This should make modes clearer (not)
Here's a chart of the ecclesiastical musical modes that shows the simple mathematics on which the modes are based. You could draw this diagram in any key. I chose the key of C because it's the key with which most people are most familiar. You can move from one mode to the next by flattening a single note. The left-to-right ordering of the modes is critical . This is the unique ordering that works. The diagram is intended to capture the really simple algebra underlying the modes. The algebra is so simple that you could write a 20-line computer program that would generate any mode from any other.
Note that the Lydian mode appears at either end of the chart. You start with C Lydian on the left and come full circle to B Lydian on the right. If you printed out the chart and rolled it into a cylinder (so that the two instances of the Lydian mode are superimposed) you would find that the modes are cyclical. One turn around the circle goes from Lydian all the way around back to Lydian while dropping the key by a semitone. (In this case from C Lydian to B Lydian)
Moving right on the chart requires flattening one tone of the scale. Moving from the top tier (heptatonic modes) to the middle tier (hexatonic modes) requires omitting one tone of the scale. Moving from the second tier to the bottom tier gives the pentatonic modes, each of which is missing two of the seven tones of the scale.
Twelve turns around the circle would bring you from C Lydian to C Lydian an octave lower.
(Click on the image to enlarge, or right-click and open it in a new window. Click on the enlarged image to produce a REALLY large image.)
The neatest thing about doing this chart was rediscovering the mathematical order that underlies the modes. There are other equivalent formulations (Bertrand Bronson's elegant but highly cryptic mode star, in particular), but this one is intended for a dulcimer-playing audience once the tunings and scale starts are added.
This is probably a totally geeky, eyes-glazed-over kind of exercise -- and apologies if so -- but it may have potential as a teaching aid in a class on the modes.
(This is copyrighted only because I haven't finished it and because itdoes in fact contain errors. Take it with a grain of salt and please don't reproduce it or pass it on. I'll release it under a Creative Commons license when I am more confident about it. At the moment I'm adding dulcimer tunings and scale starts to the diagram, so that you can trace a path through the chart and figure out which tuning to use for a given mode.)