Pythagoras was wrong: there are no universal musical harmonies, study finds
According to the Ancient Greek philosopher Pythagoras, ‘consonance’ – a pleasant-sounding combination of notes – is produced by special relationships between simple numbers such as 3 and 4. More recently, scholars have tried to find psychological explanations, but these ‘integer ratios’ are still credited with making a chord sound beautiful, and deviation from them is thought to make music ‘dissonant’, unpleasant sounding.
But researchers from the University of Cambridge, Princeton and the Max Planck Institute for Empirical Aesthetics, have now discovered two key ways in which Pythagoras was wrong.
Their study, published in Nature Communications, shows that in normal listening contexts, we 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 co-author, Dr Peter Harrison, from Cambridge’s Faculty of Music and Director of its Centre for Music and Science.
The researchers also found that the role played by these mathematical relationships disappears when you consider certain musical instruments that are less familiar to Western musicians, audiences and scholars. These instruments tend to be bells, gongs, types of xylophones and other kinds of pitched percussion instruments. In particular, they studied the ‘bonang’, an instrument from the Javanese gamelan built from a collection of small gongs.
“When we use instruments like the bonang, Pythagoras's special numbers go out the window and we encounter entirely new patterns of consonance and dissonance,” Dr Harrison said.
“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.”
“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’.
The researchers created an online laboratory in which over 4,000 people from the US and South Korea participated in 23 behavioural experiments. Participants were played 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 over 235,000 human judgments.
The experiments explored musical chords from different perspectives. Some zoomed in on particular musical intervals and asked participants to judge whether they preferred them perfectly tuned, slightly sharp or slightly flat. The researchers were surprised to find a significant preference for slight imperfection, or ‘inharmonicity’. Other experiments explored harmony perception with Western and non-Western musical instruments, including the bonang.
Instinctive appreciation of new kinds of harmonyThe researchers found that the bonang’s consonances mapped neatly onto the particular musical scale used in the Indonesian culture from which it comes. These consonances cannot be replicated on a Western piano, for instance, because they would fall between the cracks of the scale traditionally used.
“Our findings challenge the traditional idea that harmony can only be one way, that chords have to reflect these mathematical relationships. We show that there are many more kinds of harmony out there, and that there are good reasons why other cultures developed them,” Dr Harrison said.
Importantly, the study suggests that its participants – not trained musicians and unfamiliar with Javanese music – were able to appreciate the new consonances of the bonang’s tones instinctively.
“Music creation is all about exploring the creative possibilities of a given set of qualities, for example, finding out what kinds of melodies can you play on a flute, or what kinds of sounds can you make with your mouth,” Harrison said.
“Our findings suggest that if you use different instruments, you can unlock a whole new harmonic language that people intuitively appreciate, they don’t need to study it to appreciate it. A lot of experimental music in the last 100 years of Western classical music has been quite hard for listeners because it involves highly abstract structures that are hard to enjoy. In contrast, psychological findings like ours can help stimulate new music that listeners intuitively enjoy.”
Exciting opportunities for musicians and producersDr Harrison hopes that the research will encourage musicians to try out unfamiliar instruments and see if they offer new harmonies and open up new creative possibilities.
“Quite a lot of pop music now tries to marry Western harmony with local melodies from the Middle East, India, and other parts of the world. That can be more or less successful, but one problem is that notes can sound dissonant if you play them with Western instruments.
“Musicians and producers might be able to make that marriage work better if they took account of our findings and considered changing the ‘timbre’, the tone quality, by using specially chosen real or synthesised instruments. Then they really might get the best of both worlds: harmony and local scale systems.”
Harrison and his collaborators are exploring different kinds of instruments and follow-up studies to test a broader range of cultures. In particular, they would like to gain insights from musicians who use ‘inharmonic’ instruments to understand whether they have internalised different concepts of harmony to the Western participants in this study.
ReferenceR. Marjieh, P.M.C. Harrison, H. Lee, F. Deligiannaki, & N. Jacoby, ‘Timbral effects on consonance disentangle psychoacoustic mechanisms and suggest perceptual origins for musical scales’, Nature Communications (2024). DOI: 10.1038/s41467-024-45812-z
The tone and tuning of musical instruments has the power to manipulate our appreciation of harmony, new research shows. The findings challenge centuries of Western music theory and encourage greater experimentation with instruments from different cultures.
