Tonic (E) with its common chord, (C, E, G) and the Dominant (G), on which is taken the chord of seventh and ninth (G, B, D, F, A) in the ratios 4:5:6:7:9. To distinguish these scales, I have called the first the Triple Diatonic, and the last the Double Diatonic. Assuming the tonic or the key-note, as C, with 48 vibrations, the two scales will stand as follows:

(With common Chord on C: and Chord of 7 and 9 on G.)

THE FOLLOWING DIAGRAM EXHIBITS THE MONOCHORD, WITH THE DERIVATION OF THE TWO DIATONIC SCALES, AND THE PROPORTIONAL LENGTHS OF THEIR STRINGS OR PIPES, THE PRIMARY HAVING 10000 EQUAL PARTS.

It thus appears that the fourth and sixth notes may be taken differently in intonation; and that this is done, can be easily observed by giving attention to singers. The Triple Diatonic has but three

different intervals; namely, 8:9*, 9:10 and 15:16. The Double Diatonic has, in addition, two others; namely, 20:21*, and 7:8; and in combinations its variety is greatly superior to the Triple Scale, whose chords and intervals are rather duplicates of one

another.

And the remarkable fact is, that this Double Diatonic, which no theorist has defined, is more in practical use than the Triple, which stands in all the elementary books. A familiar example of the former is the "O dolce Concento" of MOZART, and the principal movement of the "Dead March in Saul" of HANDEL. The melody of the "Hundredth Psalm" is in the Triple Diatonic. The two scales often interchange, and an example of this is to be found near the close of "O dolce Concento," where for a single measure the dominant seventh and ninth yield, to admit the fourth and sixth of the Triple Scale.+

If it be feared that the distinctions which have been described as belonging to the scale will complicate it for those learning to sing or play, let it be added, that singers naturally observe them all, and need have no other instruction than to hear the sounds given by their teacher. What is here set down is of interest to him who wishes to know what is, and what ought to be, done. It may not be necessary for the singer to be even told the dimensions of any of his

* In view of the numerous names required, and to supply those needed for these unnamed intervals, I have proposed (at least for mathematical and theoretical uses) names derived from the ratio. A fifth then will be "two-three," a major-tone "eight-nine,” a diatonic semitone "fifteen-sixteen," the interval, (unnamed) between the third of the scale and the dominant seventh, is the "twenty twenty-one." The next interval (dom. seventh to fifth) is the “seven-eight." This proposition as yet needs the approval of other theorists. The desideratum is accuracy and clearness.

† Not to disfigure these mathematical pages with musical types, I have chosen examples with which every one is familiar. Every composition will furnish others. If a choice is to be made in the examples of the profuse employment of the prime seventh, there may be taken any of the vocal scores of HAYDN, MOZART, or ROSSINI.

intervals; and it perhaps does no harm (except to the one who utters the falsehood) to say that all intervals are compounded of "semitones" or artificial twelfths of the octave.

It is true that when all the four primes have furnished their numerous chords and intervals, we shall have assembled a large number of notes, and it is not impossible that those unacquainted with music may fear that the number will be unmanageable, and prefer the compromises and limitations of temperament. As the experiment has been practically made, such persons may be assured that the musician can most easily produce his desired effects, when he has the full and abundant materials which the system of just intonation gives him.

The singers and players upon the free instruments, of their own accord, use the true intervals to the best of their ability; and in spite of the tempered instruments with which they are sometimes obliged to join. It is for men of science to indicate to the makers of imperfect instruments the way to perfect them; and to withhold their approval from players, who, from indolence or incapacity, only make a pretence of interpreting the music of the great masters. There are wanted no more apologies for, or speculations upon, the choice of temperaments; that subject has long ago been exhausted; and nothing more can be done than is now done with twelve tempered notes in the octave. When some economical astronomer shall propose to reduce the bulk and expense of the Nautical Almanac, by sacrificing that accuracy which gives it priceless value, the men at Greenwich will regard him as the scientific musician will, at a future day, look on those who would restrict him to the meagre and barbarous systems of temperaments of twelve notes.

NOTES AND QUERIES.

1. Least Common Multiple. On page 396, Vol. I., it is asked, whether the rule for finding the least common multiple, which says, divide by any number, will give the same result as the rule which says, divide by any prime number. My answer is, not always. If we divide the given numbers by any prime number which will divide two or more of them without a remainder, we shall, by following the rule, always get their least common multiple; if we divide by any number which will divide two or more of them, and follow the rule, we shall get a common multiple, but not always the least. The reason is plain; but may be most clearly seen by means of an example.

Find the least common multiple of 30, 63, 66, 300.

The first solution gives a result three times too large, as it should. For when we divided by 6, the factor 3 common to all the numbers was taken out of only three of them. The least common multiple of several numbers is composed of all the factors not common to them, and of all the factors common to two or more of them, and of no others. Any common factor raised to the highest power to which it is found in either of the given numbers, must enter into the least common multiple, otherwise the supposed multiple would not be divisible by this number. On the other hand,

if any common factor, raised to a higher power than it is found in any of the given numbers, enters into the common multiple, then such multiple will not be the least, as is seen from the first solution; since 3 raised to the third power is found in the multiple 207900, which is one higher than it is found in either of the given numbers.-TEACHER.

2. On page 363, Vol. I., we find that 17 horses were devised to three sons, as follows: To the first, ; to the second,, and to the third,; and in the distribution by the Judge, the first got 9, the second 6, and the third 2. The question is, Was the distribution just?

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Now it is evident that the testator intended to give the whole of the 17 horses to his three sons; and this he thought he had done; when, in fact, according to the terms of the will, he had only disposed of ths of the horses, there being undisposed of. For, and only make ths. But the rule is, to interpret wills so as to effectuate the intention of the testator, when that can be ascertained. In this case, it is plain he intended to give them all the horses, and named the proportions they should have. All that is necessary, then, to effectuate his intention, would be to say, As the whole portion of the horses actually devised to his three sons by the letter of the will is to the whole number of horses intended to be devised to them all, so is the particular proportion actually devised to any of the sons to the number or portion intended to be devised to that particular son.

So that the distribution is proved to be just, without resorting to the Judge's horse for aid in the interpretation of the will.