California State University, Northridge Discovering The

CALIFORNIA STATE UNIVERSITY, NORTHRIDGE

DISCOVERING THE MATHEMATICS IN MUSIC

A project submitted in partial satisfaction of the requirements for the degree of Master of Arts in Secondary Education by Elisabeth S. Bates

June, 1981 The Project of Elisabeth S. Bates is approved:

J;ajrnes Cunningham, ;f:h. Dn

California State University, Northridge

ii TABLE OF CONTENTS

Page List of Tables . iv

List of Figures v Abstract . . vi

Introduction .. . 1

'Mathe~aticians Study Mathematics and Music . 3 Scales and Temperament . 16

References 46 Additional References for Interested Readers 47

Appendix A - Glossary of Terms . 48 Appendix B - Answers to Selected Exercises so

iii LIST OF TABLES

1. Commonly Used Intervals of the Pythagorean Scale ...... 34 2. Comparison of Musical Description with Mathematical Interpretation of Intervals 39

iv LIST OF FIGURES

Page Figure 1. Monochord ...... 5 Figure 2 . Strings which produce a fourth . . . 19 Figure 3. Strings which produce a fifth . . . . 19 Figure 4. Strings which produce an octave 20 Figure 5. Strings which produce a fourth . . . 27 Figure 6. Strings which produce an octave . . . 27 Figure 7. Strings which produce a double octave 27

Figure 8 . Strings which produce the octave of the fourth ...... 28 Figure 9 . Strings which produce the fifth of the octave ...... 28 Figure 10. Scale clock (circle of fifths) . . . 31 Figure 11. Frequency ratios of the Pythagorean

scale . . • ...... 33

v ABSTRACT

DISCOVERING THE MATHEMATICS IN MUSIC by Elisabeth S. Bates Master of Arts in Secondary Education ~ .

This project is designed to acquaint the secondary school student with the use of mathematiesinmusic. Specifically, a summary of the history of mathematics in music and a careful development of the musical scale are presented. The student is first asked to build a monochord, sj.milar to the one used by Pythagoras. With the use of the monochord, the student will discover, guided by appropriate exercises, some of the physical properties of a taut string. The student will discover the octave ratio of 2:1; the ratio of a fourt~ 4:3; and the ratio of a fifth, 3:2.

vi Anecdotes and comparisons are used to suggest to the student that there is a resemblance between the studies of music and mathematics. The student will be led to understand that musicians and mathematicians have a great deal in common.

Next, the Pythagorean scale is developed using three different methods. The first uses a product of ratios to determine the ratios of the intervals in the scale. The student is asked, through a series of exercises, to find the ratios. The second method, probably used by

Pythagoras, encourages the student to develop the scale using a succession of fifths. The third method, attributed to Euclid, utilizes the division of the monochord to derive ratios and the tones of the scale.

By comparison of frequency ratios, the student observes the inconsistency in the Pythagorean scale, specifically, the Greek Comma. He then works through the development of the scale we now use, the equal tempera ment scale, by calculating frequency ratios of tones in that scale. Finally, the student is encouraged to further investigate the mathematics in music.

vii Introduction

Many high_ school students who are not particularly interested in mathematics are highly interested in music.

They are surprised and fascinated by the many relationships between two such "unlike" subjects. Studying these rela tionships helps them gain a different perspective on mathematics. With a return to basics, it is possible that music will no longer be an option for some students, and exposure to music through mathematics·may be the only exposure they get. Educators are attempting to approach teaching with a right-brained emphasis as well as with the traditional left-brained emphasis. The combination of music and mathematics provides a good balance of these emphases.

While much has been written on the subject of mathe matics and music, very little of it is practical for use in the high school classroom. It is my intention in this project to assemble facts in a way that is understandable to high school students, and to provide exercises that will reinforce the facts while reviewing math skills and concepts.

This project is designed to be used as a mini-course.

It may be used by students for individual study, or th~

1 2

teacher may wish to use it as a classroom activity, It

assumes very little musical knowledge. A student with a

strong grasp of decimals and fractions should be able to

do most exercises, On the other hand, the study is quite

open~ended, providing many opportunities for an advanced math student to explore beyond the pages of this paper.

The student is urged to make use of the glossary in

the appendix. An answer sheet is provided for use at

the teacher's discretion.

Let's discover the mathematics in music. Mathematicians Study Mathematics and Music

Five thousand years ago, a Sumarian artist crouched in the semi-darkness of a nearly completed tomb. He was sketching pictures on the wall of the queen's vault. The pictures depicted important activities of the day. Remem- bering the celebration of the evening before, he hummed as he'sketched the eleven-string lyre used to entertain the queen. Already finished were the pictures of the

figures clapping curved sticks togeth~r. He paused as he

recalled the rhythmic pattern of the clicking sticks. Not far away, a Sumarian mathematician was surveying a farmer's field. He kept track of his calculations with crude marks on a stone. Did he recognize any connection between his work and that of the music-minded artist? Perhaps not. But we know that music and mathematics are related in many ways. Years later, a Greek named Pythagoras (Sixth Century, B.C.) was looking for ways in which numbers were related to music. He had }earned from his teachers that the ratio o£ the lengths of strings forming constant intervals was

\~he same. for instance, the two tones of an octav~ are I \\played by strings whose lengths have a ratio of 2;1. \ . ! \Because, in addition to music, nunbers seem to have a part \ .

3 4

in many other relationships, Pythagoras and his associates decided that there was something very special about numbers. In his experiments with ratios and music, Pythagoras used a monochord (see Figure 1). Exercise 1.1 Define "ratio." Write the octave ratio three ways.

Exercise 1.2 Build a simple monochord.

Exercise 1.3 Pythagoras discovered two ways to increase the pitch, which means to make· a higher sound. Use the monochord to help you discover what the two ways

are. (Hint: A fret is a section on the neck of a guitar over which the strings pass. When we "press a fret," we depress the string with our finger until it touches the wooden neck. Haw does this change the sound of the plucked string? Another hint: When we tune a guitar, we turn the pegs to tighten the strings. How does this change the sound of the plucked string?) You, like Pythagoras, found that when you increase

the weight (tension) on a string, ~he pitch is raised. You know that when we tighten a string, then pluck it, a higher sound is produced. Tightening the string has the same effect as increasing the weight on the monochord. 5

Figure 1. Monochord Q is a movable bridge. QR is set in motion. W is weight. T is a wheel. Physicists have studied this situation, and they have

learned that the tension (or weight) on a string is related

to the sound produced by the string. In particular, the

frequency (number of vibrations per second) is directly proportional to the square root of tension.

