Here is a knot I have invented, or possibly reinvented (I have come to kind of expect that). For now I call it the "Balance Knot". I know some of you will hate this knot, but my hope is that some of you will find it interesting.
The balance knot is self-untying. That is, when the standing part(s) are untensioned, the knot falls apart. Some rope have a sort of memory though, and will need some shaking for the untying to take place. It is also close to ultimate in simplicity.
It can be used both for an eyeknot and for a bend.
Balance Loop
This is the easiest one to tie. Arrange the rope as you can see in the attachments. Hold both the standing part and working part in one hand, and pull away from the eye. Let the two strands slide through your fingers, such that when the working part leaves your fingers it is well tensioned. The working part should then swing out to approximately 90 degrees. If it is less (leaning away from the eye), then it is good. If it is more (leaning towards the eye), you should retie the knot. Exercise and you will get the feel for it.
If the standing part is not tensioned, you can use your free hand to give it tension. Remember, of course, to keep that tension after you have tied the knot.
The crossing knot closest to the eye could take on any of four mirrored forms. It doesn't matter much. It is the other crossing knot that takes the load.
Balance Bend
Arrange the rope as you can see in the attachments. Use both hands to hold the standing part and working part on each side of the knot-to-be. Pull your hands away from each other, sliding the strands through your fingers, keeping everything tensioned. For best results, let the standing parts slide less than the working parts. When the working parts swing out, they should get an angle of approximately 90 degrees or less. Expect this to require some exercise to get right.
For both knots, they are easiest to tie if the standing part(s) are somewhat tensioned already.
Be careful not to tie the crossing knots too close together. Instead of self-untying, your knot may instead transform into reversed half hitches.