What is Linear Arithmetic Synthesis?
L.A. synthesis is a form of audio synthesis born in the late 1980`s. To be precise 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 with respect to human perception. Therefore L.A. synths used a combination of sampled attack transients and simple digital oscillators. The oscillators had only saw-tooth and pulse wave-forms 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 “. In addition to this, Subtractive synthesis and Digital sampling are both forms of linear synthesis. L.A. synthesis is basically hybrid of some of the concepts found in these technologies.
Roland D-50
Earlier mentioned D-50 synthesizer by Roland was a completely digital synthesizer. It creates sound thru 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 (addition), 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. This is because in that time it was one of the most difficult acoustic instruments to synthesize. You can brake the piano sound into two major 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 vibration of the string. After the initial attack the piano has a much longer decay and the harmonic emphasis shifts on the resonance of the soundboard. To make this more complicated, some of this characteristics differ from one point of the keyboard to another.
Piano Imitation Troubles
Traditional forms of sound synthesis that predated the L.A. synthesis had trouble to imitate the piano sound for several reasons. For one, few envelopes have been able to produce the sharp initial transient followed by the appropriate decay. Secondly, the harmonic content is so complex that no amount of manipulating of the traditional wave-forms would match the piano’s harmonic spectrum. Finally, as soon as you get close to synthesize a piano at one range of the keyboard, the timbre is typically nowhere close in other ranges.
The linear arithmetic approach actually allows user to control the individual sonic elements. It produces the sound by combining these individual elements together. Going back 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 the 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, as well as sequences of inharmonious wave cycles. Thus , L.A. synthesis offered the realistic sounds of a sampler with the control and creativity of a synthesizer.
Additional Resource & Source Texts
https://books.google.rs/books?id=ajr4_nS3X0gC&lpg=PP2&ots=_J10YkBjGc&dq=linear%20arithmetic%20sound%20synthesis&lr&hl=sr&pg=PP2#v=onepage&q&f=false
