Friday, August 31, 2007

WHY 17 PATTERNS?


"Renaissance choral music should sound like wallpaper." -- Richard K. Winslow

At first, I thought that Winslow's quip was all wrong, reducing gloriously detailed counterpoint to flat and monotonous wall hangings. But a bit of research into wallpaper patterns or groups quickly demonstrates that wallpaper is much more than Muzac for tired eyes.

There are 17 possible wallpaper groups, which are two dimensional repetitive patterns. Wallpaper groups are more complex than frieze groups (which repeat in only one dimension) and less complex than three-dimensional crystallographic or space groups. The groups are classified according to their symmetries, and once you understand the symmetries, even the most mathematically resistant can follow the proof that there can only be 17.

If I were a more social type, I'd rush out to invite 16 other composers to join me in composing a set of 17 pieces, each one based upon one of the 17 wallpaper patterns. The patterns can be usefully extended through the choice of colors, something which immediately suggests a variety of scoring patterns in a musical translation. I don't know if they'd be better served by a vocal ensemble or by Morton Feldman's ensemble of flute/percussion/piano (with doublings). In any case, the title would have to be Why 17 Patterns?

5 comments:

Elaine Fine said...

Hey, I'd do it, but I'm one of the more mathematically resistant people who doesn't understand the math. Could you spell it out for me (or provide a link that does)?

If this becomes a project, I'll take the one on the second row, the third from the left.

Daniel Wolf said...

Elaine,

The Wikipedia entry on "wallpaper groups" is quite clear, as is this page: http://xahlee.org/Wallpaper_dir/c5_17WallpaperGroups.html

This is more technical: http://www.clarku.edu/~djoyce/wallpaper/

This page has some cool animations for each pattern: http://www.clowder.net/hop/17walppr/17walppr.html

And just for for their beauty, here are the 17 as found in traditional Japanese textiles: http://mathmuse.sci.ibaraki.ac.jp/pattrn/PatternE.html

Your favorite pattern is group pmg. It has two rotation centers of 180 degrees, and reflects in one direction only, with the glide reflections perpendicular to the axes of reflection and the centers of rotation lying on the axes of reflection.

Elaine Fine said...

Thanks! This is extremely cool, and something I have NEVER thought about.

Samuel Vriezen said...

Daniel, did you ever hear Tom Johnson's "Tileworks" series? One-voice hocqueting canons - which means, one-dimensional tiling of course, not yet wallpaper. But the idea of extending this is attractive. Like a mixture between Tom's "Tileworks" and his "Eight patterns for eight instruments" (to me, one of his most moving works)

Thomas D said...

I still think the quip was all wrong. Renaissance vocal music is precisely *not* regular, in the sense of repeating itself, however much intellect was required for its construction.

And like (almost?) all other vocal music, that of the Renaissance accommodates or even invites characteristically vocal and text-based means of expression. Not to mention vocal ornamentation.

Counterpoint as intellectual discipline is only the bare beginning. The fact that composers took (say) the 14 Stations of the Cross as subjects rather than the 17 wallpaper groups, or whatever was the equivalent in 16th-century geometry, should tell us something.