Geometry Reveals Universal Property of Music

[DavidRoss]

Researchers at the Institute for Logic, Language and Computation (ILLC) of the University of Amsterdam have discovered a universal property of musical scales. Until now it was assumed that the only thing scales throughout the world have in common is the octave.

The many hundreds of scales, however, seem to possess a deeper commonality: if their tones are compared in a two- or three-dimensional way by means of a coordinate system, they form convex or star-convex structures. [...] To their surprise, they discovered that all traditional scales produced star-convex patterns.

http://www.sciencedaily.com/releases/2011/03/110325102008.htm

[Cato]

Researchers at the Institute for Logic, Language and Computation (ILLC) of the University of Amsterdam have discovered a universal property of musical scales. Until now it was assumed that the only thing scales throughout the world have in common is the octave.

The many hundreds of scales, however, seem to possess a deeper commonality: if their tones are compared in a two- or three-dimensional way by means of a coordinate system, they form convex or star-convex structures. [...] To their surprise, they discovered that all traditional scales produced star-convex patterns.

http://www.sciencedaily.com/releases/2011/03/110325102008.htm

Talk about the Music of the Spheres! 

I was reminded of a scene in Contact, where nobody can figure out the engineering plans, sent by aliens, for an intergalactic spaceship, until an old engineer/billionaire folds the plans into 3-dimensional units.

Many thanks for the article: what exactly such a phenomenon means - other than an interesting coincidence, a mathematical curiosity - awaits deeper contemplation.

And what about the 3% which did NOT form such a pattern?  What are they?  Do they still form something?

[Scarpia]

The paper is here.

http://staff.science.uva.nl/~ahoningh/publicaties/convex_scales.pdf

The crux of it is the scales are approximated by the nearest whole number ratio, and the matrix is defined as the powers of the prime numbers needed to make up those ratios.   One axis is the number of factors of 2, the second axis is the number of factors of 3, the third axis is the number of factors of 5.   So 5/4 maps to (-2,0,1) because it is 2 to the power -2, 3 to the power 0, 5 to the power 1.  The axis for 2 is normally not shown because it corresponds to the octave, which is present in all scales anyway.  Intervals defined by simple ratios (4/3, 3/2, 5/4) are located near the center of the matrix and intervals defined by complicated ratios (225/128, 27/25) are located farther out (See Fig 1 in the link).  The fact that the scales correspond to convex or star-convex patters means the group of ratios found in the scale form a blob in the center of the matrix, rather than having isolated points near the edges.   It is a neat graphical representation of the scales, but the fact that the scales are mostly convex or star convex essentially means that scales are made from simple frequency ratios like 4/3, 3/2, etc. rather than 36/25, 25/18, etc.   I don't find that so surprising.

[Cato]

The paper is here.

http://staff.science.uva.nl/~ahoningh/publicaties/convex_scales.pdf

The crux of it is the scales are approximated by the nearest whole number ratio, and the matrix is defined as the powers of the prime numbers needed to make up those ratios.   One axis is the number of factors of 2, the second axis is the number of factors of 3, the third axis is the number of factors of 5.   So 5/4 maps to (-2,0,1) because it is 2 to the power -2, 3 to the power 0, 5 to the power 1.  The axis for 2 is normally not shown because it corresponds to the octave, which is present in all scales anyway.  Intervals defined by simple ratios (4/3, 3/2, 5/4) are located near the center of the matrix and intervals defined by complicated ratios (225/128, 27/25) are located farther out (See Fig 1 in the link).  The fact that the scales correspond to convex or star-convex patters means the group of ratios found in the scale form a blob in the center of the matrix, rather than having isolated points near the edges.   

It is a neat graphical representation of the scales, but the fact that the scales are mostly convex or star convex essentially means that scales are made from simple frequency ratios like 4/3, 3/2, etc. rather than 36/25, 25/18, etc.   I don't find that so surprising.

That was my reaction after skimming through the complete essay, which - oddly in my opinion - does not contain a few samples of the scales, and does not list that 3% of uncooperative ones.  One would expect some sort of symmetrical shape from a series showing symmetry.  Even things exhibiting symmetry might have a deeper secondary asymmetry, and vice-versa.

Bezier curves:  http://flash-creations.com/notes/astb_outline3d.php 

But even Bezier curves can be created that are asymmetrical.

[71 dB]

Maybe this star-convexity is related to tuning practices of acoustical instruments? Maybe non-star-convex scales would be very difficult to tune?

[some guy]

"Universal property of music" seems a very different proposition from "universal property of musical scales."

Besides, "universal property" sounds very much like "lowest common denominator" to me, the least interesting thing about anything.

[Scarpia]

Maybe this star-convexity is related to tuning practices of acoustical instruments? Maybe non-star-convex scales would be very difficult to tune?

I think that is the most sensible thing you have ever posted on this site.     But what about vocal music?

[eyeresist]

This Changes Everything.

[DavidRoss]

Did they include the chromatic scale?

Yes, that was among the nearly one thousand traditional scales from around the world, all of which fit the pattern.