Example: If the weight on the monochord is increased

by 4, how is the frequency affected? What effect does

this produce on the sound?

Solution: The square root of the tension is 2,

(14 = 2), so the frequency is multiplied by two.

(The frequency is doubled.) The sound is higher when

the frequency increases.

Exercise 1.4 If you increase the weight on the

monochord by 9, how is the frequency

affected? What effect does this pro

duce in the sound?

Exerc:ise 1.5 If you increase the weight on the

monochord by 16, how is the frequency

affected? What effect does this pro

duce?

As you can see, one way to make a higher sound with a string is to increase the weight, or the tension, on the string.

A second way to increase the pitch of a sound is to shorten the length of a string. When a guitar player places his finger on the fret, he shortens the string. 7

When the shortened string i~ plucked, the sound produced is higher than that of the unobstructed string. Around

300 B.C., Euclid observed that when a short string vibrated rapidly, a high pitch is pro~1ced. The long string on a bass viol vibrates slowly to produce a low pitch. The longer the string, the fewer times it vibrates in a given time. A physicist would say that the frequency of vibra tion varies inversely as the length of a string.

Exercise 1.6 On a guitar or autoharp, experiment with lengths of strings and the sound they produce. First, find two strings, one of which is twice as long as the other. Strings such as these, whose

lengths have a ratio of 2:1, produce a musical interval called an octave.

An example of an octave is from C to C'. C' represents the C above middle

C on the piano. An octave includes 8

consecutive tones: C-D-E~F-G-A+B-C' in our example. To determine what musjcal interval is formed when the ratio of the stririgs is 2:3, remember

that ~ musical interval is named for the number of tones it includes. An

interval whose ratio is 2:3 is the interval from C to G, or C-D-E-F-G. How many tones does that interval

include? The interval whose ratio is

2:3 is a fifth.

Exercise 1.7 What musical interval 1s formed when

the ratio of the length of strings is

3:4? The interval from C to F is

one such ratio.

Exercise 1.8 Find two strings whose ratio is made

up of large numbers, 10:11) for example.

How do these strings sound when played

together?

Pythagoras made the same discoveries which you just made. Later, ratios of 4:5 and 5:6 were added, and corre- spending intervals were named, But Pythagoras observed that the smaller the numbers in the ratio, the better the tones in the interval seemed to sound. The way two tones sound together is called consonance. (See Appendix for a glossary of t~rms.) The smaller the numbers in the ratiq the better the consonance. The reason for this is a puzzle to mathematicians and physicists. In their search for a solution to the puzzle, they have solved rnany oth~r problems. We shall learn some of the~ later.

Eventually a scale was developed by Pythagora~ which is similar to the one upon which we base our music. The

Pythagorean scale was improved upon by mathematicians

Pierre Ramee (1515~1572) and Marin Mersenne (1588~1648). 9

In the seventeenth century, Brook Taylor (1685-1731), known for the Taylor Series which is studied in a course in calculus, discovered that a tone produced by a tuning fork could be graphed as a sine wave--a trigonometric function~

For many yearst Taylor and others occupied themselves with the problem of describing mathematically the motion of strings. After eighty years, the solution was found in

Fourier's series. Jean Fourier (1768-1834) found that musical sound can be described as the sum of sine terms of the form y = a sin bx. Another mathematician, Leonhard

Euler (1707-1783) tried to explain why a given piece of music pleases the ear, and further, why a particular piece of music will please one person and displease anothar.

:More recently, mathematical calculations have i.nproved musical instruments such as the flute and the organ, thanks to a brilliant physicist, Helmholtz. But yon will study many of these things in advanced mathematics courses, and we are getting ahead of ourselv·es l

Periods of great musical developne:;_t -:oincide with great math development (Coxeter~ 1962). For example, though their music was not written down, we know that the

Greeks had songs and hymns for every occasion. These were incorporated into the chants of the Christian church, and many are still sung as part of the Catholic liturgy. At the same time, the Greeks were developing mathematics to 10 the extent that we use, almost unchanged, the geometry that they used. To illustrate the theory that great musicians and great mathematicians often exist in the same era, consider that Italian musician, Giovanni Palestrina (1525-1594) was composing great works while a little later Italian mathematician, Girolamo Saccheri (1667-1733) was working with non-Euclidean geometry. Further, JohannS. Bach (1685-1750), Ludwig Von Beethoven (1770-1827), Franz Schubert (1797-1828), Johannes Brahms (1833-1897), and Richard Wagner (1813-1883), superstar German composers, do not outshine G. W. Leibniz (1646-1716), Carl F. Gauss

(1777-1855), and G. I~. B. Riemann (1826-1866), 3uperstaT mathematicians~ And they all lived about the same time. Meanwhile, in France, we find musicians Camille Saint-Saens (1835-1921), Claude Debussy (1812-1918), and Maurice Ravel (1875-1936) excelling along with mathematicians A. L. Cauchy (1789-1857), H. Poincare (1854-1912), and Elie Cartan. Russian musical genius Peter llyich

Tchaikovsky ( 184 0-18 9 3) \vas writing concertos ·~.;hile Russian mathematical genius N. l. Lohachevski (1713-1856) was discovering mathematics. And in England we learn that composer Henry Purcell {1659-1795) was a contemnorary of the brilliant Isaac Newton (1642-1727).

