Journal of Modern Education Review, ISSN 2155-7993, USA August 2020, Volume 10, No. 8, pp. 597–603 Doi: 10.15341/jmer(2155-7993)/08.10.2020/007 © Academic Star Publishing Company, 2020 http://www.academicstar.us
Musical Vectors and Spaces
Candace Carroll, J. X. Carteret (1. Department of Mathematics, Computer Science, and Engineering, Gordon State College, USA; 2. Department of Fine and Performing Arts, Gordon State College, USA)
Abstract: A vector is a quantity which has both magnitude and direction. In music, since an interval has both magnitude and direction, an interval is a vector. In his seminal work Generalized Musical Intervals and Transformations, David Lewin depicts an interval i as an arrow or vector from a point s to a point t in musical space. Using Lewin’s text as a point of departure, this article discusses the notion of musical vectors in musical spaces. Key words: Pitch space, pitch class space, chord space, vector space, affine space
1. Introduction
A vector is a quantity which has both magnitude and direction. In music, since an interval has both magnitude and direction, an interval is a vector. In his seminal work Generalized Musical Intervals and Transformations, David Lewin (2012) depicts an interval i as an arrow or vector from a point s to a point t in musical space (p. xxix). Using Lewin’s text as a point of departure, this article further discusses the notion of musical vectors in musical spaces.
t
i
s
Figure 1 David Lewin’s Depiction of an Interval i as a Vector
Throughout the discussion, enharmonic equivalence will be assumed. Let K be a field. For our purposes, we will assume that K is the field of real numbers. A vector space is a set S with two operations defined, addition and scalar multiplication, over a field. A scalar is an element of the field K. In this case, a scalar is a real number. A vector in n-space is an ordered n-tuple of real numbers. This means that a harmonic interval or chord with n tones, hereafter referred to as an n-chord, is a vector in Rn. Let us use integer notation for the scale, as is customary. We can write an n-chord as an ordered list. For example: C = <0, 4, 7> We can also write the chord vertically as column vector, or horizontally as a row vector:
Candace Carroll, Lecturer of Mathematics, M.A.T. in Mathematics, Gordon State College; research areas: mathematics and music theory. E-mail: [email protected].
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0 C = [4], or C = [0 4 7] 7 2. Pitch Space
Musical pitch space can be represented visually using a number line:
A A# B C4 C# D D# Figure 2 Pitch Space as a Number Line
Let V denote pitch space, and let u, v be vectors in V. Let the operation of addition of two musical vectors be defined componentwise:
u + v =
Figure 3 J. S. Bach, Jesu, Joy of Man’s Desiring, BWV 147, mm. 9–13
The opening chord is <11, 7, 2, −5>. By adding an interval vector containing the number of semitones that each voice moves, we arrive at the next chord: 11 1 12 2 14 0 14 7 0 7 2 9 −3 6 [ ] + [ ] = [ ] + [ ] = [ ] + [ ] = [ ] 2 2 4 −7 −3 2 −1 −5 −3 −8 2 −6 −4 −10 Je - su, joy of Figure 4 Jesu, Joy of Man’s Desiring, mm. 9–10 as Vector Addition
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Since vector addition is commutative and associative, the addition of musical vectors is an additive abelian group with 0 as the identity. We define a second operation of multiplication by a scalar, or real number r: ru. Closure holds because ru ∈ V for any r. Note that if r is a non-integer, this simply means we have a value that expressed in terms of cents. Let r, s be scalars. Multiplication of real numbers is associative: r(su) = rs(u) Distributivity follows from the properties of real numbers. That is, r(u + v) = ru + rv and (r + s)u = ru + su. Multiplication by the unit 1 is the identity. That is, 1u = u Hence, all of the vector space axioms are satisfied, and pitch space is a vector space. Let W be a vector space. A subset S of the vector space W is a basis for W if any vector w ∈ 푊 is uniquely represented as a linear combination w = r1w1 + r2w2 + … + rkwk, where w1, w2, …, wk are distinct vectors in S and r1, r2, …, rk are scalars. The dimension of a vector space W, denoted dim(W), is the cardinality of is bases. In other words, the dimension of a vector space W is the number of vectors in its basis. The vector space Rn has as its standard basis the basis e = {e1 = <1,0,0, …,0>, e2 = <0,1,0,…,0>, …, en = <0,0,0, …, 1>. Since there are n vectors n n in the basis, the dimension of R is n. An n-chord is a vector in R . This means that any n-chord x =
3. Pitch Class Space
Pitch class space can be represented visually using a circle:
Figure 5 Pitch Class Space as a Circle
Recall that, mathematically speaking, pitch classes are congruence classes. With pitch class space, we assume octave equivalence. In other words, if a scale has m tones, then two pitches an octave apart would be congruent modulo m. Let {x} denote the pitch class x, and let k be an integer. We can take k{x} = {kx}
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Let X denote pitch space, which is a vector space over a field K. Let A denote pitch class space. Since scalar multiplication is not defined for every k in X, pitch class space is not a vector space. However, pitch class space is an affine space. In an affine space, there is no distinguished point that serves as an origin. Any vector space may be considered as an affine space by “forgetting the zero vector”. Let a, b ∈ X. Let p, q ∈ A. Define addition p + a ∈ A for any vector a ∈ X and any element p ∈ A. The affine space axioms are as follows: (1) p + 0 = p (2) (p + a) + b = p + (a + b) (3) For any q ∈ A, there exists a unique vector a ∈ X such that q = p + a We will now show that pitch class space is an affine space. Let dim(X) = n, so that a musical vector in X has n tones. (1) Since X is a vector space, it contains the zero vector 0. Then
p + 0 =
=
4. Chord Space
So far, we have viewed chords as vectors in pitch space. Now let us consider chords as points in chord space. We can represent chord space visually by using a diagram such as the Tonnetz. The Tonnetz is formed by horizontally “unrolling” the circle of fifths.
