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


Waves    Music theory

Tuning systems    Beat frequencies    Equal temperament    Just and equal tuning    History of music and physics    Cents    Frequency sensitivity   
Instruments    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   
Overtones    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
Oscillators    Damping    Resonance    Resonance strings    Viola d'amore    Frequency precision    Uncertainty principle    Comparision of methods    Margin for error on a string
Timescales    Pitch recognition    Ear resonance time    Onset of a note    Comparison of timescales    Wave impedance   
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   
Loudness    Sound pressure    Sound loudness in decibels    Perceived loudness    Soundproofing
Technology    Loudspeakers    Amplitude    Maximum power   

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   
Strings    String tension    Wave speed    Tensile strength    Max frequency of a string    Min frequency of a string    Inharmonicity
Gases    Ideal gas    Kinetic energy    Sound speed    History    Newton's calculation    Dark energy
Musical eras and composers    History of instruments    Large instruments    Historical figures

Conservation of momentum and energy    Angular momentum   

History of kung fu    Balance    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   

Noise    Flying cars    Electric cars    Problems    Solutions    Labs    Appendix    Glossary of musical terms    Doppler effect    Units   
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.

The fortuitousness of the 12-tone scale

          Red: equal temperament           Orange: just intonation

The notes for 12-tone equal temperament coincide well with the note of just intonaton.

The most resonant notes in the 12-tone equal temperament scale are the fourth and the fifth and these are particularly close to their just-intonation counterparts.

The frequency ratio between a fourth and a fifth in just-temperament is

R  =  (3/2) / (4/3)  =  9/8  =  1.125
In a 12-tone equal-tempered scale the frequency ratio of a whole step is
R  =  2(2/12)  =  1.122
which is nearly the same as the ratio between a fourth and a fifth. This is why the 12-tone scale works so well. If you try any number other than 12 it doesn't work. This is why the 12-tone scale is the most useful for writing harmony.

Tunings exist that use numbers different from 12, such as for Indian, Thai, and Arabic music. These tunings can generate exotic melodic structure but they are less useful for harmony than the 12-tone scale.

The 12-tone scale is natural in the sense that it doesn't have any "free parameters". The choice of the number "12" emerged naturally from the positions of the resonant notes. It is also "fortuitous" in that the values of Z are so small.

Soccer is an example of a "natural sport". The rules are simple and if you change the parameters (such as field size, number of players, etc) the game is essentially the same.

American football requires "fine tuning". In order for the sport to make sense you need a large rulebook. It also has lots of "free parameters" because there are many different ways the rules could be constructed.

The chess player Edward Lasker once said:

"While the Baroque rules of Chess could only have been created by humans, the rules of Go are so elegant, organic, and rigorously logical that if intelligent life forms exist elsewhere in the universe, they almost certainly play Go."

The rules of chess are an example of "fine tuning" and there are lots of free parameters (the moves allowed by each piece).

Naturalness and fine tuning in physics:     The Multiverse and the Higgs boson     The anthropic principle


Dfferences in pitch are often expressed in "cents". A half step corresponds to 100 cents and the limit of human sensitivity is 10 cents. The above table on just and equal tuning shows the difference between the two systems in cents.

Interval      Frequency ratio       Cents

  0 cents     20/12  = 1               0
  1 cent      21/1200= 1.0006          1
 10 cents     21/120 = 1.0058         10
Half step     21/12  = 1.0595        100
Whole step    22/12  = 1.1225        200
Fifth         27/12  = 1.498         700
Octave        212/12 = 2            1200

I  =  Note index, where I=1 is a half step, I=2 is a whole step, and I=12 is an octave
C  =  Cents
   =  I/100
F  =  Frequency ratio
   =  2I/12
   =  2C/1200

C  =  1200 ln(F) / ln(2)
If F has the form
F = 1 + Z        where Z << 1
C  =  1200 ln(1+Z) / ln(2)
   ~  1200 Z / ln(2)
   ~  1731 Z

For example, the frequencies for a fifth are
Equal tuning:   Fe  =  27/12 = 1.4983
Just tuning:    Fj  =  3/2    = 1.5000
These frequencies have the ratio
F  =  Fj / Fe
   =  1.00113
Z  =  F - 1
   =  .00113
C  =  2.0
The frequencies for just and equal tuning differ by 2 cents.
Frequency sensitivity

The frequency ratio of a half step is

21/12 = 1.059
Human are capable of detecting a change in frequency of 1/10 of a half step, which corresponds to a frequency ratio of
21/120 = 1.0056
To appreciate a 12-tone scale one must have precision that is tangibly smaller than a half step. Humans are well within this bound.

For example, for the notes on an A-string with a frequency of 440 Hertz,

I  =  Index of a note.  I=0 for the tonic and I=12 for the octave
F  =  Frequency of a note on the A-string
   =  440 * 2I/12
f  =  Smallest frequency greater than 440 Hertz for which "f" sounds
      from "F"
   ~  443 Hertz
R  =  Characteristic frequency ratio for human sensitivity
   =  f / F
   ~  1.0058

1-R  =  .0058  =  1/173

Note  I      F

 A    0    440     Open A-string
       .1  442.5   Largest frequency that sounds indistinguishable from 440 Hertz
 Bb   1    466     Half step
 B    2    494     Whole step
 C    3    523
 C#   4    554
 D    5    587     Perfect fourth
 Eb   6    622     Tritone
 E    7    659     Perfect fifth
 F    8    698
 F#   9    740
 G   10    784
 G#  11    831
 A   12    880     Octave

If we start from a frequency of 880 Hertz then the frequencies are

Note  I      F

 A    0    880
       .1  885    Largest frequency that sounds indistinguishable from 880 Hertz
 Bb   1    932    Half step
 B    2    988    Whole step


Range of instruments

Green dots indicate the frequencies of open strings.

An orchestral bass and a bass guitar have the same string tunings.

The range of organs is variable and typically extends beyond the piano in both the high and low direction.

Stringed instruments

A violin, viola, cello, and double bass
String quartet

Violin and viola
Electric guitar

Strings on a violin

Strings on a viola or cello

Violin fingering
Strings on a guitar

Wind and brass instruments


French horn

In a reed instrument, a puff of air enters the pipe, which closes the reed because of the Bernoulli effect. A pressure pulse travels to the other and and back and when it returns it opens the reed, allowing another puff of air to enter the pipe and repeat the cycle.

              String   Baroque    Classical  Modern
              quartet  orchestra  orchestra  orchestra

First violin    1        6          12         16
Second violin   1        4          10         14
Viola           1        4           8         12
Cello           1        4           8         12
Bass                     2           6          8
Flute                    2           2          4
Oboe                     2           2          4
Clarinet                             2          4
Bassoon                  2           2          4
Trumpet                  2           2          4
French Horn              2           2          4
Trombone                                        4
Tuba                                            2
Harpsichord              1           1
Timpani                  1           1          1


Violins, violas, and cellos are tuned in fifths. String basses, guitars, and bass guitars are tuned in fourths. Pianos are tuned with equal tuning.

