Considering Golden Section Proportionality in Popular Music

Six Pieces by Jacob Collier

Authors

  • Dave Collins Independent Scholar

DOI:

https://doi.org/10.5429/2079-3871(2026)v16i1.6en

Keywords:

Golden section, Jacob Collier, musical proportion, time-based analysis, music composition

Abstract

The Golden Section, sometimes referred to as the “golden ratio” or “golden mean”, has been referenced in many studies in the arts, serving as a mathematically-based aesthetic criterion for form, structure, proportionality, and balance. In the domain of musical composition, scholars have noted the relationship between the golden section and musical structure; I will briefly illustrate how these studies have focused on proportions and structures in classical music only, which has used a methodology of counting bar numbers in a musical score. I follow this by demonstrating that golden section proportionality may not only be confined to classical music but can also be applied to popular music. I introduce an analytical method that measures the elapsed or clock time of the sounding music rather than examining a musical score. In doing so I use the concept of a temporal “key event point”, illustrating this in six works by Jacob Collier.

References

Bibliography

Adams, C. 1996. Erik Satie and Golden section Analysis. Music and Letters 77 (2): 242–252. https://doi.org/10.1093/ml/77.2.242 DOI: https://doi.org/10.1093/ml/77.2.242

Arnheim, R. 1966. A Review of Proportion. In Gyorgy Kepes Ed. Module, Symmetry and Proportion. London: Studio Vista: 218-230.

Atlas, A.W. 2003. Stealing a Kiss at the Golden section: Pacing and Proportion in the Act 1 Love Duet of La Bohème. Acta Musicologica 75 (2): 269-291. http://www.jstor.org/stable/25071221

Benjafield, J.G. 2010. The Golden Section and American Psychology, 1892–1938. Journal of the History of the Behavioral Sciences 46 (1): 52-71. DOI: https://doi.org/10.1002/jhbs.20409

Bonds, M.E. 2010. The Spatial Representation of Musical Form. The Journal of Musicology (27): 265-303. DOI: https://doi.org/10.1525/jm.2010.27.3.265

Boselie, F. 1984. The Aesthetic Attractivity of the Golden Section. Psychological Research, 45 (4): 367-375. DOI: https://doi.org/10.1007/BF00309712

Boykan, M. 2004. Silence and Slow Time: Studies in Musical Narrative. Oxford: The Scarecrow Press. https://www.jstor.org/stable/4487490

Brackett, D. 2023. Interpreting Popular Music: With a New Preface by the Author. University of California Press.

Broman, F. (2024) Jacob Collier // Bridge Over Troubled Water (Transcription) [Online video]. Available from: https://youtu.be/wHdoTVJgMjM. Accessed 25 November 2025.

Burnard, P. 2016. The Practice of Diverse Compositional Creativities. In Dave Collins Ed. The Act of Musical Composition. Routledge: 111-138.

Chen, Z.X., Wang, Z.Z., Sun, Y. and Bei, F.L. 2011. Discussion on Microcosmic Derivation of Biological Golden Section Phenomena from DNA Geometric Structure and Snow Flower Genesection. Interdisciplinary Sciences: Computational Life Sciences 3(1): 31-35. https://doi.org/10.1007/s12539-011-0060-2 DOI: https://doi.org/10.1007/s12539-011-0060-2

Collier, J. –

a. LOGIC SESSION BREAKDOWN: “Moon River” [Online video]. Available from: https://youtu.be/9d4-URyWEJQ. Accessed 25 November 2025.

b. LOGIC SESSION BREAKDOWN:“All Night Long (feat. Take 6)” [Online video]. Available from: https://youtu.be/UAMc4K_NXsA. Accessed 24 November 2025

The Audience Choirs of Little Blue [Online video]. Available from: https://youtu.be/E1JKr3zNqgw. Accessed 24 November 2025

LOGIC SESSION BREAKDOWN: “Bridge Over Troubled Water (feat. John Legend & Tori Kelly)” [Online video]. Available from: https://youtu.be/I8DWlis-MEY. 24 November 2024.

Collins, D. –

, Investigating Computer-Based Compositional Processes: A Case Study Approach. PhD. University of Sheffield, United Kingdom.

A Synthesis Process Model of Creative Thinking in Music composition. Psychology of music 33 (2): 193-216. DOI: https://doi.org/10.1177/0305735605050651

Collins, D. & Dunn, M. –

Problem Solving Strategies and Processes in Musical Composition: Observations in Real-Time. Journal of Music, Technology and Education, 2 (2).

