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Why do a capella singers go flat…? The mathematics of tuning systems in music Colin Graham @ColinTGraham

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Some points to consider Division of the octave

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Some points to consider Division of the octave Number of pitches to be used

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Some points to consider Division of the octave Number of pitches to be used Tuning of the pitches

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Western tuning systems Equal temperament Based on dividing into 12 equal parts

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Western tuning systems Equal temperament Pythagorean tuning Based on dividing into 12 equal parts Based on perfect fourths and fifths

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Western tuning systems Equal temperament Pythagorean tuning Just intonation Based on dividing into 12 equal parts Based on perfect fourths and fifths Based on whole- number ratios

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Fundamentally harmonic…

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A matter of ratios… 5/12 = 5/6 = 5/3 = 10/3 = 20/3

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A matter of ratios… 5/12 = 5/6 = 5/3 = 10/3 = 20/3 Get used to it, if you want to read about tuning theories!

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A matter of ratios… 5/12 = 5/6 = 5/3 = 10/3 = 20/3 Get used to it, if you want to read about tuning theories!

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What to choose? Equal Tempered Just intoned Pythagorean tuning 466.27/4 = 457.816/9 = 465.1 392.03/2= 392.4 329.65/4 = 327.181/64 = 331.1 261.6 Frequencies in Hertz based on A = 440Hz

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Now the fun begins!

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Musical. Review String – = 2L –Frequency of overtones 1 st : 2 × fundamental frequency 2 nd : 3 × fundamental frequency 3 rd : 4 × fundamental frequency.

Musical. Review String – = 2L –Frequency of overtones 1 st : 2 × fundamental frequency 2 nd : 3 × fundamental frequency 3 rd : 4 × fundamental frequency.

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