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Sunday, 26 February 2012

What is the Theory of Everything? (Part 1)

How can a theory describe everything? Surely we already have theories for most things? And anyway, what actually is everything?

In this post, we will be explaining the concept of a Theory of Everything. What does it consist of, and what are the main challenges surrounding the formation of one? In future posts, we will be looking at the main candidates which could, one day, become the Theory of Everything.


So lets start with the definition of a "Theory of Everything". If you search it on dictionary.com, you get "a theory intended to show that the electroweak, strong, and gravitational forces are components of a single quantized force." and Wikipedia says "A Theory of Everything (ToE) or final theory is a putative theory of theoretical physics that fully explains and links together all known physical phenomena, and predicts the outcome of any experiment that could be carried out in principle." So which is right? Well it turns out that both mean the same thing. There are only four basic interactions in the universe, these we briefly touched on in our post on the standard model particles. That means that everything that ever happens, is due to these forces. The reason you are pulled towards the floor is gravity, the reason you don't actually sink through the floor is electromagnetism, the reason the atoms inside of you don't break apart is the strong force, and, though slightly less relatable, the reason that we can carbon-date objects, or that the sun itself exists in the way it does, is the weak force. Therefore, the result of any experiment can, in theory, be predicted, if we take into account all four of these, as well as the properties of the particles involved. A ToE will unite all these interactions in a way that means that this sort of prediction can be made.

A youtuber called MinutePhysics, who does brilliant videos along the same lines as our posts, recently posted this video. We recommend checking out his channel.

So what are the fundamental interactions that need to be described? We have already covered Electromagnetism and the strong and weak force briefly, and a few weeks ago we posted two posts going into depth on gravity. So lets recap: There are four main forces, or interactions, which I listed above and they all do different things. The one that we can relate to the most is gravity. It is the force that pulls us to earth, and causes us to orbit the sun. Electromagnetism is what causes electricity, the grouping of atoms into molecules, so every material we see, touch and use every day, and also, why magnets work (credit to Steven Spencer on twitter for the link). The other two forces, the strong and weak force, are only applicable at the atomic level. The strong force holds protons (and the quarks inside of them) together in an atom, where their like charges should cause them to repel each other, and the weak force causes quarks to change flavour through the emitting and absorbing of W and Z bosons. If the references to particles are confusing, you may want to check back to our post "What are the subatomic particles?".
It was found that, at high energy states, like right after the big bang, electromagnetism combines with the weak force, to make the electroweak force.


The electroweak force, and the strong force are described in a part of quantum mechanics called the Standard Model. They are described at a very small scale, using fields and carrier particles. However, problems arise when you try to integrate gravity into the Standard Model. At these small scales, gravity is so weak that it is practically irrelevant. Our most accurate theory of gravity is Einstein's theory of relativity. However, Einstein describes gravity as twisted space-time, not as a field with a carrier particle. Therefore, for gravity to be united with the standard model, we need to radically re-think what gravity is. What we just talked about is shown in this diagram:

Theory of Everything
Gravitation
Electronuclear force (GUT)
Strong force
Electroweak force
Weak force
Electromagnetism
Electricity
Magnetism

Source:Wikipedia

As we can see, the next stage in finding the Theory of Everything is trying to unify the strong force with the electroweak force to create the electronuclear force. This would be a Grand Unified Theory (GUT), where all three interactions of the standard model merge. You can think of it as one step down from a ToE, and many of the theories that aim to lead to a ToE, would also lead to a GUT. The GUT would only set in at very high energies, around 16 GeV (Gigaelectronvolts). This is a lot larger than it is possible for us to reach at the moment, so a GUT remains theoretical at least for now.

Overall the main objective, at the moment, is to obtain a theory of gravity, that describes it in terms of quantum mechanics. However, with many new strains of particle physics emerging, like supersymmetry and dark matter, a completed ToE could be very different to what we think.

