Physics 10: Sound and Music
New York University
Dr. Jay Maron

Section 1:  Physics of music (this section)
Section 2:  Materials and elasticity
Section 3:  Anatomy
Section 4:  Music performance
Section 5:  Physics

Section 1: Physics of music


Wave equation    Music theory Tuning systems    Beat frequencies    Equal temperament    Just and equal tuning    History of music and physics    Cents    Frequency sensitivity   

Stringed instruments    Winds and brass    Range of orchestral instruments        Sitar    Indian tuning    Guitar frets   

Major and minor modes    Diatonic modes    Melodic modes    Circle of fifths    Transposing keys    Core keys    Raga   

Wave interference    Standing waves    String overtones    Reeds    Half-open pipe    Open pipe    Closed pipe    Drums    Chladni experiment    Voice    2D & 3D resonators    Normal modes    Guitar overtones    Plucked string    Spectrum    Timbre    Nyquist frequency    Fourier transform    Example spectra

Damping    Resonance    Resonance strings    Viola d'amore    Frequency precision    Uncertainty principle    Comparision of methods    Margin for error on a string
Frequency precision

Pitch recognition    Ear resonance time    Onset of a note    Comparison of timescales    Wave impedance    Doppler effect   
Musical history

Musical eras and composers    History of instruments    Large instruments    Historical figures

Loudness    Sound pressure    Sound loudness in decibels    Perceived loudness    Soundproofing

Loudspeakers    Amplitude    Maximum power   


Problems    Solutions    Labs   
Section 2: Elasticity


Wave types    Hooke's law    Young's modulus    Tensile strength    Toughness    Deformation    Poisson ratio    Hardness   

Gems    Alloys    High-performance materials    Engineering    Elements    History of metallurgy   

String equation    String tension    Wave speed    Tensile strength    Max frequency of a string    Min frequency of a string    Inharmonicity

Ideal gas    Kinetic energy    Sound speed    History    Newton's calculation    Dark energy
Section 3: Anatomy

The senses

The ear    Cochlea    Basilar membrane   
Color vision    Blackbody radiation    Diffraction    Visual resolution    Aural resolution    Visual brightness    Visual sensitivity    Comparison of senses    Blue whales    Echolocation   

Evolution    Tetrapods    Atlas and axis vertebra    Breathing cycle    Energy conservation    Spine    Brain    Eyes    Heart    Larynx    Nervous system    Muscle structure    Skeleton    Sleep    Fuel    Muscles    Joint motion    Head    Neck    Shoulders    Arms    Pelvis    Legs    List of muscles    Nerves   
Section 4: Music performance

Balance    History of kung fu    Instrument balance    Sword    Angular momentum    Frames of reference    Speed    Eyes    Damage control    Fundamentals    Phase lock    Standard violin technique    Videos    Styles

Algorithms    Supercomputing    Many-body theory    Resolution    Optimization    Annealing    Scale similarity    Critical damping    Convolutions    Differential equations    Interpolation    Green's function    Noise cascade   
Section 5: Physics

Units    Conservation of momentum and energy    Angular momentum   

Python programming    Python tools for music analysis


Wave speed

Frequency and period

The properties of a wave are

F  =  Frequency   (1/seconds)
W  =  Wavelength  (meters)
V  =  Wavespeed   (meters/second)
T  =  Period      (seconds)        (The time it takes for one wavelength to pass by)
A  =  Amplitude                    (half the distance between the top and bottom crests of the wave)
Wave equations:
F W = V

F T = 1


A train is like a wave.

Length of a train car  =  Wavelength  =  W  =  10 meters
Speed of the train     =  Wavespeed   =  V  =  20 meters/second
Cars per second        =  Frequency   =  F  =   2 Hertz
Car time               =  Period      =  T  =  .5 seconds 

Speed of sound in air

Your ear senses changes in pressure as a wave passes by

Speed of sound at sea level    =  V  =   340 meters/second
Frequency of a violin A string =  F  =   440 Hertz
Wavelength of a sound wave     =  W  =   .77 meters  =  W/F
Wave period                    =  T  = .0023 seconds

Speed of a wave on a string

A wave on a string moves at constant speed and reflects at the boundaries.

