PHY2464 - The Physical Basis of Music
PHY
-2464
PHY-2464
Physical Basis of Music
Presentation
Presentation 99
Strings
Strings and
and Scales
Scales
Adapted
Adapted from
from Sam
Sam Matteson’s
Matteson’s
Unit
2
Session
Unit 2 Session 21
21
Sam
Sam Trickey
Trickey
Feb.
Feb. 9,
9, 2005
2005
PHYPHY-2464
Pres. 9 Strings and Scales
Outlook •
Start with a reminder about Temperament (and
tempering): the plan of intonation compromise
that permits multiple scales, especially in fixed
tuned instruments
•
Explore the Harmonic series in strings
•
Characterize basic physics of stringed
instruments
PHY2464 - The Physical Basis of Music
PHYPHY-2464
Pres. 9 Strings and Scales
Scales constructed from rational fractions of frequencies can
not all be compatible.
Tonic C 1
264 Hz 0¢
4
2/3 264 0 Hz
0¢
2
D 9/8 297 Hz 204¢
204¢
5
3/4 297 0 Hz
Key
of
C
Major
Key
of G Major
3
E 5/4 317 Hz 316¢
316¢
6
5/6 330 13Hz
13Hz
Difference
4
F 4/3 352 Hz 498¢
¢
498
th
5
G
3/2
396
Tonic7 1 15/16
396 371 0-4Hz
♯
Hz
F 64/45 375 Hz
4♯
th
rd
7
B 15/8 475
3 Tonic
5/4 495
1 396 20Hz
0 Hz
5
G 3/2 396 Hz 702¢
702¢
rd
3 Hzin884¢
the¢ Key of
2 G major
9/8 446 6 Hz
6
A 5/3 The
440
884
3
5/4 495 20Hz
7
B 15/8will
475
Hz 1018¢
1018
“howl
” ¢if the instrument
4
8 vo C 2
528
1200¢
1200¢in C major
isHz
tuned
.4/3 528 0 Hz
PHYPHY-2464
Pres. 9 Strings and Scales
Extremes of Tempering
Equal tempering - the ratio of adjacent
semitone frequencies is 1.059463….
Pythagorean temperament (“Just” temperament): the ratios of adjacent semitone
frequencies always are rational fractions
(ratios of integers)
PHY2464 - The Physical Basis of Music
PHYPHY-2464
Pres. 9 Strings and Scales
Scales are not as intuitive as we might
suppose:
http://www.paedia
.com/miscellaneous_inventions.htm
htm#E_O
#E_O
http://www.paedia.com/miscellaneous_inventions.
PHYPHY-2464
Pres. 9 Strings and Scales
FACT: A musical instrument comprises four
primary subsystems:
(1) a mechanical energy source;
(2) a frequency generator;
(3) a frequency filter system;
(4) an antenna.
♩♪♫
~
~
f1 f2 f3 f4
fn
PHY2464 - The Physical Basis of Music
PHYPHY-2464
Pres. 9 Strings and Scales
FACT: A stringed instrument uses standing waves
on a string to provide the frequency generation.
♩♪♫
~
f1 f2 f3 f4
~
fn
String Standing Waves
PHYPHY-2464
Pres. 9 Strings and Scales
FACT: A Standing Wave results from interference
of counter-reflecting waves.
Fundamental Mode
f1 = vstring / λ1 = vstring / 2L
L = λ1/2
Node
Node
Antinode
λ1 /4
λ1 /4
PHY2464 - The Physical Basis of Music
PHYPHY-2464
Pres. 9 Strings and Scales
FACT: The distance between neighboring nodes &
antinodes is ¼ λ. [ dN-A = ¼ λ ]
Second Harmonic
f2 = vstring / λ2 = vstring / L
L = λ2
Node
Node
Antinode
λ2 /4
PHYPHY-2464
λ2 /4
Node
Antinode
λ2 /4
λ2 /4
Pres. 9 Strings and Scales
• So, the fundamental frequency of oscillation of a
string is determined by its length and its sound
speed.
• Pythagoras had discovered that the frequency was
inversely proportional to the length of the string.
f1 /f2 = L2 / L1
• What deterrmines the sound speed?
PHY2464 - The Physical Basis of Music
PHYPHY-2464
Pres. 9 Strings and Scales
Determination of the wave velocity of a string 2 T (y/x)
m
=
µx
T
T
y
x
y=½at2
a = F/m
F = 2 T y/x
∴ y = ½ (2 T y/x) t 2/m
(x /t )2 = T/ (m/x)
vstring = √ (T/ µ)
PHYPHY-2464
Pres. 9 Strings and Scales
Frequency of nth harmonic on a string under
tension T with linear density µ :
fn = [2n/(4L)] × vstring
fn = [n/(2L)] × √(T/ µ)
PHY2464 - The Physical Basis of Music
PHYPHY-2464
Pres. 9 Strings and Scales
Stating the obvious - the modes that will be present
are those that are excited.
♩♪♫
~
f1 f2 f3 f4
~
Energy source can excite
selected modes
PHYPHY-2464
Pres. 9 Strings and Scales
The Guitar is a fretted stringed instrument
that is plucked or strummed.
Tuning pegs
Neck
Frets
Strings (6)
Body
Bridge
fn
PHY2464 - The Physical Basis of Music
PHYPHY-2464
Pres. 9 Strings and Scales
Density of strings sets pitch range of each string
Denser wire: lower pitch
Less dense wire:
higher pitch
PHYPHY-2464
Pres. 9 Strings and Scales
Frets shorten the string to raise the pitch.
Lo
L1
Fret
1
2
3
4
5
Lo / Ln
1.06
1.12
1.20
1.27
1.33
PHY2464 - The Physical Basis of Music
PHYPHY-2464
Pres. 9 Strings and Scales
The tuning pegs change the string tension.
Tuning Pegs
PHYPHY-2464
Pres. 9 Strings and Scales
The strings are attached to the body and sound
board by a flat bridge.
Bridge
Strings
PHY2464 - The Physical Basis of Music
PHYPHY-2464
Pres. 9 Strings and Scales
The coupling of the string to the sound board will
accentuate or attenuate various frequencies.
♩♪♫
~
f1 f2 f3 f4
fn
~
Sound Board, cavity, tone hole, etc.
filter the harmonics
PHYPHY-2464
Pres. 9 Strings and Scales
The response of the body and air cavity
determine the filtering of the harmonics.
Guitar
Neck
in cross section
Sound hole
Bridge
Truss rod
Body
PHY2464 - The Physical Basis of Music
PHYPHY-2464
Pres. 9 Strings and Scales
Vibration Modes of Guitar Body:
Holographic vibration images
of Guitar body
PHYPHY-2464
Pres. 9 Strings and Scales
The guitar ts an evolving instrument.
Classical bracing and configuration http://www.musicfolk
.com/docs//Features/Feature_Bracing.htm
htm
http://www.musicfolk.com/docs//Features/Feature_Bracing.
Modern bracing and configuration –
Work of Michael Kasha, Distinguished Prof. of
Chemistry and Biochem. at FSU
http://www.jthbass.com/kasha.html
http://www.boazguitars
.com/clarita
clarita..htm
http://www.boazguitars.com/