Spiral array model (original) (raw)

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Mathematical model used in music theory

In music theory, the spiral array model is an extended type of pitch space. A mathematical model involving concentric helices (an "array of spirals"), it represents human perceptions of pitches, chords, and keys in the same geometric space. It was proposed in 2000 by Elaine Chew in her MIT doctoral thesis Toward a Mathematical Model of Tonality.[1] 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 (symbolic and audio[2][3]), pitch spelling,[4][5][6][7] tonal segmentation,[8][9] similarity assessment,[10] and musical humor.[11] The extensions and applications are described in Mathematical and Computational Modeling of Tonality: Theory and Applications.[12]

The spiral array model can be viewed as a generalized tonnetz, which maps pitches into a two-dimensional lattice (array) structure. The spiral array wraps up the two-dimensional tonnetz into a three-dimensional lattice, and models higher order structures such as chords and keys in the interior of the lattice space. This allows the spiral array model to produce geometric interpretations of relationships between low- and high-level structures. For example, it is possible to model and measure geometrically the distance between a particular pitch and a particular key, both represented as points in the spiral array space. To preserve pitch spelling, because musically A# ≠ Bb in their function and usage, the spiral array does not assume enharmonic equivalence, i.e. it does not fold into a torus. The spatial relationships between pitches, between chords, and between keys agree with those in other representations of tonal space.[13]

The model and its real-time algorithms have been implemented in the tonal visualization software MuSA.RT[14][15] (Music on the Spiral Array . Real-Time) and a free app, MuSA_RT,[16] both of which have been used in music education videos[17][18] and in live performance.[19][20][21]

Structure of the spiral array

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Spiral array model: pitch class, major/minor chord, and major/minor key helices

The model as proposed covers basic pitches, major chords, minor chords, major keys and minor keys, represented on five concentric helices. Starting with a formulation of the pitch helix, inner helices are generated as convex combinations of points on outer ones. For example, the pitches C, E, and G are represented as the Cartesian points P(0), P(1), and P(4) (see definitions in next section), which outline a triangle. The convex combination of these three points is a point inside the triangle, and represents their center of effect (ce). This interior point, C M(0), represents the C major chord in the spiral array model. Similarly, keys may be constructed by the centers of effect of their I, IV, and V chords.

Equations for pitch, chord, and key representations

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Generating of a major key representation as the center of effect of its I, IV, and V chords, which are in turn generated as the center of effect of their defining pitches.

Generating of a minor key representation as the center of effect of its i, iv/IV, and V/v chords, which are in turn generated as the center of effect of their defining pitches.

In Chew's model, the pitch class helix, P, is represented in parametric form by:

P ( k ) = [ x k y k z k ] = [ r sin ⁡ ( k ⋅ π / 2 ) r cos ⁡ ( k ⋅ π / 2 ) k h ] {\displaystyle P(k)={\begin{bmatrix}x_{k}\\y_{k}\\z_{k}\\\end{bmatrix}}={\begin{bmatrix}r\sin(k\cdot \pi /2)\\r\cos(k\cdot \pi /2)\\kh\end{bmatrix}}} {\displaystyle P(k)={\begin{bmatrix}x_{k}\\y_{k}\\z_{k}\\\end{bmatrix}}={\begin{bmatrix}r\sin(k\cdot \pi /2)\\r\cos(k\cdot \pi /2)\\kh\end{bmatrix}}}where k is an integer representing the pitch's distance from C along the line of fifths, r is the radius of the spiral, and h is the "rise" of the spiral.

The major chord helix, CM is represented by:

C M ( k ) = w 1 ⋅ P ( k ) + w 2 ⋅ P ( k + 1 ) + w 3 ⋅ P ( k + 4 ) {\displaystyle C_{M}(k)=w_{1}\cdot P(k)+w_{2}\cdot P(k+1)+w_{3}\cdot P(k+4)} {\displaystyle C_{M}(k)=w_{1}\cdot P(k)+w_{2}\cdot P(k+1)+w_{3}\cdot P(k+4)}

where w 1 ≥ w 2 ≥ w 3 > 0 {\displaystyle w_{1}\geq w_{2}\geq w_{3}>0} {\displaystyle w_{1}\geq w_{2}\geq w_{3}>0} and ∑ i = 1 3 w i = 1 {\textstyle \sum _{i=1}^{3}w_{i}=1} {\textstyle \sum _{i=1}^{3}w_{i}=1}.