Exercise 1.9 Wltat might explain the theory that musicians and mathematicians make advancements simultaneously in their respective fields? Exercise 1.10 Research and write a short report on two of the named contemporaries--one a musician and one a mathematician. Share your findings with your class- mates. There are many resemblances between mathematics and music. _...Composing of music compares to the discovery of mathematical ideas. Both music and mathematics are abstract. They can be written down 1n a universally accepted notation which was developed 'in the Renaissance. 1sic and mathematics are both very precise. Think about the effect of changing G# to Gb 1n a melody, and compare it ot the effect of changing -3 to +3 in a sum. (A sharp,

#, raises the tone a helf-step; a flat, b, lowers the tone a half-step.) In music a small change in a symbol can make a big change 1n melody. In mathematics, a small change in symbol can make a big change in a sum

(f~athematicians are well aware of the connections be tween math and music. One of these mathematicians was James Joseph Sylvester (1814-1897): Sylvester's sense of the kinship of mathematics to the finer arts found frequent expression in his writp ings. Thus, in a paper on Newton's rule for the discovery of imaginary roots of algebraic equations, 12

he asks in a footnote, "May not Music be described as the Mathematic of sense, Mathematic as Music of the reason? Thus the musician feels Mathematic, the mathematician thinks Music--Music the dream, Mathematic the working life--each to receive its consummation from the other when the human intelli gence, elevated to its perfect type, shall shine forth glorified in some future Mozart-Dirichlet or Beethoven-Gauss--a union already not indistinctly foreshadowed in the genius and labors of a Helmholtz!" (Bell, 1937, p. 404). Since there is a relationship between music and mathe matics, you may wonder whether there is a correlation be tween musicians and mathematicians. In other words, are there many mathematicians who are accomplished musicians, and vice versa? Samuel Tolansky, the English physicist, believes that mathematicians appreciate music because all cultured people appreciate music. Many mathematicians are amateur musicians, but not many musicians take an interest in mathematics. The mathematician, Albert Einstein (1879-1955), also played the violin. After a charity concert, a critic wrote about his performance, "He did tolerably well, but I cannot understand how his name comes to be so well known all over the wo!'lcl." G. W. Leibniz (1646-1716) wrote in a letter in 1'712, "Music is a hidden exercise in arithmetic, of a mind unconscious of dealing with numbers" (Archibald, 1924, p. 1). J. Sylvester called himself "the Mathematical Adam" because of the many mathematical terms he invented. He used "mathematic" to denote the science itself, in the way that we speak of music. He referred to music and mathematics "not merely as having arithmetic for their common parent but as similar in the habits and affections'' (Archibald, 1924, p. 2). From the days of Pythagoras (c. 540, B.C.) for nearly 2000 years, music was regarded as a mathematical science. In the Middle Ages, the study of music was limited to the mathematics of the subject. Students 'of music did not learn how to sing or play an instrument; they studied the theory and physics of music. However, music as an art did play a part in the lives of some mathematicians. WilliamHerschel, an astronomer, was a violinist, oboist, organist, conductor, and published composer. Henri Poincare (1854-1912) occupied his leisure with good music, and Emile Lemoine (1840-1912) held concerts of chaniber mus i.e at his Paris home .. Geometer Janos Bolyai (1812-1860), an excellent violinist, enjoyed a good duel. Once he agreed to a series of duels? on the condition that he be allowed to play a violin solo every two duels. Perhaps the music steadied his nerves; at any rate, he was the victor in thirteen consecutive duels! Friedrich T. Shubert was a 14 9 •

Russian astronomer and mathematician who played piano,

flute, and violin. His great-granddaughter, Sophie

Kovalevsky, who had no musical talent, was said to be

willing to part with her talent for mathematics, if only

she could sing!

Niels Abel (1802-1829) was interested in the· mathe

matical problems that music suggested. Once he was

observed paying close attention to a piano performer. As

it turned out, he was not particularly interested in the

music~ but he was trying to figure a relationship between

the number of times each key was struck by certain fingers!

(Archibald, 1924, p. 3-4).

LaGrange (1736-1813) liked music because while

listening, he could work out math problems mentally, obliv

ious to his surroundings. He liked the music that inspired

him to math discoveries. (Many students agree with

LaGrange, listening to music while they do homework. Per haps you can suggest why!)

Sergei Tanaieff, a pupil of musicians Rubenstein and

Tchaikovsky, was a prominent Russian composer. He used

algebraic symbolism and formulae in his lectures and work on musical counterpoint.

Yet, in spite of all these examples, there is little statistical evidence of a relationship between mathematical abilities and musical abilities. There is, however, a 15 great relationship between music and mathematics, as we shall now see. Scales and Temperament

According to Greek mythology, Orpheus was the poet musician son of Apollo. Orpheus possessed a strange and wonderful lyre, whose music was so powerful that it charmed even inanimate objects. The story goes that a group of jealous women decided to rid the world of Orpheus. They tried everything, but the lovely music of the lyre somehow protected Orpheus from death. It was only after the music of the lyre was drowned out by the screams of the £Vil women that Orpheu~ was finally killed. Supposedly, this marvelous lyre played only four notes (C, F, G, C', for the musician). These four notes compose the intervals of declamation. Listen to a person telling a story. When a question is asked, the voice rises a fourth, C to F. (Remember: To determine a fourth, count

4 cons~cutive notes; C~D-E-F.) When a word is emphasized, the voice goes up another step to the fifth, C to G.

And when ending a ~;to-ry, the voice drops a fifth. So you see how well-suited Orpheus's lyre ~as to his need for story-telling, You may also see how the intervals of declamation may have led Pythagoras to investigate and develop a musical scale. The ancient Greeks limited their music to the three notes, C, F, G, and possibly the octave, C'. The

16 17 scale of the Chinese and Scots contains C, D, F, G, and A (white keys on a piano). The Southern Asians and East Africans use that same scale, but one-half tone up (the black keys on a piano). You may have learned to play a tune on the piano using that "black key" scale, sung with the words of the old nursery rhyme, "Peter, Peter, Pumpkin Eater." From the Peer Gynt Suite by Edvard Grieg, the melody, "Morning," is based upon the same black key scale. By the sixth century, the Greeks had increased their scale to include F, C, G, D, A, E, and B, and this was the scale adopted by Pythagoras. The music of the ancients was homophonic, which means, in unison. There was no harmony, so they did not have to worry about how two notes sounded together. Christian music was homophonic until the eleventh century, and today the music of Oriental and Asiatic nations is still homo phonic. The Arabs, Persians, and Javanese use no harmony, so they have no need for a consonant scale (a scale in which the notes, when played together, sound pleasing). A basic characteristic of polyphonic music is two tones are sounded at the same time. Aristotle did not like polyphonic music, because, he said, "Both tones are concealed, oae by the other," and he found it as confusing as two speakers speaking at once (Jeans, 1953, p. 160). 18

He was accustomed to hearing homophonic music, and like many others of his time, he found it difficult to adjust to polyphonic music.

All primitive peoples who have advanced beyond homo phonic music have s6ales in which the intervals (composed of two different tones of the .scale) are consonant. More basic than the requirement of consonance is the fact the octave, an interval of eight tones, is fundamental in the music of all peoples! In developing a scale, we shall see how important the octave is.

We first need to understand that between a tone and its octave, there are an infinite number of tones.

