As part of Harman International’s research into defining new target frequency responses for both insert and circumaural/supra-aural (over-ear/on-ear) headphones, led by Sean Olive, it has developed equations which can be used to calculate a Predicted Preference Rating (PPR), based on the headphone’s frequency response as measured at the DRP (drum reference point) of an artificial ear. For the measurements used in developing this metric, Harman used the same GRAS artificial ear hardware as HTL employs.

Harman’s two equations, for insert and circumaural/supra-aural headphones respectively, are applied to the error response (measured frequency response minus Harman target response) and have similar but not identical forms. That for insert headphones is:

where A, B, C and D are constants, 's' is the standard deviation of the error response, 'g' is the absolute value of the gradient (slope) of the logarithmic regression line fitted to the error response, and 'm' is the mean error of the error response. (Harman uses different abbreviations for these quantities.) For circumaural/supra-aural (over-ear/on-ear) headphones the equation is a little simpler:

as including mean error did not improve the fit of the model.

The values of A, B, C and D for insert headphones are 68.685, 3.238, 4.473 and 2.658, and 's', 'g' and 'm' are defined over the frequency ranges 20Hz to 10kHz, 20Hz to 10kHz, and 40Hz to 10kHz respectively. For circumaural/supra-aural headphones the values of A, B and C are 114.49, 12.62 and 15.52, and 's' and 'g' are both calculated over the frequency range 50Hz-10kHz. (The higher LF limit in this case is because variations in earpad sealing can make the measured frequency response at lower frequencies too variable.)

Note that if a headphone is ‘perfect’ – ie, if its response exactly matches the Harman target – then 's', 'g', and 'm' are all zero and the maximum PPR values are therefore 69 and 114 respectively (to the nearest integer, as usually stated). So it is incorrect to refer to PPR values as percentages, and values for insert and circumaural/supra-aural headphones are not directly comparable. To circumvent this, HTL's PPR scores are quoted as both 'raw' figures (direct from Harman's equations), and as percentage figures to allow comparison across all headphone types. The results are presented in this form:

**84/82 ≡ 73%/72% (L/R)**

where the mathematical symbol ≡ means 'is equivalent to'.

The equations assume logarithmically spaced data points along the frequency axis, Harman having used 1/48th-octave data for its measurements and development of the PPR equations. So the first step in calculating PPR values from FFT response data, where the frequency spacing is linear, is interpolation to frequencies matching those of Harman’s target. The two responses can then be subtracted to create the error response, and truncated to the appropriate frequency range.

From this point the calculation of PPR is straightforward using standard equations for 's', 'g', and 'm':

where

But there is a wrinkle, in that the equation for 'g' stated in Harman’s papers (references below) is instead:

I have queried this with Harman but not received a reply. The PPR values quoted in HTL’s test results are calculated using the standard equation for 'g'.

**REFERENCES**

S Olive, T Welti and O Khonsaripour, "A Statistical Model That Predicts Listeners’ Preference Ratings of In-Ear Headphones: Part 1 – Listening Test Results and Acoustic Measurements", Audio Engineering Society 143rd Convention, October 2017 (available here)

S Olive, T Welti and O Khonsaripour, "A Statistical Model That Predicts Listeners’ Preference Ratings of In-Ear Headphones: Part 2 – Development and Validation of the Model", Audio Engineering Society 143rd Convention, October 2017 (available here)

S Olive, T Welti and O Khonsaripour, "A Statistical Model that Predicts Listeners’ Preference Ratings of Around-Ear and On-Ear Headphones", Audio Engineering Society 144th Convention, May 2018 (available here)