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A look at Abstract Algebra and Music Theory
Transcript of A look at Abstract Algebra and Music Theory
The Chromatic Scale
A scale is a set of notes ordered by pitch
The chromatic scale contains all notes in an octave (Wright 2009)
These notes are all a semitone apart (Wright 2009)
There are 12 notes (Wright 2009)
All other scales contain only seven unique notes, and are created using the sequence 1,1,1/2,1,1,1,1/2 starting at a base note (Zhang 2009).
The base note can be any note
1 is a whole step (2 semitones), 1/2 is a half step (1 semitone)
A C scale would yield C D E F G A B C
A F scale would yield F G A Bb C D E F
Notice how the scales return to the original base note, because of this we can consider these groups cyclic (Wright 2009)
An octave is an interval 12 semitones apart (Wright 2009)
When notes are an octave apart they are considered equivalent both mathematically and musically
Musically: notes sound the same
Mathematically: same integer mod 12
By Allison LaFleur
Understanding intervals and octaves are essential to understanding basic music theory
The chromatic scale can be represented by the integer mod 12
The chromatic scale is an isomorphic group to Z mod 12
The chromatic scale and all other scales can be represented as permutations
In context of music theory and it's relation to math an interval is simply defined as the "distance" between two notes
In music the standard interval is called a semitone (Zhang 2009)
A semitone is the interval between two keys on a keyboard (Zhang 2009)
Names of different intervals
The basic intervals are the half step, whole step and octave (Wright 2009)
Visual Representation of Octaves
Understanding the concept of octaves is essential to understanding the concept of scales!
A side note: Accidentals and Equivalent notes
The Chromatic Scale and Modular Numbers
Each note in the chromatic scale can be assigned an integer from 0 to 11, or in other words can represent the integers mod 12 (Johnson 2009).
From here it is a small leap to representing the chromatic scale as Z mod 12.
The Chromatic Scale as a Group
To have a group the given needs to have identities, inverses, associativity and closure (Benson 2008).
We know Z mod 12 is a group from class
The chromatic scale has an identity of C (0), and an inverse of -n when n is a note, and is associative and closed under addition (Johnson 2009)
We use addition for scales because adding can be considered the same as adding semitones to the notes (Benson 2008)
Benson, D., 2008: Music: A Mathematical Offering. Cambridge University Press, 517 pp.
Johnson, C. M., 2009: Introducing Music Theory through Music. Math. Teach., 103, 116-122.
Wright, D.,2009: Mathematics and Music. American Mathematical Society, 176 pp.
Zhang, A., 2009: The Framework of Music Theory as Represented with Groups. University of Washington. 26 pp.
The Chromatic Scale and Z mod 12 as an isomorphism
To be isomorphic two groups need to be one to one and onto
Because we set up the notation of the chromatic scale so there was exactly the same number of elements, meaning there are no leftover elements in each set, meaning the two groups are onto
They are also considered one to one because there is exactly one integer corresponding to each note making these two groups isomorphic
The Chromatic Scale as a Permutation
This is a permutation of the chromatic scale as it increases by one semitone (Benson 2008)
Scales as Permutations
Because of the cyclic nature of a scale we can easily represent them as a permutation, just as the chromatic scale was (Wright 2008).
We just have to remember that since the non chromatic scales do not have all the notes (and therefore integers) the permutation will be smaller, and different for each scale.
This concept creates different key signatures
EXAMPLE: F scale - F G A Bb C D E F = 5 7 9 10 0 2 4 5 as a permutation is