Pythagorean tuning

music

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June 23, 2026

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Pythagorean tuning is a musical scale system originating in ancient Greece, attributed to the philosopher and mathematician Pythagoras. This system is based on mathematical ratios, specifically using intervals derived from natural fifths. The core idea is that musical harmony can be understood through numerical proportions.1234

Key intervals in Pythagorean tuning include the octave (2:1 ratio), the fifth (3:2 ratio), and the fourth (4:3 ratio). These ratios were discovered empirically, for example, by observing the lengths of vibrating strings. While this system was foundational for Western music theory, it led to slight inaccuracies in some intervals, particularly when used for polyphonic music, which eventually contributed to the development of other tuning systems like equal temperament.2563

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tuning and temperament
Classic tuning systems
…in the Middle Ages, one, Pythagorean tuning, makes all the fifths perfectly consonant. As a result, all the major thirds and major sixths are sharp (too wide) by 22 cents (a cent is 1/1200 of an octave) or by the ratio of 81:80. This amount is called a comma of…
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…in the Middle Ages, one, Pythagorean tuning, makes all the fifths perfectly consonant. As a result, all the major thirds and major sixths are sharp (too wide) by 22 cents (a cent is 1/1200 of an octave) or by the ratio of 81:80. This amount is called a comma of…
Read More

musical sound
Division of the pitch spectrum
Pythagorean tuning provides uniformity but not the chords. Just tuning, based on the simpler ratios of the overtone series, provides the chords but suffers from inequality of intervals. Meantone tuning provides equal intervals but gives rise to several objectionable chords, even in simple music. All…
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Pythagorean tuning provides uniformity but not the chords. Just tuning, based on the simpler ratios of the overtone series, provides the chords but suffers from inequality of intervals. Meantone tuning provides equal intervals but gives rise to several objectionable chords, even in simple music. All…
Read More

comma
In Pythagorean tuning, the intervals of the scale are derived from natural fifths (the interval that occurs between the second and third tones of the harmonic series). When natural fifths are constructed in sequence, as from C to G to D. . ., a circle back…
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In Pythagorean tuning, the intervals of the scale are derived from natural fifths (the interval that occurs between the second and third tones of the harmonic series). When natural fifths are constructed in sequence, as from C to G to D. . ., a circle back…
Read More

stringed instrument
The production of sound
…tend to play in the Pythagorean rather than the well-tempered system.
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…tend to play in the Pythagorean rather than the well-tempered system.
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