Department of mathematics and informaticsFaculty of Sience

University of Kragujevac

MATHEMATHICS AND MUSIC

Professor: Student:Radmila Štajn Suzana Đorđević

File number: 11/2012

Mathemathics and music Suzana Đorđević 11/2012

1. INTRODUCTION

Mathematics is involved in some way in every field of study known to mankind. In fact, it could be argued that mathematics is involved in some way in everything that exists everywhere, or even everything that is imagined to exist in any conceivable reality. Any possible or imagined situation that has any relationship whatsoever to space, time, or thought would also involve mathematics.

          Music is a field of study that has an obvious relationship to mathematics. Music is, to many people, a nonverbal form of communication, that reaches past the human intellect directly into the soul. However, music is not really created by mankind, but only discovered, manipulated and reorganized by mankind. In reality, music is first and foremost a phenomena of nature, a result of the principles of physics and mathematics.

Manu musicians often use mathematics to understand music. Mathematics is "the basis of sound" and sound itself "in its musical aspects... exhibits a remarkable array of number properties", simply because nature itself "is amazingly mathematical".Though ancient Chinese, Egyptians and Mesopotamians are known to have studied the mathematical principles of sound, the Pythagoreans of ancient Greece are the first researchers known to have investigated the expression of musical scales1 in terms of numerical ratios2, particularly the ratios of small integers. Their central doctrine was that "all nature consists of harmony arising out of numbers".

From the time of Plato, harmony was considered a fundamental branch of physics, now known as musical acoustics. Early Indian and Chinese theorists show similar approaches: all sought to show that the mathematical laws of harmonics and rhythms were fundamental not only to our understanding of the world but to human well-being. Confucius, like Pythagoras, regarded the small numbers 1,2,3,4 as the source of all perfection.

To this day mathematics has more to do with acoustics than with composition, and the use of mathematics in composition is historically limited to the simplest operations of counting and measuring. The attempt to structure and communicate new ways of composing and hearing music has led to musical applications of set theory, abstract algebra and number theory. Some composers have incorporated the golden ratio and Fibonacci numbers3 into their work.

1 A scale is any set of musical notes ordered by fundamental frequency or pitch.

2 A ratio is a relationship between two numbers of the same kind.

3 The Fibonacci numbers or Fibonacci series or Fibonacci sequence are the numbers in the following integer sequence: 0,1,1,2,3,5,8,13,21,34,55,89,144…

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Mathemathics and music Suzana Đorđević 11/20122. MUSIC

It is a difficult task to properly define the word "music", since many individuals have quite different opinions. According to many musicians, music is sound that is organized in a meaningful way with rhythm, melody, and harmony. This is what they consider the three dimensions of music. This definition would exclude such things as "rap music", which has rhythm but has virtually no melody or harmony. I perceive "rap" to be poetry, that is spoken rhythmically with a minimum musical element at best. There are other things that pass as music, such as the works of John Cage, that fail to meet their definition of music. However, many people consider things to be music that they do not. The only definition of music that could be universally agreed upon, then, is that music is any sound, or any combination of sounds, of any kind, that someone, somewhere, enjoys listening to.

3. SOUND

To understand what music is by this definition, we must understand what sound is. Olson defines sound as an "alteration in pressure, particle displacement, or particle velocity which is propagated in an elastic medium, or the superposition of such propagated alterations creating the auditory sensation that is interpreted by the ear". In English, sound is a form of energy that is perceived by our ears. Sound is produced when a medium, usually air, is set into motion by any means whatsoever (Olson). We spend our lives surrounded by the earth's atmosphere, which exerts a pressure on everything in it. At sea level this air pressure, or barometric pressure, is about 15 pounds force per square inch. The actual value of the atmospheric pressure at any given place changes a little from time to time, but its value at any given time is called the ambient pressure. Small but rapid changes in the ambient pressure produce sensations in the ear which we call sound (Backus). Our ears transform these pressure variations into a form our brains can understand, known as the sense of hearing

