Archive for the ‘Music’ Category

Perishing lack of algebraic technique

January 11, 2022

When I was young, I learnt the violin from a teacher who was a great fan of the playing of Ruggiero Ricci, and not of the most famous violinist of the time, Yehudi Menuhin. I remember one of his comments on Menuhin, though I don’t remember the context: “Perishing lack of bow technique”. No doubt he would have used a stronger word if there wasn’t a child present. Now it seems to me that a great many famous physicists suffer in a similar way from a “perishing lack of algebraic technique”. Of course, this is a minority view, and most people are quite happy to accept the playing of Yehudi Menuhin without making a critical appraisal of his bow technique.

If your violin playing is larded with plenty of continuous (geometrical) vibrato (of the wave-function) and portato (of the mixing angles), the listener may be carried away with the beauty of your playing, but be unable to distinguish the elementary particles (notes) of the music, or distinguish between up bows and down bows (spins). You may be an expert at prestidigitation of the left hand, but if you haven’t got the right hand (algebraic) bow techniques of flying spiccato and the rest, you’ll never be able to play like Paganini, and you’ll never be able to distinguish all the discrete quantum states.

A discreet spiccato that distinguishes between up and down bows when crossing strings (from left to right or right to left) is a sine qua non of violin playing. Those violinists who think that the difference between up and down bows is a mere matter of convention (and I know quite a few violinists who do seem to think this!) do not understand the violin. Those string theorists who think the strings vibrate of their own accord are in complete denial of the fact that without the algebraic technique of the bow, there would be no music of the violin (or of the spheres). They should take a few lessons in bowing from the algebraic masters of the art.


The quadrivium

June 17, 2021

The ancient Greeks considered mathematics to consist of four parts, usually translated as arithmetic, geometry, music and astronomy. The four-fold division reflects two dichotomies, between the discrete (arithmetic and music) and the continuous (geometry and astronomy) on the one hand, and the static (arithmetic and geometry) and the dynamic (music and astronomy) on the other.

In modern times, arithmetic has expanded to include algebra, number theory, combinatorics, and other parts of discrete mathematics. Geometry has expanded to include calculus, analysis, topology and other parts of continuous mathematics. Astronomy has expanded to include general relativity, cosmology and other parts of continuous physics. Music has expanded to include quantum mechanics, particle physics and other parts of discrete physics.

Unfortunately, the neat classification of the ancient Greeks has been destroyed: modern physicists try to describe music with geometry, instead of arithmetic. It can’t be done. Discrete physics requires discrete mathematics, and cannot be adequately described by continuous mathematics. The ancient Greeks understood the difference between the discrete and the continuous. Modern physicists clearly do not.

Mathematics and music

April 23, 2021

As a mathematician who is also a musician, I have been asked countless times why mathematics and music go together. My answer for many years has been one that I know people do not want to hear: both require many thousands of hours of relentless and often soul-destroying practice to do well. Anyone who has the personality to deal with that has the potential to do well at both. Of course, what most people want to hear is that it is a special gift, because that lets them off the hook of examining their own incompetence in both fields. While that may be part of it, the real point is that to do well in any human endeavour requires dedication and hard work. And it is the particular nature of the hard work that seems to me to connect mathematics and playing a musical instrument: both are absolutely unforgiving, and the clear distinction between right and wrong means that most of the practice consists in repeatedly getting things wrong, which many people find hard to deal with.

And in both cases, a faultless technique is both essential and irrelevant at the same time. In mathematics, just as much as in music, a faultless technique without imagination is sterile: I have come across many mathematicians in my career whose technique is far better than mine, but who have no imagination, and cannot play the mathematics. Just as I have come across many musicians in my career whose technique is far better than mine, but whose music-making has no soul. The technique is not an end in itself, it is a means to an end. In music, technique is there to make something difficult and delicate sound easy and assured, so the listener can be transported to a place of beauty and imagination. We do not want music to sound difficult – we already know it is. The same is true of mathematics. A real piece of mathematics sounds like a piece of music – you may not understand a word of it, but it transports you out of this world into a place you can hardly imagine. And a true virtuoso makes it all sound so easy, as years and years of practice appear to slip effortlessly off the chalk onto the blackboard. The goal of the real mathematician is for the listener to say at the end of it: that’s easy, why didn’t I think of that? Too many mathematicians these days revel in a formidable technique, and play a kind of atonal post-modernist music that no-one wants to listen to. The mark of a true master is playing something really new in C major.

