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Numbers-Based Musical Harmony

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Treatise : Numbers-Based Musical Harmony

In the West, the study of musical harmony through ratios of integers goes back as far as Pythagoras, some 2,500 years ago. This approach, which was superseded in the 19th century by the subdivision of the octave into twelve equal parts, is now enjoying renewed interest.

The Treatise synthesizes some recent knowledge on this topic by means of a complete and coherent harmonic modulation method. The Treatise begins with simple basic notions to build toward the use of a vast repertoire of modulating harmonic sequences described strictly with integers available in the Supplements.

Consult the Treatise
Tip: Be sure to view the table of contents of the documents!


Supplement: 3-limit →
Supplement: 5-limit →
Supplement: 7-limit →
Supplement: 11-limit →
Supplement: 13-limit →
Supplement: 17-limit →
Supplement: 19-limit →
Supplement: 23-limit →
Supplement: Pitches
Supplement: Dissonances
Supplement: Modulating Factors
Supplement: Melodic Intervals



Lecture (with english subtitles)

Webinar resources: Just Intonation, Tunings and Temperaments

organized by the Canadian League of Composers