The Mathematics of Music 1 The Mathematics of Music and the Statistical Implications of Exposure to Music on High Achieving Teens Kelsey Mongeau Practical Applications of Advanced Mathematics Amy Goodrum
The Mathematics of Music 2 Abstract Music and mathematics are related through a variety of factors. Most of the features of music can be described or modeled through a mathematical formula including the frequency of the sound and waves produced and the timing of the piece. There is some debate as to whether these attributes of music can increase intelligence and whether or not exposure in early life may lead to greater intelligence and higher achievement. While these mathematical patterns can be clearly found within musical pieces, this study does not show adequate evidence as to whether or not exposure to music affects intelligence.
The Mathematics of Music 3 Background All musical patterns have a basis in music. From sound to rhythm to more complex patterns of chords and composition, mathematics can be applied. In some cases it is believed that exposure to music can lead to greater intelligence. Sound Sound is a measure of vibrations of air. It can be measured in amplitude, frequency, duration and spectrum (Benson, 2007). Amplitude is the measure of loudness and can be quantitatively determined by the size of the peak of the sound wave. The larger the wave is the louder the sound. Amplitude is measured in decibels. The hearing threshold is 0 db. The perceived pain threshold is approximately 60 db although this varies slightly from person to person (Hass, 2003). Frequency is pitch. It is measured in Hertz (Benson, 2007). All notes are based off of the A above middle C, which is keyed at a frequency of 440 Hz (Fauvel, Flood, and Wilson, 2003). Each octave is twice the frequency of the preceding octave; therefore the A below middle C is 220 Hz and the A two octaves above that is keyed at 880 Hz (Heimiller, 2002). Duration is simply the length of time for which a note is sustained. Spectrum is timbre. This quality of sound is the unique tone of the sound similar to the way two human voices do not sound exactly alike. It can be determined by the shape of the sound wave (Benson, 2007). With this information sound can be used to tune notes. The oldest known system of tuning a scale is the Pythagorean tuning system which originated in terms of its mathematical explanation, around 550 BC. This scale is based off of whole number rations where a note vibrates with a ratio of frequencies of 2 to 1 with the octave below it. The fifth of a chord, for
The Mathematics of Music 4 example a G with a root note of C vibrates at a rate of 3 to 2. This is the perfect fifth. The octave of that C is then separated by a ratio of 4 to 3 from the G. This is the perfect fourth. To create the note F, which is one step down from G the frequency of the octave of C is divided by the perfect fifth to get a ratio of 4 to 3 from the original C. When both G and F are reduced by the perfect fourth, the second tone of the scale, in this case a D, is found. With this, the order of the other notes can be found. The interval between a whole step is 9/8 while the interval between a half step is 256/248 (Fauvel, Flood, and Wilson, 2003). Rhythm and Tempo All rhythms are fractions based off of the time signature and the length of the duration of a note. In the time signature, the top number represents the number of beats per measure while the bottom number represents which note gets the beat. For example, in basic time, there are 4 beats per measure and the quarter note gets the beat. As a quarter note in this case is one beat, the duration of the other notes can be assigned. A whole note is 1/(1/4) or 4 beats; a half note is (1/2)/(1/4) or 2 beats: an eighth note is (1/8)/(1/4) or ½ of a beat. This pattern continues where the top number is always the fraction represented by the note name and the bottom number is the note that the time signature has designated as representing one beat. The pattern could be considered infinite although one will most likely stop after no smaller than a 64 th note (Cossaboom, 1973). Tempo is measured in beats per minute. Therefore in time, = 60 is equal to 60 beats per minute or 1 beat per second. This is also based off time signature as the same note given the beat in time signature gets the beat when noting tempo (Cossaboom, 1973)
The Mathematics of Music 5 Trigonometry in Music Trigonometric Functions can be applied to various parts of music. Sound waves can be modeled through functions of sine. This equation is c sin(2πvt + Ф) where frequency = v, peak amplitude = c, and phase = Ф. If you then take the frequency of the note produced, for example the A above middle C, and plug it into the equation it produces the equation for that note. In this case the note can be modeled as c sin(880πv + Ф). When modeling the sound waves of two notes being played together, while ignoring amplitude and phase, the wave can be modeled as For example the A above middle C at 440 Hz and a slightly flat A above middle C at 432 Hz, would be written as This can be heard as a pulsing sound that is created when the two notes played are played together (Benson, 2007). Increasing Intelligence through Musical Exposure There is some debate as to whether the exposure to mathematical patterns found in music can actually be used to increase intelligence. Proponents of the so called Mozart Effect claim that listening to classical music, particularly from an early age, can increase I.Q. (Rusin, 2006). The original study of this theory was conducted on a group of college undergraduates who showed significant short term increase in spatial intelligence. However, since then there has been only limited success with similar experiments. Little testing has been done to prove that other styles of music do or do not increase intelligence and despite common belief that listening to classical music in infancy will lead to greater intelligence at a later stage in life, no studies have been performed on infants to corroborate this belief. There is no consistent concrete evidence presently to confirm the existence of the Mozart Effect (Plucker, 2007).
