Steampunk phonetics, continued

In Alexander J. Ellis's 1873 article "On the Physical Constituents of Accent and Emphasis", he asserted that there are "four principal matters to be considered in a sound-curve, which will be here called length, pitch, force, and form". Yesterday I quoted his oddly labored explanation of length, by which he means what we would now generally call "duration". We can skip his equally-labored explanation of pitch — it's correct, as we'd expect from the man who introduced and named the cent as a unit of measure for pitch intervals, but otherwise its main point of interest is his adherence to  the rarely-used "philosophical pitch" standard, which has middle C at 256 Hz, and therefore C in all other octaves at frequencies of powers of two. What Ellis has to say about force, however,  is an interesting mixture of science and error.

Here's what he wrote:

The greater the force of the disturbance of the air, the further will the style depart from its position of rest, and hence the greater will be the amplitude, or greatest distance, measured from the medial line, of any point in the sound-curve corresponding to one complete double vibration. The square on this line measures the sensation of loudness produced, and will be called the force. Hence if one amplitude is 3 times as large as another, the force and the loudness of the first sound will be 9 times that of the second. If the first amplitude be to the second as 3 to 2, the forces are as 9 to 4. There is therefore no absolute, but only a relative, standard of force. And force is theoretically independent of both length and pitch. Practically more power is sometimes (by no means always) necessary to produce a great pitch than a small one, but it does not follow that the force of the sound will be greater. Practically a sound of great pitch is often more penetrative than one of small pitch, though the force of the latter may he greater. Compare cricket chirps, and deep organ pipes.

The squared value of what we'd now call the "sound waveform" produces what we would now call "intensity" or "power" (or at least this produces a number to which power is proportional). And rather than talking about the square of "any point in the sound-curve corresponding to one complete double vibration", we'd specify the mean value of the square of those measurements. And the ratio of the waveform's power to the power of a similar waveform at the threshold of hearing is what we now call "sound power level", usually abbreviated as SPL. More specifically, this ratio is generally cited on a log scale, in decibels, as ten times log10 of the ratio. (For a more elaborate version of this story, see "The dormitive virtue of root-power quantities" 8/29/2013.)

But in 1860, a dozen years before Ellis's article, Gustav Fechner had published Elemente der Psychophysik, in which he proposed and justified what has been called Fechner's Law, namely that

In order that the intensity of a sensation may increase in arithmetical progression, the stimulus must increase in geometrical progression.

This implies that "the intensity of a sensation" should grow as the log of the physical stimulus intensity:

Loudness ∝ log(Intensity)

And a century later, S. S. Stevens refined this general idea ("To honor Fechner and repeal his law", Science 1961), presenting data from various perceptual domains, including his work on loudness ("The Measurement of Loudness", JASA 1955), which concluded that

This paper reviews the available evidence (published and unpublished) on the relation between loudness and stimulus intensity. The evidence suggests that for the typical listener the loudness L of a 1000‐cycle tone can be approximated by a power function of intensity I, of which the exponent is log₁₀2. The equation is: L = kI^0.3. Intensity here is assumed to be proportional to the square of the sound pressure.

Thus revising Ellis according to Fechner, in place of Ellis'

…if one amplitude is 3 times as large as another, the force and the loudness of the first sound will be 9 times that of the second. 

we would indeed have a proportional increase of 9 to 1 in sound intensity, but this would correspond to a loudness increase of log10(9)-log10(1) = 0.954, so that the more intense sound would be a bit less than twice as loud as the less intense one. And similarly revising Ellis according to Stevens, we would have

L1 ∝ (1^2)^0.3 = 1 L2 ∝ (3^2)^0.3 ≅ 1.933

so that a factor of 3 increase in intensity again corresponds to a bit less than a factor of 2 increase in loudness, not a factor of 9 increase in loudness.

And similarly for Ellis'

If the first amplitude be to the second as 3 to 2, the forces are as 9 to 4.

Revised according to Fechner, we get log10(9)-log10(4) = 0.352, implying that an amplitude ratio of 3/2, corresponding to an intensity ratio of 9/4, corresponds to a loudness relation of about 1.352 to 1, not 9 to 4 (which is 2.25 to 1). And using Stevens' revision, we get

L1 ∝ (2^2)^0.3 ≅ 1.516 L2 ∝ (3^2)^0.3 ≅ 1.933

again giving a loudness relation of about 1.933/1.516 = 1.275, not 9/4.

So Ellis's discussion of force is an interesting combination of knowledge of acoustic physics (because he knows to square the waveform values to get intensity or power) and ignorance of acoustic psychophysics (because he thinks that subjective loudness scales directly with physical power). This suggests that as of 1873, Fechner's work was not well known in British scientific circles.

But Ellis's mistake about loudness is a small matter, compared to Thomas Edison's amazingly confused ideas about acoustics in general and speech in particular, as of the period five or six years later when he invented the phonograph. See Patrick Feaster, "Speech acoustics and the keyboard telephone: Rethinking Edison's discovery of the phonograph principle", ARSC Journal 2007 — more about this in a later post.