Introduction to the Physics of Music
Defining Music
Music is an arrangement of sounds. Sounds are defined by their frequency spectrum and length. They are made of fundamental and harmonic waves, which together define timbreāthe unique tonal quality that distinguishes one instrument or voice from another.
The Duodecimal System
European music uses a duodecimal (or dozenal) system in which an octave is divided into 12 parts. The scale formed by the 12 tones played consecutively is called the chromatic scale. When you reach the end of the scale, you hit the octave.
An octave is the doubling of a frequency. The note A 440Hz is the same as A 880Hz, except higher in pitch. By analogy, if A 440Hz was a dark green, A 880Hz would be a lighter shade of the same green.
Chords
Using notes, we build chords. Strictly speaking, chords need two notes to exist, but the term mostly refers to three-note chords. In three-note chords, there are three main categories: minor, major, and suspended.
Temperament
We call temperament the division of the octave. As we have seen, equal temperament divides the octave into equal parts, where each step is equal to the next one. The 12-tone equal temperament has become the standard in modern musical instruments.
However, equal temperament is a compromise and does not obey the laws of physics. We need an alternative. Using mathematics, we can build a better temperament, which is called just intonation. Just intonation uses ratios of perfect numbers to divide the octave:
| Interval | Ratio | Just Intonation (Hz) | Equal Temperament (Hz) |
|---|---|---|---|
| Unison | 1/1 | 220.000 | 220.000 |
| Minor second | 16/15 | 234.667 | 233.082 |
| Major second | 9/8 | 247.500 | 246.942 |
| Minor third | 6/5 | 264.000 | 261.626 |
| Major third | 5/4 | 275.000 | 277.183 |
| Perfect fourth | 4/3 | 293.333 | 293.665 |
| Augmented fourth | 7/5 | 308.000 | 311.127 |
| Perfect fifth | 3/2 | 330.000 | 329.628 |
| Minor sixth | 8/5 | 352.000 | 349.228 |
| Major sixth | 5/3 | 366.667 | 369.994 |
| Minor seventh | 16/9 | 391.111 | 391.995 |
| Major seventh | 15/8 | 412.500 | 415.305 |
| Octave | 2/1 | 440.000 | 440.000 |
| Minor ninth | 16/15 | 469.333 | 466.164 |
| Major ninth | 9/4 | 495.000 | 493.883 |
| Perfect eleventh | 8/3 | 586.667 | 587.330 |
| Minor thirteenth | 8/5 | 704.000 | 698.456 |
| Major thirteenth | 10/3 | 733.333 | 739.989 |
| Double octave | 4/1 | 880.000 | 880.000 |
Using these ratios, we correct equal temperament by detuning the notes inside each chord. Just intonation provides pure sounding intervalsāthey are consonant and eliminate the resonances that occur with equal temperament.
Just intonation is a dynamical system and therefore difficult to implement into a physical device, because corrections may or may not happen to notes depending on when they appear in the composition.
Interesting Facts
People tend to naturally follow just intonation when singing or playing instruments that allow pitch correction, such as violin, cello, and brass instruments to some extent. For example, vocal ensembles always tend to use just intonation.
Conclusion
Just intonation as superior tuning method helps fix music. It respects the human body, like incandescent light bulbs do. If Bitcoin fixes money, just intonation does the same for music and it brings a new breathe into modern composition.