Steampunk phonetics, continued

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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 = kI0.3. 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 to 1, not 9 to 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.



  1. DCBob said,

    August 24, 2015 @ 9:55 am

    It's always interesting to watch admirable early pioneers puzzling their way through the thickets, especially Henry Higgins his own self. One of my prized possessions is a near-pristine copy of Ellis' "Part V. Existing Dialectal as Compared with West Saxon Pronunciation" complete with maps. His delineation of English dialect regions on the basis of distinctive features is just amazingly good work.

    [(myl) And for others not so bibliographically fortunate, there are now good digital versions available for free, e.g. here and here.]

  2. Guy said,

    August 24, 2015 @ 12:10 pm

    I'm afraid I don't quite understand how this is something other than a matter of convention. Is there some verifiable (as true or false) meaning in the assertion that people perceive loudness in logarithmic relation to the amplitude?

    [(myl) Yes — I'm afraid you're just displaying your ignorance here. Please read either of the S.S. Stevens articles that I've linked to, or any other account of 150 years of psychophysical experimentation (for example this one or this one), for a description of the several experimental paradigms, with literally thousands of replications and publications, that all point to this general conclusion. (Though Stevens argues for a power law as a more appropriate functional form — and as Luce and others have added various caveats and extensions — the nature of the relationship between physical intensity and experimental measurements of physiological or psychological responses is not at all just a "matter of convention".)]

    I mean it's certainly true that a variation of a constant number of Watts per square meter is more obvious in a small sound than a large sound, but it's also true that a variation of 5 mm is much more obvious in a pebble than in a mountain. Does this mean we percieve length logarithmically? It wouldn't normally occur to me to interpret a claim that some physical phenomenon corresponds to some perception as asserting a mathematically linear assertion because it is in no way clear to what it would even mean to assign numbers to a strength of a perception except in relation to some measuring convention, whatever mathematical form that convention takes.

  3. Guy said,

    August 24, 2015 @ 3:26 pm


    (Before I begin my main reply, you seem to be assigning the quieter noise a value of 1, and the louder noise a value of 1+ the common logarithm of the ratio, so you are saying a sound that is ten times as loud as another is always subjectively twice as loud, and that the ratio between the sounds measured in decibels is not the subjective ratio that is perceived. Is this an error? Under your math, if sound B is twice as subjectively loud as sound A and sound C is twice as subjectively loud as sound B, then sound C has a square intensity that is 100 times that of sound A and so sound C is only three times as subjectively loud as sound A, not four, which seems incoherent. Did you mean to say that the perceived ratio in loudness would depend on the value of the sounds in relation to the threshold of hearing, so that 50dB is 1.25 times as loud as 40dB, but 60dB is only 1.20 times as loud as 50dB, even though the ratio between the square amplitudes is the same? Under your interpretation the ratio of perceived loudness between a 50dB sound and a 40dB sound equals that between a 60dB sound and a 50dB sound, which would contradict your point that perception is logarithmic.)

    If I understand the linked papers correctly, then, the claim that sensation is logarithmic in relation to the physical phenomenon is essentially the claim that when people are asked to "rate" the sensation on a scale, you get geometric rather be than arithmetic progression in the category boundaries. If that's what it means then it seems to be both meaningful and true, but I think it also unfair to describe Ellis' account as error. If it means something else or more than that, then I suppose I missed something. I don't deny that I'm mostly ignorant in this field, so I understand that it may seem annoying that I seem to be questioning what I know is a widely accepted fact in the literature. But I want to emphasize that I'm not claiming that it's false, or even that it's meaningless, rather I'm suggesting that what meaning it does have is dependent on a specialized technical interpretation of "ordinary" language – "how loud is it?" that doesn't necessarily correspond to how that language might be interpreted outside of the relevant field in some contexts – including perhaps the context in which Ellis was writing.

    I would expect that if people were asked to rate the "bigness" of items on a scale of 1-10, with items ranging from fleas to flies to cats to people to houses, you would get a roughly logarithmic classification, as my examples reflect. I think it would be unfair, though, on this basis, to say that it is "error" to say that our perception of size is the space that an object occupies as inferred from the light that enters our eyes (or tactile data from our fingers) – at least in a context where that sense of "perceived bigness" is not well-established. If someone asked me to say when something is "twice as loud" as something else, I would probably interpret that as asking me to say when it is as loud as two copies of that sound are being produced (whether I would be able to accurately discern when that is is another matter), but if I were asked not to interpret "twice as loud" that way but rather in terms of the subjective strength of my perception, I would be confused about how to interpret that, because I'm not sure what it means, subjectively, for a subjective experience of sound to be twice as intense. Under the logarithmic interpretation, that would presumably mean that the ratio between the threshold of hearing and the quieter sound equals the ratio between the quieter sound and the louder one. Maybe I'm unusual and most people feel they have a clear idea of what "twice as subjectively intense" means. But in any event, I wouldn't call it an "error" to use terms like "twice as loud" in the sense of "as loud as two things making that noise" rather than "twice the subjective intensity of the experience", whatever experimental subjects might interpret that to mean, as long as the speaker is speaking in a context where loudness doesn't have the specialized meaning of "experimental subjects rated this noise with a number twice as big as the other".

