Linear arithmetic synthesis
Linear Arithmetic synthesis, or LA synthesis, is a term that was invented by the Roland Corporation when they released their ground-breaking D-50 synthesizer in 1987.
It was the first keyboard to combine a synthesizer with sampled waveforms that could be modified.
The following is from a Roland manual: "LA synthesis involves a great many technological advances resulting not only in a superior sound quality, but also an improved ease of programming."
Linear arithmetic makes one think of simple math, such as addition, and indeed, the primary way that LA synthesis created sounds was by adding sampled waveforms to oscillator generated waveforms.
In order to save memory, which was quite expensive in 1987, the engineers at Roland recorded/sampled different instruments and sounds, then used the recordings/samples to make PCM (Pulse-Code Modulation) samples/waveforms which were stored in the unit's memory. These PCM samples/waveforms used very little memory because they usually lasted less than one second. This brief period of time was enough, because the PCM waveforms were used only1 for the short attack part of the sound, while the sound wave generators, which needed no memory, were used for the long lasting sustain part of the sound. Thus, LA synthesis offered the realistic sounds of a sampler with the control and creativity of a synthesizer, all at an affordable price.
The PCM samples/waveforms could be modified with a pitch envelope and a Time Variant Amplifier. Waveforms from the sound wave generators, could be further modified with Time Variant Filters for cutoff frequency, and resonance. These modified waveforms were called Partials.
Two Partials grouped together created a Tone. Tones could be modified using up to three LFOs (Low Frequency Oscillators), a pitch envelope, chorus, a programmable equalizer, and on-board effects such as reverb.
Two Tones grouped together created a Patch. A Patch is the sound that's heard when a key is pressed. However, a Patch is not actually a sound. Instead, a Patch is just the parameter values (for the Partials and Tones) stored in the keyboard's flash memory that tell all of the sound generation circuitry how to make the sound for that Patch.
1Some of the PCM waveforms in the D-50 could be played in a loop to create a sound that would last as long as a key was being pressed.