er re 
On Perfect Musical Intonation. 75 
13. A Scale is a series of sounds, obtained from the above har- 
monic relations, arranged in the order of their acuteness, and it 
contains the notes required for the melodies and harmonies of 
the composition for which it is used. 
14. The piaronic scaLe is composed of seven distinct notes, 
(the eighth, being the octave, is regarded as a repetition of the 
first.) It is formed by combining the chords of perfect fifth, 
(2:3,) and major third, (4:5,) and it contains all the intervals 
and chords which have been named, with the exception of those 
derived from the perfect seventh. Assuming C as a key-note, 
this scale, in the vibrations of its several notes, stands as follows: 
C D E FW G 
Key-note, Second. Third. Fourth. Fifth. Sixth, Seventh. Octave. 
js Se 8: 230 Soe: | a6 2 40.» 45 : @& 
Mpj. T. Min. T. s. Maj. T. Min.T. Maj. T. Ss. 
15. On examining the relations of these numbers, we find 
the intervals which separate the several notes of the scale. The 
ratio of 24:27 or 8:9, gives the interval between the first and 
second notes, which is called major tone. The ratio of 27:30 
or 9:10, gives the interval between the second and third notes, 
which is called minor tone. The interval by which the major 
tone exceeds the minor is called comma, whose ratio is 80:81. 
The ratio of 30:32 or 15: 16, expresses the interval between the 
third and fourth of the scale, and it is called diatonic semitone or 
simply semitone, diatonic being understood. From the fourth to 
the fifth, and from the sixth to the seventh, is the same as from 
the key-note to the second, i. e., major tone. From the fifth to 
the sixth is the same as from the second to the third, i. e., minor 
tone, and from the seventh to the eighth is the same as from the 
third to the fourth, i. e., diatonic semitone. 
_ 16. Besides the diatonic semitone, there are nsed in composi- 
ion two others, resulting from taking the diatonic semitone from 
either tone. If it be subtracted from the major tone, it leaves 
What is called the chromatic semitone, and if taken from the minor 
one, there remains the grave chromatic semitone, comma less 
than the chromatic. 
17. If from any note, as key-note, there be taken a perfect fifth 
and major third, the three notes sounded together will produce a 
common chord. These notes are, in their vibrations, as these num- 
bers, viz.: 4:5:6. If to this, the perfect seventh (4:7) be added, 
we have the chord of the seventh, expressed thus, 4:5:6:7. 
Adding the octave, the chord becomes 4:5:6:7:8. To this 
may be added the major ninth (4:9), producing this chord, viz. : 
4:5:6:7:8:9. And finally, by adding the tenth, (or octave and 
Major third ), we obtain this chord, viz.: 4:5:6:7:8:9: 10. 
ding, below the key-note, the double-octave, its octave and 
twelfth, we have what may be called the full chord of the tenth, 
