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https://texercises.com/exercise/pythagorean-comma/
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Exercise:
abcliste abc Explain why an arbitrary number of just fifths will never add up to an eger number of octaves. abc On a piano keyboard a sequence of fifths corresponds to octaves. Calculate the ratio of these ervals asing just fifths. This deviation is known as em Pythagorean comma. abcliste

Solution:
abcliste abc Since a just fifth is given by the frequency ratio : any number of fifths corresponds to a ratio of an odd and an even eger. This can never be simplified to be an even eger. A number of octaves always corresponds to an even eger. abc fifths correspond to a ratio I_ leftfracright^Nf and octaves to a ratio I_ leftright^No The ratio is thus I_:I_ frac^Nf^Nftimes ^No frac^Nf^NfoP resultratio With equally tempered fifths seven half-tones there is no such problem: left^/right^ ^ abcliste
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Exercise:
abcliste abc Explain why an arbitrary number of just fifths will never add up to an eger number of octaves. abc On a piano keyboard a sequence of fifths corresponds to octaves. Calculate the ratio of these ervals asing just fifths. This deviation is known as em Pythagorean comma. abcliste

Solution:
abcliste abc Since a just fifth is given by the frequency ratio : any number of fifths corresponds to a ratio of an odd and an even eger. This can never be simplified to be an even eger. A number of octaves always corresponds to an even eger. abc fifths correspond to a ratio I_ leftfracright^Nf and octaves to a ratio I_ leftright^No The ratio is thus I_:I_ frac^Nf^Nftimes ^No frac^Nf^NfoP resultratio With equally tempered fifths seven half-tones there is no such problem: left^/right^ ^ abcliste
Contained in these collections:
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Attributes & Decorations
Branches
Acoustics
Tags
equal temperament, interval, just scale, scale
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Difficulty
(1, default)
Points
0 (default)
Language
ENG (English)
Type
Calculative / Quantity
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Decoration
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