phonToDifferenceLimens

A function that can be used in Formulas.

## Syntax and semantics

`phonToDifferenceLimens` (`x`)
  convert a sensation level `x` in phon units into an intensity difference limen level.

## Definition

`phonToDifferenceLimens` (phon) = 30 · ((61/60) phon − 1)

## Related function

`phonToDifferenceLimens` is the inverse of `differenceLimensToPhon`.

## 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) (10 log (I / I0)) − 1) = 30 · (1.0167SL − 1)

### Links to this page

- Formulas 5. Mathematical functions
- Functions