The Mathematics of Music 6 Research Question How do mathematics and music correlate and what are the implications of music as they affect high achieving teens? Methods To answer the first part of my question, namely the correlations of mathematics and music I conducted the research found above in the background section to determine several branches of music that contain aspects of mathematics. I then analyzed the first four bars of a simple arrangement of Carol of the Bells found in Appendix A to model the waves which represent the chords. To do this I plugged in the frequencies found in Appendix B of the two notes in the chord into the equation. This became the graph found in Appendix C of the model of the sound wave. For the second part of my question, specifically the effect of musical exposure on high achieving teens I conducted a survey, which can be found in Appendix D. I created this survey through Survey Monkey, a free online site that allows the surveyor to create and distribute it to through the internet. I then created a Facebook group through which I invited my fellow Summer Ventures students, some friends, and some students from the North Carolina School of Science and Math to take the survey and asked some Summer Ventures students the questions directly. Results Modeling the Music
The Mathematics of Music 7 The following equations represent the chords heard while listening to this arrangement of Carol of the Bells found in Appendix A: A table of all frequencies used may be found in Appendix B and a graph of each equation may be found in Appendix C: C and Eb C and D C and C Bb and Eb Bb and D
The Mathematics of Music 8 Bb and C Ab and Eb Ab and D Ab and C G and Eb
The Mathematics of Music 9 G and D G and C Survey of High Achieving Students In the study of high achieving students found it was found that all participants surveyed had a GPA higher than 3.5 and an average math grade higher than 85. Of the 62 participants who responded to the survey 98.39% listen to music at least once a day and 83.87% stated they had been exposed to music since the age of 5 or younger. The majority of participants listen to rock music at a rate of 62.90% of respondents, followed by pop which had a 59.68% following. Conclusions/Recommendations/Extensions Mathematics is present in a wide variety of areas in music. Through my results it is shown that music does affect higher intelligence as 83.87% have been exposed to music from an early age. However, based off of the limitations of my survey including the small sample
The Mathematics of Music 10 population and the lack of a population representative of all levels of intelligence, I believe that these results are inconclusive. In this paper I was limited by a several factors. In my research into the mathematical properties of music and my analysis of music based off of these properties, I was unable to use the many properties of music that can be described through calculus as I do not yet understand calculus. Furthermore, I was limited to a very small population to survey which was made up of Summer Ventures students, my friends and some of the students from the NC School of Science and Math therefore I had only a population of high achieving students to base my results off of and not a whole population in which I could compare the results of high achieving students to those who were not a high achieving. For greater correspondence between the mathematical patterns of the music and the results of the survey, I would have liked to describe many different mathematical patterns found in several different genres of music. However, because of time limitations and the necessity to confirm copyright laws, I was unable to do this. This would be an excellent study to perform given a more diverse sample population, such as a high school, a more expansive knowledge of higher level mathematics and more time.
The Mathematics of Music 11 References Benson, D. (2007). Music: A mathematical offering. New York, New York: Cambridge University Press. Cossaboom,S. (1973). Fundamentals of music theory. Boston, Massachusetts: Crescendo Publishing Company Fauvel, J., Flood, R., and Wilson, R. (2003). Music and mathematics: From Pythagoras to fractals. New York, New York: Oxford University Press. Hass, J. (2003). What is amplitude. Retrieved from http://www.indiana.edu/~emusic/acoustic/ amplitude.htm Heimiller, J. (2002). Where math meets music. Retrieved from http://www.musicmasterworks. com/wheremathmeetsmusic Plucker, J. (2007). Human intelligence: Mozart effect. Retrieved from http://www.indiana.edu/~ intel/mozarteffect2.shtml Rusin, D. (2006) Mathematics and music. Retrieved from http:www.math.niu.edu/~rusin/uses- math/music/
The Mathematics of Music 12 Appendices Appendix A Source: (2006). 8notes.com: Carol of the bells. Retrieved from http://www.8notes.com/scores/ 9648.asp Appendix B Note Frequency (Hz) C 524.731 Bb 468.363 Ab 418.050 G 392.734 Eb 312.889 D 293.941 C 262.365 Source: Link, J. (1977). The mathematics of music. Baltimore, Maryland: Gateway Press, Inc.
The Mathematics of Music 13 Appendix C C and C and C Eb C and D
The Mathematics of Music 14 Bb and Bb and C Eb Bb and Ab and Eb D Ab and D
The Mathematics of Music 15 Ab and G and D C G and C G and Eb Appendix D Survey 1. What is your gender? a. Male b. Female 2. What is the range of your weighted GPA? (round to the nearest 10th)
The Mathematics of Music 16 a. 5.0+ b. 4.5-4.9 d. 3.5-3.9 e. 3.5- c. 4.0-4.4 3. What is your average math grade? a. 95+ b. 90-94 c. 85-89 d. 80-84 e. 75-79 f. 74-4. What styles of music do you listen to most often? (more than one choice possible) a. Pop b. Classical c. Rock e. Rap f. Country g. Other (please specify d. Jazz 5. On average, how often do you listen to music? a. 10+ hours a day b. 8-9 hours a day c. 6-7 hours a day d. 4-5 hours a day e. 2-3 hours a day f. one hour a day h. every few days i. once a week j. every few weeks k. once a month l. less than once a month m. never g. less than 1 hour a day 6. When do you listen to music on a regular basis? (more than one choice possible) a. While riding in a car or other form of transportation b. While doing homework/studying c. During free time
The Mathematics of Music 17 d. While falling asleep at night e. While practicing a sport or getting exercise f. While dancing g. At a concert h. When playing it with a group 7. If you listen to music while studying, what style of music do you listen to most often, and how often do you do so, i.e. occasionally, often, always, etc? 8. Please detail how you have been exposed to music. a. Since what age? b. Do you play any musical instruments? If so, which ones? c. Do your parents/ guardians play any instruments? d. Do you sing? If so, in what setting? e. Do you dance? If so, in what setting?