Planeación de suministros médicos en el área de

What became collectively known as theories of pitch for some authors (for example,

Rossing et al. (2002) [64], Stevens & Davis (1983) [79]), ortheories of hearing for others (for

example, Gulick (1971) [26]), refer to theories on how the ear physiologically resolves sound,

that is to say, when air is excited, how do our brains perceive it as a sound.

It is customary to refer to the ear’s ability to discriminate pitches in terms of the largest amount a frequency can deviate from itself and still be considered as the same tone.

For example, if we take a tone of frequency fbase and we start frequency–modulating that

tone up or down (call itfmod), at some point a listener will identify fmod as a different tone.

The difference between these two (fmodnon–inclusive, of course) is what is known as thejust

noticeable difference (jnd) or in older literature as difference limen.

Basic physiology is needed to facilitate further discussion on theories of pitch. A mechanical sound wave enters the ear canal causing the eardrum to vibrate. The vibrations on the eardrum are conducted through the middle ear via the ossicles—three tiny bone structures, the last of which are the stapes. The stapes, in turn, oscillate the oval window which signify the beginning of the cochlea in the inner ear. The cochlea is filled with fluid, and the sound mechanical vibrations are now hydraulic mechanical vibrations. The cochlea has a membrane in it, called the basilar membrane, which takes these hydraulic pressures and transforms them into electrical pulses, firing with specific neurological patterns in what’s known as the organ of Corti. The leap from the mechanical to the electrical happens by means of hair cells (celia) getting bent from the hydraulic pressure on the membrane which causes them to fire the electrical pulses. These pulses are then migrated into the brain via the auditory nerve, and electrochemical synapses are involved so that a sound perception is formed.

There are two classical theories of pitch, one known as the frequency theory and the

other known as the timetheory. There are a number of modern theories, as well. The next

two subsections, briefly talk about both and their relevance to this dissertation.

3.2.1 Classical Theories of Pitch

Thefrequency(orplace) theory of pitch has to do with where on the basilar membrane

monumental work on cochlear experiments. The basilar membrane is divided into 24 regions,

called critical bands, with each region being about 1.3 mm long and containing about 1300

neurons; each band acts as a unit for collecting sound data. The membrane itself is wider and loose at one end (close to the oval window) and narrow and stiff towards the other end (apex). Low frequency tones excite the wide portion of it and high frequency tones excite the narrow portion of it. The critical bands themselves have different frequency discrimination limits as well. The bands close to the wide, loose part have wider limits, thus making pitch discrimination of low–frequency tones less accurate than bands that pick up higher pitch tones at the other end of the membrane which happen to have smaller frequency deviation limits around the band’s center frequency point. In other words, jnd is a function of at least

one physical factor dictated by physiology, the frequency of the tone1_{. This theory views the}

ear as a spectrum analyzer.

The time (or periodicity) theory of pitch wants a time–series analysis applied to the

firing pattern of the electrophysiological impulses in the organ of Corti (Gulick (1971) [26]).

Shouten was not convinced that the frequency theory explains well–known phenomena like

the case of themissing fundamental 2, the ear’s ability to resolve the fundamental frequency

and the brain’s ability to process that information and “know” what the pitch of a tone is, even when the fundamental frequency was physically intentionally left out of the musical

complex tone3 (Rossing et al. (2002) [64]). Shouten called this missing lower part of the

spectrum the “residue,” and in a series of monumental experiments (in the first half of the

1940’s) convinced the scientific community that it is not merely the physiological place on

the basilar membrane that matters, but there must be some way for the brain to process the electrophysiological impulses from the organ of Corti and further into the brain. This

pattern analysis was done in time, hence the name of the theory.

This means that there must exist a centralized unit in the brain that processes the signal in both domains. This centralized unit must be selectively using either frequency (spectra) or time (autocorrelation) data depending on the situation. It has been suggested that lower tones are processed primarily by the time domain analyzer and that higher tones are processed by frequency analyzers, with checks and balances between the two (modern theory). Echolocation, for example, could have an effect on how the time–frequency analyzers

weigh input/output from one another. An interesting field, called auditory computation has

since risen dealing with the mathematical neuromodeling of acoustic nerve firing patterns, but which in general encompasses a wide variety of models, from sound to perception. A short account of this fascinating field will be provided below.

1_{An interesting psychoacoustics experiment would be to quantify vibrato for lower and higher tones.} One would expect that since the ear is less fine–tuned at the lower frequency span, that vibrato would be wider there compared to higher tones (directly proportional to the bandwidths). I also always thought that another consequence of this physiology would be that singing voice would be prone to more vibrato at the lower spectrum compared to instruments like violin at the same low frequencies, because a violinist can use his finger to “correct” this paradoxical limitation of her brain, but a vocalist has only but his ear to correct his ear. The signal does not pass through another “self–correcting” loop.

2_{This phenomenon was demonstrated about 100 years before by Seebeck (1841), but the paper is in} German which I unfortunately do not speak. Shouten’s work is what made this case widely known in the world of psychophysics.

3_{Complex here refers to many (hopefully harmonic) partials, as opposed to a pure tone which is only} one wave with nothing above it.

3.2.2 Modern Theories of Pitch

Logic would suggest that from smaller axioms, lemmas, or theorems a scientist should be looking for a higher–level, unifying, or universal theory that would encompass the smaller parts of knowledge units and generalize them in as much simplicity as it is possible. At the same time (a second criterion, if you will) these general theories should be based on observa- tion. In optics, the once distinct doctrines of corpuscle (Newton) and wave (Huygens) light theory were unified to what now explains most reality as the dual nature of light. Professor A. Einstein moved towards this unifying direction, albeit unsuccessfully, by attempting a “unified field theory,” a theory that would bring gravity, nuclear forces, and electromagnetic

theory under one umbrella4_.

It is, therefore, very unclear why modern theories of pitch perception (mainly those originating in the area of cognitive psychology) fixated on theories that not only do not seem to follow the logical idea of parsimonious unification and generalization, but also do

not explain all of the phenomena we observe in empirical data (Rossing et al. (2002) [64]).

Since psychoacoustics is a vehicle for understanding the results of this dissertation and not the main topic of it, modern pitch theories en masse will not be discussed. One,

however, will be outlined, that of Moore (2003) [45]. It follows Occam’s razor and explains

most observations, compared to competing theories.

Loven (2009) [36] is probably one of the sources that simplify and tailor Moore’s

theory (Moore (2003) [45]) in a practical way that fits our needs. It is a three–layer theory

with frequency, time, and adjustment as its three layers. The first two layers have two levels each and the single–level final is a consideration of other variables to adjust the different weights for the first two layers in addition to considering some new information. The theory

is summarized in Figure (3.1).