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**The geometry of music revealed!**The

*red*lines connect notes that are a

**major third**apart. The

*green*lines connect notes that are a

**minor third**apart. The

*blue*lines connect notes that are a

**perfect fifth**apart.

Each triangle is a chord with three notes, called a

**triad**. These are the most basic chords in Western music. There are two kinds:

A

**major triad**sounds happy. The major triads are the triangles whose edges go

*red-green-blue*as you go around clockwise.

A

**minor triad**sounds sad. The minor triads are the triangles whose edges go

*green-red-blue*as you go around clockwise.

This pattern is called a

**tone net**, and this one was created by David W. Bulger. There's a lot more to say about it, and you can read more in this Wikipedia article:

http://en.wikipedia.org/wiki/Neo-Riemannian_theory

and this great post by Richard Green:

https://plus.google.com/u/0/101584889282878921052/posts/bgpNTT8WqHx

The symmetry group of this tone net is important in music theory, and if you read these you'll know why!

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- The observation about color being a projective vector presumably means that

if you (say) double or triple all the spectral intensities, the color

remains the same. This is true over several orders of magnitude, but the

examples I gave show that that is not true for very high intensities (which

tend to look white unless the colors are spectrally pure) or very low

intensities (grayscale.)

"clash," and there appears to be a regularity to them, like that found in a

color wheel.

There is, but it doesn't seem to have anything to do with small rational

ratios of frequency, or approximations to that, as it does in music.Mar 23, 2016 - +Robert Dawson

"The observation about color being a projective vector presumably means that if you (say) double or triple all the spectral intensities, the color remains the same..."

Thanks for the clarification. Here is the more complete quote:

"Thus the colors with their various qualities and intensities fulfill the axioms of vector geometry if addition is interpreted as mixing; consequently, projective geometry applies to the color qualities."

Weyl explores his business further:

"Mathematics has introduced the name isomorphic representation for the relation which according to Helmholtz exists between objects and their signs. I should like to carry out the precise explanation of this notion between the points of the projective plane and the color qualities [...] the projective plane and the color continuum are isomorphic with one another. Every theorem which is correct in the one system Σ_1 is transferred unchanged to the other Σ_2. A science can never determine its subject matter except up to an isomorphic representation. The idea of isomorphism indicates the self-understood, insurmountable barrier of knowledge. It follows that toward the 'nature' of its objects science maintains complete indifference. This for example what distinguishes the colors from the points of the projective plane one can only know in immediate alive intuition [...]"

Concerning your other points, it seems reasonable to remember that our sensory organs have natural limitations, which skew our perceptions under various conditions.

How to tease out what's really happening "out there" as opposed to what we experience?

My approach has been to consider the regularities, including the one you mention. One of my favorite lines from all of science comes from Newton:

"Truth is ever to be found in the simplicity, and not in the multiplicity and confusion of things."

So saying, I think it best to give a plausible account of the simplest cases and proceed from there to see whether the model can account for more particular issues.

I've only just begun to do so regarding sound, but would underline the symmetries in play.

Beyond that, it seems clear enough that the ratios you cite are telling us something fundamental and quite fascinating -- and so with the vast body of mathematics & physics concerning vibrating strings and membranes.

I've tried to pull my thoughts together in this paper, which is pitched at about the level of a SciAm article.

http://bit.ly/2IKsQW [PDF]Mar 24, 2016 - This is one more application of Grunbaum triangulations.Jul 22, 2017
- What's a Grunbaum triangulation - a triangulation of a torus by equilateral triangles?Jul 22, 2017
- A Grunbaum coloring is coloring the edges (lines) of a triangulation red, blue, and green so that each triangle meets all the three colors. For details and applications, see: ajc.maths.uq.edu.au - ajc.maths.uq.edu.au/pdf/67/ajc_v67_p119.pdfJul 22, 2017
- +Serge Lawrencenko - oh, that sounds more interesting.Jul 22, 2017

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