There are many more kinds of harmony out therePeter HarrisonAndrew Otto via Flikr under a CC licenseA man playing a bonang
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New report into Turkey-Syria earthquakes uncovers deficiencies in building structures and construction shortcuts were the main cause of casualties
A new, independent field investigation into the aftermath of the Turkey-Syria earthquakes has found that a drive for profit has pushed all players within the construction industry to take shortcuts, with building stock primarily made of Reinforced Concrete (RC) structures, being the main cause of the casualties.
Findings show that deficiencies were also recorded among even the newest building stock. This is despite established technical know-how, state-of-the-art building codes and rigorous building regulations.
The longitudinal study report published here today by the Institution of Structural Engineers for EEFIT, was co-led by Cambridge's Professor Emily So, Professor of Architectural Engineering and Director of the Cambridge University Centre for Risk in the Built Environment (CURBE). Some of the findings include:
- The drive for profit pushes players within the construction industry to take shortcuts. The auditing and quality control mechanisms embedded in the legal and bureaucratic processes should be strengthened to ensure code compliance. The legalisation of non-compliant buildings through amnesties cannot continue.
- Critically, despite established technical know-how, state-of-the-art building codes and rigorous building regulations, deficiencies in Reinforced Concrete (RC) structures were found even in the newest building stock. This demonstrates that seismic resilience is not only a technical problem in Turkey, but one that demands a multi-sectoral and interdisciplinary dialogue, scrutinising the regulatory system, bureaucracy, the legal and political backdrop within which the construction sector operates in Turkey.
- Building stock is primarily composed of Reinforced Concrete structures, which were therefore the main cause of the casualties. The team saw problems with such structures across their whole lifecycle from design to implementation and post-occupancy stages. The structures therefore did not withstand the seismic pressures.
- A review of building stock and infrastructure is critical to understand risk levels for future earthquakes. Lack of publicly available data is a big problem in Turkey, hindering not only a robust inquiry into damage and associated building characteristics, but also reliably establishing the risk profiles for future events.
- Debris management and demolishment practices have not fully recognised the potential of mid-/long-term environmental and public health implications. Field observations and contacts in the affected communities show that they are already affected by the poor air quality. The Compulsory Earthquake Insurance (CEI) is a system that was put in place in Turkey following the 1999 earthquakes to provide monetary reserves to fund the management of future disasters. The extent to which these funds have been used and how resources have been allocated remain unclear.'
Read the full report and findings here.
Professor So says: “The 2023 Türkiye and Syria earthquakes were truly tragic, hitting an already fragile population, including migrants. Our field work and remote analysis revealed many issues, including the issue of non-compliant buildings with little seismic resilience. Building code compliance needs to be strengthened.”
EEFIT - a joint venture between industry and universities - gathered a team of 30 global experts to assess the damage and develop suggestions to reduce future impacts and vulnerabilities. They studied the science, engineering and data related to the earthquakes including geotechnics, the structural and infrastructure impact, and the relief response and recovery.
The Earthquake Engineering Field Investigation Team (EEFIT), co-led by Professor Emily So, today publishes its findings and recommendations.
Our field work and remote analysis revealed many issues, including the issue of non-compliant buildings with little seismic resilience. Building code compliance needs to be strengthened.” Professor Emily SoEEFITA partially-collapsed building in the aftermath of the Turkey-Syria earthquakes in 2023.
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Professor Dame Carol Black DBE is awarded Dame Commander of the Order of the British Empire (GBE) for public service. Black was Principal of Newnham College from 2012-2019 and formerly a Deputy Vice-Chancellor of the University of Cambridge.
She said: “I am absolutely delighted to have been made a Dame Grand Cross of the Order of the British Empire. This award comes almost 20 years after I received a DBE for services to medicine and recognises the progress being made to tackle some of the most entrenched and interrelated problems in society – poverty, homelessness, unemployment, and drug dependency. My heartfelt thanks go to everyone who has helped and supported me, and to those individuals doing great work on the frontline to change culture and practice.”
