Music and Physics

Pre-scriptum (dated 26 June 2020): These posts on elementary math and physics have not suffered much the attack by the dark force—which is good because I still like them. While my views on the true nature of light, matter and the force or forces that act on them have evolved significantly as part of my explorations of a more realist (classical) explanation of quantum mechanics, I think most (if not all) of the analysis in this post remains valid and fun to read. In fact, I find the simplest stuff is often the best. 🙂

Original post:

My first working title for this post was Music and Modes. Yes. Modes. Not moods. The relation between music and moods is an interesting research topic as well but so it’s not what I am going to write about. 🙂

It started with me thinking I should write something on modes indeed, because the concept of a mode of a wave, or any oscillator really, is quite central to physics, both in classical physics as well as in quantum physics (quantum-mechanical systems are analyzed as oscillators too!). But I wondered how to approach it, as it’s a rather boring topic if you look at the math only. But then I was flying back from Europe, to Asia, where I live and, as I am also playing a bit of guitar, I suddenly wanted to know why we like music. And then I thought that’s a question you may have asked yourself at some point of time too! And so then I thought I should write about modes as part of a more interesting story: a story about music—or, to be precise, a story about the physics behind music. So… Let’s go for it.

Philosophy versus physics

There is, of course, a very simple answer to the question of why we like music: we like music because it is music. If it would not be music, we would not like it. That’s a rather philosophical answer, and it probably satisfies most people. However, for someone studying physics, that answer can surely not be sufficient. What’s the physics behind? I reviewed Feynman’s Lecture on sound waves in the plane, combined it with some other stuff I googled when I arrived, and then I wrote this post, which gives you a much less philosophical answer. 🙂

The observation at the center of the discussion is deceptively simple: why is it that similar strings (i.e. strings made of the same material, with the same thickness, etc), under the same tension but differing in length, sound ‘pleasant’ when sounded together if – and only if  – the ratio of the length of the strings is like 1:2, 2:3, 3:4, 3:5, 4:5, etc (i.e. like whatever other ratio of two small integers)?

You probably wonder: is that the question, really? It is. The question is deceptively simple indeed because, as you will see in a moment, the answer is quite complicated. So complicated, in fact, that the Pythagoreans didn’t have any answer. Nor did anyone else for that matter—until the 18th century or so, when musicians, physicists and mathematicians alike started to realize that a string (of a guitar, or a piano, or whatever instrument Pythagoras was thinking of at the time), or a column of air (in a pipe organ or a trumpet, for example), or whatever other thing that actually creates the musical tone, actually oscillates at numerous frequencies simultaneously.

The Pythagoreans did not suspect that a string, in itself, is a rather complicated thing – something which physicists refer to as a harmonic oscillator – and that its sound, therefore, is actually produced by many frequencies, instead of only one. The concept of a pure note, i.e. a tone that is free of harmonics (i.e. free of all other frequencies, except for the fundamental frequency) also didn’t exist at the time. And if it did, they would not have been able to produce a pure tone anyway: producing pure tones – or notes, as I’ll call them, somewhat inaccurately (I should say: a pure pitch) – is remarkably complicated, and they do not exist in Nature. If the Pythagoreans would have been able to produce pure tones, they would have observed that pure tones do not give any sensation of consonance or dissonance if their relative frequencies respect those simple ratios. Indeed, repeated experiments, in which such pure tones are being produced, have shown that human beings can’t really say whether it’s a musical sound or not: it’s just sound, and it’s neither pleasant (or consonant, we should say) or unpleasant (i.e. dissonant).

The Pythagorean observation is valid, however, for actual (i.e. non-pure) musical tones. In short, we need to distinguish between tones and notes (i.e. pure tones): they are two very different things, and the gist of the whole argument is that musical tones coming out of one (or more) string(s) under tension are full of harmonics and, as I’ll explain in a minute, that’s what explains the observed relation between the lengths of those strings and the phenomenon of consonance (i.e. sounding ‘pleasant’) or dissonance (i.e. sounding ‘unpleasant’).

Of course, it’s easy to say what I say above: we’re 2015 now, and so we have the benefit of hindsight. Back then –  so that’s more than 2,500 years ago! – the simple but remarkable fact that the lengths of similar strings should respect some simple ratio if they are to sound ‘nice’ together, triggered a fascination with number theory (in fact, the Pythagoreans actually established the foundations of what is now known as number theory). Indeed, Pythagoras felt that similar relationships should also hold for other natural phenomena! To mention just one example, the Pythagoreans also believed that the orbits of the planets would also respect such simple numerical relationships, which is why they talked of the ‘music of the spheres’ (Musica Universalis).

We now know that the Pythagoreans were wrong. The proportions in the movements of the planets around the Sun do not respect simple ratios and, with the benefit of hindsight once again, it is regrettable that it took many courageous and brilliant people, such as Galileo Galilei and Copernicus, to convince the Church of that fact. 😦 Also, while Pythagoras’ observations in regard to the sounds coming out of whatever strings he was looking at were correct, his conclusions were wrong: the observation does not imply that the frequencies of musical notes should all be in some simple ratio one to another.

Let me repeat what I wrote above: the frequencies of musical notes are not in some simple relationship one to another. The frequency scale for all musical tones is logarithmic and, while that implies that we can, effectively, do some tricks with ratios based on the properties of the logarithmic scale (as I’ll explain in a moment), the so-called ‘Pythagorean’ tuning system, which is based on simple ratios, was plain wrong, even if it – or some variant of it (instead of the 3:2 ratio, musicians used the 5:4 ratio from about 1510 onwards) – was generally used until the 18th century! In short, Pythagoras was wrong indeed—in this regard at least: we can’t do much with those simple ratios.

Having said that, Pythagoras’ basic intuition was right, and that intuition is still very much what drives physics today: it’s the idea that Nature can be described, or explained (whatever that means), by quantitative relationships only. Let’s have a look at how it actually works for music.

Tones, noise and notes

Let’s first define and distinguish tones and notes. A musical tone is the opposite of noise, and the difference between the two is that musical tones are periodic waveforms, so they have a period T, as illustrated below. In contrast, noise is a non-periodic waveform. It’s as simple as that.

