Introduction to Pitch Bracket Notation

Pitch bracket notation is a new music notation that reveals the beauty and elegance of any melody in a simple system of lines and dots.

Music notation has been passed down to us through the centuries, having incorporated many innovations along the way. The most important of these innovations are the staff and metre. The vertical position of a note, relative to the lines and spaces of the staff, determines its pitch. The horizontal position of a note along with the note’s value (i.e.. 1/4, 1/8, 1/16), determines its timing. These two visual dimensions correspond to the musical dimensions of pitch and time. In a typical melody notes are played in sequence, one note beginning at the end of the previous note. So you might say that the time dimension of music notation is note-to-note relative. The pitch of a note, however, is not relative to the pitch of the previous note. Rather, it is relative only to the staff lines. Musicians learn to recognize common pitch intervals between notes but only by referencing the staff lines. So you might say that the pitch dimension is note-to-staff relative.

Consider for a moment what music notation would look like if the pitch dimension were note-to-note relative. Start by drawing a horizontal line, the melody line. Notes are drawn as dots on the melody line. Melody lines have a particular pitch associated with them just like the staff lines, but this pitch may be changed by drawing a pitch bracket. Figure 2 shows a melody line with the same nine notes from Figure 1. The pitch increases one scale step at a time, from C to G, and then back down to C. The melody line starts at the first scale step which is C in this case. Each parenthesis bracket raises or lowers the melody line by one scale step. The opening brackets add one step while the closing brackets subtract one step. The matching opening and closing brackets show the symmetry between rising and falling notes.

Just as the parenthesis bracket adds or subtracts one scale step, higher valued pitch brackets add or subtract multiple scale steps. The angle bracket adds/subtracts two steps and the square bracket adds/subtracts three steps. Again, the direction of the bracket, opening or closing, determines whether to add or subtract. The short names for these three brackets are b1, b2, and b3. The number indicates how many steps to add or subtract. As such, the b4, b5, and b6 brackets add/subtract 4, 5, and 6 steps. Their shapes are similar to b1, b2, b3 except they have a filled-in appearance. The final pitch bracket, b7, raises or lowers a melody line by an octave. Figure 3 shows all of the brackets.

Figure 3

The shape of the b2 bracket appears to have two “sides”, that is to say, the two lines that make up the shape. The b3 bracket has three “sides”. The b1 bracket has only one “side”, a single bent line. The b4, b5, and b6 brackets have similar shapes to the smaller brackets except they are “filled in”. Bracket pairs with similar shapes add to 7. For example, the b2 and b5 brackets are angle brackets and they add to 7. The octave bracket, b7, resembles a combined b3 and b4 bracket.

Figure 4

The three melody lines in Figure 4 each represent the melody, “C E G C”, which is a simple triad with the octave. The first melody line uses multiple b1 in sequence to add up to larger intervals; two b1 add to a b2, three b1 add to a b3. If this melody is rewritten in scale degrees, “1 3 5 8”, then the melody line can be understood as the following three equations.

1 + b1 + b1 = 3

3 + b1 + b1 = 5

5 + b1 + b1 + b1 = 8

The second melody line uses b2 and b3 brackets rather than multiple b1. These higher brackets simplify the melody line and visually distinguish the 3rd interval from the 4th interval. The equations for this line are:

1 + b2 = 3

3 + b2 = 5

5 + b3 = 8

The third melody line uses an unusual sequence of brackets. While not as obvious as the first or second melody lines, the notes still have the same pitch. The melody line can be interpreted as raising to the 5th, then dropping down a 3rd, and then later raising a 4th to the octave. The equations are then:

1 + b4 = 5

5 – b2 = 3

5 + b3 = 8

When speaking about intervals, it is important to note that the pitch bracket value is one less than the corresponding interval. For example, a b2 bracket changes the melody line by an interval of a 3rd. This is obvious when writing equations involving pitch brackets

1 + b2 = 3

Sharps and flats are used in music notation to raise or lower the pitch of a note by a half step, a semitone. A vertical slash through a note lowers it a semitone and a small cross through a note raises it a semitone. These modifiers resemble a minus sign and a plus sign. Figure 5 shows the semitones from C to G, using flats in the first line and sharps in the second.

An example of real music written in pitch bracket notation is now appropriate. Beethoven’s 5th symphony, probably the most famous piece of music in the history of music, can be instantly recognized by only the first four notes. The four notes in scale steps are “5 5 5 3”.

Figure 6

This theme is repeated in variation throughout the first movement of the symphony. The next four notes are “4 4 4 2”:

Figure 7

In both melody lines the first three short notes are followed by a long note a third below. The outer brackets show that the second melody is just the first melody lowered by a step. The following melody lines show the first section of this symphony. The minor scale is indicated by the “min” on the first line.

The next example is from Bach’s Cello Suite number 1. The melody is more complex here but the theme is again repeated in variation form. The first line in scale degrees is “1 5 (3 2 3) 5 (3) 5” where numbers in parenthesis are an octave higher. The next two lines are “1 6 (4 3 4) 6 (4) 6” and “1 7 (4 3 4) 7 (4) 7”. Several patterns can be recognized across lines. All but the first note tends to climb one step at a time (b4, b5, b6 and then b7). The fourth note tends to dip down a step from the surrounding two notes.

Pitch brackets operate on melody lines by adding or subtracting a number of scale steps. If arithmetic can be applied to pitch brackets, why not algebra? The three simple melody lines in Figure 10 are each represented by variables x, y and z. Adding an integer to a variable raises the whole melody line. Multiplying a variable by an integer multiplies each bracket value by that integer. Adding two variables combines their brackets at corresponding points along the melody line. Pitch bracket algebra is ideal for expressing melodies which are variations of a theme. In Figure 10, the fourth line is constructed by an algebraic expression including x, y, and z.

Figure 10

If we return to Beethoven’s 5th symphony in Figure 8, we can now apply pitch bracket algebra to reconstruct each line from the two, very basic variables, x and y. In fact, most lines only need the x variable. The two final lines are trivial and do not require any mathematics.

In traditional music notation, the staff lines are given a particular pitches. Melody lines, on the other hand, can change pitch to conform to the melody being played. Pitch brackets modify the pitch of the melody line by adding or subtracting scale steps. There are many ways to write the same melody with pitch brackets. As long as the notes fall on the melody line with the desired pitch then the melody line is correct.

The mathematical nature of this notation is clear when using the bracket names, b1 through b7, in simple arithmetic equations. By assigning melody lines to variables, algebra can be applied to entire melody lines. Then the melodic components that underlie a piece of music can be analyzed, compared with other melodic components, or even recycled to compose new music.

Pitch bracket notation does not yet attempt to represent rhythm or timing. The author has explored many possible solutions but at this time none are satisfactory. Another problem which will need to be addressed is polyphonic music. Music with multiple monophonic lines in parallel (i.e. symphonic music) can be represented with multiple parallel melody lines. But for complex polyphonic music such as piano music, a new solution will be necessary.

Even with these omissions, pitch bracket notation has introduced a powerful new way to communicate and think about music. Analogies can be drawn in many fields including linear algebra, electrical circuits, topology, and the visual arts. I hope to explore many of these analogies on this website as well as providing many examples.

Cory Gledhill