But there is more to this than just unifying the forces. Could we, in theory, if we had a TOE, unlimited time with unlimited computing power, create a model of the universe, from start to end? If every little interaction could be explained in one theory, why can't the life of the universe, which can be broken down to a large number of these interactions, be described by this theory? If it was the case that everything was determined at the very beginning, does this mean that free will doesn't exist? It is true that for every interaction between particles we would be able to predict the results, however, the problem arrises when we actually have to predict when these interactions would happen. For this we have to look at bit into quantum mechanics.

When we look at the movement of particles, we can only ever predict the probability of it being in one place. When an electron moves from one position to another, it does so in anyway that it possibly can. This means that whilst the electron is moving, it is everywhere in the whole universe at once. This is an incredibly foreign concept, but one that we have to accept. This means that we cannot predict the course the universe will take, but only the probability of each particle being in any position. This then leads onto free will, and there are two positions on this. Either we do not have free will and our actions, and the events around us occur because of these probabilities, or we do have free will. At our macro-scale, we are able to influence the probabilities of things going on around us. The philosophical debate about free will has raged for millennia, and it seems, will continue for many more, even once we have a Theory of Everything.


In a couple of weeks we will be talking about The Theory of Everything, but then considering the leading theories, and the problems they face, that could become the ToE.

We hope you enjoyed this post. If you have anymore questions, you can follow us on twitter, @theaftermatter, email us at contactus@theaftermatter.com or search "The Aftermatter"on Facebook. We really like hearing your feedback or just talking about the posts or other physics and maths. We hope you enjoyed this post.

Ned Summers.

Check out our last two posts:
What are Imaginary Numbers? - What is a imaginary number? Surely a number, by definition, is a unit of measurement, and if it is, how can a number not exist?
What is Gravity? (2/2) - What actually makes up gravity? And how does it fit in with our other understanding of the universe?

What are we posting about next:
What are Artificial Intelligence Composers? - How are computers managing to use algorithms to create music that sounds like it was written by a regular human?

Sunday, 19 February 2012

What Are Imaginary Numbers?

There is a lot of confusion around imaginary numbers. What exactly are they? If they're not "real," what's the point of them? Surely you can't just invent numbers? The Aftermatter shall explain all this and more...


So what exactly is an imaginary number? It is most commonly represented by the letter "i," or in electronics as "j" (to avoid confusion with Current, which also has symbol "I" or "i"). "i" is normally known as the "imaginary unit," and is a constant in algebra. So, some examples of imaginary numbers would be:
i, 5i, -7i, 3.59i

In short, an imaginary number is any real number multiplied by the imaginary unit.

But what exactly is the imaginary unit?

It is defined as:
i2=-1 (or i=√-1)


The problem that mathematicians hit was that they could find no real number whose square was negative. If you square a positive number, the result is positive,  and if you square a negative number, the result is also positive -  as in 52=25 and -52=25,  and if you square zero then the answer is zero. Up until the 1500's mathematicians just considered equations such as x=√-1 to have no solutions. This meant that certain quadratic equations had no solutions, such as x2+1=0, but with the invention of imaginary numbers, it becomes ±i.

You may still consider this absurd, but if you think about it, they are no more "imaginary" than negative numbers! You can't actually have -3 cows in the real world, for example, but they are very useful for certain calculations, and so we use them. The name "imaginary" is quite misleading, it was actually coined by Descartes and used as a derogatory term about these numbers, trying to explain how nonsensical they were. Descartes, of course, was wrong, but unfortunately the name has stuck.

So, the square root of a negative number is simply the square root of the positive equivalent, multiplied by i. This is because we can express √-x as √x×√-1, which is equivalent to √x × ±i. Therefore, √-x = ±i√x. For example, √-4 = √4√-1 = i√4 = ±2i.

This is very hard to visualise; how big is i? Is it greater than 1, or less than 1? Where does it fit in the number line?

