• # Question: Do you now anything about the connection between maths and music?

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Asked by on 16 Jun 2020. This question was also asked by .
• Sophie Carr answered on 16 Jun 2020:

Hi, this is a great question. Both are certainly creative subjects and I know quite a few mathematicians who enjoy playing music (I played the trombone and double bass all through school and sixth form) There is a lot of maths and physics in the sound waves made by musical instruments starting with the fact that sound is made by something that is vibrating (e.g. a string on a violin) and for musical notes we can “tune” then we want the vibrations to be at a constant frequency (how often the vibration occurs) that is easily controlled. In instruments such as brass, string or basically those which aren’t electronic the vibration is made by a standing wave and for electronic instruments by an electrical circuit. Waves are really good fun mathematically so I’d suggest you have a look!

• Chris Budd answered on 16 Jun 2020:

Maths has a lot to do with music. My favourite connection is in the music of the scale. It was recognised by the Greeks that musical notes sound good when played together if their frequencies are related by simple fractions. So, in the scale of C, the chord C:C (the octave) is in the ratio 1:2, the chord C:G (the perfect 5th) is in the ratio 2:3 (or 1.5) and the chord C:E is in the ratio 4:5. So that is a nice link between fractions and music. Pythagoras (of triangle fame) then constructed the first scale (the Just Scale) which had notes in the ratio 1:9/8:5/4:4/3:3/2:5/3:15/8:2.
This was used until the 18th Century when key board instruments came along. These had to be tuned so that they sounded good in all scales. This meant a change in the construction of the scale to become the equal tempered scale. This divides the octave up into 12 equal semitones, and the ratio of the frequencies of these is the twelth root of 2 .. which is the number which when multipled by itself 12 times gives 2. This number is 1.054946. and it plays a huge role in music. Without it we would not have the modern scale. So, when you are practising your scales you are doing maths : )

• Katy Tant answered on 16 Jun 2020: last edited 16 Jun 2020 8:56 am

I do research in ultrasound (sounds which lie beyond the limits of human hearing). Sound waves are just vibrations of particles in air. The types of sound these vibrations make is dependent on their length scale. So large length scales make low sounds (long, slow vibrations=low frequency) and small length scales make high sounds (short, fast vibrations=high frequency). Objects, depending on their size and the material they’re made of, have a natural frequency at which they vibrate very easily at. So, if we have a series of objects with different sizes then we’ll be able to make a range of sounds by exciting those objects at their natural frequency. So this is what happens with a guitar. We have 6 strings of different thicknesses which range from lower sounds (thick strings) to higher sounds (thin strings). By plucking these we can excite the natural frequencies. And we can make an even broader range of sounds by shortening the strings (holding them against the neck of the guitar). To study sound waves we require a whole range of mathematical techniques, some of which we begin to learn about in secondary school – for example, calculus, trigonometry and geometry!

• Maja Popovic answered on 16 Jun 2020:

There are many connections between maths and music, several are already answered.

One which I find very interesting is that we can basically hear the square root of 2: it is the interval called “tritone” (because it has three whole tones), “augmented fourth”, or diminished fifth”. The difference between the frequencies of the two tones is square root of 2.

For me, it is the very pleasant and interesting sound combination.

• Richard Pinch answered on 16 Jun 2020:

I see there have already been some good answers about the mathematics found in music, so I just want to add a personal comment. People who know I’m a mathematician often ask me whether I’m “good at music”. Unfortunately, not, if you mean can I play (no) or sing (badly). I like to listen to early classical music, though. The music of JS Bach, for example, has patterns in it that are like patterns found in geometry.

• Hannah Speed answered on 16 Jun 2020:

I find the golden ratio really interesting, in a lot of artistic areas

https://www.classicfm.com/discover-music/fibonacci-sequence-in-music/

Also, I’ve just been starting to learn guitar recently and find it really interesting how different intervals sound…
– good and major (happy)
– “experimental” – painful to listen to!

…and that it’s 3rds and 5ths that are repetitively used.

• Hannah West answered on 17 Jun 2020:

Interesting question. I won’t repeat what everyone else has said but I will add that I know people working at the intersection of maths and music. I have a friend who is using machine learning to use computers to create music. They give the computer music written by humans and the computer writes music based on this. Fascinating.

• Arick Shao answered on 18 Jun 2020: last edited 18 Jun 2020 12:24 am

There are many connections between maths and music.

Sounds are mathematically described by functions that one thinks of as waves. (These waves model vibrations in physical media.) Then, certain properties of these wave functions will correspond to how it sounds to you.

If you know a bit of trigonometry, then one basic example is the sine wave, f(x)=A*sin(w*x). Here:
– A is the “amplitude” of the wave, and it describes how loud the sound is. (If A is big, then the sound is loud.)
– w is the “frequency” of the wave, and it describes the pitch of the sound. (If w is big, then the sound is very high-pitched.)
If you playing these sine waves on a computer, then you will hear a continuous beep.

Moreover, the “shape” of the your wave function corresponds to the quality of the sound (or “timbre”, in musical terms) – for instance, whether it sounds like a human voice, a beep, or a trumpet.

There is also interesting maths behind which frequencies/pitches sound good together and which do not. Chris Budd gave a nice answer along this direction!