Sound SynthesisSound Synthesis

Sound SynthesisSound Synthesis

CE 476 Music & ComputersCE 476 Music & Computers

Additive SynthesisAdditive Synthesis

• We add together different soundwaves sample-by-sample to create a new sound, see Applet 4.3.

• We can add spectral envelopes to a number of partials. By this way, we can impose a different amplitude trajectory for each harmonic, independently making each harmonic louder and softer over time, see Applet 4.4.

sin(4*x) + sin(x)+cos(x/2), x=0..2*Pi

sin(5*x) + sin(x), x=0..2*Pi

Changing Sound Parameters

• The quality of a synthesized sound can be improved by varying its parameters (partial frequencies (harmonics), amplitudes, and envelope) over time.

• These soundfiles are examples of sentences reconstructed with sine waves. Soundfile 4.2 is the sine wave version of the sentence spoken in Soundfile 4.3, and Soundfile 4.4 is the sine wave version of the sentence spoken in Soundfile 4.5.

Amplitude envelope shape

Attacks, Decays, and Time Evolution in Sounds

• Additive synthesis has some drawbacks. One serious problem is that while it’s good for periodic sounds, it doesn’t do as well with noisy or chaotic ones.

• For instance, creating the steady-state part (the sustain) of a flute note is simple with additive synthesis (just a couple of sine waves), but creating the attack portion of the note, where there is a lot of breath noise, is nearly impossible.

• Therefore, we have to synthesize a lot of different kinds of information: noise, attack transients, and so on.

• The ear and brain are much more interested in things like attacks, decays, and changes over time in a sound. That’s bad news for all that additive synthesis software, which doesn’t handle such things very well.

Subtractive Synthesis:Subtractive Synthesis: FiltersFilters

• A filter is a function that takes in a signal and gives back some sort of transformed signal.

• Usually, what comes out is "less" than what goes in.

• In additive synthesis, we start with simple sounds and add them together to form more complex ones.

• In subtractive synthesis, we start with a complex sound (like noise) and subtract, or filter out, parts of it.

Cutoff Frequency

• A filter is defined as the point at which the signal is attenuated to 0.707 of its maximum value (which is 1.0).

• That is, the power of a signal is determined by squaring the amplitude: sqrt(0.5) = 0.707, i.e, which gives the amplitude of a signal at 0.707 of its maximum value, it is also called half-power.

Low-pass Filter Characteristics

Transition Band

• The area between where a filter "turns the corner" and where it "hits the bottom" is called the transition band.

• The steepness of the slope in the transition band is important in defining the sound of a particular filter.

• If the slope is very steep, the filter is said to be "sharp"; conversely

• If the slope is more gradual, the filter is "soft" or "gentle."

Transition Band

Four Basic Type of Filters

http://music.columbia.edu/cmc/musicandcomputers/chapter4/04_03.php

See Applet 4.5 and 4.6

IIR and FIR Filters

• Digital filters are classified in two groups as the technical approach:

• FIR: Finite impulse response,– what comes out uses a finite number of samples, and

a sample only has a finite effect. By delaying a signal and then averaging the delayed signal and the nondelayed one.

• IIR: Infinite impulse response. – If we delay, average, and then feed the output of that

process back into the signal, we create a IIR filter. The feedback process actually allows the output to be much greater than the input. These filters can, as we like to say, "blow up."

FIR FILTERS

Example FIR Filter

Example:Filter Type: Low Pass Sampling Frequency: 2000 Hz Cut off Frequency: 460 Hz Filter Length (# weights): 21M - filter order, it is always equal to the number of taps minus 1 ft - Normalised transition frequency.

Response of the Example FIR Filter

Example:Filter Type: Low Pass Sampling Frequency: 2000 Hz Cut off Frequency: 460 Hz Filter Length (# weights): 21M - filter order, it is always equal to the number of taps minus 1 ft - Normalised transition frequency.

IIR FILTERS

Subtractive Sound Synthesis

• Sound generators used in subtractive synthesis have large spectra. Therefore, we need to subtract from the sound.

• Mostly filter banks are used to subtract desired harmonics or partials and put them together to create the desired sound.

Sound GeneratorsSound Generators

• Noise generators– White noise– Pink noise

• Oscillators– Sine– Square– Sawtooth– Triangle

• Audio files and sampled sounds– Sounds being produced by live sources in real time

WHITE NOISEWHITE NOISE

• One of the most commonly used source of sounds for subtractive synthesis is white noise

• Definition: A sound that contains all audible frequencies, whose spectrum is essentially flat, i.e., Amplitudes of individual frequencies (harmonics) are randomly distributed.

