# Tuning pitch brackets: part 2

The Pythagorean tuning system prevailed in Europe until the early 16th century. Pythagorean tuning uses only pitch intervals of the octave (2:1) and the perfect 5th (3:2). The purpose of any tuning system is to divide an octave into a sequence of scale steps. The number of steps and their sizes, measured in ratios of pitch, vary from one tuning system to the next. Here is the Pythagorean method:

- Start at 1. The octave is then 2.
- Raise by perfect 5ths (multiply by 3/2) and record the pitch ratios.
- 1
- 3/2
- 9/4
- 27/8
- 81/16

- For each note pitch above the octave, lower the note by octaves until it fits between 1 and 2. That means dividing by 2.
- 1
- 3/2
- 9/8
- 27/16
- 81/64

- Sort the pitches numerically and you now have a tuned scale.
- 1
- 9/8
- 81/64
- 27/8
- 3/2

Only the first five notes above are tuned, so this is a pentatonic scale. You can continue raising by 5th if you want a diatonic scale or chromatic scale. The problem with the Pythagorean method of tuning is the major 3rd has a harsh dissonant sound which we can retrospectively call “out of tune”. The large integral ratio, 81/64, is an indication of the complexity of this scale pitch compared to, for example the ratio for the 5th, 3/2. New methods of tuning emerged in the 16th century which would correct the major 3rd dissonance along with all the scale pitches. The resulting major diatonic scales and the triad would eventually bring about the development of modern harmony.

Now, in pitch bracket notation, the sequence of 5ths looks like this:

Each of these pitch ratios is drawn on a separate melody line below. If the pitch ratio raises above the first octave (i.e. 2), then apply the octave pitch bracket until the pitch fits the first octave.