states having failed to produce the needed funds, Congress in February, 1781, begged the states for authority to levy a duty of five per cent. ad valorem on all goods, wares and merchandise imported after May 1st of that year, and a like duty of prizes and prize goods condemned in any of the state Courts of Admiralty, and use the money to pay the debts already contracted on the faith of the United States. Nine states consented, one recalled its assent, two refused to take any action and, as the consent of all was necessary, the power was not given.

After the peace of 1783 Congress submitted to the states three amendments to the Articles of Confederation. One granted authority to lay specific duties on tea, coffee, cocoa, sugar, molasses, pepper, and liquors imported, and levy five per cent. ad valorem on all other articles brought in from abroad, the money to be used to pay the debts. Another pledged the states to raise by taxation $1,500,000 each year to meet the cost of government. The third granted power to regulate trade with foreign countries. Again the states refused and the debts remained unpaid until the first Congress assembled under the Constitution. In that instrument are embodied many results of the disastrous financial experience of the Continental Congress. The states are forbidden to issue Bills of Credit, or make anything but gold and silver legal tender for debt; the Congress is forbidden to issue Bills of Credit, and is given power to levy taxes, duties, imposts and excises to pay the debts and provide for the common defense and general welfare. Then at last provision was made. Loan office certificates, lottery prize certificates, issued by the loan offices, quartermaster, commissary, hospital, marine certificates, final settlements with the soldiers, were funded at their face value in interest bearing stock; but the Continental currency then out was redeemed at one cent on the dollar.

THE MUSICAL INTERVAL

BY HAROLD C. BARKER

Assistant Professor of Physics

I have here a group of eight small pieces of wood, alike in length and breadth, different in thickness. Dropping one of them upon the table, a sound is produced, a noise, as all hearing it would probably say, of a character quite familiar and fully anticipated. Dropping another, the effect is similar. But now I will drop all of them in succession, in the order of their thickness. At once you recognize the major scale in the series of sounds, which by reason of their association now seem quite musical.

The experiment shows to what notable extent our ears will tolerate an intrinsic quality of sound that is unsatisfactory or even unpleasant, if only we detect among the sounds the familiar and pleasing musical intervals. I have it in mind to spend this hour in briefly indicating the physical character of these musical relationships among tones, and to suggest how they may be made matters for definite measurement.

The names of some of the intervals are familiar in literature; octaves, fifths, major and minor thirds and even less well known intervals are named in non-technical prose and in poetry. Browning, in one short poem, mentions four different intervals.

The dictionary will tell us that a musical interval is the difference in pitch between two tones, whether sounded successively, or simultaneously.

Now sounds always proceed from vibrating bodies, and the physical characteristic of the vibration to which the pitch of the sound sensation corresponds is its frequency. A particular pitch means a definite frequency. This tuning fork for example makes 256 complete vibrations in one second-we say, its frequency is 256 vibrations per second-and its pitch may be denoted by the phrase "scientific middle C."

Now here is another whose frequency is 512 vibrations per second-just twice as great-and its pitch is just one octave higher, as we can all recognize upon hearing, unless unfortunately tone deaf, or unfamiliar with the elements of music.

The physicist, moreover, has found that any note and its octave correspond to frequencies in the ratio of one to two regardless of the absolute magnitude of frequency. In other words the musical interval called the octave a certain difference in pitch-is represented physically, not by a definite difference in frequency, but by a definite ratio of frequencies.

And so of all other musical intervals. These four forks have frequencies 256, 320, 384, 512 vibrations per second, numbers in the ratios 1:::2, and, touching them with the bow, we have plainly the intervals of a major chord; the major third, the perfect fifth and the octave. Were we to begin with any other frequency than 256, we must still preserve the same ratios if we would build up a major chord.

What I have said implies that we have means of measuring frequency. How can we say for instance that this fork makes 256 vibrations per second?

