Music is an unexpected alliance of “rivers” in our head and body of very ancient and very modern forces that reflect the way we operate inside and the social/cultural milieu we swim in and share.

Music reflects motion and travel, and obstacles giving us resistance, and the sight of objects promising us rewards.

Music reflects a sense of the effort it takes to do these things — where to turn, when to wait, how hard to work to scale a steep terrain, how to proceed carefully. The underlying meaning of music comes from how to do these things, not how to describe a scene but how to react emotionally to interaction with a scene. This traces back to our animal nature, but the process is superimposed on our experience of aesthetic emotion, a product of our civilization and humanity, and our spatial sense of traversal and transformation.

So we have at least several forces at work in our experience of music:

• Animal emotion and its tie to effort and life-preserving traverse. (Kevin Behan basing his ideas on animal emotion researcher, Jaak Panksepp).
• Experience of inner tension and release as projected by a spectrum of harmonic relationships in a melody. (Carol Krumhansl and others)
• Aesthetic emotion, a refinement and enjoyment of complex and transforming relationships between artistic elements. (Virtually spatial; rhythmic; textural)

Melody, I and others believe, is experiencing the effects of such a path.

Path Mathematics:  Mathematically, the trace of  line intervals resulting from moving and traversing along from significant point to significant point, is called a graph. The theory of constructing these and analyzing them is called graph theory. The points are called nodes and the connecting intervals are called edges. The theory is  applied to choices of paths, often along the edges of a solid 3D figure such as a tetrahedron (4 equal sides) or more complex shapes, such as a dodecahedron (12 sides).

Somewhere within our cognitive apparatus, I believe, may be a capacity to carry on such activities — to construct, store, remember, and transform such graph paths. Our “river” components catalyze it — travel, effort, animal emotion, and aesthetic emotion.

Gilles Baroin, in his dissertation at the University of Toulouse (2011), shows how to construct such graph paths to represent melodies, using quaternion objects and computer representation to create a picture of visual turning and observation of a melody graph. It does not directly incorporate perceptual experimentation involving music (psychophysics), but does show how a path system can be built around quaternion objects and relationships.

For several quotes and references from researchers, go to page Rivers and Paths. 

By Herb Klitzner
Copyright 2015

Can a brilliant but controversial math system virtually sink from sight? Isn’t math permanent, fixed, self-evident? No, it’s not.

Quaternion math is such a system — celebrated around the 1870s and 1880s at some of the finest universities such as Harvard — this system was almost forgotten in the 20th Century after the theory of vector spaces was extracted and fashioned from parts of quaternion elements and thinking, but was then re-encountered in 1985-1995 because it solved many new problems occurring in the fields of aerospace, computer graphics, and signal/image processing, especially those involving the processes of rotation, orientation-determining, and filtering.

Ask three scientists at random if they have ever heard of quaternions — the chances are you will get “no” as the answer from all three. The exception is the mathematical physics community. But even there, ask them to name several new examples of their use after 1985, or even 1935. For some it will be difficult.

In the future, I personally believe, quaternions will be used in a variety of ways (they are already used in well over 25 different applications) — researchers will especially will leverage the quaternions’ 4D structure to analyze a variety of topics in human cognition and in the neuroscience of music.

A modest number of great researchers in cognitive psychology and AI, over the last hundred years, have already used quaternions and their mathematical family, the hypercomplex numbers, to investigate the nature of thought, motion, and personality  — these scientists include Jean Piaget, Karl Pribram, Ben Goertzel, and others.

This diverse site and exploring blog — including history, culture, math design, future speculation, and new research interpreting and highlighting — will help you explore and fill in this remarkable story for yourself.

I provide generous lists of references, links, and commentary in each major cultural/mathematical topic and sub-topic, and provide interesting and informative quotes from many of these papers to give you the real flavor and essence of their value.

Topics and techniques I may highlight in future blogs include:

• briefly explaining math object constructions based on the unit hypersphere in a 4D space, creating objects such as melodic 4D directed paths (Gilles Baroin), stereographic projections of dodecahedrons (A. Ocneanu), etc.
• exploring the possible 4D nature of melody and melodic rotation in the brain
• the role of Ralph Waldo Emerson in advocating the strong development of pure math in young America
• possible knowledge storage and transfer based on the projective Fano plane within octonion integrated “collections” of quaternions
• also, in neuroscience, I would like to explore  and sketch the relevance of 4D processes and quaternions to various possible conceptual functions of, and connections between, the parietal lobe (object assembly and enhancement), the prefrontal cortex (planning/control), the occipital lobe (vision), and the thalamus, as suggested by some researchers, such as Jerath & Crawford, Arnold Trehub, Marina Korsakova-Kreyn, Daniel Wolpert, Daniela Dentico, and Ben Goertzel.

It will be interesting to see where this leads. I thought about many of these questions in graduate school 40 years ago, and recently decided to return to these questions after pursuing a long career in computer science and educational research. Your companionship and ideas are appreciated.