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
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 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
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
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.
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:
A chromatic scale contains all 12 notes. A chromatic scale with a tonic of "C" looks like:
Wikipedia: Clefs Musical intervals Chromatic scale Major scale Minor scale Octave Perfect fifth Perfect fourth
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 1/2Because 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.
If we double both frequencies then it also sounds like an octave. The shape of the blue wave is preserved.
Color Frequency Wavelength Orange 440 Hertz 1/2 Red 880 Hertz 1/4When 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 = 2If 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,
F_{1} = 1 F_{2} = 2For a fifth (playing an A and an E),
F_{1} = 1 F_{2} = 3/2
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 integersthen 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 resonantThe 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.
If two notes are out of tune they produce dissonant beat frequencies.
Frequency of note #1 = F_{1} Frequency of note #2 = F_{2} Beat frequency = F_{b} = F_{2} - F_{1}For the beats to not be noticeable, F_{b} 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.
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.
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 = 2^{I/12}For the tonic,
F = 2^{0/12} = 1For the octave,
F = 2^{12/12} = 2The frequency ratio between two adjacent pitches is
Frequency ratio = 2^{(I+1)/12} / 2^{I/12} = 2^{1/12} = 1.059which is independent of I.
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 0In equal tuning, the frequency ratio of an interval is
Frequency ratio = 2^{(Index/12)} Where "Index" is an integerEqual 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.
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 developed. 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
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.
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 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 = 1.125In a 12-tone equal-tempered scale the frequency ratio of a whole step is
R = 2^{(2/12)} = 1.122which 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 2^{0/12} = 1 0 1 cent 2^{1/1200}= 1.0006 1 10 cents 2^{1/120} = 1.0058 10 Half step 2^{1/12} = 1.0595 100 Whole step 2^{2/12} = 1.1225 200 Fifth 2^{7/12} = 1.498 700 Octave 2^{12/12} = 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 = 2^{I/12} = 2^{C/1200} C = 1200 ln(F) / ln(2)If F has the form
F = 1 + Z where Z << 1then
C = 1200 ln(1+Z) / ln(2) ~ 1200 Z / ln(2) ~ 1731 ZFor example, the frequencies for a fifth are
Equal tuning: F_{e} = 2^{7/12} = 1.4983 Just tuning: F_{j} = 3/2 = 1.5000These frequencies have the ratio
F = F_{j} / F_{e} = 1.00113 Z = F - 1 = .00113 C = 2.0The frequencies for just and equal tuning differ by 2 cents.
The frequency ratio of a half step is
2^{1/12} = 1.059Human are capable of detecting a change in frequency of 1/10 of a half step, which corresponds to a frequency ratio of
2^{1/120} = 1.0056To 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 * 2^{I/12} f = Smallest frequency greater than 440 Hertz for which "f" sounds indistinguishable 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 ...
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.
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.
Hertz Violin E 660 = 440*1.5 Violin A 440 Violin D 293 = 440/1.5 Violin G 196 = 440/1.5^{2} 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 * 1.5^{2} 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 82When 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"
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 A G Violin F# Trumpet Alto E Guitar Clarinet D C Viola Tenor Bb A G Baritone F# Horn E Trombone Bass D C Cello Bb Bassoon A G F 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 voiceString basses and bass guitars have the same string tuning.
For guitars, tenors, basses, and bass guitars, the tuning is an octave lower
than written.
The viola d'amore has 7 playing strings and 6 resonance strings.
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.
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.
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
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 = 9/4The 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 = 1.6666If 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 = 1.688If the note "E" is played with equal tuning with the G-string as the tonic,
E = 2^(9/12) = 1.682All three values for the E are different. Musicians have to develop a sensitivity for this.
Red: Equal tuning Green: Just tuning Orange: Pythagorean tuningIndian 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.
The notes in an A-minor mode are
A Octave G F E Perfect fifth D Perfect fourth C B A TonicThere 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 BC D EF G A BC D EF G A BC D EF G A BC D EF G AA "*" 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) A B CD E F GA # # #In an A-minor scale there are no flats or sharps. In an A-major scale the sharped notes are C#, F#, and G#.
