Logical Notation is a system I've designed for keeping track of, and playing chess variants in text form (although you may find keeping a board nearby is helpful.) The system was originally designed for use with Courier Chess, but it will work on any square chessgame of infinite size (and cube, if you feel like adding a few more numbers). Of course stating my system is better than the existing ones is quite arrogant but it is extensible.

The Basics: All you really need to, say, play a game in text-based format. The first thing you'll need to know is that each row and column is numbered. Assuming you are looking at the board from the vantage point of white, the numbers go like this.
 1 2 3 4 5 6 7 8 9 10 11 12
You will, obviously, add or subtract numbers from here, depending.

Each square is repesented by row.column, thus 4.6 would be part of this board's center.

Movement is represented by an exclamation point. Thus, advancing white's pawn on 7.2 on space would look like this: 7.2 ! 6.2

The end of one player's turn is repesented by a question mark (the wisdom of which will soon become apparent) and the end of a turn with a period, so thus, if black parrots white: 7.2 ! 6.2 ? 2.2 ! 3.2.

Symbols showing what the moving piece is, check, or if it captures are superfluous and thus omitted.

What of games with multiple moves? Probably the most familar is the FIDE Chess castling. Multiple moves may be separated by a comma. For example, if white were to castle on king's side, it would look like this: 8.5 ! 8.7, 8.8 ! 8.6 ? blah blah blah.

Commenting: Okay, so you understand the movement, but you want to explain a concept to someone. Well, that's what I'm here for.

A plus sign (+) can be used to indicate a solid move. A minus sign (-) can be used to indicate a poor move. A combination of both can be used if you want to indicate a move could go either way, with the one with more prevalence going first. Comments begin and end with two forward slashes (//).
Putting it All Together: Here is the infamous FIDE Chess Fool's Mate in Logical Notation (Line breaks are not necessary, but certainly easier on the eyes).

7.4 ! 6.4 - //This move is weak and the beggining of the Fool's Mate// ? 2.3 ! 3.3 + //He sees the weak opening already//.
7.3 ! 5.3 - ? 1.4 ! 4.1.
//The game has ended in only two moves!//

The could also work for cube-shaped variants, I suppose, if you added a third number.

The only real advantages are the intuitive feel of the form and the fact it transcends language.
If you're interested in older (and admittedly more established) forms, check out:
Descriptive Notation
Algebraic Notation

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