Section 1: Physics of music (this section)
Section 2: Materials and elasticity
Section 3: Anatomy
Section 4: Music performance
Section 5: Physics
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Units Conservation of momentum and energy Angular momentum
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.