Universal Harmony (Harmonie Universelle)

string, that is to say, of the first division of the Fifth, for the ratio of the former is five to four The ratio 5:4 defines the Major Third., and of the latter six to five. 75. This sentence completes a thought from the previous page regarding the mathematical division of musical intervals.

XXIX. To determine if the two preceding Thirds The Major and Minor Thirds. are Consonances, and by how much the major is sweeter than the minor. 76.

XXX. To determine if the Thirds and their Octave-repetitions term: Repliques (The same musical interval played in a higher or lower register, such as a tenth being a repetition of a third). are sweeter than the Fourth and its repetitions. 76.

XXXI. To determine if the two Sixths, of which the major is five to three and the minor is eight to five, are Consonances. 78.

XXXII. To explain how much the preceding Hexachords term: Hexachordes (In this context, Mersenne refers to the intervals of a Sixth). are more or less pleasing than the Thirds. 79.
Corollary. Why the Fourth is not as good against the Bass as the Thirds or the Sixths. 81.

XXXIII. Why there are only seven or eight simple Consonances. 82. See the moral reflections. Mersenne frequently relates mathematical truths to "moralities" or spiritual lessons to show the harmony between nature and the soul.

XXXIV. To determine in how many ways each Consonance and ratio can be divided: how the Arithmetic, Harmonic, and Geometric means are found, and what their differences and properties are. 90.

XXXV. To provide all the Arithmetic and Harmonic divisions of all the Consonances that exist within the span of four Octaves, which comprise the twenty-ninth [key] of the Keyboard of Spinets term: Epinettes (A small, rectangular keyboard instrument of the harpsichord family).; and all the ways of composing for three, four, or several other parts, which are used on each syllable. 93.

XXXVI. To demonstrate that the sweetest and best division of Consonances is not Harmonic, as has been believed until now, but Arithmetic: and that this division is the cause of the sweetness of the said divisions. 97.

XXXVII. Given two or more divisions of a Consonance, to determine how much one is sweeter than the other; and what is the best division of each Consonance, if one considers all the ratios it can sustain according to the laws of Music. 99.

XXXVIII. To explain what each Consonance supposes Implying what the ear or mind expects to hear following a specific interval to feel satisfied. above or below to produce good effects; that is, what presents itself to the imagination to perfectly satisfy the hearing when one plays a Consonance on an Instrument or performs it with voices. 102.

XXXIX. To explain through practical musical notes what has been shown through numbers; and the true reasons for the suppositions. 103.

XL. To give the radical terms The simplest mathematical ratios, such as 2:1 or 3:2. of the first hundred Consonances and the first fifty Dissonances. 106.

Propositions 14. of the book of Dissonances.

I. To determine if there are Dissonances, and if they are necessary to Music. 113.

II. To explain all the Semitones and the Diesis term: Dieses (A very small musical interval, smaller than a semitone, used in specialized tuning systems). which are used in Music considered in its greatest perfection. 114.

III. To explain the ratios of the simple Dissonances that serve Music. 118.

IV. Dissonances can be divided Arithmetically, Harmoni-