Phase: Audible or not?

"The frequency-domain representation of periodic sounds was studied by the scientists Ohm, Helmholtz, and Hermann in particular. Ohm stated that changes in the phase spectrum, although they altered the waveshape, did not affect its aural effect. Helmholtz developed a method of harmonic analysis with acoustic resonators. According to these studies, the ear is "phase-deaf", and timbre is determined exclusively by the spectrum. ... Such conclusions are still considered essentially valid -- for periodic sounds only because Fourier series analysis-synthesis works only for those."
--Risset, 1991. p. 10

"It was Ohm who first postulated in the early nineteenth century that the ear was, in general, phase deaf, Risset and Mathews (1969). Helmholtz validated Ohm's claim in psycho-acoustic experiments and noted that, in general, the phase of partials within a complex tone (of three or so sinusoidals) had little or no effect upon the perceived result. Many criticisms of this view ensued based on faulting the mechanical acoustic equipment used, but Ohm's and Helmholtz' observations have been corroborated by many later psycho-acoustic studies.

The implications of Ohm's acoustical law for sound analysis are that the representation of Fourier spectral components as complex-valued elements possessing both a magnitude and phase component in polar form is largely unnecessary and that most of the relevant features in a sound are represented in the magnitude spectrum. This view is, of course, a gross simplification. There are many instances in which phase plays an extremely important role in the perception of sound stimuli. In fact, it was Helmholtz who noted that Ohm's law didn't hold for simple combinations of pure tones. However, for non-simple tones Ohm's law seems to be well supported by psycho-acoustic literature, Cassirer (1944)."
--Casey, 1998. p. 82.

"Fourier claimed in his writing that any periodic continuous signal could be represented by the sum of an infinite number of sine and cosine waves. This elegant description of periodic signals was later exploited by the 19th century physicist Herman Helmholtz (Helmholtz 1877). His view of the ear was that of a "frequency analyzer" based primarily on Fourier's mathematical theorem, Ohm's physical definition of a simple tone and the existence of a resonator in the cochlea, capable of accomplishing sound analysis. According to Helmholtz's theory, the cochlea behaved like a spectral analyzer analogous to the Fourier transform. He believed that the cochlea resonated at specific locations along the basilar membrane (Carterette and Friedman 1978), each tuned to specific frequencies. Helmholtz also claimed that the spectral magnitude components, and not the phase components, were the sole factors contributing to the perception of musical tones. ... [T]he importance of phase in perceiving musical sounds was demonstrated by Clark (Clark, Luce, Abrams, Schlossberg and Rome 1963), who clearly showed that in the absence of phase information, acoustic waveforms sounded unrealistic. This may be partly attributed to the fact that the highly transient onset part of a signal stores a great deal of phase information. Helmholtz's theory works well in ideal situations when a signal is periodic. However, real-life sounds are only quasi-periodic and vary considerably."
-- Park, 2000. Chapter 1

There is obviously much confusion as to whether phase is audible.

We've got Risset saying that the ear is phase-deaf for periodic sounds only. Casey says just the opposite, that "for simple combinations of pure tones" the phase information is important aurally. Park says that it works for periodic signals but not for quasi-periodic sounds. So.... as I said, there's some confusion. I aimed to make a few observations of my own. I had always heard (from stereo magazines, mostly) that phase wasn't important *except* in the cases of the 'attack' portion of complex sounds such as brass instruments. (The concern of stereo magazines is thus: does it matter if you hook up your speakers with reversed polarity? It would seem that the answer ideally should be no, but due to the capacitors and resistors involved in the speaker itself, some of which are "one-way" or have differing output depending on which direction the current takes through the component in question, differences are indeed often perceived in real-world equipment. The "correct polarity" sounds realistic, while the incorrect polarity introduces distortion into the signal. Somewhat ironically (at least to my tastes), this means that the answer to our question of whether phase is audible, becomes indeterminate essentially due to the Heisenberg Uncertainty Principle. We can have fun manipulating sound files on our computers all day, but what do these sound files sound like? The answer to this question is, they sound like nothing. You can't hear a thing until you have manipulated that signal enough to transform it into a physical vibration in the air which can be perceived by your ears. The very act of observing (or a necessary step in the observation process) alters what you're trying to observe. Unfortunately, there seems to be no way around this. No natural instrument can reverse phase polarity on demand, so any manipulated signal must be played through loudspeakers, which are inherently limited and will introduce distortion of the signal. Something to keep in mind. I was misled by an apparent particularity of my playback system in preparing this presentation (see below).)

Observations

It would appear as though the phase differences are indeed audible. Subtle distinctions can be heard among these 6 examples. When I focus on specific parts of the sound, I cannot tell a difference -- however, my initial reaction to each example is that it's different somehow. A slight shift in overall pitch or perhaps some concept of liveliness -- I can't put my finger on what seems different, but

It appears as though what I was hearing earlier was a result of my particular audio equipment. Upon further listening with Mr O'Donnell's equipment, I could no longer hear any differences.

You may want to normalize these files to bring them up to readily audible levels on your system.

See next section for a simplified version of the formula I used to generate these sound files. (Not that you couldn't figure it out.) I used a simple spreadsheet to generate the random numbers and splice together the formulas, although a C program could easily do the same.

Random Phases in a 200-Hz tone with 3 harmonics at amplitudes of 1/n

Example of function to create these sound files:

1*x*sin(2*pi*t*1*f+0)+0.5*x*sin(2*pi*t*2*f+0)+0.333333*x*sin(2*pi*t*3*f+0)

where
x=.3 (I found that (under all encountered circumstances) this setting would keep the maximum amplitude from reaching the point of clipping)
f=120 or 200 (frequency)
the 0's indicate that this is the initial, pseudo-triangle-wave form. The random phase information would take the place of the 0's for the random phase forms.

For the earlier 20-harmonic waveforms, there would be 20 summands rather than 3.

3sines.wav	3sines-random1.wav
3sines-random2.wav	3sines-random3.wav

Phases of Harmonics (in radians)
harmonic	3sines	3sines-random1	3sines-random2	3sines-random3
1	0	5.04924	3.18175	3.29374
2	0	5.31725	3.47728	3.83017
3	0	0.231593	2.47091	4.32364

Stereophonic Phase Inversion with a Pseudo-Triangle Wave

Waveform	Wave files
	trianglereg.wav triangle-rightinverted.wav
	3sinesreg.wav 3sines-rightinverted.wav

Playing with Dither

References