Now, from previous posts, you know we can write any period function as the sum of a potentially infinite number of simple harmonic functions, and that this sum is referred to as the Fourier series. I am just noting it here, so don’t worry about it as for now. I’ll come back to it later.

You also know we have seven musical notes: Do-Re-Mi-Fa-Sol-La-Si or, more common in the English-speaking world, A-B-C-D-E-F-G. And then it starts again with A (or Do). So we have two notes, separated by an interval which is referred to as an octave (from the Greek octo, i.e. eight), with six notes in-between, so that’s eight notes in total. However, you also know that there are notes in-between, except between E and F and between B and C. They are referred to as semitones or half-steps. I prefer the term ‘half-step’ over ‘semitone’, because we’re talking notes really, not tones.

We have, for example, F–sharp (denoted by F#), which we can also call G-flat (denoted by Gb). It’s the same thing: a sharp # raises a note by a semitone (aka half-step), and a flat b lowers it by the same amount, so F# is Gb. That’s what shown below: in an octave, we have eight notes but twelve half-steps.

Let’s now look at the frequencies. The frequency scale above (expressed in oscillations per second, so that’s the hertz unit) is a logarithmic scale: frequencies double as we go from one octave to another: the frequency of the C4 note above (the so-called middle C) is 261.626 Hz, while the frequency of the next C note (C5) is double that: 523.251 Hz. [Just in case you’d want to know: the 4 and 5 number refer to its position on a standard 88-key piano keyboard: C4 is the fourth C key on the piano.]

Now, if we equate the interval between C4 and C5 with 1 (so the octave is our musical ‘unit’), then the interval between the twelve half-steps is, obviously, 1/12. Why? Because we have 12 halve-steps in our musical unit. You can also easily verify that, because of the way logarithms work, the ratio of the frequencies of two notes that are separated by one half-step (between D# and E, for example) will be equal to 21/12. Likewise, the ratio of the frequencies of two notes that are separated by half-steps is equal to 2n/12. [In case you’d doubt, just do an example. For instance, if we’d denote the frequency of C4 as f0, and the frequency of C# as f1 and so on (so the frequency of D is f2, the frequency of C5 is f12, and everything else is in-between), then we can write the f2/fratio as f2/f= ( f2/f1)(f1/f0) =  21/12·21/12 = 22/12 = 21/6. I must assume you’re smart enough to generalize this result yourself, and that f12/fis, obviously, equal to 212/12 =21 = 2, which is what it should be!]

Now, because the frequencies of the various C notes are expressed as a number involving some decimal fraction (like 523.251 Hz, and the 0.251 is actually an approximation only), and because they are, therefore, a bit hard to read and/or work with, I’ll illustrate the next idea – i.e. the concept of harmonics – with the A instead of the C. 🙂

Harmonics

The lowest A on a piano is denoted by A0, and its frequency is 27.5 Hz. Lower A notes exist (we have one at 13.75 Hz, for instance) but we don’t use them, because they are near (or actually beyond) the limit of the lowest frequencies we can hear. So let’s stick to our grand piano and start with that 27.5 Hz frequency. The next A note is A1, and its frequency is 55 Hz. We then have A2, which is like the A on my (or your) guitar: its frequency is equal to 2×55 = 110 Hz. The next is A3, for which we double the frequency once again: we’re at 220 Hz now. The next one is the A in the illustration of the C scale above: A4, with a frequency of 440 Hz.

[Let me, just for the record, note that the A4 note is the standard tuning pitch in Western music. Why? Well… There’s no good reason really, except convention. Indeed, we can derive the frequency of any other note from that A4 note using our formula for the ratio of frequencies but, because of the properties of a logarithmic function, we could do the same using whatever other note really. It’s an important point: there’s no such thing as an absolute reference point in music: once we define our musical ‘unit’ (so that’s the so-called octave in Western music), and how many steps we want to have in-between (so that’s 12 steps—again, in Western music, that is), we get all the rest. That’s just how logarithms work. So music is all about structure, i.e. mathematical relationships. Again, Pythagoras’ conclusions were wrong, but his intuition was right.]

Now, the notes we are talking about here are all so-called pure tones. In fact, when I say that the A on our guitar is referred to as A2 and that it has a frequency of 110 Hz, then I am actually making a huge simplification. Worse, I am lying when I say that: when you play a string on a guitar, or when you strike a key on a piano, all kinds of other frequencies – so-called harmonics – will resonate as well, and that’s what gives the quality to the sound: it’s what makes it sound beautiful. So the fundamental frequency (aka as first harmonic) is 110 Hz alright but we’ll also have second, third, fourth, etc harmonics with frequency 220 Hz, 330 Hz, 440 Hz, etcetera. In music, the basic or fundamental frequency is referred to as the pitch of the tone and, as you can see, I often use the term ‘note’ (or pure tone) as a synonym for pitch—which is more or less OK, but not quite correct actually. [However, don’t worry about it: my sloppiness here does not affect the argument.]

What’s the physics behind? Look at the illustration below (I borrowed it from the Physics Classroom site). The thick black line is the string, and the wavelength of its fundamental frequency (i.e. the first harmonic) is twice its length, so we write λ1 = 2·L or, the other way around, L = (1/2)·λ1. Now that’s the so-called first mode of the string. [One often sees the term fundamental or natural or normal mode, but the adjective is not necessary really. In fact, I find it confusing, although I sometimes find myself using it too.]

We also have a second, third, etc mode, depicted below, and these modes correspond to the second, third, etc harmonic respectively.

For the second, third, etc mode, the relationship between the wavelength and the length of the string is, obviously, the following: L = (2/2)·λ= λ2, L = L = (3/2)·λ3, etc. More in general, for the nth mode, L will be equal to L = (n/2)·λn, with n = 1, 2, etcetera. In fact, because L is supposed to be some fixed length, we should write it the other way around: λn = (2/n)·L.