Violin E      660      =  440*1.5
Violin A      440
Violin D      293      =  440/1.5
Violin G      196      =  440/1.52

Viola  A      440      Same as a violin A
Viola  D      293
Viola  G      196
Viola  C      130

Cello  A      220      One octave below a viola A
Cello  D      147
Cello  G       98
Cello  C       65

String bass G  98      =  55 * 1.52
String bass D  73      =  55 * 1.5
String bass A  55      3 octaves below a violin A
String bass E  41      =  55 / 1.5

Guitar E      326
Guitar B      244
Guitar G      196
Guitar D      147
Guitar A      110      2 octaves below a violin A
Guitar E       82
When an orchestra tunes, the concertmaster plays an A and then everyone tunes their A strings. Then the other strings are tuned in fifths starting from the A.

A bass guitar is tuned like a string bass.

The viola is the largest instrument for which one can comfortably play an octave, for example by playing a D on the C-string with the first finger and a D on the G-string with the fourth finger. Cellists have to shift to reach the D on the G-string.

According to legend Bach used a supersized viola, the "Viola Pomposa"

Low note

Singers typically have a range of 2 octaves. The low note for each instrument is:

    Strings   Winds      Brass      Voice

D             Piccolo
C             Flute                 Soprano
Bb            Oboe
G   Violin
F#                       Trumpet    Alto
E   Guitar    Clarinet
C   Viola                           Tenor
G                                   Baritone
F#                       Horn
E                        Trombone   Bass
C   Cello
Bb            Bassoon
E   Bass
D                      Tuba

Treble clef:  Violin, flute, oboe, clarinet, saxophone, trumpet, French horn, guitar,
              soprano voice, alto voice, tenor voice.
Alto clef:    Viola
Base clef:    Cello, bass, bass guitar, bassoon, trombone, tuba, timpani,
              baritone voice, bass voice
String basses and bass guitars have the same string tuning.

For guitars, tenors, basses, and bass guitars, the tuning is an octave lower than written.


Viola d'amore

The viola d'amore has 7 playing strings and 6 resonance strings.

Instruments of Indian classical music


A sitar has 6 or 7 playing strings and 11 or more sympathetic strings.

There is no standard tuning for sitar strings. An example tuning is to set the playing strings to {C, C, G, C, G, C, F} and the sympathetic strings to {C, B, A, G, F, E, E, D, C, B, C}

The fret positions can be tuned.

The bridge is curved so that the contact point between the string and the bridge is not sharp, which has the effect of transferring energy between the string modes.

Sarod and Sitar




The surbahar is typically tuned 2 to 5 whole steps below the sitar.

The tanpura does not play melody but rather supports and sustains the melody of another instrument or singer by providing a continuous harmonic drone.

Electric sitar

Guitar frets

Guitars frets are set by equal tuning.

L   =  Length of an open A-string
    =  .65 meters
T   =  Wave period
F   =  Frequency of the A-string
    =  220 Hertz
V   =  Speed of a wave on the A-string
    =  2 L F
    =  2 * .65 * 220
    =  286 meters/second
I   =  Index of a fret
    =  1 for B flat
    =  2 for B
    =  3 for C, etc.
f   =  Frequency of note I
    =  F * 2^(I/12)
X   =  Distance from the bridge to fret I
    =  V / (2 f)
    =  V / (2 F) * 2^(-I/12)
    =  L * 2^(-I/12)

 I  Note   X     L-X

 0   A    .650   .0
 1   Bb   .614   .036
 2   B    .579   .071
 3   C    .547   .103
 4   C#   .516   .134
 5   D    .487   .163
 6   Eb   .460   .190
 7   E    .434   .216
 8   F    .409   .241
 9   F#   .386   .264
10   G    .365   .285
11   Ab   .344   .306
12   A    .325   .325

Flexibility of just tuning

The frequency of a note depends on context. Suppose a set of viola strings is tuned in fifths so that the frequencies are

G  =  1
D  = 3/2
A  = 9/4
The G-string has been normalized to have a frequency of 1. There are several possibilities for assigning the pitch of the "E" on the D-string.

If the note "E" is chosen to resonate with the G-string its frequency is

E  =  5/3  =  1.6666
If the note "E" is chosen to resonate with the "A-string" then it is placed a perfect fourth below the A.
E  =  (9/4) / (4/3)  =  27/16  =  1.688
If the note "E" is played with equal tuning with the G-string as the tonic,
E  =  2^(9/12)  =  1.682
All three values for the E are different. Musicians have to develop a sensitivity for this.

Indian tuning

Red:    Equal tuning
Green:  Just tuning
Orange: Pythagorean tuning
Indian music has two separate tones for each half step, one from just tuning and the other from Pythagorean tuning. For the tonic and the fifth these tones are the same for both tunings. There are 22 tones in total.


Major and minor modes

The notes in an A-minor mode are

A    Octave
E    Perfect fifth
D    Perfect fourth
A    Tonic

There is a half step between the B and C and another half step between the E and F. All other intervals are whole steps.

If the notes of the minor scale are arranged depicting the whole and half steps then it looks like:

* oo o oo o * oo o oo o * oo o oo o * oo o oo o *          (Minor scale)


A "*" indicates the tonic and an "o" denotes a note in the scale. Each successive "*" denotes an octave. Four octaves are depicted.

The minor mode has the following properties:

There are no instances of 2 half-steps in a row.
Each half step is at least 2 whole steps from another half step.
There are no gaps larger than a whole step.
There are 8 notes spanning the octave.

A major scale has the same properties. The notes in a major scale look like:

* o oo o o o* o oo o o o* o oo o o o* o oo o o o*       (4 octaves of a major scale)

    #    # #

In an A-minor scale there are no flats or sharps. In an A-major scale the sharped notes are C#, F#, and G#.
Diatonic scales

A set of 7 diatonic scales (or "modes") follow from a compact and natural set of definitions. k

A diatonic scale consists of a set of notes such that:
(*) The tonic and octave are both included
(*) There are 8 notes including the tonic and octave
(*) Steps larger than a whole step are forbidden
(*) There must be at least 2 whole steps separating each half step,
       including octave periodicity

This implies:
The scale has 2 half steps and 5 whole steps.
The half steps are separated by 2 whole steps in one direction and 3 whole steps
in the other direction.
There is exactly 1 tritone.
There are 7 modes that satisfy the definition.

The first seven modes in this figure are the diatonic modes. The upper staff is a standard treble clef and the bottom note in each mode is an "A". The Aeolian mode corresponds to the minor mode and the Ionian mode corresponds to the major mode. In addition to the major and minor modes there are 5 additional modes.

The modes are ordered so that they grow progressively "sharper" as you move to the right.

Each mode differs by exactly one note from its adjacent modes.

The lower staff is a "geometric clef" where the vertical position of each note corresponds to its pitch. The bottom line is the tonic, the top line is the octave, and the middle line is the tritone. Adjacent lines are separated by a whole step. This representation is designed to visually bring out the pitch of each note. In the leftward diatonic modes the notes are shifted toward the tonic and the rightward diatonic modes the notes are shifted toward the octave.