Spectrum of White NoiseSpectrum of White Noise

Sonogram of White Noise

PINK NOISEPINK NOISE

• Pink noise, in contrast to white noise, has a spectrum whose energy decays as frequency rises. More precisely, the attenuation in pink noise is 3 dB per octave.

• Pink noise is also called 1/f noise to indicate that the spectral energy is proportional to the reciprocal of the frequency.

Spectrum of Pink NoiseSpectrum of Pink Noise

Spectrum of a Sine WaveSpectrum of a Sine Wave

Spectrum of a 440 Hz Sine WaveSpectrum of a 440 Hz Sine Wave

PPseudo-random generatorsseudo-random generators

• white noise is generally produced using random number generators. The resulting waveform contains all of the reproducible frequencies

• Random number generators use

• mathematical procedures that are not precisely random but they generate series that repeat after some number of events.

PPseudo-random sample generatorsseudo-random sample generators

• These generate random values at a given frequency with a constant value until it is time to generate the next sample. This results in a waveform

• Example of a 100 Hz noise generator: the random value is repeatedly output for a period equal

• to 1/100 of a second, after which a new random value is computed. If the sampling rate were 48,000 Hz, for example, each random value would be repeated as a sample 48,000 / 100 = 480 times.

PPseudo-random sampleseudo-random sample

OSCILLATORS AND OTHER OSCILLATORS AND OTHER SIGNAL GENERATORSSIGNAL GENERATORS

• We use classic waveforms such as the sine wawe, square wave, the sawtooth wave, and the triangle.

• Thes waveforms, when geometrically perfect, contain an infinite number of frequency components.

• However, the infinity causes nasty problems when producing digital sound, since an audio interface cannot reproduce frequencies above half of its sampling rate.

• Undesired components are almost always non-harmonic.

BBand-limited oscillatorsand-limited oscillators

• To avoid the problem with the undesired frequency components band-limited oscillators are used

• Such oscillators, which produce the classic waveforms, are built so that their component frequencies never rise above half of the sampling rate

Waveshaping SynthesisWaveshaping Synthesis

• Waveshaping technique turns simple sounds into complex sounds. We can take a pure tone, like a sine wave, and transform it into a harmonically rich sound by changing its shape.

• A guitar fuzz box is an example of a waveshaper. An unamplified electric guitar sound is fairly close to a sine wave. But the fuzz box amplifies it and gives it sharp corners.

• Waveshaper generally have much more energy in their higher-frequency harmonics, which gives them a "richer" sound.

How to WavesahpeHow to Wavesahpe

• transfer functions are used to transfrom one waveform into another, e.g.,

• That is, to pass a simple sine wave, x = sin(wt), varying from -1.0 to 1.0 through this waveshaper, then we get

y = x3 = sin3(wt)

:= y ( )f x x3

y ( )sin t 3

Triangle:=

Sinewave

Trianglewave

( )sin t ( )sin t 5

Normalized WNormalized Waveshaper aveshaper Output Output

• If we use X^3 as the output of X for the input range -1 to 1 than we can get vary high output values, we can keep the output between -1 to 1, by another transfer function:

yx

1 x

1.5 ( )sin t 0.5 ( )sin t 5

Sinewave

Squarewave

1.5 ( )sin t 0.5 ( )sin 3 t 5Trianglewave

Sinewave

Square-like ewave

Sinewave

1.5 ( )sin t 0.5 ( )sin 5 t 5

Chebyshev PolynomialsChebyshev Polynomials

• A transfer function is often expressed as a polynomial:

( )f x d0 d1 x d2 x2 d3 x

3 dn xn

The highest exponent n of this polynomial is called the "order" of the polynomial. Order 2 results in a doubling of the pitch. So a polynomial of order 2 produces strong second harmonics in the output.

Chebyshev Polinomial as Chebyshev Polinomial as Frequency MultiplierFrequency Multiplier

• If you input a sine wave of amplitude 1.0, you get out a sine wave whose frequency is N times the frequency of the input wave.

• If the amplitude of the input sine wave is less than 1.0, then you get a complex mix of harmonics.

• Generally, the lower the amplitude of the input, the lower the harmonic content. This gives musicians a single number, sometimes called the distortion index, that the musican can tweak to change the harmonic content of a sound.

• If you want a sound with a particular mixture of harmonics, then you can add together several Chebyshev polynomials multiplied by the amount of the harmonic that you desire.

Example Example Chebyshev polynomialsChebyshev polynomials

Formula for generating Chebyshev polynomials:

Table-Based WaveshapersTable-Based Waveshapers

• We generally precalculate these polynomials and put the results in a table. Then when we synthesize a sound, we just take the value of the input sine wave and use it to look up the answer in the table.