Consonance and Dissonance: The Mathematics of Musical Harmony
The distinction between consonant (pleasant, stable) and dissonant (tense, unstable) musical intervals has both mathematical and psychological foundations rooted in the harmonic series and the way the human auditory system processes complex sounds. Consonant intervals correspond to simple frequency ratios where the harmonics of the two notes align or complement each other, while dissonant intervals create frequency ratios that result in complex beating patterns and auditory roughness.
The most consonant intervals correspond to the simplest frequency ratios found early in the harmonic series: - Unison (1:1) - Perfect consonance, identical frequencies - Octave (2:1) - Strong consonance, second harmonic relationship - Perfect fifth (3:2) - Strong consonance, third harmonic relationship - Perfect fourth (4:3) - Moderate consonance, fourth harmonic relationship - Major third (5:4) - Mild consonance, fifth harmonic relationship - Minor third (6:5) - Mild consonance, sixth harmonic relationship
Hermann von Helmholtz's theory of consonance and dissonance, developed in the 19th century, attributes these perceptions to the interaction between harmonics of different notes. When two notes are played simultaneously, their respective harmonic series create combination tones and beating patterns that can enhance or detract from the perceived consonance.
Critical bandwidth theory provides a modern understanding of dissonance based on the frequency resolution of the human auditory system. The ear analyzes sound through a bank of overlapping bandpass filters with bandwidths that increase with frequency. When two pure tones fall within the same critical bandwidth (approximately 1/3 octave), they create roughness and dissonance. When they are separated by more than a critical bandwidth, they are perceived as distinct and potentially consonant.
The critical bandwidth Δf can be approximated by: Δf = 25 + 75(1 + 1.4f²)^0.69
Where f is frequency in kHz. This relationship explains why the same interval (in terms of frequency ratio) can sound more or less dissonant depending on the register in which it's played—intervals that sound consonant in higher registers may sound muddy and dissonant in lower registers where critical bandwidths are narrower.
Sensory dissonance, based on the roughness created by beating between closely spaced frequency components, differs from musical dissonance, which incorporates cultural and stylistic factors. A major seventh interval (15:8 ratio) creates significant sensory dissonance due to its complex frequency relationships, but it functions as a stable chord tone in jazz harmony. Conversely, a perfect fourth (4:3 ratio) has low sensory dissonance but creates musical tension in certain harmonic contexts.
Combination tones add another layer of complexity to consonance and dissonance perception. When two loud pure tones are played together, the nonlinear response of the auditory system creates additional frequencies not present in the original sound:
- Difference tone: f₂ - f₁ - Summation tone: f₂ + f₁ - Higher-order combinations: 2f₁ - f₂, 2f₂ - f₁, etc.
These combination tones can enhance consonance when they align with harmonic series relationships or create additional dissonance when they conflict with the intended harmony. The effect is most noticeable with loud, pure tones but also occurs with complex musical tones.
Just intonation tuning systems are based on pure frequency ratios derived from the harmonic series, creating perfectly consonant intervals within specific keys but limiting modulation to other keys. Equal temperament tuning, used on modern pianos and most Western instruments, slightly adjusts all intervals except the octave to enable playing in any key with equal facility. The equal-tempered semitone ratio is:
r = 2^(1/12) ≈ 1.0595
This compromise means that no interval except the octave is perfectly in tune according to harmonic series ratios, but all intervals are close enough to sound acceptably consonant while enabling unlimited modulation between keys.