The ultimate nightmare

March 7, 2021

For ten years I have been questioning all the assumptions of theoretical physics. Or so I thought. Never in my darkest nightmares did I ever seriously consider the possibility that the theory of special relativity could be wrong. This morning I woke up from this most dreadful nightmare, and slowly realised it wasn’t a nightmare, it is reality. The Lorentz group does not describe the universe we live in. Panic! Be afraid, be very afraid.

No, don’t panic. The Lorentz group works perfectly well in 3 spacetime dimensions. It’s just that it doesn’t work in 4 spacetime dimensions. So it’s perfectly fine in a flat part of the universe like the Solar System, where all you need is a small perturbation from 3-dimensional spacetime. It is perfectly fine for electromagnetism and quantum mechanics, and for particle physics as long as you consider only one vertex of the Feynman diagram at a time. It only goes wrong when you have two vertices in the Feynman diagram. The evidence that it is wrong in these situations has been building gradually for more than a century, at least since 1884, when Lord Kelvin gave a lecture in which he pointed out that the Milky Way does not contain enough stars to hold itself together. In other words, the evidence against the Lorentz group was already there 20 years before Lorentz’s paper.

How can this be? This is worse than a nightmare. This is a calamity. This is not a crisis. This is a disaster. This asteroid didn’t fly past harmlessly at a distance of 10 million miles. This asteroid will come back in 8 years at a distance of 20 thousand miles. We haven’t got long to get the theory right so that we know exactly where this asteroid will be. The current theory is only an approximation. We’ve got to get it exactly right. As I said, don’t panic, but be very, very afraid.

The particle physics experiments that demonstrate that the 4-dimensional theory is wrong are many and various. The chirality of the weak force, the three generations of fermions, the oscillations of neutrinos, the CP-violation of neutral kaons, the anomalous magnetic moment of the muon, the list goes on and on. Every new experimental anomaly results in a sticking plaster on the previous theory. The standard model consists of a few solid pieces of 2-dimensional metal sheets, stuck together with half a ton of sticky tape. When the asteroid hits, it will be smashed to smithereens. We haven’t got long to invest in the nuts and bolts that we need to bolt these pieces of metal together properly. Let’s get on with it fast. We don’t want to be caught napping again, as we were with Covid-19.

Apocalypse now, or in 2029? Don’t let us take our eyes off the ball (I mean asteroid). We know that our theory isn’t good enough to predict the motion of spacecraft on flybys of Earth accurately enough. We must get a correct theory as soon as possible. Don’t leave it until it is too late.

The parable of the rhinoceros and the unicorn

February 1, 2021

The origin of the concept of a unicorn is lost in the mists of time, but there is a plausible theory that it goes back to a time when the rhinoceros was widespread and well-known to well-informed people. Over the course of time, either the rhinoceros disappeared, or the story spread out of Africa to places where the rhinoceros was unknown, so that theory diverged further and further from observation, and the theoretical unicorn replaced the experimental rhinoceros in the minds of the people. The use of rhinoceros horn in Chinese medicine indeed suggests an ancient trade between China and Africa, in which the supposed characteristics of the rhinoceros gradually mutated in a long-distance game of Chinese whispers.

Much the same seems to have happened in fundamental physics, where the experimentalists are busy studying the rhinoceros in detail, and the theorists are saying, haven’t you found a unicorn yet? Go and look harder!

Let us go back to the time of the first sightings of the rhinoceros:

People knew what to expect of horned animals: they had two horns, one on the left and one on the right. These were normal animals, that people were used to and happy with. In Dirac’s theory, the left and right horns combined into a single “Dirac horn” that explained how the two were connected by being joined to the skull of the animal. Everyone was happy in this Garden of Eden.

Then suddenly daring explorers/experimentalists came back with stories of a strange animal with a horn in the middle, on its nose. No-one knew what to make of it. Theorists surmised that the left horn had moved to the middle, and the right horn had disappeared. But theorists only studied the skeletons of dead animals, which were always lying on their side, so the left horn became the down quark, the right horn became the up quark, and the nose-horn became the strange quark. The animal was given a name, originally just “strange animal” and then “Lambda baryon” in honour of the fact that the Greek letter Lambda is a diagram of its horn, and “barys” meaning “heavy”.

