Universal Harmony (Harmonie Universelle)

VIII. Explaining the utility of the two previous systems, and the origin of all their intervals. 162.
IX. Explaining the degrees of the system of 25 strings and 24 intervals to the Octave which contains the 3 Genera, following the thought of Salinas. 163. Francisco de Salinas (1513–1590) was a renowned Spanish music theorist and organist whose work influenced Mersenne's understanding of mathematical tuning.
X. To determine if any string or degree is missing in the figure of the previous proposition, or in the systems of the 5th and 6th propositions, and if one should add some degrees to perfect Music. 166. Where one sees the Octave divided into 32 sounds.
XI. Explaining the system of Fabius Colomna, which he divides into 59 sounds, or 38 intervals; and simultaneously the monochord A one-stringed instrument used by theorists to measure musical intervals mathematically. that he uses, and all its divisions. 167. Fabio Colonna was an Italian scholar who invented the "Sambuca Lincea," a 17th-century keyboard instrument capable of playing microtonal music.
XII. Explaining the simplest and easiest system of all those in which one can begin all sorts of notes and pieces of music, transposed on any string or to any tone one wishes; and simultaneously the Enharmonic One of the three ancient Greek musical scales, using very small intervals smaller than a semitone. system, or the mixture of the 3 Genera The three types of scales used in ancient music: Diatonic (standard), Chromatic (using semitones), and Enharmonic (using quarter-tones).. 170.
XIII. Explaining the Diatonic, Chromatic, and Enharmonic genera, and the common genus of the Greeks, in their simplicity. 172.
XIV. Explaining all the species of Fourths, Fifths, and Octaves that can be used in the Diatonic genus. 176.
XV. That one can establish more than 7 species of Octaves in Music. 180.
XVI. Explaining the 12 modes of the Practitioners, and showing that one can establish 72. 181. Musical "modes" were the predecessors to modern scales (like major and minor). Mersenne explores how these can be mathematically expanded.
XVII. Determining what the modes of the ancients were. 185.
XVIII. Explaining the force and properties of each tone, and of the modes, and the manner of knowing from which mode or tone a given melody original: "Chant-donné" — a pre-existing melody, like a Gregorian chant, used as a basis for a musical composition. is; and showing that there are only 7 different modes or tones. 187.
XIX. Determining if one can reduce the tones and modes to B-natural original: "b quarre" and B-flat original: "b mol"; and showing how to sing without any mutation original: "muance" — the complex system of shifting between hexachords (six-note patterns) used by Renaissance singers. other than those of these two keys. 190. See the first two propositions of the 6th Book of Composition, where one is taught to sing without mutations.
XX. Determining if the 7 species of Octaves and the 12 modes are found in the Chromatic and Enharmonic genera. 194.

28 Propositions of the 4th Book of Composition.

I. Determining if simple solo recitals, performed by a single voice, are more agreeable than when the same thing is sung in 2 or more parts. 197.
II. Determining if a song in three parts is more agreeable than in two. 201.
III. Determining if the Bass is the foundation and the principal part of Music, and for what reasons. 207.
IV. Explaining how many other parts of music there can be, what the Tenor original: "Taille", the Counter-tenor original: "Hautecontre", and the Treble original: "Dessus" consist of, and which is the most excellent part of the four. 211. Corollary. On the Music of the Platonists.
V. All the ways of passing from one consonance A combination of notes that sounds pleasant or stable. to another can be related to the principal movements used in composition, namely to conjunct movement, made by conjunct, disjunct, similar, and contrary degrees. 216.
VI. When one of the parts holds steady and continues the same sound, the other...