major triad

See here for a more comprehensive reckoning.

While explaining a major triad as scale degrees 1, 3, and 5 is accurate and simple, it is lacking. Major triads are not strictly limited to this combination. For example, degrees 4, 6, and 1 also form a major triad, as well as 5, 7, and 2, and if we want to get fancy, we can even dare to steal from other keys and use b2, 4, and b6. Therefore a more comprehensive definition is one that includes the intervals within the triad that construct it, because these intervals do not rely on the context of a key.

First things first:
A triad is a set of three notes in a neat little stack of thirds. That is, the distance between the bottom note and the middle note is a third, and the distance between the middle note and the top note is a third. In turn this means that the distance from the bottom note to the top note will always be a fifth. On a staff, triads spelled as chords (all three notes played at the same time) will look like snowmen: three circles on top of each other, the same distance apart (all on lines or all on spaces). It should be noted that triads do not have to be chords. They can be written (or played) linearly (as in a melody), or even out of order, and still be considered triads.

Triads come in different qualities: major, minor, augmented, and diminished. What determines the quality of the triad is the quality of the thirds within it. If there are two thirds, and each can be major or minor, then you can see how there are four possible triads to be made. One would consist of two major thirds: this is augmented. One would consist of two minor thirds - diminished. But you can also have one of each. In these cases, the triad is named after the quality of the lower third; therefore, if the bottom third is minor, and the top third is major, that creates a minor triad. And finally, if the bottom third is major, and the top third is minor, we have our major triad.

One of the interesting things about both major and minor triads is their geometry, particularly their resemblance to the Pythagorean triangle with sides in the ratio 3-4-5.

Consider that an octave is divided into 12 semitones. For the sake of example, let us start from the note C (this is an arbitrary choice - these ideas are just as true starting from Gb, or A of course).

If we divide the octave using the numbers 3-4-5 (that is, taking intervals of 3, 4 and 5 semitones, or minor 3rd, major 3rd and perfect 4th) we get the notes C-Eb-G and then C one octave up. This is the minor triad with the octave added (simply, a pure minor chord).

If we take 4-3-5, we get C-E-G-C, the major.

The combinations 5-4-3 and 5-3-4 yield the notes C-F-A and C-F-Ab respectively - which are the second inversions of the major and minor chords on the related key of F (the fourth or subdominant).

Continuing the idea of relating geometry to types of chord, we may note that a 4-4-4 triangle (equilateral triangle) corresponds to an augmented chord/triad (C-E-G#). A diminished 6th chord (C-Eb-F#-A), being a series of minor thirds, relates to the shape 3-3-3-3, or a square.

Note that the sides of squares and equilateral triangles cannot be distinguished from each other by length. Analogously, the dim 6 and aug chords don't really belong to any key - you can invert (rotate) them and they sound the same, in terms of intervals (not, of course in absolute pitch).

Music is indeed mathematics, at least in an analytical sense. In a practical sense, it can be something much more, of course. Perhaps that is why it functions so well as a bridge between the logical and super-logical mind.