What does it imply for the frequencies? We know that the speed of the wave – let’s denote it by c – as it travels up and down the string, is a property of the string, and it’s a property of the string only. In other words, it does not depend on the frequency. Now, the wave velocity is equal to the frequency times the wavelength, always, so we have c = f·λ. To take the example of the (classical) guitar string: its length is 650 mm, i.e. 0.65 m. Hence, the identities λ1 = (2/1)·L, λ2 = (2/2)·L, λ3 = (2/3)·L etc become λ1 = (2/1)·0.65 = 1.3 m, λ2 = (2/2)·0.65 = 0.65 m, λ3 = (2/3)·0.65 = 0.433.. m and so on. Now, combining these wavelengths with the above-mentioned frequencies, we get the wave velocity c = (110 Hz)·(1.3 m) = (220 Hz)·(0.65 m) = (330 Hz)·(0.433.. m) = 143 m/s.

Let me now get back to Pythagoras’ string. You should note that the frequencies of the harmonics produced by a simple guitar string are related to each other by simple whole number ratios. Indeed, the frequencies of the first and second harmonics are in a simple 2 to 1 ratio (2:1). The second and third harmonics have a 3:2 frequency ratio. The third and fourth harmonics a 4:3 ratio. The fifth and fourth harmonic 5:4, and so on and so on. They have to be. Why? Because the harmonics are simple multiples of the basic frequency. Now that is what’s really behind Pythagoras’ observation: when he was sounding similar strings with the same tension but different lengths, he was making sounds with the same harmonics. Nothing more, nothing less.

Let me be quite explicit here, because the point that I am trying to make here is somewhat subtle. Pythagoras’ string is Pythagoras’ string: he talked similar strings. So we’re not talking some actual guitar or a piano or whatever other string instrument. The strings on (modern) string instruments are not similar, and they do not have the same tension. For example, the six strings of a guitar strings do not differ in length (they’re all 650 mm) but they’re different in tension. The six strings on a classical guitar also have a different diameter, and the first three strings are plain strings, as opposed to the bottom strings, which are wound. So the strings are not similar but very different indeed. To illustrate the point, I copied the values below for just one of the many commercially available guitar string sets.  It’s the same for piano strings. While they are somewhat more simple (they’re all made of piano wire, which is very high quality steel wire basically), they also differ—not only in length but in diameter as well, typically ranging from 0.85 mm for the highest treble strings to 8.5 mm (so that’s ten times 0.85 mm) for the lowest bass notes.

In short, Pythagoras was not playing the guitar or the piano (or whatever other more sophisticated string instrument that the Greeks surely must have had too) when he was thinking of these harmonic relationships. The physical explanation behind his famous observation is, therefore, quite simple: musical tones that have the same harmonics sound pleasant, or consonant, we should say—from the Latin con-sonare, which, literally, means ‘to sound together’ (from sonare = to sound and con = with). And otherwise… Well… Then they do not sound pleasant: they are dissonant.

To drive the point home, let me emphasize that, when we’re plucking a string, we produce a sound consisting of many frequencies, all in one go. One can see it in practice: if you strike a lower A string on a piano – let’s say the 110 Hz A2 string – then its second harmonic (220 Hz) will make the A3 string vibrate too, because it’s got the same frequency! And then its fourth harmonic will make the A4 string vibrate too, because they’re both at 440 Hz. Of course, the strength of these other vibrations (or their amplitude we should say) will depend on the strength of the other harmonics and we should, of course, expect that the fundamental frequency (i.e. the first harmonic) will absorb most of the energy. So we pluck one string, and so we’ve got one sound, one tone only, but numerous notes at the same time!

In this regard, you should also note that the third harmonic of our 110 Hz A2 string corresponds to the fundamental frequency of the E4 tone: both are 330 Hz! And, of course, the harmonics of E, such as its second harmonic (2·330 Hz = 660 Hz) correspond to higher harmonics of A too! To be specific, the second harmonic of our E string is equal to the sixth harmonic of our A2 string. If your guitar is any good, and if your strings are of reasonable quality too, you’ll actually see it: the (lower) E and A strings co-vibrate if you play the A major chord, but by striking the upper four strings only. So we’ve got energy – motion really – being transferred from the four strings you do strike to the two strings you do not strike! You’ll say: so what? Well… If you’ve got any better proof of the actuality (or reality) of various frequencies being present at the same time, please tell me! 🙂

So that’s why A and E sound very well together (A, E and C#, played together, make up the so-called A major chord): our ear likes matching harmonics. And so that why we like musical tones—or why we define those tones as being musical! 🙂 Let me summarize it once more: musical tones are composite sound waves, consisting of a fundamental frequency and so-called harmonics (so we’ve got many notes or pure tones altogether in one musical tone). Now, when other musical tones have harmonics that are shared, and we sound those notes too, we get the sensation of harmony, i.e. the combination sounds consonant.

Now, i’s not difficult to see that we will always have such shared harmonics if we have similar strings, with the same tension but different lengths, being sounded together. In short, what Pythagoras observed has nothing much to do with notes, but with tones. Let’s go a bit further in the analysis now by introducing some more math. And, yes, I am very sorry: it’s the dreaded Fourier analysis indeed! 🙂

Fourier analysis

You know that we can decompose any periodic function into a sum of a (potentially infinite) series of simple sinusoidal functions, as illustrated below. I took the illustration from Wikipedia: the red function s6(x) is the sum of six sine functions of different amplitudes and (harmonically related) frequencies. The so-called Fourier transform S(f) (in blue) relates the six frequencies with the respective amplitudes.

In light of the discussion above, it is easy to see what this means for the sound coming from a plucked string. Using the angular frequency notation (so we write everything using ω instead of f), we know that the normal or natural modes of oscillation have frequencies ω = 2π/T = 2πf  (so that’s the fundamental frequency or first harmonic), 2ω (second harmonic), 3ω (third harmonic), and so on and so on.

