Outside inspiration
Other concepts for inspiration
– mean just wanted to get basic idea and check out how it can be use in sound design/ music.
…
Overview
1) Bifurcation (Magnetism) – Topology
– Slope fields
2) Orbits – 2a – Keplerian motion
3) Structural mechanic – Euler Bernoulli beam theory, ..
5) Quantum mechanic – just scratching the surface with topic like: simple harmonic oscillator, kinetic/ potential energy..
6-7) FFT and Wavelets.
8) Spherical/ polar coordinates system (instead of Cartesian) – Spherical harmonics..
1) Bifurcation theory, Slope fields and heat decipation
Bifurcation theory
Term bifurcation was coined by Henri Poincaré in 1885.
It is changing of topological structure of group of curves.
Changing topology is generally stretching, twisting, crumpling, and bending without making hole in ..
Slope Fields
Is graphical representation of the solutions to a first-order differential equation.
The slope field of dy/dx=x2-x-2, with the blue, red, and turquoise lines being (x3/3)-(x2/2)-2x+4, (x3/3)-(x2/2)-2x, and (x3/3)-(x2/2)-2x-4, respectively.
Heat equations
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2) Planets, galaxy disk and super-galactic plane orbits
2a) Kepler planets movement
-..
3) Structural mechanics
3a) Euler–Bernoulli beam theory
..
Cantilevered beam
5) Quantum Harmonic Oscillator
Just 4 inspiration…
It has smth to do w/ Schrödinger equation of quantum mechanics –
– 8 lines represent quantized energy fields – as discovered by Max Planck energy levels can have in (quantum physics/ quantum osc.) only discrete levels: can not be changed continuously, but only in so called quantum jumps. Also energy can not be 0, but there is some min state…
ψ (psi – small italic 𝜓) – is a wave function – each field has additional half of waveform.
5a) Standing wave
Static wave can be seen in right pic (in quantum harmonic osc. – G)
Multivariate B-Spline Curves
6) Fourier analysis (Transform, Series, Descriptors)
A) FOURIER ANALYSIS GENERAL CONCEPT
FA is based on calculus and derivates as signal processing tool – btw synthesizer actually mean producing synthetics signal.
For just curves it has IMHO little direct use since its generally standard useful frequency range starts on about 20 Hz (audio signal).
Its Fast algorithm FFT has IMHO also limited use. It rather serves for science task like for X-ray satellites used in ALMA – Hi-Tec X-rays dishes in Atacama desert. X-ray has freq up to 30 PHz (Peta Hz – 10^15 ) and it could not be processed like visible electromagnetic radiation we called light, since there is no computer that would handle it. It can be done only thru FFT.
FFT (same like wavelets) are also used in compressing video footages…
FA is basically frequency analysis – it view signal thru its components. It was discovered by Jospeh Fourier in 1822, but since it is processing intensive, it has its first real use with first PCs – so roughly in 60s numerous “re-newed” algorithms emerged, that are still used as core technique for signal processing in Hi-tech..
This is summary of vid on ytb: https://youtu.be/spUNpyF58BY (“But what is the Fourier Transform? A visual introduction”)
– tone is frequency in pressure of the air (transmitted from speaker to ear) – mean freq is also negative (like from 1pi to 2pi).
Sound signal is adding of this sound pressures. This is quite hard to process, specially when two inverse sounds cancel each other – you can easily put together inverse saw waves and get silence, but used this technique for tweaking real audio signal is generally (4Q 2022) almost impossible.
L: Peaks add up → high pressure.
R: Two inverse max levels canceling out signal.
FA can decompose signal into its pure tone (sin waves) components in 3 views: 1) Signal identify by Intensity (x)/ Time (y), 2) winding shown on circle and 3) Frequency with center of mass (x)/ Winding frequency (y).
Example of the concept idea: 2 seconds interval of a 4 and half seconds long (Time) signal with intensity 3 BPS (2 sec → 6 peaks (see bellow)) is wrapped into a circle: one can see, that far from center mean high pressure and center is 0 pressure.
That mean, that wrapping has 0,5 cycles/ sec – called winding frequency.
Left pic: displayed are 2 cycles (“bars”) – 6 beats in 1 second (that mean already mentioned 0,5 cycles/ sec winding freq).
If freq of signal match freq of a winding – graph will get special characteristic – see pic on right -: Center of mass will move toward edge of the graph.
Spike of center of mas can also be adjusted by shifting intensity – than it is also clear, that first spike in start is created just by shift (left pic)
If intensity would be 2 BPS, same would happen with winding freq 2 (right pic)
Sound with two tones would work the same:
This would almost works also inversely – so you can basically identify certain unwanted tone in the signal and than filter it out (add particular inverse wave)..
