The Mathematics of Music
Jack H. David Jr.

Austin Community College
Spring 1995
MATH 1513.5097

The Mathematics of Music

          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.

I. Music

          It is a difficult task to properly define the word "music", since many individuals have quite different opinions. My personal definition, is that music is sound that is organized in a meaningful way with rhythm, melody, and harmony. This is what I 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 my definition of music. However, many people consider things to be music that I 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.

II. 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" (3). 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 3). 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 18-19). Our ears transform these pressure variations into a form our brains can understand, known as the sense of hearing

III. Frequency

          Energy can exit in different forms such as electrical, magnetic, or mechanical energy, as in the case of sound (Rossing 31); however, regardless of the form of energy involved, the flow of energy will either be a steady flow, such as direct current (DC) electricity, or it will pulsate or vibrate in waves at different speeds. This pulsation of the energy flow is called oscillation (Moravcsik 23).

          The number of times the wave of energy completes a cycle of oscillation in one second is called its frequency. Frequency is measured in cycles per second or Hertz (Hz). Sound, as perceived by human ears, is the range of frequencies of about 16 to 16,000 Hz. The higher the frequency is, in this audible range, the "higher" the pitch is, as perceived by our ears. As the frequency of the energy increases beyond the range of human hearing, it eventually becomes radio waves, then light waves as perceived by our eyes, and then X- rays, gamma rays, etc. Figure 1 is a table of frequencies and some of their specific applications (Grob 717).

          The philosophical distinction between music and noise may be a bit fuzzy, but scientifically, there is a well-defined difference. If there is a mixture of a very large number of audible frequencies, such that the ear cannot perceive any specific frequencies or tones, the result is noise. However, if the sound is created by a constant oscillation at a given frequency, our ears would perceive the sound as a specific pitch, or musical note. A good example of this is the human voice; if you were to recite the words to "Twinkle Twinkle Little Star" in your normal speaking voice, the audible sounds produced by your vocal chords would be within a range of audible frequencies but not any one specific frequency. If, however, you articulated the same exact words, but with exact frequencies for each syllable, you would now be singing! The human voice has now become a musical instrument.

          If you could actually see your vocal chords vibrating, it would be very similar to the vibration of a guitar string. The low E string on a guitar, when plucked, would oscillate 82.407 complete cycles in one second, thus the sound produced would have a frequency of 82.407 Hz (Olson 48). If our eyes were able to see the actual movement of the string, we would see that the string would move to one side, then back to where it started, then to the other side, then back to the starting point again. It would complete this cycle 82.407 times each second, which of course is too fast for our eyes to see. What we might see, however, is an optical illusion of three blurry images, a stronger image at the midpoint, and two fainter images to the left and right. If we were to pluck the low E string on a bass guitar, the frequency would be an octave lower (41.203 Hz) and the visual effect would probably be easier to detect because the bass guitar string is vibrating at half of the speed of the guitar string.

IV. Amplitude

          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. I have personally been given a citation by the "sound police" in Austin, Texas for exceeding legal decibel limits in a bar I was performing in on 6th street. The local cops now carry hand held decibel meters and write tickets to offenders.

V. Sine Waves

          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 40). 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 115).

VI. Pitch Standard

          Before I attempt to discuss harmonics, timbre, and the musical scale, I need to explain the pitch standard. Today, the standard of tuning is such that A4 or the A above middle C is equal to 440 Hz, with all the other notes of the chromatic scale at frequencies relative to their position in the scale. This has not always been the case. There are pipe organs in different parts of Europe with A4 tuned anywhere from 374 to 567 Hz. A tuning fork reportedly used as a pitch standard by Handel vibrated at 422.5 Hz and this standard was generally accepted for a period of about 200 years, which included the lives of Bach, Haydn, Mozart, Beethoven and their contemporaries. In 1859, a commission appointed by the French government, which included Berlioz, Meyerbeer, and Rossini, selected 435 Hz as a standard. In the early 20th century, there was a movement to establish tuning based on C, where all the C's would be powers of 2, such as 128, 256, 512, etc., which would place A4 at about 431 Hz. Although it was "theoretically perfect" on paper, such a radically different standard sounded "out of tune" to the musicians, and this idea did not last very long. In 1939, an International Conference in London unanimously adopted 440 Hz as the standard frequency for A4, and this standard is almost universally used by musicians all around the world today. In fact, the United States Bureau of Standards broadcasts an exceedingly precise 440 Hz tone on its shortwave radio station WWV for checking local standards (Rossing 112).

