The power spectrum of each sample had the expected shape, with the maximum spike at the fundamental frequency and smaller spikes at the overtones.  I took data both close to the microphone and further away.  Figure 11 is the power spectrum of an A at 880 Hz with player 1 on flute A only about 5 inches from the microphone.  The y-axis is normalized by the maximum spike which occurs at 2f, 2640 Hz (E).  If this E were the prominent frequency, a listener would most likely hear an E instead of an A at 880 Hz.  Figure 9 is the same note played at a distance (9 feet) from the microphone.  The power spectrum shows the maximum peak at the fundamental frequency, 880 Hz, which is what the listener hears.  In order to analyze the same frequencies that a listener hears, all of the following data was taken as a point source, approximately 9 feet from the microphone.
     I took all of my data without vibrato.  Vibrato is a pulse created in the sound by the player.  The pulse causes the frequencies to oscillate around the actual value. Figure 12 is an A at 440 Hz played by player 1 on flute A without vibrato.  In Figure 13, player 1 used vibrato.  The spikes appear at the same frequencies in each graph.  However, as shown in Figures 14 and 15, close-ups of the overtone with maximum relative amplitude (2f or 1320 Hz), the spikes in the note with vibrato are thicker.  This is due to the oscillations around the frequency.
     The higher notes (higher frequencies) have a smaller harmonic series on a power spectrum than the lower notes.  Figures 3, 9, and 10 are power spectra for three different octaves of A, 440 Hz, 880 Hz, and 1760 Hz, respectively, all played by player 1 on flute A.  Figure 9 shows that player 1 only has 2 overtones for 880 Hz.  The first overtone, 2f, is 1760 Hz (A), and has a high relative amplitude.  This spike is an average of 20% of the fundamental frequency.  The spike at 3f, 2640 Hz (E) is very small (less than 1% of the fundamental). Figure 10 shows that for an A at 1760 Hz, which is the third octave A on a flute, there really is no harmonic series present.  There is a very tiny spike at 3520 Hz, an octave above, but it is much less than 1% of the fundamental.
    I also compared the power spectra of 4 different octaves of the note B, 247 Hz, 494 Hz, 988 Hz, and 1976 Hz.  Figures 16, 17, 18, and 19 show these plots, respectively.  247 Hz is the lowest note that a flute with a B foot joint can play.  As seen in these graphs, the lower notes have more overtones from the harmonic series than do the higher notes.  This is the same pattern that was seen in the different octaves of A.
    The amplitudes and the number of overtones that appear vary depending on the player and on the flute.  A comparison of the power spectra of three different players all playing the same note on the same flute shows that the player has a greater effect over the frequencies produced by the flute than any characteristic of the instrument.  To be consistent, I used A at 440 Hz for the comparisons between flutists and between flutes.  As seen in Figure 3, player 1 produces the first 4 overtones of the harmonic series of A at 440 Hz.  The overtone with the greatest relative amplitude compared to the fundamental is 3f (2200 Hz), which is an E.  As shown in Figure 4, Player 2 creates the first 5 overtones with the relative amplitudes decreasing with each subsequent overtone.  Player 3 produces the first 6 overtones, as seen in Figure 5.  In this plot, the spike at 3f has a very high relative amplitude compared to the fundamental, such that in some of the samples, it is actually almost equal to the fundamental.  Figure 21 shows the average relative percentages for each overtone of the fundamental frequency for the three players on flute A.  For the frequency values and corresponding notes of each overtone, see Table 1.
    The previous comparison showed that the player has a greater effect on the power spectrum of a note than does the flute.  The next comparison is player 1 playing an A at 440 Hz on flutes A, B, C, and D.  Figure 3 is player 1 on flute A, as described in the previous comparison.  Figure 6 shows the power spectrum for player 1 on flute B.  The overtones in this plot have similar relative amplitudes compared to Figure 3. Figure 7 shows the power spectrum of player 1 on flute C.  The relative amplitudes decrease as the frequency increases. Figure 8 is the power spectrum of player 1 on flute D.  The overtones in this plot have low relative amplitudes.  Player 1 seems to always create only four overtones, except on flute D, where 6f has a very small peak.  Figure 22 shows the average relative percentages for each overtone of the fundamental frequency for player 1 on the four different flutes.
    As seen in the previous data, the flute player has a greater effect on the resulting harmonic series for each note, than does the instrument.  The greatest uncertainty in the frequency measurements for each note is due to the human factor.  Each player can change the resulting frequencies by changing her embouchure, how much air she uses, and any other factor that is part of playing a wind instrument.

Abstract
Introduction
Experimental Setup
Results
Discussion
Conclusions
References