means of which a definite time-unit is established, called the beat or pulse. When one hums Yankee Doodle, he also find a pulsation forming groups, but in this case it is one loud and one soft. These tunes, then, represent two kinds of grouping — by two and by three — and such grouping is not peculiar to these tunes, for there are many others that fall into the same two classes, showing that it is possible to form ideas of time-units independently of any tune. Such effects we shall call ideas relating to pulse-grouping, the first of the three classes into which structural ideas will be divided.

If one hums America again, he will notice that, though the first three tones exactly coincide with the first three pulses, the fourth tone extends over the pulse and the fifth tone is shorter than the pulse, while in Yankee Doodle most of the tones are just twice as fast as the pulse. Here we have ideas of tone-duration, based on very simple yet exact arithmetical proportions. We can think of tones lasting two pulses or one pulse or of two tones in one pulse, without having to think of any definite tune. Thus we perceive ideas of tone-duration, forming the second class of structural ideas.

Turning again to America and Yankee Doodle, the tones are not arranged in a haphazard way, but the pitches selected are in certain definite relations to each other. This relationship is known as the key. We can think of a tone as being in a key, independently of any tune, thus forming the third class of structural ideas. The combination of these three classes of ideas in a musically logical way, expressed in a series of sounds, gives, not a general idea, but a specific, definite tune.

Turning to the characters that awaken the first class of ideas — tho'se of pulse-grouping — one finds at the beginning of every tune two figures arranged one above the other. The upper figure agrees with the pulse grouping: in America three; in Yankee Doodle, two. Taking the fourth tone in America and the fifth in Yankee Doodle, we find that both are preceded by a vertical bar, which indicates that these tones fall upon a strong pulse. Thus the vertical bar groups pulses into measures, and the figure at the beginning states how many pulses there are in each measure : whether it be one strong and two weak in America; or one strong and one weak in Yankee Doodle. It is obvious that, if the notes in these measures are to represent the time of just three or just two pulses, the relation of the note's length to the pulse must be defined. This is done by the lower figure at the commencement of the tune, the four showing that the time-duration used is to be represented by a

quarter-note; hence in America three quarter-notes or their equivalent will fill the measure and in Yankee Doodle two quarter-notes or their equivalent. In some tunes the figure two or the figure eight may be the lower one, showing that the pulse is either a half or an eighth-note. Pulse-grouping is further suggested to the eye by so arranging the notes in the measure both by spacing and by connecting lines as to suggest the pulse to which they belong.

Turning to the characters that awaken the second class of ideas — those of relative tone-duration — we find that this is accomplished through the shape of the notes themselves. Notes have heads, stems and a mark called the flag, often going from the stem of one note to the stem of another. A circular note without a stem represents a whole note; such a note with a stem, a half-note; a note with a black head and a stem, a quarter-note; when to such a note is added one flag or bar, it represents an eighth-note; and the addition of a flag or bar doubles the denomination. Most pieces of music require not tones only, but silences, to complete the design. In order to be able to mark the relative duration of these silences as they occur in the measures, each one of the above notes has a corresponding rest; the whole rest is an oblong black mark generally attached to but under the third line of the staff; the half-rest is a similar mark above the line; the 'quarter-rest is somewhat like an abbreviated two, or often like a seven turned the wrong way I ; while the eighth rest is like the figure seven; by adding more heads to this last stem, we represent the rests of the higher denominations. Besides these notes, marks are used, such as ties, combining notes or rests in one long tone equivalent to their united value. A dot may be placed after a note, adding half the value of the note it follows. A second dot may be added, adding half the value of the first dot, or a dot may be placed over the note with a curve over it, meaning that the note shall be prolonged an indefinite length of time, generally from two to four times its regular length.

The characters that suggest the third class of ideas — those dealing with differences in pitch — consist first of a series of five lines called the staff, with short lines added above and below as required, called leger-lines. By placing the heads of the notes on the lines and the spaces adjoining them, the relation of the pitches of these notes to each other is suggested to the eye. If the piece of music requires very low and very high tones, a number of staves may be placed one over the other and united by a brace at the beginning; or the figure