Golden Section Proportionality in the Music of DZihan and Kamien. Dancecult: Journal of Electronic Dance Music Culture 13 (1): 101-128. DOI: https://doi.org/10.12801/1947-5403.2021.13.01.07

Covach, J. (2005). Form in Rock Music: A Primer. In D. Stein Ed. Engaging Music: Essays in Music Analysis. Oxford University Press: 65-76.

Dalos, A. 2017. Ligeti and the beginnings of Bartók Analysis in Hungary. In György Ligeti's Cultural Identities. Routledge: 139-148. DOI: https://doi.org/10.4324/9781315592411-9

Donin, N. 2016. Empirical and Historical Musicologies of Compositional Processes: Towards a Cross-Fertilisation. In Dave Collins Ed. The Act of Musical Composition. Routledge: 1-26.

Elam, K. 2001. Geometry of Design: Studies in Proportion and Composition. Princeton Architectural Press.

Epstein, D. 1995. Shaping Time: Music, the Brain, and Performance. Schirmer Books.

Erickson, B. 1986. Art and Geometry: Proportioning Devices in Pictorial Composition. Leonardo, 19 (3): 211-221. DOI: https://doi.org/10.2307/1578239

Evans, B. 2005. Foundations of a Visual Music. Computer Music Journal 29 (4): 11-24. DOI: https://doi.org/10.1162/014892605775179955

Everett, W. (2000) Expression in Pop-Rock Music. Garland Publishing.

Frings, M. 2002. The Golden Section in Architectural Theory. Nexus Network Journal 4: 9-32. DOI: https://doi.org/10.1007/s00004-001-0002-0

Frith, S. ed., 2004. Popular music: Critical concepts in media and cultural studies (vol. 2). Psychology Press.

Gielo-Perczak, K. 2001. The Golden Section as a Harmonizing Feature of Human Dimensions and Workplace Design. Theoretical Issues in Ergonomics Science 2 (4): 336-351. DOI: https://doi.org/10.1080/14639220110109669

Goodman, N. 1978. Ways of Worldmaking. Hackett. DOI: https://doi.org/10.5040/9781350928558

Green, C. D. 1995. All that Glitters: A Review of Psychological Research on the Aesthetics of the Golden section. Perception 24 (8): 937-968. DOI: https://doi.org/10.1068/p240937

Holm-Hudson, K. (2010). A Study of Maximally Smooth Voice Leading in the Mid-1970s Music of Genesis. In Sounding Out Pop: Analytical Essays in Popular Music. University of Michigan: 99-123.

Howat, R. 1993. Debussy in Proportion: A Musical Analysis. Cambridge University Press.

Koh, H. J. 2015. The Application of Practical Geometry and the Golden Ratio n Product Design. PhD. University of Sussex, United Kingdom.

Konečni, V.J. –

The Golden Section: Elusive, but Detectable. Creativity Research Journal 15 (2/3): 267-275. DOI: https://doi.org/10.1080/10400419.2003.9651419

On the “Golden Section”, Visual Arts Research (1): 76-87.

Kramer, J. D. –

New Temporalities in Music. Critical Inquiry 7(3): 539-556. DOI: https://doi.org/10.1086/448114

Studies of Time and Music: A Bibliography. Music Theory Spectrum, (7) 72-106. DOI: https://doi.org/10.2307/745881

The Time of Music. Schirmer.

Postmodern Concepts of Musical Time. Indiana Theory Review 17 (2): 21-61.

Langer, S. 1953. Feeling and Form. Charles Scribner’s Sons.

Lee, J. (2019) Jacob Collier - Moon River (Transcription) [Online video]. Available from: https://youtu.be/KhyyjRcrn84. Accessed 11 June 2025.

Lendvai, E. 1966. Duality and Synthesis in the Music of Béla Bartók. In Georgy Kepes Ed. Module, Proportion, Symmetry, Rhythm. Studio Vista: 173-193.

Lendvai, E. 1979. Béla Bartók: An Analysis of his Music. Kahn & Averil.

Lessing, L. (2021) Don’t You Know (Transcription) [Online video]. Available from: https://youtu.be/rrmcU8NvCHw. Accessed 11 June 2025.