Meanwhile, the discrete theory of three types of horn was overtaken by the continuous theory that the horns could in principle move from one place to another, and that until you actually observed the animal, it was impossible to tell where the horns actually were. That is to say, the discrete internal symmetry of the animal had been replaced by a continuous “internal symmetry”, which makes no sense. Clearly the continuous “symmetries” are not symmetries at all, but describe the animal thrashing about as it is caught in the explorers’ net. Of course, if all you can see is the horns, you may be forgiven for making this incorrect deduction.

Now if a real live animal such as a bull (called a “proton” on account of its charge, distinguished from the very similar cow or neutron, which has no charge, but is heavier, on account of being pregnant with an electron) is studied by means of its horns, it is not difficult to observe that the mass of the horns makes up no more than about 1% of the mass of the animal. Theorists have come up with the idea that these horns move about all over the body of the animal, and become virtual horns with much larger mass, but which cannot be observed directly, since they annihilate with virtual anti-horns before you can see them. I don’t know about you, but I don’t buy this theory.

There is some experimental support for this theory, however. Not much, but some. The rhinoceros horn is known to be made essentially of compressed hair, which lends some (!) support to the theory that the rhinoceros has internal spherical symmetry, and it is only when you interact directly with the horn (or it interacts with you) that you can tell where it actually is.

Come now, enough of these fairy stories. We all know that these animals have only a bilateral symmetry between left and right. And we all know that even the symmetry between left and right does not apply to the internal symmetries. So that when you look at the internal structure of the cow and the bull, you have to take account of this chirality.

The theory of the “weak interaction” is based on the idea that the cow and the bull are different aspects of the same species of animal, which I think is an idea we can all accept. The propagation of the species occurs when the neutron gives birth to an electron, and an “anti-neutrino”. The analogy works better if we think of this anti-neutrino disappearing into the sky rather as a neutrino appearing out of the sky, and perhaps as a kind of “soul” to give to the baby animal, to denote the fact that where there was one animal, there are now two.

Anyway, while the theorists are busy speculating, the explorers are out there finding more and more strange animals. The narwhal, for example, also has a single horn, which looks a good deal more like the theoretical unicorn horn than the experimental rhino horn. Good news for those theorists who predicted the existence of a narwhal. Although its experimental properties, like living in the sea, are rather different from those predicted. Gell-Mann predicted the existence of the Omega baryon, with a strangeness of 3 (or -3, since strangeness seems to be regarded as a negative property). Good news, when the explorers found an elephant with a huge trunk and two massive tusks.

These days, there is not a lot of point in predicting new particles. Explorers find lots of new animals every year, and although they are interesting, they don’t tell us anything fundamentally new. There is perhaps just one more worth mentioning, and that is the animal that gives mass to the sperm whale, predicted to be a giant squid. And yes, eventually the giant squid (Higgs boson) was found, although it was somewhat smaller than most predictions said it had to be.

Well, I am not a zoologist, and I cannot tell you everything about the particle zoo. I just sing a song of the Carnival of the Animals for your musical entertainment,

Creative boredom

January 13, 2021

Covid lockdowns have had far-reaching consequences in all sorts of ways. I have spent far more time online than I would otherwise have done, and read all sorts of things I would not normally have read. One of the things I read today was a report that suggests that lockdowns have made children much more bored than they were before. This is not in any way a surprise. The surprise is that the report suggests this is a good thing. I like ideas like this, that challenge our assumptions and make us think. In a nutshell, the thesis is that boredom stimulates creativity. Once you put it like that, it seems obvious. And when they suggest that children (pre-pandemic) did not have time to be bored, and that this was a bad thing because it stifled their creativity, this is something I can understand.

It doesn’t only apply to younger children, but to older children at least into their 60s (as I can personally testify) and probably older. It is when you get bored with your ways of combatting boredom that the creativity starts. This creativity can take many forms, of course, and much of it will be of no interest to anybody else. But it keeps one amused. No-one will be interested in the time I have spent trying to understand the rhythmic and melodic structure of Mandinka music, and the sense of achievement I felt when I suddenly realised that the rhythm that I was trying to fit into 12/8 was actually in 4/4, divided as (3+3+3+3+4)/16 in the melody and as (3+3+2)/8 in the bass, nor in the fact that I was then able, with a lot of practice, to make a plausible imitation of the resulting aural ambiguity of 5 beats against 4 against 3, and turn what had been a meaningless sequence of notes into a dance, or at least a song. This is an explanation that my Gambian teacher cannot give me, because it is rooted in my European classical musical education and heritage. But it enables me to relate my mathematical approach to the desired practical outcome in the musical performance.

