Physics 1251
The Science and Technology
of Musical Sound
Unit 2
Session 21 MWF
Musical Scales and Strings
Physics 1251
Unit 2 Session 21
Scales and Strings
Foolscap Quiz:
If A4 is the tonic of an A-major scale (just
intonation) and has a fundamental frequency
of 440 Hz, what is the frequency of (1) the
major third C4♯, (2) the “perfect” fifth E4, and
(3) the octave A5 ?
The frequency of the major third is 5/4 ftonic = 550 Hz.
The frequency of the “perfect” fifth is 3/2 ftonic = 660 Hz.
The frequency of the octave is 2 ftonic = 880 Hz.
Physics 1251
Unit 2 Session 21
Scales and Strings
1′ Lecture:
•
•
•
•
A Just Tempered Scale sets the frequencies of
the notes in the scale at precise ratios of whole
numbers.
Not all chords can be in tune with Just
Temperament.
The Equal Tempered Scale is the compromise
that sets all semitones an equal interval apart
(100¢, frequency ratio:1.05946).
The frequency of vibration of a string is
inversely proportional to its length.
Physics 1251
Unit 2 Session 21
Scales and Strings
What is a scale?
•
“Gamut” {Note “G-Clef”}
♩
♩
♩
♩
♩
♩
♩♩
Do Re Mi Fa So La Ti Do
C-major
♩
♩
♯
♩
♩
♩
♩
♩♩
Do Re Mi Fa So La Ti Do
G-major
Guido d’Arezzo: “gamma ut→gamut” • Solfeggio
G is “Do” in the G-scale
Physics 1251
Unit 2 Session 21
Scales and Strings
What is a scale?
•
Solfege
Ti
Do
La
Fa
Do
Re
So
Mi
Zoltan Kodaly
Physics 1251
Unit 2 Session 21
Scales and Strings
The Circle of Fifths:
633.6
950.4
440
A
[tonic]
352
528
792
2:3
[major 3 rd]
[perfect 5th]
528
792
1188
Physics 1251
Unit 2 Session 21
Scales and Strings
A Scale is constructed from Integer Ratios of
Frequencies! Not all scales can be ‘tuned.”
Tonic
2
3
4
4♯
5
6
7
8 vo
C 1
264 0
D 9/8 297 204 ¢
E 5/4 317 316 ¢
F 4/3 352 498 ¢
F ♯ 64/45 375
G 3/2 396 702 ¢
A 5/3 440 884 ¢
B 15/8 475 1018 ¢
C 2
528 1200 ¢
4
5
6
2/3 264 0 Hz
3/4 297 0 Hz
5/6 330 13Hz
7 15/16
Tonic 1
2
9/8
3
5/4
4
4/3
371 -4 Hz
396 0 Hz
446 6 Hz
495 20Hz
528 0 Hz
Physics 1251
Unit 2 Session 21
Scales and Strings
A Scale is constructed from Integer Ratios of
Frequencies! Not all scales can be ‘tuned.”
Tonic
2
3
Difference
5 4th
th
7 4♯
5
6
7
8 vo
C 1
264 0
4
2/3 264 0 Hz
D 9/8 297 204 ¢ 5
3/4 297 0 Hz
Key of C Major
Key of G Major
E 5/4 317 316 ¢ 6
5/6 330 13Hz
FG 3/2
4/3 396
352 498Tonic
¢
1 396
0 Hz
♯
7 15/16 371 -4 Hz
F 15/8
64/45 375
B
475
3 rd 5/4 495
20Hz
G 3/2 396
702 ¢ Tonic 1 396 0 Hz
rd
The 3 in the Key
2 of G
9/8 446 6 Hz
A 5/3 440 884 ¢
major
3 if the
5/4 495 20Hz
B 15/8
475with
1018“howl”
¢
is tuned
4 major.
4/3 528 0 Hz
Cpiano
2
528
1200 ¢ in C
Physics 1251
Unit 2 Session 21
Scales and Strings
80/20It
is impossible to tune perfectly scales
in all keys using the same frequencies
and just temperament.
80/20The
Equal Tempered Scale sets each
semitone exactly 100 ¢ apart or at a
ratio of 1.05946….
