| F | 139 | 32 | 130 || 74760 | 321,03 || 30,9 | 11,9 | 57,8 | +-----+------+------+-----++-------+----------++-------+------+------+ | E♯ | 154 | | 78 || 76874 | 312,20 || 33,2 | | 33,5 | +-----+------+------+-----++-------+----------++-------+------+------+ | E | 149 | 74 | 110 || 80085 | 299,68 || 30,8 | 25,2 | 45,3 | +-----+------+------+-----++-------+----------++-------+------+------+ | E♭ | 110 | 13 | 54 || 83608 | 287,05 || 21,7 | 4,1 | 21,5 | +-----+------+------+-----++-------+----------++-------+------+------+ | D♯ | 154 | 53 | 78 || 85868 | 279,50 || 29,6 | 17,0 | 30,0 | +-----+------+------+-----++-------+----------++-------+------+------+ | D | 144 | 61 | 112 || 89480 | 268,21 || 26,5 | 18,5 | 41,1 | +-----+------+------+-----++-------+----------++-------+------+------+ | D♭ | 180 | 50 | || 93342 | 257,12 || 32,0 | 14,8 | | +-----+------+------+-----++-------+----------++-------+------+------+ | C♯ | 156 | 78 | 82 || 95920 | 250,20 || 26,6 | 22,0 | 28,0 | +-----+------+------+-----++-------+----------++-------+------+------+ | C | 156 | 46 | 143 ||100000 | 240,00 || 25,8 | 12,8 | 47,5 | +-----+------+------+-----++-------+----------++-------+------+------+ PROPOSITION VIII. To compare the harmoniousness of the foregoing system with that of several others, which have been most known and approved. The aggregate of dissonance, heard in any tempered concord, is as its temperament (Prop. I.) when its frequency of occurrence is given, and as its frequency of occurrence, when its temperament is given: hence, universally, it is as the product of both. The whole amount of dissonance heard in all the concords of the same name must consequently be as the sum of the products of the numbers denoting their temperaments, each into the number in Table IV. denoting its frequency. These products, for the scale of Huygens which divides the octave into 31 equal parts, of which the tone is 5 and the semi-tone 3; for the system of mean tones, and for Dr. Smith's system of equal harmony, compared with the scale of the last proposition, (cutting off the three right-hand figures) stand as follows: TABLE VIII. +--------------------+---------+------------+-----------+----------+ | Systems. |Huygen's.|Dr. Smith's.|Mean Tones.|New Scale.| +--------------------+---------+------------+-----------+----------+ | Dissonance { Vths | 825 | 945 | 850 | 786 | | of the { IIIds | 121 | 382 | 0 | 240 | | { 3ds | 1049 | 629 | 944 | 683 | +--------------------+---------+------------+-----------+----------+ | Total | 1995 | 1956 | 1794 | 1709 | +--------------------+---------+------------+-----------+----------+