Physics 1251 The Science and Technology of Musical Sound Unit 2 Session 21 MWF Musical Scales and Strings Unit 2 Session 21 MWF Musical Scales and Strings

Physics 1251Physics 1251The Science and The Science and

Technology of Musical Technology of Musical SoundSound

Physics 1251Physics 1251The Science and The Science and

Technology of Musical Technology of Musical SoundSound

Unit 2Unit 2

Session 21 MWFSession 21 MWF

Musical Scales and Musical Scales and StringsStrings

Unit 2Unit 2

Session 21 MWFSession 21 MWF

Musical Scales and Musical Scales and StringsStrings

Physics 1251Physics 1251 Unit 2 Session 21Unit 2 Session 21Scales and StringsScales and Strings

Foolscap Quiz: Foolscap Quiz: If AIf A44 is the tonic of an A-major scale (just is the tonic of an A-major scale (just intonation) and has a fundamental intonation) and has a fundamental frequency of 440 Hz, what is the frequency of 440 Hz, what is the frequency of (1) the major third Cfrequency of (1) the major third C44♯, (2) ♯, (2) the “perfect” fifth the “perfect” fifth E E44, and (3) the octave , and (3) the octave AA55 ? ?The frequency of the major third is The frequency of the major third is 5/4 f5/4 ftonictonic = =

550 Hz.550 Hz.

The frequency of the “perfect” fifth is The frequency of the “perfect” fifth is 3/2 f3/2 ftonictonic = = 660 Hz660 Hz..

The frequency of the octave is The frequency of the octave is 2 f2 ftonictonic = 880 Hz. = 880 Hz.


11′ Lecture:′ Lecture:• A A Just Tempered ScaleJust Tempered Scale sets the frequencies sets the frequencies

of the notes in the scale at precise of the notes in the scale at precise ratios of ratios of whole numberswhole numbers..

• Not all chords can be in tune with Just Not all chords can be in tune with Just Temperament.Temperament.

• The The Equal Tempered ScaleEqual Tempered Scale is the is the compromise that sets all semitones compromise that sets all semitones an an equal intervalequal interval apart (100¢, frequency apart (100¢, frequency ratio:1.05946).ratio:1.05946).

• The frequency of vibration of a string is The frequency of vibration of a string is inversely proportional to its length.inversely proportional to its length.


What is a scale?What is a scale?• ““Gamut” {Note “G-Clef”}Gamut” {Note “G-Clef”}

♩♩♩♩♩♩♩♩♩♩♩♩♩♩♩♩

Do Re Mi Fa So La Ti Do Re Mi Fa So La Ti Do Do

♯♯♩♩♩♩♩♩♩♩♩♩♩♩♩♩♩♩

Do Re Mi Fa So La Ti Do Do Re Mi Fa So La Ti Do C-C-major major

G-G-major major

Guido d’Arezzo: “gamma utGuido d’Arezzo: “gamma ut→gamut” • →gamut” • SolfeggioSolfeggio

G is “Do” in the G-scaleG is “Do” in the G-scale


What is a scale?What is a scale?• SolfegeSolfege

DoDoReRe

MiMi

FaFaSoSo

LaLaTiTi

DoDo

Zoltan Zoltan KodalyKodaly

AA

BBCC

DDEE

FFGG

AABB

CCDD

EE

FFGG

AA

BB

CCDD

EEFF

GG


The Circle of Fifths:The Circle of Fifths:

[tonic][tonic]

[major 3 [major 3 rdrd]]

[perfect 5[perfect 5thth]]4:5

4:5

352352

440440

5285282:32:3

5:65:6

528528

633.633.66

792792792792

950.950.44

11811888


A Scale is constructed from Integer A Scale is constructed from Integer Ratios of Frequencies! Not all scales Ratios of Frequencies! Not all scales can be ‘tuned.”can be ‘tuned.”

Tonic Tonic CC 1 264 0 1 264 0

2 D 9/8 297 204 ¢2 D 9/8 297 204 ¢

3 E 5/4 317 316 ¢3 E 5/4 317 316 ¢

4 F 4/3 352 498 ¢ 4 F 4/3 352 498 ¢

44♯ F ♯ F ♯♯ 64/4564/45 375 375

5 G 3/2 396 5 G 3/2 396 702702 ¢ ¢

6 A 5/3 440 884 ¢6 A 5/3 440 884 ¢

7 B 15/8 475 1018 ¢7 B 15/8 475 1018 ¢

8 8 vovo C 2 528 1200 ¢ C 2 528 1200 ¢

4 2/3 264 4 2/3 264 0 0 HzHz

5 3/4 297 5 3/4 297 0 0 HzHz

6 5/6 330 6 5/6 330 1313HzHz

7 15/16 371 7 15/16 371 -4 -4 HzHz

Tonic 1 396 Tonic 1 396 0 0 HzHz

2 9/8 4462 9/8 446 6 6 HzHz

3 5/4 495 3 5/4 495 20Hz20Hz

4 4/3 528 4 4/3 528 0 0 HzHz


A Scale is constructed from Integer A Scale is constructed from Integer Ratios of Frequencies! Not all scales Ratios of Frequencies! Not all scales can be ‘tuned.”can be ‘tuned.”