Almost because Center of mas is complex value- it is 3D and rotate around x and y axis and so combines real and imaginary numbers (called complex numbers) based on Euler´s formula. To add winding into complex plane equation e^2piit (2nd pic) t is multiply by frequency (3rd pic).
Img. and real numbers would have similar trajectory for center of mass (4th pic).
While complex number traditionaly rotate counter clockwise, in FFT standard is CW – so u add “-” into equation: e^-2pift. Than to get it winding, you multiply e by g(t) – 1st pic.
Write it in fnct, you can average points (2nd pic) or even better write limits 3rd pic, or as expected – write integral (4th pic):
But it can be simpler)) – time has actually repeating content: so u don´t have devide by time interval (t2 – t1).
While in sound design integral usually have interval, typical FFT consider all possible values and goes from minus infinity to infinity and that when troubles with Uncertainty principle come into play.
FA in sound
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Signal approach
– as mentioned, FA can be most directly used in audio-rate
Piano
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Brass
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Trumpet
Trumpet (gsu.edu)
– trumpet can follow harmonic series up to tenth partial, but generally brass instruments produce HS with offset fundamental called pedal tone (according to organ).
–Speech
–Speech
String
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Violin
Double stop or Tartini tones
– as will be mention in Etudy when two tone are played, overall frequency will be difference between them. This phenomenon will be more distinct, if two tones share same partial(s) and hence will be more hearable in (well) tempered (not equal like are now (Q1 2023) in basically all DAWs) scales.
Major third Two octaves below the low note
Minor third One octave + one fifth below the high note
Perfect fourths Two octaves below the high note
Perfect fifths One octave below the low note
Major sixth One fifth below the low note
Minor sixth Major sixth below the low note
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b) Fast Fourier Transform
– FFT is names in honor to Joseph Fourier – it is algorithm of Fourier analysis, published in 1965 Cooley-Tukey – it first use was supposed to detect underground nuclear explosions tests, since it could decompose signal of the explosion and hence determine how strong and deep underground the explosion was.
A2 – Rieman Zeta function
c) Fourier series
One of the limitation of FFT is that is not best at making hard breaks – in paradox, this is from where most of musicians FFT know – creating square wave from sins of even harmonics..
Convergenece form
– ..
Amplitude-phase form
Sine-cosine form
Exponential form
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d) Fourier descriptor
some interesting shape was presented on: Fourier Art – inspired by paper: Zahn, Charles T.; Roskies, Ralph Z., “Fourier Descriptors for Plane Closed Curves” Computers, IEEE Transactions on , vol.C-21, no.3, pp.269,281, March 1972
7) Wavelets/ wavelet transform
Wavelets use similar system as FFT, but
It is also based on using splines..
sss
8) Spherical harmonics – using Legendre polynomials and Laplace transform
– spherical concepts brings Fourier analysis one step further – as shown in Orthogonal fncts, add 90° axis is equivalent adding one dimension.. and it using this propagation on otherwise Fourier transform same-like principles ..
Intro
– spherical coordinate system is 3D space defined by: radial distance (r), polar angle (θ – theta) and zenith direction (φ – phí (také Phase).
– polar coordinate system is 2D: θ/ φ .
history
– fds
a) Orthogonal fnctions
– fds
…
b) Legendre polynomiaLS
– discovered 1782.
– they were probably not that decisive for graphic, but were further used in Spherical harmonics and I think similar system of rotation in additional dimension, that is also used in FFT and particles science..
jjj
Fnction of particular Ps
-sdf
Looks like can be easily done in BW.
Theta (θ) – is in gonimetry symbol for radian – value range 0-1 Tau (𝜏) . 1 is equal to 2pi (6,28)). In here I guess it is the same as for t (probably suggesting unit of time).
c) LaPlace transform
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La Place
– Laplace transform – its first concept were in parallel with Fourier Analysis described in 1714 by Pierre-Simon Laplace – but it wider use come mainly after WWII.
– unlike FA it works in 6D system…))
La Place
–D) SPHERICAL HARMONICS
Spherical harmonics are based on orthogonal fncts, Legendre polynomial and Laplace transforms..
The rotation of variables has similar princip as in FFT ..
Equation for acustics
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Equation for quantum mechanics
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Solid harmonics
– are solutions of the Laplace equation in spherical polar coordinates
Bessel function
– are solutions of the Laplace equation in spherical polar coordinates
Kaiser–Bessel window
– are solutions of the Laplace equation in spherical polar coordinates
Solid harmonics
– are solutions of the Laplace equation in spherical polar coordinates
9) Electricity
Is related to Phase spaces, Phase modulation etc…
Electricity (sigsaly.xf.cz)