Physics 1251
Unit 2 Session 21
Scales and Strings
A Comparison of the Just and Equal
Tempered Scales:
Just:
Equal Tempered:
Tonic C 1
264 0
0¢ 1.000 261.6
2nd
D 9/8 297 204 ¢ 200¢ 1.122 293.7
3rd
E 5/4 317 316 ¢ 400¢ 1.260 329.6
4th
F 4/3 352 498 ¢ 500¢ 1.335 349.2
5th
G 3/2 396 702 ¢ 700¢ 1.498 392.0
6th
A 5/3 440 884 ¢ 900¢ 1.682 440
7th
B 15/8 475 1018 ¢ 1100¢1.889 493.8
8 vo C 2
528 1200 ¢ 1200¢ 2.000 523.2
-2
-3
13
-3
-4
0
-1
-5
Physics 1251
Unit 2 Session 21
Scales and Strings
Listen to Pachelbel’s Canon in Just and
Equal Temperament:
Microtonal URL
Physics 1251
Unit 2 Session 21
Scales and Strings
Musical Notation
♯
♭♩♭♩
♭
♩
♩♭♩
♩♭♩
♯♩
♩
♯♩ ♩
♯
♩
♩ ♯♩
G4♯ B4 C5♯
F
♯
E 4
A♯
D4♯ 4
A3♭ C ♯
E3♭
4
D3♭ F3 G ♭
3
B3♭
C3
♯♩
4
Physics 1251
Unit 2 Session 21
Scales and Strings
On the standard Keyboard the sharps ♯
and flats ♭ are enharmonic (sound the
same frequency and pitch.)
In some systems this is not the case.
Physics 1251
Unit 2 Session 21
Scales and Strings
Examine a standard Guitar:
Note that each succeeding fret shortens the
string by a constant fraction.
Physics 1251
Unit 2 Session 21
Scales and Strings
What determines the fundamental
frequency of oscillation of a string?
•Pythagoras discovered that the frequency was
inversely proportional to the length of the string.
•f1 /f2 = L2 / L1
•Why?
Physics 1251
Unit 2 Session 21
Scales and Strings
The fundamental frequency of oscillation of a
string is fixed by the time required for a
round trip of a string wave:
L
f = [1/2L] ‧ vstring
Physics 1251
Unit 2 Session 21
Scales and Strings
Guitar frets shorten string to raise pitch.
Fret
1
2
3
4
5
Lo / Ln
1.06
1.12
1.20
1.27
1.33
Physics 1251
Unit 2 Session 21
Scales and Strings
What determines the velocity of a wave on a
string?
2 T (y/x)
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)
Physics 1251
80/20Velocity
Unit 2 Session 21
Scales and Strings
of a wave on a string:
vstring = √T/ μ
T is the tension in the string [N].
μ is the mass per unit length [kg/m]
“Rope Race Demonstration”
Physics 1251
80/20f
Unit 2 Session 21
Scales and Strings
= [1/(2L)] ‧ vstring
f = [1/(2L)] ‧ √(T/ μ)]
Tuning pegs
More Tension:
Raises pitch
Physics 1251
80/20f
Unit 2 Session 21
Scales and Strings
= [1/(2L)] ‧ √(T/ μ)]
Denser wire: lower pitch
Less dense wire:
higher pitch
Physics 1251
Practice:
Unit 2 Session 21
Scales and Strings
80/20f
= [1/(2L)] ‧ √(T/ μ)]
What is the frequency of a guitar string that is 65 cm
long, has a mass density μ = 0.005 kg/m and is
under a tension of T = 182 N?
f = [1/(2L)] ‧ √(T/ μ)]
f = [1/ (2(0.65 m)) ]√(182 N/ 0.005 kg/mμ)
f = 146.8 Hz
Physics 1251
Unit 2 Session 21
Scales and Strings
Why do you think that are the frets in this
guitar oddly spaced?
Microtonal music can be
played on this instrument.
Physics 1251
Summary:
•
•
•
Unit 2 Session 21
Scales and Strings
A Just Tempered Scale sets the
frequencies of the notes in the scale at
precise ratios of whole numbers..
The Equal Tempered Scale is the
compromise that sets all notes an
equal interval apart (100¢, frequency
ratio:1.05946).
f = [1/(2L)] ‧ √(T/ μ)]