1. A BRIDGE BETWEEN MUSIC AND MATHEMATICS
Oladimeji Ogunyemi* and Tong Wu#
Department of Mathematics
College of Science and Technology
Texas Southern University
3100 Cleburne Street, Houston, TX 77004
ABSTRACT
Both mathematics lovers and music lovers can cross the thin line of separation
between their interests through knowledge. Music is mathematical in nature
because of its inherent patterns and mathematics is musical in reasoning through
creative manipulations of numbers. This work’s intention is to mathematically
interpret one of the key tools of western music, “the circle of fifths.” Information
present here will bridge the divide between music and mathematics in a manner
that is comprehensible to readers with basic knowledge of music or mathematics.
Additional objective would be to make known how musical problems are linked to
challenges that arise in mathematics. Both worlds have strong relationship
encompassing more than 2000 years, around the time when Pythagoras
discovered the relationship between the length of vibrating string and its frequency.
Many mathematicians had turned to music for inspiration and some musicians like
Igor Stravinsky and Beethoven used mathematics creatively in some of their
works. Additionally, generation and propagation of musical sound
(frequency/pitch) have been proven mathematically through physics.
INTRODUCTION
It is amazing that as humans, we are embodiment of skills and abilities and through
these gifts; man has conquered and explored lots of fallow fields with outstanding
results. Many of the discoveries of man came from borrowed ideas from various
fields. A typical example is the exploration of outside world, which is never left to
the astronauts but so many talented professionals with diverse backgrounds
(biology, mathematics, chemistry, acoustic, nuclear power, electrical, computer,
electronics etc.) came together for a common goal.
For many years, scholars (scientists and engineers) have been indulging in
borrowing ideas from one another. Recently, majority has turned to nature for
inspiration while others continue in the path of unconventionality in order to break
new grounds. This sort of endeavor is very crucial in eliminating latency among
scholars whose ability to think outside the box has been crowded with superficial
conclusions about the relevance of some certain subjects to human development.

2. Music is an escape from mathematics to some people because their minds, as
feelings, have been prevailed upon by [3] “mathophobia.” Likewise, some brainy
mathematicians considered themselves to be novice when it comes to musical
notes embedded in “five lines and four spaces,” this might be a sign of [4]
“melophobia.”
Scientifically, some researchers came up with findings to support the notion that
same part of the brain is engaged whenever math lovers look aesthetically at
mathematical formulae and when musicians appreciate musical works.
Additionally, it has been discovered that the aspect of the brain used for some
mathematical operations can be enhanced through engaging in music
performance. These are just few examples to show how closely related these two
worlds are even in our subconscious minds. If these two subjects (mathematics
and music) are not in conflict in our brain, but rather aid the development of one
another. Therefore, separation between mathematics and music as imposed by
our minds, can be considered as irrelevant within the scope of this paper.
Scholars whose opinions serve as the building blocks for present researches have
long established the connectivity between music and mathematics. Considering
some of the quotes relating to both music and mathematics, they evidently clarified
any dispute that might arise to disregard such interdependency of both
mathematics and music.
For emphasis sake, [1] Quadrivium: Boethius gave us idea about how primitive the
cordiality music and mathematics have both enjoyed. As early as the 6 B.C, music,
geometry, arithmetic and astronomy were regarded by Boethius as the “four-ways
of life.” [2] Such grouping should never be taken for granted because abstract
intellectualism is duly engaged while dealing with the duo. Medically, the spatial-
temporal (ST) reasoning is said to be a common tool for solving both musical and
mathematical problems and this reasoning could be enhanced for the benefit of
mathematics while engaging in music and vice versa. Recently, the emergence of
the term “Mozart Effect,” is another indication to the relevance of music to the
development of critical reasoning ability of children when they are exposed at early
age to classical music.
[1] James Joseph Sylvester said in 1860, “May not Music be described as the
Mathematic of Sense, Mathematics as the Music of reason? The soul of each the
same! Thus the musician feels Mathematic, the mathematician thinks Music, —
Music the dream, Mathematic the working life, — each to receive its consummation
from the other.”
[1] Claude Debussy, c. 1900, a French impressionist made a remarkable statement
as follows, “Music is the arithmetic of sounds as optics is the geometry of light.”

