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 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!

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