    Likewise if someone asked me to judge whether, given three volumes, the spacing between them is equal. I'm not sure how I would interpret that. Going back to "size" (which we can think of as a vague category eliding the distinctions between length, area and volume), I can think of situations where I would label one object as being "intermediate" logarithmically rather than arithmetically (for example if asked to find a boat that was at the "midpoint" between a canoe and a cruise ship, I might very well pick a yacht or something, and I doubt that when asked to find the midpoint between a cat and a flea, many people would pick something nearly exactly half the length – or volume – of a cat, rather they would likely take something smaller, though I may be wrong). This doesn't indicate anything about the strength of the perception, though, it relates to how I categorize the mental objects I create to represent sensory stimuli that are ultimately highly complex, just as the perception of a "single" sound is really an amalgamation of a highly complex set of data that can be simplistically thought of as a number of "holistic" measures such as pitch (ultimately derived from greatest common divisor of constituent frequencies) volume (ultimately a function of amplitude/power output/logarithm thereof as measured on whatever numeric scale) and "quality" (a grab-bag of salient features of the data set). But maybe the study of qualities of what I would call (likely I artfully when I could use better terminology from the literature if I were familiar with it) post-perception mental objects, such as the idea of a canoe as constructed from memories of sight, touch, and other sense data, is considered to be a part of the realm of perception.

    I feel like this is similar to how people sometimes say that the brain "flips" the inverted image on the back of the retina to be in "the correct orientation", which talks about the mental image as if it has some orientation in physical space (one that needs to be "correct", for some reason). It's true that we correctly associate the mental stimulus coming from our left and projected on the right of the retina as being to the left in our mental image, according to our mental images notion of "left", but it's at best highly misleading, and at worst highly confused, to describe the existence of this correct association as "flipping the picture back", as though the fact that the image is inverted on the retina creates extra work that the brain has to do. But if the formation of the correct association is conventionally described in metaphorical terms as though the mental image has an orientation in physical space, then such language may be fine in contexts where that kind of metaphorical usage is well-established and listeners understand it with the correct meaning.

  4. Guy said,

    August 24, 2015 @ 3:44 pm

    Sorry, where I wrote above "a sound that is ten times as loud as another is always subjectively twice as loud" I should have written "a sound that has ten times the square amplitude of another is always subjectively twice as loud". Similar substitutions should be made anywhere else that I make similar formulations. I don't mean to argue in favor of the position that subjective intensity is linear with square-amplitude. And such formulations, in addition to being sloppily vague, could create the impression that that's what I'm arguing.

  5. D.O. said,

    August 24, 2015 @ 4:18 pm

    I will pile on the first (parenthetical) part of Guy's comment. Indeed, based on Fechner the ratio of loudnesses is log(I2/I0)/log(I1/I0) (I2 and I1 are 2 signals and I0 is threshold). And you must establish the threshold to make it work. Stevens gets (I2/I1)^0.3 independent of threshold.

    It's also funny that Stevens claims that his power 0.3 proposal is, in fact, log2. Gotta read the paper to find out how he hit upon that explanation.

  6. D.O. said,

    August 24, 2015 @ 4:39 pm

    Actually, Stevens discusses intricacies of experiments to determine loudness scale, including such factors as what sorts of dials to use and people's bias toward mid-range (40-60db) loudness. Of course, it cannot be the last word on the subject.

  7. Guy said,

    August 24, 2015 @ 5:41 pm

    In the Science paper by Stevens (which has a broken link – missing the "S"). I find interesting the contrast between the "category" and "subjective magnitude" tests interesting. If the shape of the curve varies with the granularity of the scale – or perhaps with the establishment of endpoints – it seems reasonable to say that the curves we are looking at reflect post-perception judgments about how to categorize and classify, not immediate measures of intensity of perceptions. It would be interesting to see how these curves look with measures of "bigness", where it's relatively clear (I think??!?) that our judgments are "post-perception" in the sense I was suggesting before.

    [(myl) Modern psychophysics also includes neurophysiological measurements, as diagrammed in the Ehrenstein & Ehrenstein chapter:


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