4. FREQUENCY AND HARMONY

A musical scale is a discrete set of pitches4 used in making or describing music. The most important scale in the Western tradition is the diatonic scale5 but many others have been used and proposed in various historical eras and parts of the world. Each pitch corresponds to a particular frequency, expressed in hertz (Hz), sometimes referred to as cycles per second (c.p.s.). A scale has an interval of repetition, normally the octave. The octave of any pitch refers to a frequency exactly twice that of the given pitch. Succeeding superoctaves are pitches found at frequencies four, eight, sixteen times, and so on, of the fundamental frequency. Pitches at frequencies of half, a quarter, an eighth and so on of the fundamental are called suboctaves. There is no case in musical harmony where, if a given pitch be considered accordant, that its octaves are considered otherwise. Therefore any note and its octaves will generally be found similarly named in musical systems (e.g. all will be called doh or A or Sa, as the case may be). When expressed as a frequency bandwidth an octave A2–A3 spans from 110 Hz to 220 Hz (span=110 Hz). The next octave will

4 A pitch is a perceptual property that allows the ordering of sounds on a frequency-related scale.

5 A diatonic scale -from the Greek διατονικός, meaning "[progressing] through tones", also known as the heptatonic prima.

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Mathemathics and music Suzana Đorđević 11/2012span from 220 Hz to 440 Hz (span=220 Hz). The third octave spans from 440 Hz to 880 Hz (span=440 Hz) and so on. Each successive octave spans twice the frequency range of the previous octave.

Because we are often interested in the relations or ratios between the pitches (known as intervals) rather than the precise pitches themselves in describing a scale, it is usual to refer to all the scale pitches in terms of their ratio from a particular pitch, which is given the value of one (often written 1/1), generally a note which functions as the tonic6 of the scale. For interval size comparison cents7 are often used.

Common nameExample

name Hz

Multipleof fundamental

Ratiowithin octave

Centswithin octave

Fundamental A2, 1101x 1/1 = 1x 0

Octave A3 2202x 2/1 = 2x 1200

2/2 = 1x 0

Perfect Fifth E4 3303x 3/2 = 1.5x 702

Octave A4 4404x 4/2 = 2x 1200

4/4 = 1x 0

Major Third C♯5 5505x 5/4 = 1.25x 386

Perfect Fifth E5 6606x 6/4 = 1.5x 702

Harmonic seventh G5 7707x 7/4 = 1.75x 969

Octave A5 8808x

8/4 = 2x 12008/8 = 1x

The exponential nature of octaves when measured on a linear frequency scale.

This diagrams presents octaves as they appear in the sense of musical intervals, equally spaced.

5. AMPLITUDE6 The tonic is the first scale degree of a diatonic scale and the tonal center or final resolution tone

7 The cent is a logarithmic unit of measure used for musical intervals.- 3 -

Mathemathics and music Suzana Đorđević 11/2012

In addition to the frequency, or pitch, of the vibrating sting, there is another factor to consider. The harder you pluck the string, the further the string vibrates, creating a greater amount of energy, which is perceived by our ears as being louder; this is known as the amplitude of the sound wave.

          Amplitude, or loudness of sound is measured in decibels (dB). Human ears begin to perceive sound at a decibel level of about 5 dB; this is called the threshold of hearing. At about 130 dB the sound amplitude level is actually high enough to overload our human limitations and, in effect, hurt our ears; this is known as the threshold of pain (Pierce 109). A detailed scale of decibel levels is given in figure 2.They have personally been given a citation by the "sound police" in Austin, Texas for exceeding legal decibel limits in a bar they werw performing in on 6th street. The local cops now carry hand held decibel meters and write tickets to offenders.

5. SINE WAVES- 4 -

Mathemathics and music Suzana Đorđević 11/2012

If we were to graph the wave of a single perfect musical note of a specific frequency on a XY axis, with X being frequency and Y being amplitude, the result would look something like figure 3, a wave which rises and falls sinusoidally with time, and is called, simply, a sine wave. The sine wave is the most perfect type of sound wave, and usually exits only in the laboratory, or in the sound wave produced by a tuning fork (Pierce 41). In fact, when a tuning fork is vibrating, the motion of the prongs is sinusoidal. The simple experiment shown in figure 4 demonstrates this. One prong of the fork is provided with a light pointed stylus as shown. A glass plate is coated with a layer of soot or other material that will yield a fine line when the tip of the stylus is drawn across it. The fork is then set into vibration and the vibrating stylus is drawn across the plate by moving the fork in the direction of the arrow. The stylus then inscribes a line in the coating which is found to have the shape of the sine wave (Backus 30).

          Most acoustically produced sound waves are not perfect sine waves because of harmonics and other factors. Also, the actual shape of the wave can be changed electronically, as in the case of synthesizers and other electronic musical instruments. Some examples of these waves are shown in figure 5 (Rossing 354).

