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Mathematics and Music 1 Relationships between Mathematics and Music Si On Lim Summer Venture of Science and Mathematics: Mathematics Mrs. Amy Goodrum July 19, 2003 Mathematics and Music 2 Abstract Many people do not realize the mathematical side of music often. Many think that mathematics and music exist separately. However, mathematics and music are linked closely together. From simple additions to complicated functions, mathematics are involved in every part of music. Note lengths, frequencies of sound, writing music with measures and time signatures, and harmonic means are mathematics that is established in music. Throughout this essay, relationships between mathematics and music are explained. Mathematics and Music 3 Relationships between Mathematics and Music Mathematics and music now take up great parts of our society. Without mathematics, we cannot accomplish many advantages that we enjoy right now. Buildings, houses, cars, and much more involve mathematics. Every little part of our lives involves mathematics. As well as mathematics, music influences our lives in many ways. It calms or stirs our mind. It helps us in expressing or dealing with our emotions. It also makes up an important part of our economy. Both mathematics and music seem to be essential to our lives. However, somehow people find difficulties relating one to the other. One of the reasons is because of their obvious roles. For instance, from working at their jobs to grocery shopping, everyone has to know mathematics. Also, students are compelled to learn mathematics in schools. But, only musicians study music, and not all students necessarily learn music in school. Moreover, mathematics consists of signs, notations, numbers, and equations. Music contains sound and notations that are very different from mathematic notations. People rarely realize that they use mathematics and music together. No one studies mathematics to become a musician or vise versa. In spite of the misconception, mathematics and music relate closely to each other, even though much of their relationship have been forgotten since the Middle Ages (Beers, 1998). Although many do not consider relationships between mathematics and music, they relate to one another through connections between lengths of notes, time-signatures, frequencies of sound, and golden numbers. Many investigators research relationships between music and academic achievements, especially mathematics. Now, it is known that students who engage in music activities are most likely to perform better on academics, especially in mathematics, than those who are not involved in music. For example, in Rhode Island, researchers have studied “test art” groups of Mathematics and Music 4 eight public schools. The test art class has had under-achiever kindergartners who have received music and visual arts training. After several months, these students have taken standardized tests, and the result has shown that the students are caught up with regular students in reading and even exceeded their classmates in mathematics (American Music Conference, 2003). Additionally, they have shown improvement in their behavior. Another study by universities in Georgia and Texas have illustrated that middle school and high school students who participate in musical-instrument courses score higher on standardized tests (American Music Conference, 2003). Numerous other studies are being conducted to find relationships between music learning and academic performance, especially in mathematics, and prove that music training helps students and others in improving mathematics and other skills. In addition, a famous theory called the “Mozart Effect” states that children who are exposed to classical music in their early ages are more likely to receive high scores in mathematics tests (Rusin, 2002). Already, investigations exemplify relationships between mathematics and music. One of the most basic mathematical concepts in music is length of notes. Many different lengths exist in music. With different length of notes, different types of music are created. However, all of the note lengths can be identified by using simple divisions. A very basic note is a whole note, which generally gets four beats. The rest of the notes can be found by dividing this whole note. Dividing a whole note into halves gives two half notes, which are usually played two beats long. Dividing a half note produces two quarter notes, which are usually one beat, and so on Ewer (2002) demonstrates this idea with the following diagram: Whole note Half notes Quarter notes Eighth notes Mathematics and Music 5 Addition also applies when these notes have dots beside them, which are named dotted notes. When a note has a dot beside it, half of the original note value is added to the note length. For example, a dotted quarter note is one-and-a-half beats long (assuming that the quarter note equals one beat) because the dot gives the quarter note a half of one beat: = 1+ ½ = 1½ (Ewer, 2002). Yet, not all whole notes equal four beats. In music, a notation called time signature decides how many beats a note gets, and it involves some type of mathematics. Here are some common time signatures: