This document discusses the application of abstract algebra concepts to music theory. It explains the circle of fifths, which is a geometric representation of the relationships between the 12 pitch classes in the chromatic scale. The circle of fifths allows musicians to understand and describe these relationships. The document also discusses Pythagoras' contributions to understanding consonance and dissonance in music theory through identifying numerical ratios between musical intervals. Pythagoras recognized that more consonant intervals had ratios of smaller integers, like the unison, fourth, and fifth intervals.

1. An Application of Abstract
Algebra to Music Theory

2. The Chromatic Scale

3. A Closer Look at these Intervals

4. Group Structure on the Set of
Chromatic Intervals

7. The Circle of Fifths
• In music theory, the sequence generated by ascending the
chromatic scale by fifths is called “The Circle of Fifths.”
• The circle of fifths is a
geometric representation of
the relationships among the
12 “pitch classes,” or set of all
pitches that are whole octaves
apart.
• Musicians and composers
use the of fifths to understand
and describe these
relationships.

8. Pythagoras and Consonance
• In music theory, a consonance (as opposed to dissonance) is a harmony, chord, or
interval considered stable. It is a combination of notes that sound pleasant to most
people when played at the same time.
• Consonance can be defined by the ratio of frequencies between pitches, and the
ratios of lower simple integers are generally more consonant than those that are
higher. Different tuning systems create variations in frequency ratios between
intervals.
• It is said that Pythagoras and/or his followers were the first to draw attention to
the fact that musical intervals could be expressed as numerical ratios and that
more consonant intervals had ratios of small integers.
• In Pythagorean tuning, the unison, fourth, and fifth intervals have ratios 1/1, 4/3,
and 3/2 respectively, corresponding to the generators 1, 5, and 7 in our set. These
intervals are the most consonant of all the intervals in the chromatic scale.
• Although the Pythagorean tuning system is generally not in use today, the fifth
interval is still considered the most consonant after the unison and octave
intervals.