Table of Propositions

Propositions of the 1st Book of Instruments.

I. To determine how many species of sounds and musical instruments there are. 1.
II. To explain the matter and the manner in which the strings of instruments are made. 3.
III. To determine if harmonic instruments were made in imitation of voices, or if the intervals of voices were regulated by those of instruments; and if Art can perfect nature, or vice versa; and if one should judge artificial things by natural ones. 7.
IV. Which is the most pleasant sound of all instruments, and which instrument should be used to regulate harmonic intervals. 9. Wherein one sees Ptolemy’s Monochord. The Monochord is an ancient scientific instrument with a single string and a movable bridge, used to measure the mathematical ratios of musical intervals. Claudius Ptolemy was a 2nd-century scholar who defined many of these ratios.
V. To demonstrate all the divisions of the Monochord, and consequently the entire science of music. 16.
VI. To demonstrate that the Monochord divided into 8 equal parts contains all the Consonances. 19. Consonances are intervals that sound "stable" or pleasant to the ear, such as the octave or perfect fifth.
VII. To explain the simplest division of a string to make it produce Consonances and the Diatonic steps. 20. The "Diatonic" scale is the standard seven-note scale (like the white keys on a piano).
VIII. To explain the intervals, both Consonances and Dissonances, found in the remainders of the Monochord string after marking the Diatonic steps. 21.
IX. To explain all the Consonances and Dissonances of the Monochord and the Perfect System, whether comparing the whole string to the parts forming the Diatonic, Chromatic, and Enharmonic steps, or comparing each step or sound with the entire string or its remainder. In this way, the Monochord and the Harmonic System are considered here in every way that can serve Harmony. 22.
X. To divide all sorts of strings, or straight lines, into as many equal parts as one wishes, without changing the opening of the compass by chance?. 25. See also the 17th proposition of the 4th book which follows.
XI. To determine the number of Aspects by which the stars look upon the earth, and the Consonances to which they correspond. 27. In the 17th century, "Aspects" referred to the angular distances between planets in astrology; Mersenne is connecting the "harmony" of the heavens to the "harmony" of music.
XII. To explain the figure of a particular Monochord and all its divisions. 32.
XIII. To explain the difference and distance from one Consonance or Dissonance to another by means of the Monochord; and the manner of dividing the same string half by half to make all sorts of Consonances and Dissonances. 35.
XIV. To explain another Monochord of equality, for dividing the neck of the Lute, Viol, Cittern, and all other instruments played in 12 equal semi-tones, and for making the Diapason and the tuning of Spinets and organs. 37. "Monochord of equality" refers to what we now call Equal Temperament, where the octave is divided into twelve identical steps. "Diapason" here refers to the standard pitch or the range of an octave. See the 6th and 7th propositions of the 2nd book, and the 9th of the 4th book following.
XV. To determine how much the intervals of this Monochord of equality are smaller or larger than those of the Monochord following just proportions: and if the ear can perceive the differences. 39. "Just proportions" (Just Intonation) is a tuning system based on whole-number ratios, which sounds purer but makes it difficult to play in different keys.
XVI. What is the force of all sorts of strings, of whatever length