maths and music

Western music has its roots in the harmonics discovered by Pythagoras – himself a keen musician – over 2000 years ago.  Pythagoras noticed that certain string ratios would produce sounds that were in harmony with each other.   The simplest example is illustrated above with an electric guitar.  When a string is played, and then that same string pressed half-way along its length (in the guitar’s case the 12th fret), then we get the same note – this is a whole octave.

If you were to then half the distance again you would get another octave (which explains why guitar frets get smaller and smaller near the base of the instrument – the frets mark ratios relative to the whole string).

The ratio 1: 1/2 shows the ratio of an octave.  A full length string: half length string.  We can multiply both sides by 2 to remove the fraction to get, 2: 1.  This is the octave ratio.

All the other harmonies that are the basis of Western music can also be understood through similar ratios.  The chord sequence E, A, B – which is the standard progression for blues and modern music comprises of the base note (in this case E), along with the perfect fourth (A) and the major fifth (B) of the base note.

Looking at the guitar fret we can see that the perfect fourth (A), which occurs on the fifth fret, has the ratio 1: 3/4.  That is 1 whole string: 3/4 of the whole string.  We can simplify this to get 4:3.

The major fifth (B) which occurs on the seventh fret has the ratio 1: 2/3 which simplifies to 3:2.

The other most likely note used in the key of E would then also be either G (the minor third) which has a ratio of 6:5, or G sharp (the major third) which has a ratio of 5:4).

maths and music 2

It’s interesting that we find these particular whole number ratios pleasing to listen to – indeed these are the notes that often sound “right” when playing through songs.  It’s also helpful to look at the circle of fifths – which shows all notes which are in the ratio 3:2 with each other.  Moving around the circle again produces music which sounds nice.  For an example of this, starting at C, the progression C,G,D,A,E is the one used by Jimmy Hendrix in the classic song, Hey Joe

maths music 3

There are lots of other areas to explore when looking at the relationship between maths and music – one of which is looking at how we can model the wave frequencies of notes using modified sine/cosine curves.  The IB have included a piece of coursework on this as an example for the new exploration topics.

Another interesting exploration is looking at the strange properties of the Harmonic Sequence – which is the sequence 1, 1/2, 1/3, 1/4… This sequence like many of those found in music is said to be in harmonic progression .  There are some interesting paradoxes related to the harmonic sequence – and a variety of methods of proving that the sum of this sequence (the series) actually diverges to infinity – even though you would intuitively expect it to converge.  The video below provides a taster on this topic:

If you liked this post you might also like:

Synesthesia – Do Your Numbers Have Colour? What happens when 2 senses get cross-wired in the brain.

Wau: The Most Amazing Number in the World? A great video by Vi Hart – see if you can spot the twist!

Screen Shot 2018-03-19 at 5.29.06 PM

IB Revision

Screen Shot 2018-03-19 at 4.35.19 PM

If you’re already thinking about your coursework then it’s probably also time to start planning some revision, either for the end of Year 12 school exams or Year 13 final exams. There’s a really great website that I would strongly recommend students use – you choose your subject (HL/SL/Studies if your exam is in 2020 or Applications/Analysis if your exam is in 2021), and then have the following resources:

Screen Shot 2018-03-19 at 4.42.05 PM.pngThe Questionbank takes you to a breakdown of each main subject area (e.g. Algebra, Calculus etc) and each area then has a number of graded questions. What I like about this is that you are given a difficulty rating, as well as a mark scheme and also a worked video tutorial.  Really useful!

Screen Shot 2019-07-27 at 10.02.40 AM

The Practice Exams section takes you to ready made exams on each topic – again with worked solutions.  This also has some harder exams for those students aiming for 6s and 7s and the Past IB Exams section takes you to full video worked solutions to every question on every past paper – and you can also get a prediction exam for the upcoming year.

I would really recommend everyone making use of this – there is a mixture of a lot of free content as well as premium content so have a look and see what you think.