phonToDifferenceLimens

A routine for converting sensation level in phons into intensity difference limen level, the inverse of differenceLimensToPhon.

### Formula

 phonToDifferenceLimens (phon) = 30 · ((61/60) phon – 1)

### Derivation

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 / I0) –0.072

In this formula, I is the intensity of the sound in Watt/m2, I0 is the intensity of the auditory threshold (i.e. 10–12 Watt/m2 at 1000 Hz), and ΔI is the just noticeable difference.

Boersma (1998: 109) calculates a difference-limen scale from this. Given an intensity I, the number of difference limens above threshold is

 ∫I0I dx ΔI(x) = (1 / 0.463) ∫I0I dx I0–0.072 x0.072–1
 = (1 / (0.463·0.072)) ((I/I0)0.072 – 1)

The sensation level in phon is defined as

 SL = 10 log10 (I/I0)

so that the number of difference limens above threshold is

 (1 / (0.463·0.072)) (10(0.072/10)(10log(I/I0)) – 1) = 30 · (1.0167SL – 1)