See my store for how to get Inversions (my book of upside down lettering), Masters of Deception (recent art book featuring many illusionary artists), and my 2005 Brainteasers calendar.

More about my math dance educational work, including my book of activities for K-12 teachers Math Dance.

See my inversions lettering designs online.

Here is my canon by augmentation on the theme of the Musical Offering.

Here are the PowerPoint slides from my talk.

Post your photos on the A5 web site.

Here is the diagram of the generalized keyboard (this is also in the PowerPoint slides). Moving to the right gently upwards takes you up a whole step (a frequency interval of ((3/2)^2)/2) while moving to the right sharply upwards takes you up a half step (a frequency interval of ((3/2)^7)/16). By the way, narrowing the slice also gives you the diatonic scale (the white hexagons) and the pentatonic scale (the black hexagons), while widening it gives you the next few "good" equal-tempered scales, which start with 19 and 23.

And where to get the manufactured generalized keyboard.

The theory behind this is: two tones produce the interval of an octave if the ratio of their frequencies is 2/1. The octave is the simplest and most basic interval. The next most basic musical interval is the perfect fifth: a frequency ratio of 3/2. Ideally a musical scale should include: perfect octaves, perfect fifths, and be composed of a series of small intervals all the same size.

Unfortunately it is impossible to satisfy all three of these constraints. So equal-tempered scales (ones in which the smallest interval is always the same frequency ratio) compromise by mistuning the fifths slightly while keeping the octaves perfect. Most western instruments today are tuned to equal temperament, though continuous instruments such as violin, trombone and voice can produce other scales.

Equal temperament became all the rage during the industrial revolution, when musical instruments became mass produced. Before then, music was commonly played in other tuning systems such as "just" intonation and the "well-tempered" tuning (which is close enough to equal temperament that music can be played in all possible keys, a fact Bach trumpeted (?) in The Well-Tempered Clavier).

There is a backlash of musicians who continue to play and compose music in other tuning systems, especially "just" intonation. (Shades of noneuclidean geometry.) The charm of such music is that scales starting on different notes are not all the same, thus modulation produces a real change in the harmonic landscape. Some music such as Indonesian gamelan and Indian classical music have always used tuning systems radically different from equal temperament.

Arbitrarily dividing the octave into different numbers of equal intervals doesn't always produce a good scale. 12 happens to be an excellent number for equal temperament, not as you might think because 12 is so beautifully composite, but because (3/2)^12 = 531441/4096 just happens to be extremely close to a power of 2. (3^12=531441, while 2^19=524288.) The next good approximation is (3/2)^19.