Mathematicians have been fascinated by music for thousands of years, studying the patterns displayed by notes, chords, and scales to interpret its many mysteries. Many discoveries later, most concepts are well understood, starting with the basis of sound waves – check out last week’s post. Mathematics can also be used to explain why some sounds are so harmonious and others make us stick our fingers in our ears…
When played together, two notes that an are an octave apart will produce a sound our brains interpret pleasantly – think the first two notes of ‘Somewhere over the Rainbow’, like variations of the same note. This is thanks to the relationships between their frequencies. By definition (as discussed last week), the frequency of the higher note is exactly twice that of the lower and so the sound waves, or vibrations, arrive at the ear in perfect synchronisation.
Clearly notes an octave apart fit together well; however, with no discordance at all, they fit almost too well. For example, playing a Middle and High C simultaneously can sound slightly hollow, arguably boring. Considering instead a ‘fifth interval’ by changing the high C to a middle G, the two notes instead complement each other well (think the first two notes of the Star Wars theme). Middle G has a frequency that is 1.5 times larger than that of C and so the sound waves will coincide in a regular pattern, something our brain interprets positively, and tells us that these sounds work well together.
Even more pleasing is adding a third note, a middle E, to this pair to create a C major chord. The new frequency is 1.25 times that of middle C, meaning every 4 time periods (a time period being the time for one complete wave/vibration) of middle C will coincide with 5 for E and 6 for G. This balance between variety and consistency is what creates the sweetly harmonious, happy sound of the C Major Chord, the most common basis for a tune in all music from Mozart to Metallica.
E and G are not the only notes between the middle and high C. There are 11 in total: some have a pleasing sound when played with a C (such as G) but others produce an unpleasant and uncomfortable one. As discussed, this is related to the ratio between the frequencies and these frequencies are defined, as with most things, using mathematics. A system called equal tempering is used to space all twelve notes of an octave, breaking it up into twelve equal ‘semi-tones’. Since the octave between middle and high C (and all octaves) represents multiplying the frequency by a factor of 2, each semi-tone must be equivalent to multiplying by the, about 1.059464. For example, middle C sharp, a semitone above C, will have a frequency that is 1.05946 times that of C. A middle G is 7 semitones above C and so by definition has a frequency that is 1.49830, approximately 1.5. Therefore, to determine the frequencies that will coincide well with a note it is simply a case of considering the powers of
Clearly maths is a useful tool and language to define music and understand its different properties. Mathematicians have started to take this a step further, seeing the musical results when translating mathematical concepts into sound. Next week we explore exactly this, taking the example of the visually beautiful fractals but representing them instead with music!