Small Angle Tuning

• ET : Equal temperament.
• MIDI [st.] : MIDI note number in 12 ET semitone.

This table demonstrates the tuning shift caused by applying the small angle approximation to the Magic Circle oscillator.

The exact Magic Circle oscillator uses a coefficient k = 2 sin(ω / 2) to rotate exactly by ω radians per step, where ω = 2π ftarget / fs.

If we use the small angle approximation sin(x) ≈ x, the coefficient simplifies to k = ω. This introduces a pitch drift, making the generated frequency increasingly sharper than the target frequency.

The actual angular frequency θ' generated by the approximated system is derived from the rotation matrix trace: 2 cos(θ') = 2 - k2. Solving for the actual generated frequency factual yields:

factual = (fs / π) arcsin(π ftarget / fs)

Note that the oscillator becomes unstable if k > 2, which occurs when the target frequency exceeds fs / π.