Tonic Tonic CC 1 264 0 1 264 0

2 D 9/8 297 204 ¢2 D 9/8 297 204 ¢

3 E 5/4 317 316 ¢3 E 5/4 317 316 ¢

4 F 4/3 352 498 ¢ 4 F 4/3 352 498 ¢

44♯ F ♯ F ♯♯ 64/4564/45 375 375

5 G 3/2 396 5 G 3/2 396 702702 ¢ ¢

6 A 5/3 440 884 ¢6 A 5/3 440 884 ¢

7 B 15/8 475 1018 ¢7 B 15/8 475 1018 ¢

8 8 vovo C 2 528 1200 ¢ C 2 528 1200 ¢

4 2/3 264 4 2/3 264 0 0 HzHz

5 3/4 297 5 3/4 297 0 0 HzHz

6 5/6 330 6 5/6 330 1313HzHz

7 15/16 371 7 15/16 371 -4 -4 HzHz

Tonic 1 396 Tonic 1 396 0 0 HzHz

2 9/8 4462 9/8 446 6 6 HzHz

3 5/4 495 3 5/4 495 20Hz20Hz

4 4/3 528 4 4/3 528 0 0 HzHz

Key of C MajorKey of C Major Key of G Major Key of G Major DifferenceDifference

5 5 thth G 3/2 396 G 3/2 396 Tonic 1 396 Tonic 1 396 0 Hz0 Hz

7 7 thth B 15/8 475 B 15/8 475 3 3 rdrd 5/4 495 5/4 495 20Hz20Hz

The 3The 3rdrd in the Key of G in the Key of G major with “howl” if the major with “howl” if the

piano is tuned in C piano is tuned in C major.major.


80/2080/20It is impossible to tune perfectly It is impossible to tune perfectly scales in all keys using the same scales in all keys using the same frequencies and just temperament.frequencies and just temperament.

80/2080/20The Equal Tempered Scale sets The Equal Tempered Scale sets each semitone exactly 100 ¢ apart each semitone exactly 100 ¢ apart or at a ratio of 1.05946….or at a ratio of 1.05946….


A Comparison of the Just and Equal A Comparison of the Just and Equal Tempered Scales:Tempered Scales:

Just:Just:Tonic Tonic CC 1 264 0 1 264 0

22ndnd D 9/8 297 204 ¢ D 9/8 297 204 ¢

33rdrd E 5/4 317 316 ¢ E 5/4 317 316 ¢

44thth F 4/3 352 498 ¢ F 4/3 352 498 ¢

55thth G 3/2 396 G 3/2 396 702702 ¢ ¢

66thth A 5/3 440 884 ¢ A 5/3 440 884 ¢

77thth B 15/8 475 1018 ¢ B 15/8 475 1018 ¢

8 8 vovo C 2 528 1200 ¢ C 2 528 1200 ¢

Equal Tempered:Equal Tempered:

0¢0¢ 1.000 261.6 1.000 261.6 -2 -2

200¢200¢ 1.122 293.7 1.122 293.7 -3-3

400¢400¢ 1.260 329.6 1.260 329.6 1313

500¢500¢ 1.335 349.2 1.335 349.2 -3-3

700¢700¢ 1.498 392.0 1.498 392.0 -4-4

900¢900¢ 1.682 1.682 440440 0 0

1100¢1100¢1.889 493.8 1.889 493.8 -1-1

1200¢1200¢ 2.000 523.2 2.000 523.2 -5-5


Listen to Pachelbel’s Canon in Just Listen to Pachelbel’s Canon in Just and Equal Temperament:and Equal Temperament:

Microtonal URL

http://www.io.com/~hmiller/music/warped-canon.html

Musical NotationMusical Notation


♩♩♯♯♩♩♯♯♩♩♯♯♩♩♩♩♯♯♩♩♩♩

♯♯♩♩

♭♭♩♩♭♭♩♩♭♭♩♩♩♩♩♩♭♭♭♭♩♩♩♩

BB44CC55♯♯

DD44

♯♯

EE44FF44♯♯

GG44♯♯AA44♯♯

CC44♯♯

♯♯

EE33♭♭DD33

♭♭FF33

AA33♭♭

BB33♭♭GG33♭♭CC33


On the standard Keyboard the On the standard Keyboard the sharps sharps ♯ ♯ and flats and flats ♭ ♭ are are enharmonic (sound the same enharmonic (sound the same frequency and pitch.)frequency and pitch.)