Well, we need to stop thinking of a single "number line" as such. The imaginary numbers have their own number line, which is at right angles to the "real" number line. This creates a number 'plane' rather than a number line; the crossing lines have brought the number lines into 2 dimensions. This plane is known as the 'complex plane,' and is shown in the diagram below.


As you can see from the diagram, no matter how much you add imaginary numbers, you will never get to a real number, and no matter how much you add real numbers, you will never get to an  imaginary number. In short, we can treat different and imaginary numbers as different dimensions. For example, increasing the height of a rectangle is completely independent of the width, and vice versa.

The powers of i have some rather interesting properties. They are shown below...
i0=1 (as is with any number to the power of 0)
i1=i (any number to the power of 1 is itself)
i2=-1 (this is the definition of i in the first place)
i= i2×i = -i (you can substitute the i2 with -1)
i= i2×i= -1×-1 = 1 (you can substitute both i2 for -1)
i5=i (1×i, we're back to where we started!)
i6=-1
etc.

So as you can see, the powers of i actually repeat through the pattern 1, i, -1, -i. But why does this happen? How can powers go round in a circle? Surely this violates all mathematical logic?

As you have probably noticed, imaginary numbers are quite different to real numbers. If we consider the complex plane, then addition of a real number is moving left or right on the graph, and multiplying by a real number is scaling by that number from the origin. This idea is quite familiar to us. If you think about it there is nothing unusual about this. But with imaginary numbers, addition is sliding up or down, and multiplication is actually rotation about the origin! When you multiply by i, you are actually rotating the number 90° counter-clockwise. This explains the powers of i; when it repeats, you have literally travelled in a circle!



There is where the mathematical joke comes from: What is 8×i? Answer? ∞!


(Rotate by 90°? Get it? No? Never mind...)


But what about numbers that fall on neither of these number lines? What are they?

These are known as complex numbers, numbers that have both a real and imaginary part, such as "3 + 6i." Complex numbers are of the form "a + bi", where a and b are both real numbers. The diagram below shows their position on the complex plane.

There are some important things to note about this. Firstly, technically speaking, all real and imaginary numbers are complex. For real numbers, b happens to be 0, and for imaginary numbers, a is 0. However, All complex numbers are not imaginary or real, 3+6i sits on neither line but in the middle of the plane.

Secondly, how do we measure "size" in the complex plane? How big is 3+6i? We call this the "magnitude," and the magnitude of a number z is expressed as |z|. The magnitude is the length of the line from the origin to the red dot on the diagram. This is the hypotenuse of a right angled triangle with sides 3 and 6, so by Pythagoras' Theorem we can find the magnitude.


We can also find the angle that the point makes with the Real number line and the origin, by finding the angle of the right angled triangle using trigonometry.


Now, adding complex numbers is very simple, simply add the two "a" terms, and add the two "b" terms, then you have your answer! For example, (3+6i)+(2+4i) = (3+2)+(6+4)i = 5 + 10i. All very intuitive, still only sliding, albeit in two dimensions.

Multiplication and division, however, are a little more complicated. 

Firstly, you have to find the magnitudes of both complex numbers. Secondly, you find the θ angle as we did above, the one the point formes with the Real line and the origin. The magnitude of the new number is the product of the two old magnitudes, and to find the new θ angle we add the two angles together. 

Let's take an example:
(3 + 6i)×(4+3i)
And then to add the angles...




We can check this by multiplying out the brackets fully...

The magnitude of this would be...

which agrees with our original method.

And θ would be the following...
(Note that the convention is for θ to be treated as the angle formed by the
 +∞ end of the real line, the origin, and the complex point in question. 
Anti-clockwise is considered positive. Thus, a negative imaginary number 
would have θ of 270° or -90°. This is why we have the "180-79.7" rather 
than simply 79.7).

So we can see that this multiplication method works. Division is merely the opposite: you divide by the magnitudes, and subtract the angles.

But why should we care? It's nice maths, but the numbers are imaginary, it doesn't affect the real world right? 