3. METHODS
The major conceptualization of this work is based on the in-depth investigation of
the "circle of fifth." For many years, musicians have utilized this chart in deciphering
the theory of music. To average musicians, the answers to their quest lie in what
they can perceive physically. But to those thinking abstractly and mathematically,
the details go beyond what their physical eyes could perceive. Their interest is
never distracted from the patterns as exhibited in the chart and their pursuance is
to formulate relationship between all the parameters or elements of the chart. The
circle of fifths (figure 1) became the main drawing board of this project and a guide
for both worlds represented here and is regarded as the foundation on which we
build our bridge. Transformation of the circle of fifths (figure 2) into mathematical
functions has opened up a unique avenue for the comprehension of music by
average scholars with basic understanding of mathematics. Likewise, it enables
scholars with musical knowledge to view themselves as operating mathematically
through music.
MODULE 1
ACCIDENTALS (SHARP AND FLAT)
A sharp raises the pitch of note by semitone (the shortest interval with respect to
pitch between two successive notes), while a flat reduces the pitch by semitone.
Mathematically, the sharp (#) denotes plus (+) and the negative (-) is denoted by
flat (♭). This is quite basic for both mathematical and musical world. From the
diagram below (figure 2), two successive white keys (B - C and E –F) or from a
black key to white key (F – F#/G♭) and vice versa (F#/G♭ – F) encompasses a
semitone. A tone is the combination of two semitones (C –D, E – F#, E ♭- F and
A – B).

4. KEY SIGNATURE
Figure 1 (Chromatic circle of fifths with 15 divisions)
Interestingly, the circle at this point is divided into 15 equal parts, locating the points
around the circumference of the circle using De Moivre’s Theorem for calculating
the nth root of a complex number to locate.
[5]
2 2
cos( ) sin , where 0... 1k
k k
w i k n
n n
     
    
 
Where n= number of divisions = 15
k = 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15 and θ= 0
Considering the alphabets at the right side of the above figure 1, from the top
shows the respective key signatures; number(s) of sharps and flats. The values of
k from 1 – 7 are for the keys with sharp #), while 8 -14 show the flat (♭) keys. It is
worth to note that the number of sharps representing the key signature is
increasing from 1 – 7 but the number for the flat representing the major keys with
flat sign is decreasing as we move from k = 8 (C♭) to K = 14 (F major).
The key of C major has neither sharp nor flat sign, therefore, it is represented by
point k = 0. G is a fifth from C with point k = 1 (one sharp). Remember that we are
moving in fifth as we move counterclockwise from C major (K=0) then k increases,
signifying increase in number of sharps. Accordingly, from point k = 8 (C♭), the
number of flats decreases in the counterclockwise direction.
In summary, the value of K indicates the number of sharps and flats, which is the
The Circle of Fifths Major Keys & Minor Keys
C
G
D
A
E
B
F#
C#
Cb
Gb
Db
Ab
Eb
Bb
F
a
e
b
f#
c#
g#
d#
a#
ab
eb
bb
f
c
g
d
B D
C
E
F
G
A
F#
C#
Ab
Bb
Eb
Gb
Db
Cb
b
d
c
e
f
g
a
eb
a#
d#
g#
c# f#
ab
bb

5. key signature.
DIATONIC SCALE
In the previous section, key signatures were introduced with regards to number of
accidentals. This concept is limited without specifically showing the affected notes
(sharpened or flattened notes of the scale). The 15 divisions became paramount
for proper indication of how the application of sharps and flats affects the pitch of
the fundamental note. Also, the main importance is to show the mathematical
relationships of numbers of sharps/flats as related to the key signatures.
Understanding how accidentals (sharps/flats) make up the key signatures,
prepared the way forward into knowing musical intervals, chords, melody writing,
modulation and transposition.
Again, the starting point is the C major diatonic scale for illustration because it has
no sharps and flats (C D E F G A B C). This scale originated from the Pythagoras’s
ratio but was later developed into the equal tempered divisions. It has fixed
intervals between all of its 8 elements, denoting an octave scale. The tonal interval
between two successive notes of the scale is as follows;
C – D = tone (T)
D – E = tone (T)
E – F = semitone (t)
F – G = tone (T)
G – A = tone (T)
A – B = tone (T)
B – C = semitone (t)
These intervals can be summarized as follows:
Tone Tone Semitone Tone Tone Tone Semitone
C – D D – E E – F F – G G – A A – B B – C
do : re re : mi mi : fa fa : soh soh : la la : ti ti : do
1st – 2nd
degree
2nd – 3rd
degree
3rd – 4th
degree
4th – 5th
degree
5th – 6th
degree
6t h – 7th
degree
7th – 8th
degree
Table 1
All major diatonic scales of the western music are manipulated using the
accidentals (sharps/flats) so as to maintain the above pattern as necessitated by
the equal tempered division. This pattern can be modeled mathematically using
figure 1. The main importance is to generate all the major scales and even going
further indicating the affected (sharpened/flattened) note(s) of each scale.
Referencing the right hand side column of figure 1, we know C major is K = 0, while
k = 1 is G major (fifth from C). What we know from previous section is that key of