For the human ear, audible sounds have a range of frequency from say sixteen vibrations per second to possibly 40,000 vibrations per second, so that it is evident that we cannot directly count vibrations with watch in hand, and thus determine frequency. Indirect methods must be used, some of which I shall now describe briefly.

By attaching a stylus to a vibrating body, as for example a fork, and drawing it over a glass surface coated with lamp black or chalk, easily removed by the point of the stylus, we can oblige the fork to write, as it were, its chronology. A wave trace is obtained, and by simply counting the number of waves in the distance traversed in a known time, the frequency will be known, inasmuch as there will be one complete wave for each complete vibration of the fork.

To illustrate this method rather crudely I have here a falling fork apparatus. The rate of fall can be regulated by the counterweight, and the time corresponding to any given part of the

fall can be determined. Releasing the trigger, the fork falls with its carriage, and we have a faithful wave trace upon the chalk coated glass.

A second method makes use of the toothed wheel of Savart. Here is a Savart wheel, a vertical cylinder studded with horizontal rings of projecting teeth placed at equal intervals around the circumference. The cylinder is rotated, and a card so held that each tooth in turn engages and releases the card-that is, snaps the card-as it passes. For each revolution the card will snap as many times as there are teeth in the particular ring, and if this number be multiplied by the number of revolutions per second, the product is the frequency of the note emitted. For example, there are forty-eight teeth in the uppermost ring of this instrument, and if it be rotated at ten revolutions per second, the frequency is 480 vibrations per second.

This particular Savart wheel has been arranged to give the successive tones of a major scale through an entire octave, when rotated at a constant speed.

Now, to determine an unknown frequency, it is necessary only to produce a sound of the same pitch, as determined by the ear, with the wheel.

The principle involved in the use of Seebeck's siren is similar. A disk, perforated with holes placed at equal distances around a circumference, is rotated, and a blast of air directed against the face of the disk from a nozzle (or nozzles). As each hole in the series reaches the position of the nozzle, a puff of air passes through, and, at a proper speed, these successive impulses produce an audible tone whose pitch depends upon the rate of rotation and the number of holes around the circumference. As with the Savart wheel it is necessary only to produce a tone in unison with that produced by the vibrating body whose frequency is to be determined. The number of holes is known, the rate of rotation, in revolutions per second, is noted, and the product of the two numbers is the required frequency.

These briefly described methods will suffice to show that we are in position to establish standards of pitch or frequency. Tuning forks, such as I have here, made by the famous Koenig,

These

of Paris, are excellent examples of such standards. standards may be either fixed, that is, of invariable frequency, or adjustable, with a frequency variable at will between certain limits. With the help of such standards, unknown frequencies may be determined by comparison in a great variety of ways.

In theory, the simplest way would consist in finding by selection or adjustment a standard fork of the same pitch, as shown by the unison of the two sounds, or by a resonance effect. I happen to know that these two forks before me have the same pitch; if I did not, the unison of their respective tones would convince me, and if in addition, the vibration of the one will set up vibration in the other, as we see, the resonant effect proves equal frequency.

But it is not necessary to secure unison. Here are two forks, one a standard of 512 vibrations per second, the other slightly different. When sounded together, we hear a periodic variation of intensity, due to interference of the two trains of waves, called by musicians beating. If not too frequent, the beats may be counted, and it can very simply be shown that the difference in frequency of the two tones is represented by the number of beats per second. If the ear does not tell us which of the two frequencies is the greater, that is, which pitch is higher, a physical test must be made, by slightly loading the unknown fork, say with a bit of wax, thus decreasing its frequency. If originally below the standard, it will now be still more so, and the number of beats per second increased. If originally higher, a slight load will bring it nearer to unison with the standard, thus decreasing the number of beats per second.

Moreover, there is a beautiful optical method by which certain ratios of frequency may be recognized, and a comparison with a standard of quite different pitch effected, but I cannot take the time to enter upon its description.

To conclude this part of the subject I will show you this instrument called the Stern "Ton Variator." It is essentially a cylinder of variable length, with an arrangement by which