A [musical] chord of four distinct tones, none of which is an octave transposition of another, is a tetrad.
"Common-practice" (classical) music-theory is based on triads, but jazz and blues use chords which in theory are never smaller than tetrads. Blues is the more straightforward of the two, in which all chords are one of the functions I, IV, or V, and are all of the dominant-7th type. In jazz theory, there are considered to be five main types of tetrads: diminished, half-diminished, minor, dominant, and major. Below is a table showing the interval-string (is) for all 5 types and their inversions, in 12-edo tuning:
12-edo interval strings of the 5 main chord types (tetrads) and their inversions
chord-members are listed in ascending pitch order, from left to right
(parentheses indicate the last interval which completes the octave)
diminished
Root 3rd 5th 7th Root 3rd 5th 7th
Position:
Root 3 3 3 (3)
1st-inv 3 3 3 (3)
2nd-inv 3 3 3 (3)
3rd-inv 3 3 3 (3)
half-diminished
Root 3rd 5th 7th Root 3rd 5th 7th
Position:
Root 3 3 4 (2)
1st-inv 3 4 2 (3)
2nd-inv 4 2 3 (3)
3rd-inv 2 3 3 (4)
minor
Root 3rd 5th 7th Root 3rd 5th 7th
Position:
Root 3 4 3 (2)
1st-inv 4 3 2 (3)
2nd-inv 3 2 3 (4)
3rd-inv 2 3 4 (3)
dominant
Root 3rd 5th 7th Root 3rd 5th 7th
Position:
Root 4 3 3 (2)
1st-inv 3 3 2 (4)
2nd-inv 3 2 4 (3)
3rd-inv 2 4 3 (3)
major
Root 3rd 5th 7th Root 3rd 5th 7th
Position:
Root 4 3 4 (1)
1st-inv 3 4 1 (4)
2nd-inv 4 1 4 (3)
3rd-inv 1 4 3 (4)
The two less-common types are found only in the harmonic-minor scale, and are permutations containing the same interval-vector as 'major':
12-edo interval strings of the 2 secondary chord types (tetrads) and their inversions
minor/major-7th
Root 3rd 5th 7th Root 3rd 5th 7th
Position:
Root 3 4 4 (1)
1st-inv 4 4 1 (3)
2nd-inv 4 1 3 (4)
3rd-inv 1 3 4 (4)
augmented/major-7th
Root 3rd 5th 7th Root 3rd 5th 7th
Position:
Root 4 4 3 (1)
1st-inv 4 3 1 (4)
2nd-inv 3 1 4 (4)
3rd-inv 1 4 4 (3)
Below are links to download .pdf files of 12-edo diagrams of the tetrads in all the major and minor keys of traditional 31-tone musical notation:
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