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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
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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)
What is cyclic group ?
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.
Important Results :
Fret Board of a Guitar
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.
Key concepts in music :
Frequency & Pitch
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.
Octaves in Music
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.
Theory of Cyclic Groups helps in mysteries of Music :
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.
Why should doubling the frequency give a similar sound sensation ?
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.
Doubling Effect of Frequency in Light
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.
Absolute & Relative Pitch :
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.
Constructing the Cyclic Group :
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.
The Big Question
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.
The Big Question
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 K (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) K (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.
The Big Question
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.
Reference
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