A dropping pin and a shout in the street

In a quiet room you can hear a pin drop; in a noisy street you may not catch someone shouting at you from across the road. The contrast holds two facts. First, whether a change is noticed depends on the background: the same increment of sound stands out against silence and is lost against noise. Second, the ear works over a range of intensities spanning many orders of magnitude, so it cannot respond equally to every physical increment.

The nineteenth-century German physiologist Weber was the first to state the first fact as a law. In his work on touch and weight he found that the smallest detectable difference in weight was not a fixed number of grams but proportional to the standard weight; and that weights were discriminated more finely when they were lifted than when they rested on a still hand (Weber 1834). His successors wrote the rule as

$$\frac{\Delta I}{I} = k,$$

where $I$ is the intensity of the standard, $\Delta I$ the just noticeable difference (JND) and $k$ the Weber fraction. To say that a pin can be heard in a quiet room is to say that when the background $I$ is small, a small $\Delta I$ suffices.

Where Weber’s law comes from: internal noise

Why should the discriminable difference grow in proportion to intensity? Signal detection theory gives a natural explanation (volume 26, Chapter 4 introduced it; only its simplest form is used here). The nervous system’s internal representation of an intensity $I$ is not a fixed number but a noisy one: a random variable with mean $I$ and standard deviation $\sigma(I)$. Two stimuli can be told apart reliably only when the difference between their representations clearly exceeds the noise. The JND is therefore proportional to the noise: $\Delta I \approx \sigma(I)$ (taking 76 % correct in a two-interval “which is stronger?” task, that is $d' = 1$).

If the standard deviation of the noise is proportional to intensity, $\sigma(I) = cI$, Weber’s law follows. But the nervous system also has a “floor” of noise $\sigma_0$ that does not depend on the stimulus, and Poisson-like noise that grows with the square root of intensity (fluctuations in counts of photons or nerve impulses, for example). Adding the three variances:

$$\sigma^2(I) = \sigma_0^2 + qI + c^2I^2 .$$

The program below computes the JND under two combinations of noise: model A has only floor noise ($\sigma_0 = 1$) and multiplicative noise ($c = 0.05$); model B adds Poisson-like noise ($q = 1$). The units are illustrative. At the end the program checks the calculation with 40,000 simulated two-interval trials.

# Weber's law from internal noise (illustrative units). Internal response R ~ N(I, s(I)^2)
# with s^2 = s0^2 + q*I + (c*I)^2: additive, Poisson-like and multiplicative noise.
# 2AFC "which is stronger?": threshold dI = the step giving 76 % correct (d' = 1).
import math, random
from statistics import NormalDist
PHI = NormalDist().cdf
TARGET = PHI(1 / math.sqrt(2))                      # 0.760: 2AFC with d' = 1

def sd(i, s0, q, c):
    return math.sqrt(s0 ** 2 + q * i + (c * i) ** 2)

def jnd(i, s0=1.0, q=0.0, c=0.05):
    lo, hi = 0.0, 10 * sd(i, s0, q, c) + 1
    for _ in range(80):                             # bisection on percent correct
        d = (lo + hi) / 2
        pc = PHI(d / math.hypot(sd(i, s0, q, c), sd(i + d, s0, q, c)))
        lo, hi = (d, hi) if pc < TARGET else (lo, d)
    return hi

if __name__ == "__main__":
    levels = [0.1, 1, 10, 100, 1e3, 1e4, 1e5]
    print("intensity  | A: additive + multiplicative | B: plus Poisson-like term (q = 1)")
    print("           |      dI      dI/I            |      dI      dI/I   dI/sqrt(I)")
    for i in levels:
        a, b = jnd(i), jnd(i, q=1.0)
        print(f"{i:10.1f} | {a:9.2f} {a / i:8.3f}            | {b:9.2f} {b / i:8.3f}"
              f" {b / math.sqrt(i):8.2f}")
    k = jnd(1e6) / 1e6                              # Weber fraction at the top
    ia = jnd(0.0) / k                               # floor: dI at I = 0 equals k * Ia
    xs = [10 ** (e / 20) for e in range(-40, 101)]
    lin = max(k * (x + ia) / jnd(x) - 1 for x in xs)
    quad = max(abs(k * math.hypot(x, ia) / jnd(x) - 1) for x in xs)
    print(f"model A: k = {k:.4f}, Ia = {ia:.1f} (s0 / c = 20). Largest error of"
          f" dI = k (I + Ia): {lin:+.1%};")
    print(f"  of dI = k sqrt(I^2 + Ia^2): {quad:.1%} (the two differ only near I = Ia)")
    rng = random.Random(1834)
    for i in (1.0, 1000.0):
        d = jnd(i)
        ok = sum(rng.gauss(i + d, sd(i + d, 1, 0, 0.05)) > rng.gauss(i, sd(i, 1, 0, 0.05))
                 for _ in range(40000))
        print(f"check by simulation at I = {i:6.0f}: {ok / 40000:.3f} correct"
              f" (target {TARGET:.3f})")
# intensity  | A: additive + multiplicative | B: plus Poisson-like term (q = 1)
#            |      dI      dI/I            |      dI      dI/I   dI/sqrt(I)
#        0.1 |      1.00   10.008            |      1.33   13.294     4.20
#        1.0 |      1.00    1.003            |      1.69    1.690     1.69
#       10.0 |      1.13    0.113            |      3.63    0.363     1.15
#      100.0 |      5.23    0.052            |     11.61    0.116     1.16
#     1000.0 |     51.31    0.051            |     60.73    0.061     1.92
#    10000.0 |    512.99    0.051            |    523.15    0.052     5.23
#   100000.0 |   5129.85    0.051            |   5140.10    0.051    16.25
# model A: k = 0.0513, Ia = 19.5 (s0 / c = 20). Largest error of dI = k (I + Ia): +40.7%;
#   of dI = k sqrt(I^2 + Ia^2): 0.6% (the two differ only near I = Ia)
# check by simulation at I =      1: 0.763 correct (target 0.760)
# check by simulation at I =   1000: 0.758 correct (target 0.760)

