Music: where physics and philosophy collide

Sound is the compression and expansion of air. Musical notes are produced when something (usually an instrument) vibrates and produces sound. This vibration is a wave, meaning it has the related properties frequency and wavelength. Frequency corresponds to the pitch of the sound. Wavelength is the distance taken up by the wave’s motion, and is inversely proportional to frequency, meaning higher frequencies have smaller wavelengths and vice versa. In short, larger vibrating parts produce lower pitch.

The Greek mathematician Pythagoras supposedly discovered this relationship between pitch and size. Given a string of some length, he found that a string of 1/2 that length produces a pleasant sound an octave higher, and a string of twice the length produces sound an octave lower. Similarly, a ratio of 2/3 produces a fifth, 3/4 a fourth, 4/5 a major third… In short, the ratio of two frequencies determines an interval, and lower integer ratios tend to produce more pleasing sounds. By combining these string lengths of different ratios Pythagoras divided the octave into a number of intervals, forming a musical scale similar to the modern major scale.

The reason why integer ratios make pleasing sounds lies in overtones. Most things that vibrate can do so in several modes, called harmonics or overtones. This is a picture of how a string vibrates in multiple modes to produce harmonics. The first mode typically vibrates the strongest, producing the fundamental pitch of the note.

Consonance (more pleasing sounds) and dissonance (less pleasing sounds) are the result of a physiological (caused by the human body) reaction to two notes played simultaneously. This is a picture of how the harmonics of a consonant interval align versus a dissonant interval. In consonant intervals more overtones align because the two fundamental pitches are a simple ratio apart, whereas less overtones align in dissonant intervals. In short, the better two notes “fit” in one another the more pleasing they are to the ear.

While intervals, consonance, and dissonance exist naturally (thanks to physics!), the way these are used in musical scales and musical harmonies is up to the musician. Dissonances that were considered ugly in the Baroque and Classical eras became beautiful and even preferred from the Romantic era onward. There isn’t a single unresolved tritone in Bach while many modern pieces are littered with them. The aesthetic of music (its methods and objectives) combines both objective (fact-based) and subjective (opinion-based) ideas. Music theory explains how every piece of music uses a unique combination of the objective and subjective properties of sounds to form its own musical aesthetic.