Proposition VII.

To determine the temperaments and beats of all the concords, together with the values of the diatonic and chromatic intervals, and the lengths and vibrations per second of a string producing all the sounds, of the system resulting from the last proposition.

The temperaments of all the concords are easily deduced from Table V. The Vth CG, for example, has its lower extremity lowered 12, and its upper extremity 14. Hence it is flatter by 2 than at first, and consequently its temperament=156. The temperaments of all the concords, thus calculated, will be found in the 2d, 3d, and 4th columns of Table VII.

Having ascertained the temperaments, the value of the diatonic and chromatic intervals may be found. The Vth CG being flattened 156, and the Vth FC 139, the major tone FG must be diminished 156 + 139, or be = 4820. By thus fixing the extent of one interval after another, from the temperaments of either of the different kinds of concords, as is most convenient, the intervals in question will be found to have the values exhibited in Table VI.

Let the numbers in this table be added successively, beginning at the bottom, to the log. of 240, the number of vibrations per second of the tenor C, (see Rees's Cyc. Art. Concert Pitch,) and the numbers corresponding to these logarithms will be the vibrations in a second, of a string sounding the several degrees of the scale. They are shown in col. 6, Table VII.

Since the length of a string cæteris paribus is inversely as its number of vibrations, the lengths in col. 5 may be deduced from the vibrations in col. 6; or more expeditiously, by subtracting the numerical distances from C of the several degrees in Table VI. from O, and taking the corresponding numbers, from the table of logarithms. These numbers, when used as logarithms, must be brought back to the decimal form, agreeably to Scholium 2. Prop. I.

To find the number of beats made in a second by any concord, it is only necessary to take from col. 5 the numbers belonging to the degrees which terminate that concord, and to multiply them crosswise into the terms of its perfect ratio. The difference of the products will be the number of beats made in a second. The 3 last columns contain the beats made by each of the concords, in 10 seconds.

TABLE VI.

C	———	———	———	C
	2998	2998	———	B
	2998	2998	1772
B	———	———	———	B
	1831		3033
B	———	4813
			———	A
	2982		1780
A	———	———	———	A
	1871		3030
A	———	4839
			———	G
	2968		1809
G	———	———	———	G
	1814		———	F
G	———		1798
		4820	———	F
	3006		1824
F	———	———	———	F
			———	E
	2988	2988
			1777
E	———	———	———	E
	1870		3028
E	———
		4818	———	D
	2948		1790
D	———	———	———	D
	1835		3018
D	———	4827
			———	C
	2992		1809
C	———	———	———	C

TABLE VII.

Bases	Temperaments of the			Lengths of String.	Vibrations in a Second.	Beats in 10 S. of the
Bases	Vths	IIIds	3ds	Lengths of String.	Vibrations in a Second.	Vths.	IIIds.	3ds.
B			77	51431	466,64			43,4
B	154	76	93	53574	447,98	47,4	39,0	57,8
B	147	35	97	55880	429,49	43,5	17,7	57,4
A	156		78	57448	417,77	45,1		46,2
A	153	71	107	59852	400,99	42,5	33,5	59,4
A	154	9		62487	384,08	40,4	4,0
G	151	76	75	64177	373,97	39,1	32,9	39,2
G	132	39	97	66907	358,71	32,9	16,3	48,1
F			101	68778	348,95			48,5
G		56		69760	344,03		21,9
F	154	76	83	71685	334,80	36,0	29,2	38,5
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