Now, there’s no reason to assume that all of the sinusoidal functions that make up our tone should have the same phase: some phase shift Φ may be there and, hence, we should write our sinusoidal function  not as cos(ωt), but as cos(ωt + Φ) in order to ensure our analysis is general enough. [Why not a sine function? It doesn’t matter: the cosine and sine function are the same, except for another phase shift of 90° = π/2.] Now, from our geometry classes, we know that we can re-write cos(ωt + Φ) as

cos(ωt + Φ) = [cos(Φ)cos(ωt) – sin(Φ)sin(ωt)]

We have a lot of these functions of course – one for each harmonic, in fact – and, hence, we should use subscripts, which is what we do in the formula below, which says that any function f(t) that is periodic with the period T can be written mathematically as:

You may wonder: what’s that period T? It’s the period of the fundamental mode, i.e. the first harmonic. Indeed, the period of the second, third, etc harmonic will only be one half, one third etcetera of the period of the first harmonic. Indeed, T2 = (2π)/(2ω) = (1/2)·(2π)/ω = (1/2)·T1, and T3 = (2π)/(3ω) = (1/3)·(2π)/ω = (1/3)·T1, and so on. However, it’s easy to see that these functions also repeat themselves after two, three, etc periods respectively. So all is alright, and the general idea behind the Fourier analysis is further illustrated below. [Note that both the formula as well as the illustration below (which I took from Feynman’s Lectures) add a ‘zero-frequency term’ a0 to the series. That zero-frequency term will usually be zero for a musical tone, because the ‘zero’ level of our tone will be zero indeed. Also note that the an and bn coefficients are, of course, equal to an = cos Φand b= –sinΦn, so you can relate the illustration and the formula easily.]

You’ll say: What the heck! Why do we need the mathematical gymnastics here? It’s just to understand that other characteristic of a musical tone: its quality (as opposed to its pitch). A so-called rich tone will have strong harmonics, while a pure tone will only have the first harmonic. All other characteristics – the difference between a tone produced by a violin as opposed to a piano – are then related to the ‘mix’ of all those harmonics.

So we have it all now, except for loudness which is, of course, related to the magnitude of the air pressure changes as our waveform moves through the air: pitch, loudness and quality. that’s what makes a musical tone. 🙂

Dissonance

As mentioned above, if the sounds are not consonant, they’re dissonant. But what is dissonance really? What’s going on? The answer is the following: when two frequencies are near to a simple fraction, but not exact, we get so-called beats, which our ear does not like.

Huh? Relax. The illustration below, which I copied from the Wikipedia article on piano tuning, illustrates the phenomenon. The blue wave is the sum of the red and the green wave, which are originally identical. But then the frequency of the green wave is increased, and so the two waves are no longer in phase, and the interference results in a beating pattern. Of course, our musical tone involves different frequencies and, hence, different periods T1,T2, Tetcetera, but you get the idea: the higher harmonics also oscillate with period T1, and if the frequencies are not in some exact ratio, then we’ll have a similar problem: beats, and our ear will not like the sound.

Of course, you’ll wonder: why don’t we like beats in tones? We can ask that, can’t we? It’s like asking why we like music, isn’t it? […] Well… It is and it isn’t. It’s like asking why our ear (or our brain) likes harmonics. We don’t know. That’s how we are wired. The ‘physical’ explanation of what is musical and what isn’t only goes so far, I guess. 😦

Pythagoras versus Bach

From all of what I wrote above, it is obvious that the frequencies of the harmonics of a musical tone are, indeed, related by simple ratios of small integers: the frequencies of the first and second harmonics are in a simple 2 to 1 ratio (2:1); the second and third harmonics have a 3:2 frequency ratio; the third and fourth harmonics a 4:3 ratio; the fifth and fourth harmonic 5:4, etcetera. That’s it. Nothing more, nothing less.

In other words, Pythagoras was observing musical tones: he could not observe the pure tones behind, i.e. the actual notesHowever, aesthetics led Pythagoras, and all musicians after him – until the mid-18th century – to also think that the ratio of the frequencies of the notes within an octave should also be simple ratios. From what I explained above, it’s obvious that it should not work that way: the ratio of the frequencies of two notes separated by n half-steps is 2n/12, and, for most values of n, 2n/12 is not some simple ratio. [Why? Just take your pocket calculator and calculate the value of 21/12: it’s 20.08333… = 1.0594630943… and so on… It’s an irrational number: there are no repeating decimals. Now, 2n/12 is equal to 21/12·21/12·…·21/12 (n times). Why would you expect that product to be equal to some simple ratio?]

So – I said it already – Pythagoras was wrong—not only in this but also in other regards, such as when he espoused his views on the solar system, for example. Again, I am sorry to have to say that, but it is what is: the Pythagoreans did seem to prefer mathematical ideas over physical experiment. 🙂 Having said that, musicians obviously didn’t know about any alternative to Pythagoras, and they had surely never heard about logarithmic scales at the time. So… Well… They did use the so-called Pythagorean tuning system. To be precise, they tuned their instruments by equating the frequency ratio between the first and the fifth tone in the C scale (i.e. the C and G, as they did not include the C#, D# and F# semitones when counting) with the ratio 3/2, and then they used other so-called harmonic ratios for the notes in-between.

Now, the 3/2 ratio is actually almost correct, because the actual frequency ratio is 27/12 (we have seven tones, including the semitones—not five!), and so that’s 1.4983, approximately. Now, that’s pretty close to 3/2 = 1.5, I’d say. 🙂 Using that approximation (which, I admit, is fairly accurate indeed), the tuning of the other strings would then also be done assuming certain ratios should be respected, like the ones below.

So it was all quite good. Having said that, good musicians, and some great mathematicians, felt something was wrong—if only because there were several so-called just intonation systems around (for an overview, check out the Wikipedia article on just intonation). More importantly, they felt it was quite difficult to transpose music using the Pythagorean tuning system. Transposing music amounts to changing the so-called key of a musical piece: what one does, basically, is moving the whole piece up or down in pitch by some constant interval that is not equal to an octave. Today, transposing music is a piece of cake—Western music at least. But that’s only because all Western music is played on instruments that are tuned using that logarithmic scale (technically, it’s referred to as the 12-tone equal temperament (12-TET) system). When you’d use one of the Pythagorean systems for tuning, a transposed piece does not sound quite right.

The first mathematician who really seemed to know what was wrong (and, hence, who also knew what to do) was Simon Stevin, who wrote a manuscript based on the ’12th root of 2 principle’ around AD 1600. It shouldn’t surprise us: the thinking of this mathematician from Bruges would inspire John Napier’s work on logarithms. Unfortunately, while that manuscript describes the basic principles behind the 12-TET system, it didn’t get published (Stevin had to run away from Bruges, to Holland, because he was protestant and the Spanish rulers at the time didn’t like that). Hence, musicians, while not quite understanding the math (or the physics, I should say) behind their own music, kept trying other tuning systems, as they felt it made their music sound better indeed.