Equivalently, we can define an infinite sequence of whole and half steps where the notes look like

o o oo o o oo o oo o o oo o oo o o oo

There are 7 unique choices of tonic which correspond to the 7 diatonic modes.

o o oo o o oo o oo o o oo o oo o o oo

I = Ionian D = Dorian P = Phrygian L = Lydian M = Mixolydian A = Aeolian l = Locrian I = Ionian

This can be thought of as a "sequence of tritone avoidance" since it contains only 1 instance of 3 consecutive whole steps.

The tritone defines a natural sequence for the modes that we'll call the "Diatonic sequence". When the modes are arranged this way, adjacent modes differ by exactly one pitch and distant modes can be continuously connected by the tritone sequence.

In the following table an orange dot indicates a note involved in a tritone.

The Dorian mode is symmetric under inversion.

The Mixolydian, Dorian, and Aeolian modes are at the center because their fourths and fifths are not part of a tritone. The Lydian and Locrian modes are at the edges because they are missing either a major fourth or a major fifth, the two most resonant notes with the tonic.

Inverted Lydian     = Locrian
Inverted Ionian     = Phrygian
Inverted Mixolydian = Aeolean
If an instrument is tuned in fifths there is a symmetry between modes and strings. Transposing up one string corresponds to transposing up one mode and transposing down one string corresponds to transposing down one mode.

For an instrument tuned in fourths, transposing up one string corresponds to transposing down one mode.

Melodic modes

In the diatonic modes there must be at least 2 whole steps separating each half step. If we relax this condition and allow half steps to be separated by only one whole step then another set of modes appears with the sequence:

o o o oo oo o o o oo oo o o o oo oo o o o oo oo o

This sequence has 7 unique choices of tonic hence there are 7 modes, the "melodic modes". They contain 2 tritones whereas the diatonic modes contain 1.

The melodic modes can be arranged into a natural sequence that parallels the diatonic modes. The melodic modes are depicted in the above figure, which is equivalent to the table below.

Notes           Mode              Sharpness

o o o oo o oo   Lydian                3
o o oo o o oo   Ionian                2       Major mode
o o oo o oo o   Mixolydian            1
o oo o o oo o   Dorian                0
o oo o oo o o   Aeolian              -1       Minor mode
oo o o oo o o   Phrygian             -2
oo o oo o o o   Locrian              -3

o o o o oo oo   Lydian sharp 5        4
o o o oo oo o   Lydian/Mixolydian     2
o oo o o o oo   Melodic minor         1
o o oo oo o o   Mixolydian/Aeolian    0
oo o o o oo o   Dorian/Phrygian      -1
o oo oo o o o   Aeolian/Locrian      -2
oo oo o o o o   Locrian flat 4       -4

"Sharpness" reflects whether the notes are stacked toward the octave (lots of sharps, or positive sharpness) or toward the tonic (lots of flats, or negative sharpness).

If the notes in a mode are assigned integers I where I=0 corresponds to the tonic and I=12 corresponds to the octave, the sharpness is

Sharpness  =  Sum over all notes of (I - 6)

If you start with a diatonic mode and move one of the tritone pitches by a half step then you get an adjacent diatonic mode. If you change a non-tritone pitch by a half step then you get a melodic mode.

To move from a melodic mode to an adjacent melodic mode you need to change 2 notes.

The melodic modes have a natural ordering that parallels the diatonic modes. The melodic modes can be thought of as alternate pathways for shifting between diatonic modes.

This figure shows the connectivity between modes. White lines connect modes that are one note apart. If you change one note in a mode then you change the sharpness, which is why there are no horizontal lines in the figure.

Relative major and minor keys

Top = C-major.   Bottom = A-minor.   Lines indicate shared notes.

The keys of A-minor and C-major share the same notes. A-minor is the "relative minor" of C-major and C-major is the "relative major" of A-minor. These two keys also have no sharps or flats.

Circle of fifths

If you start from the key of "A" and transpose up a fifth then you are the key of "E", and transposing down a fifth puts you in the key of "D". The interval of a "fifth" forms a sequence which repeats itself after 12 intervals. The following sequence starts at the bottom at "A" and rises in fifths until at the top it it returns to "A".

B flat
E flat
A flat

This is the circle of fifths expressed as key signatures on a treble clef. Major keys are in red capital letters and minor keys are in green lower-case letters. The keys of A-minor and C-major are at the top of the circle because they have no sharps or flats. At the bottom of the circle, E flat is equivalent to D sharp.

Plane of keys and modes

The mode "A minor" has tonic "A" and sharpness "-1".

Raising the tonic by a fifth to "E" while keeping the notes unchanged produces a mode with sharpness "-2".

Lowering the tonic by a fifth to "D" while keeping the notes unchanged produces a mode with sharpness "0".

In general, transposing up a fifth is equivalent to increasing the sharpness by 1 and transposing down a fifth is equivalent to decreasing the sharpenss by 1. The following figure expresses this equivalence.

Each row corresponds to a choice of tonic and is labeled with white letters. Going up by 1 row corresponds to raising the tonic by a fifth. The top row is identical to the bottom row.

Each tonic has 7 diatonic modes arranged horizontally by sharpness. Modes with sharpness "-3" are red, modes with sharpness "-2" are orange, etc. Minor modes are yellow and major modes are blue.

"Sharps" indicates the numbers of sharps that are written on the clef (negative sharps corresponds to flats). Every mode in the same column has the same number of sharps on the clef.

This figure contains all possible diatonic modes with all possible tonics.

All modes in the same column have the same notes.

Transposing up a fifth corresponds to moving up one dot and transposing down a fifth corresponds to moving down one dot.

Increasing the sharpness by 1 corresponds to moving right one dot and decreasing the sharpness by 1 corresponds to moving left one dot.

Violins are tuned in fifths and can conveniently transpose in fifths. Guitars are tuned and fourths and can conveniently transpose in fourths. Transposing up a fifth corresponds to transposing down a fourth.

If you start from a minor mode and increase the tonic by 3 half steps then you arrive at a major mode with the same notes. For example, C-major is the "relative major" of A-minor and A-minor is the "relative minor" of C-major.


Chopin's "24 Preludes, Op. 28" for piano covers all 12 major and minor keys by circumnavigating the circle of fifths. The keys are ordered as

A-minor   (same notes as C-major)
E-minor   (same notes as G-major)
B-minor   (same notes as D-major)

Path of keys in Chopin's 24 Preludes, Op. 28

Core keys

The most commonly used keys are indicated by dots with white edges. They tend to cluster vertically around the strings of a violin because they are the most convenient for a violin to play. This forms a sweet spot in the circle of fifths.

The clef is designed so that the core keys have few sharps or flats. Keys distant from the core keys have lots of sharps or flats.

The core keys are connected harmonically because they have many notes in common and because they are nearby in terms of fifths. It would be awkward to use a major key with 4 flats or a minor key with 4 sharps because these are far from the center from the cluster.

