Operators and Algorithms in Yamaha FM Synthesis

Here’s an extended and detailed technical section on how operators and algorithms work across Yamaha FM chips, written in the same technical tone as the report. This can be appended right after the “Cross-Family Technical Notes” section.

1. Fundamentals of FM (Frequency Modulation) Synthesis

At its core, FM synthesis creates complex waveforms by using one oscillator (the modulator) to modulate the frequency of another oscillator (the carrier). In Yamaha’s implementation, each operator functions as a self-contained oscillator unit with its own parameters and envelope generator. By chaining these operators together, a single “voice” or “channel” can produce an extremely wide range of timbres.

Operators are then arranged in algorithms — predefined signal flow diagrams that determine how operators are connected. Within those algorithms, operators play two roles:

• Carrier: The operator whose output goes directly to the DAC (i.e., what you hear). It defines the audible pitch and volume of the final sound.
• Modulator: The operator that feeds into another operator rather than going to the output. It modulates the phase (frequency) of the next operator, changing its timbre by introducing harmonics.

Because all operators are identical in design, an operator can be a carrier in one algorithm or a modulator in another — its function depends only on how it’s connected.

1.1 Each Operator Contains:

• Phase Generator (PG) – computes the instantaneous phase (angle) of the oscillator.
• ADSR Envelope Generator (EG) – defines the amplitude over time (attack, decay, sustain, release).
• Sine Lookup Table (or waveform selector) – converts the phase value to an output waveform (usually a sine wave, but more waveforms exist in later chips like OPL2/OPZ/OPL3).
• Modulation Input – accepts a phase offset from another operator’s output (the modulator).
• Amplitude Scaling and Key Scaling – modify output amplitude depending on note frequency and velocity (to simulate acoustic response).

Thus, each operator is a small, independent digital synthesizer that can act as either a modulator or a carrier depending on its position in the algorithm.

2. Operator Routing: Algorithms

The algorithm defines how operators are connected within a single channel (voice). An algorithm specifies which operators modulate which others, and which operators feed the final output. This routing determines whether the voice sounds simple and harmonic (few modulators) or complex and metallic (deep modulation chains).

This image shows the 8 different configuration of the OPN - OPM chips

2.1 Example – 4-Operator Algorithm (YM2151 / YM2612 / YM2608)

Let’s label operators OP1 → OP4.

• Algorithm 1 (Stacked / Serial)

OP4 → OP3 → OP2 → OP1 → Output

• Deepest modulation chain. Produces complex, evolving, inharmonic spectra (great for metallic and bell sounds).
• Algorithm 2 (Parallel)

OP4 → Output
OP3 → Output
OP2 → Output
OP1 → Output

• Each operator contributes independently to the output. Produces layered or additive sounds, ideal for pads or organ tones.
• Algorithm 3 (Mixed)

OP4 → OP3 → Output
OP2 → OP1 → Output

• Two independent 2-operator FM pairs summed together. Useful for creating hybrid tones (bass + harmonic layer).

Yamaha’s 4-op chips typically include 8 selectable algorithms (some up to 32, depending on model). Each algorithm defines a unique modulation topology, altering harmonic complexity and brightness.

3. Frequency and Modulation Mechanics

3.1 Modulation Depth

The output of a modulator alters the phase increment of its carrier, creating sidebands around the carrier’s base frequency. The modulation index (determined by the modulator’s amplitude) controls the number and amplitude of these sidebands — effectively, how “bright” or “complex” the resulting sound becomes.

In Yamaha chips, this is controlled primarily by:

• Total Level (TL) – sets the overall output amplitude of an operator.
• Modulation Sensitivity / Output Level Scaling – adjusts how much an operator affects its carrier.
• Feedback – feeds an operator’s output back into its own phase input (self-modulation), generating additional harmonics.

3.2 Feedback

Feedback is applied in specific algorithms where an operator modulates itself. It’s typically implemented on the first operator (OP1) and controlled by a feedback register (FB) per channel. Increasing feedback intensity increases harmonic richness but also distortion — similar to analog oscillator sync effects.

4. Envelope Generator (EG) Behavior

Every operator’s amplitude follows a four-stage ADSR envelope, but Yamaha divides this into four rate and four level parameters:

Certain chips add features like:

• Rate Scaling (RS): higher notes shorten envelope times.
• Key Scaling Level (KSL): higher notes reduce operator amplitude, simulating natural acoustic damping.
• Amplitude Modulation Enable (AM): allows LFO tremolo modulation.

Together, these parameters make FM synthesis dynamically expressive and capable of acoustic-like behavior.

5. Frequency Control and Key Scaling

Each operator’s frequency is determined by:

• The base note frequency of the channel (set by registers FNUM + BLOCK).
• The operator’s multiplier (MUL), a small integer (1–15) that scales the frequency relative to the carrier.

Thus:

Operator frequency = Channel frequency × Multiplier

By giving modulators and carriers different multipliers, one can generate harmonic or inharmonic relationships (e.g., bell tones, electric pianos, brass).

6. Differences Between 2-Operator and 4-Operator Systems

In 2-op chips, each “voice” is effectively a single modulator-carrier pair — simple but efficient. 4-op chips allow multi-layer modulation, enabling dynamic, evolving timbres such as electric pianos, brass, and bells.

7. LFO, Vibrato, and Tremolo

All Yamaha FM chips include one or more Low Frequency Oscillators (LFOs) that can modulate:

• Pitch (vibrato)
• Amplitude (tremolo)

Parameters usually include:

• LFO Frequency
• Amplitude Modulation Depth (AMD)
• Frequency Modulation Depth (PMD)
• Per-channel enable flags.

Later chips (e.g., YM2414 OPZ) introduced two separate LFOs for more dynamic modulation.

8. The “Channel 3 Trick” (Special Mode)

On certain OPN chips (e.g., YM2612, YM2203), Channel 3 can run each of its 4 operators at an independent frequency instead of sharing a common base frequency. This allows:

• Inharmonic or bell-like tones
• Manual detuning effects
• Experimental sounds not possible on normal channels.

It’s a powerful feature often used in Sega Genesis and PC-98 compositions for metallic textures or complex percussion sounds.

9. Mathematical Overview (simplified)

In mathematical terms, a Yamaha FM operator outputs:

where

• \(f_c\) : carrier frequency
• \(f_m\) : modulator frequency
• \(I\) : modulation index (proportional to modulator amplitude)
• \(A(t)\) : amplitude envelope

Stacking multiple operators means that ( \(f_m\) ) becomes a nested sine wave function — leading to very rich spectra with hundreds of sidebands, controlled entirely by the envelope generators.

10. Summary

• Operators are the atomic sound generators — oscillators with envelopes.
• Algorithms define their modulation topology.
• Feedback and modulation index control harmonic richness.
• 4-op chips enable deeper modulation hierarchies for more expressive tones.
• 2-op chips trade richness for efficiency, ideal for simpler sound systems.

Together, these principles form the backbone of Yamaha’s FM sound — a synthesis method that remains one of the most efficient and expressive forms of digital tone generation ever created.