Linear Arithmetic (LA) Synthesis

What Is Linear Arithmetic Synthesis?

L.A. synthesis is a form of audio synthesis developed in the late 1980's. The first synthesizer that offered this new and unique form of synthesis was Roland's D-50. The theory behind it rests upon the idea that the attack transient of the sound is its most important part concerning human perception. Therefore, L.A. synths combined sampled attack transients and simple digital oscillators. The oscillators had only saw-tooth and pulse waveforms in order to generate the sustained part of the sound. The fact that those parts were cross-modulated was used to coin the term "Linear Arithmetic. "Subtractive synthesis and Digital sampling are both forms of linear synthesis. L.A. synthesis is a hybrid of some of the concepts found in these technologies.  

Roland D-50

The earlier mentioned D-50 synthesizer by Roland was utterly digital. It creates sound through a series of incredibly complex internal calculations. These calculations were primarily addition and subtraction. On the D-50, sounds are created by combining partials and tones (extra), removing unwanted tones (subtraction), and using the ring modulation (sum and difference).  

Piano Synthesis

Characteristics of the piano sound can easily describe the advantages of this type of audio synthesis. It was one of the most challenging acoustic instruments to synthesize. You can break the piano sound into two significant elements. These elements are the initial attack and decay that follows. The initial attack can then be subdivided into additional components. These components are the high transient of the hammer hitting the string and the many complex harmonics representing the string's vibration. After the initial attack, the piano has a much longer decay, and the harmonic emphasis shifts to the resonance of the soundboard. To simplify this, some characteristics differ from one keyboard point to another.  

Piano Imitation Troubles

Traditional forms of sound synthesis that predated the L.A. synthesis had trouble imitating the piano sound for several reasons. For one, few envelopes have produced the sharp initial transient followed by the appropriate decay. Secondly, the harmonic content is so complex that no amount of manipulation of the traditional waveforms would match the piano's harmonic spectrum. Finally, as soon as you get close to synthesizing a piano at one range of the keyboard, the timbre is typically nowhere near in other ranges. The linear arithmetic approach allows the user to control the individual sonic elements. It produces the sound by combining these particular elements. Returning to the piano sound, we could say that the PCM element is the best for the overall harmonic structure and the initial attack of the hammers. However, you can simulate the decay of the string vibrations by using subtractive synthesis elements.     

Looped Samples

On top of that, L.A. synthesis in the D-50 synthesizer incorporated a third aspect. It has a suite of single-cycle sampled waveforms. Sound could now have three components: An attack, a body made from a subtractive synth sound (saw, square, or pulse wave through a filter ), and the embellishment of one of many looped samples. The looped samples also contained a collection of totally synthetic waves derived from additive synthesis and sequences of inharmonious wave cycles. Thus, L.A. synthesis offered the realistic sounds of a sampler with the control and creativity of a synthesizer.