Frequency of a violin A-string          =  F                  =   440 Hertz
Length of a violin A-string             =  L                  =   .32 meters
Round trip time up and down the string  =  T  =  2L/V  = F-1  =.00227 seconds
Speed of a wave on a violin A-string    =  V  =  F/(2L)       =   688 meters/second

Music theory

Musical notation

The notes in a treble and bass clef

The "A" at the center of the treble clef has a frequency of 440 Hertz. We will use this "A" as a reference. There are 12 notes between this "A" and the "A" an octave above it, where each note is separated by a half step.

Note    Note       Notes in an     Notes in an     Name of
index   letters    A-major scale   A-minor scale   interval

 0      A              A               A           Tonic
 1      A# or Bb                                   Minor second  =  Half step
 2      B              B               B           Major second  =  Whole step
 3      C                              C           Minor third
 4      C# or Db       C#                          Major third
 5      D              D               D           Perfect fourth
 6      D# or Eb                                   Tritone
 7      E              E               E           Minor fifth
 8      F                              F           Minor sixth
 9      F# or Gb       F#                          Major sixth
10      G                              G           Minor seventh
11      G# or Ab       G#                          Major seventh
12      A              A               A           Octave

"A#" stands for "A sharp" and "Bb" stands for "B flat".

The bottom note is the "tonic" and the "interval" is the distance between the tonic and the given note.

When describing intervals we will usually refer to the note index rather than the interval name or the note letter. A change of index of 1 is a half step and a change of index of 2 is a whole step. An octave is 12 half steps.

A-major scale
A-minor scale

The choice of "A" for the tonic is arbitrary. We could have used any of the other 11 notes. If the tonic is "D" then the notes in a major and minor scale are:

D-major scale
D-minor scale

A chromatic scale contains all 12 notes. A chromatic scale with a tonic of "C" looks like:

Chromatic scale with C as the tonic

Wikipedia:     Clefs     Musical intervals     Chromatic scale     Major scale     Minor scale     Octave     Perfect fifth     Perfect fourth

Tuning systems


If two notes are played at the same time then we hear the sum of the waveforms.

If two notes are played such that the frequency of the high note is twice that of the low note then this is an octave. The wavelength of the high note is half that of the low note.

Color       Frequency       Wavelength

Orange      220 Hertz           1
Red         440 Hertz          1/2
Because the red and orange waves match up after a distance of 1 the blue note is periodic. This makes it easy for your ear to process.

Orange = 220 Hertz          Red = 440 Hertz   (octave)          Blue = Orange + Red

If we double both frequencies then it also sounds like an octave. The shape of the blue wave is preserved.

Orange = 440 Hertz          Red = 880 Hertz   (octave)          Blue = Orange + Red

Color       Frequency       Wavelength

Orange      440 Hertz          1/2
Red         880 Hertz          1/4
When listening to two simultaneous pitches our ear is sensitive to the frequency ratio. For both of the above octaves the ratio of the high frequency to the low frequency is 2.
440 / 220  =  2
880 / 440  =  2
If we are talking about frequency ratios and not absolute frequencies then for simplicity we can set the bottom frequency equal to 1. Hence for an octave,
F1 = 1       F2 = 2
For a fifth (playing an A and an E),
F1 = 1       F2 = 3/2

Gallery of intervals


Orange = 1 Hertz          Red = 2 Hertz   (The note "A")          Blue = Orange + Red

Perfect fifth

Orange = 1 Hertz          Red = 3/2 Hertz    (the note "E")

Perfect fourth

Orange = 1 HertzA          Red = 4/3 Hertz    (the note "D")         

Major third

Orange = 1 Hertz          Red = 5/4 Hertz    (the note "C#")         

Minor third

Orange = 1 Hertz          Red = 6/5 Hertz    (the note "C")         


Orange = 1 Hertz          Red = 2½ Hertz    (the note "D flat")         

The octave, fifth, fourth, major third, and minor third are all periodic and sound harmonious.

The tritone is not periodic and sounds dissonant.

If two notes in an interval have frequencies such that

Frequency of top note  /  Frequency of bottom note  =  I / J       where I and J are small integers
then the summed note will be periodic. The smaller the integers I and J, the more noticeable the periodicity and the more harmonious the interval. This is why fifths and fourths sound more resonant than thirds.