One of these ‘other systems’ is the so-called ‘good’ temperament, which you surely heard about, as it’s referred to in Bach’s famous composition, Das Wohltemperierte Klavier, which he finalized in the first half of the 18th century. What is that ‘good’ temperament really? Well… It is what it is: it’s one of those tuning systems which made musicians feel better about their music for a number of reasons, all of which are well described in the Wikipedia article on it. But the main reason is that the tuning system that Bach recommended was a great deal better when it came to playing the same piece in another key. However, it still wasn’t quite right, as it wasn’t the equal temperament system (i.e. the 12-TET system) that’s in place now (in the West at least—the Indian music scale, for instance, is still based on simple ratios).

Why do I mention this piece of Bach? The reason is simple: you probably heard of it because it’s one of the main reference points in a rather famous book: Gödel, Escher and Bach—an Eternal Golden Braid. If not, then just forget about it. I am mentioning it because one of my brothers loves it. It’s on artificial intelligence. I haven’t read it, but I must assume Bach’s master piece is analyzed there because of its structure, not because of the tuning system that one’s supposed to use when playing it. So… Well… I’d say: don’t make that composition any more mystic than it already is. 🙂 The ‘magic’ behind it is related to what I said about A4 being the ‘reference point’ in music: since we’re using a universal logarithmic scale now, there’s no such thing as an absolute reference point any more: once we define our musical ‘unit’ (so that’s the so-called octave in Western music), and also define how many steps we want to have in-between (so that’s 12—in Western music, that is), we get all the rest. That’s just how logarithms work.

So, in short, music is all about structure, i.e. it’s all about mathematical relations, and about mathematical relations only. Again, Pythagoras’ conclusions were wrong, but his intuition was right. And, of course, it’s his intuition that gave birth to science: the simple ‘models’ he made – of how notes are supposed to be related to each other, or about our solar system – were, obviously, just the start of it all. And what a great start it was! Looking back once again, it’s rather sad conservative forces (such as the Church) often got in the way of progress. In fact, I suddenly wonder: if scientists would not have been bothered by those conservative forces, could mankind have sent people around the time that Charles V was born, i.e. around A.D. 1500 already? 🙂

Post scriptum: My example of the the (lower) E and A guitar strings co-vibrating when playing the A major chord striking the upper four strings only, is somewhat tricky. The (lower) E and A strings are associated with lower pitches, and we said overtones (i.e. the second, third, fourth, etc harmonics) are multiples of the fundamental frequency. So why is that the lower strings co-vibrate? The answer is easy: they oscillate at the higher frequencies only. If you have a guitar: just try it. The two strings you do not pluck do vibrate—and very visibly so, but the low fundamental frequencies that come out of them when you’d strike them, are not audible. In short, they resonate at the higher frequencies only. 🙂

The example that Feynman gives is much more straightforward: his example mentions the lower C (or A, B, etc) notes on a piano causing vibrations in the higher C strings (or the higher A, B, etc string respectively). For example, striking the C2 key (and, hence, the C2 string inside the piano) will make the (higher) C3 string vibrate too. But few of us have a grand piano at home, I guess. That’s why I prefer my guitar example. 🙂

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Riemann surfaces (II)

Pre-scriptum (dated 26 June 2020): the material in this post remains interesting but is, strictly speaking, not a prerequisite to understand quantum mechanics. It’s yet another example of how one can get lost in math when studying or teaching physics.

Original post:

This is my second post on Riemann surfaces, so they must be important. [At least I hope so, because it takes quite some time to understand them. :-)]

From my first post on this topic, you may or may not remember that a Riemann surface is supposed to solve the problem of multivalued complex functions such as, for instance, the complex logarithmic function (log z = ln r + i(θ + 2nπ) or the complex exponential function (zc = ec log z). [Note that the problem of multivaluedness for the (complex) exponential function is a direct consequence of its definition in terms of the (complex) logarithmic function.]

In that same post, I also wrote that it all looked somewhat fishy to me: we first use the function causing the problem of multivaluedness to construct a Riemann surface, and then we use that very same surface as a domain for the function itself to solve the problem (i.e. to reduce the function to a single-valued (analytic) one). Penrose does not have any issues with that though. In Chapter 8 (yes, that’s where I am right now: I am moving very slowly on his Road to Reality, as it’s been three months of reading now, and there are 34 chapters!), he writes that  “Complex (analytic) functions have a mind of their own, and decide themselves what their domain should be, irrespective of the region of the complex plane which we ourselves may initially have allotted to it. While we may regard the function’s domain to be represented by the Riemann surface associated with the function, the domain is not given ahead of time: it is the explicit form of the function itself that tells us which Riemann surface the domain actually is.”

Let me retrieve the graph of the Riemannian domain for the log z function once more:

For each point z in the complex plane (and we can represent z both with rectangular as well as polar coordinates: z = x + iy = reiθ), we have an infinite number of log z values: one for each value of n in the log z = ln r + i(θ + 2nπ) expression (n = 0, ±1, ±2, ±3,…, ±∞). So what we do when we promote this Riemann surface as a domain for the log z function is equivalent to saying that point z is actually not one single point z with modulus r and argument θ + 2nπ, but an infinite collection of points: these points all have the same modulus ¦z¦ = r but we distinguish the various ‘representations’ of z by treating θ, θ ± 2π, θ ±+ 4π, θ ± 6π, etcetera, as separate argument values as we go up or down on that spiral ramp. So that is what is represented by that infinite number of sheets, which are separated from each other by a vertical distance of 2π. These sheets are all connected at or through the origin (at which the log z function is undefined: therefore, the origin is not part of the domain), which is the branch point for this function. Let me copy some formal language on the construction of that surface here:

“We treat the z plane, with the origin deleted, as a thin sheet Rwhich is cut along the positive half of the real axis. On that sheet, let θ range from 0 to 2π. Let a second sheet  Rbe cut in the same way and placed in front of the sheet R0. The lower edge of the slit in Ris then joined to the upper edge of the slit in R1. On  R1, the angle θ ranges from 2π to 4π; so, when z is represented by a point on R1, the imaginary component of log z ranges from 2π to 4π.” And then we repeat the whole thing, of course: “A sheet Ris then cut in the same way and placed in front of R1. The lower edge of the slit in R1. is joined to the upper edge of the slit in this new sheet, and similarly for sheets R3R4… A sheet R-1R-2, R-3,… are constructed in like manner.” (Brown and Churchill, Complex Variables and Applications, 7th edition, p. 335-336)

The key phrase above for me is this “when z is represented by a point on R1“, because that’s what it is really: we have an infinite number of representations of z here, namely one representation of z for each branch of the log z function. So, as n = 0, ±1, , ±2, ±3 etcetera, we have an infinite number of them indeed. You’ll also remember that each branch covers a range from some random angle α to α + 2π. Imagine a continuous curve around the origin on this Riemann surface: as we move around, the angle of z changes from 0 to 2θ on sheet R0, and then from 2π to 4π on sheet Rand so on and so on.

