The document discusses cyclic groups and their application to music theory. It defines a cyclic group as a group where every element can be expressed as a power of a single generator element. Important properties of cyclic groups are discussed, such as the order of a cyclic group being equal to the order of its generator. The concepts of pitch, frequency, and octaves in music are explained. It is noted that doubling the frequency results in the same note one octave higher. Viewing the ratios of frequencies as a group under multiplication reveals the connection to cyclic groups - the interval of an octave can be divided into 12 parts corresponding to the ratios generated by raising the frequency to powers of the ratio for a perfect fifth.

1. Luckshay Batra
luckybatra17@gmail.com

2.  What is a Cyclic Group ?
 Important Results of Cyclic Group
 Key Concepts in Music
 How the Theory of Cyclic Groups helps in solving
mysteries of Music

3.  A cyclic group is a group all of whose
elements are powers of a particular element
‘a’; that is , a group G is called a cyclic group
if ∃ an element a∊G such that every element
of G can be express as a power of ‘a’.
 In this case ‘a’ is called generator (primitive
element) of G.
 We express this fact by writing G=<a> or
g=(a)

4.  Order of a cyclic group is equal to the order
of its generator.
 A subgroup of a cyclic group is cyclic.
 If G is finite group, then order of any element
of G divides order of G.
 Any two cyclic group of same order (finite or
infinite) are isomorphic.

6.  Note : A sign used in musical notation to represent the relative
duration and pitch of a sound (♪, ♫).
For example: the whole song "Twinkle, twinkle, little star" can be
played using six different notes: C, D, E, F, G and A.
 Pitch: Pitch is the music that is heard & is usually associated with
frequencies of vibrations . Pitch is based on human sensation. In
music the pitch of a note means how high or low a note is . The
pitch of a note can be measured in a unit called Hertz. It has a
specific frequency.
 Frequency: Frequency of a note is the number of times the
corresponding wave repeats every second. It is measured
quantitatively.

8.  Octave: The interval between the first and last notes is an
octave. For example, the C Major scale is typically written C D
E F G A B C, the initial and final C's being an octave apart.

9.  Doubling the frequency : If the frequency of one note
is double the frequency of another note, the two notes sound
similar in a strong sense. Musicians say that two such notes
are separated by one octave, with the shriller (high-
frequency) version being one octave higher. If we're singing a
tune and reach a very shrill note, it's customary for us to
move an octave down to continue the song. Similarly, if we
reach a note that's too basal, we customarily move one octave
higher to continue the song.
If we assign the frequency of any note to be the identity,
then 2n times the frequency of the note is also the identity
,whenever n ∊ ℤ.

10. This is best viewed by thinking of frequency in terms of
the number of repetitions per second.
Suppose one note corresponds to a frequency of 760
repetitions per second, and another corresponds to
1520 repetitions per second.
Then, if the both notes are sounded simultaneously,
every repetition of the lower note corresponds to every
second repetition of the higher note.
Thus, the higher note contains the lower note in this
sense, and so the sounds match up in some sense.

11. Our visible spectrum of light ranges from violet
(highest frequency) to red (lowest frequency). And
red is approximately half the frequency of violet.
That's why there is an uncanny similarity between
violet and red.

12.  As per absolute frequency , we're generally not good at
sensing absolute pitch.
It's hard to judge, by sounding a note in isolation,
where on the scale it is.
Having a sense of absolute pitch does not necessarily
make someone a good musician, although it can be very
useful.
 The power of music comes more through its use of
relative pitch - if two notes are played one after
another, we can judge what the ratio of their pitches is.
Thus, if we take a musical tune, and multiply the
frequencies of all notes by some constant factor, the
tune still sounds the same tune.

13.  Viewing as logarithms :
 The group we are interested in is essentially the group of
possible ratios of frequencies, under multiplication. This is
just the group (ℝ⁺‚*) . However, we observed that any
frequency sounds a lot like its double frequency, so we'd
like to quotient the group of positive reals by the
subgroup generated by integral powers of 2.
 It may be more convenient to view this logarithmically. By
taking logarithms to base 2, we can identify the group of
positive reals under multiplication, with the group of all
reals under addition. Further, what we now want to do is
identify any two numbers that differ by an integer.

14.  Why does the interval of an octave get divided into
12 parts?
 Why there are seven natural notes and some sharp
notes?
 The answer lies in a somewhat mathematically imperfect
numerical coincidence. Fortunately, the mathematical
imperfection of this coincide is too small for our ears to
notice, and so we can enjoy music .
To understand the answer, we need to step back from
the logarithms and go back to looking at frequency
ratios.

15.  What happens if one frequency is 3/2
times another ?
Every second repetition of the higher frequency
note corresponds to a third repetition of the lower
frequency note.
Thus, even though neither note contains each
other, there is a harmony between them that makes
them particularly pleasing to the ear when
combined.
This combination is termed the perfect fifth in
music.

16.  What if we start with a frequency, and
keep multiplying it by 3/2 ?
We're never going to strictly get back to an equivalent
frequency again. That's because (3/2)n = (2)m has no positive
integer solutions. However, it turns out that (3/2)12 ̴ (2)7 ,
and the ratio is so close as to be practically indiscernible to
the human ear.
Back to the language of logarithms, this translates to
log2(3/2) ̴ (7/12) .
Thus, when we're looking at multiples of log(3/2), we cover
all multiples of (1/12).
The perfect fifth is thus, up to a reasonable approximation, a
generator of a cyclic group of order 12, that divides the
octave into twelve equal parts.

17.  http://groupprops.subwiki.org/wiki/Cyclic_groups
_in_music
 The Fascination of Groups by F. J.
Budden, Cambridge University Press, Chapter 23
(Groups and music).
 Joseph A. Gallian , Contemporary Abstract Algebra ,
Narosa Publishing House , 4th edition , 2002.