# Extended scales

The notation of melodies that use a definite scale has been introduced using examples with a fairly small number of scale degrees of which the tonic is the lowest (see Scales and melody). In much melodic music, however, the tonic is not the lowest pitch, and the pitch range may extend over more than one octave. We will deal with each of these situations in turn.

An example of a tune in which the tonic is not the lowest pitch is the familiar “Happy Birthday” song. Here the tonic is in the middle of the pitch range: the conclusive-sounding final note is neither the lowest nor the highest in the song.

In this situation, pitch lines can be drawn below the tonic in the same way that they are drawn above it, spaced in proportion to the intervals between the scale degrees they represent. The only new question is what cents values to write at the beginning of these lower pitch lines. They could be given as negative values indicating intervals below (rather than above) the tonic; but a graph with both positive and negative values might make it harder to perceive the pitch relationships among the various scale degrees. Global notation therefore offers a different solution.

Remembering that pitches an octave apart are treated as “equivalent” in most musical cultures, a pitch one octave below the tonic pitch that appears in the melody would also be the tonic. Even if this pitch does not actually occur in the music, we can treat it as a second reference pitch and measure the pitches of the lower scale degrees relative to it. Since an octave is equal to 1200 cents, if there is a scale degree (say) 200 cents below the tonic, then the interval between it and this (imaginary) lower tonic is 1200 – 200 = 1000 cents; so the figure 1000 is written at the beginning of the relevant pitch line.

In this way, the relative pitch for each scale degree is stated as an interval in cents above the next lower tonic pitch, even if that pitch does not actually occur in the music. (This song also illustrates another common situation: a melody that does not begin on the first beat of a bar. The score could begin where the melody does, on the last beat of the bar, but the meter is more easily perceived when the score begins with a full bar, as here.)

The advantage of specifying all scale degrees with positive cents values now becomes apparent: since the lowest and the highest pitch lines have the same cents value, it is clear that they are an octave apart. Stated as a general principle: Any two pitch lines with the same cents value represent pitches one or more octaves apart—or, as music theorists call them, pitches of the same “class.”

In melodies that extend over a wider range of pitch, pitches of the same class usually occur in more than one octave. For example, in Harold Arlen and Yip Harburg’s song “Over the Rainbow,” the tonic pitch and the two scale degrees below it (at 1100 and 900 cents) all occur in two different octave positions. Global notation handles this by simply adding more pitch lines, some of which will have the same cents values as others (namely those which represent the same pitch class). When the tonic pitch occurrs in more than one octave, its pitch line is drawn the same way—bold and extended—in each octave. (Notice how the melody line gives an intuitive picture of the shape of each phrase and shows how the whole melody is built out of the two ideas in the first and second bars.)

Clearly, if the music ranges very widely in pitch, we could finish up with an awful lot of pitch lines, which might be difficult to read. Next we’ll look at some ways of avoiding that.

Next: Wide pitch range