In fact, without the discovery of imaginary numbers our world would be very different. Imagine a world without radio waves. Obviously, radio wouldn't exist. Neither would our interview this week on Lauren Laverne's WebChat on Radio 6 (check it out at 43 minutes in). You'd turn on your TV and get nothing but black and white fuzz. There would be no WiFi. There would be no 3G. There would be no GPS. We couldn't have so many planes in the sky because air traffic control would be virtually impossible. We would be stuck in the dark ages of information.

The equations that allow devices to filter out the correct signal from other conflicting frequency (this conflict is known as impedance) is heavily reliant upon complex numbers. Without them, these devices would be unable to isolate a single one of the two waves below, and would only receive a garbled mismatch of the two waves. When you spread this across millions of possible frequencies, without being able to isolate one, there would be no discernible information at all.


In addition, in order to encode audio files such as MP3's from an analogue audio source, it is necessary to find the discrete Fourier Transform of the sound waves, to digitise the track so that it can be read by computers. This once again uses complex numbers.

Complex numbers are almost ubiquitous in computing. In order to calculate the movement of 3D graphics, we actually need two more dimensions of imaginary numbers, called j and k, to fully describe where the object is and what it is doing. This proves the point: we could have as many "dimensions" of imaginary numbers as we want to define: the original imaginary numbers were defined because one extra dimension was useful. Quaternions were defined because two more were found to be useful as well. At some point, use may be found for even more dimensions, and by the Point-Line-Plane postulate, we could theoretically create an infinite amount of number lines, and an infinite amount of numerical dimensions. Numbers without units really are quite abstract: As long as it is mathematically consistent, you can pretty much define anything that may be useful to you.

In conclusion, we hope you now have a better understanding of what imaginary numbers are, and also what they are used for. It can seem like an absurd idea at first, but just as the greatest minds struggled at first with 0 and the negative numbers when they were first proposed, you will eventually understand, and we hope we have helped you. The fact that you can define new sets of numbers does make us question how "real" our real numbers actually are; in fact, it can be argued that without units, real numbers are just as abstract as imaginary ones!


We hope you enjoyed this post. If you have anymore questions, you can follow us on twitter, @theaftermatter, email us at contactus@theaftermatter.com or search "The Aftermatter"on Facebook. We really like hearing your feedback or just talking about the posts or other physics and maths. We hope you enjoyed this post.

Theo Caplan.

Check out our last two posts:
What is Gravity? (2/2) - What actually makes up gravity? And how does it fit in with our other understanding of the universe?
What is Harmony? - Following on from our post What is Music?: How can physics and maths describe why some notes work with each other, and some don't?

What are we posting about next
What is the "Theory of Everything"? - What are the leading theories that unite quantum mechanics and Einstein's general relativity to describe the actions of everything in our universe?

Sunday, 12 February 2012

What is Gravity? (2/2)

A couple of weeks ago, I wrote the first post on gravity. We found out what the fundamental rules that stated what objects in a gravitational field did. But there is a bigger question underlying all of it. What actually is gravity? How does it work?

In the first post, we looked at the first two modern analyses of gravity, by Galileo and Newton. Galileo majorly revolutionised view of gravity at the time that he was alive. Up until then, it was commonly believed that the speed at which something travels at when in free fall, was governed by mass. A three kilogram weight would fall faster than one that weighed only one kilogram. Galileo performed experiments and proved that, in fact, that all objects would fall accelerating uniformly at the same rate. The only reason that a hammer and a feather fall at different speeds is due to air resistance (Video in first post). Then we moved onto Newton, who formulated equations, specifically F=G m1m2r2, that described the force that two objects of mass m1 and m2 at a distance of r from each other exert on one another. Using this equation, and some of Newton's other laws of motion, we worked out what happened if everyone in China jumped at the same time. This was a brief summary, and I recommend you check out the original post before reading this one.