and . The number at the bottom represents which note gets one beat. In this case, a quarter (1/4) note gets one beat, and the top number represents how many beats one bar, or a measure, contains. In case of , there are two beats in one bar. Therefore, in this bar, note lengths should add up to two beats. Again, music requires simple mathematics: (Ewer, 2002) To understand relationships between mathematics and music in better ways, learning scale intervals is important. An interval is a distance between a low note and a high note. In music, intervals can be categorized into two different tones: semitone, or half step, and whole tone, or whole step. A semitone is the smallest interval on instruments; a whole tone occurs when two different notes are two semitones apart. These tones make notes and two different types of scales: major and minor. In major scale, step patterns are: whole-whole-semi-whole- whole-whole-semi. In minor scale the patterns are: whole-semi-whole-whole-whole-semi- whole. Using these names and patterns, early musicians have named notes and intervals as shown in Figure 1. Musicians and investigators use these scales in different mathematical ideas. Mathematics and Music 6 Numbers of half step between Interval name Note names in C major scale. two notes starting at C (Tonic note, or base note: C) 1 Minor 2nd (m2) C#, Db 2 Major 2 nd (M2) D 3 Minor 3rd (m3) D#, Eb 4 Major 3rd (M3) E 5 Perfect 4th (P4) F 6 Tritone (TT) F#, Gb 7 Perfect 5th (P5) G 8 Minor 6th (m6) G#, Ab 9 Major 6th (M6) A 10 Minor 7th (m7) A#, Bb 11 Major 7th (m7) B 12 Octave C Figure 1: Interval Chart (Sommer, 2003) When two notes are played together, they either make dissonant or consonant sounds. Dissonant happens when these two sounds are unpleasant to hear, and consonant produces nice sounds. For instance, in Figure 1, when D, or major 2 nd , and D#, or minor 3 rd, are played together, they make a dissonant sound. When C, or a tonic note in this case, and G, a perfect 5 th, are played together, they make consonant sound. Of course, mathematics are involved here also. Ancient Greeks have discovered that only integer multiples of frequency of a given note can be combined to make consonants. For example, a given note of frequency 220 Hertz (Hz) makes a consonant sound when played with notes of frequency 440Hz, 660Hz, 880Hz, and so on. Simple multiplication can produce a fine sound (Beer, 1998). Ratios also have been a vital factor of music. In Ancient Greek, Pythagoras discovered the relationship between harmonic ratios and proportions. He realized that when a blacksmith hit an anvil with his hammer, it created different tones, according to the weight of the hammer. So, Pythagoras studied this in depth and discovered that different proportions of string length created different proportions of frequencies, especially consonant tones. He divided a string into halves and compared the sounds of the original string and the divided string. He found that the divided Mathematics and Music 7 string was exactly an octave higher than the original one. He also noticed that the divided string vibrated twice as fast as the original one (figure 2). Therefore, a ratio of 1:2 created an octave. He also divided the string into thirds and quarters, which created a perfect fifth (2:3) and a perfect fourth (3:4). He stated that 12 is the “most divisible” small number; therefore, he expressed the ratios in 12 (figure 3). Therefore, according to Pythagoras, the most ideal musical number is 12 (Sabine, 2000). Figure 2: Octave (Sabine, 2000) 1:1 12:12 Unison 1:2 6:12 Octave 2:3 8:12 Perfect fifth 3:4 9:12 Perfect fourth Figure 3: Note ratios (Sabine, 2000) However, tuning instruments with Pythagoras’s ratio is a little inaccurate, so it is called the Pythagoras comma. A ratio of an octave is 1:2. Raising the ratio with factor of 2 gives another ratio of a different octave in whole numbers. However, if one uses the perfect fifth ratio of 3:2 and raises it with factor of 2, it will give the frequency ratio of a different octave. These cannot appear as whole numbers; therefore, cannot make 1:2 ratios when compared to the original ratio. Therefore, another tuning system called equal-temperament has been developed, as shown in figure 4, by mathematician and musician Mersenne. This tuning system compromises Pythagoras’ mathematical tuning system. Mersenne has developed the equation Mathematics and Music 8 (122)n = f (Moiseiwitsch, n.d.). Now, an equal-temperament tuning system is used more widely, even though human ears prefer “pure” Pythagoras interval tuning system. Notes Pythagorean scale Tone Equal-temperament scale Tone C 1 1 Whole tone Whole tone D 9/8=1.125 ( 2) =1.1224… 12 2 Whole tone Whole tone E 81/64=1.2656… ( 2) =1.2599… 12 4 Semitone semitone F 4/3=1.3333 ( 2) =1.3348… 12 5 Whole tone Whole tone G 3/2=1.5 ( 2) =1.4983… 12 7 Whole tone Whole tone A 27/16=1.6875 ( 2) =1.6817 12 9 Whole tone Whole Tone B 243/128=1.8984… (122)11 =1.8877… Semitone semitone C 2 2 Figure 4: Tuning Chart (Moiseiwitsch, n.d.) Harmonic mean is used to obtain an interval number between two numbers. It is defined as 2ab/a+b = x with two different numbers, a and b. Harmonic mean is reciprocal of the arithmetic mean, where