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Different tuning systems (just intonation, Pythagorean, equal temperament, meantone, and non-Western systems) produce different sonorities and harmonic relationships; understanding these distinctions explains compositional choices and performance practice in different musical traditions. Contemporary microtonality deliberately exploits these alternative tuning systems for new expressive and structural possibilities.
From your study of pitch and frequency you know that musical intervals correspond to frequency ratios: an octave is a 2:1 ratio, a perfect fifth is 3:2, a major third is 5:4. These pure ratios arise from the overtone series — the natural harmonics that sound objects produce — and they are what the ear perceives as maximally consonant. A tuning system is a decision about which frequency ratios to use for the notes of a scale. The problem is that pure ratios are mathematically incompatible with each other at scale, and every tuning system is a different compromise.
Here is the fundamental conflict, which your knowledge of logarithms and rational numbers will help you see precisely. Stack twelve perfect fifths (each 3:2): you move upward by (3/2)12. Stack seven octaves (each 2:1): you move upward by 27 = 128. If the circle of fifths truly closed — if twelve fifths equaled seven octaves — these two numbers would be equal. But (3/2)12 = 531441/4096 ≈ 129.746, not 128. The ratio 531441/524288 (≈ 1.01364) is the Pythagorean comma: the small gap by which a twelve-fifth stack overshoots seven octaves. Every tuning system is a strategy for distributing this comma across the scale.
Pythagorean tuning keeps all fifths pure (3:2) and accepts that the major thirds will be wide — specifically the ratio 81:64 rather than the pure 5:4. This produces a characteristic bright, tense sound for thirds, which suits monophonic melody but creates beating in sustained chords. Just intonation instead tunes the major thirds pure (5:4) and accepts that some fifths must be impure. The result is maximally consonant harmony in one key but uneven intervals that make modulation problematic. Meantone temperament — dominant from the Renaissance through the Baroque — splits the difference: slightly narrow fifths (each reduced by one quarter of the syntonic comma) produce pure or near-pure major thirds, enabling rich polyphony in common keys while relegating certain remote keys to extreme dissonance.
Equal temperament takes the comma and distributes it uniformly: all twelve fifths are slightly narrow by the same amount, making each semitone exactly the twelfth root of 2 (21/12 ≈ 1.05946). Every key is equally in tune — and equally slightly out of tune relative to pure ratios. The logarithm is central here: the cent is defined as one hundredth of an equal-tempered semitone, or (21/12)1/100 = 21/1200. Any interval can be expressed in cents by taking 1200 × log₂(frequency ratio). A pure perfect fifth is 701.96 cents; an equal-tempered fifth is exactly 700 cents — a difference of less than 2 cents, imperceptible to most listeners but nonzero. Understanding these systems reveals that Western tonal harmony as we know it rests on a practical compromise accepted in the 18th century, and that composers working in microtonality — using intervals between the twelve equal-tempered pitches — are not departing from tradition but returning to its full mathematical richness.
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