The illustration above also illustrates the meaning of a branch point. Imagine yourself walking on that surface and approaching the origin, from any direction really. At the origin itself, you can choose what to do: either you take the elevator up or down to some other level or, else, the elevator doesn’t work and so then you have to walk up or down that ramp to get to another level. If you choose to walk along the ramp, the angle θ changes gradually or, to put it in mathematical terms, in a continuous way. However, if you took the elevator and got out at some other level, you’ll find that you’ve literally ‘jumped’ one or more levels. Indeed, remember that log z = ln r + i(θ + 2nπ) and so ln r, the horizontal distance from the origin didn’t change, but you did add some multiple of 2π to the vertical distance, i.e. the imaginary part of the log z value.

Let us now construct a Riemann surface for some other multiple-valued functions. Let’s keep it simple and start with the square root of z, so c = 1/2, which is nothing else than a specific example of the complex exponential function zc = zc = ec log z: we just take a real number for c here. In fact, we’re taking a very simple rational number value for c: 1/2 = 0.5. Taking the square, cube, fourth or  nth root of a complex number is indeed nothing but a special case of the complex exponential function. The illustration below (taken from Wikipedia) shows us the Riemann surface for the square root function.

As you can see, the spiraling surface turns back into itself after two turns. So what’s going on here? Well… Our multivalued function here does not have an infinite number of values for each z: it has only two, namely √r ei(θ/2) and √r ei(θ/2 + π). But what’s that? We just said that the log function – of which this function is a special case – had an infinite number of values? Well… To be somewhat more precise:  z1/2 actually does have an infinite number of values for each z (just like any other complex exponential function), but it has only two values that are different from each other. All the others coincide with one of the two principal ones. Indeed, we can write the following:

w = √z = z1/2 =  e(1/2) log z e(1/2)[ln r + i(θ + 2nπ)] = r1/2 ei(θ/2 + nπ) = √r ei(θ/2 + nπ)

(n = 0, ±1,  ±2,  ±3,…)

For n = 0, this expression reduces to z1/2 = √r eiθ/2. For n = ±1, we have  z1/2 = √r ei(θ/2 + π), which is different than the value we had for n = 0. In fact, it’s easy to see that this second root is the exact opposite of the first root: √r ei(θ/2 + π) = √r eiθ/2eiπ = – √r eiθ/2). However, for n = 2, we have  z1/2 = √r ei(θ/2 + 2π), and so that’s the same value (z1/2 = √r eiθ/2) as for n = 0. Indeed, taking the value of n = 2 amounts to adding 2π to the argument of w and so get the same point as the one we found for n = 0. [As for the plus or minus sign, note that, for n = -1, we have  z1/2 = √r ei(θ/2 -π) = √r ei(θ/2 -π+2π) =  √r ei(θ/2 +π) and, hence, the plus or minus sign for n does not make any difference indeed.]

In short, as mentioned above, we have only two different values for w = √z = z1/2 and so we have to construct two sheets only, instead of an infinite number of them, like we had to do for the log z function. To be more precise, because the sheet for n = ±2 will be the same sheet as for n = 0, we need to construct one sheet for n = 0 and one sheet for n = ±1, and so that’s what shown above: the surface has two sheets (one for each branch of the function) and so if we make two turns around the origin (one on each sheet), we’re back at the same point, which means that, while we have a one-to-two relationship between each point z on the complex plane and the two values z1/2 for this point, we’ve got a one-on-one relationship between every value of z1/2 and each point on this surface.

For ease of reference in future discussions, I will introduce a personal nonsensical convention here: I will refer to (i) the n = 0 case as the ‘positive’ root, or as w1, i.e. the ‘first’ root, and to (ii) the n = ± 1 case as the ‘negative’ root, or w2, i.e. the ‘second’ root. The convention is nonsensical because there is no such thing as positive or negative complex numbers: only their real and imaginary parts (i.e. real numbers) have a sign. Also, these roots also do not have any particular order: there are just two of them, but neither of the two is like the ‘principal’ one or so. However, you can see where it comes from: the two roots are each other’s exact opposite w= u2 + iv= —w= -u1 – iv1. [Note that, of course, we have w1w = w12 = w2w = w2= z, but that the product of the two distinct roots is equal to —z. Indeed, w1w2 = w2w1 = √rei(θ/2)√rei(θ/2 + π) = rei(θ+π) = reiθeiπ = -reiθ = -z.]

What’s the upshot? Well… As I mentioned above already, what’s happening here is that we treat z = rei(θ+2π) as a different ‘point’ than z = reiθ. Why? Well… Because of that square root function. Indeed, we have θ going from 0 to 2π on the first ‘sheet’, and then from 2π0 to 4π on the second ‘sheet’. Then this second sheet turns back into the first sheet and so then we’re back at normal and, hence, while θ going from 0π  to 2π is not the same as θ going from 2π  to 4π, θ going from 4π  to 6π  is the same as θ going from 0 to 2π (in the sense that it does not affect the value of w = z1/2). That’s quite logical indeed because, if we denote w as w = √r eiΘ (with Θ = θ/2 + nπ, and n = 0 or ± 1), then it’s clear that arg w = Θ will range from 0 to 2π if (and only if) arg z = θ ranges from 0 to 4π. So as the argument of w makes one loop around the origin – which is what ‘normal’ complex numbers do – the argument of z makes two loops. However, once we’re back at Θ = 2π, then we’ve got the same complex number w again and so then it’s business as usual.

