Phase coherence at 8kHz in cardioid pairs
X/Y, ORTF, and Blumlein on the same cymbal, run through a 2k–6k complex coherence analysis. ORTF wins, but not for the reason in the textbooks.
Three stereo pairs on the same overhead position, same cymbal, same room, same take. Run them through a 2k–6k complex coherence analysis. The textbook answer is that ORTF wins because the time-of-arrival difference matches the human head and gives the most localisable image. The textbook is right about the result and wrong about the mechanism.
The setup
A single 18-inch Sabian HHX Evolution crash, hit at the center, recorded with:
- X/Y — two Schoeps MK4 capsules, 90°, coincident.
- ORTF — same capsules, 110°, 17cm spacing.
- Blumlein — two Royer R–122 ribbons, 90°, coincident.
All three pairs identically suspended at 1.4m above the cymbal, identical preamp gain (Grace m801, +42dB), identical converters, identical signal chain. The cymbal was hit by a robotic arm I borrowed from a friend who builds drumming machines for kinetic art. The hits were repeatable to within 0.4dB across 30 strikes.1
What complex coherence shows
Magnitude-squared coherence is the standard tool and it is the wrong tool for stereo. It tells you whether the two channels share a signal; it does not tell you the phase relationship of that shared signal. Two channels can be perfectly coherent and 180° out of phase — the coherence number is 1.0, the mono sum is silent.
Complex coherence keeps the phase term. The result is a number in the unit disc rather than the interval [0,1]. The angle is the phase relationship, the magnitude is the textbook coherence.
The result
Between 2kHz and 6kHz — the band where most of the cymbal’s character lives — the complex coherence:
- X/Y: magnitude 0.94, phase ± 3°. Mono-compatible. Image narrow.
- ORTF: magnitude 0.71, phase rotating with frequency, mean ~22°. Image wide. Mono-compatible because the magnitude never inverts.
- Blumlein: magnitude 0.62, phase oscillating between +30° and –40°. Image widest. Mono sum dips 4–6dB at 4.7kHz — the figure-of-eight nulls.
The interesting result is ORTF’s phase rotation. It is monotonic with frequency from 2k to 5.5k. This is not an accident: it is the small time-of-arrival delay (about 80μs for an off-center cymbal hit) showing up as a linear phase term. A linear phase term is a delay, and a delay across two ears is what tells you where the sound came from.2
Why the textbook is right
ORTF gives the best image. This is true.
Why the textbook is wrong
The textbook reason is “inter-channel level differences from the cardioid pattern, plus inter-channel time differences from the 17cm spacing, sum to a head-like cue.” The actual reason is that the cardioid pattern’s 8kHz lobing is, by accident, the inverse of what the spacing introduces — they cancel each other above 5kHz and the result is a delay-only image with no level imbalance. Above 6kHz, ORTF is closer to a pure delay-stereo than X/Y is.
This matters when you change the angle. The classical 110° ORTF is not the optimal angle on every cardioid — it is optimal on the Schoeps MK4. Try the same setup with a pair of KM184s and the right angle is closer to 95°, because the KM184’s polar pattern is tighter at 8kHz. The textbook does not tell you this.