Laura Gallant Spring 2007 Steven Errede Physics 199POM Project: Analyzing Ensembles Music is a fundamental part of my life. As a music major I am often surrounded by music or musical concepts. The sounds of music are in my tests, my lectures, and my practice. Music can easily become commonplace and easy to take for granted. Upon taking the Physics of Music and Instruments course I was excited to have the opportunity to actually see what goes on in the air around me each time I hear a single note. Chamber music is an important aspect of the musical world. For my project I was interested in actually seeing the harmonic structure that we had talked about in Physics 199 lecture 1. I analyzed recordings of two ensembles with which I played this year. I played with a trio made up of oboe, cello and violin and with a quartet of two violins, viola and cello. According to John Backus 2, the first 12 harmonics above the fundamental C2 would sound like this: C2, C3, G3, C4, E4, G4, Bb4, C5, D5, E5, F#5, and G5. This intervallic pattern remains the same with any starting note. In analyzing the two recordings I looked for this pattern. I began by analyzing the trio. I recorded two friends and myself who had played as a trio all semester. I recorded us playing a C; the cello on C3, the violin and oboe on C4. I analyzed the harmonics using the MATLab computer program. The program created a chart of each harmonic I chose, measuring the frequency, phase and amplitude. For the
2 purposes of this paper I am looking at the frequency, or the number of vibrational cycles completed in a second. This defines the pitch of the note. I chose the 11 loudest harmonics and as the 12 th harmonic I chose a high pitched, oddly shaped harmonic out of curiosity. (See graph below: arrows mark harmonics chosen.) (MATLab also calls the fundamental pitch a harmonic.) The pattern of harmonics is very similar to Backus pattern from above. In looking at the average frequencies of the harmonics I came up with the following sequence: C3, C4, G4, C5, E5, G5, C6, E6, G6, an unclear harmonic close to A6#, C7, and the oddly shaped harmonic closest to C8.
3 The second harmonics of both fundamentals are louder than the rest. Because the bow/bowing of stringed instruments is not linear the second harmonic is generally the loudest harmonic. The C4 has a high amplitude because it is both the second harmonic of C3 and the fundamental played by both the violin and oboe. (See graph below.) C3 C4 G4 C5 E5 G5 C6 E6 G6 A6# C7 C8 The pattern follows that of the pattern described by Backus. The patterns overlap because two patterns are used, one beginning on C3 the other on C4, thus harmonics are shared by both harmonic groups. I was impressed to notice that the pitches that make up the pitch C are the same pitches used in a C Major chord. Perhaps this explains why the particular pitches which make up this chord, C, E and G, sound good together.
4 I next analyzed the recording of the quartet. The recording I chose was a recording of a piece my friend Matt wrote for his freshman composition course. The piece is entitled String Quartet in C minor. I chose to analyze only the opening note of the quartet and cut this from the recording. In this opening beat the cello plays a C2 (two octaves below middle C) and a G2, the viola plays a C3 and a G3, the second violin plays a G3 and a G4, and the first violin plays a C4. I analyzed this cut using the MATLab program. Because this recording involves 4 instruments playing several pitches the harmonic structure is more complicated than the harmonic structure of the trio. I decided to analyze 11 harmonics that looked to be major pitches of low frequencies. (See graph below)
5 The first pitch I identified was a C3; the C2 is not distinctly present on the analysis. I originally thought that because the cello is playing two notes the performer hit both notes, but rolled to the G2 without holding the C2 for very long. However, I then realized neither the C2 nor the G2 were showing up prominently. Through discussing this situation with Professor Errede 1 I realized an interesting phenomenon was occurring. The typical microphone used for recording would have a frequency response that drops quickly after 100 hertz. Because both C2 and G2 have frequencies under 100 hertz, they would not be easily detected and analyzed on the graph. The two low pitches are possibly represented the tiny peaks before the C3. The next pitch appears to be a G3, then C4, G4, C5, D5, G5, B5, D6, an unusual harmonic closest to F6, and another harmonic not easily fit, which looks closest to F6#. The higher the harmonic, the harder it is to get an accurate analysis. These two notes are included in the graphs of the analysis below, but are difficult to define. When choosing the opening beat of the piece, I had anticipated a pattern similar to that of a C, since C was the lowest note. However, I realized that there are more Gs than Cs in this beat, and three Gs are played as an open string (open strings, or strings played without finger placement, ring out more), while only two Cs are open strings. Thus the Gs are the largest harmonics, and the C harmonics are quieter, as demonstrated in the graph below.
6 C3 G3 C4 G4 C5 D5 G5 B5 D6 F6 F6# It is interesting to note that the G4 was the loudest of the harmonics. Two G3s were played, making G4 the second harmonic. That combined with the fact that the fundamental G4 was also played by the second violin makes G4 the loudest pitch. Note again the pronounced second harmonic because of the bow s nonlinearity. It is interesting to note that because both Cs and Gs were played by the instruments there are different sequences of harmonics occurring. I have created a graph, with the help of Professor Errede, and information from Backus book, demonstrating the overlapping qualities of the harmonics. From this chart, one can see how the harmonics are shared by many fundamentals when various instruments play. For the sake of space and because of the complexity involved with higher harmonics, I have given the number of the harmonic in relation to it s fundamental for only the second harmonic. I did not
7 include the two highest harmonics because of their odd shape. I also included the cello fundamentals not analyzed by MATLab. C2 (66Hz): Fundamental of cello(a) G2 (100Hz): Fundamental of cello(b) C3 (133Hz): Fundamental of viola(a), second harmonic of cello(a) G3 (200Hz): Fundamental of viola(b) and violin II(a), second harmonic of cello(b), harmonic of cello(a) C4 (263Hz): Fundamental of violin I, second harmonic of viola(a) harmonic of cello(a), G4 (392Hz): Fundamental of violin II(b), second harmonic of violin II(a) and viola(b), harmonic of cello (a) and (b), viola(a) C5 (524Hz): Second harmonic of violin I, harmonic of cello(a), viola(a) D5 (596Hz): Harmonic of cello(a) and (b), viola(b), violin II(a), G5 (780Hz):)Second harmonic of violin II(b), harmonic of cello(a) and (b), viola(a), viola(b), violin II(a), violin I B5 (984Hz): Harmonic of cello(b), viola(b), violin II(a) D6 (1182Hz): Harmonic of cello(a?) and (b), viola(a) and (b), violin II(a) and (b) I was excited to finally see through these analyses what I have been playing for all these years. Learning about the harmonic structure of these pitches gave me a greater appreciation for what happens each time I plunk a note at a piano or play my violin. It has given me greater awareness of why I hear music the way I do, and appreciation for the amazing world in which we live where an aspect I take for granted involves a complicated occurrence of events.
8 Resources 1 Errede, Steven. Physics 199: The Physics of Music and Instruments lectures, University of Illinois. Urbana, IL. Spring 2007. 2 Backus, John. The Acoustical Foundations of Music. 2 nd ed. New York: W. W. Norton & company, 1977. MATLab computer program www.mattthomascomposer.com for String Quartet in C minor score. Special thanks to Natalie and Sarah for volunteering their time to record, Matt for use of his String Quartet in C minor score and recording, and Ryan for his information about cello techniques.