Anyway, Newton's developments into gravity were so revolutionary that they stood for about 200 years until they were added too. This addition came from the mind of one the most brilliant theoretical physicists ever. A man, born in March 1879, called Albert Einstein. He had an ability to look at an everyday situation and to see that there was more to it then there seemed. This lead to his Theory of Relativity, which is split into two parts. Einstein's Special Theory of Relativity looks into the way that our space-time is structured, although at the time that he wrote it, he did not consider there to be unified space-time, but more on that later. Special relativity is where we get the famous formula E=mc2. It also radically changed our understanding of time and what happens to things as their velocity increases. I wrote post on it and a proof that an object moving very quickly will be experiencing slower time than a stationary objects, and you can read that here.
But what we are more interested in this post is Einstein's General Theory of Relativity. It is a theory of gravity, and the most recent and accurate we have so far. What astronomers had noticed at the time was that Mercury's orbit was changing over time. The shape remained the same, but the ellipse itself moved, an effect called apsidal precession. This is shown in this animation:
Newton's theory only went far enough to describe the original elliptical shape, but could not explain this precession. Further more, Einstein was intrigued by the observations of Newton and Galileo. They had both confirmed that two objects with different masses would be pulled on by a different force within a gravitational field. He realised that this was a strange result, any object would accelerate at the same rate towards the source of gravity, but not with the same force and he set out to find out how. 

With Special Relativity, Einstein adapted Newton's laws of motion to work with objects moving at very high velocities near the speed of light. However, these only worked with an object at constant velocity, this is because one of the axioms that the theory is based on only affects objects that are not accelerating. However, when he did a thought experiment two years after publishing his Special Theory of Relativity,  he realised that someone in free fall and someone accelerating due to a force other than gravity would not be able to tell the difference. In other words, gravity is simply an acceleration.  The fact the gravity is equivalent to mass meant that there was also a link between mass and gravity and, through the equation E=mc2, gravity and energy.  If we think about this, it makes sense. Imagine a rocket moving very quickly. One of Einstein's conclusions in Special Relativity says that as an object's speed increases so does its mass. Therefore, this fast moving object has a lot of mass due to it having a lot of energy and therefore, it has a large gravitational pull.

His biggest revelation came after this though. Einstein spent ten years working on this theory and one year after he started a past teacher of his, Hermann Minkowski, came up with a different interpretation of the paper. He concluded that we did not live in three dimensions of space and then a completely separated dimension of time, but instead we lived in four dimensions of Space-time. I go into a little bit more detail about space-time in this post. This interpretation was essential for General Relativity as it enabled on of the most radical changes to gravitational theory since Galileo first analysed it. He did not describe gravity as two objects pulling  on one another. Instead, he said that an object's mass caused it to bend the space-time around it. An object with a large mass would bend it more. When something is being pulled by an object's gravity, it is actually moving along a straight line in this curved space time, but from our perspective, it seems to curve around the centre of gravity. It is easy to imagine this bending and curving if we imagine the phenomenon to occur in two spacial dimensions rather than its usual three and with out the temporal dimension:
Now you can see that an object's path on this plane would be severely distorted. However, from the eye of the person in the object, they would just seem to be pulled towards the object in the centre. It is a bit more difficult to imagine curved space time in its full four dimensions, but the best analogy I came across is imagining putting your finger in a still pool of water. The water ripples and your view of the bottom is distorted. Of course, this still does not really show what curved space-time is really like, but it gives you some idea. So as an object's mass increases, so does the effect it has on its surrounding space-time. One thing that Einstein's theory confirmed was the existence of black holes. Objects of near infinite density that curve space-time so much than nothing can escape. Now Einstein had established a link between space-time and acceleration, he had a theory that worked for all objects, whether at a constant speed or accelerating. In fact, he found that gravity affected time just as speed did. When an object was close to an object with a large mass, time slows down for it. That is describe as gravitational time dilation and is explained in this post. There is a lot more to talk about in the realms of relativity, however, I need to move on!

