Scales and the Maths of Music

12 November 2006

Interview with

Tim Gowers, Cambridge University


Kat - We want to find out about the maths and music. Can you tell us a little about the basics of music? What is a scale, what are notes, and how do they fit together?

Tim - A scale is any sequence of notes that goes up in smallish steps. You can get major scales, or a minor scale.

Kat - What is different about the minor scale?

Tim - For some reason that I can't really explain, major scales seem to make you feel happy but minor don't.

Kat - Does a major scale sound happy and minors sad in all cultures or is it just us here in Cambridge?

Tim - That would be an interesting experiment that should be done and possibly has but I don't know.

Kat - In Western classical music, we have a very defined idea a major or a minor scale, is this true in other cultures?

Tim - Even in Western music, jazz uses scales which are not major or minor.

Kat - These are modal aren't they?

Tim - Some of them are modal and some are more complex. In jazz this is the diminished scale. Some classical musicians call it the octatonic scale. You make it by going up alternating by a semi-tone then a tone. It sounds strange as it has a lot of chords in it called diminished fifths which sound strange. There is a mathematical reason for this. None of the ratios between the two frequencies are close to a whole number ratio of one another.

Chris - So we like notes that are a whole number ratio of one another's frequency?

Tim - Yes. For example, two notes that are an octave apart sound nice together and one is exactly twice the frequency of the other. In fact the ratio of 2:1 seems to make us perceive it as the same.

Kat - If you play two notes a semitone apart, it sounds to my ears horrible. Has what is considered as horrible changed over time?

Tim - It has changed a lot. A long time ago people even found things based on major thirds as dissonant, and everything was based on fifths and octaves.

Chris - You are the Rouse Ball Professor of Mathematics at Cambridge University. Is it a hobby or part of your research?

Tim - It definitely isn't part of my research, as I think the maths involved in music isn't very complex. So I would say that the fact I am a mathematician means that I have a certain way of looking at music.

Kat - It has been said that mathematical people are better at music or vice versa.

Tim - I like to say both mathematics and music are dealing with abstract ideas; so it's like saying does being good at football and cricket go together? In some ways yes, as if you are good at one you are probably reasonably coordinated, but it doesn't necessarily follow.


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