Figure 6 The Circle of Fifths
The rows of the Tonnetz are shifted horizontally so that the major third and major sixth is above each tone, and the minor third and minor sixth is below.
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Figure 7 The Tonnetz
Within a given octave, each scale degree is unique. This means that if we fix a starting point, there exists a unique vector from one chord to the next chord in a chord progression. For example, consider Frédéric Chopin’s E Major Mazurka, op. 6, no. 3:
I
IV V I IV V I Figure 8 Frédéric Chopin, E Major Mazurka, op. 6, no. 3, mm. 91 – 98
# C# G
IV I V RL RL
A E B
LRLR
Figure 9 Mazurka Chord Progression as Vectors in Chord Space
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Beginning with measure 94, the E Major Mazurka features the familiar I-IV-V-I chord progression.1 The progression is repeated again in measures 96–98, ending the piece. This movement away from the tonic to the predominant, from the predominant to the dominant, and from the dominant back to the tonic is depicted using vectors on the Tonnetz (see Figure 9). In measure 95 of the Mazurka, the E major chord is transformed into the A major chord, the predominant, via the neo-Riemannian transformation RL. Next, the A major proceeds to the dominant of B major via LRLR. Finally, the dominant B major chord is restored back to the tonic E major, again via RL. Once again, since scalar multiplication is not defined for every k in X, chord space is not a vector space. However, chord space is an affine space. Although the Tonnetz gives us a convenient way to visualize chords in triadic chord space, we can give a general proof for m-dimensional chord space. Let X denote pitch space, which is a vector space. Let dim(X) = m, so that a musical vector in X has m n-chords in the progression. Let A denote m- dimensional chord space. Let x, y ∈ X. Let p, q ∈ A. Define addition p + x ∈ A for any vector x ∈ X and any element p ∈ A. In order to prove that chord space is an affine space, we must once again verify the three affine space axioms: (1) Since X is a vector space, it contains the zero vector 0. Then
p + 0 =
=
5. Conclusion
Since a musical interval has both magnitude and direction, a musical interval is a vector. Harmonic intervals, or chords, from a piece of music can be added together using vector addition, in the natural way. If the operation of scalar multiplication is also defined, then musical pitch space forms a vector space. While pitch class space and chord space are not vector spaces, it is true that each of these musical spaces is an affine space.
References Cohn, Richard (2012). Audacious Euphony, New York, NY: Oxford University Press. Bergstrom Tony, John C. Hart and Karrie Karahalios (2007). “Isochords: Visualizing structure in music”, in: Proceedings of Graphics Interface 2007, May 2007, pp. 297−304, doi: 10.1145/1268517.1268565. Lax Peter D. (2007). Linear Algebra and Its Applications, Wiley & Sons, Inc. Lewin David (2012). Generalized Musical Intervals and Transformations, Oxford University Press. Meeùs N. (December 2018). “Harmonic vectors and the constraints of tonality,” Music Theory Online, Vol. 24, No. 4, doi: 10.30535/mto.24.4.11.
1 6 Some might hear the IV chord in mm. 95 and 97 as ii5. I am taking a Schenkerian analysis approach, and hearing the chord progression as T-PD-D-T.
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Bach Johann Sebastian (1991). Jesu, Joy of Man’s Desiring, BWV 147, Orem, UT: Jackman Music Corporation. Chopin Frédéric (1833). E Major Mazurka, op. 6, no. 3, Leipzig: Kistner.
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