Luchese, D. 2001. Come un Meccanismo di Precisione: The Third Movement of Ligeti’s Second String Quartet. In Reza Saranghi and Slavi Jablan Ed. Bridges: Mathematical Connections in Art, Music and Science; Conference Proceedings 2001. Winfield, Kansas: Bridges Conference: 37-46. https://archive.bridgesmathart.org/2001/bridges2001-37.pdf. Accessed: 15 November 2021.

Lüttge, U. and Souza, G.M. 2019. The Golden section and Beauty in Nature: The Perfection of Symmetry and the Charm of Asymmetry. Progress in Biophysics and Molecular Biology 146: 98–103. http://dx.doi.org/10.1016/j.pbiomolbio.2018.12.008 DOI: https://doi.org/10.1016/j.pbiomolbio.2018.12.008

Macchiusi, I. 2017. Knowing is Seeing: The Digital Audio Workstation and the Visualization of Sound. PhD. York University, Toronto. https://yorkspace.library.yorku.ca/xmlui/handle/10315/34478

Maconie, R. 2005. Other Planets: The Music of Karlheinz Stockhausen. Lanham, UK: Scarecrow Publishing.

Martin, B. –

(2019) Jacob Collier: Moon River – Harmonic Analysis [Online video]. https://youtu.be/UTDHylJZMX. Accessed 14 November 2025.

(2021). Jacob Collier: Home Is - Harmonic Analysis [Online video]. https://youtu.be/ulB0mfsbNdY. Accessed 14 November 2025.

Mellers, W.H. 1973. Twilight of the Gods: The Beatles in Retrospect. Viking Books.

McManus, I.C. and Weatherby, P. 1997. The Golden Section and the Aesthetics of Form and Composition: a Cognitive Model. Empirical Studies of the Arts 15 (2): 209-232. DOI: https://doi.org/10.2190/WWCR-VWHV-2Y2W-91EE

McWhinnie, H. J. 1986. A Review of the Use of Symmetry, the Golden Section and Dynamic Symmetry in Contemporary Art. Leonardo, 19(3): 241-245. DOI: https://doi.org/10.2307/1578244

Middleton, R., Ed. 2000. Reading Pop: Approaches to Textual Analysis in Popular Music. Oxford University Press, USA. DOI: https://doi.org/10.1093/oso/9780198166122.001.0001

Moore, A. –

A Review of Selected Research on the Golden Section Hypothesis. Visual Arts Research 13(1): 73-84. http://www.jstor.org/stable/20715640

Analysing Popular Music. Cambridge University Press.

Murtinho, V. 2015. Leonardo’s Vitruvian Man Drawing: A New Interpretation Looking at Leonardo’s Geometric Constructions. Nexus Network Journal 17: 507-524. DOI: https://doi.org/10.1007/s00004-015-0247-7

Oldershaw, R. L. 1982. The Preferred Pitch Angle of Spiral Galaxies: Mathematical and Physical Implications. Monthly Notes of the Astronomical Society of Southern Africa 41: 42-46. https://ui.adsabs.harvard.edu/abs/1982MNSSA..41...42O

Persson, R.S. and Robson, C. 1995. The Limits of Experimentation: on Researching Music and Musical Settings. Psychology of Music, 23 (1): 39-47. DOI: https://doi.org/10.1177/0305735695231003

Phillips, M. E. 2019. Rethinking the Role of the Golden section in Music and Music Scholarship. Creativity Research Journal 31 (4): 419-427. http://dx.doi.org/10.1080/10400419.2019.1651243 DOI: https://doi.org/10.1080/10400419.2019.1651243

Powell, N. W. 1979. Fibonacci and the Golden mean: Rabbits, Rumbas, and Rondeaux. Journal of Music Theory 23 (2): 227-273. https://www.jstor.org/stable/843726 DOI: https://doi.org/10.2307/843726

Raghubir, P. and Greenleaf, E. A. 2006. Ratios in Proportion: What Should the Shape of the Package be? Journal of Marketing 70 (2): 95-107. DOI: https://doi.org/10.1509/jmkg.70.2.095

Reuter,H. (2021). Pop as Process: The Digitalisation of Groove, Form and Time. Dancecult: Journal of Electronic Dance Music Culture 13 (1): 2-21. DOI: https://doi.org/10.12801/1947-5403.2021.13.01.01

Reybrouck, M. 1996. Gestalt concepts and music: Limitations and possibilities. In Joint International Conference on Cognitive and Systematic Musicology. Springer Berlin Heidelberg: 57-69. DOI: https://doi.org/10.1007/BFb0034107

Rofe, M. D. –

Shostakovich and the Russian Doll: Dimensions of Energy in the Symphonies. PhD. University of York, United Kingdom. https://etheses.whiterose.ac.uk/11069/

Dimensions of Energy in Shostakovich. Abingdon, UK: Routledge.

