## Pythagorean tuning

Pythagorean tuning, or 3-limit tuning, also allows ratios including the number 3 and its powers, such as 3:2, a perfect fifth, and 9:4, a major ninth. 12-tone Pythagorean temperament is based on a stack of intervals called perfect fifths, each tuned in the ratio 3:2, the next simplest ratio after 2:1. Starting from D for example (D-based tuning), six other notes are produced by moving six times a ratio 3:2 up, and the remaining ones by moving the same ratio down:

E♭–B♭–F–C–G–D–A–E–B–F♯–C♯–G♯

This succession of eleven 3:2 intervals spans across a wide range of frequency (on a piano keyboard, it encompasses 77 keys). Since notes differing in frequency by a factor of 2 are given the same name, it is customary to divide or multiply the frequencies of some of these notes by 2 or by a power of 2. The purpose of this adjustment is to move the 12 notes within a smaller range of frequency, namely within the interval between the base note D and the D above it (a note with twice its frequency). This interval is typically called the basic octave (on a piano keyboard, an octave has only 12 keys).

The tetractys (Greek: τετρακτύς) is a triangular figure consisting of ten points arranged in four rows: one, two, three, and four points in each row, which is the geometrical representation of the fourth triangular number. As a mystical symbol, it was very important to the secret worship of Pythagoreanism.