4:5:6
4:5:6 is the classical major triad, and can also be referred to as the Ptolemaic major triad. It is found on the I (1⁄1), IV (4⁄3), and V (3⁄2) of Ptolemy's intense diatonic scale (Zarlino), which is perhaps the most common 5-limit diatonic.
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However, there are other 5-limit diatonic scales which don't have the Ptolemaic major triad occurring in all the same places. For instance, Aura is known to use a diatonic major scale in which this chord only occurs on the I and V scale degrees while using a Pythagorean major triad (that is, 64:81:96) on the IV. Compared to its Pythagorean counterpart, the Ptolemaic major triad sounds like it's more easily tonicized, a fact which Aura exploits in order to help stabilize Ionian mode in fixed pitch diatonic scales. Because of these properties, the Ptolemaic major triad has earned its status as a bread-and-butter chord in 5-limit harmony.
There are a number of possible tetrads which can be reasonably built off of this triad, such as 36:45:54:64 and 20:25:30:36 in the 5-limit, as well as 4:5:6:7 in the 7-limit and 32:40:48:55 in the 11-limit.