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Pythagoras was wrong: scientists have discovered that there are no universal musical harmonies

Their research, published in Nature Communications, shows that in normal listening contexts, we do not necessarily prefer chords that are in these exact mathematical relationships but prefer small deviations from them

A band of bouzouki players from the Balkans. Illustration:
A band of bouzouki players from the Balkans. Illustration:

A new study challenges traditional Western music theory by showing that the appreciation of harmony is not necessarily subject to mathematical relationships, as previously thought, but can be enhanced by slight imperfections and the use of non-Western instruments. The research reveals a wider spectrum of musical consonance (harmony) and dissonance (disharmony). This discovery, made through large-scale behavioral experiments, suggests that there is much to be learned from the musical instruments and theories of other cultures, and opens up new possibilities for creativity and musical enjoyment.

According to the ancient Greek philosopher Pythagoras, 'consonance' - a combination of sounds pleasing to the ear - is created by special relationships between simple numbers like 3 and 4. Recently, researchers have tried to find psychological explanations, but these whole number ratios are still attributed to the beauty of a chord, and deviation from them is considered to create 'dissonant' music, sounds that are unpleasant to hear.

Research from the University of Cambridge, Princeton and the Max Planck Institute for Empirical Aesthetics has revealed two main ways in which Pythagoras was wrong.

Their research, published in Nature Communications, shows that in normal listening contexts, we do not necessarily prefer chords that are in these exact mathematical relationships.

“We prefer small amounts of deviation. We like a bit of imperfection because it gives life to sounds, and it appeals to us," said co-author Dr Peter Harrison, from the Faculty of Music at the University of Cambridge and director of the Center for Musicology there.

The researchers also found that the role of these mathematical relationships disappears when considering certain musical instruments that are less familiar to Western musicians, audiences and researchers. These instruments tend to be bells, gongs, types of xylophones and other percussion instruments. In particular, they studied the 'boneng', a Yawanese gamelan musical instrument made from a collection of small gongs.

"When we use tools like the Bonang, the special Pythagorean numbers disappear and we encounter entirely new patterns of consonance and dissonance," said Dr. Harrison.

"The shape of some percussion instruments means that when you strike them, and they resonate, their frequency components do not respect traditional mathematical relationships. When that happens we find that interesting things happen.”

"Western research has focused so much on familiar instruments of an orchestra, but other musical cultures use instruments that, because of their shape and physics, are considered what we call 'non-harmonic'.

The researchers created an online study in which over 4,000 people from the US and South Korea participated in 23 behavioral experiments. The participants were played chords and were asked to give each a numerical pleasantness rating or to tune and adjust certain notes in the chord to make it more pleasant. The experiments generated over 235,000 opinions.

The experiments explored musical chords from different perspectives. Some focused on certain musical intervals and asked participants to judge whether they preferred them perfectly tuned, slightly sharp, or slightly weak. The researchers were surprised to find a significant preference for slight imperfection, or 'aharmonicity'. Additional experiments investigated the perception of harmony with western and non-western musical instruments, including the bonang.

The researchers found that the bonang's consonances fit nicely with the particular musical scale found in the Indonesian culture from which it comes. These consonances are not reproducible on a Western piano, for example, because they would fall between the tonal intervals of the traditional scale.

"Our findings challenge the traditional idea that harmony can only be one way, that the chords must reflect these mathematical relationships. "We show that there are many more types of harmony out there, and that there are good reasons why other cultures have developed them," said Dr. Harrison.

Importantly, the study suggests that its participants - who are not professional musicians and do not know Javanese music - were able to instinctively appreciate the new consonances of the bonang sounds.

"Making music is all about exploring the creative possibilities of a given set of features, for example, finding what kinds of tunes you can play on a flute, or what kinds of sounds you can make with your mouth," Harrison said.

"Our findings suggest that if different tools are used, it is possible to develop a completely new harmonic language that people appreciate instinctively, they don't need to learn it to appreciate. Much of the experimental music in the last 100 years of Western classical music has been difficult for listeners because it involves very abstract structures that are difficult to enjoy. Conversely, psychological findings like ours can help encourage new music that listeners instinctively enjoy."

Exciting opportunities for musicians and producers

Dr. Harrison hopes the research will encourage musicians to try unfamiliar instruments and see if they offer new harmonies and open up new creative possibilities.

"A lot of pop music today tries to combine Western harmony with local melodies from the Middle East, India and other parts of the world. It can be more or less successful, but one of the problems is that the tunes can sound dissonant if played with western instruments.

"Musicians and producers may be more successful in combination if they take our findings into account and consider changing the 'timbre', the quality of the sound, by using specially selected real or synthetic instruments. So they can really get the best of both worlds: harmony and local scale systems.”

Harrison and his collaborators are studying different types of musical instruments and are planning follow-up studies to examine a wider range of cultures. In particular, they want to gain insights from musicians who use 'non-harmonic' instruments to understand whether they have internalized different concepts of harmony compared to the Western participants in this study.

for the scientific article

More of the topic in Hayadan:

12 תגובות

  1. Another response to Riki - the comparative calibration (calibration is my translation of temperament) was not created for the purpose of passing scales - in the 16th century they did not think in terms of scales and in terms of a phrase in one or another tonic, or a phrase that goes from scale to scale. I recommend Elam Rotem's excellent early music sources channel on YouTube for more. The equal is a very convenient method for stringed instruments with frets. Amazingly, it was Prescovaldi, whose organ music sounds exceptional with meantone temperament a quarter note (not to be missed), actually recommended equal.