In some systems this is not the case.In some systems this is not the case.


Examine a standard Guitar:Examine a standard Guitar:

Note that each succeeding fret Note that each succeeding fret shortens the string by a constant shortens the string by a constant fraction.fraction.


What determines the fundamentalWhat determines the fundamental

frequency of oscillation of a frequency of oscillation of a string?string?

•Pythagoras discovered that the Pythagoras discovered that the frequency was inversely proportional to frequency was inversely proportional to the length of the string.the length of the string.

•ff1 1 /f/f22 = L = L2 2 / L/ L11

•Why?Why?


The fundamental frequency of oscillation The fundamental frequency of oscillation of a string is fixed by the time required of a string is fixed by the time required for a round trip of a string wave:for a round trip of a string wave:

f = [1/2L] f = [1/2L] ‧ ‧ vvstringstring

LL


Guitar frets shorten string to raise Guitar frets shorten string to raise pitch.pitch. LL

oo

LL11

Fret Fret LLo o / L/ Lnn

11 1.061.0622 1.121.1233 1.201.2044 1.271.2755 1.331.33


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

y = ½ a t y = ½ a t 22

a = F/ma = F/m

F = 2 T y/xF = 2 T y/x

∴ ∴ y = ½ (2 T y/x) y = ½ (2 T y/x) t t 22/m/m

(x /t )(x /t )22 = T/ (m/x) = T/ (m/x)

vvstringstring = = √ (T/ √ (T/ μ)μ)

xxyy

TT TT

m= μ

x

m= μ

x

2 T (y/x)2 T (y/x)


80/2080/20Velocity of a wave on a string:Velocity of a wave on a string:

vvstringstring = = √T/ √T/ μμ

T is the tensT is the tension in the string [N].

μ is the mass per unit length [kg/m]μ is the mass per unit length [kg/m]

““Rope Race Demonstration”Rope Race Demonstration”


80/2080/20f = [1/(2L)] f = [1/(2L)] ‧‧ v vstring string

f = [1/(2L)] f = [1/(2L)] ‧ √(T/ ‧ √(T/ μ)]μ)]Tuning pegsTuning pegs

More More Tension:Tension:

Raises pitchRaises pitch


80/2080/20f = [1/(2L)] f = [1/(2L)] ‧ √(T/ ‧ √(T/ μ)]μ)]

Denser wire: lower Denser wire: lower pitchpitch

Less dense wire: Less dense wire: higher pitchhigher pitch


Practice:Practice: 80/2080/20f = [1/(2L)] f = [1/(2L)] ‧ √(T/ ‧ √(T/

μ)]μ)]What is the frequency of a guitar string that is 65 What is the frequency of a guitar string that is 65

cm long, has a mass density μ = 0.005 kg/m cm long, has a mass density μ = 0.005 kg/m

and is under a tension of T = 182 N?and is under a tension of T = 182 N?f = [1/(2L)] f = [1/(2L)] ‧ √(T/ ‧ √(T/ μ)] μ)]

f = [1/ (2(0.65 m)) ]f = [1/ (2(0.65 m)) ]√(182 N/ 0.005 kg/m√(182 N/ 0.005 kg/mμ) μ)

f = 146.8 Hzf = 146.8 Hz


Why do you think that are the frets in Why do you think that are the frets in this guitar oddly spaced?this guitar oddly spaced?

Microtonal music can be Microtonal music can be played on this instrument.played on this instrument.


Summary:Summary:• A A Just Tempered ScaleJust Tempered Scale sets the sets the

frequencies of the notes in the frequencies of the notes in the scale at precise scale at precise ratios of whole ratios of whole numbersnumbers....

• The The Equal Tempered ScaleEqual Tempered Scale is the is the compromise that sets all notes compromise that sets all notes an an equal intervalequal interval apart (100¢, apart (100¢, frequency ratio:1.05946).frequency ratio:1.05946).

• f = [1/(2L)] f = [1/(2L)] ‧ √(T/ ‧ √(T/ μ)]μ)]

Documents

Physics 1251 The Science and Technology of Musical Sound Unit 2 Session 21 MWF Musical Scales and Strings Unit 2 Session 21 MWF Musical Scales and Strings