Pythagoras and Music


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Presentation for the Festival of Ideas
October 25, 2008

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  • Pythagoras and Music

    1. 1. <ul><li>Festival of Ideas </li></ul><ul><li>25th October 2008 </li></ul>Pythagoras a nd Music Alison Ainley Paul Jackson
    2. 2. Pythagoras <ul><li>Born Samos ?580-578 BC </li></ul><ul><li>Founder of mysterious group in Croton, Southern Italy </li></ul><ul><li>Died Metapontum ?500-490 BC. </li></ul>
    3. 3. Pythagoras <ul><li>Famous as mathematician, but not until 500 years after his death </li></ul><ul><li>In the ancient world he was more famous for his views about the immortal soul and his strict rules about how to live </li></ul>
    4. 4. Life in Pythagoras’ Group <ul><li>Inner and outer circle of very loyal followers – ‘the Pythagoreans’: </li></ul><ul><li>Mathematikoi – (inner group) – lived in one house, studied mathematics, no possessions, vegetarian diet, included men and women </li></ul><ul><li>Akousmatikoi – (outer group) – lived in the town, could have possessions and eat meat, but had a lot of complex rules to keep. Could only listen to Pythagoras from behind a screen </li></ul>
    5. 5. Mathematikoi <ul><li>Lived a simple life: study, religion and rituals, shared meals and exercise </li></ul><ul><li>Vow of silence – ‘words can be dangerous’ </li></ul><ul><li>Belief in peace and harmony </li></ul><ul><li>Importance of music – hymn singing in rituals, music of lyre to cure illness and to help sleep (Orphic tradition – Orpheus) </li></ul>
    6. 6. Mathematics <ul><li>Pythagoras’ Theorum: ‘the square on the hypoteneuse is equal to the sum of the squares on the other two sides’ – did he even work on this? </li></ul><ul><li>Idea that maths is the key to the universe – maths is everywhere! </li></ul>
    7. 7. More Mathematics <ul><li>Mathematics means order, pattern and stability </li></ul><ul><li>It is clear and consistent and has rules </li></ul><ul><li>It is abstract and logical </li></ul><ul><li>Pythagoras’ Tree </li></ul>
    8. 8. Mathematics and Music <ul><li>Pythagoras heard blacksmiths striking different sized anvils and producing different notes – in harmony </li></ul><ul><li>He realised there was a mathematical explanation – ratios! </li></ul>
    9. 9. Geometry, Astronomy and Harmonics <ul><li>Geometry </li></ul><ul><li>Geometry studies abstract patterns, principles and rules also found in nature – sea shells, ferns etc. </li></ul>
    10. 10. Astronomy <ul><li>Astronomy studies movements of stars and planets - perfect curved lines and spheres </li></ul><ul><li>The ancient Greeks could measure these patterns mathematically </li></ul>
    11. 11. Harmonics <ul><li>Harmonics studies the relation of mathematical ratios in music </li></ul>
    12. 12. Music and Mathematics <ul><li>Musical concords – octave, 4 th and 5 th – correspond to ratios: 2:1, 3:2, and 4:3 </li></ul><ul><li>Numbers 1 + 2+ 3 + 4 = 10 </li></ul><ul><li>Numbers 1 to 4 are special source of wisdom </li></ul><ul><li>10 is the total and therefore has special status - ancient Greek tetrakys </li></ul>
    13. 13. <ul><li>The stars and planets had their own patterns and ratios and moved in harmony </li></ul><ul><li>‘ The music of the spheres’ – cosmic harmony and balance = perfection </li></ul>
    14. 14. Balance in Body and Soul <ul><li>Elements of body and soul can be calculated and balanced to achieve harmony: ‘tuning’ the 4 elements or humours </li></ul>
    15. 15. Pythagoras <ul><li>Pythagoras saw special significance in numbers and mathematics </li></ul><ul><li>We can still see these patterns today in music and the natural world </li></ul>
    16. 16. Pythagoras and Music A quick lesson in sound <ul><li>Vibration of air (no air = no sound) </li></ul><ul><li>Range of hearing from 20 vibrations per second to 20,000 </li></ul><ul><li>The faster the vibrations, the ‘higher’ we hear the sound </li></ul><ul><li>Frequency (pitch) often expressed as ‘Wavelength’ </li></ul>
    17. 17. Ratios <ul><li>Ratios express relationships, not absolute numbers </li></ul><ul><li>Ratios express proportion </li></ul>Example: A line 8 times longer than another would exhibit a ratio of: 8:1
    18. 18. Sound and the Harmonic Series ‘ Natural’ sound producing instruments (pipes, ‘strings’) have a tendency to produce multiple sounds in proportion to their length: 1, ½, ⅓, ¼, ⅕, ⅙, etc.
    19. 19. The Harmonic Series in Music <ul><li>Musical notes can be expressed as: </li></ul><ul><li>Pitch names (C, D, E, etc) </li></ul><ul><li>Frequencies (vibrations per second = Hertz (Hz)) </li></ul><ul><li>Ratios (e.g.) ‘middle C’ is twice the frequency of the ‘C’ below </li></ul><ul><li>OR </li></ul><ul><li>‘ G’ is 1½ times the frequency of the ‘C’ below – a ratio of 3:2 </li></ul><ul><li>Octave Equivalence </li></ul>
    20. 20. Intervals and Scales <ul><li>Intervals based on simple ratios (1:1, 2:1, 3:2, 4:3) </li></ul><ul><li>These ratios approached ‘Perfection’ </li></ul><ul><li>The name remains today: </li></ul><ul><li>Perfect Unison, Perfect Octave, Perfect 5 th , Perfect 4th </li></ul><ul><li>Early music preferred these intervals, giving birth to notions of ‘Harmony’ </li></ul>
    21. 21. Organum 3:2
    22. 22. A Problem Appears… Early, on, it, became, apparent, that, there, was, a, problem, with, the, Pythagorean, method, of, making, notes, and, scales, from, the, Harmonic, Series. This, became, known, as, the, Pythagorean…
    23. 23. Oh Dear! More Maths…. Scales constructed by the simplest ratio that produces a different note (3:2)
    24. 24. Harmony <ul><li>Solutions to the problem of Pythagorean tuning resulted in the prominence of the ‘third’ note of the scale </li></ul><ul><li>This note has defined modern sense of harmony </li></ul><ul><li>Major and minor chords </li></ul><ul><li>These systems would have been unacceptable to Pythagoras </li></ul><ul><li>Modern senses of ‘Harmony’ are at odds with their Pythagorean origin </li></ul>
    25. 25. Music of the Spheres? The Sun Globular Cluster