Keys in the Bach sonatas for violin:

Keys in the Vivaldi "L'Estro Armonico concerti for violin"
D-major   (occurs twice)
A-minor   (occurs twice)

Circle of tritones

To be a diatonic mode the mode must contain the tonic. If we relax this condition then 5 new modes appear and they can be ordered by a tritone sequence. Each mode has 1 tritone which is denoted by an orange dot.

The modes in the center are the diatonic modes and the modes at the edges are new modes that don't contain the tonic. Mode "+6" is equivalent to mode "-6".

Tritone torus

The circle of fifths and the circle of tritones form a torus.

The top row is equivalent to the bottom row and the left edge is equivalent to the right edge.

The circle of fifths is in the vertical direction and the circle of tritones is in the horizontal direction.

The following are examples of toruses. A torus can be created by connecting the opposite edges of a chessboard


Two surfaces are topologically equivalent if they can be connected by a continuous deformation. For example, a coffee mug is topologically equivalent to a torus.

Examples of surfaces that are topologically inequivalent.

Mobius strip
Double torus
Triple torus
Knotted torus


North Indian raga

North Indian raga on a geometric clef

Define "gap size" as being 1 for a half step, 2 for a whole step, etc.

Let a "doublet" be a set of 2 notes separated by a half step and let a "triplet" be a sequence of three notes all separated by half steps.

Suppose a pitch set has 8 notes, including the tonic. The pitch set will have 1, 2, or 3 tritones. Most have more than 1.

If there are no gaps larger than 2, less than 4 doublets, and no triplets, then the possible sequences are

oo oo o oo  o
oo oo  oo o o
oo oo o oo  o
oo oo  oo o o
oo o oo  oo o
Each sequences has 3 doublets and 2 tritones and each is asymmetric.

If triplets are allowed then the following sequence appears, which has 1 tritone and is asymmetric.

ooo oo o o  o
There are 7 choices of tonic and so this sequence generates 7 scales. It is asymmetric and so its inversion also generates 7 scales. The inversion: o o o oo ooo The Indian raga contain all the diatonic and melodic modes except the Locrian and Locrian-flat-4 modes, the modes with the most flats.


If a wave is linear then it propagates without distortion.

Wave interference

If a wave is linear then waves add linearly and oppositely-traveling waves pass through each other without distortion.

If two waves are added they can interfere constructively or destructively, depending on the phase between them.

Two speakers

If a speaker system has 2 speakers you can easily sense the interference by moving around the room. There will be loud spots and quiet spots.

The more speakers, the less noticeable the interference.

Noise-cancelling headphones use the speakers to generate sound that cancels incoming sound.

Online tone generator

Standing waves

Two waves traveling in opposite directions create a standing wave.

Waves on a string simulation at


Whan a wave on a string encounters an endpoint it reflects with the waveform preserved and the amplitude reversed.

Overtones of a string

Standing waves on a string
Standing waves on a string
Notes in the overtone series

Notes in the overtone series

When an string is played it creates a set of standing waves.

L  =  Length of a string
V  =  Speed of a wave on the string
N  =  An integer in the set {1, 2, 3, 4, ...}
W  =  Wavelength of an overtone
   =  2 L / N
F  =  Frequency of the overtone
   =  V/W
   =  V N / (2L)

N = 1  corresponds to the fundamental tone
N = 2  is one octave above the fundamental
N = 3  is one octave plus one fifth above the fundamental.
Audio: overtones

For example, the overtones of an A-string with a frequency of 440 Hertz are

Overtone  Frequency   Note

   1         440       A
   2         880       A
   3        1320       E
   4        1760       A
   5        2200       C#
   6        2640       E
   7        3080       G
   8        3520       A

Wikipedia: Overtones

Overtone simulation at

Overtones of a half-open pipe

Overtones of a half-open pipe
Airflow for the fundamental mode

In the left frame the pipe is open at the left and closed at the right. In the right frame the pipe is reversed, with the left end closed and the right end open. Both are "half-open pipes".

An oboe and a clarinet are half-open pipes.

L  =  Length of the pipe
   ~  .6 meters for an oboe
V  =  Speed of sound
N  =  An odd integer having values of {1, 3, 5, 7, ...}
W  =  Wavelength of the overtone
   =  4 L / N
F  =  Frequency of the overtone
   =  V / W
   =  V N / (4L)

The overtones have N = {1, 3, 5, 7, etc}

A cantilever has the same overtones as a half-open pipe.

N=1 mode
N=3 mode

Overtones of an open pipe

Overtones of an open pipe
Airflow for the fundamental mode

A flute and a bassoon are pipes that are open at both ends and the overtones are plotted in the figure above. In this case the overtones have twice the frequency as those for a half-open pipe.

L  =  Length of the pipe
V  =  Speed of sound
N  =  An odd integer having values of {1, 3, 5, 7, ...}
W  =  Wavelength of the overtone
   =  2 L / N
F  =  Frequency of the overtone
   =  V / W
   =  V N / (2L)

Overtones of a pipe that is closed at both ends

Airflow for the fundamental mode
Airflow for the N=2 mode
A string is like a closed pipe

A string has the same overtones as a closed pipe.

A closed pipe doesn't produce much sound. There are no instruments that are closed pipes. A muted wind or bass instrument can be like a closed pipe.

Modes 1 through 5 for a closed pipe.

Mode 1
Mode 2
Mode 3
Mode 4
Mode 5

Overtones for various instruments


An instrument of length L has overtones with frequency

Frequency  =  Z * Wavespeed / (2 * Length)
Z corresponds to the white numbers in the figure above.

An oboe is a half-open pipe (open at one end), a flute is an open pipe (open at both ends), and a string behaves like a pipe that is closed at both ends.

If a violin, an oboe, and a flute are all playing a note with 440 Hertz then the overtones are

Violin      440, 2*440, 3*440, 4*440, ...
Oboe        440, 3*440, 5*440, 7*440, ...
Flute       440, 3*440, 5*440, 7*440, ...

Drum modes




The fundamental mode is at the upper left. The number underneath each mode is the frequency relative to the fundamental mode. The frequencies are not integer ratios.

In general, overtones of a 1D resonator are integer multiples of the fundamental frequency and overtones of a 2D resonator are not.

Wikipedia: Virations of a circular membrane

The Chladni experiment

Chladni's original experiment

In 1787 Chladni published observations of resonances of vibrating plates. He used a violin bow to generate a frequency tuned to a resonance of the plate and the sand collects wherever the vibration amplitude is zero.

Modes of a vibrating plate

Chladni modes of a guitar

Vocal modes

A "formant" is a vocal resonance. Vowels can be identified by their characteristic mode frequencies.