Similarly, in the number system, there are infinitely many numbers between 1 and 10. We must somehow decide which tones will make up our scale.

We will begin constructing our scale with the three intervais we first discussed--the fourth, the fifth, and the octave. A musical interval is the distance in pitch between two notes. The names for intervals, fourth, fifth, and octave, refer to the number of scale steps from the lower to the higher note. For example, from C to F, or

C-D-E-F, spans 4 tones; therefore, the interval from C to F is called a fourth. Similarly, C-G is a fifth because it is an interval of 5 scale steps, C-D-E~F-G. Likewise, the octave is an interval from C to C', using eight scale steps, C-D-E-F-G·A-B-C'. 19

You have seen from the monochord that the ratio of the lengths of strings which form a fourth are 3:4. The longer string, depicted in Figure 2, when plucked produces a tone named C. The shorter string in Figure 2 produces

a tone named F. The ratio of the length of the shorter string to the length. of the longer string is 3:4. In

other words, the shorter string is 314 the length of the longer string.

I I sounds F I I I ____ sounds C Figure 2. Strings which produce a fourth.

When played together, the two strings produce a pleasant sounding interval called a fourth. When two strings are such that their lengths form a ratio of 2:3, the strings form an interval called a fifth. In Figure 3, the shorter string is 2 units long, the longer string is 3 units long. When plucked, the strings produce a fifth. The shorter string is 213 the length of the longer string.

I sounds G I I sounds C Figure 3. Strings which produce a fifth. 20

Finally, strings whose lengths have a ratio of 1:2 sound an interval called an octave (see Figure 4). The shorter string is 112 the length of the longer string.

______I sounds C' I sounds C Figure 4. Strings which produce an octave.

To construct our scale, it is necessary to determine the ratio of the interval between two consecutive tones. For our interval between two consecutive tones, let us choose the interval from F to G. This interval is called a second. Notice that F is the higher tone of a fourth and G is the higher tone of a fifth. Exercise 2.1 In the figure, the longest string sounds a C, the shortest sounds a G) and the other sounds an F. I I I I I I I I I I I I c 12/12 I I I I I I I I I F 9/12 = 3/4 I I I I I I I I· G 8112 = 2/3 What is the ratio of the shortest string to the middle string? The shortest string is what fractional

part of the middle str~ng? Mathematically speaking, 21

213 = (x) ( 314) , x = ? (x expresses the ratio of the lengths of strings of a second). Musicians have gotten into the habit of referring to the ratio of the lengths of the two strings of an interval as simply the ratio of the interval, We shall adopt that tradition.

Exercise 2.2 Let the unit string produce the tone named c when it is plucked. If we compare the length of the unit string with a string of the same length, the ratio is 1:1, alsd written 111. In Exercise 2.1, we found that the inter

val between two consecutive tones (C~D, for instance) has the ratio 8:9. To obtain the string which produces

the next tone above C, we take 819 of

the unit string, (819) (1) =: ? Eight-ninths expresses the ratio of a second.

_...... ;..._I I I / I I I I D 8/9 I I 1./ I'; '.'_L_ c 919 Exercise 2.3 The interval from C to E includes three consecutive tones. The third tone, E, is sounded when a string

819 the length of the D-string is plucked. Taking 8/9 of the unit string produces D. Taking 8/9 of the 8/9 produces the next consecutive tone, E. The ratio of the interval from C to E is found by multiplying:

(1) (8/9) (8/9) = ? Exercise 2.4 Complete the table. INTERVAL Second Third

NOTES From C ... C D E -~------RATIOS 1/1 ? ?

Fourth Fifth Sixth Seventh Octave

F G A B c

3/4 2/3 ? ? ? Did you realize that not all intervals in our scale can be found by taking powers of 8/9? For instance, the ratio for a fourth, 3/4 f (8/9)(8/9)(8/9). It will be helpful to determine what fractions are multi,lied to obtain the intervals in our scale; To get the idea, look at the following sequence, SEQUENCE: 1 1 2 6 24 120 MULTIPLIERS: (1) (2) (3) (4) (5) (6) Each number in the sequence is found by multiplying the previous number by a consecutive integer: 23

(1) (1) = 1 (1) (2) -- 2 (2) (3) = 6 (6) (4) = 24 (24) (5) = 120, etc, Exercise 2.5 Using the sequence of fractions found

for the intervals in Exercise 2.4,

determine the numbers that are multi~ plied to obtain consecutive fractions in the sequence.

SEQUENCE: 1 8/9 64/81 3/4 2/3 16/27 ...

(? )' MULTIPLIER: ( 8/9) (8/9) I.. (?)

SEQUENCE (continued): 16/27 128/243 2/1

?'\ MULTIPLIER (continued): ( • J (?) The term, "tone," in music is used in two ways. It refers first of all to a sound of well-defined pitch or quality. The note is then a written symbol of the tone that we hear. The term, "tone,n is also used in the meaning of whole tone (that is, a major second) as opposed to half tone or semi-tone. In this way, the ratio 8:9 refers to a whole tone, while 243:256 represents what is known as a semi-tone. The scale from C to C 1 yields the following pattern (W: whole tone or whole step, S: semi-:-tone):

c D E F G A B C' w w s w w w s 24

You can see how this pattern is similar to the one you

~ompleted in Exercise 2.5.

Exercise 2.6 What arithmetic operation do we use to

obtain a musical interval from the

previous interval?

The table which you completed in Exercise 2.4 gives us the ratios of the intervals in a commonly used scale, a diatonic scale. A diatonic scale is one which primarily uses whole steps. Pythagoras arrived at this scale by a dif£erent method than you did. Because he attached importance to the interval of a fifth, believing it related to the astronomical order of tbe sun and planets, he derived the scale from a succession of fifths.

Exercise 2.7 Pythagoras used seven letters to

represent tones in his scale, A, B,

C, D, E, F, and G. Tones an

octave higher are represented like

1 1 1 1 1 1 1 this: A , B , C , D , E , F , and G •

Another octave higher: A", B", C", D", E", pn, and G". Begin with the

fundamental C, middle C on the piano.