two different numbers, a and b, are defined as a+b/2 = x. Substitute x, a, and b with their inverses, 1/x, 1/a, and 1/b. From that, the following formula can be obtained: 1/x = 1/2(1/a+1/b). Then, solve for x, which gives a formula of harmonic mean (Rameau, 1971). This mean can be used to find notes in the intervals. For example, there is a ratio of an octave which is 1:2. Using the harmonic mean, one obtains 4/3, which gives 2: 4/3: 1. And 4/3 is also the ratio of perfect fourth, which is in between tonic tone and an octave (Rameau, 1971). Music involves trigonometry as well. Sound is created by something that vibrates. Therefore, when a string is plucked, the string vibrates and produces sound. The sound can be expressed as waves by functions of sine: y = A sin (2 pi f t), where t is time, f is frequency, and Mathematics and Music 9 A is amplitude (Rossa, 1996). The graph below is middle C, which has a frequency of 256 Hz per second. Figure 5: Middle C frequency graph (Rossa, 1996) To get a frequency of C that is an octave higher than one in Figure 5 above, one has to double the frequency because the ratio of an octave is 1:2. So, substituting 512 Hz as frequency into the function will produce a graph of C that is an octave higher than middle C as shown in figure 6. Figure 6: Frequency of one octave high C (Rossa, 1996) Sound gets more complex when something produces a sound of different frequencies and loudness together. A violin string or an oboe cavity vibrates with more than one frequency: Mathematics and Music 10 Figure 7: Graph of more than one frequency. (Rossa, 1996) Also, in music, more than one note can be played. Therefore, sound becomes more complex when three notes are played together at the same time. The following figure is a graph of two Cs that are one octave apart (512 Hz and 1024 Hz) and a G (1536 Hz). Figure 8: Graph of middle C, Octave C, and Octave G (Rossa, 1996) Once more, functions of trigonometry are involved in music, showing that mathematics and music are closely related. It is known that Mozart may have used the golden mean for his music. Golden mean is the ratio of two unequal parts of a segment when the segment is divided into two unequal parts. It is defined as 1/x = x/(1-x). It is also about 0.61803, and its reciprocal is 1.61803 (Quentin, Mathematics and Music 11 n.d.). The golden mean is also related to Fibonacci numbers because the ratios of two adjacent Fibonacci numbers are close to the golden mean. However, it was found that most of Mozart’s sonatas were divided into two parts exactly at the golden mean. The famous song “Hallelujah” chorus in Handel’s Messiah also contains the golden mean. The piece consists of 94 measures, and the trumpet solo “King of Kings” happens in measures 57 to 58, which are about 8/13 of the whole piece (Beer, 1998). No one knows if this happened accidentally or was planned. However, this presents different ideas of the relationship between music and mathematics. This text has illustrated that mathematics and music are not separate areas. Music involves different areas of mathematics: arithmetic, geometry, and trigonometry. Simple addition and division create great music. Accordingly, learning music improves mathematic skill. It is clear that mathematics and music relate through academic performances, length of music notes, time signatures, sounds, tuning systems, and music compositions. However, it is not often that people connect mathematics and music to each other. Unfortunately, music is not taught in some schools. Hopefully, this text will change someone’s thought about mathematics and music, and someday we will find that mathematics and music are easily linked. Mathematics and Music 12 References Moiseiwitsch, B. (n.d.). Art, mathematics and music. Retrieved July 17, 2003, from <http://www.qub.ac.uk/mp/amtpt/pers/moiseiwitsch/AMM/AMM.htm> American Music Conference. (2003). Research briefs: Did you know? Retrieved July 17, 2003, from <http://www.amc-music.com/research_briefs.htm> Rameau, J. (1971). Treatise on harmony. (P. Gossett, Trans.). New York: Dover. Rusin, D. (2002). Mathematics and music. Retrieved July 17, 2003, from <http://www.math.niu.edu/~rusin/papers/uses-math/music/> Beer, M. (1998). How do mathematics and music relate to each other? Retrieved July 17, 2003, from <http://perso.unifr.ch/michael.beer/mathandmusic.pdf> Sommer. S. (2003). Math and music. Retrieved July 17, 2003, from <http://www.geocities.com /isymjazz/MathandMusicPage1> Rossa, R. (1996). Trigonometry and music. Retrieved July 17, 2003, from <http://www.csm. astate.edu/music.html> Rueffer, (n.d.). Basic music theory and math. Retrieved July 17, 2003, from <http://www.geocities.com/ctempesta04/index> Sabine, D. (2000), Mathematics and music. Pythagoras. Retrieved July 17, 2003, from <http://www.davesabine.com/music/mathematics.asp?action=pythagoras> Pythagoras: Music and space. (2002). Retrieved July 17, 2003, from <http://www.aboutscotland.com/harmony/prop.html> Quentin, B. (n.d.). The golden section. Retrieved July 17, 2003, from <http://brian.quentin.students.noctrl.edu/webproject.htm Mathematics and Music 13 Ewer, G. (2002). Easy music theory. Retrieved July 17, 2003, from <http://www.musictheory.halifax.ns.ca/>