2D and 3D resonators

Standing waves on a string have the form

L  =  Length of string
N  =  An integer >= 1
X  =  Position along the string
H  =  Height of the standing wave as a function of X

H  =  sin(π N X / L)
Modes in 1D
Modes in 2D

Suppose a resonator has multiple dimensions. For example, a square is like a 2D string and a cube is like a 3D string. If a resonator consists of a cubical volume of air then the modes are

L  =  Side length of the cube
Nx =  An integer >= 1 representing the mode number in the X direction
Ny =  An integer >= 1 representing the mode number in the Y direction
Nz =  An integer >= 1 representing the mode number in the Z direction
N  =  SquareRoot(Nx2 + Ny2 + Nz2)
H  =  Height of the standing wave as a function of X, Y, and Z
V  =  Wave velocity
W  =  Wavelength of mode (Nx,Ny,Nz)

H  =  sin(π Nx X / L) * sin(π Ny Y / L) * sin(π Nz Z / L)
The frequency of a mode {Nx, Ny, Nz} is proportional to N.
F  =  N V / (2L)

For simplicity we set V/(2L) = 1 so that
F  =  N  =  SquareRoot(Nx2 + Ny2 + Nz2)
For example, the modes of a 1D string are
 Nx     N

 1      1
 2      2
 3      3
 4      4
...    ...

The modes of a 2D square are
  Nx  Ny               N

  1   1       SquareRoot( 2) = 1.41
  1   2       SquareRoot( 5) = 2.24
  2   1       SquareRoot( 5) = 2.24
  2   2       SquareRoot( 8) = 2.83
  1   3       SquareRoot(10) = 3.16
  3   1       SquareRoot(10) = 3.16
  2   3       SquareRoot(13) = 3.61
  3   2       SquareRoot(13) = 3.61
  1   4       SquareRoot(17) = 4.12
  4   1       SquareRoot(17) = 4.12
  3   3       SquareRoot(18) = 4.24
  2   4       SquareRoot(20) = 4.47
  4   2       SquareRoot(20) = 4.47
 ... ...              ...

Modes in 2D

Orange dots correspond to (Nx,Ny) pairs and the length of the red lines corresponds to N.

The modes of a 3D cube are

  Nx  Ny  Nz               N

  1   1   1     SquareRoot( 3) = 1.41
  1   1   2     SquareRoot( 6) = 2.45
  1   2   1     SquareRoot( 6) = 2.45
  2   1   1     SquareRoot( 6) = 2.45
  1   2   2     SquareRoot( 9) = 3.00
  2   1   2     SquareRoot( 9) = 3.00
  2   2   1     SquareRoot( 9) = 3.00
  1   1   3     SquareRoot(11) = 3.32
  1   3   1     SquareRoot(11) = 3.32
  3   1   1     SquareRoot(11) = 3.32
  2   2   2     SquareRoot(12) = 3.46
  1   2   3     SquareRoot(14) = 3.74
  1   3   2     SquareRoot(14) = 3.74
  2   1   3     SquareRoot(14) = 3.74
  2   3   1     SquareRoot(14) = 3.74
  3   1   2     SquareRoot(14) = 3.74
  3   2   1     SquareRoot(14) = 3.74
 ... ... ...            ...

These are the mode frequencies for various resonators, with the frequencies normalized so that the fundamental frequency is unity. The size of each dot is equal to the square root of the number of modes at that frequency.

"String", "square", and "cube" correspond to the resonators discussed above and "circle" and "sphere" are discussed below.

As the dimensionality increases the number of modes increases. Singing involves a 3D resonator, which is why there are so many vocal formants.

The 2D resonators (square and circle) have similar spectra and the 3D resonators (cube and sphere) have similar spectra.

If the dimensionality is larger than 1 then there can be multiple modes with the same frequency.

For large N we can approximate the number of modes as:

Dimension        Number of mode with N < M

    1                         M
    2            (1/4)   π    M2
    3            (1/8) (4π/3) M3

In 2D the number 1/4 represents a quadrant of the plane an in 3D the number 1/8 represents an octant of a volume.

Drum modes

A circular drum and a square drum have similar spectra. The fundamental mode of a drum is

Z  =  Membrane tension in Newtons/meter    =   2000 Newtons/meter for a typical typani
M  =  Membrane density in kg/meter2        =   .26  kg/meter2 for a typical tympani
D  =  Membrane diameter                    =   .6   meters for a typical tympani
F  =  Fundamental mode frequency           =   112  Hertz for a typical tympani
   =  .766 SquareRoot(Z/(MD))

Drum modes in order of increasing frequency are


The following python script calculates the mode frequencies of a drum, normalized so that the fundamental frequency is 1.

>>> from scipy.special import jn_zeros   # Compute the zeros of the Bessel function
>>> jn_zeros(0,4)/jn_zeros(0,1)          # Compute the first 4 monopole modes
>>> jn_zeros(1,4)/jn_zeros(0,1)          # Compute the first 4 dipole modes
>>> jn_zeros(2,4)/jn_zeros(0,1)          # Compute the first 4 quadrupole modes

The mode frequencies for a circle and sphere are plotted above.

Whispering gallery

The whispering gallery in St. Paul's Cathedral has the same modes as a circular drum.

Whispering gallery waves were discovered by Lord Rayleigh in 1878 while he was in St. Paul's Cathedral.

St. Paul's Cathedral
St. Paul's Cathedral
U.S. Capitol
A mode in a circular chamber
Grand Central Station

The interior of a football is a spherical resonator.

Ultraviolet catastrophy

These are the classical and quantum predictions for the radiation intensity at 2000 Kelvin. The divergence of the classical prediction at high frequency is the "Ultraviolet catastrophy". The problem is resolved by quantum mechanics. In classical mechanics the thermal energy is the same for each mode and in quantum mechanics the energy depends on frequency. This eliminates the divergence at high frequency.

The behavior of the classical blackbody spectrum as a function of frequency is analogous to the modes of the 3D resonator plotted above.

Thermodynamic equilibrium

For a system in thermodynamic equilibrium each degree of freedom has a mean energy of .5 K T, where K is Boltzmann's constant.

If the modes of a resonator are mechanically connected and if the resonator has infinite time to evolve then each mode will have the same mean energy. The larger the dimension, the more modes a resonator has and the more energy it can store. In 3D the number of modes can be quite large.

Normal modes

Overtones are ubiquitous in vibrating systems. They are usually referred to as "normal modes".

Guitar overtones

Guitar overtones in relation to the positions of the frets

Table of fret values for each overtone

Guitar tuning

You can increase the pitch by pulling the string sideways. This increases the string tension, which increases the wavespeed and hence the frequency.

If you are playing a note on a guitar using a fret, you can change the frequency of the note by bending the string behind the fret.

Tension  =  Tension of a string
D        =  Mass per meter of the string
V        =  Speed of a wave on the string
         =  (Tension/D)½
L        =  Length of the string
T        =  Wave period of a string (seconds)
         =  2 L / V
F        =  Frequency of a string
         =  1/T
         =  V / (2L)

Plucked string

The vibration of the string depends on where it is plucked. Plucking the string close to the bridge enhances the overtones relative to the fundamental frequency.

Plucked at the center of string
Plucked at the edge of the string

A bow produces a sequence of plucks at the fundamental frequency of the string.


As a sound waves travels back and forth along the clarinet it forces the reed to vibrate with the same frequency.

In a brass instrument your lips take the function of a reed.

Bernoulli principle

In the figure, as the flow constricts it speeds up and drops in pressure.

