Beneath the beauty of music lies some interesting mathematics, from Fourier transforms of waveforms to ratios of frequencies. In this blog post, we’ll be discussing frequency ratios and tuning in particular!
The most simple ratio is the 1:1 ratio (perfect unison); that is, two sounds with the same frequency will sound at the same pitch. There is also the 2:1 ratio (perfect octave), the most consonant interval. Multiplying the frequency by 2 will always give a pitch an octave above, so the 4:1 ratio will be a perfect fifteenth (2 octaves above) and so forth. This means that if you play one tone at 100 Hz and another at 400 Hz, you will hear two tones separated by an interval of 2 octaves.
The just intonation (or pure intonation) tuning system utilizes similarly simple ratios for other common intervals. For example, the 3:2 ratio is the perfect fifth (the interval from C going up to G). The 4:3 ratio is the perfect fourth, and the 5:4 ratio is the major third.
However, our current twelve-tone musical system does not function very well when using these simple ratios. There are many intricacies with this tuning system that can result in some “out of tune” sounds and the music drifting away from the original pitch. One example is in a comma, which is the interval between a note being tuned in two different ways. For example, the syntonic comma is the 81:80 ratio.
In modern music, equal temperament is used. In our twelve-tone system, that means the difference in frequencies in a semitone is the twelfth root of 2. A perfect fifth is 7 semitones up, thus the frequency difference is 7 times the twelfth root of 2, which roughly approximates 3/2. This system allows us to play in any key equally by having all intervals slightly out of tune from their just counterparts.