Temperament, Tuning, and Timbre -- the underrated trinity in music

 The trinity here is not about the Christianity faith, but similar to the idea of the Christian doctrine of it defines God as being one God existing in three coequal, the core and fundamental definition, temperament, tuning, and timbre (TTT) are the basic building blocks of music, yet, underrated and even neglected in our music education system.

TTT correlates with each other. A note is played, it carries out a series of frequencies, the harmonic series. The series multiples from the fundamental frequency with a series of integers (1f 2f 3f 4f 5f 6f 7f 8f 9f 10f 11f 12f 13f...). The loudness of individual particles, and different patterns, creates different timbres. View the series in a different way, the combination of the integers creates intervals. The combination of the beginning of the series sounds more harmonious and vice versa; simple ratios sound pure and vice versa. The pure perfect 5th interval is based on the ratio of 2:3 (2nd and 3rd harmonics at the same time), Pytheogorous used it to find new notes, called the Pythagorean tuning; since the Pythagorean comma, this tuning cannot generate perfect octave, different methods to adjust the individual interval to achieve a perfect octave called temperament. (common temperament is 12 notes in an octave, but can be in any integer.)

The temperament and tuning affect the timbre back. The 1/4 meantone temp. makes the third interval sounds pure, emphasizing the 5th harmonics of the series. While Pytheogorous tuning sounds better of the fifth, the 3rd harmonics. Equal temperament sounds fine on fifth and suitable to modulate into any key. Thus, even with the same notes and intervals, when two and more notes are played, with different tuning or temperament, the distances of the fundamentals may slightly vary, the harmonics particles combine differently.

String instruments tend to adjust with just intonation (tuning) but piano and percussions set the instruments in equal temperament nowadays. Thus, even if they are playing the same chord or interval, it sounds different, beyond the timbre of the instruments, the combination of the frequency.

Understanding the temperament and tuning is important for a composer is fundamental of expression in music, 

the composer must obtain the character of his piece, the building up of an emotion, and the strength of expression, from sources quite other than the creative powers of the tuning hammer or cone.

Marpurg (1776)

Marpurg, F. W.(1776) “Versuch über die musikalische Temperatur”. Quoted in Grove, “Temperaments”, no. 8.