VII. Harmonics

          As stated earlier, the actual sound waves that we hear from natural sources are rarely perfect sine waves (Moravcsik 115). There is an interesting phenomena that applies to audio frequencies and to other frequencies as well. In addition to the primary frequency, called the fundamental, there are also other higher frequencies called overtones, or more precisely, harmonics, that are produced by the natural sound source, but at a much lower amplitude. These harmonics are integer multiples of the fundamental (Moravcsik 115). If we were to label the fundamental frequency "f", then the value of the harmonics would be 2f, 3f, 4f, 5f, etc. For example, if we were to choose A1, which has a frequency value of 55 Hz, as the fundamental frequency, f, then the harmonics would be 2f=110=A2, 3f=165=E3, 4f=220=A3, 5f=275=C#4, etc. Each of these harmonics would have intervals that would get closer and closer together. The integer multiples that are powers of 2, such as 2f=110, 4f=220, 8f=440, etc., would each sound an octave higher. The harmonic series is composed of the following frequency ratios: 2:1, 3:2, 4:3, 5:3, 5:4, 6:5 ,8:5, etc. with any two frequencies an octave apart having a ratio of 2:1 (Olson 38). Figure 6 represents the fundamental A1, and its first 9 harmonics. The powers of 2, or the octaves, will continue to appear as the series continues. These harmonics continue into virtual infinity above the fundamental, but become weaker in amplitude the higher they get.

VIII. Timbre

          The first obvious question in regards to the harmonic series is "can human ears hear these frequencies above the fundamental?"; the answer is yes... and no. Our ears perceive these frequencies, but only in the way that they affect the tone color of the fundamental, and that perception is often very subtle. Different factors affect which harmonics have the greatest amplitude. In the case of acoustical musical instruments, the actual materials that produce the sound, such as metal, wood, plastic, fiberglass, etc., and the density of those materials affect the harmonics. Other factors include the angles of acoustically reflective surfaces, and the method of attack, such as the bowing of a violin string, the striking of a piano string with a hammer, etc. When different harmonics are emphasized or de- emphasized, different instruments such as a flute, violin, and clarinet could play the same exact note, but would each sound very different. This unique tone color of each instrument, which is created by emphasizing or de-emphasizing different parts of the harmonic series is called its timbre (Rossing 114). Pipe organs that were built centuries ago were designed to make it possible for the player to change the timbre of the instrument by manipulating the harmonics through the use of stops and drawbars that controlled the amount of airflow to the different pipes. If just the right amount of air was sent to the pipes that were tuned to specific harmonics and those pipes then sounded their respective pitches at a given amplitude, the result would be a change in the timbre of the instrument. The human voice also produces different timbres from person to person, because of variance in vocal chords, oral cavity, facial bone structure, etc. Today, electronic musical instruments such as synthesizers, can artificially manipulate harmonic emphasis, and even the shape of the wave, in ways that could never be done naturally, yielding a variety of very interesting new timbres.

IX. Musical Scale

          In the process of the development of music, the first step was to select from the infinite variety of audio frequencies possible, the limited series to be used. The series of notes so selected is called a scale (Wood 171). A scale is sort of a "musical ladder" that climbs from a starting note to a note one octave higher. This can be done in a virtually infinite number of ways. The notes of these scales are then used to create melodies and harmonies. In diverse cultures all over the world, many different musical scales are used, many of which sound very strange to western ears. For our purposes, I will only consider the basic diatonic scales used in western music. These scales originated in Europe, and are still the basis for all western music today, including American music.

          There are a variety of different scales used in western music, all of which are notes of the basic chromatic scale, which is a series of 12 half-steps or semitones in each octave. There are 7 scales called modes that use different combinations of half-steps and whole- steps (two half-steps) to climb to the octave in 8 notes, with the 8th note being a note that is one octave above the 1st note. The most common of these 7 scales are the Ionian or Major scale and the Aolian or Minor scale.

          The origin of our own major and minor scales can be traced with fair certainty to the music of the ancient Greeks. Music undoubtedly played an important part in the life of the Greek people. Plato assigned to it a prominent role in education, maintaining that it was effective in producing a certain inner harmony which other subjects of education failed to give. It is said that all Greek citizens had some training in music and were able to take part in the music which accompanied public functions. Unfortunately the actual music has been almost completely lost, only a few fragments survive. The Greek contributions to the theory of music, on the other hand, have on the whole, been well preserved in a mass of writings, especially those coming from the followers of Pythagoras. The records of musical culture show that there was considerable development as early as 1200 B.C. (Wood 173).