Sandresky, M.V. 1981. The Golden section in Three Byzantine Motets of Dufay. Journal of Music Theory 25 (2): 291-306. http://dx.doi.org/10.2307/843653 DOI: https://doi.org/10.2307/843653

Scotto, C., Smith, K., and Brackett, J. 2019. The Routledge Companion to Popular Music Analysis: Expanding Approaches. Routledge, New York. DOI: https://doi.org/10.4324/9781315544700

Spicer, M. (2010). “Regatta de Blanc”. In Sounding Out Pop: Analytical Essays in Popular Music. University of Michigan: 124-153.

Spicer, M. and Covach, J. 2010. Sounding Out Pop: Analytical Essays in Popular Music Analysis. University of Michigan.

Stravinsky, I. 1970. Poetics of Music in the Form of Six Lessons. Vol. 66. Harvard University Press. DOI: https://doi.org/10.2307/j.ctv1n3x177

Strachan, R., 2017. Sonic Technologies: Popular Music, Digital Culture and the Creative Process. Bloomsbury Publishing USA. DOI: https://doi.org/10.5040/9781501310652

Tagg, P. –

Musicology and the semiotics of popular music. Semiotica 66 (1/3): 279-298.

Analysing Popular Music: Theory, Method and Practice. In Richard Middleton Ed. Reading Pop: Approaches to Textual Analysis in Popular Music. Oxford University Press: 71-103.

Tatlow, R. 2006. The Use and Abuse of Fibonacci Numbers and the Golden section in Musicology Today. Understanding Bach 1: 69-85. https://www.bachnetwork.org/ub1/tatlow.pdf. Accessed: 15 November 2021.

Trowell, B. 1979. Proportions in the Music of Dunstable. Proceedings of the Royal Musical Association 195: 100-141. http://dx.doi.org/10.1093/jrma/105.1.100 DOI: https://doi.org/10.1093/jrma/105.1.100

Verba, E. 2012. The Golden Ratio in Time-Based Media. Journal of Arts and Humanities 1 (1): 56-68. https://www.theartsjournal.org/index.php/site/article/download/6/6. Accessed: 15 November 2021.

Warner, T. 2006. “Quantitative listening: Temporal Proportion in Recordings of Mozart’s Piano Sonatas.” In 7th WSEAS International Conference on Acoustics & Music: Theory & Applications. Cavtat, Croatia.

Winterson, J. 2024. Music and Maths. Huddersfield, UK: University of Huddersfield Press.

Withington, A. 2020. Just Intonation: A Basis for Enhancing Choral Intonation. The Choral Journal 60 (7): 26-41.

Ying, Y. and Yue, S. 2021. Research on Visual Communication Graphic Design Information System Based on Computer Simulation. In Journal of Physics: Conference Series, vol. 1952 (2): 022032. IOP Publishing. DOI: https://doi.org/10.1088/1742-6596/1952/2/022032

Zagorski-Thomas, S. 2014. Musical Meaning and the Musicology of Record Production. Black Box Pop: Analysen populärer Musik, 38.

Zeising, A. –

Neue Lehre van den Proportionen des meschlischen Körpers. Leipzig: Rudolph Weigel.

Aesthetic Research. Frankfurt a.M: Verlag von Heidinger John and Company.

Zeng, L. and Wang, G. 2009. Modeling Golden section in Plants. Progress in Natural Science 19(2): 255-260. http://dx.doi.org/10.1016/j.pnsc.2008.07.004 DOI: https://doi.org/10.1016/j.pnsc.2008.07.004

Žuvela, S. 2011. The Golden Section as a Source of Consistency in 20th Century Music. Arti Musices 42(2): 274-280.

Discography

Collier, J. –

Djesse Vol. 1. Geffen Records, December, Europe.

Djesse Vol. 2. Geffen Records, July, Europe.

Djesse Vol. 4. Hajanga Records, February, Europe.

Snarky Puppy, 2016. Family Dinner – Volume 2. Universal Music Classics, February, USA.

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Published

15-05-2026

Issue

Section

Articles – Open Section