P  =  Pressure
V  =  Fluid velocity
H  =  Height
g  =  Gravity  =  9.8 meters/second2
D  =  Fluid density
The bernoulli principle was published in 1738. For a steady flow, the value of "B" is constant along the flow.
B  =  P  +  .5 D V2  +  D g H
If the flow speeds up the pressure goes down and vice versa.

A wing slows the air underneath it, inreasing the pressure and generating lift. In the right panel, air on the top of the wing is at increased speed and reduced pressure, causing condensation of water vapor.

Angle of attack
Lift as a function of angle of attack

Lift incrases with wing angle, unless the angle is large enough for the airflowto stall.

A turbofan compresses the incoming airflow so that it can be combusted with fuel.

In a reed instrument, a puff of air enters the pipe, which closes the reed because of the Bernoulli effect. A pressure pulse travels to the other and and back and when it returns it opens the reed, allowing another puff of air to enter the pipe and repeat the cycle.

Vocal chords

The vocal tract is around 17 cm long. For a half-open pipe this corresponds to a resonant frequency of

Resonant frequency  =  WaveSpeed / (4 * Length)
                    =  340 / (4*.17)
                    =  500 Hertz
One has little control over the length of the vocal pipe but one can change the shape, which is how vowels are formed. Each of the two vocal chords functions like a string under tension. Changes in muscle tension change the frequency of the vibration.

Male vocal chords tend to be longer than female vocal chords, giving males a lower pitch. Male vocal chords range from 1.75 to 2.5 cm and female vocal chords range from 1.25 to 1.75 cm.

When air passes through the vocal chords the Bernoulli effect closes them. Further air pressure reopens the vocal chords and the cycle repeats.

The airflow has a triangle-shaped waveform, which because of its sharp edges generates abundant overtones.

Waves: sine, square, triangle, sawtooth
Creating a triangle wave from harmonics
Creating a sawtooth wave from harmonics

Audio file: Creating a triangle wave by adding harmonics.

                     Lung pressure (Pascals)

Passive exhalation          100
Singing                    1000
Fortissimo singing         4000
Atmospheric pressure is 101000 Pascals.

For a lung volume of 2 liters, 4000 Pascals corresponds to an energy of 8 Joules.

Singers, wind, and brass musicians train to deliver a continuous stable exhalation. String musicians train locking their ribcage in preparation for delivering a sharp impulse.


A spectrum tells you the power that is present in each overtone.

The first row is the waveform, the second row is the waveform expanded in time, and the third row is the spectrum. The spectrum reveals the frequencies of the overtones. In the panel on the lower left the frequencies are 300, 600, 900, 1200, etc. In the panel on the lower right there are no overtones.

Spectrum of a violin G string

A quality instrument is rich in overtones.

A waveform can be represented as an amplitude as a function of time or as an amplitude as a function of frequency. A "Fourier transform" allows you to go back and forth between these representations. A "spectrum" tells you how much power is present at each frequency.

Fourier transform simulation at

Music analysis software such as "Audacity" can evaluate the spectrum.


Every instrument produces sound with a different character. The sound can be characterized either with the waveform or with the spectrum

In the following plots the white curve is the waveform and the orange dots are the spectrum.

Plucked violin



Nyquist frequency

Sampling a wave at the Nyquist frequency

Suppose a microphone samples a wave at fixed time intervals. The white curve is the wave and the orange dots are the microphone samplings.

F    =  Wave frequency
Fmic =  Sampling frequency of the microphone
Fny  =  Nyquist frequency
     =  Minimum frequency to detect a wave of frequency F
     =  2 F

In the above figure the sampling frequency is equal to the Nyquist frequency, or Fmic = 2 F. This is the minimum sampling frequency required to detect the wave.

This figure shows sampling for Fmic/F = {1, 2, 4, 8, 16}. In the left panel the wave and samplings are depicted and in the right panel only the samplings are depicted.

The top row corresponds to Fmic=F, and the wave cannot be detected at this sampling frequency.

The second row corresponds to Fmic=2F, which is the Nyquist frequency. This frequency is high enough to detect the wave but accuracy is poor.

For each successive row the value of Fmic/F is increased by a factor of 2. The larger the value of Fmic/F, the more accurately the wave can be detected.

Human hearing has a frequency limit of 20000 Hertz, which corresponds to a Nyquist frequency of 40000 Hertz. If you want to sample the highest frequencies accurately then you need a frequency of at least 80000 Hertz.

Overtones can generate high-frequency content in a recording, which is why the sampling frequency needs to be high.

Fourier transform

The Fourier transform takes A(T) as input and gives you the coefficients C(F) and S(F).

The "spectrum" gives you the energy as a function of frequency.

The largest useful frequency F in the Fourier transform is the Nyquist frequency.


The spectrum reveals the overtones of a pitch.

In the following plots the white curve is the waveform as a function of time and the orange dots are the spectrum as a function of frequency.

Sine wave
Musical pitch with overtones
Distorted sine wave

The sine wave has all its power at one frequency. A musical pitch is rich in overtones.

The distorted sine wave gains overtones at higher frequencies. Distortion always adds overtones.

The spectrum tells you how much energy is present at each frequency.

Smooth structure
Rugged structure

The smooth structure has power at low frequencies and the rugged structure has power at higher frequencies.

Triangle wave
Square wave

The spectrum tends to work well for smoothly-varying functions and it tends to work poorly for jagged functions. For jagged structure the overtones don't give you much information.

Gibbs phenomenon

If the function is smooth then the Fourier transform can be a useful representation of the function. If the function contains sharp jumps then the Fourier transform fails.

Suppose a shock wave passes by, which is a sharp jump in pressure. Such a wave looks like a "step function".

If you model a step function as a Fourier series the result is poor.

10 modes
50 modes
250 modes

This is the "Gibbs phenomenon". No matter how many Fourier modes you use the function always overshoots and oscillates.

Missing fundamental

The top panel shows a sound with a frequency of 100 Hertz and with all overtones present. In the bottom panel the 100 Hertz and 200 Hertz components have been subtracted from the sound, but the 100 Hertz periodicity is still evident in the waveform. Our ears can sense the fundamental frequency iven if the fundamental overtone is absent.

Graphic equalizer

A graphic equalizer allows you to amplify or suppress specific frequency bands.

A Fourier transform can function like a graphic equalizer. For example, transform the waveform A(T) to the C(F) and S(F) coefficients, change the coefficients according to your taste, and then transform back to the waveform A(T).


Wikipedia:     Harmonic oscillator     Q factor     Resonance     Resonance

Hooke's law for a spring

A force can stretch or compresses a spring.

A spring oscillates at a frequency determined by K and M.

Frequency = Squareroot(K/M) / (2 π)

T     =  Time
X     =  Displacement of the spring when a force is applied
K     =  Spring constant
M     =  Mass of the object attached to the spring
Force =  Force on the spring
      =  - K X      (Hooke's law)

Solving the differential equation:
Force  =  M * Acceleration

- K X  =  M * X''

This equation has the solution
X  =  sin(2 π F T)
F = SquareRoot(K/M) / (2 π)

Wikipedia: Hooke's law


Damped spring

Vibrations of a damped string

After a string is plucked the amplitude of the oscillations decreases with time. The larger the damping the faster the amplitude decays.

