This time of year is full of music, with carol singers out in force and the ubiquitous Christmas “hits” playing on loop in every store, but a recent discovery has brought music to mind in a different way…
The music of the spheres – Pythagorean theory of the universe
The ancient Greeks described music in terms of a seven-note scale, to which have been assigned the letters A through G, where the main intervals could be expressed as simple mathematical ratios between the first four integers, with: octave (eighth)=2:1, fifth=3:2, fourth=4:3 (the counting system was inclusive, of the original numbers, which is how to get eight notes in a seven-note system – the fourth and fifth reduce to three and four as numbers, which then do add up to seven). These ratios harmonise musically as well as mathematically ie they are pleasing to the mind and the ear.
However, when extending this tuning however, a problem arises since no stack of 3:2 intervals (perfect fifths) will fit exactly into any stack of 2:1 intervals (octaves). Although there is a suggestion that the Pythagoreans were unaware of this, the philosopher Arthur Schopenhauer, described the problem:
“…thus, a perfectly pure harmonious system of tones is impossible not only physically, but even arithmetically. The numbers themselves, by which the tones can be expressed, have insoluable irrationalities,”
– The World as Will and Representation, Volume I, §52, E.F.J. Payne translation, Dover Publications, 1966, p.266
By considering a scale to consist of 12 equal intervals, a solution to stacking of the fifths can be found from the 12th root…if the seventh octave is 128:1 or 27, then the fifth is 27/12 or 1.4983:1. This is close to, but not the same as 3:2 (1.5). This “equal semitones” approach was developed simultaneously in Europe and China in the 16th century.
“Middle C” is the reference point for all the keys on a piano, and for the musical scale in general. “A above Middle C” is the benchmark note for which there is a definition in terms of physics, with a sound frequency now set at exactly 440 Hz.
This tuning approach suits other intervals better…while the Pythagorean approach works well tonally for fifths – a 3:2 ratio having a pleasing sound, other intervals such as thirds – 81:64 for major thirds and 32:27 for minor thirds sound much less pleasing, and as a result, Pythagorean tuning is rarely found after around 1510.
Pythagoras was the first to identify that the pitch of a musical note is in inverse proportion to the length of the string that produces it.
According to Dr George N. Gibson of the University of Connecticut Pythagoras “got lucky”: he did not actually study the frequencies that made up pleasing intervals and the musical scale – he just made observations about the lengths of the strings that made intervals and scales, and by coincidence, the frequency of a string has a simple relationship to its length. So, all the conclusions he reached about ratios of lengths of strings for different intervals, also apply to the ratios of the frequencies in the intervals. Unfortunately, this is not true for the tension in a string, or for the length of vibrating bars such as those on a xylophone.
This woodcut was made in the Middle Ages to explain illustrate Pythagorean ratios and how they applied to musical instruments:
The lower right panel shows flutes whose lengths correspond to the Pythagorean ratios. This works because, like a string, the frequency of an air column is simply related to its length. However, the panel on the lower left is problematic since the weights on the ends of the strings change their tension – as the frequency of the strings is not simply related to the tension, weighted strings will not sound in accordance with Pythagorean intervals. The examples in the upper panels are even more complicated, but bells, water glasses, and anvils also would not produce the correct intervals.
Unfortunately, the reverence for ancient Greek philosophers meant their theories were widely applied without any testing to see if the theories applied to any given situation….as Gibson puts it in his notes, “had anyone bothered to build any of the instruments (except for the flute) in the ratios prescribed in the woodcut, they would have found that the intervals were all wrong.”
It was actually Vincenzo Galilei, Galileo Galilie’s father, who first noticed that there was something wrong with the woodcut, which is interesting considering how Pythagoras’ theories about music and astronomy were to be challenged by his son.
Greek astronomers had identified certain “fixed” stars whose relative position in the sky did not change through the seasons. They also noted that there were “wanderers” or planets which moved relative to the background stars. To explain these observations, the astronomers theorised that the fixed stars were attached to a large black sphere that defined the edge of the universe, while the planets were attached to moving spheres, with each planet on its own sphere. In order for the fixed stars to be visible through these spheres, the Greeks believed they were made of crystal.
The known “planets” at the time were the moon, Mercury, Venus, the sun, Mars, Jupiter, and Saturn…seven, in total, so there were seven crystal spheres. This chimed with the seven notes in the musical scale discovered by Pythagoras, so the Greeks combined the two ideas into the concept of the “Music of the Spheres”.
Pythagoras proposed that the Sun, Moon and planets all emit their own unique hum based on their orbital revolution, and that the quality of life on Earth reflects the tenor of celestial sounds which were imperceptible to the human ear.
This idea dominated astronomy for the next 15 centuries, and many great physicists were convinced that universe had an order that was musical in nature. The astronomer Johannes Kepler became obsessed with trying to fit the orbits of the planets to a musical scale, and in the process discovered his eponymous laws of planetary motion.
Recent discoveries herald the Earth’s own music
While the music of the spheres may have been generally discounted, it turns out the Earth actually does have its own hum. This was confirmed in 1998 when a research team found the Earth constantly generates a low-frequency vibrational signal in the absence of earthquakes.
Since then, seismologists have proposed a number of different theories to explain the existence of this vibration, from atmospheric disturbances to the movement of ocean waves over the sea floor. A 2015 paper found that two marine factors are primarily responsible for the imperceptible planetary drone: the ebb and flow of ocean waves reaching the seafloor, and the vibrations caused by the collision of ocean waves add up to produce the hum. The result is a strange ultralow frequency that resonates almost identically all over the planet.
The vibration has been measured using seismometers on land, but now a new study has measured the vibrations beneath the sea, and determined that Earth’s natural vibration peaks at several frequencies between 2.9 and 4.5 millihertz – about 10,000 times smaller than the lower hearing threshold of the human ear, which is 20 hertz.
Closer to home and more familiar is the hum of electricity
The electricity hum (also called the “mains hum”) is more familiar, but what most of us don’t think about is that this hum has its own musical pitch associated with the frequency of the system. In a 50 Hz system such as the UK, this pitch is close to a G on the musical scale, whereas in the 60 Hz US system it is almost exactly halfway between A♯ and B.
This mains hum can be heard in audio equipment, and also from power grid equipment such as transformers. The intensity of the hum is a function of the applied voltage.
Hums can also appear at the frequency harmonics, though with a much lower intensity. In the case of a 50 Hz current, there could be humming at 100 Hz, 200 Hz, and so on, up until very high frequencies creating an entire spectrum of electrical hum with sound both at the original pitch, and higher octaves.
The hum is generally considered to be a nuisance, particularly in electrical musical instruments, however it does have some uses…forensic analysts use a technique called Electrical Network Frequency which allows them to validate audio recordings by comparing how the frequency changes in the background mains hum to a pre-existing database. This allows them to identify when a recording was created and help detect any edits in the recording.
Science has come a long way since the ancient Greeks theorised about a fundamental connection between music and astronomy, and yet we find that our planet does in fact have a natural hum, and we are also surrounded by a man-made humming from our electricity system. It’s not quite as romantic as the harmony of the spheres, but it’s curious nonetheless.
I would like to thank everyone who has read and commented on my blog over the past year, and wish you all a very merry (and musical!) Christmas, and a happy and prosperous New Year!