If the note "A" is played together with the notes of the 12-tone scale the result is

Note  Interval      Frequency   Result

 A    Unison          1.000     Strongly resonant
 Bb   Minor second    1.059     Dissonant
 B    Major second    9/8       Resonance barely noticeable
 C    Minor third     6/5       Weakly resonant
 C#   Major third     5/4       Weakly resonant
 D    Fourth          4/3       Strongly resonant
 Eb   Tritone          1.414    Dissonant
 E    Fifth           3/2       Strongly resonant
 F    Minor sixth      1.587    Weakly resonant
 F#   Major sixth     5/3       Weakly resonant
 G    Minor seventh    1.587    Dissonant
 G#   Major seventh    1.888    Dissonant
 A    Octave           2        Strongly resonant
The notes {Bb, B, Eb, G, Ab} cannot be expressed as a ratio of small integers and so they sound dissonant when played together with an A.

Beat frequencies: consequences of playing out of tune

If two notes are out of tune they produce dissonant beat frequencies.

Frequency of note #1  =  F1
Frequency of note #2  =  F2
Beat frequency        =  Fb  =  F2 - F1
For the beats to not be noticeable, Fb has to be less than one Hertz. On the E string there is little margin for error. Vibrato is often used to cover up the beat frequencies.

Examples of beat frequencies

Orange = A          Red = A

Orange = A          Red = 1.03 A

Orange = A          Red = 1.06 A

Orange = A          Red = 1.09 A

The more out of tune the note, the more pronounced the beat frequencies. In the first figure, the notes are in tune and no beat frequencies are produced.

If you play an octave out of tune you also get beat frequencies.

An octave played in tune
Orange = A          Red = 2 A

An octave played out of tune
Orange = A          Red = 2.1 A

Equal temperament

If you want to divide the octave into 12 pitches such that the interval between each pitch is equal, the pitches have the form

I  =  An integer where 0 corresponds to the tonic and 12 corresponds to the octave.
F  =  Frequency of the pitches
   =  2I/12
For the tonic,
F  =  20/12  =  1
For the octave,
F  =  212/12 =  2
The frequency ratio between two adjacent pitches is
Frequency ratio  =  2(I+1)/12 / 2I/12
                 =  21/12
                 =  1.059
which is independent of I.
Tuning systems

The notes on the A-string of a violin
          Red: equal temperament           Green: just intonation           Orange: Pythagorean tuning

Note  Index  Interval       Equal  Just tuning   Major  Minor   Pythagorean    Cents
                            tuning               scale  scale     tuning

A       0    Unison         1.000  1.000 = 1/1     *      *     1/1   = 1.000    0
Bflat   1    Minor second   1.059                             256/243 = 1.053
B       2    Major second   1.122  1.125 = 9/8     *      *     9/8   = 1.125  + 9
C       3    Minor third    1.189  1.200 = 6/5            *    32/27  = 1.185  -16
C#      4    Major third    1.260  1.250 = 5/4     *           81/64  = 1.266  +14
D       5    Fourth         1.335  1.333 = 4/3     *      *     4/3   = 1.333  + 2
Eflat   6    Tritone        1.414                             729/512 = 1.424
E       7    Fifth          1.498  1.500 = 3/2     *      *     3/2   = 1.500  - 2
F       8    Minor sixth    1.587  1.600 = 8/5            *   128/81  = 1.580  -14
F#      9    Sixth          1.682  1.667 = 5/3     *           27/16  = 1.688  +16
G      10    Minor seventh  1.782                         *    16/9   = 1.778
Aflat  11    Major seventh  1.888                  *          243/128 = 1.898
A      12    Octave         2.000  2.000 = 2/1     *      *     2/1   = 2.000    0
In equal tuning, the frequency ratio of an interval is
Frequency ratio  =  2(Index/12)        Where "Index" is an integer
Equal tuning is based on equal frequency ratios. Just tuning adjusts the frequencies to correspond to the nearest convenient integer ratio. For example, in equal tuning, the frequency ratio of a fifth is 1.498 and just tuning changes it to 1.500 = 3/2.

For the 12 tone scale, equal tuning and just tuning are nearly identical.

The major and minor modes favor the resonant notes.

Cents refers to the difference between just tuning and equal tuning. 100 Cents corresponds to a half step and 1 cent corresponds to .01 half steps.

