The Pythagoreans noticed that nice simple ratios of string length made nice sounds together.
Using an understanding that 1:2 and 2:3 were good ratios, start with a length and keep reducing it to 2/3 of itself. Each time that took the length under 1/2 they doubled it to get back within range.
The scale on a piano does something clever : the ratio (interval) between any adjacent points on the scale is equal. If you play any note, twelve points higher will be exactly an octave on.
The reader is invited to investigate changes (or permutations) in the ringing of church bells, illustrated by braid diagrams showing the order in which the bells are rung.
Suppose you are a bellringer. Can you find the changes so that, starting and ending with a round, all the 24 possible permutations are rung once each and only once?
Bellringers have a special way to write down the patterns they ring. Learn about these patterns and draw some of your own.
An article for students and teachers on symmetry and square dancing. What do the symmetries of the square have to do with a dos-e-dos or a swing? Find out more?