The Mathematics of Tuning Systems
5 days ago
- #music-theory
- #mathematics
- #tuning-systems
- Leibniz's quote on music as unconscious counting.
- Different tuning systems suit different musical styles, evolving with mathematical innovations.
- Piano keyboard pattern: groups of 2 and 3 black notes, repeating every 12 notes (7 white notes).
- White notes form a 7-note scale; black notes form another useful scale; together, a 12-tone scale.
- Frequency ratios between notes are fundamental, with simple ratios like 3/2 (fifth) being musically significant.
- Historical note naming by Boethius, starting with A, evolving to C major as the 'vanilla' scale.
- Introduction of 'musica ficta' (false notes) outside the 7-tone system, now accepted as black keys.
- 12-tone equal temperament divides the octave into 12 equal parts, the most common tuning system since the 19th century.
- Pythagorean tuning favors rational frequency ratios, leading to the 'Pythagorean comma' discrepancy.
- The tritone ('devil in music') represents mathematical challenges with irrational numbers in tuning.
- Just intonation offers simpler frequency ratios for harmony, using a hexagonal grid to derive scales.
- Just intonation scales can avoid certain dissonances but introduce complexity with multiple semitone sizes.
- Major triads (frequency ratios 4:5:6) are fundamental in music, beautifully realized in just intonation.
- Comparison between just intonation and equal temperament highlights the latter's phase drift in chords.
- Historical glitches in music theory: syntonic comma (81/80) and lesser diesis (128/125) affect tuning.
- The future of music tuning systems is open, with computers enabling exploration of new mathematical models.