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 ooThere 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 D PL M A lII = 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.
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 = AeoleanIf 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.
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.
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.
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".
A D G C F B flat E flat A flat C# F# B E A
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.
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
C-major
A-minor (same notes as C-major)
G-major
E-minor (same notes as G-major)
D-major
B-minor (same notes as D-major)
etc.
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:
G-minor
B-minor
A-minor
D-minor
C-major
E-major
Keys in the Vivaldi "L'Estro Armonico concerti for violin"
D-major (occurs twice)
D-minor
G-minor
G-major
E-minor
E-major
A-minor (occurs twice)
A-major
F-major
B-minor
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".
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.
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 oEach 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 oThere 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.
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.
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.
Two waves traveling in opposite directions create a standing wave.
Waves on a string simulation at phet.colorado.edu
Whan a wave on a string encounters an endpoint it reflects with the waveform
preserved and the amplitude reversed.
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
Overtone simulation at phet.colorado.edu
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.
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)
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.
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, ...
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
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.
A "formant" is a vocal resonance. Vowels can be identified by their characteristic mode frequencies.
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)
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 N_{x} = An integer >= 1 representing the mode number in the X direction N_{y} = An integer >= 1 representing the mode number in the Y direction N_{z} = An integer >= 1 representing the mode number in the Z direction N = SquareRoot(N_{x}^{2} + N_{y}^{2} + N_{z}^{2}) H = Height of the standing wave as a function of X, Y, and Z V = Wave velocity W = Wavelength of mode (N_{x},N_{y},N_{z}) H = sin(π N_{x} X / L) * sin(π N_{y} Y / L) * sin(π N_{z} Z / L)The frequency of a mode {N_{x}, N_{y}, N_{z}} is proportional to N.
F = N V / (2L)For simplicity we set V/(2L) = 1 so that
F = N = SquareRoot(N_{x}^{2} + N_{y}^{2} + N_{z}^{2})For example, the modes of a 1D string are
N_{x} N 1 1 2 2 3 3 4 4 ... ...The modes of a 2D square are
N_{x} N_{y} 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 ... ... ...
Orange dots correspond to (N_{x},N_{y}) pairs and the length of the red lines corresponds to N.
The modes of a 3D cube are
N_{x} N_{y} N_{z} 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 M^{2} 3 (1/8) (4π/3) M^{3}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.
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/meter^{2} = .26 kg/meter^{2} 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.
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.
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.
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.
Overtones are ubiquitous in vibrating systems. They are usually referred to as "normal modes".
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)
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.
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.
P = Pressure V = Fluid velocity H = Height g = Gravity = 9.8 meters/second^{2} D = Fluid densityThe bernoulli principle was published in 1738. For a steady flow, the value of "B" is constant along the flow.
B = P + .5 D V^{2} + D g HIf 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.
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.
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 HertzOne 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.
Audio file: Creating a triangle wave by
adding harmonics.
Lung pressure (Pascals) Passive exhalation 100 Singing 1000 Fortissimo singing 4000Atmospheric 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.
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 phet.colorado.edu
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.
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 FIn 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.
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.
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.
The smooth structure has power at low frequencies and the rugged structure has power at higher frequencies.
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.
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.
This is the "Gibbs phenomenon". No matter how many Fourier modes you use the function always overshoots and oscillates.
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.
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
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)where
F = SquareRoot(K/M) / (2 π)Wikipedia: Hooke's law
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 T_{damp}= Characteristic timescale for vibrations to damp q = "Quality" parameter of the string = Characteristic number of oscillations required for the string to damp = T_{damp} / TIn the above figure,
q = T_{damp} / T = 4The smaller the damping the larger the value of q. For most instruments, q > 100.
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 π^{2} q^{2})]^{½} ~ 1 if q>>1A 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.
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 F_{w} = Characteristic frequency width for resonance If |f-F| < F_{w} then the resonator vibrates If |f-F| > F_{w} then the resonator doesn't vibrateResonance simulation at phet.colorado.edu
Wind can make a string vibrate (The von Karman vortex).