A fifth above Cis G (£-D-E-F-Q). In the following sequence,· circle every

fifth note. © D E F @A B c' @) E I p 1 G I@ B I

11 C" D"@ f!i G" A"@ C D" E" I ® 25

·~· G"' A"' B'''~l Now use the circled notes to complete

the succession of fifths: C G D 5th 5th 5th-5th-5th-5th-5th-

Having found this succession of fifths, we, like

Pythagoras, now use these fifths to derive a scale. Using the succession of fifths, reduce D' and A' an octave each to D and A respectively. Similarly, r.educe E" and B" two octaves each to E and B. We have C G D A E B, or in order, C D E · G A B. To determine the missing tone, F, we could take a fifth below the fundamental C and raise it up an octave. We illustrate this method as follows:

D E F G ABC'$E' F' G' B' C" G" t §) D_j3 F" A"Cf ~c t i ; ~ I In Exercise 2.4, we obtained the tones of the diatonic scale by multiplying ratios. Let's use the method of multiplying ratios to verify Pythagoras's method of obtaining a scale by fifths. Using 2/3 as the ratio of a fifth, to find the ratio of D', two fifths above C, we would multiply: (2/3)(2/3) =? Then, to lower D' an octave, we multiply its ratio by 2/1.. To raise an octave from low tone to high tone, multiply b·y 1/2; to reduce from high tone to low tone, multiply by 2/l. Thus, to obtain the ratio of D: (2/3)(2/3)(2/1) = 8/9

Eight-ninths is the ratio of a second (Exercise 2.4). The 26

other tones of the scale can be derived by multiplying the correct ratios. Then the tones can be lowered an octave by multiplying by 2. This is where the octave ratio shows its importance. Exercise 2.8 Using the method of multiplying ratios, rlerive the other ratios in the diatonic scale. Example: The ratio which represents E is:

(2/3) (2/3) (2/3) (2/3) (2/1)

(2/1) = 64/81

To rais~ C four fifths to E, multiply (2/3)(2/3)(2/3)(2/3} To lower the tone two octaves

multiply by (2/1)(2/1).

The ratio which represents G is ?

The ratio which represents A is ? B ?

F ? Euclid wrote a paper on music entitled Theory of Intervals, which contains 19 theorems (Archibald, 1924, p. 9). Most of the theorems have to do with the division of a string, or monochord. Let us say a string on the monochord produces a tone which we call A, ·the fundamental tone. Intervals in music do not have to be measured from

C. For example, from A to D, A-B~C-D, is a fourth. 27

Euclid divided the A-string as in Figure 5. The fourth has the ratio 3:4. So D, the fourth above A, is produced by taking 314 of A.

I I I A 414 I I D 314 Figure 5. Strings which produce a fourth.

Using the same string, A, and the same division, the octave, A', whose ratio is 112, is established by taking two parts of the four parts of the unit string (Figure 6).

I I I A 414 I A' 214 = II 2 ------~------Fifture 6. Strings which produce an octave.

The double octave, A", is established by taking one part of the four parts. Multiplying ratios, (112){1/2) = 114 (Figure 7).

I I I A 414 N' 1/4 Figure 7. Strings which produce a double octave.

Next, Euclid divided that part of the string which produced D into two equal parts (Figure 8). 28

I I I A 414 I I D 314 I ? (112) (314) = ? Figure 8. Strings which produce the octave of the fourth.

Exercise 2.9 Using Figure 8, name the interval.

produced by taking 112 of the D-string. (Octave of the .) What is the letter name of the tone produced

by taking 112 of the D-string? Euclid then divided the original.string, A, into two equal parts, and one of these parts was divided into three, of which two parts were taken (Figure 9).

I A

I I A' = ll 2 I ? (213) (112) = ? Figure 9. Strings which produce the fifth of the octave.

Exercise 2.10 In Figure 9, besides the octave of A, the of the octave was -·---- produced. Its letter name is ? Exercise 2.lla Using diagrams like Figures 5-9,

indicate methods by which Euclid might have arrived at the other tones of the scale. To determine B, we count 29

by fifths: A is the unit string,

A-E is a fifth, E-B' is a fifth.

So, 2/3 of the A-string produces E,

and 2/3 of the B-string produces B' .

Represent the A-string this way: I I A 3/3 Next, represent the E-string. Finally,

represent the B'-string. What part of

the A-string is the B'-string? What

is the ratio of that interval?

2.llb Similarly, use fifths (ratio 2:3) to

find F.

Z.llc Use fourths (ratio 3:4) to find G.

2.lld Use fourths to find C.

2.lle Perhaps you can find other ways to

derive these tones.

We have seen how the diatonic scale can be derived in three ways: by using Euclid's method of division of the monochord, by using Pythagoras' method of a succession of fifths, and by using products of ratios.

There were several objections to Pythagoras's diatonic scale. One objection was that little variety was possible in melody because of the lack of semi-tones, or half tones. In others words, there were no G-sharps, no

D-flats. On a piano, there would be no black keys, because the interval from a white key to a black key 30

sounds a semi-tone! A musician working with the

Pythagorean scale would be a little like a mathematician

working with the natural number system (1, 2, 3, ... ).

Sooner or later, the mathematician needs fractions. And

sooner or later, the musician needs semi-tones. Another

objection to the Pythagorean scale arose when composers

began to use polyphonic music, where two tones.were sounded

simultaneously, and the tones on the Pythagorean scale did

not sound well together. Other problems arose with the

Pythagorean scale when it was discovered to be inconsistent

mathematically by a margin of error called the Pythagorean

Comma, also known as the Greek Comma.· Musically, this

error caused dissonance, or unpleasant sound, when two

or more tones were sounded together.

In order to understand the problem of the Pythagorean

Comma, you need to be able to use another term, frequency.

Euclid observed that when a string was plucked, it vibra

ted. A long string vibrates slowly; a short string vi brates rapidly. Frequency refers to the number of vibrations of ~ string in a unit of time. It was dis covered that the frequency of vibration of a string is related to its length. For example, string X is 2/3 as long as string Y, and string X vibrates 3/2 as fast as string Y. The frequency ratio is the reciprocal of the interval ratio. The frequency of vibration determines the sound--pitch--of the tone. 31

Now let's look at an example of inconsistency in the Pythagorean scale. We know that the octave produces the best concord. This means that the octave is the interval most pleasing to listen to. The next most plea sant consonance is the fifth, which has a frequency ratio of 3 to 2, sometimes w·ri tten in decimal form, 1. 5 to 1, for ease in computation. Beginning at C, by multiplication we increase the frequency of vibration by 3/2. The next note would then be G (a fifth). In 12 such steps, we would-arrive at the seventh C above the starting C. The intervals named make up a scale clock, so called because it shows the relationship between the ·scales (Figure 10). c

Figure 10. Scale clock (circle of fifths).