So that will help you to understand why this Riemann surface is said to have two complex dimensions, as opposed to the plane, which has only one complex dimension.

OK. That should be clear enough. Perhaps one question remains: how do you construct a nice graph like the one above?

Well, look carefully at the shape of it. The vertical distance reflects the real part of √z for n = 0, i.e. √r cos(θ/2). Indeed, the horizontal plane is the the complex z plane and so the horizontal axes are x and y respectively (i.e. the x and y coordinates of z = x + iy). So this vertical distance equals 1 when x = 1 and y = 0 and that’s the highest point on the upper half of the top sheet on this plot (i.e. the ‘high-water mark’ on the right-hand (back-)side of the cuboid (or rectangular prism) in which this graph is being plotted). So the argument of z is zero there (θ = 0). The value on the vertical axis then falls from one to zero as we turn counterclockwise on the surface of this first sheet, and that’s consistent with a value for θ being equal to π there (θ = π), because then we have cos(π/2) = 0. Then we go underneath the z plane and make another half turn, so we add another π radians to the value θ and we arrive at the lowest point on the lower half of the bottom sheet on this plot, right under the point where we started, where θ = 2π and, hence, Re(√z) = √r cos(θ/2) (for n = 0) = cos(2π/2) = cos(2π/2) = -1.

We can then move up again, counterclockwise on the bottom sheet, to arrive once again at the spot where the bottom sheet passes through the top sheet: the value of θ there should be equal to θ = 3π, as we have now made three half turns around the origin from our original point of departure (i.e. we added three times π to our original angle of departure, which was θ = 0) and, hence, we have Re(√z) = √r cos(3θ/2) = 0 again. Finally, another half turn brings us back to our point of departure, i.e. the positive half of the real axis, where θ has now reached the value of θ = 4π, i.e. zero plus two times 2π. At that point, the argument of w (i.e. Θ) will have reached the value of 2π, i.e. 4π/2, and so we’re talking the same w = z1/2 as when we started indeed, where we had Θ = θ/2 = 0.

What about the imaginary part? Well… Nothing special really (as for now at least): a graph of the imaginary part of √z would be equally easy to establish: Im(√z) = √r sin(θ/2) and, hence, rotating this plot 180 degrees around the vertical axis will do the trick.

Hmm… OK. What’s next? Well… The graphs below show the Riemann surfaces for the third and fourth root of z respectively, i.e. z1/3 and z1/4 respectively. It’s easy to see that we now have three and four sheets respectively (instead of two only), and that we have to take three and four full turns respectively to get back at our starting point, where we should find the same values for z1/3 and z1/4 as where we started. That sounds logical, because we always have three cube roots of any (complex) numbers, and four fourth roots, so we’d expect to need the same number of sheets to differentiate between these three or four values respectively.

In fact, the table below may help to interpret what’s going on for the cube root function. We have three cube roots of z: w1, wand w3. These three values are symmetrical though, as indicated be the red, green and yellow colors in the table below: for example, the value of w for θ ranging from 4π to 6π for the n = 0 case (i.e. w1) is the same as the value of w for θ ranging from 0 to 2π for the n = 1 case (or the n = -2 case, which is equivalent to the n = 1 case).

So the origin (i.e. the point zero) for all of the above surfaces is referred to as the branch point, and the number of turns one has to make to get back at the same point determines the so-called order of the branch point. So, for w = z1/2, we have a branch point of order 2; for for w = z1/3, we have a branch point of order 3; etcetera. In fact, for the log z function, the branch point does not have a finite order: it is said to have infinite order.

After a very brief discussion of all of this, Penrose then proceeds and transforms a ‘square root Riemann surface’ into a torus (i.e. a donut shape). The correspondence between a ‘square root Riemann surface’ and a torus does not depend on the number of branch points: it depends on the number of sheets, i.e. the order of the branch point. Indeed, Penrose’s example of a square root function is w = (1 – z3)1/2, and so that’s a square root function with three branch points (the three roots of unity), but so these branch points are all of order two and, hence, there are two sheets only and, therefore, the torus is the appropriate shape for this kind of ‘transformation’. I will come back to that in the next post.

OK… But I still don’t quite get why this Riemann surfaces are so important. I must assume it has something to do with the mystery of rolled-up dimensions and all that (so that’s string theory), but I guess I’ll be able to shed some more light on that question only once I’ve gotten through that whole chapter on them (and the chapters following that one).  I’ll keep you posted. 🙂

Post scriptum: On page 138 (Fig. 8.3), Penrose shows us how to construct the spiral ramp for the log z function. He insists on doing this by taking overlapping patches of space, such as the L(z) and Log z branch of the log z function, with θ going from 0 to 2π for the L(z) branch) and from -π to +π for the Log z branch (so we have an overlap here from 0 to +π). Indeed, one cannot glue or staple patches together if the patch surfaces don’t overlap to some extent… unless you use sellotape of course. 🙂 However, continuity requires some overlap and, hence, just joining the edges of patches of space with sellotape, instead of gluing overlapping areas together, is not allowed. 🙂

So, constructing a model of that spiral ramp is not an extraordinary intellectual challenge. However, constructing a model of the Riemann surfaces described above (i.e. z1/2, z1/3, z1/4 or, more in general, constructing a Riemann surface for any rational power of z, i.e. any function w = zn/m, is not all that easy: Brown and Churchill, for example, state that is actually ‘physically impossible’ to model that (see Brown and Churchill, Complex Variables and Applications (7th ed.), p. 337).

Huh? But so we just did that for z1/2, z1/3 and z1/4, didn’t we? Well… Look at that plot for w = z1/2 once again. The problem is that the two sheets cut through each other. They have to do that, of course, because, unlike the sheets of the log z function, they have to join back together again, instead of just spiraling endlessly up or down. So we just let these sheets cross each other. However, at that spot (i.e. the line where the sheets cross each other), we would actually need two representations of z. Indeed, as the top sheet cuts through the bottom sheet (so as we’re moving down on that surface), the value of θ will be equal to π, and so that corresponds to a value for w equal to w = z1/2 = √r eiπ/2 (I am looking at the n = 0 case here). However, when the bottom sheet cuts through the top sheet (so if we’re moving up instead of down on that surface), θ’s value will be equal to 3π (because we’ve made three half-turns now, instead of just one) and, hence, that corresponds to a value for w equal to w = z1/2 = √r e3iπ/2, which is obviously different from √r eiπ/2. I could do the same calculation for the n = ±1 case: just add ±π to the argument of w.

