

Planck time is not a “tick” in the universe; events are not aligned on tick boundaries. Rather, it is a duration that is the smallest that can be measured as being before or after other times.

If you measure the time of some event, there will be uncertainty as to the exact measurement. The tightest measurement possible (when the conjugate value is completely uncertain) is Planck’s time.

So, given a piece of music of finite length, you draw your notes and measures with fuzzy lines having some tolerance to them. You can then always choose a rational number such that the needed time points where you draw your theoretical lines are always within the tolerances of the measured lines.

Answer 6

As a practical answer: with completely computerized music (through probably not in traditional DAWs, which try to save you from "bad decisions" like this), you can come up with music that is not technically irrational, but so close to it, the human ear could certainly not tell the difference (e.g. if you wanted pi/4, it could easily give you 3.14159265358979323846264338327950288419716939937510/4). Whether you'd want to do that, I leave to your artistic vision.

Answer 7

An amusing question.

More relevant than Planck's constant is, as leftaroundabout mentions, Heisenberg's uncertainty principle: you could never measure a piece of music accurately enough to say whether the time signature was irrational.

There is also a mathematical problem. Even if we move to a dream universe in which space and time are infinitely divisible, exact measurements are possible, and we live forever; we still could never say whether a piece of music was in π / 4 time or just a very close rational approximation to it.

Irrational numbers arise in music elsewhere: intervals. In a just tempered scale a perfect fifth has a frequency ratio of 1.5. So, the waveform of a perfect fifth will repeat every 2 wavelengths of the lower note. In a well tempered scale, the interval (in the dream universe that I just mentioned) would have a ratio of 2^(7/12) which is irrational. So, its waveform will never repeat. It will start looking very much like the just tempered perfect fifth but gradually the phase of the two notes will shift. It will never quite repeat but it will come arbitrarily close to doing so.

Topo morto mentions irrational meters which was interesting and new to me. These are not irrational in the mathematical sense; they just don't have a denominator which is a power of 2 as is usual. Despite being a mathematician, this terminology does not upset me: I don't feel a need to impose mathematical definitions of terms on to other disciplines. Anyway, even within mathematics, terms might be defined quite differently in different contexts or from author to author.

Answer 8

The 'Planck's constant' reference is just thrown in to sound clever, to dazzle us with science.

Experimental composers seem to use non-standard time signatures to indicate tempo relationships - the sort of thing where three quintuplet quavers in the old tempo add up to a double-dotted minim in the new one. And I'm only slightly exaggerating!

My feeling is the same as when CERN discover yet another fundamental particle - yes, your analysis may come close to explaining the observed facts, but there MUST be a simpler way to describe it!

Answer 9

Generally, the composer tries to make it easier for the player to understand and play. Time signatures like 3927/1250 would certainly not be used in regular music, due to the fact that

1. It unnecessarily complicates things
2. There is no need
3. It makes it extremely hard or impossible for the player to play (therefore needing an electric music software to play)

Lastly, no matter what the numerator of the time signature is, the denominator must be a power of 2. This is because 1 represents a semibreve, 2 represents a minim, 4 represents a crotchet, 8 represents a quaver, 16 represents a semiquaver, and so on.

Since experimental music chooses sometimes to ignore these rules, they may contain some of these exotic time signatures, but the whole purpose would just to fit in the theme of experimentation. However, they would all use an electronic software. Although irrational units of time (for the extreme experimenters) do not exist because of Plank's constant as mentioned in the video, obviously one experimenter could develop a piece with an approximation of the irrational number.

There is some music with non-integer numerators in time signatures, however they are simple to understand and easy to play.

For example, the time signature (4 1/2)/4 (without parenthesis, of course) dictates measures of four quarter notes then an eigth note. An example of a piece utilizing this is Touch Piece, for piano, by Gardner Read.

Answer 10

Interestingly no one thinks about what the two numbers in a time signature mean. The second note tells us which note is 1 beat. The first number tells us, how many beats are in one bar. Thus the meter 2:4, 2:8 2:16 are all the same. Clearly also 2:3 would be the same meter, and finally even 2:Sqrt(2) is the same meter.

If we want an irrational time signature, the first number must be an irrational number.

Now lets make an example with an irrational time signature.: Sqrt(2):4, with a tempo 60bpm. Thus one bar is Sqrt(2) seconds long - which makes perfect sense. Btw.: Plancks constant has nothing to with that problem.

Conclusio: There are irrational time signatures.