has acquired the name of "key-note.") But, with a very small extension of musical desires, we find that other notes are required. Having sounded any note, perhaps we desire to associate with it the Third above it. 5 We must multiply the fraction for the Third, or, by the fraction for the note. This, applied to the Third 5 5 25 5 5 25 and Sixth, gives x or and X or in which 16' 4 3 12 the numbers are too large (Art. 87); applied to the Fourth, not excessively large, and which falls well between the Sixth and the Octave. This proportion 15 8 is therefore adopted as Seventh, with the letter b. If we desire to 4 associate with any note its Fourth, whose factor is; ap 4. 16 plied to the Fourth or it gives (which we may con 9 3 sider as a "flat Seventh"); applied to the Fifth or it 2 (which when we have found a note preferable for adoption as Second we may consider as a "flat Second"). If we desire to associate with any note its Fifth; the appli 9 cation to the Fifth gives, or (as referred to the Octave) 9 8' 4 This falls well between c and e; and the note which is an octave below it is adopted in the same place in the first Octave as Second, with the letter D. The 5 Fifth applied to the Sixth gives, or (referred to the 5 Octave) which is the Third or e. Thus we find that 4 9 15 only two new notes, namely and are to be inserted in our series; and it now stands thus : The reason for the term Octave is now obvious, The scale which we have thus obtained is called the "Major Scale," or sometimes the "Diatonic Scale." It is universally recognized as the foundation of Music. 6 8 Sometimes the Minor Third and Minor Sixth, and , are substituted for the Major Third and Major Sixth, producing the "Minor Scale." These two new notes, though well connected together, are not well related to the other notes; and they produce a partially discordant music, of peculiar character, usually melancholy. 93**. Systems of application of Logarithms to the expression of musical intervals. Professor Pole, in an essay attached to Sir F. A. Gore Ouseley's Treatise on Harmony, has given the logarithms of the proportions of vibrations of different notes to those of c. Those for the simple scale above are D E G C b 00000 05115 09691 12494 17609 22185 27300 ·30103 These numbers possess the convenience of being connected with the ordinary system of logarithms, but they do not offer facility for extension. We are permitted by Sir John Herschel to explain a system proposed by him which possesses that advantage. It consists in using such a modulus that the logarithm of 2 is 1000. Thus the logarithms of the proportions of the vibrations to those of c are C D E F G 8 b d e, &c. 0 170 322 415 585 737 907 1000 1170 1322, &c. It is seen here that, with the exception of the figure representing a multiple of 1000, the number correspond ing to each nominal letter is the same in every octave; and that, in successive octaves, the numbers increase successively by 1000. This is probably the most convenient logarithmic scale (assuming the octave-interval as the fundamental interval for music) that can be devised. 94. Remarks on the intervals between successive notes; extension of the Scale; appropriation of numbers of vibrations and of lengths of waves to the different notes. If we divide the number for each note by the number for the next preceding note, we find the following series of proportions: Considering the fractions attached to 1 as measuring the intervals of the notes, it is seen that there are two small equal intervals (E to F, and b to c); and five large intervals, nearly equal. Although the large intervals are not double of the small ones, yet in common ich schwebende imperator language the larger intervals are called tones and the 12 scale, would give for the successive notes of the diatonic scale с D E F G & b с d e, &c. 0 167 333 417 583 750 917 1000 1167 1333, &c. We imagine that this system would fail at every critical point of harmony. In order to remove all denominators of fractions, and to give to each of the numbers which are associated with the notes a magnitude that represents a physical truth, we shall multiply all the numbers of Article 93* by 480. The number thus produced for c is 480. The received number of vibrations in a second of time for the counter-tenor C (see Article 85) is 528 = 11 10 × 480. Therefore, each of the numbers which we shall now exhibit represents the number of vibrations of air made 10 of a second of time, corresponding to the note 11 to which that number is attached. We shall take this opportunity of adding another system of numerical elements corresponding to the different notes. In Article 30 we have shewn that |