Composer Judith Weir CBE, Honorary Fellow and alumna of King’s College, Cambridge, is also awarded Dame Commander of the Order of the British Empire (GBE), for services to music. Weir is Master of the King’s Music, having been appointed by Queen Elizabeth II in 2014, and has twice written a specially-commissioned carol for the college’s A Festival of Nine Lessons and Carols.
GBE is the highest rank in the Order, and rarely awarded, to recognise the most exceptional and sustained service to the UK. Since its creation in 1917, fewer than 80 women have been awarded a GBE.
Organist, conductor and broadcaster Anna Lapwood, Janeway Director of Music at Pembroke College, Cambridge, is awarded Member of the Order of the British Empire (MBE) for services to music. Lapwood reaches a huge audience through her concerts and via social media with over 1 million followers across all platforms. Her passion for the organ is matched by her mission to support girls and women in music.
She said: “When you work as a musician, so much of what you do isn't quantifiable or finite - your work on a certain piece is never 'finished', and your playing is always changing and developing. Receiving this award feels like something concrete - a deeply significant moment in my musical journey.”
Gerard Grech, former CEO and Founder of Tech Nation, is awarded Commander of the Order of the British Empire (CBE) for services to the technology sector. He joined Cambridge Enterprise this year to lead a new flagship ‘Founders Initiative’ that will support University founders to make an even greater impact on the world in the technology and software sectors.
Grech said: “I’m honoured to have been recognised for my contribution to the growing success of the UK’s tech and startup sector which is increasingly creating globally important tech and science-backed companies, from my time at Tech Nation. This honour is also recognition of the founders, ecosystem experts, investors, policy makers, and my colleagues who generously shared their knowledge and insights to support the UK’s most ambitious tech entrepreneurs. I would like to extend my sincere thanks to them all for all their hard work.”
Dr Sabesan Sithamparanathan, Enterprise Fellow at Girton College, Cambridge, and former student in the University’s Department of Engineering, is awarded Officer of the Order of the British Empire (OBE) for services to innovation technology. As Founder & President of PervasID he pioneered the world’s most accurate battery-free, real-time location tracking technology which is now in use by several NHS trusts, the largest aircraft manufacturers, airlines and blue-chip retailers.
He said: “I am absolutely delighted; this is a great honour and testament to the hard work and innovation of the entire team at PervasID. Our products offer a national and international benefit and we will continue to pioneer technology that has a wider value to society as a whole.”
Professor Ann Prentice OBE, Honorary Senior Visiting Fellow at the University’s MRC Epidemiology Unit, is awarded Commander of the Order of the British Empire (CBE) for services to British and Global Public Health Nutrition. A former director of the MRC Elsie Widdowson Laboratory (previously MRC Human Nutrition Research) at Cambridge, and programme leader of the MRC Nutrition and Bone Health Research Group at Cambridge, she was also head of the calcium, vitamin D and bone health research team at MRC Unit The Gambia. Her research is focused on life-course nutritional requirements for population health, with an emphasis on calcium and vitamin D, and encompasses the nutritional problems of both affluent and resource-limited societies.
She said: “I am delighted to receive this honour on behalf of all the people, in this country and worldwide, who have worked with me to improve our understanding of the links between nutrition and health.”
Dr Gillian Tett, Provost at Kings College, Cambridge, is awarded Officer of the Order of the British Empire (OBE) for services to Economic Journalism. Tett, a former student at the University, is currently Chairman of the US Editorial Board and America Editor-at-Large of the Financial Times. She became the 45th Provost at King’s College in October 2023, and is renowned for her warnings ahead of the financial crisis of 2008.
Tett said: “I am deeply honoured to receive an OBE - and hope this helps to champion the importance of British intellectual capital, both in journalism and higher education. Thank you to everyone who has helped me in my career!”
Joan Winterkorn is awarded Member of the Order of the British Empire (MBE) for services to heritage and culture. She is an expert on archives and literary and historical manuscripts, and was formerly in the antiquarian and rare book trade. In Cambridge she played a vital role in enabling the Churchill Archive Centre to acquire the papers of Sir Winston Churchill and Lady Thatcher and the University Library to gain those of Siegfried Sassoon and Dame Margaret Drabble. In 2019 she received the honorary degree of Master of Arts from the University.
Academics and staff associated with the University of Cambridge feature in the 2024 list, which recognises the achievements and service of people across the UK, from all walks of life.
Senate house
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