6. RHYTHM

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Mathemathics and music Suzana Đorđević 11/2012

          In most music, a given note generally is not sounded for more than one second, therefore they are confronted with the transition from one note to another as time goes by. The relative time durations of such notes determine what in musical language is called rhythm. Our body, through its various periodic functions, such as heartbeat, breathing, wake-and-sleep cycles, etc., has its own rhythms, and so music, which also has a rhythm, seems natural to us (Moravcsik 110-110). Rhythm is the whole feeling of movement in music, with a strong implication of both regularity and differentiation. Thus, breathing (inhalation vs. exhalation), pulse (systole vs. diastole), and tides (ebb vs. flow) are all examples of rhythm. Rhythm and motion may be analytically distinguished, the former meaning movement in time and the latter movement in space (Apel 729).

          The standard of measurement in musical time is the beat. The beat is not a fixed length of time; it can be long or short according to the character of the particular musical composition. The nature of the beat is commonly experienced by most persons when listening to music. For example, when walking to the accompaniment of a military march, your footsteps mark off equal measurements of time, which can be considered as beats (Ottman 55-56). Beats are usually grouped into sets of 2, 3 or 4 called bars or measures. These measures follow each other in time as a repeating pattern of beats. The first beat of each measure is usually stronger or accented, to establish the beginning of each measure, i.e. ONE two three four. Other beats of the measure are often accented as well; for example, rock-&-roll is distinguished by accenting 2 and 4 (one TWO three FOUR). This organization of beats into measures is called meter.

          If each measure has 4 beats, a note value that would fill the entire time value is called a whole note. A note that is one-half of that value, 2 of which would fill the time space of the measure is called a half note. A note that is one-fourth of the value of the whole note, 4 of which would fill the time space of the measure is called a quarter note; in this case, this would be an example of 4/4 time or 4 beats per measure with the quarter note being equal to one beat. This designation number of beats per measure and which note value equals one beat is called the time signature.

          Even as measures are divided into beats, beats are then sub- divided into smaller pieces. For example, half of the value of a quarter note is a eighth note. This sub-division continues using powers of 2, i.e. 16th notes, 32nd notes and sometimes even 64th notes. This form of time division into powers of 2 is called simple meter.

          There is another type of meter in which beats are sub-divided into 3 equal parts called compound meter. The same note values are used but with the addition of a dot behind the note; a dot adds one- half the value of the note it follows, so if a quarter note equals 2 eighth notes, a dotted quarter note equals 3 eighth notes. The most common example of compound meter is 6/8 time, which actually has two beats per measure with a dotted quarter note being equal to one beat (ONE two three FOUR five six). Even in simple meter, any given beat can be divided into three equal parts; this is known as a triplet. Figure 10 is an table of simple time signatures and figure 11 is a table of compound time signatures.

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Mathemathics and music Suzana Đorđević 11/2012

10. TEMPO

Tempo is nothing more than the speed of the beat. For many years, Italian words were used to indicate tempo such as largo (broad), lento (slow), adagio (at ease), andante (walking), moderato (moderate), allegro (fast), and presto (very fast). The problem is that these designations were open to personal interpretation, and were therefore sort of ambiguous. The common practice today is to use metronomic markings, or beats per minute. For example, there are 60 seconds in each minute; if the tempo was such that a beat equaled one second, and each quarter note got one beat, the tempo would be would be quarter note = MM 60. Most musical compositions fall in the range of MM 60-80, which is about the speed of human heartbeats or moderate walking (Apel 836-837). Of course, the tempo can easily be twice that fast if the music is intended for dancing, especially the music of those younger folks that are still full of energy!

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Mathemathics and music Suzana Đorđević 11/2012

10. CONCLUSION

          There are many examples of the relationship of music and mathematics which I have managed to identify by reading some text of different authors. This subject could be expanded into a doctoral dissertation of hundreds of pages, and I am quite sure that someone, somewhere has already done just that. I chose the subject because I wanted to learn something relevant to my career field; I honestly feel that I have succeeded in that goal. It could be argued that music is, in fact, a branch of mathematics. My final conclusion is that music is a unique blend of mathematics, physics, and many other fields of our lives. And finally, mathemathics has been used for centuries to describe, analyze, and create music.

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Mathemathics and music Suzana Đorđević 11/2012

10. LITERATURE

http://jackhdavid.thehouseofdavid.com/papers/math.html 23.12.2012 00:09Chambers' Twentieth Century Dictionary, 1977, p. 1100Imogen Holst, The ABC of Music, Oxford 1963, p.100

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