T    =  Time for one oscillation of the string
Tdamp=  Characteristic timescale for vibrations to damp
q    =  "Quality" parameter of the string
     =  Characteristic number of oscillations required for the string to damp
     =  Tdamp / T
In the above figure,
q = Tdamp / T = 4
The smaller the damping the larger the value of q. For most instruments, q > 100.

Damping of a string for various values of q

The above figure uses the equation for a damped vibrating string.

t    =  Time
X(t) =  Position of the string as a function of time
T    =  Time for the string to undergo one oscillation if there is no damping
q    =  Quality parameter, defined below
        Typically  q>>1
F    =  Frequency of the string if there is no damping
     =  1/T
Fd   =  Frequency of string oscillations if there is damping
     =  F Z
Z    =  [1 - 1/(4 π2 q2)]½
     ~  1  if  q>>1
A damped vibrating string follows a function of the form: (derived in the appendix)
X  =  exp(-t/(Tq)) * cos(Zt/T)
The consine part generates the oscillations and the exponential part reflects the decay of the amplitude as a function of time.

For large q, the oscillations have a timescale of T and the damping has a timescale of T*q. This can be used to measure the value of q.

q = (Timescale for damping)  /  (Time of one oscillation)
For example, you can record the waveform of a vibrating string and measure the oscillation period and the decay rate.

Wikipedia: Damping


If you shake a spring at the same frequency as the oscillation frequency then a large amplitude can result. Similarly, a swing can gain a large amplitude from small impulses if the impulses are timed with the swing period.

Suppose a violin A-string is tuned to 440 Hertz and a synthesizer produces a frequency that is close to 440 Hertz. If the synthesizer is close enough to 440 Hertz then the A-string rings, and if the synthesizer is far from 440 Hertz then the string doesn't ring.

This is a plot of the strength of the resonance as a function of the synthesizer frequency. The synthesizer frequency corresponds to the horizontal axis and the violin string has a frequency of 440 Hertz. The vertical axis corresponds to the strength of the vibration of the A-string.

A resonance has a characteristic width. The synthesizer frequency has to be within this width to excite the resonance. In the above plot the width of the resonance is around 3 Hertz.

F  =  Frequency of the resonator
f  =  Frequency of the synthesizer
Fw =  Characteristic frequency width for resonance

If  |f-F|  <  Fw          then the resonator vibrates
If  |f-F|  >  Fw          then the resonator doesn't vibrate
Resonance simulation at

Wind can make a string vibrate (The von Karman vortex).

Tacoma Narrows bridge collapse

The Tacoma Narrows bridge collapse was caused by wind exciting resonances in the bridge.

Resonant strength

The larger the value of q, the stronger the resonance. The following plot shows resonance curves for various values of q.

If q>>1 then

Amplitude of the resonance  =  Constant * q

You can break a wine glass by singing at the same pitch as the glass's resonanant frequency. The more "ringy" the glass the stronger the resonance and the easier it is to break.

Resonant width

The width of the resonance decreases with q. In the following plot the peak amplitude of the resonance curve has been set equal to 1 for each curve. As q increases the width of the resonance decreases.

T      =  Time for one oscillation of the string
Tdamp  =  Characteristic timescale for vibrations to damp
q      =  Characteristic number of oscillations required for the string to damp
       =  Td / T
F      =  Frequency of the resonator
       =  1/T
f      =  Frequency of the synthesizer
Fw     =  Characteristic frequency width for resonance  (derived in appendix)
       =  F / (2 π q)

If  |f-F|  <  Fw          then the resonator vibrates
If  |f-F|  >  Fw          then the resonator doesn't vibrate

If q>>1 then

Width of the resonance  =  F / (2 π q)
Overtones can also excite a resonance. For example, if you play an "A" on the G-string of a violin then the A-string vibrates. The open A-string is one octave above the "A" on the G-string and this is one of the overtones of the G-string.

The strings on an electric guitar are less damped than the strings on an acoustic guitar. An acoustic guitar loses energy as it generates sound while an electric guitar is designed to minimize damping. The resonances on an electric guitar are stronger than for an acoustic guitar.

Coupled oscillators

Oscillators that are mechanically connected can transfer energy back and forth between them.

String overtones

If you place your finger lightly on the string at the point of the green dot then you can chose which mode appears.

If you pluck close to the bridge then overtones are favored.

You can change the frequency of the fundamental mode. If you place two fingers on the string, one firmly at the left green dot and the other lightly at the right green dot, then you can excite the 4th mode. The left dot can be placed wherever you wish and then the position of the right dot is fixed by the choice of overtone.

Sympathetic string resonances (resonance tuning)

Resonances can be used to calibrate tuning. If you play a note on a violin then it can excite resonances on the open strings. The pitch of the note is chosen to maximize the resonances.

The strings on a violin, arranged from low to high frequency, are G, D, A, and E. The notes on the strings are:

Strings on a violin

The note at the left of a string correspond to an open string.

If you use your finger to play an "E" on the A-string it resonates with the open E-string because both notes have the same frequency.

In this figure the finger is places at the "E" on the A-string, which is marked by the green dot. The figure shows the fundamental modes that are excited on the E and A strings.

This is an example of a resonator and a driver. The open E-string is the resonator because it has a fixed pitch and the E on the A-string is the driver because its pitch can be changed by moving the finger. To calibrate your tuning you can vary the position of your finger and listen for the ring of the E-string and find the position that maximizes the ring.

This figure lists some possibilities for resonance tuning, where e ach set of 4 strings corresponds to one of the possibilities. The first set corresponds to the above example and the 2nd and 3rd sets are similar examples.

In the 4th set an A is played on the E string, which has twice the frequency of the open A-string. The A on the E string resonates with the second mode on the open A-string. The 5th and 6th sets are similar examples.

In the 7th set an E is played on the D-string, and the second mode of this note resonates with the open E string. The 8th set is a similar example.

The following table shows the notes on a violin, where notes that resonate with open strings are colored in green.

Violin Resonant notes are colored in green

Guitar resonant notes are colored in green

Differential equation for resonance
T     =  Time
X(T)  =  Vibration of a string as a function of time
X'    =  Time derivative of X
X''   =  Second time derivative of X
F     =  Frequency of the string
q     =  Characteristic number of oscillations for damping to quell the vibration
The differential equation for a harmonic oscillator is
X'' =  - 4 π2 F X
which is solved by
X  =  cos(2 π F T)
X oscillates with a frequency of F.


The differential equation for a harmonic oscillator with damping is

X'' =  - 4 π2 F X  - (2 F / Q) X'
which is solved by
X  =  exp(- F t / q) cos(2 π F Z t)
Z  =  [1 - 1/(4 π2 q2)]½

If the string is forced by a driver with frequency f then the string vibrates at this frequency. The amplitude for vibrations is

A  =  Amplitude for vibrations when the string is forced with frequency f
F  =  Resonant frequency of a string
Fw =  Characteristic width of the resonance

A-2  =  F4 / q2  +  π2 (f2 - F2)2
The amplitude reaches its peak when F=f. In this case,
A  =  q / F2
The amplitude of a resonance is proportional to q.