    All the best to everyone.

  2. This is not new at all, Bakshi Bekesy received a Nobel Prize in 1961 for discoveries in the functioning of the basilar membrane and then they did new studies, in their theoretical basis there is a different distinction between consonance and dissonance, but instead it is about a dimension of tonal fusion and a dimension of tonal coherence (which is closer to consonance and dissonance) ), and these (and others such as the distinctness of the pitch) depend on a variety of parameters, with specific coherence depending on the distance of the frequency from the two ends of the vertebra or the specific component (each such component is sensitive to a frequency range) within the basilar membrane that converts the mechanical energy into nerve pulses.

  3. The author of the article is not accurate. The equator was born in the 16th century many years after the death of Pythagoras. The comparative direction was created for technical reasons only for an easy transition between scales and transpositions. Pythagoras did come up with the idea, but in his time it was not realized at all. Even today beyond the direction of the equator there are several hundred types of direction.

  4. What a trick this Pythagoras is.
    And we learned about the trial of this criminal for years.
    Why did he get so many years in court?
    The Pythagorean Theorem is a well-known case

  5. I didn't check who wrote it, and I don't want to offend, but sorry, you made a salad, and the main claim is not true at all.
    The rational relationship between frequencies is natural to the ear, and is the basis of harmonies and scales in all cultures. The rational relationship is expressed in every acoustic instrument, in overtones.
    The attraction to dissonances is another matter. Dissonances are intervals that are further along the harmonic series. They create an expectation of a solution, and the expectation, even if it is not solved, causes a certain sharpness that makes the musical complex interesting and exciting.
    If you are looking for an example of irrational relationships in music, you can point to the compared direction. In acoustic instruments the compared direction is approximate, and not completely compared, and therefore maintains approximately rational relations. In the measure and in general of the artificial use of sounds, the more you try to achieve a comparable direction, the less natural it is.
    In other cultures, as much as acoustic instruments are played, nature takes over and maintains harmonious relationships. also in quarter tones.
    It's on one leg.
    Trust Pythagoras and listen to musicians and not computers.

  6. It is not clear how the research "challenges" anything in the Western view. Small deviations, meaning an imperfect direction as it was before the classical period. Musicians like Gardiner try to reproduce this in their performances. The significant discoveries here seem to be: 1 There is a universal preference for an unmatched direction. 2. Overtones created in such a direction can create slightly different tonal systems in non-western cultures. But this is a hypothesis, lacking information.

  7. Pythagoras was not wrong, but the mathematicians who came after him, were wrong and have been misleading generations of students.
    Here is an example of a terrible mistake of the mathematicians, who thought that all circles have a single pi number, and it is a little bigger than 3.14

    Recommendation for receiving the Nobel Prize, according to 2 criteria accepted for the prize.
    1: Extraordinary research,
    2: Initial invention of equipment or technology.

    The extraordinary research disproves an ancient mathematical truth of thousands of years, which said:

    For any size of a circle - (from a diameter of zero mm, to a diameter of infinite mm)
    There is a single pi number and it is slightly larger than 3.14

    It should be remembered that pi number of a circle is obtained,
    From the millimeter length of the circumference of the circle (partial) the millimeter length of the diameter of the circle.

    If the idea of ​​a single pi number is correct, then the following equation follows from it.
    The number expressing the ratio of the lengths of the diameters of two selected circles,
    must be (exactly equal) to the number expressing the ratio of the lengths of the circumferences of the circles.

    But if the idea of ​​the only number is not true, then the number expressing the ratio of the lengths of the diameters of two chosen circles, (is not equal) to the number expressing the ratio of the lengths of their circumferences.
    Obviously, this inequality will be tiny, but it will exist without a shadow of a doubt.

    Since there is no mathematical calculation capable of deciding between the two options, I came up with the idea of ​​inventing a mechanical device - which would decide between the two options by way of a practical experiment, which includes a very, very precise measurement.
    20 years passed before I invented this device, and it was named the scope.
    The measurement description of the scope appears in the video.

    The scope works with actual circuits that appear in very precise steel cylinders.
    The diameter of the rolls is 120 mm and 2 mm.
    The ratio of the lengths of the diameters is 60, and the circumference revealed that the ratio of the lengths of the circumferences is 59.958
    This result means that the scope is a very accurate measuring device.
    This result also means that the idea of ​​a single pi number suitable for all circles is incorrect.

    The scope experiment was published in 2017, and not a single mathematician was found who accepted it.
    Nor was there a single mathematician who was willing to repeat the scope experiment.
    The mathematicians were probably offended by the rejection of the idea of ​​a single pi number suitable for all circles, and they chose to mock and disparage the circumference experiment.

    This is how mathematics lost the possibility of imparting to science a new geometry of circles.
    In this geometry, each size of a circle has a private pi number, and an infinity of these private pi numbers are in a very narrow numerical range between 3.14 and 3.16. In this geometry, there is also a formula, which allows a transition from a millimeter diameter of a circle, to a private pi number of the circle.

    But mathematics preferred to ignore the circumference experiment and its dramatic result, and continued to teach students the lie of a single pi number that fits all circles.

    A Nobel Prize can dispel the lie of a single pi number, and it will also change the study of mathematics and geometry throughout the world. Anyone can recommend receiving a Nobel Prize.

    The scope experiment and its result deserve a Nobel Prize.
    A. Asbar

  8. The context for Pythagoras is silly. Obviously the music they played at the time and the musical taste were different from today. It is possible that what he said was true for his time.

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