Exercise 2.12 Complete the series of fifths.

(Remember: Each fifth includes 5

tones.) 32

Steps 1 2 3 4 5 6 7 8 9 10 11 12

C G (C)

Exercise 2.13 The frequency ratio of each step on the scale clock increases by a factor of 1.5 = 3/2. If our starting C has a frequency ratio of 1/1, we could determine the frequency ratio of the fourth step of the scale clock by 4 calculating (3/2) = 5.1625. In this way, calculate the frequency of vibration of the twelfth step on the scale clock. The twelfth step on the scale of fifths clock is also the seventh C above the starting C. We arrived there by taking 12 steps of fifths, but it is possible to arrive at this C in another way-~seven steps of octaves! We can do this by taking seven steps in each of which we increase the frequency of vibration by 2/1 (the frequency vibration of an octave). For example, the frequency ratio of the fourth . 4 octave C could be found by calculating (2/1) = 16.

Exercise 2.14 Calculate the freq~ency of vibration of the seventh octave C·, using 2/1 as a factor to increase an octave. Exercise 2.15 By division, compare the answers you

obtained in Exercises 2.13 and 2.14.

This number is known as the comma of

Pythagoras, or the Greek Comma.

If we refer to the table from Exercise 2.4 and compute

the reciprocal of each ratio there, we will obtain frequency ratios of the tones in the original Pythagorean scale (Figure 11). (Remember that frequency ratios are reciprocals of interval ratios.)

From C to ... C D E F G A B c

Ratio 1/1 9/8 81/64 4/3 3/2 27/16 243/128 2/1

Figure 11. Frequency ratios of the Pythagorean

scale.

Because the octave, the fourth and the fifth (having frequency ratios of 2/1, 4/3, and 3/2, respectively) were the most pleasant-sounding intervals, smaller integers were believed to make better consonance. There£ore, the ratio for E, 81/64, became 5/4 = 80/64. The ratio for A, 27/16, became 5/3 = 25/15. In time, other intervals evolved as composers sought variety in their musical compositions. The intervals most commonly used in early composition using the Pythagorean scale are shown in Table

1 •

For many years, composers used the Pythagorean scale, with modifications here and there. In composing for piano Table 1 Commonly Used Intervals of the Pythagorean Scale

Minor Major Perfect Perfect Minor Major Interval Unison Third Third Fourth Fifth Sixth Sixth Octave From C to ... c Eb E F G Ab A C' Ratio 1/1 6/5 5/4 4/3 3/2 8/5 5/3 2/1

v~ +:>. 35

and organ, howeve~ some difficulties with the scale were encountered. One of these problems concerns the fact that on a piano Ab and G# are the same tone, and should have the same frequency ratio, We shall see how in the Pythagorean scale the two tones have a different frequency ratio. Exercise 2.16 Using Table 1, determine the ratio of the interval from C to G*. C to E is a third, E to G# is a third. Remember to use the ratios of the intervals as factors, and find their product. Exercise 2,17 From Table 1, what is the ratio of the interval from C to Ab? Exercise 2.18 Divide the ratio in Exercise 2.17 by the ratio obtained in Exercise 2,16. This represents the frequency ratio between G# and Ab, But since G# and Ab are the same note, the frequency

ratio from G# to Ab should be 1:1 1 the identity number! The amount you calculated in Exercise 2.18 provides a difference in sound which is easily perceptible to a musician. (And it's close to the Greek Comma!) Here's another example: Let's compare the whole tones from F to G, and Eb to F. 36

Exercise 2.19 Calculate the frequency ratio for the interval F to G. (4/3)(?) = 3/2. Exercise 2.20 Calculate the frequency ratio for the ,interval Eb to F (use Table 1). Exercise 2.21 Divide the ratio obtained in Exercise 2.19 by the ratio obtained in Exercise 2.20. (Express the results as a decimal to the nearest ten thou- sandth.) Since the intervals are both whole tones, their ratio should be 1/1, but the decimal you found should be familiar. What do we call it? You have seen from your calculations that the Pythagorean scale involved frequency ratios which were not always consistent. The musical effect of these inconsis tencies was unpleasant to the ear. Before we investigate the scale which was invented to avoid these difficulties, let's take a look at one way in which Greek music may have had a role in the development of pure mathematics. There is a similarity between the development of musical fre quency ratios and the use of logarithms in mathematics. Logarithms were developed by mathematicians to save time and labor in calculations. By using logarithms, multiplication becomes a problem of addition, and division becomes a problem of subtraction. 37

In the statement 25 = 32, the 2 is called the base, the 5 is called the exponent. We can express the relation 25 = 32 in another form by using the word logarithm, abbreviated log. Instead of saying "2 exponent 5 is 32," we can say, "the logarithm of 32 to the base 2 is 5," 5 written log232 = 5. The expression 2 = 32 is written in exponential notation; log232 = 5 is written in logarithmic notation. The logarithm of a positive number relative to another· given positive number as the base is the exponent that indicates the power to which the base must be raised to equal the number. Here are some examples: 2 log10100 = 2 means the same as 10 = 100 2 49 2 7 49 log7 = means the same as = 3 log5125 = 3 means the same as s = 125 Here is an example of the use of exponents: 2 3 5 10 X 10 = 100 X 1000 = 100,000 = 10 Using logarithms, we write

log10100 + log101000 = 2 + 3 = 5 = log10100000 The operation of multiplication in exponential notation becomes the operation of addition in logarithmic notation. Another example is, 3 5 8 The arithmetic statement z X z = 8 X 32 = 256 = z can be interpreted in logarithmic notation:

logz8 + log232 = 3 + 5 = 8 = log22S6 38

Even though the concept of logarithms wasn't fully developed until the sixteen hundreds by Napier and Briggs, there was a hint of it in the use of mathematics to develop musical frequency ratios. Look at the table on the following page. You see that in music addition of intervals is interpreted as multiplication of frequency ratios. This compares with addition of logarithms, interpreted as multiplication in exponential notation. Exercise 2.22 Discover at least two other examples of intervals and verify them mathe matically, as in'the table on the following page. (Hint: Obtain the musical interval first, then use corresponding ratios for mathematical interpretation. A fifth = major

third+ minor third. 3/2 = (?)(?) Exercise 2.23 Another interesting problem which can be approached from Greek music might be stated this way, ''To find the first four notes of a scale, determine· all possible ways of

decomposing the ratio ~/3 (musical fourth) into the product of three ratios of the form (n + 1)/n." Table 2 Comparison of Musical Description with Mathematical Interpretation of Intervals Musical Description Mathematical Interpretation Using Intervals Using Ratios

Octave = fifth + fourth 2/1 = (3/2) (4/3) C to C' = (C to G) + (G to C') 2 Octave = 2 fourths + whole tone 2/1 = (4/3) (9/8) C to C' = (C to F) + (F to Bb) + (Bb to C') Octave = fourth + third + minor third 2/1-= (4/3)(5/4)(6/5) C to C' = (C to F) + (F to A) + (A to C')

VI 1.0 40

For example, (9/8)(10/9)(16/15) = 4/3. Find two more ways of decomposing

the ratio 4/3 into the product of

three ratios of the form (n + 1)/n.