The width of the resonance occurs for a frequency f such that

F4 / q2  =  π2 (f2 - F2)2

Let f = F + Fw,    where |Fw| << F

f2 - F2  =  F2 + 2 F Fw + Fw2 - F2
           ~  2 F Fw
F4 / q2  ~  4 π2 F2 Fw2
Fw  ~  F / (2 π q)
The width of the resonance Fw is proportional to F and inversely proportional to q.

Precision for frequency and time

Uncertainty principle

Suppose you measure the frequency of a wave by counting the number of crests and dividing by the time.

T  =  Time over which the measurement is made
N  =  Number of crests occurring in a time T
F  =  N/T
dF =  Uncertainty in the frequency measurement
   =  1/T
Suppose the number of crests can only be measured with an uncertainty of +-1. The uncertainty in the frequency is dF = 1/T. The more time you have to observe a wave the more precisely you can measure the frequency.

The equation for the uncertainty in a frequency measurement is

dF T  >=  1

Quantum-mechanical uncertainty principle
h  =  Planck's constant
   =  6.62e-34 Joule seconds
Q  =  Particle momentum
W  =  Particle wavelength
   =  h / Q
F  =  Particle wave frequency
E  =  Particle energy
   =  h F
dE =  Uncertainty in the particle energy
dF =  Uncertainty in the particle frequency
dT =  Time interval
Uncertainty principle for particle energy:
dE dT >= h / (4π)

Using dE = h dF,

dF dT >= 1 / (4π)

Beat frequencies

If two notes are played simultaneously then the pitches of the notes can be calibrated by listening for beat frequencies.

T     =  Duration of a note
Fbeat =  Frequency resolution for just-intonation
      =  1/T

This is also the precision limit for measuring relative frequencies using just-intonation.

Precision for measuring frequency

F     =  Frequency of a note
T     =  Duration of a note in seconds
q     =  Quality parameter for the resonator
      =  Characteristic number of times a resonator oscillates before losing
         its energy to damping
Fhear =  Frequency width for human perception
      =  .006 F
      =  F / 170
Fres  =  Frequence width for a resonance at a frequency of F
      =  F / (2 π q)
Fbeat =  Frequency width for detecting beat frequencies
      =  1 / T
Fjust =  Frequency resolution for just-intonation
      =  Fbeat
      =  1 / T
Func  =  Frequency precision from the uncertainty principle
      =  Frequency precision for a computer tuner
      =  1 / T
The larger the value of F, the more difficult it is to play just-intonation.

For low frequencies your ear is more precise than just-intonation.

For high requencies your ear is less precise than just-intonation.

The characteristic frequency for which the ear is equally precise as just-intonation is F=170 Hertz.

If q is large then resonances are sharper than just-intonation. If q is small then just-intonation is sharper than resonances.

Margin for error on a violin string
X  =  Length of a violin string
   =  .32 meters
x  =  Length of the active part of the string between the finger and the bridge.
F  =  Frequency of the open string
   =  660 Hertz for an E string
f  =  Frequency of the note being played by the finger
I  =  Index of the note being played.
   =  0 for an open string
   =  1 for a half step
   =  12 for an octave
D  =  Distance between the peg end of the string and the finger
   =  X - x

f x = F X = Constant

f = F 2I/12
x = X 2-I/12
D = X (1 - 2-I/12)

If I=1,   D = 18 mm
If I=.1,  D = 1.8 mm
If F=660 and f=661, x/X = 660/661 D = .48 mm There is little margin for error on an E-string.

A viola C-string has a frequency of 130 Hertz.

If F=130 and f=131,
x/X =  130/131
D   =  2.9 mm

Note duration

The lower the frequency of the note, the longer it takes to sense its pitch.

F     =  Frequency of a note
T     =  Duration of a note
Fhear =  Frequency resolution for human hearing
      =  F/170
Func  =  Frequency resolution from the uncertainty principle
      =  1/T
Thear =  Duration of a note for which Fhear=Func
      =  170/F
Frequency resolution is limited by either by Fhear or by Func, whichever is larger.
If  (T < Thear)  then the precision is limited by the uncertainty principle.
If  (T > Thear)  then the precision is limited by the ear.
For a given frequency F, the values for Thear and Tres are (Tres is defined below)

     F     Tres   Thear

    55     .49     3.1
   110     .25     1.6
   220     .12      .77
   440     .061     .39
   880     .031     .19
  1760     .015     .10

For low-frequency notes it takes a long time for the pitch to develop.

Resonance time of the ear

The resonators in the ear have a characteristic quality parameter which can be estimated from the frequency resolution of the ear.

F     =  Frequency of a note
T     =  Duration of a note
q     =  Quality parameter for the resonators in the ear
Fhear =  Frequency resolution of the ear
      =  F / 170
Thear =  Characteristic timescale for the ear to sense pitch
      =  1 / Fhear
      =  170 / F
Fres  =  Frequency resolution of a resonator
      =  F / (2 π q)
Tres  =  Time for a resonator to activate
      =  q * (Duration of one cycle of the resonator)
      =  q / F
If we set Fhear=Fres then
F / 170  =  F / (2 π q)

q  =  170 / (2 π)
   =  27
The activation time for the resonators in the ear is
Tres  =  q / F
      = 27 / F

Thear / Tres = 2 π

If (T < Tres) then the ear resonators are not fully activated and the note sounds less loud than if it had been played longer.

If (T > Tres) then the ear resonators are fully activated.

Start time of a note

When you start playing a note it takes a minimum of one wave period for the tone to stabilize and it usualy takes longer. The lower the frequency of the note the greater the challenge in starting the note quickly.

F       =  Note frequency
Tstart  =  Minimum start time of a note
        =  1/F
Stabilizing the start of a note


Summary of timescales:

Tstart =  Minimum start time of a note
       =  1/F
Tres   =  Time for the resonators in the ear to activate
       =  27/F
Thear  =  Duration of a note for which Fhear=Func
       =  170/F

Between Tstart and Tres the organ of Corti is amplifying the resonances. After Tres the organ of Corti is narrowing the resonances to refine the frequency measurement.
Wave impedance

Because the diameter of a whip tapers gradually, wave energy can be transmitted from the handle to the tip. If the diameter were to change abruptly then wave energy would be reflected at the transition, such as in the animation below.

Wave reflection at a sudden change of impedance

The speed of a water wave slows down as it approaches shallow water, increasing the wave amplitude.

A bow is tapered from the tip to the frog to prevent abrupt reflections of wave energy. A tuba is flared at the end to smooth the transition from the air inside the instrument to the air outside.

Slow motion baseball pitch

In a baseball pitch, the motion starts from the feet and then progresses to the hips, the torso, the shoulder, the upper arm, the lower arm, the wrist, and then to the fingers. This maximizes the speed that can be delivered by the fingers.

Sound energy does not transmit well between air and water because of the abrupt change in density.

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