6. G major has just one sharp but we can’t really tell what note has the sharp. When
k = 1 the note with sharp will be k = 14 (F), meaning in the key of G major, the
sharpened note is F and that is why playing the scale of G major on the piano will
include one black key (G, A, B, C, D, E, F#, G). For D(k = 2) major, fifth from G,
2 sharps. The sharpened notes are k = 14 (F#) and k = 0 (C#). A major (k = 3), 3
sharps will have the following notes at point k = 14, k = 0 and k = 1 sharpen (F#,
C# and G#). At this point, we can summarize all the remaining key signatures with
sharps using the above model:
k = 4: k = 14, k = 0, k = 1 and k = 2
k = 5: k = 14, k = 0, K = 1, k = 2 and k = 3
k = 6: k = 14, k = 0, K = 1, k = 2, k = 3 and k = 4
k = 7: k = 14, k = 0, K = 1, k = 2, k = 3, k = 4 and k = 5.
The key signatures with the flats are quite different in their mathematical model;
the following examples will help us in understanding their inherent pattern. But
before that, I will like to point out here that a 5th up has similar effect as a 4th down;
the 5th up from C is G and the 5th up from F is C. The 4th down or back from C is G
and the 4th down or back from C is F. Both 5th and 4th are invertible because they
sum up to 9 (nine). Other intervals summing up to 9 up are 3rd and 6th, 2nd and 7th,
unison and 8th.
Therefore, it is very clear that both F and G have common relationship to C
(harmonic series). Key of G major has 1 sharp (#), while key of F major is the
starting scale for the flats with a single flat as its signature. But what we know about
F major is limited to having just 1 flat, as we can’t really tell what note is flattened.
When k = 14 the note with the flat will be k = 13 (B), meaning in the key of F major
the flattened note is B and that is why playing the scale of F major on the piano will
include one black key (F, G, A, B♭, C, D, E, F). For B♭ major, fourth from (k =
13; 2 flats. The flattened notes are k = 13 (B♭) and k = 12 (E♭). E♭ major (k =
12); 3 flats will have the following notes at point k = 13, k =12 and k = 11 flattened
(B♭, E♭ and A♭). At this point, we can also summarize all the remaining key
signatures with flats using the above model:
k = 12: k = 13, k = 12 and k = 11
k = 11: k = 13, K = 12, k = 11 and k = 10
k = 10: k = 13, K = 12, k = 11, k = 10 and k = 9
k = 9: k = 13, K = 12, k = 11, k = 10, k = 9 and k = 8
k = 8: k = 13, K = 12, k = 11, k = 10, k = 9, k = 8 and k = 7.

7. MODULE 2
INTERVALS AND CHORDS
The first seven letters of the English alphabet used in defining musical notes are
A, B, C, D, E, F and G. The music staff helps in differentiating notes of same letter
but different pitches by vertically positioning them at a specific distance apart. This
discrete distance with respect to the bottom or reference note is called interval.
Any music interval is a multiple of the smallest step between any two or more
vertically spaced notes.
Counting in music is solely limited to the positive integers (natural numbers or
counting numbers); starting from 1 (one). But mathematics does the same thing
engaging both positive and negative integers.
C
C#/
D♭
D
D#/
E♭
E F
F#/
G♭
G
G#/
A♭
A
A#/
B♭
B C
0 1 2 3 4 5 6 7 8 9 10 11 12
Table 2
For mathematical correctness, the 12 divisions of the scale of C major are counted
from zero (chromatic scale - movement of half step /semitone between successive
notes). The ascribed figures (table 2) also depict the semi tonal interval from the
1st degree of the scale C (0). For example, D (major 2nd) is 2 and G (perfect 5th) is
2 and 6 semitones away from C (counting from left to right).
Figure 2. Color code: Black = black keys of the keyboard and blue = the white
keys.
[5]
C
12 -Equal tempered tones
A#/Bb
A
G
F
D
E
F#/Gb
C#/Db
G#/Ab
D#/Eb
B

