Alina Abraham

1. University of Auckland ICL School of Business, Auckland, NZ The Corelli School of the Arts, Browns Bay, NZ The Zone of Musical Creativity: Harmonic Series Structures, from Pictorial Representation to a Method of Teaching and Coaching for Music Performance Alina Abraham

2. Glossary (1) • The Zone… …that state of mind where everything clicks, everything is easy, where your actions are effortless, and when your results are up to or even exceed your previous expectations. The Zone is quite simply being in the perfect state of mind for a given performing situation, resulting in an optimal level of performance… (Gorrie, 2009, p.21) … the flow experience… focus - based on repetition and mastery (Custodero, 2008 cit. Czikszentmihalyi; http://www.ted.com/talks/mihaly_csikszentmihalyi_on_flow.html; Goleman,1995) … meditation… create a new reality (Werner, 1996)

3. Glossary (2) • Visualisation…- uses symbolic thinking • Result of using visual strategies, e.g., mindmapping; use of graphics – visual props; • Association of an abstract concept with presented or own visual image TimeDance, an algebra of movement… - Daniel Belton, Michael Norris, 2012 http://www.scoop.co.nz/stories/CU1209/S00478/time-dance-an-algebra-of-movement.htm http://vimeo.com/52898697 • Pictorial representation – uses iconic thinking Association of abstract concept with suggested (reversed metaphor), images, or own. • http://www.michaelnorris.info/orchestralworks/lonelymargins.html

4. Abstract • For any pitched sound in nature, there is an inner structure of that sound displaying a series of harmonics, or partials, that vibrate with different frequencies; this series is called harmonic series (HS) (Backus, 1977). • Due to the mathematical underpinnings of those partials’ frequencies, the pitched sound may be visualized with a Cartesian (logarithmic) spiral design (Novac, Charalambides, 2003). • This paper discusses how various melodic and rhythmic structures that originate in HS might be visualized in pictorial ways in order to: • Explain elements of form in a musical piece and facilitate students' understanding of compositional processes in music • Enhance memorization in music performance and develop such visualizations into a method of teaching in music and coaching for music performance 

5. 1. THS - linear representation The Harmonic Series (THS)of a pitched sound Acoustics: The harmonic series (THS) is a mathematical model in which the strictly increasing series of ‘n’ oscillators represents multiples of the frequency of a fundamental sound, e.g., 1f, 2f, 3f … (Backus, 1977) The linear representation of THS (colour coded) shows how the odd harmonics generate geometric progressions with the ratio r = 2, i.e. {1, 2, 4, 8, 16…}; {3, 6, 12, 24…}; {5, 10, 20…}; {7, 14}. The relation between harmonics follows the golden mean principles Fibonacci Series 1,2,3,5,8 evidences the Major Chord

6. 2. THS - geometric representation The Harmonic Series (THS)of a pitched sound • In the geometric representation of the harmonic series (THS) • the geometric progressions with the ratio r=2 represent radii starting from the twelve nodal points of the total chromatic scale of the well temperate system. • the 12 nodal points – i.e., pitches of the total chromatic, also represent the division of the circle in 12 areas – as described by Descartes [37, 38] – which validtes the philosophical Pythgorean concept as a mathematical one, too • in a logarithmic spiral the angle between radii is a constant of 2100 [17]; the pressure in each point of the 3D model is constant; Galaxies, mineral, vegetal, human cochlea follow this path. (Bekesy, 1960; Gupta, 1999) Pythagoras’“Circle of Fifths”

7. Visualisation of arithmetic and geometric patterns in music - A Steiner approach / holistic learning / Goethe The History of the Project – Dunedin, NZ/UNSW, 2003 Continuing from Schenker: Two harmonic series places a perfect fifth apart complete the total chromatic of the tonal quadrant Cooke, D, 1959 - In 1959, D. Cooke explains the Western world music development as result of aural perception development following the natural resonance laws (HS). Schenker, 1935 – The Chord of Nature (Harmonics 1-6); subdominant not included in THS

