In the last section we looked at the Harmonic Series and the harmonics that it consists of. Below we will be looking at how music temperament and scales relate to this system. Music temperament is “the tuning of an instrument, that is determining the exact pitches to be sounded. Since pitch also depends on frequency, differences in tuning are created by differences in frequency.”

1. Frequency being a physical quantity equal to the number of events in a given time. In music, this means the rate at which an instrument vibrates, measured in cycles per second or Hz.

2. Pitch is dependant on its frequency, that is the number of vibrations per second of a musical sound.

The most prevalent temperaments in Western Music have been the Pythagorean System, Just Intonation, the Mean Tone System, and, since 1880, Equal Temperament. Understanding temperament systems is a very complex undertaking but below we will look at the basics of the different systems and briefly discuss the advantages and disadvantages of the different systems.

There are quite a number of ways of looking at how intervals are generated from the harmonic series, which is how just intonation, and to a lesser degree the Pythagorean systems work. Graham Breed explains intervals in terms of “adding intervals”, “Adding intervals means multiplying frequency ratios. So, adding a major and minor third gives 6/5*5/4=3/2, a fifth. A fourth is the difference between a fifth and an octave, so 2/(3/2)=4/3. A tone is the difference between a fourth and fifth. (3/2)/(4/3)=9/8. But, a major third is two tones. 9/8*9/8 is not 5/4. So, there are two different tones in just intonation. 9/8 is a major tone, and a minor tone is 10/9. That comes from (5/4)/(9/8)=(10/9)” (these intervals are for just intonation). Pitch can also be looked at in a mathematical way (which I’m not versed in this method well enough to explain) by which “Scientists have devised a standard unit for measuring the size of perceived intervals resulting from two frequencies vibrating at a given ratio. This unit is called a cent because it equals 1/100th of a half step. A half step is the smallest interval between two notes on the piano. There are 12 half steps in an octave, and so one octave = 1200 cents. By definition”.

This means that all of our normal intervals on the modern piano are divisible by 100 cents.
For example, what musicians call a
half-step (C up to Db) = 100 cents whole step (C to D) = 200 cents minor third (C to Eb) = 300 cents major third (C to E) = 400 cents perfect fourth (C to F) = 500 cents augmented fourth (diminished fifth, C to F#) = 600 cents perfect fifth (C to G) = 700 cents minor sixth (C to Ab) = 800 cents etc.

There is a rather complicated formula for figuring out how many cents large an interval is:
Divide 1200 by the logarithm of 2. If you use base 10 logarithms (any base is permitted), 1200/log 2 = 3986.3137… For any ratio n/p, the number of cents in the interval is
log (n/p) x 1200/log 2
If you’re using log 10, then cents = log (n/p) x 3986.3137…” Kyle Gann

The unit of measurement of cents as a means for looking at frequency will come into play later when we compare musical temperaments. One last thing to point out is that, all these ratios can be much easier heard from different sized musical strings, chimes, metal rods, midi synthesizers, etc, but I’m giving you a pretty thorough introduction to this very complicated subject.