Huh? You’ll probably wonder what I am trying to say here. Well, what I am saying here is that plot of the surface gives us the impression that we do not have two separate roots w1 and won the (negative) real axis. But so that’s not the case: we do have two roots there, but we can’t distinguish them with that plot of the surface because we’re only looking at the real part of w.

So what?

Well… I’d say that shouldn’t worry us all that much. When building a model, we just need to be aware that it’s a model only and, hence, we need to be aware of the limitations of what we’re doing. I actually build a paper model of that surface by taking two paper disks: one for the top sheet, and one for the bottom sheet. Then I cut those two disks along the radius and folded and glued both of them like a Chinese hat (yes, like the one the girl below is wearing). And then I took those two little paper Chinese hats, put one of them upside down, and ‘connected’ them (or should I say, ‘stitched’ or ‘welded’ perhaps? :-)) with the other one along the radius where I had cut into these disks. [I could go through the trouble of taking a digital picture of it but it’s better you try it yourself.]

Wow! I did not expect to be used as an illustration in a blog on math and physics! 🙂

🙂 OK. Let’s get somewhat more serious again. The point to note is that, while these models (both the plot as well as the two paper Chinese hats :-)) look nice enough, Brown and Churchill are right when they note that ‘the points where two of the edges are joined are distinct from the points where the two other edges are joined’. However, I don’t agree with their conclusion in the next phrase, which states that it is ‘thus physically impossible to build a model of that Riemann surface.’ Again, the plot above and my little paper Chinese hats are OK as a model – as long as we’re aware of how we should interpret that line where the sheets cross each other: that line represents two different sets of points.

Let me go one step further here (in an attempt to fully exhaust the topic) and insert a table here with the values of both the real and imaginary parts of √z for both roots (i.e. the n = 0 and n = ± 1 case). The table shows what is to be expected: the values for the n = ± 1 case are the same as for n = 0 but with the opposite sign. That reflects the fact that the two roots are each other’s opposite indeed, so when you’re plotting the two square roots of a complex number z = reiθ, you’ll see they are on opposite sides on a circle with radius √r. Indeed, rei(θ/2 + π) = rei(θ/2)eiπ = –rei(θ/2). [If the illustration below is too small to read the print, then just click on it and it should expand.]

The grey and green colors in the table have the same role as the red, green and yellow colors I used to illustrated how the cube roots of z come back periodically. We have the same thing here indeed: the values we get for the n = 0 case are exactly the same as for the n = ± 1 case but with a difference in ‘phase’ I’d say of one turn around the origin, i.e. a ‘phase’ difference of 2π. In other words, the value of √z in the n = 0 case for θ going from 0 to 2π is equal to the value of √z in the n = ± 1 case but for θ going from 2π to 4π and, vice versa, the value of √z in the n = ±1 case for θ going from 0 to 2π is equal to the value of √z in the n = 0 case for θ going from 2π to 4π. Now what’s the meaning of that?

It’s quite simple really. The two different values of n mark the different branches of the w function, but branches of functions always overlap of course. Indeed, look at the value of the argument of w, i.e. Θ: for the n = 0 case, we have 0 < Θ < 2π, while for the n = ± 1 case, we have -π < Θ < +π. So we’ve got two different branches here indeed, but they overlap for all values Θ between 0 and π and, for these values, where Θ1 = Θ2, we will obviously get the same value for w, even if we’re looking at two different branches (Θ1 is the argument of w1, and Θ2 is the argument of w2).

OK. I guess that’s all very self-evident and so I should really stop here. However, let me conclude by noting the following: to understand the ‘fully story’ behind the graph, we should actually plot both the surface of the imaginary part of √z as well as the surface of the real part of of √z, and superimpose both. We’d obviously get something that would much more complicated than the ‘two Chinese hats’ picture. I haven’t learned how to master math software (such as Maple for instance), as yet, and so I’ll just copy a plot which I found on the web: it’s a plot of both the real and imaginary part of the function w = z2. That’s obviously not the same as the w = z1/2 function, because w = z2 is a single-valued function and so we don’t have all these complications. However, the graph is illustrative because it shows how two surfaces – one representing the real part and the other the imaginary part of a function value – cut through each other thereby creating four half-lines (or rays) which join at the origin.

So we could have something similar for the w = z1/2 function if we’d have one surface representing the imaginary part of z1/2 and another representing the  real part of z1/2. The sketch below illustrates the point. It is a cross-section of the Riemann surface along the x-axis (so the imaginary part of z is zero there, as the values of θ are limited to 0, π, 2π, 3π, back to 4π = 0), but with both the real as well as the imaginary part of  z1/2 on it. It is obvious that, for the w = z1/2 function, two of the four half-lines marking where the two surfaces are crossing each other coincide with the positive and negative real axis respectively: indeed, Re( z1/2) = 0 for θ = π and 3π (so that’s the negative real axis), and Im(z1/2) = 0 for θ = 0, 2π and 4π (so that’s the positive real axis).

The other two half-lines are orthogonal to the real axis. They follow a curved line, starting from the origin, whose orthogonal projection on the z plane coincides with the y axis. The shape of these two curved lines (i.e. the place where the two sheets intersect above and under the axis) is given by the values for the real and imaginary parts of the √z function, i.e. the vertical distance from the y axis is equal to ± (√2√r)/2.

Hmm… I guess that, by now, you’re thinking that this is getting way too complicated. In addition, you’ll say that the representation of the Riemann surface by just one number (i.e. either the real or the imaginary part) makes sense, because we want one point to represent one value of w only, don’t we? So we want one point to represent one point only, and that’s not what we’re getting when plotting both the imaginary as well as the real part of w in a combined graph. Well… Yes and no. Insisting that we shouldn’t forget about the imaginary part of the surface makes sense in light of the next post, in which I’ll say a think or two about ‘compactifying’ surfaces (or spaces) like the one above. But so that’s for the next post only and, yes, you’re right: I should stop here.