Physics 1251 The Science and Technology of Musical Sound

1
Physics 1251The Science and Technology of
Musical Sound

• Unit 2
• Session 21 MWF
• Musical Scales and Strings

2
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.
3
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 ratio1.05946).
• The frequency of vibration of a string is
  inversely proportional to its length.

4
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
Do Re Mi Fa So La Ti Do
C-major
G-major
Guido dArezzo gamma ut?gamut Solfeggio G is
Do in the G-scale
5
Physics 1251 Unit 2 Session 21 Scales and
Strings

• What is a scale?
• Solfege

Do
Ti
La
So
Fa
Mi
Re
Zoltan Kodaly
Do
6
Physics 1251 Unit 2 Session 21 Scales and
Strings

• The Circle of Fifths

45
56
23
7
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 C 1 264 0
• 2 D 9/8 297 204
• 3 E 5/4 317 316
• 4 F 4/3 352 498
• 4? F ? 64/45 375
• 5 G 3/2 396 702
• 6 A 5/3 440 884
• 7 B 15/8 475 1018
• 8 vo C 2 528 1200

8
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 C 1 264 0
• 2 D 9/8 297 204
• 3 E 5/4 317 316
• 4 F 4/3 352 498
• 4? F ? 64/45 375
• 5 G 3/2 396 702
• 6 A 5/3 440 884
• 7 B 15/8 475 1018
• 8 vo C 2 528 1200

• Key of C Major Key of G Major Difference
• 5 th G 3/2 396 Tonic 1 396
  0 Hz
• 7 th B 15/8 475 3 rd 5/4 495
  20Hz

The 3rd in the Key of G major with howl if the
piano is tuned in C major.
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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.

10
Physics 1251 Unit 2 Session 21 Scales and
Strings

• A Comparison of the Just and Equal Tempered
  Scales
• Just
• Tonic C 1 264 0
• 2nd D 9/8 297 204
• 3rd E 5/4 317 316
• 4th F 4/3 352 498
• 5th G 3/2 396 702
• 6th A 5/3 440 884
• 7th B 15/8 475 1018
• 8 vo C 2 528 1200

Equal Tempered 0 1.000 261.6 -2 200
1.122 293.7 -3 400 1.260 329.6 13 500
1.335 349.2 -3 700 1.498 392.0 -4 900
1.682 440 0 11001.889 493.8 -1 1200
2.000 523.2 -5
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Physics 1251 Unit 2 Session 21 Scales and
Strings

• Listen to Pachelbels Canon in Just and Equal
  Temperament

Microtonal URL
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Physics 1251 Unit 2 Session 21 Scales and
Strings

• Musical Notation

??
?
??
??
??
?
??
C5?
G4?
B4
F4?
?
E4
A4?
??
??
D4?
??
?
??
??
?
A3?
E3?
C4?
D3?
F3
G3?
B3?
C3
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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.

14
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.

15
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?

16
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
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Physics 1251 Unit 2 Session 21 Scales and
Strings

• Guitar frets shorten string to raise pitch.

Fret Lo / Ln 1 1.06 2 1.12 3 1.20 4 1.27 5
1.33
Lo
L1
18
Physics 1251 Unit 2 Session 21 Scales and
Strings

• What determines the velocity of a wave on a
  string?

2 T (y/x)
y ½ a t 2 a F/m F 2 T y/x ? y ½ (2 T
y/x) t 2/m (x /t )2 T/ (m/x) vstring v (T/ µ)
T
T
y
m µx
x
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Physics 1251 Unit 2 Session 21 Scales and
Strings

• 80/20Velocity of a wave on a string
• vstring vT/ µ
• T is the tension in the string N.
• µ is the mass per unit length kg/m

Rope Race Demonstration
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Physics 1251 Unit 2 Session 21 Scales and
Strings

• 80/20f 1/(2L) ? vstring
• f 1/(2L) ? v(T/ µ)

Tuning pegs
More Tension Raises pitch
21
Physics 1251 Unit 2 Session 21 Scales and
Strings

• 80/20f 1/(2L) ? v(T/ µ)

Denser wire lower pitch
Less dense wire higher pitch
22
Physics 1251 Unit 2 Session 21 Scales and
Strings

• Practice 80/20f 1/(2L) ? v(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) ? v(T/ µ) f 1/ (2(0.65 m))
v(182 N/ 0.005 kg/mµ) f 146.8 Hz
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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.
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Physics 1251 Unit 2 Session 21 Scales and
Strings

• Summary
• 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 ratio1.05946).
• f 1/(2L) ? v(T/ µ)