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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 This approach, which was superseded in the 19th century by the subdivision of the octave into twelve equal parts, is now enjoying renewed interest.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!

### Supplements

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