8. 2 2
cos( ) sin , where 0... 1k
k k
w i k n
n n
     
    
 
n = 12 and θ= 0
k = 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11
In this section, the angle of the circle has been divided into 12 parts, representing
the 12 divisions (equal tempered) of an octave of a musical scale (C D E F G A B
C). One division has a value of 30o in both clockwise and counterclockwise
directions, which is equivalent to a semitone. From the above diagram, C has an
angular value of 0o but the interval of C to D from our previous section is a tone
that gives an angular equivalent of 60o. The tables below summarize the diatonic
intervals of the major and minor scales.
Table 3
MAJOR CHORD (TRIAD)
From the top section of table 3, all of the music intervals can be generated and
formulated. Just as it has be pointed out in module 1 that the interval between all
of the notes of both major and minor diatonic scales are fixed, unless alterations
are made through accidentals (# or ♭). For the scope of the paper, triad (chord)
with three elements) are the focus. Triad is made up of root, third and fifth (Roman
numerals: I, III, V) but comes in different configurations depending on the
requirment. For instance, a major triad of any scale is made up of the root, major
3rd (4 semitones from root) and perfect 5th (7 semitones from root). This can be
interpreted from above table for the C major scale as, C = root, E = major 3rd (4 x
300 = 1200) and G = perfect 5th (7 x 300 = 2100).
Therefore, a major triad will always have θ+00, θ+1200 and θ+2100 as respective
intervals between its three elements (root, root - 3rd and and root - 5th). This
configurationis constant regardlessofthe key orscale ofconsideration. Interestingly,
0 1 2 3 4 5 6 7 1 2 3 4 5 6 7
Tonic Sofa Intervals C Major G Major D Major A Major E Major B Major F#
Major C#
Major F Major Bb
Major Eb
Major Ab
Major Db
Major Gb
Major Cb
Major
1 do Unison 0 210 60 270 120 330 180 30 150 300 90 240 30 180 330
2 re Major 2nd 60 270 120 330 180 30 240 90 210 0 150 300 90 240 30
3 mi Major 3rd 120 330 180 30 240 90 300 150 270 60 210 0 150 300 90
4 fa P4 150 0 210 60 270 120 330 180 300 90 240 30 180 330 120
5 soh P5 210 60 270 120 330 180 30 240 0 150 300 90 240 30 180
6 la Major 6th 270 120 330 180 30 240 90 300 60 210 0 150 300 90 240
7 ti Major 7th 330 180 30 240 90 300 150 0 120 270 60 210 0 150 300
8 do P8 360 570 420 630 480 690 540 390 510 660 450 600 390 540 690
Tonic Sofa Intervals a minor e minor b minor f#
minor c#
minor g#
minor d#
minor a#
minor d minor g minor c minor f minor bb
minor eb
minor ab
minor
1 la Unison 270 120 330 180 30 240 90 300 60 210 0 150 300 90 240
2 ti Major 2nd 330 180 30 240 90 300 150 0 120 270 60 210 0 150 300
3 do minor 3rd 0 210 60 270 120 330 180 30 150 300 90 240 30 180 330
4 re P4 60 270 120 330 180 30 240 90 210 0 150 300 90 240 30
5 mi P5 120 330 180 30 240 90 300 150 270 60 210 0 150 300 90
6 fa minor 6th 150 0 210 60 270 120 330 180 300 90 240 30 180 330 120
7 soh Minor 7th 210 60 270 120 330 180 30 240 0 150 300 90 240 30 180
8 la P8 270 480 690 540 390 600 450 660 420 570 360 510 660 450 600
Mathematical Module in Degrees
Angular Intervals
Major Keys
Minor Keys
Number of sharps Number of flats