The weights "w" affect 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 controls 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 helix, Cm is represented by:

C m ( k ) = u 1 ⋅ P ( k ) + u 2 ⋅ P ( k + 1 ) + u 3 ⋅ P ( k − 3 ) {\displaystyle C_{m}(k)=u_{1}\cdot P(k)+u_{2}\cdot P(k+1)+u_{3}\cdot P(k-3)} {\displaystyle C_{m}(k)=u_{1}\cdot P(k)+u_{2}\cdot P(k+1)+u_{3}\cdot P(k-3)}

where u 1 ≥ u 2 ≥ u 3 > 0 {\displaystyle u_{1}\geq u_{2}\geq u_{3}>0} {\displaystyle u_{1}\geq u_{2}\geq u_{3}>0} and ∑ i = 1 3 u i = 1. {\displaystyle \sum _{i=1}^{3}u_{i}=1.} {\displaystyle \sum _{i=1}^{3}u_{i}=1.}

The weights "u" function similarly to the major chord.

The major key helix, TM is represented by:

T M ( k ) = ω 1 ⋅ C M ( k ) + ω 2 ⋅ C M ( k + 1 ) + ω 3 ⋅ C M ( k − 1 ) {\displaystyle T_{M}(k)=\omega _{1}\cdot C_{M}(k)+\omega _{2}\cdot C_{M}(k+1)+\omega _{3}\cdot C_{M}(k-1)} {\displaystyle T_{M}(k)=\omega _{1}\cdot C_{M}(k)+\omega _{2}\cdot C_{M}(k+1)+\omega _{3}\cdot C_{M}(k-1)}

where ω 1 ≥ ω 2 ≥ ω 3 > 0 {\displaystyle \omega _{1}\geq \omega _{2}\geq \omega _{3}>0} {\displaystyle \omega _{1}\geq \omega _{2}\geq \omega _{3}>0} and ∑ i = 1 3 ω i = 1 {\textstyle \sum _{i=1}^{3}\omega _{i}=1} {\textstyle \sum _{i=1}^{3}\omega _{i}=1}.

Similar to the weights controlling how close constituent pitches are to the center of effect of the chord they produce, the weights ω {\displaystyle \omega } {\displaystyle \omega } control the relative effect of the I, IV, and V chord in determining how close they are to the resultant key.

The minor key helix, Tm is represented by:

T m ( k ) = ν 1 ⋅ C m ( k ) + ν 2 ⋅ ( α ⋅ C M ( k + 1 ) + ( 1 − α ) ⋅ C m ( k + 1 ) ) + ν 3 ⋅ ( β ∗ C m ( k − 1 ) + ( 1 − β ) ⋅ C M ( k − 1 ) ) . {\displaystyle T_{m}(k)=\nu _{1}\cdot C_{m}(k)+\nu _{2}\cdot (\alpha \cdot C_{M}(k+1)+(1-\alpha )\cdot C_{m}(k+1))+\nu _{3}\cdot (\beta *C_{m}(k-1)+(1-\beta )\cdot C_{M}(k-1)).} {\displaystyle T_{m}(k)=\nu _{1}\cdot C_{m}(k)+\nu _{2}\cdot (\alpha \cdot C_{M}(k+1)+(1-\alpha )\cdot C_{m}(k+1))+\nu _{3}\cdot (\beta *C_{m}(k-1)+(1-\beta )\cdot C_{M}(k-1)).}

where ν 1 ≥ ν 2 ≥ ν 3 > 0 {\displaystyle \nu _{1}\geq \nu _{2}\geq \nu _{3}>0} {\displaystyle \nu _{1}\geq \nu _{2}\geq \nu _{3}>0} and ν 1 + ν 2 + ν 3 = 1 {\displaystyle \nu _{1}+\nu _{2}+\nu _{3}=1} {\displaystyle \nu _{1}+\nu _{2}+\nu _{3}=1} and 0 ≥ α ≥ 1 {\displaystyle 0\geq \alpha \geq 1} {\displaystyle 0\geq \alpha \geq 1} and β ≥ 1 {\displaystyle \beta \geq 1} {\displaystyle \beta \geq 1}.

