
A routine for converting sensation level in phons into intensity difference limen level, the inverse of differenceLimensToPhon.
phonToDifferenceLimens (phon) = 30 · ((61/60)^{ phon} – 1) 
In first approximation, humans can detect an intensity difference of 1 phon, i.e. if two sounds that differ only in intensity are played a short time after each other, people can generally detect their intensity difference if it is greater than 1 phon.
But the sensitivity is somewhat better for louder sounds. According to Jesteadt, Wier & Green (1977), the relative difference limen of intensity is given by
DLI = ΔI / I = 0.463 · (I / I_{0})^{ –0.072} 
In this formula, I is the intensity of the sound in Watt/m^{2}, I_{0} is the intensity of the auditory threshold (i.e. 10^{–12} Watt/m^{2} at 1000 Hz), and ΔI is the just noticeable difference.
Boersma (1998: 109) calculates a differencelimen scale from this. Given an intensity I, the number of difference limens above threshold is
∫_{I0}^{I} dx ΔI(x) = (1 / 0.463) ∫_{I0}^{I} dx I_{0}^{–0.072} x^{0.072–1} 
= (1 / (0.463·0.072)) ((I/I_{0})^{0.072} – 1) 
The sensation level in phon is defined as
SL = 10 log_{10} (I/I_{0}) 
so that the number of difference limens above threshold is
(1 / (0.463·0.072)) (10^{(0.072/10)(10log(I/I0))} – 1) = 30 · (1.0167^{SL} – 1) 
© ppgb, December 15, 2002