Returning to the subject of scales: recall that

there were problems with inconsistencies in the frequency ratios of the Pythagorean scale. To avoid these diffi-

culties, a scale was suggested in 1482 which came to be called the equal temperament scale, so-called because each interval in the scale was based on the same frequency ratio. In this scale, the Greek Comma is distributed equally over the twelve intervals which make up the circle on the scale clock face. Since the error in frequency rat~os is about a quarter of a semi-tone, and there are twelve intervals (fifths), it is necessary to lower the upper tone of each interval about a 48th of a semi-tone, (1/4) (1/12) = 1/48. In the equal temperament scale, there are 12 semi-tones in an octave (rather than

Pythagoras's eight tones), and because the octave ratio is 2:1, the frequency ratio of a semi-tone is the twelfth 1 12 root of 2 or 2 / = 1.05946. In the equal temperament scale, no interval but the octave can be expressed using a frequency ratio mad~ up of integers~

Exercise 2.24 The frequency ratio of a fifth, . . f 7 . . . 27/12 cons1st1ng 0 . sem1-~0nes, IS .

Using your calculator, compute the 41 I •

frequency ratio of a fifth in the

equ .. ~:;. ·temperament scale. (Hint:

2 7 !J 2 = ( 2 1 I 12 ) x.)

Exercise 2.25 Looking at it another way, the twelve

steps around the clock face in fifths

represent 7 octaves, whose frequency

is 2 7 = 128. So each fifth must be a twelfth root of 128: 128l/lZ 7 12 = 1.4983 = 2 / (fro~ Exercise 2.23) .· Verify that 1281112 ~ 27/lZ by

factoring 128.

Exercise 2.26 Compare the valu~ of a fifth in the

equal temperament scale with the

value of a fifth in the Pythagorean

scale.

Exercise 2.27 Calculate the major third, 4 semi-

tones on the equal temperament scale.

Exercise 2.28 Compare the values of the thirds

in the equal temperament scale and

the Pythagorean scale.

The frequency ratios based on the twelfth root of two were calculated by the French mathematician, Mersenne,

and published in 1636. But the scale using 12 semi-tones wasn't used until late in the seventeenth century when an organ was tuned to the equal temperament scale. J. S.

Bach strongly advocated the equal temperament scale and had 42 a clavichord and harpsicord tuned to it. He even composed 48 preludes and fugues to show that compo~itions using the equal temperament scale could be played in all keys without disagreeable chords. English pianos adapted the equal temperament scale, also known as the well-tempered scale, in the mid- nineteenth century, and it is now in use in keyed instruments. Few people notice that the intervals are not in perfect tune (remember that the frequency ratios of intervals of this scale are not perfect integral ratios as in the Pythagorean scale). It is interesting that the human voice sings, and the violin· plays what sounds best to the human ear, and what they play is different from what is played by the instrument tuned to the equal temperament scale (Jeans, 1953, p. 176). Around the middle of the seventeenth century, Nicholas Mercator proposed an improvement on the equal tempered, 12-tone scale. His idea incorporated a 53-note tempered 17 53 scale in which the third is z 1 = 1.2490, and the fifth is 231 / 53 = 1.49997. These frequency ratios closely approximate those of the Pythagorean scale. Exercise 2.29 What is the error in the 53-note major third? (CompaTe with Pythagorean frequency ratio, Figure 11.) 43

Exercise 2.30 What is the error in the 53-note

major fifth? (Now you can se~ why it has been called the "perfect" scale.) Presumably, Mercator arrived at the number 53 by solving for x 2x = 1.s = 3/2 or 2(2x) = 3 or 2x+l ·= 3

Takin~ the logarithms of both sides of the equation~ we have log 2Cx+l) = log'3 or (x + l)log 2 = log 3 or x(log 2) + log 2 = log 3 or x(log 2) = log 3 log 2 which yields X = (log 3)/(log 2) 1 or X = 1760913/3010300 = 7/12 = 31/53 (Coxeter, 1962, p. 19) Exercise 2.31 Using a calculator, find the value of z1153 . Of what does this decimal remind you? Still another scale which was developed and used from 1500 to 1700 is one called the mean ton_e __scal_~._ Taking four intervals of fifths from C on the clock face, we arrive atE (Figure 10). 44

Exercise 2.32 Caiculate the frequency ratio of the

interval from C to the E four steps

above, by fifths.

The frequency ratio which you found is greater than the "pleasurable" ratio of 5:1. The solution which brought about the mean tone scale was to diminish the four intervals of fifths so that the frequency of one interval . 1/4 is the fourth root of 5 or 5 = 1.49527 (as compared with 1.5 in the Pythagorean scale). Difficulties with the m~an tone scale arose in such intervals as from G# to Eb, which was 3/8 of a semi-tone more than the exact fifth.

These intervals produced horrible howling sounds which came to be known as "wolf-tones." In spite of special keys, special instruments, and special tuning to the mean tone scale, the scale failed to gain wide acceptance

(Jeans, 1953, p. 172).

Basically, we have two requirements that must be met in the construction of a scale. First, there must be simple intervals, that is, as many tones as possible must be pleasing to the ear when they are sounded together. Second, we must have freedom to shift keys

(Saunders, 1948, p. 40).

Although the purists object to the 12-tone scale, saying, "It hurts our ears," because the intervals are not perfect, the expert piano tuners tune the low notes low (or flat) and the high notes high (too high to fit the 45

scale). So, on the piano, the scale has fifths which

are true and octaves which are "spread" greater than 2:1.

