One of the things that the book does not cover in Musical Materials is tuning systems. Our c

One of the things that the book does not cover in Musical Materials is tuning systems. Our current tuning system is called 12-tone equal temperament. It is based on the simple mathematical formula, the 12th root of 2. The octave (2) is divided into 12 equal parts. But in Western music history, there have been many different tunings: Pythagorean, Mean-Tone, Well Temperament and finally 12 Equal. Pythagorean tuning was the first systematized tuning in the West, originating around the 6th century B.C. by Pythagoras. This tuning and other derivations of that tuning was used in Ancient Greece and though the Medieval period, roughly 2000 years. During the Renaissance, with the frequent use of 3rds and 6ths, they had to adjust the 3rds and 6ths because they were out of tune. They discovered that not only the 3rds and 6ths needed to be adjusted, but the other scale steps needed to be adjusted as well. This was called Mean-Tone tuning. Mean-Tone was used from the Renaissance through the Baroque Period. During the Baroque Period, as harmony and keys expanded, Mean-Tone tuning did not suffice, and they continued to make all kinds of adjustments to the tuning. Johann Sebastian Bach's solution was to adjust the tuning so most of the scale steps were in a more equal configuration, called Well-Temperament. But the ultimate solution was utilized around the beginning of the Classical Period, 12-Tone Equal Temperament.

I've included a link to a movie about the development and history of these tunings. Please watch.