Two related questions: Are the seven independent axes (dimensions) of the octonian space relatable to the adult capacity of STM (short term memory – number of independent objects simultaneously retainable in the mind)?
Can octonions serve as a mathematical Rosetta stone for cognitive and brain modeling by linking the world of algebra (operators, e.g. for mental rotation) to the world of geometry (relations, membership, and clustering), by exploiting the fact that octonions are composed of algebraic structures (7 quaternion sets) that also map onto a simple projective geometry structure (the Fano plane) that is organized as a system of 7 lines and 7 points?
I think I get what this question is about - how consonance and dissonance interact as if apprehended as a 3D mnemonic, and given that our tonal system of Western music is set up as 7 discrete tones and an octave per scale arrangement - so if that is the limit of parameters then most pop tunes should fit. If consideration is given to a fully chromatic 12 tone system then deferring to Nicholas Slominsky's Thesarus of Melodic Patterns and Scales offers a much wider "vista" for serious evaluation of how each of his interpolations of tones and extrapolations of possible synthetic scales acts on the listener. Then there is also the matter of the composition process, especially when one considers a little known dictum of Hans Eisler: "A well constructed melody, when heard once, will be recognized upon a second hearing [days, months, years later]." - which brings LTM into the phenomena, meaning that Eisler's idea is more about a complex right brain process, and as such is as much applicable to nursery rhymes or Bartok's late string quartets.
Davis -- Thanks for the information on Slominsky's thesaurus and the quote from Hans Eisler. I've been very interested in his music, ideas, and history since about 2006, as part of a 20-year eclectic music-mixes exploration I started in 1994. I paired him with Frank Sinatra and others, and it worked well.
LTM is part of the story, yes, and partly applicable to non-musical material as well, but there seems to be a superior memory quality with music -- many aphasic patients can sing text but not say it alone. There seems to be something qualitatively different in the binding of melody that lets it bypass some kinds of stroke damage.
The part of the brain that seems to be involved in multisensory synthesis of image, according to Arnold Trehub (The Cognitive Brain), is the inferior parietal lobe.
My interest in quaternion and octonion math as models for aspects of cognition and music, including STM, coincides with veteran AI researcher/mathematician Ben Goertzel (referenced below)). This also connects with projective geometry (John Baez) by dropping the identity element in octonions-- the roles of containers and the contained can be swapped.
Goertzel, B. (2007)
Mirror Neurons, Mirrorhouses,
and the Algebraic Structure of the Self. Institute for Ethics and Emerging Technologies.
http://ieet.org/index.php/IEET/print/2595
Attached are two parts of a presentation I made on the history and culture of quaternions. The second file focuses on music and includes several quotes from Arnold Trehub, including the 4D attributes of "autaptic cells" in the parietal lobe -- able to hold a series of versions of the moving present.
Herb
Thanks, Herb Klitzner. Especially thank you for the PDFs - they really help me understand your intriguing question. An esteem PhD friend and I have been musing over your idea about quarternions. I will give the matter more study and confer with my man. thank you so much for opening an area of inquiry that is of great interest to me.
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