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Harmonics
& Music
By Stephanie Tacit
Grade 11 Physics
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)
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
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
The first seven harmonics of a string
musical note
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
Consonance and Dis.
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)
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
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
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
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
The Harmonic Series
To hear the harmonic series, visit this
website:
http://www.philtulga.com/harmonics.ht
ml
Bibliography
Definition of a harmonic
http://dictionary.reference.com/browse
/harmonic
Phythagoras Information
http://www.davesabine.com/Music/Arti
cles/PythagorasMathematicalTheoru
minMusic/tabid/169/Default.aspx
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
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.