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Chad Stearns added section_Method_for_frequency_determination__.tex
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\section{Method for frequency determination}
Recordings were collected from the Javanese Gamelan instruments at Arizona State University with the help and permission of Dr. Ted Solis. Starting ten minutes before a Gamelan Ensemble class, Dr Solis and I recorded two short samples, one from the Saron Barung, and one from the Saron Demung. During each sample Dr Solis would twice play through each of the 6 bars of Slendro scale, ascending from the bottom, and letting each tone ring for about 2 seconds. After each tone, Dr Solis would silence the prior bars to ensure that each bar sounded unaccompanied by the other bars.
I used the audio software Audacity to determine the frequencies of each tone in the Slendro scale. Rather than rely on Audacitys frequency determination processes, which I have found to be unreliable, I determined each frequency by a guess and check process using my ear. For each tone in the Slendro scale, I would generate a sine wave of comparable volume, and arbitrary frequency. I repeatedly played the slendro tone, and the sine wave concurrently, while adjusting the frequency of the sine wave until it was perceptually identical to that of the slendro tone. The process was is analogous to how two violinists might tune the strings of one violinists violin, to the strings on the other violinists violin, however instead of strings, it was between a recording of a Saron, and a sine wave of a known frequency.
\subsection{Margin of error in determining Slendro Frequencies}
To detect if the Gamelan recording, and the sine wave were in tune, I would observe whether or not there was ‘beating’ between the two sounds. Beating is the auditory phenomenon of two tones very close in pitch, oscillating between being in and out of phase with each other\footnote{When two tones of the same frequency are in phase with each other, their amplitudes at any time t are the same, and therefore combine when sounded together. Putting aside how sound reflects in a physical environment, a listener hearing two tones with the same frequency and amplitude perceives the tone twice as load than if he heard one tone at the frequency. When two tones of the same frequency are out of phase, their amplitudes at any time t are opposite, and therefore cancel out when sounded together. A listener would hear silence if two tones of the same frequency but out of phase were sounded.}.
The closer two tones are to each other in frequency, the longer it takes for a beat to occur between them. As the difference between two frequencies approaches zero, the duration of a beat approaches infinity. Because beats happen on such slow time scales, they are individually perceivable and countable.
If a human being detects frequency differences by the presence of beating, then, roughly speaking, a beat of a duration greater than twice the duration of the tone sounding cannot be detected by a human being. Each bar in the samples was sounded for about 2 seconds, which puts an upper limit on a human listeners capacity to detect frequency differences. The duration of beating for two nearby frequencies can be determined by the following equation
