The Mathematical Problem with Music, and How to Solve It
Formant・22 minutes read
Sound is vibrations in air pressure, perceived by our ears, with frequency measured in Hertz and musical tones are made of pure tones called harmonics. Different tuning systems like Pythagorean tuning and Just Intonation affect the smoothness of note combinations and transposition challenges in music, with modern technology aiding in overcoming these challenges.
Insights
- Pythagoras discovered the mathematical foundation of musical tones and perception, emphasizing the significance of frequency ratios in creating harmonious melodies and intervals.
- Different tuning systems like Pythagorean tuning, Just Intonation, and Equal Temperament offer unique approaches to addressing the challenges of transposition and note combinations, with each system balancing advantages like perfect transposition and harmonious chords against drawbacks such as deviations from natural ratios and loss of key distinctiveness.
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Recent questions
What are musical tones made of?
Harmonics
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Summary
00:00
Understanding Musical Tones and Frequencies
- Sound is vibrations in air pressure, perceived by our ears, with frequency measured in Hertz.
- Musical tones are made of pure tones called harmonics, with frequencies as multiples of the base tone.
- Melodies are sequences of tones with specific frequency ratios, perceived as similar if ratios match.
- Intervals in music, like the octave and fifth, are determined by frequency ratios.
- Octave equivalence makes tones an octave apart sound similar, crucial in music perception.
- Pythagoras discovered the mathematical basis of musical tones and perception.
- Musical instruments face the challenge of producing a wide range of notes with limited keys.
- Pythagorean tuning uses octaves and fifths in simple ratios for musical notes.
- Just Intonation, a historical tuning system, retains some Pythagorean ratios while introducing new ones.
- Transposition, changing the key of a melody, poses challenges in different tuning systems.
15:03
"Harmonics and Transposition in Musical Tuning"
- Pythagorean ratios are close to Just ratios, with small deviations in the third, sixth, and seventh notes.
- A small difference of 1.25 percent can be heard between Pythagorean and Just notes under good listening conditions.
- Just Intonation aims to create better-sounding note combinations, like major chords, compared to Pythagorean tuning.
- Harmonics play a crucial role in the smoothness of note combinations, with Just Intonation eliminating clashes.
- Transposition challenges arise in Just Intonation due to unequal intervals between notes, unlike in Pythagorean tuning.
- Meantone temperament, like quarter-comma meantone, addresses transposition issues by adjusting intervals based on geometric means.
- Equal temperament divides the octave into 12 equal intervals, allowing for perfect transposition but with slightly off ratios compared to Just Intonation.
- The equal temperament's advantage lies in its ability to transpose melodies to any key seamlessly.
- Downsides of equal temperament include noticeable deviations from Just Intonation, especially in the third note.
- The loss of unique character in keys, as seen in other historical temperaments, is a drawback of the equal temperament.
29:07
"Temperaments in Music: Historical and Modern Perspectives"
- Equal temperament allows for perfect transposition, making it popular today, while Just Intonation, played in the original key, sounds horrific due to transposition limitations. Historically informed musicians use temperaments from the time of composition, with Arab, Turkish, and Persian music utilizing microtonal intervals like quarter tones, but some prefer other microtonal temperaments like 19, 31, or 41 equal intervals. Just Intonation advocates for pure integer ratios, with modern technology helping overcome its challenges.