8. The intellectual exercise of using pictorial representation is to apply nature’s own strategies, or ‘voice’, to mind constructs, due to the structural similarities between the two. • visualization in general seems to be the support system that permits students the access to the ‘zone’ of the brain where results come naturally and effortlessly in live / public performances, in music, sports, or other – as result of repetitive work accumulated, with the possibility of transfer of skills into other areas, (Coyle, 2009; Gorrie, 2009). • The New Zealand school curricula to focus of creativity in terms of learning outcomes, with music, dance, and drama, for example. • A potential pedagogy of creativity is envisaged, and expected to be ‘life changing’ for the K–12 students (Ken Robinson, 2013!)*. • Pictorial representation visualizes familiar objects or pictures in the user’s mind**. • Spiral representation- • Same spiral is used by [15] in explaining the significance of the golden mean in Bela Bartok’s music. “Spiral”, by Stockhausen, composed in 1969 [24] is a composition exploiting a similar approach. Inspired by Stockhausen, Jill Purce describes the presence of this pattern in human consciousness, anatomy, and socio-cultural reality, with her publication in 1974 [19].

9. Pictorial visualisation and the learning process Coyle (2009) evidences the role of neuronal synapses that facilitate reasoning and memorization in the brain, and also about another substance that plays an extremely important role in memorization and reasoning in humans, that myelin. Myelin acts like an ‘auto-pilot’ mechanism in the brain that continues to work for the stabilization of information long after the initial stimulus is gone, that is for days and even weeks later. This is how reasoning and memorization are enhanced. Reasoning is enhanced by symbolic iconic connotations (Bruner, LASS). Apparent ‘informal’ learning – relaxation of the mind facilitates creativity and innovation (‘breakthrough’)*

10. Pictorial Visualisation may explain elements of form in a musical piece and facilitate students' understanding of compositional processes in music • Rondeau (ABACADA) & Bridge (ABCBA) forms; • Symmetry; applied golden section ratios; • Geometric development techniques (diminutio; elongation) • Spiraling discourse, e.g., Norris, 2001 • Harmonic Series Structures’, e.g., harmonic 3:2 the perfect fifth (harmonic mean, Zarlino, Apel 1996); Xmajor = 5 x perfect fifth; (K.Field, Mme. Butterfly) • Harmonic 4:3, the perfect fourth(geometric mean, Zarlino, Apel 1996) – • Quartal harmony (descending perfect fifths), • integration in“So What Chords” (Miles Davis/Kind of Blue, 1959) • Harmonic 5:3 – the major third (harmonic mean, ibid); Harmonic 7:6 minor third (geometric mean, ibid) – Ron Samsom, Race to space (2012) • The acoustic mode (Lydian Dominant) – the Moebius strip (C, E, G, Bb, D, F#, A) • Same principles apply to rhythm (Chris O’Connore, 2011)

11. Pictorial use of the quadrant In New Zealand music theory, a similar pictorial approach was used by Jenny McLeod in her ‘Tone Clock Theory’ treaty, after Peter Schat harmonic theory using the twelve possible triads within the twelve note chromatic system - as shown in Appendix 4 of her treaty [32].

12. Performance practice Pictorial strategies enhance memorization in music performance and develop such visualizations into a method of teaching in music and coaching for music performance  • Gorrie, 2009 - 5 step method, a 5 or 12 weeks programme • Coyle, 2009 – matching performance to pre-existing blueprint • own teaching practice • Piano- Scales… and… the butterfly model… • Hanon and… sea creatures; planets –spinning around a centre, similar to various pitches in a scale, around the tonic; • Aural training: intervals and… iconic / symbolic / enactive learning (Bruner)

13. Conclusion Music stimulates both the left and right hemispheres of the brain, and so do pictures (Corballis, 2012; Coyle, 2009) The use of a pictorial representation approach for music analysis, performance, teaching & coaching - and beyond beneficial to enhance students’ individualized learning in music, with music, and not limited to music By associating familiar images of objects or clusters of objects to new contexts, students relate in an almost subconscious, relaxed manner, structural similarities between old and new situations, and develop new understandings from old knowledge. Pictorial representation can be incorporated in all disciplines at all levels, and be contributing to the development of creative and innovative practices in education. Pictorial representation is supportive of all learning, at all levels – due to the individualized (personalized) facets of ‘what’ (content), ‘why’ (motivation), and ‘how’’ (strategies) one may use in teaching and learning