So as I said, Einstein's theory of gravity in General Relativity has never been disproven. But it poses a problem. The two theories that can describe pretty much everything are the Theory of Relativity, specifically General Relativity, and Quantum Mechanics. Quantum mechanics describes everything at a molecular level, however the one thing it cannot describe is gravity. All the other four fundamental interactions have been modified and work perfectly within quantum mechanics. They each have "force carrying" particles, or bosons, and can be described in this way. However, at sub-atomic scales, gravity, the weakest of all the fundamental interactions, has such a tiny effect, it is negligible. This means that it can not be included in quantum mechanics, and because general relativity describes gravity as a change to the fabric, instead of a field or particle, it can not be integrated with it. This is one of the only things that is stopping us from discovering a "Theory of Everything", a theory that could predict what would happen in any experiment. There are many theories of how to get around this problem, and many theories that could potentially be the theory of everything. However none of these have enough proof to be considered. I would just like to briefly talk about quantum gravity, or trying to develop a theory of gravity that describes it so that it can be integrated into quantum mechanics. One theory is that there is a virtual carrier boson, called the Graviton, that carries the force. This theory comes about because all the other fundamental interactions have been shown to have virtual bosons. The graviton would be massless, as it can affect anything, anywhere instantly. A theory we have discussed before, string theory, predicts the existence of the graviton, however, in almost every other theory it runs into large problems. Quantum gravity is a very grey area and it may be many years until a successful theory integrates gravity and quantum mechanics, though there are some theories out there, look here for a bit more information of quantum gravity. And look here for more information on the standard model particles and the force carrying particles.


We hope you enjoyed this post. Please follow us, @theaftermatter, or email us at contactus@theaftermatter.com. We really like hearing your feedback or just talking about the posts or other physics and maths. We hope you enjoyed this post.

Ned Summers.

Check out our last two posts:
What is Harmony? - Following on from our post What is Music?: How can physics and maths describe why some notes work with each other, and some don't?
What is Gravity? (1/2) - What is the history of the force that holds us onto the earth, and what would happen if everyone in China jumped at the same time?

What are we posting about next?
Imaginary Numbers - How can a number not exist? And if it doesn't exist, where and how can we use it in real life?

Sunday, 5 February 2012

What is Harmony?

A couple of weeks ago, we published a post called "What is music?" and had a flood of comments from people asking a follow up post. Here it is: (We recommend that you read the original post in order to help with you understanding of this one)



As we discussed in our last post on music, the Harmonic Series is a series of musical notes whose frequencies are integer multiples of the original note chosen, known as the fundamental frequency. Here are the first few harmonics:



The musical among you may recognise some common chords within this. This is more than a coincidence, many chords are actually defined by the Harmonic Series.

For example, the simplest chord is simply the first and second harmonic played together. This is an octave, which is arguably the most harmonious interval. However, these notes are almost too related to be musically interesting, and are taken to be virtually synonymous in the musical world. The ratio of the frequency of these notes is simply 1:2, since it is built of the first and second harmonics. This means that if we consider the ‘pockets of air’ hitting our ear as we did in the last music post, then we get the pockets lining up every time a lower frequency is played. This pattern is extremely repetitive, but that is what makes it sound natural.

When we try and build a slightly more complex chord, things become a little more interesting. Let’s take the major triad: probably the most common chord in music. If we take the chord to be a C Major triad, then the notes we need to form it are C, E and G. If we look at the Harmonic Series, we see that E and G are in fact the first two notes to appear on the Series which are not simply an octave or two octaves above C. Here we can see that C, E, and G are the 1st, 3rd , and 5th harmonics of C, as shown below. Therefore, the ratio of their frequencies is 1:3:5.


By taking the lowest common multiple of 1, 3 and 5, which is 15, we see that it does not take particularly long for the patterns in the air pockets to recur. This is much faster than more random frequency ratios, such as 1:5.3273:7.8079, which would only repeat after 78,079 units of time, much longer. This chord would not sound harmonious at all, and it is this that is at the basis of what makes something sound harmonious: frequent repetition of the patterns in the sound waves.