History of physics and music

In the 6th century BCE, Pythagoras developed a 12-tone scale based on the ratios 2/1 and 3/2. This tuning was widely used until the 16th century CE. Pythagorean tuning gives good results for fourths and fifths but poor results for thirds, and it is not possible to write contrapuntal music.

In the 2nd century CE, Ptolemy developed the "major scale", based on the frequency ratios 2/1, 3/2, 4/3, and 5/4. This scale allows for consonant thirds.

1523  Pietro Anon introduced "meantone tuning" to fix the thirds, using a
      frequency ratio of 5/4 for major thirds.  His treatise "Thoscanello de la
      musica" expanded the possibilities for chords and harmony.

1555  Amati develops the 4-string violin

1584  Equal tuning developed.  Equal tuning divides the octave logarithmically.
      The first known examples were:
      Vincenzo Galilei in 1584  (Father of Galileo Galilei)
      Zhu Zaiyu in 1584
      Simon Stevin in 1585

1585  Simon Stevin introduces decimal numbers to Europe.
      (For example, writing 1/8 as 0.125)

1586  Simon Stevin drops objects of varying mass from a church tower to demonstrate that
      they accelerate uniformly.

1604  Galileo publishes a mathematical description of acceleration.

1614  Logarithms invented by John Napier, making possible precise calculations
      of equal tuning ratios.  Stevin's calculations were mathematically sound but
      the frequencies couldn't be calculated with precision until logarithms were

1637  Cartesian geometry published by Fermat and Descartes.
      This was the crucial development that triggered an explosion of mathematics
      and opened the way for the calculus.

1672  Newton builds the first reflecting telescope and presents it to the Royal Society
A replica of Newton's telescope
Schematic of Newton's telescope
1684  Leibniz publishes the calculus

1687  Newton publishes the Principia Mathematica, which contained the calculus,
      the laws of motion (F=MA), and a proof that planets orbit as ellipses.

1722  Bach publishes "The Well Tempered Clavier".
Until ~ 1650, most keyboards used meantone tuning. This tuning gives good results if you confine yourself to a small number of keys and use few accidentals, but it can't be made to work for all keys.

J.S. Bach tuned his own harpsichords and clavichords and he customized the tuning to work in all 24 keys ("well temperament"). He demonstrated its effectiveness in his 1722 work "The Well Tempered Clavier".

Just tuning is based on integer ratios and equal tuning is based on logarithms, and there is no direct connection between them. By freak mathematical coincidence, 12-tone equal tuning gives a set of notes that are nearly identical to those for just tuning (see the above table). The correspondence is close, but not exact, and violinists use a compromise between just and equal tuning that is highly situation dependent. The Bach Chaconne in D minor is a tour de force of just intonation.

Bach Chaconne for viola

The synthesis of just and equal tuning offers rich contrapuntal possibilities, as was explored during the Baroque age by composers such as Vivaldi, Bach, and Handel.

1733  Euler develops the calculus of variations
1762  Lagrange discovers the divergence theorem, the 2D generalization of the
      fundamental theorem of calculus.
      The surface flux integral equals the volume divergence integral
1788  Lagrangian mechanics published
1821  Cauchy publishes the "epsilon-delta" definition of a limit, raising the
      level of rigor in mathematics.
1822  Fourier transform published
1828  Green's theorem. In 2D, the circulation integral equals the curl area integral
1833  Hamiltonian mechanics published
1834  Eikonal approximation developed by Hamilton
1850  Kelvin-Stokes theorem. 3D generalization of Green's theorem
1854  Riemann Integral published, the first rigorous definition of an integral
1854  Chebyshev polynomials published
1863  Helmholtz publishes "On the Sensations of Tone"
1870  Heine defines "uniform continuity"
1872  Heine proves that a continuous function on an open interval need not be
      uniformly continuous.
1872  Weierstrass publishes the "Weierstrass function", the first example of
      a function that is continuous everywhere and differentiable nowhere.
1877  Lord Rayleigh publishes "Theory of Sound"
1887  Poincare discovers the phenomenon of chaos while studying celestial mechanics
1926  WKB theory published
1935  Bourbaki textbooks published, with the aim of reformulating mathematics on
      an extremely abstract and formal but self-contained basis.  With the goal
      of grounding all of mathematics on set theory, the authors strove for rigour
      and generality.
1978  "Bender & Orszag" textbook published.  Art of blending special functions
      like Scotch.