Beyond his famous theorem, the Greek philosopher and mathematician Pythagoras was a scholar who traveled throughout the known world to try to unravel the secrets of the universe. One of his particular interests was sound. He was the one who devised the monochord after hearing the clacking of hammers in a blacksmith shop.
Obsessed with mathematically explaining intervals to obtain pleasant resonances, it was not long until he identified the eight basic notes that make up the musical scale. He had discovered the origin of harmony and then dedicated himself body and soul to finding his relationship with numbers.
It was Pythagoras who proposed the idea of “consonance,” the creation of combinations of notes that sound good together, the basis of much Western music theory. When there is tuning, nothing clashes and the sound is perceived as satisfactory. However, 2,500 years later this whole house of cards seems about to collapse.
Researchers from the universities of Cambridge and Princeton and the Max Planck Institute have discovered two key ways in which the philosopher got it wrong. As explained in an article published in the journal Nature Communications, “in normal listening contexts, we do not actually prefer the chords to be perfectly” in the mathematical proportions indicated by the Greek genius.
The pitch and tuning of musical instruments have the power to manipulate our appreciation of harmony, experts explain, challenging centuries of thesis. “We prefer slight deviations. We like a bit of imperfection because it brings life to the sounds and we find that attractive,” explains Dr. Peter Harrison, co-author of the study.
According to Pythagoras, "consonance" occurs through special relationships between simple numbers such as three and four. But the authors of this work have discovered that the role played by these mathematical links disappears when considering certain musical instruments that are less familiar to Western musicians, audiences, and scholars.
These instruments are typically bells, gongs, types of xylophones, and other types of tuned percussion tools. In particular, the bonang was studied, a basic element of Javanese gamelan (a traditional Indonesian musical group) built from a collection of metal boxes.
"When we use instruments like the bonang, the Pythagorean special numbers are lost and we find completely new patterns of consonance and dissonance," says Dr Harrison, director of the Cambridge Center for Music and Science.
“The shape of some percussion instruments means that when you hit them and they resonate, their frequency components do not respect those traditional mathematical relationships. That's when we discover that interesting things happen,” adds the expert.
The researchers created an online laboratory in which more than 4,000 people underwent 23 experiments. Participants were given chords and were invited to give each one a rating from best to worst. They also used a control to adjust particular notes to make it sound more pleasant.
“Western works have focused a lot on familiar orchestral instruments, but other musical cultures use instruments that, due to their shape and physics, are what we would call ‘inharmonic,’” Harrison adds. Hence, the study explored chords from different perspectives.
Some focused on particular musical intervals and asked participants to judge whether they preferred them perfectly in tune, slightly sharp, or slightly flat. The experts were surprised to find a significant preference for a slight imperfection or "inharmonicity." The perception of harmony with Western and non-Western musical instruments, including the bonang, was also explored.
Specialists discovered that the consonances of the bonang clearly corresponded to the particular musical scale used in the Indonesian culture from which it comes. These harmonies cannot be reproduced on a Western piano because they would fall between the cracks of the traditional scale.
“Our findings challenge the idea that harmony can only be unidirectional, that chords must reflect these mathematical relationships. We show that there are many more types of consonance and that there are good reasons why other cultures developed them,” Peter Harrison points out in a statement.
The participants in the study were not trained musicians nor were they familiar with Javanese music, and yet they were able to instinctively appreciate the tones of the bonang. "Music-making is about exploring the creative possibilities of a given set of qualities, discovering what kinds of melodies you can play on a flute or what kinds of sounds you can make with your mouth," Harrison said.
“Our findings suggest that if you use different instruments, you can unlock a completely new harmonic language that people appreciate intuitively, they don't need to study it to appreciate it. Much of the experimental classical music of the last 100 years has proven to be quite difficult for listeners because it involves very abstract structures that are difficult to enjoy. On the contrary, findings like ours can help stimulate new pieces that listeners intuitively enjoy,” he adds.
Dr. Harrison hopes his work will encourage musicians to try unfamiliar tools and see if they offer new harmonies and creative possibilities. “A lot of pop music tries to marry Western harmony with local melodies from the Middle East, India and other parts of the world. This can be more or less successful, because the notes can sound dissonant if you play them with Western instruments,” he notes.
“Musicians and producers could make that marriage work better if they considered changing the 'timbre,' the quality of tone, by using specially chosen real or synthesized instruments. Then they could really get the best of both worlds: harmony and local scale systems,” he concludes.
