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Group Structure on the Set of Chromatic Intervals
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.
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.