Spiral Array Model
Encyclopedia
In music theory
, the spiral array model is an extended type of pitch space
. It represents human perceptions of pitch
, chord
and key
in the same geometric space
, as a mathematical model involving concentric helix
es (an "array of spiral
s"). It was proposed in 2000 by Prof. Elaine Chew in her MIT doctoral thesis "Toward a Mathematical Model of Tonality
". Further research by Chew and others have produced modifications of the spiral array model, and, applied it to various problems in music theory and practice, such as key finding and pitch spelling.
The spiral array model can be viewed as an extension of the tonnetz
, which maps pitches into a two-dimensional lattice structure. Just like the tonnetz, the spiral array models higher order structures such as chords and keys in the same space as the low level structure: pitches. This allows the spiral array model to produce geometric interpretations of relationships between low and high level structures. For example, you can measure the geometric distance between a particular pitch and a particular key (both represented as points). Like the tonnetz, when applied to equal temperament
, the spiral array model folds into a torus as octaves overlap.
of points on outer spirals. For example, the pitches C, E, and G are represented as points by the cartesian coordinates C(x,y,z), E(x,y,z) and G(x,y,z). The convex combination formed by the points CEG is a triangle, and represents the "center of effect" of the three pitches. This convex combination represents the triad, or chord, CEG (the C major chord) in the spiral array model. The geometric center (or other point chosen by a weighting of the constituent points, as seen in the equations below) of the C major chord (formed by CEG) can be called the "center" of the C major chord, and assigned a point CM(x,y,z). Similarly, keys may be constructed by the centers of effect of their I, IV, and V chords.
Where k is an integer representing a semitone, r is the radius of the spiral, and h is the "rise" of the spiral
The major chord CM is represented by:
where
and 
The weights "w" effect how close the center of effect are to the fundamental, major third, and perfect fifth of the chord. By changing the relative values of these weights, the spiral array model effects how "close" the resulting chord is to the three constituent pitches. Generally in western music, the fundamental is given the greatest weight in identifying the chord (w1), followed by the fifth (w2), followed by the third (w3).
The minor chord Cm is represented by:
where
and 
The weights "u" function similarly to the major chord.
The major key TM is represented by:
where
and 
Similar to the weights controlling how close constituent pitches are to the center of effect of the chord they produce, the weights "W" control the relative effect of the I, IV, and V chord in determining how close they are to the resultant key.
The minor key Tm is represented by:
where
and 
and 
and 
.
Music theory
Music theory is the study of how music works. It examines the language and notation of music. It seeks to identify patterns and structures in composers' techniques across or within genres, styles, or historical periods...
, the spiral array model is an extended type of pitch space
Pitch space
In music theory, pitch spaces model relationships between pitches. These models typically use distance to model the degree of relatedness, with closely related pitches placed near one another, and less closely related pitches placed farther apart. Depending on the complexity of the relationships...
. It represents human perceptions of pitch
Pitch (music)
Pitch is an auditory perceptual property that allows the ordering of sounds on a frequency-related scale.Pitches are compared as "higher" and "lower" in the sense associated with musical melodies,...
, chord
Chord (music)
A chord in music is any harmonic set of two–three or more notes that is heard as if sounding simultaneously. These need not actually be played together: arpeggios and broken chords may for many practical and theoretical purposes be understood as chords...
and key
Key (music)
In music theory, the term key is used in many different and sometimes contradictory ways. A common use is to speak of music as being "in" a specific key, such as in the key of C major or in the key of F-sharp. Sometimes the terms "major" or "minor" are appended, as in the key of A minor or in the...
in the same geometric space
Space
Space is the boundless, three-dimensional extent in which objects and events occur and have relative position and direction. Physical space is often conceived in three linear dimensions, although modern physicists usually consider it, with time, to be part of a boundless four-dimensional continuum...
, as a mathematical model involving concentric helix
Helix
A helix is a type of smooth space curve, i.e. a curve in three-dimensional space. It has the property that the tangent line at any point makes a constant angle with a fixed line called the axis. Examples of helixes are coil springs and the handrails of spiral staircases. A "filled-in" helix – for...
es (an "array of spiral
Spiral
In mathematics, a spiral is a curve which emanates from a central point, getting progressively farther away as it revolves around the point.-Spiral or helix:...
s"). It was proposed in 2000 by Prof. Elaine Chew in her MIT doctoral thesis "Toward a Mathematical Model of Tonality
Tonality
Tonality is a system of music in which specific hierarchical pitch relationships are based on a key "center", or tonic. The term tonalité originated with Alexandre-Étienne Choron and was borrowed by François-Joseph Fétis in 1840...