At high intensities model A gives a constant Weber fraction of 0.051; below an intensity of 1 the JND stops at the value of 1.00 set by the floor noise, so $\Delta I/I$ rises steeply (10.0 at $I = 0.1$). This is the “correction at low intensities” of the framework: combining the Weber fraction at high intensity, $k = 0.0513$, with the JND at zero intensity gives $I_a = 19.5$, close to $\sigma_0/c = 20$.

The program also shows that the framework’s form $\Delta I = k(I + I_a)$ is only an approximation. When the variances of the two noises add, the corresponding form is $\Delta I = k\sqrt{I^2 + I_a^2}$ (within 0.6 % of the program’s exact calculation); the linear form overestimates by up to 40.7 % near $I \approx I_a$. The linear form corresponds instead to standard deviations that add directly (if the two noises were perfectly correlated, for instance). The two forms agree far from the knee; whether real data can tell them apart depends on how precisely the knee is measured.

Model B has an extra middle region: between intensities of about 10 and 100, $\Delta I/\sqrt{I}$ hardly changes (1.15, 1.16), so the JND grows as a square root; only at higher intensities does Weber’s law return. “Weber’s law” is therefore not a universal law but an approximation that holds when multiplicative noise dominates. The last two lines are the simulation check: at $I = 1$ and $I = 1000$, 40,000 two-interval trials at the computed $\Delta I$ give 0.763 and 0.758 correct, in agreement with the target of 0.760.

From Weber to Fechner: an extra assumption

In 1860, in his Elemente der Psychophysik, Fechner proposed that the magnitude of sensation could be measured by counting JNDs (Fechner 1860). His argument: if every JND is subjectively a step of the same size, $S = $ constant, then Weber’s law $\Delta I = kI$ gives $dS \propto dI/I$, and integration yields

$$S = \frac{1}{k}\ln\frac{I}{I_0},$$

where $I_0$ is the absolute threshold. This is Fechner’s law: sensation grows with the logarithm of stimulus intensity.

There are two problems with this step. The first is mathematical: replacing a finite step by a differential goes wrong when the Weber fraction is not constant; Luce and Edwards (1958) criticised the derivation systematically (Luce and Edwards 1958), while Dzhafarov and Colonius (2011) re-examined Fechner’s reasoning: the familiar derivation from Weber’s law plus equal JNDs holds only approximately, the better the smaller the Weber fraction, and Fechner had a second derivation that does not use JNDs at all, in which the logarithmic law follows if the subjective dissimilarity of two stimuli depends only on the ratio of their intensities, a principle logically independent of Weber’s law (Dzhafarov and Colonius 2011). The second problem is more basic: “every JND is subjectively the same size” is an assumption, not an observation.

The program below first counts JNDs step by step under the generalised Weber law $\Delta I = k(I + I_a)$ and compares the count with Fechner’s integral; then it asks how many JNDs a “doubling” contains near threshold; finally it compares Miller’s 0.41 dB with the sone scale of loudness.

# From Weber to Fechner, and what logarithmic units do and do not measure.
# (1) Count JNDs step by step when dI = k (I + Ia), against Fechner's integral.
# (2) Equal JNDs are not equal loudness: 0.41 dB JNDs (Miller 1947) vs sones (loudness
#     doubles per 10 dB above 40 phon, Stevens). (3) One unit of three logarithmic scales.
import math

def count_jnds(i0, i, k, ia):
    n, x = 0, i0
    while x * (1 + k) + k * ia <= i:                 # next step: x + k (x + Ia)
        x, n = x * (1 + k) + k * ia, n + 1
    return n

