Mathematical and Scientific Analysis of European and Chinese Tuning Systems
Citation: Matthew Ho, “Mathematical and Scientific Analysis of European and Chinese Tuning Systems”, American Research Journal of Physics, Vol 11, no. 1, 2025, pp. 1-7.
Abstract
Abstract
This paper presents a comparative mathematical and scientific analysis of European and Chinese tuning systems, tracing
how cultures approached the problem of dividing the octave and balancing consonance with flexibility. Using frequency
ratios and logarithmic pitch units (cents), it quantifies interval sizes in Pythagorean tuning, just intonation (5-limit),
meantone temperaments, and modern 12-tone equal temperament (12-TET), with worked examples and tables to show
the trade-offs among pure intervals, “wolf” intervals, and modulatory freedom. The study then examines traditional
Chinese theory of the 十二律 (twelve lug) and the 三分損益 (cycle of fifths) method, its pentatonic emphasis, and Jing
Fang’s near-equivalence of 53 fifths to 31 octaves—revealing deep historical parallels with Western developments.
Acoustical foundations (harmonic overtones, resonance, beating) explain why small-integer ratios sound consonant and
how tempering subtly detunes them to enable practical performance across keys. The analysis concludes that, despite
differing musical priorities—European polyphony versus Chinese pentatonic practice—both traditions ultimately
converged on equal temperament as a universal compromise between purity and versatility, highlighting the shared
interplay of mathematics, physics, and musical aesthetics. , meantone, equal temperament, twelve lug, Santen Shunyi,
cents, consonance, resonance.