Much the same applies to the way I am trying to understand theoretical physics. Physicists are unable to give me explanations in the fundamental mathematical language that I need in order to absorb the music and dance of the elementary particles, so I have to work it out for myself. It is hardly surprising that the mathematical language I come up with is not the same language that physicists use – I don’t speak Mandinka, although I have learnt a few words. As far as I understand the Mandinka musical tradition, which is admittedly not very far, the name of a piece defines only the bass line. This is the intrinsic property that is the same for all practitioners and all observers for all times. Everything else varies from time to time, from place to place, and from one performer to another, though with certain common elements that reappear repeatedly. Well, this is probably an exaggeration, because the music is vocal, so there is a melodic line that must also be invariant. But the kora does not always play the vocal line, and usually plays a semi-improvisatory and often virtuosic decorative line instead. This decorative line plays (in the jocular sense) with the bass line, shadowing it sometimes in octaves, sometimes in tenths or thirteenths, often with appoggiaturas or trills (starting on the upper note), and it is this playfulness that is most prized in the musical tradition, as far as I understand it.

But I digress. The point is to distinguish between the intrinsic physical properties of elementary particles (the bass-line, or kumbengo as it is called in Mandinka) and the decorative or extrinsic properties (the birimintingo – well, to be more precise, birimintingo means the biriminting, and “biriminting” said quickly is exactly the sound of one beat of a typical example). In the modern Western classical musical tradition, every note is regarded is intrinsic to the piece. This is a relatively new interpretation, and does not apply in many other musical traditions, where improvisation is a central part of the culture. Much the same applies in modern physics: every note has to be in exactly the place that the composer put it 50 or 100 years ago, and no alterations of any kind are permitted under any circumstances. That describes a culture that is dead or dying, not a thriving culture that grows with the development of society.

What I am trying to do with physics, as with Mandinka music, is to understand the difference between what is fundamental (kumbeng) and what is interpretation (biriminting), and then to play around with the biriminting to make a piece of music that makes sense, that one can dance to, that is surprising but beautiful, to which one can sing the song of how God really made the universe. And in both cases, the ambiguity between 3 and 4 is crucial to making sense of the music.


June 20, 2020

I have known for many years, in principle, that group theory is relevant to the art of change-ringing of bells. But when I first came across this idea at the age of 16, I did not understand the concept of a coset. Without understanding cosets, it is impossible to understand how group theory is used in bell-ringing. By the time I understood cosets, a year or two later, I had lost interest in bell-ringing.

Fast forward 45 years. A chance question from a bell-ringer about the history of groups, a week or so ago, re-kindled this lost interest. It became clear that in the translation from pure group theory to bell-ringing I had confused the elements of the group with the cosets of the group, and this confusion had persisted in my brain for 45 years without any apparent intervention from outside. Faced with the experimental evidence, however, I was forced to confront this confusion and deal with it. Of course, I am not confused about the difference between elements and cosets in abstract groups – that would be absurd. I was confused only about how these concepts map to the appropriate concepts in the application. I wouldn’t say that I now understand the mathematics of change-ringing in its entirety, but I understand the principles on which it is based. After a few days of practice, I have a repertoire of techniques that I can use to compose peals of varying lengths on various numbers of bells. Not that I can compete with hundreds of years of accumulated experience by the expert bell-ringers and composers, of course.

Two points I would like to make. First, that bell-ringers were using groups two centuries before they were defined by Galois. Admittedly these were mostly dihedral groups and symmetric groups. But, secondly, in 1814, I am told (and I have no reason to doubt this), Hudson discovered what we now call PSL(2,5) permuting six bells, and used it for composing peals. Two decades before Galois.

Now it gets interesting for group theorists. What happens for PSL(3,2) on 7 bells? Known in bell-ringing circles since 1906. Why 1906? PSL(3,2) was known to Galois, and featured in Camille Jordan’s 1870 book, but did not become generally accessible until L. E. Dickson’s book in 1901. Is this relevant? Perhaps there is a PhD for a historian of mathematics in this story. And what about PSL(2,7) on 8 bells? Before anyone gets started on the idea of using M11 on 11 bells, let me point out that you have to use it for a full peal of 11! changes, which would take you a couple of years of non-stop ringing, or at best a half-peal of 11!/2 changes. Think what might happen if someone made a mistake and you had to start again… Of course, M11 was known in the 1860s, so it is not impossible that someone might have thought of this, before discarding it as obviously ridiculous.