  1. ^ Chew, Elaine (2000). Towards a Mathematical Model of Tonality (Ph.D.). Massachusetts Institute of Technology. hdl:1721.1/9139.
  2. ^ Chuan, Ching-Hua; Chew, Elaine (2005). "Polyphonic Audio Key Finding Using the Spiral Array CEG Algorithm". Multimedia and Expo, 2005. ICME 2005. IEEE International Conference on. Amsterdam, the Netherlands: IEEE. pp. 21–24. doi:10.1109/ICME.2005.1521350. 0-7803-9331-7.
  3. ^ Chuan, Ching-Hua; Chew, Elaine (2007). "Audio Key Finding: Considerations in System Design and Case Studies on Chopin's 24 Preludes". EURASIP Journal on Advances in Signal Processing. 2007 (56561). doi:10.1155/2007/56561. Retrieved 1 Dec 2015.
  4. ^ Chew, Elaine; Chen, Yun-Ching (2005). "Real-Time Pitch Spelling Using the Spiral Array". Computer Music Journal. 29 (2): 61–76. doi:10.1162/0148926054094378. JSTOR 3681713. S2CID 905758.
  5. ^ Chew, Elaine; Chen, Yun-Ching (2003). "Determining Context-Defining Windows: Pitch Spelling using the Spiral Array" (PDF). Proceedings of the International Conference on Music Information Retrieval. Baltimore, Maryland.
  6. ^ Chew, Elaine; Chen, Yun-Ching (2003). "Mapping Midi to the Spiral Array: Disambiguating Pitch Spellings". Computational Modeling and Problem Solving in the Networked World. Phoenix, Arizona: Springer. pp. 259–275. doi:10.1007/978-1-4615-1043-7_13.
  7. ^ Meredith, David (2007). "Optimizing Chew and Chen's Pitch-Spelling Algorithm" (PDF). Computer Music Journal. 31 (2): 54–72. doi:10.1162/comj.2007.31.2.54. S2CID 17444672.
  8. ^ Chew, Elaine (2002). "The Spiral Array: An Algorithm for Determining Key Boundaries". Music and Artificial Intelligence, Second International Conference. Edinburgh: Springer. pp. 18–31. LNAI 2445.
  9. ^ Chew, Elaine (2005). "Regards on two regards by Messiaen: Post-tonal music segmentation using pitch context distances in the spiral array". Journal of New Music Research. 34 (4): 341–354. doi:10.1080/09298210600578147. S2CID 61149753.
  10. ^ Mardirossian, Arpi; Chew, Elaine (2006). "Music Summarization Via Key Distributions: Analyses of Similarity Assessment Across Variations" (PDF). Proceedings of the International Conference on Music Information Retrieval. Victoria, Canada. pp. 613–618.
  11. ^ Chew, Elaine; François, Alexandre (2007). "Visible Humour — Seeing P.D.Q. Bach's Musical Humour Devices in The Short-Tempered Clavier on the Spiral Array Space". Mathematics and Computation in Music, First International Conference, MCM 2007 Berlin, Germany, May 18–20, 2007 Revised Selected Papers. Berlin Heidelberg: Springer. pp. 11–18. doi:10.1007/978-3-642-04579-0_2.
  12. ^ Chew, Elaine (2014). Mathematical and Computational Modeling of Tonality: Theory and Applications. International Series in Operations Research & Management Science. Springer. ISBN 978-1-4614-9474-4.
  13. ^ Chew, Elaine (2008). "Out of the Grid and Into the Spiral: Geometric Interpretations of and Comparisons with the Spiral-Array Model" (PDF). Computing in Musicology. 15: 51–72.
  14. ^ Chew, Elaine; François, Alexandre (2003). "MuSA.RT: music on the spiral array. real-time". MULTIMEDIA '03 Proceedings of the eleventh ACM international conference on Multimedia. Berkeley, California: ACM. pp. 448–449.
  15. ^ Chew, Elaine; François, Alexandre (2005). "Interactive multi-scale visualizations of tonal evolution in MuSA.RT Opus 2". Computers in Entertainment. 3 (4): 3. doi:10.1145/1095534.1095545. S2CID 14391843.
  16. ^ François, Alexandre (2012). "MuSA_RT". iTunes.
  17. ^ Megan Swan (12 December 2014). See What You Hear. 3:41 minutes in. Inside the Music. Los Angeles Philharmonic.
  18. ^ Eric Mankin (20 January 2010). Engineer-Pianist Elaine Chew Talks About Using Mathematical and Software Tools to Analyze Music. 5:49 minutes in. Viterbi. University of Southern California.
  19. ^ Avril, Tom (22 September 2008). "Analyzing music the digital way—Computers have exquisite ears". Philadelphia Inquirer. Philadelphia, Pennsylvania. Retrieved 1 December 2015.
  20. ^ Hardesty, Larry (2008). "The Geometry of Sound". Technology Review: MIT News Magazine: 111. Retrieved 1 December 2015.
  21. ^ "New Resonances Festival". Wilton's Music Hall, London. 19 June 2012.