In the same way, a C7 chord would consist of C, E, G and B, which first appear as the 1st, 3rd, 5th, and 7th harmonics with C as a fundamental. This gives a lowest common multiple of 105, meaning the pattern recurs after 105 units of time, which is again very low for a lowest common multiple of 4 numbers.

Likewise, to form a minor triad, we can take the 3rd, 5th, and 15th harmonics, which gives us an E minor chord. By once again taking the frequency ratios, 3:5:15, we find their lowest common multiple once again to be 15.

So it is simple integer ratios between the frequencies which make the difference between consonance (notes sounding harmonious) and dissonance (notes sounding more jarring), but why is this the case? Neurologists think that musical frequencies cause neurons to fire in the brain with patterns that represent the wavelengths of the music. This means that harmonious chords will cause more regular patterns of neuron firing than a random collection of frequencies. Information theory states that a regular binary pattern (such as neurons firing) carries much more information that a random pattern, and neurologists think it is this greater amount of information that makes harmony appealing. They also found that people's preference for consonance has nothing to do with culture or musical training: babies and even monkeys have both been found to demonstrate significant differences in brain activity when listening to consonant or dissonant music. 

But there is a problem. The precise frequencies found in the Harmonic series form scales where there are unequal differences in pitch from one note to another. This is no problem at all, until you try to modulate, or change key. As soon as you change key, the same notes of the scale will sound horribly out of tune. For this reason, up until about 300 years ago, many instruments, such as early pianos and harps, could only play in a very limited number of keys, and key changes during a piece were virtually impossible. The solution suggested for this was 'equal temperament.' A new method of tuning was brought about, where each note was a fixed distance higher than the one below it. This allowed changes in key, but it meant that the perfect ratios for harmonious chords were now slightly off. We no longer use 'true' harmony, just a rough approximation. Equal temperament is a compromise: less harmonic integrity, and a poorer sound, in exchange for easy modulation and extreme versatility.

It is a compromise that some have begrudged. Recently, a man called Geoff Smith has invented an instrument called the 'Fluid Piano,' which is like a normal piano, but with a tuning slider on each note, which allows it to be rapidly tuned to any tuning system rather than being stuck with equal temperament. I have been lucky enough to see, play, and even compose for this instrument myself. I wrote a piece for the Fluid Piano, Kora (a West African lute-harp), and Tabla (Indian Drums). This was actually one of the first pieces ever composed for the Fluid Piano, and it is the first time that the Kora has ever played with a keyboard instrument! This is because the Kora still uses 'just intonation,' or 'true' harmony as dictated by the harmonic series. If played with a normal piano, the Piano and Kora would sound out of tune with each other, but with the Fluid Piano tuned to a true Bb scale, they sound perfect together. The piece is in the video below.



So, in conclusion, harmony is based on repeating patterns in the sound waves, and therefore firing in the neurons in our brains. We no longer use the perfect harmonies that the Harmonic Series give us, but that is a compromise we have had to make to allow modulation: a vital element of modern music. Although this post has been about harmony, it is important to remember that much of the progress made in music has been in dissonance, and how to use it in a way that is still pleasing to us despite its neurological disadvantage.



Please follow us, @theaftermatter, or email us at contactus@theaftermatter.com. We really like hearing your feedback or just talking about the posts or other physics and maths. We hope you enjoyed this post.

Theo Caplan.

Check out our last two posts:
What is Gravity? - What is the history of the force that holds us onto the earth, and what would happen if everyone in China jumped at the same time?
What is Music? - How can music be described mathematically? Why do different instruments sound different? What actually is sound? (If Stephen Fry liked it then we are sure you will!)

Next week, we will post again on gravity. We shall move on from Newton's Law of Universal Gravitation on to Einstein's General Theory of Relativity, and then Quantum Gravity. We have written posts on Einstein's relativity and how it explains that time is not as constant as we think before, if you would like to read those, click here. The post is up!