Universal Harmony, Volume I

[...continued from previous page] chord, that is to say, of the first division of the Fifth, for the ratio of the former is five to four, and of the latter six to five. 75. The "former" refers to the Major Third and the "latter" to the Minor Third.

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

XXX. To determine if Thirds and their replicates Intervals expanded by an octave, such as tenths. 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 eight to five, are Consonances. 78.

XXXII. To explain how much the preceding Hexachords Sixths are more or less agreeable 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 applications. Mersenne often draws analogies between mathematical ratios and spiritual or moral virtues.

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 are their differences and properties. 90. These are different mathematical ways to find a middle point between two numbers or frequencies.

XXXV. To provide all the Arithmetic and Harmonic divisions of all the Consonances within the range of four Octaves—which is the Twenty-ninth An interval of four octaves of the Spinet keyboard; and all the ways of composing for three, four, or several other parts, as used for 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 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, considering all the ratios it can sustain according to the laws of Music. 99.

XXXVIII. To explain what each Consonance assumes above or below to produce good effects; that is, what presents itself to the imagination to perfectly satisfy the ear when a Consonance is played on an Instrument or performed with voices. 102.

XXXIX. To explain through practical notes what has been shown by numbers; and the true reasons for these assumptions. 103.

XL. To give the radical terms The lowest possible whole numbers that express a ratio. for the first hundred Consonances and the first fifty Dissonances. 106.

14 Propositions from the Book on 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 A very small musical interval, smaller than a semitone. 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-