12-tone music is a 20th-century style of music which was pioneered by Arnold Schoenberg. This style of music uses all chromatic pitches equally. The concept of 12-tone music can be extended to 24-TET. Each pitch cannot be used again until all pitches have been used once. As a consequence, 24 tone row matrices are much larger than their 12-Tone counterparts.
The core is in the initial tone row. The tone row is the initial ordering of pitches. All pitches must be used, and none can be repeated. Once the initial tone row is created, the composer can use the pitches in this order, as well as in the following orders:
- Inverted - all intervals are flipped. A major third up becomes a major third down, for example.
- Retrograde - the order of intervals is switched. The last interval happens first, 2nd to last happens 2nd, etc.
- Inverted Retrograde - the intervals are flipped, and the order is switched.
A tone row matrix can be used to generate this information. The top of the matrix represents the prime tone row. Each subsequent row represents the prime row offset by a certain interval, where the start note is different, but all intervals are the same. The inversions, retrogrades, and retrograde inversions can be found in the following manner:
- Inverted - start on the top row and read downwards. The leftmost column represents the inversion starting from the original root pitch.
- Retrograde - start from the rightmost column and read leftwords. The bottom row represents the retrograde starting from the original root pitch.
- Inverted Retrograde - start from the bottom row and read upwards. The rightmost column represents the inverted retrograde starting from the original root pitch.