PYTHAGORAS AND THE RELATIVE FREQUENCIES IN THE MUSIC SCALE

Music was numbered by the ancients among the arts that are called liberal, that is, worthy of a free man, and among the Greeks its masters and discoverers, like those of almost all the other sciences, were always in great esteem. And by the best legislators it was decreed that it must be taught, not only as a lifelong delight but as useful to virtue, to those who were born to acquire perfection and human happiness, which is the object of the state.

To make a sound you need something that vibrates. and if you want to make musical notes, you need the vibration to have an almost constant frequency. The controlled vibration is produced by a standing wave of a stretched string. Due to it's elastic properties, the string oscillates back and forth between two gentle curves. If the length is cut in half, say by touching the string gently at its midpoint, each half will ring at twice the original frequency, or an interval of an octave above the fundamental. By touching the string at the 1/3 point, each of the three sections will produce 3x the original frequency , or an interval of one octave + a major fifth above the fundamental, etc.

This property of a stretched string was first identified by the Greeks and used by Pythagoras to work out the mathematical relationship between all tones and semi-tones. The octave was found to be a 2:1 ratio and what we today call a fifth to be a 3:2 ratio. Pythagoras concluded that all the notes could be produced by these two ratios as (3/2)*(3/2)*(1/2) gave 9/8 which is a second and so on. The problem was that after applying these ratios repeatedly he was able to move through the whole scale and end up back where he started... except that it missed by a bit, called the Pythagorean comma.
After twelve movements by a fifth (and adjusting down an octave as required) he got back to the same note but it had a frequency of 3^12 / 2^19 which is 1.36% higher in frequency than it should be. So although Pythagoras did a wonderful job he did get it slightly wrong.

The correct solution was worked out by Vicenzo Galilei(1606-1649). He had two sons. One, Michaelangelo who was a lutenist, and the other was the famous mathematician Galileo. Vincenzo Galilei was associated with Mei and the Camerata in Florence, engaged in the resurrection of ancient Greek music. In the course of time the Greeks lost the art of music and the other sciences as well, along with their domination. The Romans had a knowledge of music, obtaining it from the Greeks, but they practiced chiefly that part appropriate to the theatres where tragedy and comedy were performed, without much prizing the part which is concerned with speculation; and being continually engaged in wars, they paid little attention even to the former part and thus easily forgot it. After Italy had suffered for a long period from great barbarian invasions, some musicians, like Vincenzo, worked seriously on the resurrection of the ancient Greek music and their work should not be underestimated because without Vincenzo there may have been no Opera! He published a number of books , and engaged in arguments over the emerging iniquity of what would become "equal temperament". Concerning the latter, Vicenzo Galilei concluded that the best frequencies for do, re, mi, fa, sol, la, ti, do were resp.the proportions 1, 9/8, 5/4, 4/3, 3/2, 5/3, 15/8, 2. The proportions proposed by V. Galilei are called the Just Intonation music scale and are the most pleasing proportions for note frequencies for any one key. The differences from Pythagoras are small, so that mi is 5/4 (=1.250) rather than 81/64 (=1.266).

However when music contains modulations, that is, changes of key, then some of the notes need to change frequency. Human voices and string quartets do this adjustment automatically because they listen for the harmony. Guitars and pianos just cannot do it hence it was necessary to make a compromise. From the 16th century, Zarlino(1558) and also the Flemish mathematician, Simon Stevin(1608) in his work "Van de Spiegeling der Singconst en de onvolmaectheyt dieder int stellen der orghels ende clavesimbels ghebuert", proposed the division of the octave in 12 semi-tones of exactly equal ratios.

bach.JPG (2548 bytes)

But it was J. S. Bach(1685-1753) who popularised the system called "equitempered" in his 2 times 24 fugas "für das wohltemperierte Klavier". This equitempered system, which is used almost exclusively today, is a compromise between all keys and uses a common ratio between every semitone of 2^(1/12) or the 12th root of 2 (1.0594631)

 


This gives relative frequencies of :

                      do      re       mi       fa       sol       la       ti       do
equitempered         1.000   1.122   1.260   1.335     1.498    1.682    1.888    2.000
just intonation      1.000   1.125   1.250   1.333     1.500    1.667    1.875    2.000
Pythagoras           1.000   1.125   1.266   1.333     1.500    1.688    1.898    2.000   
OR
equitempered          1    2^(1/6)  2^(1/3)  2^(5/12)  2^(7/12)  2^(3/4)  2^(11/12)   2
just intonation       1       9/8      5/4      4/3      3/2       5/3     15/8       2
Pythagoras            1       9/8     81/64     4/3      3/2      27/16   243/128     2 

Pythagoras and his followers and later also Kepler were to consider that the musical relations or harmonies had wider application in the universe. To them everything in the universe is vibrating in tune with the larger things that contain it and we are literally living inside this giant musical instrument which is playing notes, chords and scales that only Gods can hear. . They believed that the universe is completely organised on a system of mathematical harmony and that it shows up in every branch of scientific study. In fact cycles can been found in every aspect affecting life on earth like : wars, economic fluctuations, births and deaths, climate, geophysics, animal populations, social variables, stock and commodity prices...

SOURCES

http://www.geom.umn.edu/~demo5337/Group3/hist.html
http://www.west.net/~beck/EC18-Greekto500.html
http://magna.com.au/~prfbrown/pythagor.html
http://www.showgate.com/medea/bulfinch/bull34.html
http://www-groups.dcs.st-and.ac.uk
http://www.astro.virginia.edu/~eww6n/math/Pythagoras.html
http://www-history.mcs.st-and.ac.uk/history/Mathematicians/Pythagoras.html
http://www.speakeasy.org/berry/pyth/pythtext.html
http://www.magna.com.au/~prfbrown/pythagor.html
http://www.kcbbs.gen.nz/users/rtomes/alex-ha.htm

Atlas van de algemene en Belgische geschiedenis, F. Hayt, J. Grommen, R. Janssen, A. Manet, Ed. Van In - Lier
Theorie van de Muziek, H.F. Steylaerts, Van In - Lier
Muziektheorie, Dr.J. Daniscas, Wolters Groningen 1996, 11de druk

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