Now it gets interesting for sociologists. What has been the reaction of bellringers to my quite naive and ill-informed attempts to develop the group theory they use? They reply to my emails. They suggest problems to look at. They appraise my compositions and make polite remarks about them. They forward my emails to their friends. What has been the reaction of (theoretical) physicists to my much less naive and much less ill-informed attempts to develop the group theory they use? The exact opposite.

Group theory in both bell-ringing and physics is quite straightforward, once a couple of initial hurdles are overcome. A professional group theorist can help in both endeavours. In change-ringing the important issues are well-understood, and the techniques have been well-developed over centuries, so the group-theorist’s potential influence is quite limited, but not negligible. The theories are well-tested in experiments, so there is no doubt about the exact correspondence between the two. In physics, the use of group theory goes back only one century, is not well understood, and has changed only over a time-span of about 50 years, up to the 1970s, after which no development at all has taken place. There are significant ways in which the theory does not agree with the experiments. The group-theorist’s potential influence is therefore enormous, if only the extraordinarily arrogant and dismissive attitude of theoretical physicists could change a little bit.


Adventures with a kora, 4

December 23, 2019

It is, of course, not true, despite what you might read in unreliable sources, that my musical activities are largely restricted these days to playing the kora. In public, I play viola and violin, and in private, also piano, recorder and kora. But I have been having kora lessons, on and off, from Seikou Susso, and I find this musical culture fascinating. It is difficult for me, trained in the European classical musical tradition, to accommodate myself to an aural, semi-improvisatory tradition. The fact that there is no fixed pitch is hard to deal with since I have perfect pitch. But the music itself is of a sophistication and virtuosity that is marvellous to behold. Accustomed as I am to reading rhythms off the page, I find it more difficult than I expected to analyse and internalise rhythms by ear. The stress accents in the music, as in the Mandinka language, are less obvious than in English, and it is easy to mis-read a main beat as an off beat, or an appoggiatura as being before the beat instead of on the beat. The music is typically in 4/4 on an ostinato bass or tenor line (kumbengo), but it is often designed to make the beats as rhythmically ambiguous as possible. 2+3+3 is common, but far greater subtlety is often introduced. There is a fluidity of rhythm that cannot easily be notated in Western musical notation. The treble line (birimintingo) often contains a bewildering variety of rhythmic devices and ornamentation. There is only one rule, as far as I can see – you must get to the beginning of the next bar at the right time.

Adventures with a kora, 3

January 23, 2017

After belatedly posting no.2 in this series, I thought I’d better post another update. A couple of weeks ago I tried suarta tuning (with B natural, instead of the B flat of silaba tuning, so roughly tuned to a C major scale) and started practising Jarabi, from the video tutorial I bought from the Kora Workshop. The video does not play properly even on my new laptop, so this is a frustrating experience. Nevertheless, I think I’ve got the basics, although I’ve probably misunderstood some of the subtleties of the rhythm. I’ve memorised the introduction and both versions of the kumbengo (thumb version and finger version), and can play them more or less up to speed, though not with all the right notes every time! When I’ve practised them a bit more, I’ll move on to the variations.

Adventures with a kora, 2

January 23, 2017

Oops! I forgot to post this. It’s a bit out of date now:

Eventually I was forced to replace all 21 strings. Mixing the old and new strings simply did not work, as they were very different in thickness and in tension. It took about a week of what felt like fulltime tuning to get it to the point where the strings were all compatible lengths and tensions, so it could be tuned correctly. Each adjustment of one string would require detuning all the shorter strings, and tuning them back up again. The whole process started to feel like the Towers of Hanoi! (I’m exaggerating, of course, it was more like n^2 retunings rather than 2^n.)
The new strings are a lot heavier than the old ones, and so have a much higher tension. Several of the tail loops broke and I had to make new ones out of the old strings. Anyway, to cut a long story short, the strings are now tuned up to F major, and I’ve learnt a few tunes. Kelefa Ba is the one traditionally taught first to beginners, and I reckon I can give a fairly accurate rendition of the basic version by now. No frills yet. That comes later, I suppose. I’ve been working on some more difficult ones, Kuruntu Kelefa and Alla La Ke.