Music Education or Cross Platform Term Paper

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Aristoxenos, two centuries after Pythagoras released his model, sought to discredit the standing theories held by Pythagorean devotees. In his works, he established that numbers are not relevant to music, and that music is based on perception of what one hears, not any mathematical equation. Descartes as well as Vincenzo Galilei (Galileo's father) both also discredited the music-to-math theories that formed the revolutionary basis for Pythagoras' music work, but not on the basis that music and numbers are unrelated. Rather, Galilei in particular figured that the tension of a string compared to the pitch made by that string should be the variables to create the sound ratio, not the length of that string. "Using weights to vary the tension of a string, he found that the above mentioned intervals arise for ratios of 1:1, 1:4, 4:9, and 9:16 respectively. These ratios as different from those found for length; they are more complex, and don't agree with the importance that the Pythagoreans gave to numbers from 1 to 4." (De Cheveigne 2004) the debate between followers of Pythagoras and followers of Galilei (and the many others who have contributed conflicting pitch theories) is symbolic of the questions in music theory that remains today about what precise roles, and the importance of each role, that are taken by mathematics, physics, and perception -- the laws of the universe.

One can see how the development of music and pitch theories exemplifies the cyclic nature of history and science, not only because of the continued questions regarding perception and the place of the sciences. Anyone familiar with the development of music-related software may recognize the same type of dueling theories and conflicting coding and hardware ideals among developers. However, issues surrounding the development of actual software and hardware for the purpose of pitch recognition and other music and sound related purposes will be discussed later in this literature review.

Returning to the time period at hand, using the Greek concept of pitch lead to the understanding of pitch relations as consistent with the physics of sound. Galileo measured the relationship between string length and vibration frequency, however Mersenne lengthened the strings used and was therefore able to count the individual vibrations -- a measurement which is today taken by use of technology that would have been inconceivable at the time, but which functions by a similar means as counting the vibrations by sight. Mersenne was therefore able to determine "the actual frequencies of each note of the scale. This provided a relation of pitch with number that was firmly grounded in the physics of sound." (De Cheveigne 2004)

Resonance was noted by Aristotle, and it became a concept that has been used since that time in theories on hearing, the popularity due in part to the already common notion of "like by like." (De Cheveigne 2004)

Du Verney, a theorist from the 1600s, proposed many important notes about resonance theory, and his work "concentrates several key concepts of place theory: frequency-selective response, tonotopy, and tonotopic projection to the brain." (De Cheveigne 2004) the cochlea was compared to a steel spring in order to explain resonance, suggesting that the bony spiral lamina was the source of resonance. Others later suggested that the basilar membrane had strings like a harpsichord.

The concepts relating to superposition and Ohm's law were difficult for many theorists to grasp until the 1700s. Mersenne, previous to that time, "reported that he could hear within the sound of a string, or a voice, up to five pitches.... He knew also that a string can respond sympathetically to higher harmonics, and yet he found it hard to accept that it could vibrate simultaneously at all those frequencies." (De Cheveigne 2004) it was during the eighteenth century that the terms "fundamental" and "harmonic" were first used, and also the time period when the actual physics of string vibrations -- multiple vibrations included -- were comprehended on a relatively complete level. Linear superposition was a concept introduced by Euler, which was a particularly important finding for making the simultaneous vibrations at different frequencies comprehensible.

Earlier researchers, such as Mersenne and Galialeo, thought of vibrations as periodic, but the shape of the vibrations was not taken into consideration. Mersenne, of course, had no way to observe the shape of the periodic vibrations. This was one of the concepts that revolutionized the physics of sound in the eighteenth century, and Fourier developed a theorem regarding the superposition of sinusoids in 1820 that impacted the mathematics and physics of the time period, and in the developments of the physics of sound that would follow.

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Fourier's theorem stated that a vibration might contain several sinusoidal partials, and therefore several different frequencies, depending on the shape. Ohm's law, found in 1843 but later made far more clear, stated that every pitch corresponds to a sinusoidal partial within the stimulus waveform. "Ohm's law extended the principle of linear superposition to the sensory domain....The sensation produced by a complex sound such as a musical note was 'composed' of simple sensations, each evokes by a partial. In particular, [Helmhotz, who rephrased and clarified Ohm's work,] associated the main pitch of a musical tone to its fundamental partial." (De Cheveigne 2004) Ohm therefore related pitch to the period of one of the sinusoidal partials, not to the period of the vibration as a whole.

The work done by Ohm and Helholtz directly contradicted work completed previously y Seebeck and others. Seebeck found that pitch does not depend on a particular partial. However, Helholtz was able to explain certain aspects of higher pitches and other circumstances that had been observed by previous researchers that did not fit into the work by Seebeck, and he believed strongly in Fourier's theorem and could not fathom that any work which did not allow Fourier's theorem to remain intact could be valid. The work by Ohm, Helholtz, and Seebeck have appeared in contradicting arguments in many later works, and has served to be a quandary for many researchers.

Pattern matching models assume that pitch, when the fundamental partial is missing, continues to be perceivable by the human mind because of the human ability to reconstruct patterns when a part of that pattern is missing. This answers the questions posed by many pitch researchers regarding whether or not the fundamental partial is actually the necessary correlate of pitch. If the human mind can use other parts of the pattern, such as the harmonics associated with the pitch, then the pitch may be perceivable without this necessary correlate. "This idea was prefigured by Helholtz's 'unconscious interference' and...Mill's concept of 'possibilities.' As a possible mechanism, Thurlow suggested that listeners use their own voice as a 'template' to match with incoming patterns of harmonics." (De Cheveigne 2004) Throughout the 1900s, many researchers suggested variants on the pattern matching theories, suggesting both learned -- such as Terhardt's work which introduced the virtual pitch -- and intuitive means by which the human mind perceives the pitch. Terhardt's learned model was responded to later by the introduction of Shamma and Klein's suggestion that exposure to noise could produce such learning; therefore, harmonic relations may be interpreted as a mathematical property which is discovered, rather than learned in specific.

Temporal models were generally less elaborate than resonance models; early temporal models suggested that pulse patterns were "handled" by the brain, rather than within the ears. The concept of strings (or other sound-producing things) which vibrated hitting the air many times and pitch therefore being a reflection of how many times the air punched forward from it hit the ear, was important to these early temporal models. It was in the fifth and fourth century BC that Democritus and Epicurus, respectively, first introduced this idea by stating that a sound-producing body actually emits atoms that are projected to the listener's ear. Anaxagoas, in the fifth century BC, explained that hearing was "penetration of sound to the brain," and Crotona, also in the fifth century BC, elaborated that "hearing is by means of the ears, because within them is an empty space, and this empty space resounds." (De Cheveigne 2004)

Perhaps the most notable difference between temporal and resonance models is the amount of time which is required to make a frequency measurement. Resonance involves the build-up of energy by accumulation of successive waves. Helholtz found that notes, in music, could happen at a rate of eight notes per second; based on this time-frame, Helholtz calculated what he believed to be the narrowest bandwidth for cochlear filters. Therefore, frequency resolution was not related to cochlear filters, but rather dictated by temporal resolution. "In contrast, a time-domain mechanism needs just enough time to measure the interval between two events (plus enough time to make sure that they are not both part of a larger pattern). The time required is on the order of two periods of the lowest expected frequency." (De Cheveigne 2004)

Temporal models can be seen to rely heavily on the definition of events. However, it….....

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