14. References J. Backus, The Acoustical Foundations of Music, New York, NY: W. W. Norton & Company, 1977. G. von Bekesy, Experiments in Hearing. New York, NY: Mc Graw Hill Book Company, Inc., 1960.L. S. Catrrel. (n.d.) Center for Research, Creativity, and Innovation. [Online]. Available: http://www.calarts.edu/creativity/T. Christensen, The Cambridge History of Western Music Theory. Chicago, IL: University of Chicago, 2002.L. Cohen, L. Manion, K. Morrison, Research Methods in Education, 5th Ed., London, UK, New York, NY: RoutledgeFalmer, 2002.D. Cooke, The Language of Music. Great Britain: Oxford University Press, 1959.M. Corballis, Pieces of Mind. Auckland: Auckland University Press, 2011.D. Coyle, The Talent Code. Greatness Isn't Born. It's Grown. Here's How. New York, NY: Bantam, 2009.M. C. Ghyka, The Geometry of Art and Life. New York, NY: Shee and Word, 1946.M. C. Ghyka, Philosophie et mistique du nombre. Paris, FR: Payot, 1952 (in French).J. Gorrie, Performing in The Zone. Unleash Your True Performing Potential! US: Lexington, 2009. C. L. Krumhansl, “The Psychological Representation of musical Pitch in a Tonal Context,”Cognitive psychology. Vol. 11. No. 3, pp. 346–374, 1979. C. L. Krumhansl, “Concerning the applicability of geometric models to similarity data: The interrelationship between similarity and spatial density,”Psychology Review, Vol. 85, no. 5, pp. 445–463, 1978.M. Lardon, Finding Your Zone: Ten Core Lessons for Achieving Peak Performance in Sports and Life New York: Penguin Group, 2008.E. Lendvai, Bela Bartok: An Analysis of His Music. London: Kahn & Averill, 1971.

15. References A. L. Novac, S. Charalambides, “A New Model of Perception in (Music) Acoustics: The Equiangular Spiral Pathway,” in P. Slezak (Ed.), Proceedings of the Joint International Conference on Cognitive Science with the Australasian Society for Cognitive Science. Sidney: University of New South Wales, Vol. 2, pp. 496–500, 2003.A. L. Novac, S. Charalambides (2003). “A New Model of Perception in (Music) Acoustics: The Equiangular Spiral Pathway,” in P. Slezak (Ed.), Proceedings of the Joint International Conference on Cognitive Science with the Australasian Society for Cognitive Science. Sidney: University of New South Wales, poster.R. D. Patterson, “Spiral Detection of Periodicity and the Spiral Form of Musical Scales,” In Psychology of Music, 1986, Vol 14, pp. 44–61.J. Purce, The Mystic Spiral: Journey Of The Soul. New York, NY: Thames and Hudson., 1974.R. Samsom, “The Race to Space”, Auckland, NZ: University of Auckland, 2012, Unpublished.H. Schenker, Der frei Satz. Vienna, UE: Universal Edition, 1935. W. Tatarkiewicz, D. Petsch, J. Barrett, History of Aesthetics. Thoemmes Press, 1999.M. Van Manen, Researching lived experience: human science for an action sensitive pedagogy. New York: State University of New York Press, 1997.http://classical-music-online.net/en/production/9031http://calarts.edu/creativity/focus-areashttp://musicworkings.blogspot.co.nz/2011/06/frequency-and-tempo.htmlhttp://www.creativityinplay.comhttp://www.en.wikipedia.org/wiki/Acoustic_scalehttp://www.matrix.s3.amazonaws.comhttp://www.music.sc.edu/fs/bain/atmi02/hs/biblio.htmlhttp://www.sacred-geometry.com/sacredgeometry.html;http://sounz.org.nz/resources/show/448http://www.ted.com/talks/ken_robinson_says_schools_kill_creativity.html?quote=83http://www.youtu.be/V6SaIg5eNMQhttp://youtu.be/I1A4OGiVK30http://www.youtu.be/xQrPviAAd4c; http://www.xahlee.org/SpecialPlaneCurves_dir/EquiangularSpiral_dir/equiangularSpiral.htmlhttp://xahlee.org/SpecialPlaneCurves_dir/Spiral_dir/spiral.html

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