1. Harmonics & Music By Stephanie Tacit Grade 11 Physics

2. Frequency and Pitch As frequency increases, pitch increases Frequency increases as length decreases (ƒ α 1/ℓ), tension increases (ƒ α √T), diameter decreases (ƒ α 1/d), and density decreases (ƒ α 1/ρ )

3. Standing Waves Standing wave: occurs when waves of identical amplitude and wavelength traveling in opposite directions interfere (fundamental frequency and multiples) Node: point of complete destructive interference (remains at rest) Harmonic: a single oscillation whose frequency is an integral multiple of a fundamental frequency

4. Harmonics Overtone: resulting modes of vibration when a string of vibration has numerous segments The relative intensity and number of overtones determines the quality of a musical note The first seven harmonics of a string

5. Consonance and Dis. Consonance: occurs when a harmony, chord, or interval is considered stable and pleasant to hear Dissonance: occurs when a harmony, chord, or interval is considered instable and unpleasant to hear (needs to resolve) Pythagoras established that low or simple frequency ratios are most consonant

6. Perfect consonances: perfect unison, perfect octave, perfect fourth, perfect fifth Imperfect consonances: major third, minor third, major sixth, minor sixth Dissonances: major second, minor second, major seventh, minor seventh, tri-tone (augmented fourth or diminished fifth)

7. Tri-Tones Tri-tone: an augmented fourth or diminished fifth; an extremely dissonant sound (so dissonant, it was historically associated with the devil) The ratio between a tri-tone and fundamental frequency is √2:1 (a complex ratio, resulting in dissonance) Pythagoras discovered that when an octave (not a string’s harmonic) is divided exactly in half, a tri-tone is produced; each octave has six whole tones, but three whole tones make a tri-tone

8. The Harmonic Series Harmonic series: an infinite series of tones consisting of a fundamental tone and the consecutive harmonics produced by it Pythagoras exposed the first four overtones which create consonant intervals in music harmony: the octave, perfect fifth, perfect fourth, and major third

9. The Harmonic Series Pythagoras related these intervals as ratios: 1:1 = Unison 2:1 = Octave 3:2 = Fifth 4:3 = Fourth 5:4 = Major Third

10. The Harmonic Series The intervals between the above notes in a harmonic series are as follows: perfect octave, perfect fifth, perfect fourth, major third, minor third, (flat) minor third, (sharp) major second, major second, major second, (flat) major second, (sharp) minor second **The – signs indicates a pitch is significantly lower than the written note on the staff

11. The Harmonic Series To hear the harmonic series, visit this website: http://www.philtulga.com/harmonics.ht ml

12. Bibliography Definition of a harmonic http://dictionary.reference.com/brows e/harmonic Phythagoras Information http://www.davesabine.com/Music/Arti cles/PythagorasMathematicalTheoru minMusic/tabid/169/Default.aspx

13. Bibliography Harmonics Diagrams http://cnx.org/content/m11118/latest/ http://en.wikipedia.org/wiki/Harmonic_ series_%28music%29 Consonance and Dissonance info. http://en.wikipedia.org/wiki/Consonan ce_and_dissonance Tri-tone info. http://en.wikipedia.org/wiki/Tritone

14. Bibliography Harmonic Series Diagram http://www.music.sc.edu/fs/bain/atmi0 2/hs/index-audio.html#tptsats All other information: Hirsch, Alan, David Martindale, Steve Bibla, and Charles Stewart. Physics 11. Toronto: Nelson Thomson Learning, 2002.