Table of Propositions

cally and Geometrically, as well as the Consonances. 121.
Corollary. Dissonances serve harmony, although they only enter it by accident. 122.

V. How many commas the minor and major tone contain, and in what sense one can say that the minor is larger than nine commas. 123. A "comma" is a very small musical interval, used here to measure the precise difference between notes that might otherwise sound the same to the untrained ear.
VI. To determine how many commas the Octave has. 125.
VII. Whether the false Fifth exceeds the Triton, and by how much: where several degrees and intervals that serve to understand the Diatonic genus are explained. 126. term: Diatonic (The standard musical scale consisting of seven notes).
VIII. Whether the Triton exceeds the Fourth more than the Fifth exceeds the Semidiapente. 127. term: Semidiapente (A diminished fifth, often considered a "false" or "imperfect" fifth).
IX. Two minor Thirds, which can be taken in the same place as the Semidiapente—namely from the mi of E-mi-la to the fa of B-fa, or from #mi to F-fa—are larger by a major comma than the false Fifth: consequently, they exceed the Semidiapente more than it exceeds the Triton. 128. The letters and syllables (like E-mi-la) refer to the Guidonian hexachord system used by musicians of this era to name and locate notes.
X. To determine if Dissonances are as disagreeable as Consonances are agreeable: where one sees why pain is more keenly felt than pleasure. 129.
XI. To explain the Harmonic intervals, both consonant and dissonant, which cannot be expressed by numbers. 132.
XII. From what points weights must fall to create such proportions, and chords or discords as one wishes, when they meet opposite one another. 134. Mersenne was deeply interested in the physics of sound; here he relates the speed of falling objects to musical ratios.
XIII. To demonstrate that there is no difficulty in the Theory of Music, and that everything in it is done by the simple addition or subtraction of beats of air: where one sees how sounds resemble light. 137.
XIV. To give a summary of everything that has been said in the book on Consonances and Dissonances. 139.

20 Propositions from the book of Harmonic Genera, Systems, and Modes.

I. To explain what the Diatonic genus consists of, its types, and the one used now: what the scale of Guy Aretin consists of, and what the Tetrachords of the Greeks are. 141. Guy Aretin is the French name for Guido of Arezzo, the 11th-century monk who developed the system of solmization (ut, re, mi, fa, sol, la).
II. To know whether the Diatonic degrees are more natural and easier to sing than those of the Chromatic and Enharmonic. 147.
III. The ratios of the Diatonic degrees can be explained by the length of the strings, and by the number of their beats. One sees where the minor tone and the major tone must be placed. 150. term: Beats (Here, Mersenne refers to the frequency of vibrations or "beats of air").
IV. To explain the Diatonic, Chromatic, and Enharmonic Genera so clearly that all Musicians can easily understand them and use them in their Compositions. 153.
V. To explain the use of the Octave, which contains the three aforementioned Genera. 155.
VI. To explain the same System or Diapason by beginning it with C-sol-ut. 157. term: Diapaſon (The Greek term for an Octave). C-sol-ut is the note middle C.
VII. One can begin each note of Music on each Diatonic degree of the two preceding Systems, in order to transpose all sorts of tones on the keyboard of the Organ arranged according to the Diapason. 161.