The Tacoma Narrows bridge collapse was caused by wind exciting resonances in the bridge.
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.
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 T_{damp} = Characteristic timescale for vibrations to damp q = Characteristic number of oscillations required for the string to damp = T_{d} / T F = Frequency of the resonator = 1/T f = Frequency of the synthesizer F_{w} = Characteristic frequency width for resonance (derived in appendix) = F / (2 π q) If |f-F| < F_{w} then the resonator vibrates If |f-F| > F_{w} then the resonator doesn't vibrateIf 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.
Oscillators that are mechanically connected can transfer energy back and forth between them.
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.
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:
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.
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.
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 vibrationThe differential equation for a harmonic oscillator is
X'' = - 4 π^{2} F Xwhich 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 π^{2} F X - (2 F / Q) X'which is solved by
X = exp(- F t / q) cos(2 π F Z t)where
Z = [1 - 1/(4 π^{2} q^{2})]^{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} = F^{4} / q^{2} + π^{2} (f^{2} - F^{2})^{2}The amplitude reaches its peak when F=f. In this case,
A = q / F^{2}The amplitude of a resonance is proportional to q.
The width of the resonance occurs for a frequency f such that
F^{4} / q^{2} = π^{2} (f^{2} - F^{2})^{2} Let f = F + F_{w}, where |F_{w}| << F f^{2} - F^{2} = F^{2} + 2 F F_{w} + F_{w}^{2} - F^{2} ~ 2 F F_{w}Hence
F^{4} / q^{2} ~ 4 π^{2} F^{2} F_{w}^{2} F_{w} ~ F / (2 π q)The width of the resonance Fw is proportional to F and inversely proportional to q.
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/TSuppose 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
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 intervalUncertainty principle for particle energy:
dE dT >= h / (4π) Using dE = h dF, dF dT >= 1 / (4π)
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.
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 F_{hear} = Frequency width for human perception = .006 F = F / 170 F_{res} = Frequence width for a resonance at a frequency of F = F / (2 π q) F_{beat} = Frequency width for detecting beat frequencies = 1 / T F_{just} = Frequency resolution for just-intonation = F_{beat} = 1 / T F_{unc} = Frequency precision from the uncertainty principle = Frequency precision for a computer tuner = 1 / TThe 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.
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 2^{I/12} x = X 2^{-I/12} D = X (1 - 2^{-I/12}) If I=1, D = 18 mm If I=.1, D = 1.8 mmIf 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
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 F_{hear} = Frequency resolution for human hearing = F/170 F_{unc} = Frequency resolution from the uncertainty principle = 1/T T_{hear} = Duration of a note for which F_{hear}=F_{unc} = 170/FFrequency resolution is limited by either by Fhear or by Func, whichever is larger.
If (T < T_{hear}) then the precision is limited by the uncertainty principle. If (T > T_{hear}) 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 .10For low-frequency notes it takes a long time for the pitch to develop.
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 F_{hear} = Frequency resolution of the ear = F / 170 T_{hear} = Characteristic timescale for the ear to sense pitch = 1 / F_{hear} = 170 / F F_{res} = Frequency resolution of a resonator = F / (2 π q) T_{res} = Time for a resonator to activate = q * (Duration of one cycle of the resonator) = q / FIf we set F_{hear}=F_{res} then
F / 170 = F / (2 π q) q = 170 / (2 π) = 27The activation time for the resonators in the ear is
T_{res} = q / F = 27 / F T_{hear} / T_{res} = 2 πIf (T < T_{res}) then the ear resonators are not fully activated and the note sounds less loud than if it had been played longer.
If (T > T_{res}) then the ear resonators are fully activated.
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 T_{start} = Minimum start time of a note = 1/FStabilizing the start of a note
Summary of timescales:
T_{start} = Minimum start time of a note = 1/F T_{res} = Time for the resonators in the ear to activate = 27/F T_{hear} = Duration of a note for which Fhear=Func = 170/FBetween T_{start} and T_{res} the organ of Corti is amplifying the resonances. After Tres the organ of Corti is narrowing the resonances to refine the frequency measurement.
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.
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.
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.