PHY2464 - The Physical Basis of Music
PHY
-2464
PHY-2464
Physical Basis of Music
Presentation
Presentation 77
Musical
Musical Elements:
Elements: Timing,
Timing,
Perception
Perception of
of Pitch,
Pitch, and
and Scales
Scales
Adapted
Adapted from
from Sam
Sam Matteson’s
Matteson’s
Unit
Unit 22 Sessions
Sessions 15
15 &
& 20
20
Sam
Trickey
Sam Trickey
Jan.
Jan. 31,
31, 2005
2005
PHYPHY-2464
Pres. 7 Musical Elements
Timing:
ƒ It is esthetically pleasing to have sounds made in
temporal sequence with repetitions, elaborations,
variations. What is the clock?
ƒ The human heart rate is a possible natural clock,
typically ranging 70 – 130 beats per minute.
ƒ There are other relevant aspects of human
physiology.
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PHY2464 - The Physical Basis of Music
PHYPHY-2464
Pres. 7 Musical Elements
Timing (continued):
ƒ One second is close to half the period of a oneone-meter
long pendulum: T = 2π l / g , so one second is a
reasonable human time scale (arms, legs are on scale of
one meter).
ƒ Human response time is of order 1/10ths of a second, pick
2/10ths for convenience. (Note, human motion can be
much faster, 1/20th sec.)
ƒ Thus a pleasing regular beat could range from 60 per
minute up to perhaps 300 per minute.
PHYPHY-2464
Pres. 7 Musical Elements
Timing (continued):
ƒ Esthetic considerations (WESTERN Music)
ƒ TEMPO – the basic clock rate, typically between 60
and 150 per minute, usually denoted with a quarterquarternote symbol = a number or “M.M”
M.M” and a number.
This is the “heartbeat”
heartbeat” of the music (notice a different
use of the word “beat”
beat” from that involved with “beat
note”
note” from two closely paired frequencies).
ƒ METER – Grouping of “clock ticks”
ticks” into logical units
4
called measures The time signature written as 4
means four quarterquarter-note valued beats to the measure
6
means 6 eightheighth-note beats arranged in groups of 3
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PHY2464 - The Physical Basis of Music
PHYPHY-2464
Pres. 7 Musical Elements
Timing (continued):
ƒ Esthetic considerations (WESTERN Music)
ƒ RHYTHM – sequence/pattern of emphasized and dedeemphasized notes. Extreme example: Ravel’
Ravel’s “Bolero”
Bolero”
Read and study pp 116 – 117 in “Musical Acoustics”
Acoustics”
PHYPHY-2464
Pres. 7 Musical Elements
Pitch Intervals
Perception: sensation of pitch is a human auditory
characteristic inferred from the repeat period
(frequency).
Perception: an equal ratio of frequencies sounds
like an equal difference or interval of Pitch.
An octave is the pitch interval corresponding to a
frequency ratio of 2:1.
Listen to octaves on flute. Play octaves on demo xylophone
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PHY2464 - The Physical Basis of Music
PHYPHY-2464
Pres. 7 Musical Elements
Pitch Intervals continued
A sequence of ascending frequency intervals repeated each
octave is a SCALE.
Western Music partitions the octave into 12 pitches or
semitones.
semitones.
Since ratios of frequency are perceived as differences of
pitch,
pitch, we have the following chain of inference:
„ Semitones must be in equal frequency ratios
„ 1 octave corresponds to fUpper = 2 fLower
„ 12 semitones (pitches) to one octave
→ Semitone ratio is 21/12 = 1.059463…
PHYPHY-2464
Pres. 7 Musical Elements
Pitch Intervals continued
Semitone ratio is 21/12 = 1.059463…
[A “cent”
cent” (¢) is 1/100 of a semitone or 1/1200 of an octave. 1
octave = 1200 ¢.]
Tuning an instrument to the semitone ratio is called “equal
tempering”
tempering”
Not all instruments are tuned to equal temperament. Why?
Because there are other 1212-note scales that sound better in
some sense of that term. The issue is the harmonic series
We must take a brief tour of Pythagorean ideas
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PHY2464 - The Physical Basis of Music
PHYPHY-2464
Pres. 7 Musical Elements
Pythagoras of Crotona (ca. 580-500 BCE)
[Cf. Nebuchadnezzar 605 – 562 BCE]
•Believed in “rationality
rationality “
of Nature
•Philosophy based on whole
numbers
•Discovered law of strings
Crotona
PHYPHY-2464
Pres. 7 Musical Elements
Pythagoras and the Monochord
1
5 ::3 4:2 2:1“1“Major
Perfect
Octave
Unison
Third
Fifth””
Monochord
Conclusion: Strings with lengths that are integer
ratios of each other sound “consonant.”
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PHY2464 - The Physical Basis of Music
PHYPHY-2464
Pres. 7 Musical Elements
Why does this work?
Pythagoras found Harmonics!
The harmonics must be “in tune”
tune” to avoid beats.
3 rd
5 th
Octave
Unison
Frequency
PHYPHY-2464
Pres. 7 Musical Elements
Harmonics
f1= “fundamental”
fundamental” ; 2 f1= “2nd harmonic”
harmonic”= f2 ,
n f1= “nth harmonic”
harmonic” = fn
f1
f2
f3
f4
f5
Etc.
Frequency
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PHY2464 - The Physical Basis of Music
PHYPHY-2464
Pres. 7 Musical Elements
Perception:
Harmonics may be missing but we hear the
difference frequencies as well as the harmonics.
This Fact applies even to the fundamental, which
may be missing but we “hear” it!
PHYPHY-2464
Pres. 7 Musical Elements
Auditory Demonstration #23
Houstma and Rossing
Institute for Perception Research (IPO) Eindhoven,
Eindhoven, The Netherlands
&
Acoustical Society of America
Tracks 4343-45
Listen to the Westminster Chimes with various harmonics
missing.
missing.
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PHY2464 - The Physical Basis of Music
PHYPHY-2464
Pres. 7 Musical Elements
Build a Scale from Whole Number Ratios of
Frequencies!
Reminder: a scale is a series of tones arranged in
ascending pitch.
The beginning pitch of the scale is called the
“tonic.”
In Pythagorean (or “Just”) temperament, the
frequencies of the tones of the scale are integer
ratios of each other.
PHYPHY-2464
Pres. 7 Musical Elements
Ooops! Now we have a problem
Equal tempering, the ratio of pitch
frequencies is 1.059463….
Pythagorean temperament, the ratios are
always rational fractions (ratios of
integers)
The two won’t sound the same!
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PHY2464 - The Physical Basis of Music
PHYPHY-2464
Pres. 7 Musical Elements
Summary:
• Perception of pitch is inferred from frequency
• Equal intervals of pitch are equal ratios of frequency.
• The physical octave (doubling of frequency) and
perception octave (unison pitches) therefore are
deeply related
• Western music uses 12 tones to the octave, so equal
pitch spacing corresponds to a frequency ratio of
adjacent semitones of 21/12 = 1.059463…
• The harmonic series yields a set of musically pleasing
intervals related by rational fractions.
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