When the violin, tuned to perfect fifths, or the profes

sional singer, who does not use any scale exactly, performs

with the piano, no one complains that the sound is not

pleasant. Perhaps we are not as particular as we thought.

We have seen how mathematicians from the days of

Pythagoras have been intrigued by the problems of express

ing music mathematically. First, a scale was developed

using integral ratios of three basic intervals: the

octave, whose ratio is 2/1; the fourth, whose ratio is 4/3;

and the fifth, whose ratio is 3/2. Then mathematicians

and musicians set about improving upon the Pythagorean

scale. As they worked together to solve their problems,

they discovered useful mathematics, and created beautiful

music. The result of their efforts was our own equal

temperament scale.

Yet, this is only the beginning of the story. In

the harmony and harmonics of music, you will find mathe matics, and the musical instruments we use are dependent

upon physics and mathematics for their existence. The music of our day makes use of mathematics in its rhythm

and structure. Further study of the relationships between mathematics and music prove both interesting and beneficial

as we discover the mathematics in music. References

Archibald, R. C. Mathematics and music. American Mathematical Monthly, 1924, 31, 1-25.

Bell, Eric Temple. Men of mathematics. New York: Simon and Schuster, 1937.

Coxeter, H. M. S. Music and mathematics. The Canadian Music Journal, 1962, ~' 13-24.

Jeans, Sir J. H. Science and music. Cambridge, Mass.: Cambridge University Press, 1953.

Saunders, Frederick A. Physics and music. The Scientific American, July 1948, 179, 33-41.

46 Additional References for Interested Readers

Benade, Arthur. The physics of woodwinds. The Scientific American,-1969, 203, 144-154.

Blackham, E. Donnell. The physics of the piano. The Scientific American, 1965, 213, 88-96.

Brinton, Henry. Sound--What it is and how we hear it. New York: Golden Press, 1966.

Brown, J. D. Music and mathematicians since the seventeenth century. Mathematics Teacher, 1968, ~' 783-787.

Gardner, Martin. Mathematical games. 1be Scientific American, 1978, 239, 16-31.

Helmholz, Hermann L. F. von. On the sensations of tone as a physiological basis _for __ the theory of mus~c. Translated by A. J. Ellis, London: 1863; Magnolia, Mass.: Peter Smith; New York: Dover Publications, pap er , 19 54 .

Lineback, Hugh. Musical tones. The Scientific American, 1948, ~~' 33-41. Shelling, J. C. Physics of the bowed string. The Scientific American, 1974, 230, 87-95.

47 I '

Appendix A Glossary of Terms

clavichord - a small stringed, keyed instrument, the forerunner of the spinet, harpsichord and piano concord - a combination of harmonious sounds (emphasizing the aesthetic impression of "pleasantness" rather than the technical aspect) consonance - tones sounding simultaneously, producing a pleasant effect, providing stability and repose (subjective) diatonic - designation for the major (also minor) scale as opposed to the chromatic scale. For instance, in C major, C-D-E is diatonic; c'-n-n*-E is chromatic dissonance - tones sounding simultaneously, producing an unpleasant effect, producing tension and motion by "pulling" toward a resolution in a consonance equal temperament scale - the division of the octave into twelve equal semi-tones frequency - the number of vibrations per second hemitone - the interval between the third and fourth tones, and between the seventh and eighth tones of the Pythagorean (diatonic) scale; has ratio of 256/243 homophonic - single melodic line supported chordally

interval ~ distance in pitch between two notes; names for intervals refer to the number of scale steps from lower to the higher note, as follows: C-C, unison; C-D, second; C-E, third; C-F, fourth; £-G, fifth; C-A, sixth; C-B, seventh; C-C', octave logarithm - the power to which a base must be raised to produce a given number

48 49 mean-tone system - a system of tuning which used perfect thirds and almost perfect fifths monochord - an apparatus consisting of one string which can be used to determine the frequency ra~ios of musical intervals octave - the interval embracing eight diatonic tones; interval of the ratio 2:1 pitch - referring to· the high-low quality of a musical sound; pitch is determined by the number of vibrations per second, i.e., the frequency of the tone polyphonic - "many voices," several parts

Pythagorean scale - a scale whose development is attributed to Pythagoras, in which all the tones are derived from the interval of the fifth scale - a succession of notes, normally either a whole tone or a half tone apart, arranged in ascending or descending order semi-tone - a half-tone or half-step temperament - general designation for various systems of tuning in which the intervals are adjusted timbre - the quality of tone or sound tone - a sound of well-defined pitch and quality; the interval of a second Appendix B Answers to Selected Exercises

1.1 A ratio is the comparison of two numbers by division 2: 1, 2 to 1, 2 I 1

1.4 Multiplied by 3, sound is higher. 1.6 Five 1.7 Fourth

2.1 8:9, 8/9 2.3 64:81

2.4 8/9, 64/81, 16/27, 128/243 2.5 254/256, 8/9, 8/9, 8/9, 243/256 2.6 Multiplication 2.7 A', E", B"', F"', C"" 2.8 G: 2/3 A: (2/3) (2/3) (2/3) (2/1) = 16/27 B: (2/3) (2/3) (2/3) (2/3) (2/2) (2/1) (2/1) = 128/243 F: 3/2 X 1/2 2.9 Fourth, D' 2.10 Fifth, E'

Z.lla (2/3)(2/3) = 4/9 [To lower octave, (4/9)(2/1) = ?] 2.1lb (2/3)(2/3)(2/3) = 8/27 [To lower octave, (8/27)(2/1) = ?) 2.12 D, A, E, B, C# , G# , D# , A# , E# 2.13 129.74634 2.14 128

I so ! 2.15 1.0136433 2.16 (5/4)(5/4) = 25/16 - 1.5625 2.17 8/5 = 1.6 2.18 1.024 2.19 9/8 2.20 (4/3) .. (6/ 5) -· 10/9 2.21 9/8 10/9 = 1.0125 2.22 Fifth = fourth + whole torte

3/2 = (4/3) (9/8) ~ 3/2 2, 23 (8/7) (10/9) (21/20) = (9/8) (8/7) (28/27) = (10/9)(11/10)(12/11) = 4/3

2.24 X~ 7, 1.4982764 2.25 (2 7) 12 = 2 7/12 2.21 (21112) 4 = z113 = 1.26 2.32 (3/2) 4 = 81/16 = 5.0625