". Further research by Chew and others have produced modifications of the spiral array model, and, applied it to various problems in music theory and practice, such as key finding and pitch spelling.
The spiral array model can be viewed as an extension of the tonnetz
Tonnetz
In musical tuning and harmony, the Tonnetz is a conceptual lattice diagram representing tonal space first described by Leonhard Euler in 1739....
, which maps pitches into a two-dimensional lattice structure. Just like the tonnetz, the spiral array models higher order structures such as chords and keys in the same space as the low level structure: pitches. This allows the spiral array model to produce geometric interpretations of relationships between low and high level structures. For example, you can measure the geometric distance between a particular pitch and a particular key (both represented as points). Like the tonnetz, when applied to equal temperament
Equal temperament
An equal temperament is a musical temperament, or a system of tuning, in which every pair of adjacent notes has an identical frequency ratio. As pitch is perceived roughly as the logarithm of frequency, this means that the perceived "distance" from every note to its nearest neighbor is the same for...
, the spiral array model folds into a torus as octaves overlap.
Structure of the Spiral Array
The model covering basic pitch, major chords, minor chords, major keys and minor keys comprises five concentric helixes. Starting with a formulation of the pitch spiral, inner spirals are generated by a convex combinationConvex combination
In convex geometry, a convex combination is a linear combination of points where all coefficients are non-negative and sum up to 1....
of points on outer spirals. For example, the pitches C, E, and G are represented as points by the cartesian coordinates C(x,y,z), E(x,y,z) and G(x,y,z). The convex combination formed by the points CEG is a triangle, and represents the "center of effect" of the three pitches. This convex combination represents the triad, or chord, CEG (the C major chord) in the spiral array model. The geometric center (or other point chosen by a weighting of the constituent points, as seen in the equations below) of the C major chord (formed by CEG) can be called the "center" of the C major chord, and assigned a point CM(x,y,z). Similarly, keys may be constructed by the centers of effect of their I, IV, and V chords.
- The outer helix represents pitches, a repeating sequence of 12 semitones, or pitch classes, representing the chromatic scale. Each semitone is placed a quarter rotation away from its predecessor. The order of the semitones is determined by the circle of fifthsCircle of fifthsIn music theory, the circle of fifths shows the relationships among the 12 tones of the chromatic scale, their corresponding key signatures, and the associated major and minor keys...
, so the nearest neighbor along the spirals are related by a major fifth. For example, C would be followed by G, which would be followed D, etc. As a result of this structure, and one of the important properties leading to its selection, the nearest neighbor above a semitone is a major third higher. Thus the nearest neighbors of a pitch are its perfect fifth and major third, ahead and above it, respectively. - By taking every consecutive triad along the helix, and projecting their centers of effect, a second helix is formed inside the pitch helix, representing the major chords.
- Similarly, by taking the proper minor triads and projecting their centers of effect, a third helix is formed, representing the minor chords.
- The major key helix is formed by projections of the I, IV, and V chords from points on the major chord
- The minor key helix is formed by similar projects of minor chords.
Equations
The pitch spiral P, is represented in parametric form by:Where k is an integer representing a semitone, r is the radius of the spiral, and h is the "rise" of the spiral
The major chord CM is represented by:
where
and The weights "w" effect how close the center of effect are to the fundamental, major third, and perfect fifth of the chord. By changing the relative values of these weights, the spiral array model effects how "close" the resulting chord is to the three constituent pitches. Generally in western music, the fundamental is given the greatest weight in identifying the chord (w1), followed by the fifth (w2), followed by the third (w3).
The minor chord Cm is represented by:
where
and The weights "u" function similarly to the major chord.
The major key TM is represented by:
where
and Similar to the weights controlling how close constituent pitches are to the center of effect of the chord they produce, the weights "W" control the relative effect of the I, IV, and V chord in determining how close they are to the resultant key.
The minor key Tm is represented by:
where
and and and .