Sinusoidal Additive Synthesis

We will learn more about the plucked string and higher-dimensional vibrating systems such as bars, plates, membranes, etc., in later chapters. The point of introducing the example of the plucked-string system here was to motivate the notion that sinusoids can occur in systems more complex than just the simple mass/spring/damper. 

Sinusoidal Additive Synthesis Model, allowing us to control the amplitudes and frequencies of a number of sinusoidal oscillators. 

The modal filter model can be improved and expanded to make it more expressive and physically meaningful. For example, we can use rules to 

Rules as function of strike position, re-strike, damping, etc. 

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The notion that some components of soimds are well-modeled by sinusoids, while other components are better modeled by spectrally shaped noise, further motivates the residual-excited refinement to the purely sinusoidal additive synthesis model presented in Chapter 4 (Figure 4.4). Using the Fourier transform, we ean inspect the spectrum of a sound and determine whieh... 

Informed by the technique of sines plus noise spectral modeling, we can now improve our sinusoidal additive synthesis model significantly by simply adding a filtered noise source as shown in Figure 6.13. 

Linear Synthesis. The most popular method of synthesis is so-called Additive Synthesis , where the output is a sum of oscillators. While it is commonly assumed that the oscillators produce sinusoids (Fourier synthesis), in fact, they can be any waveform. Furthermore, with static additive synthesis, a pre-mixed combination of harmonics was stored in the lookup table. Unfortunately, this doesn t permit inharmonic partials. Dynamic Fourier synthesis allows the amplitudes and frequencies of the partials to be varied relative to each other. Computationally, it is important to recognize the that updating oscillator coefficients for large numbers of oscillators can be expensive. 

This solution has the advantage of being very simple, but the drawback of being expensive in terms of calculations, even when tabulated sinusoids are used in Eq. (7.17). In fact, additive synthesis (of which Eq. (7.17) is an example) can be implemented at a much lower cost by use of the Fourier transform [Rodet and Depalle, 1992]. This last remark is a strong motivation for using the following alternative ... 

An early approach to music processing, referred to as additive synthesis [Moorer, 1977], used the sinusoidal model of a quasi-periodic music note... 

Figure 6.13. Sinusoidal additive model with filtered noise added for spectral modeling synthesis.

In prior chapters we found that spectral shape is important to our perception of sounds, such as vowel/consonant distinctions, the different timbres of the vowels eee and ahh, etc. We also discovered that sinusoids are not the only way to look at modeling the spectra of sounds (or soimd components), and that sometimes just capturing the spectral shape is the most important thing in parametric sound modeling. Chapters 5 and 6 both centered on the notion of additive synthesis, where sinusoids and other components are added to form a final wave that exhibits the desired spectral properties. In this chapter we will develop and refine the notion of subtractive synthesis and discuss techniques and tools for calibrating the parameters of subtractive synthesis to real sounds. The main technique we will use is called Linear Predictive Coding (LPC), which will allow us to automatically fit a low-order resonant filter to the spectral shape of a sound. 

Physical modeling synthesis endeavors to model and solve the physics of sound-producing systems in order to synthesize sound. Unlike sinusoidal additive and modal synthesis (Chapter 4), or PCM sampling synthesis (Chapter 2), both of which can nse one powerM generic model for any sound, physical modeling reqnires a different model for each family of sound producing object. LPC (Chapter 8) is a spectral modeling techniqne, but also has physical interpretations in the one-dimensional ladder implementation. 

Additive synthesis is deeply rooted in the theory of Fourier analysis. The technique assumes that any periodic waveform can be modelled as a sum of sinusoids at various amplitude envelopes and time-varying frequencies. An additive synthesiser hence functions by... 

Figure 3.1 Additive synthesis functions by summing up sinusoids in order to form specific waveforms...

Although additive synthesis specifically refers to the addition of sinusoids, the idea of adding simple sounds to form complex timbres dates back to the time when people started to build pipe organs. Each pipe produced relatively simple sounds that combined to form rich spectra. In a way, it is true to say that the organ is the precursor of the synthesiser. 

Figure 25-2. The formation and secretion of (A) chylomicrons by an intestinal cell and (B) very low density lipoproteins by a hepatic cell. (RER, rough endoplasmic reticulum SER, smooth endoplasmic reticulum G, Golgi apparatus N, nucleus C, chylomicrons VLDL, very low density lipoproteins E, endothelium SD, space of Disse, containing blood plasma.) Apolipoprotein B, synthesized in the RER, is incorporated into lipoproteins in the SER, the main site of synthesis of triacylglycerol. After addition of carbohydrate residues in G, they are released from the cell by reverse pinocytosis. Chylomicrons pass into the lymphatic system. VLDL are secreted into the space of Disse and then into the hepatic sinusoids through fenestrae in the endothelial lining.

The resynthesis process results from two simultaneous synthesis processes one for sinusoidal components and the other for the noisy components of the sound (Figure 3.15). The sinusoidal components are produced by generating sinewaves dictated by the amplitude and frequency trajectories of the harmonic analysis, as with additive resynthesis. Similarly, the stochastic components are produced by filtering a white noise signal, according to the envelope produced by the formant analysis, as with subtractive synthesis. Some implementations, such as the SMS system discussed below, generate artificial magnitude and phase information in order to use the Fourier analysis reversion technique to resynthesise the stochastic part. 

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