          The notes of the major scale are found in the harmonic series that I discussed earlier. The musical scale, then, is a phenomena of nature, and is nothing more than numerical ratios of the value of the frequency of the starting note. If we were to make a musical scale using only the notes in the harmonic series, we could still play some very simple melodies. The bugle, for example can play the fundamental of the harmonic series and maybe 5 or 6 of the harmonics and can play very simple tunes such as "Taps" or "Reveille". If a bugle is tuned to fundamental pitch of C, the average player could play C,G,C,E,G, and C. However, to play more complex melodies, it is obvious that more notes are needed. The trumpet, for example, has 3 valves that open different paths of tubing, which changes the fundamental pitch, and enables the trumpet player to play the harmonics of 7 different fundamentals.

X. The Pythagorean Scale

          There is a mathematical method that is used to create all the notes of the musical scale from the harmonic series; this scale is called the Pythagorean Scale. As I explained earlier, the naturally occurring harmonics are whole number multiples of the fundamental. To get the other notes of the scale, we must use fractions.

          The harmonic series gives us the ability to create intervals of perfect 5ths, the ratio of 3/2f, "f" being the starting frequency. By using these p5ths, we can create the pythagorean scale. We'll use the key of C, since the C scale only involves the white keys on the piano. If the starting note C is the frequency "f", then C an octave higher is 2f. The notes of the ascending pythagorean C major scale are C=f, D=9/8f, E=81/64f, F=4/3f, G=3/2f, A=27/16f, B=243/128f, and C=2f (Backus 138). These notes are obtained by jumping p5ths from the starting note, in this case C. In other words, 5th up from C is G, a 5th up from G is D, a 5th up from D is A, a 5th up from A is E, and a 5th up from E is B. The only note missing is F, which may be found by going a p5th down from the starting note C. Note that since these notes were obtained from the perfect 5th ratio of 3/2, each of the fractions is a power of 3 over a power of 2, with the exception of the note F, which was obtained by going a p5th down, and therefore has a ratio of 4/3, a power of 2 over a power of 3, the reciprocal of 3/2. The notes thus created are in several octave groups; if we compress these letter-name notes into one octave, we have the Pythagorean Major Scale (Backus 137-138). This system of jumping 5ths, if continued, would also yield the "in-between" notes, i.e. the black keys, and eventually end up back at C. This is called the Circle of 5th's (Figure 7).

XI. The Tempered Scale

          The Pythagorean Scale is mathematically perfect in the relationship of the notes of the scale to the starting pitch, but it was soon discovered that this perfection created serious problems. In the Pythagorean Scale we just created based on C, melodies and harmonies would sound beautiful if only those 7 notes, in different octaves, were used, i.e. the white keys, and C was always the tonal center. But suppose that we continued with the circle of p5th method and obtained the other notes, the black keys, and tried to use some other note than C as the tonal center. We would find that when we attempted to use another note as a tonal center, such as A-flat, it would sound badly out of tune. All the intervals would be perfect in relation to C, but quite imperfect in relation to A-flat. Every time you wanted to play in a different key, which had a different tonal center, you would have to re-tune the instruments, which is exactly what they had to do back in the 17th century. After an orchestra played a selection, the tuners would run up on the stage and re-tune the harpsichord and any other instrument that required a specialist to tune, to the key of the next selection. Woodwind and brass players had to play a different instrument for each key. It was a real mess until they got the idea of creating a scale that would work in every key.

          The most important thing to understand is that the Pythagorean Scale had mathematically perfect intervals, but only in relation to the starting note upon which the scale was built. Any given note would have a slightly different frequency in the different keys. If you started on C, or any other note, and traveled the circle of 5ths, using the perfect 5ths found in the harmonic series, when you arrived back at the starting note, it would not be the same pitch! Also, the half-step intervals of a chromatic scale would not be equal. What was eventually done to correct these inconsistencies was that a little was added to some intervals, and a little was shaved off of others, until the interval ratio between each of the half-step intervals was the same. This alteration is called tempering, the result being the tempered scale, or the scale of equal temperament. Equal temperament is usually said to have been invented by Andreas Werckmeister in about 1700, but there is evidence of experimentation with the idea much earlier. In the scale of equal temperament, perfect 5ths, and other intervals, are no longer mathematically perfect, according to the harmonic series, but they are so close that very few people could hear the difference. Modern musicians that have grown up hearing the tempered scale do not even notice the slight imperfections of its intervals. This compromise of intervalic relationships makes it possible to tune a piano to the tempered scale and play it in any one of the 12 major and 12 minor keys. Bach was so happy about this new scale that he wrote his now famous "The Well Tempered Clavier" in 1722, which contains pieces of music in each of the 24 different keys, which now can be played without re-tuning the instrument (Apel 835- 836).

