Also I've found that when you double the third of a major chord it sounds more minor, and when you double the third of a minor chord it sounds more major. This can also be explained by your website.

A very astute observation. When doubling the so-called third of a chord (we assume you mean a Triad), it is probably not the third at all, but more likely a primary note (root or fifth) of its shadow. For example in the progression Dm to C, one will often find the Dm in "first inversion" (third in the bass) with the F doubled. All the more reason to suspect that it is not a Dm at all, but an F6 with a doubled root and no fifth!

[note: We were so impressed by this questioner's observation that we are inserting a new page on the site called "Pete's Paradox". Look for it sometime in early August, 2003.]

On the website harmony section, it says to be cautious of using the 5th in the bass. Why is it so "dangerous" to invert the frame of a chord? When you play a 5th, it feels like the bottom note is a root, and when you play a 4th it feels like the top note is the root. It doesn't seem so dangerous to me...is there something I'm missing here?

The bass note of a chord can have very strong implications with regard to our perception of harmony. This is especially true with primary notes (root and fifth). When the 5th is in the bass, it tends to sound like a root, and gives the actual root, a 4th above, the feeling of a non-chordal tone, like what is commonly referred to as a "sus 4." Likewise, the note a sixth above the bass (the actual 3rd) sounds like a non-chordal tone of the fifth above the bass. (This is the famous "cadential six-four" chord.) This is only "dangerous" when used at the wrong place at the wrong time. In some instances it would be desirable, even necessary. For instance, take a piece of music that ends with a dominant/tonic chord progression with one measure of dominant resolving to one measure of tonic. Now replace the first half of the dominant measure with a flatter tenant chord. This will be a tonic embellishing the dominant. This tonic should definitely have the dominant root (5th of the tonic) in the bass, so as to make it clear that the true chord is the dominant, and the tonic is an embellishment, which could also be perceived as a dominant with two non-chordal tones.
So, in C major, the original progression would be
     G - - - C - - -
and with the flatter tenant it would be
     C/G - G - C - - -
If you play the first tonic, the tenant, with the tonic root C in the bass, you spoil the build up to the final tonic. If you keep the G in the bass for the final tonic, you spoil the finality of the resolution, and it sounds like another tenant waiting to go back to a dominant. So, from this illustration, hopefully it is clear that at certain places in the music, we would definitely WANT the 5th in the bass, and in certain other places, we definitely would NOT WANT the 5th in the bass. The "danger" comes from making the wrong choice at the wrong place. If you want the harmonic function of the chord to be clear, and to sound like a true chord as opposed to a tenant, it is best to avoid a prolonged 5th in the bass.
Now, to take your question one step further, one may ask "well, is that G in the bass really the fifth of a tonic C chord, or is it really the root of a dominant G chord which is embellished with 6-5 and 4-3 suspensions?" Music theorists in the academic tradition have disagreed on the answer to this question. Actually, the correct answer is that it is both, but at different levels. This is where a thorough understanding of rhythmic levels comes in handy. At a large level you may have a dominant chord with two non-chordal tones, while at a smaller level you may have a tonic to dominant progression. The ideal voice-leading is one that fits both interpretations.
Now to take your question to a more subtle level. There are certain situations that make this inverted frame (5th in the bass with root above) phenomenon particularly undesirable. They are
     1) On the chord of rest of a descending (flattening) progression
     2) At a large level rhythmic fall (beat)
     3) With no motrix in the chord voicing
In contrast, the following are situations in which the 5th in the bass disposition becomes more acceptable and even quite usable
     1) On the chord of rest of an ascending (sharpening) progression
          (especially when the fifth is followed immediately by the root - this is quite common)
     2) At a small level rhythmic rise (off-beat) (This is known as the "passing six-four chord")
     3) With a motrix in the chord voicing, as in the Secondary Disposition of Natural Canons.
Hopefully, these general observations will be of use to the composer/arranger in deciding how to handle the inverted frame. It may be helpful to remember that the disposition of voice-leading lines is an aspect of arranging, and is therefore not as fundamental as the generative processes by which the four orbit lines are created.

I just read some of the pages dealing with harmony. Let me get this straight - are you saying that chords can only progress by fifths, either up or down? That seems rather limiting. Music seems to be filled with other types of chord progressions, although I admit the progression by 5th is one of the most common.

The answer is a qualified "yes." First of all, if you look at the section on Incompleteness in the Harmony - Transformations pages, you will see that there are many ways that a progression by fifth can be "disguised" to appear as something else. Secondly, there is another way that chords can move around besides "progressing." If you see the section on Metamorphoses (especially the "Mixing"), you will find that chords are able to "morph" into other chords, fulfilling one function in relation to the chord that precedes it, and another in relation to the chord that comes after it. This opens up a world of possibilities, and not only is it not limiting, you may find it in a sense liberating.