9. the absractness of dealling with intervals has been resolved through this table
enabling the grasping of the idea of intervals in a much more tangible way.
Other types of triad are mere modification of the major either by adding or
substracting 300 to any of its elements (root, 3rd and 5th). Any major triad can turn
minor by subracting 300 from its 3rd; root, m3 (M3-300) and P5 where M3 = major
3rd, m3 = minor 3rd and P5 = perfecr 5th. Meaning that a minor triad in angular form
can be defined as thus from D major triad, D (root) = 600, F# (3rd) =1800 (root + 1200)
and A (5th) = 2700 (root + 2100). Dminor triad = D (root) = 600, F (m3) = 1500 (1800 -
300) and A (5th) = 2700. The main objective here is that a major triad can be
diffrienciated from minor triad by paying attention to the value of the 3rds of both
triads. Additional example will be (E♭) in order to show how it translates to the flat
keys. From module 1, we know E♭has the the following flats k = 13, k =12 and k =
11 flatten (B♭, E♭ and A♭), so here is the scale of E♭ (one octave):
E♭ D G A♭ B♭ C D E♭
Root M2 M3 P4 P5 M6 M7 P8
Major triad (E♭) = 900 (root), 2100 (M3) and 3000 (P5) = E♭, G, B♭
Minor triad (E♭) = 900 (root), 1800 (2100- 300) (m3) and 3000 (P4) = E♭, G♭ (or F#),
B♭
INTERVAL MAJOR MINOR TRIAD
5th P5 P5
3rd M3
M3 - 300 = minor 3rd
(m3)
Root Root Root
INTERVAL MAJOR TRIAD AUGMENTED TRIAD
5th P5
P5 + 300 = augmented
5th (aug 5th)
3rd M3 M3
Root Root Root
INTERVAL MAJOR DIMINISHIED
5th P5
P5 - 300 = diminished 5th
(dim 5th)
3rd M3
M3 - 300 = minor 3rd
(m3)
Root Root Root

10. Though musical sound can be generated randomly from table 3, such selections
will lack artistic nature of music that gives life to any piece of music. In order to
add some degree of movement in time wise, various pitches are ascribed time
(beat). For those without or with little music knowledge, the use of quarter and
half notes are recommended to get them started.
Figure 3
Translation of the English nursery rhyme into the mathematical degree format
MARY HAD A LITTLE LAMB IN D MAJOR
RESULT AND DISCUSSION
It is quite astounding that music is an embodiment of mathematical functions and
the mental algorithm used while engaging in making music either as a performer
or a composer can be broken down into mathematical modules that can help in
encouraging students and professionals of both worlds to be more effective and
creative. But understanding basic dos and don’ts of both worlds can foster such
relationship. Although, piece of music can be generated randomly from the
mathematical perspective but such random selections can be refined and made
meaningful by being conversant with some guiding rules and tested principles.
Likewise, having music skill will not translate automatically into mathematical
know-how without learning the rules that govern the area of interest in
mathematics.
Circle of Time in beats
Half
Quarter
8th
16th
16th
The whole circle
is whole note
180(1/4) 120(1/4) 60(1/4) 120(1/4) 180(1/4) 180(1/4) 180(1/2) 180(0/1) 120(1/4) 120(1/4) 120(1/2) 120(0/1)
180(1/4) 270(1/4) 270(1/2) 270(0/1) 180(1/4) 120(1/4) 60(1/4) 120(1/4) 180(1/4) 180(1/4) 180(1/2) 180(0/1)
120(1/4) 120(1/4) 180(1/4) 120(1/4) 60(1/4) 60(1/4) 60(1/2) 60(0/1)
Mary Had a Little lamb

11. CONCLUSION
A relationship that has existed for more than 200 decades can never be considered
infinitesimal. Mathematics and music are both different subjects with many facets
that established their connectivity. Such connection has been proven through the
circle of fifths whose main purpose is bridging the divide in such a way to
encourage scholars of both fields for future advancement of similar research.
ACKNOWLEDGEMENTS
Summer Undergraduate Research Program (SURP) of the College of Science and
Technology at Texas Southern University supported this study.
REFERENCES
[1] Gareth E. Roberts, (2008) Math and Music: Exploring The Connections.
http://mathcs.holycross.edu/~groberts/Talks/HCAlum08Web.pdf
[2] Cindy Zhan, (2002) The Correlation Between Music and Math: Neurobiology
Perspective.
http://serendip.brynmawr.edu/exchange/node/1869#comment-148887
[3] http://leannafearlessmarie.blogspot.com/2010/10/mathophobia.html
[4] http://common-phobias.com/Melo/phobia.htm
,
Jeffrey Rosenthal, The Magical Mathematics of Music
https://plus.maths.org/content/os/issue35/features/rosenthal/index
http://www.thirteen.org/get-the-math/the-challenges/math-in-music/see-how-the-
teams-solved-the-challenge/22/
[5] James Stewart, Lothar Redlin, Saleem Watson (2014). Precalculus:
Mathematics for Calculus, Sixth Edition. Belmont, CA. Liz Covello.