          The scale of equal temperament is a division of the octave into 12 equal intervals, called tempered half tones. Since an octave is the distance between f and 2f and that octave interval is divided into 12 equal intervals, a half tone or semitone is the frequency ratio between any two tones whose frequency is the 12th root of 2, or 1.059463 (Apel 836). So, if C=f, C#=(12th-root-of-2)*f=1.059463f, D=[(12th-root-of-2)*2]*f=1.122462f, etc. The interval between any two semitones is 12 times the logarithm on the base 2 of the frequency ratio (Olson 46-47). Now, the ratios of the C major scale are a bit more complicated, and decimals are needed. Given the starting note C=f, the tempered major scale would be: C=f, D=1.122462f, E=1.25992f, F=1.33484f, G=1.494307f, A=1.681793f, B=1.887749f, and C=2f (Olson 47). Figure 8 is table of the actual frequencies of the tempered scale in the entire 8 octave range of the piano, with ranges of musical instruments and human voices included.

XII. Psychology of the Musical Scale

          For some reason that no one really understands, there is a psychological effect upon human listeners in regards to the musical scale. The tonic pitch, or tonal center is not only the mathematical center of the scale, but is the psychological center as well. Human perception of the tonic pitch in relation to the other notes of the scale gives each note of the scale, including the tonic pitch, a distinct "personality" or identity. If we were to label each note of the major scale with a number, with the tonic pitch being 1, then the ascending major scale would be 1, 2, 3, 4, 5, 6, 7, 1. The last 1 is an octave above the first, and could also be called 8; an octave above 2 could also be called 9, etc. When individual notes of the scale are played in certain melodic sequences, such as 1, 3, 2, 4, 3, 5, 4, 6, 5, 7, 6, 8, 7, 9... the human ear anticipates the next note to be 8 or 1, the tonic pitch. In the major scale, 2, 5 and 7 are notes that the ear usually expects to be followed by 1, if they are preceded by notes in a certain sequence. The psychological pull is strongest toward the tonic pitch, but this phenomena also exists in other parts of the scale as well. Notes that make the ear expect a certain note to follow them are called tendency tones. This psychological effect extends into harmony, which is the playing of several notes together as chords, and into chord progression, or the sequence of the chords played. This tension and release effect is extremely important in the perception of music. It is my personal belief that these psychological effects are shared by other species as well. I have listened very carefully to birds singing and I can hear evidence of the what we call the musical scale. If birds did not use some form of the musical scale in their singing, it probably would not sound pleasant to human listeners.

XIII. Time

          So far, we have discussed elements of music that have some relationship to the frequency of the musical note. Frequency or pitch is the number of times the acoustical energy oscillates in one second, which is perceived by our ears as how high or low the note sounds. Timbre is the tone quality of the note, caused by emphasis of different harmonics. Amplitude is amount of acoustical energy, or how loud the note is. There is one more important area to consider, which is the duration of the note, or how long of a period of time that we hear the note.

          Figure 9 shows the mathematical relationship of time (x axis) and frequency (y axis). Graph paper that has one logarithmic axis is called semilogarithmic graph paper, and it is used quite frequently for representing quantities that are functions of frequency (Rossing 133-134). It surprised me to learn that I had been reading semilogarithmic graph paper for the past 27 years, since music notation is nothing more than frequency and time on an axis! The time factor of music falls into two main areas, rhythm and tempo.

XIV. Rhythm

          In most music, a given note generally is not sounded for more than one second, therefore we 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 51). 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.

XV.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!

XVI. The Non-Mathematical Factor

          There are obviously many aspects of music, that are directly related to mathematics and physics, and are easily explained; however, an explanation of the phenomena of music would be incomplete without briefly discussing those aspects of music that are impossible to explain rationally and are dependant upon human perception, such as the psychology of the musical scale that I discussed earlier. The subject of this paper is the relationship of music and mathematics; however, in another paper I have written, "The Two Sides of Music", I examine the non-mathematical aspects of music in much greater debth.

XVII. Conclusion

          There are many examples of the relationship of music and mathematics; I have managed to identify a few of the most important ones. 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 the unexplainable emotional right- brain human perception phenomena.

          If one person were to say that music is a set of mathematical relationships that can be explained with algebraic equations, and another person were to say that music is a gift from God, that mankind will never really totally comprehend, both of those individuals would be absolutely correct.

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