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New study questions Pythagorean’s mathematical theorem of musical ‘consonance’

These findings about the impact of instrument tuning on musical harmony appreciation challenge centuries-old Western music theory and throw the field open to more experimentation with instruments from different cultures.

By Newsd
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Pythagorean's mathematical theorem

Challenging one of the theorems of Greek philosopher Pythagoras, a new research has found that musical ‘consonance’, or the pleasant-sounding combination of notes, may not have to be in mathematical ratios to be appreciated by listeners as harmony.

These findings about the impact of instrument tuning on musical harmony appreciation challenge centuries-old Western music theory and throw the field open to more experimentation with instruments from different cultures.

According to Western music’s notion of harmony, to a listener, certain chords will sound particularly pleasant or consonant, while others will sound relatively unpleasant or dissonant, the team of researchers, including those at Max Planck Institute for Empirical Aesthetics, Germany, said in their study published in the journal Nature Communications.

Pythagoras’ concept of ‘consonance’ regards the pleasant-sounding combination of notes to be produced by special relationships between simple numbers such as 3 and 4, they said.

However, the researchers said they have showed that in contexts of normal listening, people do not actually prefer chords to be perfectly in these mathematical ratios.

”We prefer slight amounts of deviation. We like a little imperfection because this gives life to the sounds, and that is attractive to us,” said the study’s co-author Peter Harrison from Cambridge University’s Faculty of Music and Director of its Centre for Music and Science.

The researchers found that when certain musical instruments less familiar to Western musicians, audiences and scholars were thrown in the mix, the role played by these mathematical ratios and relationships disappears.

Familiar Western orchestral instruments included bells, gongs, xylophones and other kinds of pitched percussion instruments, they said.

”When we use instruments like the (Javanese) bonang, Pythagoras’s special numbers go out of the window and we encounter entirely new patterns of consonance and dissonance,” Harrison said. Bonang is built from a collection of small gongs.

”The shape of some percussion instruments means that when you hit them, and they resonate, their frequency components don’t respect those traditional mathematical relationships. That’s when we find interesting things happening,” explained Harrison.

For their study, the researchers created an online laboratory in which over 4,000 people from the US and South Korea participated in 23 behavioural experiments.

The participants were made to listen different chords and invited to give each a numeric pleasantness rating or to use a slider to adjust particular notes in a chord to make it sound more pleasant. The experiments produced a total of over 235,000 human judgments, the researchers said.

”Western research has focused so much on familiar orchestral instruments, but other musical cultures use instruments that because of their shape and physics are what we would call ‘inharmonic’,” said Harrison.

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