if __name__ == "__main__":
    k, ia, i0 = 0.10, 1.0, 1.0
    print("I / I0   JNDs counted  (1/k) ln R  ln R / ln(1+k)   [R = (I+Ia)/(I0+Ia)]")
    for i in (2, 10, 100, 1e4, 1e6):
        r = (i + ia) / (i0 + ia)
        print(f"{i:9.0f} {count_jnds(i0, i, k, ia):12d} {math.log(r) / k:12.1f}"
              f" {math.log(r) / math.log(1 + k):12.1f}")
    print("JNDs per doubling of I when Ia = 10 (Weber's law fails near threshold):")
    print("  " + "  ".join(f"{a:g}->{2 * a:g}: {count_jnds(a, 2 * a, k, 10)}"
                           for a in (1, 4, 16, 64, 1000)))
    step = 0.41                                      # dB per JND
    print("level (dB)  sones   sones gained per 0.41-dB JND")
    for level in (40, 60, 80, 100):
        s = 2 ** ((level - 40) / 10)
        print(f"{level:8d} {s:8.1f} {s * (2 ** (step / 10) - 1):14.3f}")
    print(f"40 -> 100 dB = {60 / step:.0f} JNDs; loudness grows 64-fold;")
    print(f"  one JND at 100 dB adds {2 ** 6:.0f} times the loudness of one JND at 40 dB")
    print("unit        definition                         one unit means")
    print("decibel     10 log10(P / P0)                   power x 10^0.1 = 1.259")
    print(f"magnitude   -2.5 log10(F / F0) (Pogson 1856)   flux / {100 ** 0.2:.3f}")
    print("Richter ML  log10(A / A0) (Richter 1935)       trace amplitude x 10")
# I / I0   JNDs counted  (1/k) ln R  ln R / ln(1+k)   [R = (I+Ia)/(I0+Ia)]
#         2            4          4.1          4.3
#        10           17         17.0         17.9
#       100           41         39.2         41.1
#     10000           89         85.2         89.4
#   1000000          137        131.2        137.7
# JNDs per doubling of I when Ia = 10 (Weber's law fails near threshold):
#   1->2: 0  4->8: 2  16->32: 5  64->128: 6  1000->2000: 7
# level (dB)  sones   sones gained per 0.41-dB JND
#       40      1.0          0.029
#       60      4.0          0.115
#       80     16.0          0.461
#      100     64.0          1.845
# 40 -> 100 dB = 146 JNDs; loudness grows 64-fold;
#   one JND at 100 dB adds 64 times the loudness of one JND at 40 dB
# unit        definition                         one unit means
# decibel     10 log10(P / P0)                   power x 10^0.1 = 1.259
# magnitude   -2.5 log10(F / F0) (Pogson 1856)   flux / 2.512
# Richter ML  log10(A / A0) (Richter 1935)       trace amplitude x 10

The first table shows that the JNDs counted step by step agree with the exact discrete solution $\ln R/\ln(1+k)$ (for an intensity ratio of $10^6$, 137 against 137.7), while Fechner’s integral $(1/k)\ln R$ is systematically about 5 % lower, the difference between $\ln(1.1)$ and $0.1$; all three grow with the logarithm of intensity. So as long as Weber’s law holds, counting JNDs does give a logarithmic scale, and on this point Fechner’s calculation is sound. The second line shows that matters differ near threshold: with $I_a = 10$, doubling from 1 to 2 contains not even one JND, while doubling from 1000 to 2000 contains 7. Near threshold the “JND scale” is closer to linear than logarithmic.

The second table returns to the assumption itself. From 40 dB to 100 dB there are 146 JNDs of 0.41 dB; if each JND were subjectively the same size, loudness would grow evenly. On the sone scale, however, loudness grows 64-fold over these 60 dB, and one JND at 100 dB adds 64 times as much loudness as one JND at 40 dB. The sone scale itself comes from Stevens’s magnitude estimation (Chapter 2) and is not beyond dispute; but Miller’s observation is enough to show that the number of JNDs and the magnitude of a sensation are not the same quantity.

Decibels, stellar magnitudes and the Richter scale: three logarithms, three histories

The framework says that the decibel, the stellar magnitude and the Richter magnitude share “a common psychophysical origin”. Checking their histories, the answer is no. The last table of the program lists the definitions of the three units.

Of the three logarithmic units, then, only the stellar magnitude has a forerunner in human sensory grading, and its logarithmic form was fixed by astronomers before Fechner. The decibel and the Richter magnitude are engineering conventions adopted to handle enormous ranges of values and to simplify calculation. What they share with Fechner’s law is a mathematical form, not a psychophysical origin.

Chapter summary

Weber’s law says that the JND is proportional to the intensity of the standard. A model of internal noise explains it, and also explains its failure at low intensities: floor noise stops the JND from falling near threshold, and Poisson-like noise creates a square-root region at intermediate intensities. The framework’s $\Delta I/(I + I_a) = k$ is an approximation; when the noise variances add, the corresponding form is $\Delta I = k\sqrt{I^2 + I_a^2}$, and the two differ by up to about 40 % near the knee. Fechner’s law requires the extra assumption that every JND is subjectively the same size; Miller’s observation and the loudness scale both show that this assumption fails. The logarithmic forms of stellar magnitude, the decibel and the Richter scale each have their own history; only the stellar magnitude has a forerunner in sensory grading, and it predates Fechner. The next chapter looks at how Stevens measured sensation another way and found power functions.