Lead

The lead function maps input notes to output notes by adding accidentals, i.e. sharps or flats. Unlike the bass function which transposes by shifting the entire scale, the lead function transposes in place. For any given input note, the lead function generates an output note as close as possible to the input note in terms of pitch, usually within a half step. If a consistent melodic pattern is input, the output notes vary in response to the song's chord progression, but only by small amounts. The overall shape of the input is preserved, and this quality makes the lead function ideal for soloing. The lead function doesn't have any settings.

Numbers

The numbers function purposefully limits the number of scale tones in use at any one time, in order to facilitate melodious intervals and allow consistent management of dissonance. Input notes are mapped to a four-note subset of the current heptatonic scale, called a group. Seven different groups are available, and they gradate smoothly in their tension or distance from the tonal center. Each group has two variants, one for major chords and one for minor chords. The groups are summarized in the table below.

The triad plus the seventh (1, 3, 5, 7) are chord tones, while the remaining scale tones (2, 4, 6) are tensions. The groups can be viewed as tonal palettes ranging from mostly chord tones to mostly tensions. At one extreme, group 1 - 5 contains the triad, and therefore sounds consonant, warm, or red. At the other extreme, group 2 - 6 contains all three tensions, and therefore sounds dissonant, cool, or blue. The groups in between gradually add tensions, or gradually remove them.

The group setting simply selects one of the seven groups. As long as you continue to use the same group, you're maintaining a consistent degree of dissonance relative to the chord changes. Think of the group setting as a tension knob. By changing the group setting periodically you can vary the mood of your improvisation.

The following table shows the output of the numbers function for various major and minor chords. Only the first two groups are shown. Notice that the numbers function transposes in place, inverting the group as needed to maximize continuity between different chords. Also notice that the output spans nearly two octaves for every input octave; this asymmetry is discussed further below.

Numbers origin

This setting aligns the numbers mapping function with the diatonic scale. Alignment is necessary because the numbers system is tetratonic, meaning it has only four notes per octave, unlike the diatonic scale which has seven notes per octave. Each octave of the diatonic scale spans nearly two octaves of the numbers system, and consequently they diverge rapidly from each other in terms of pitch.

Alignment occurs at a single note, called the origin. By default, the numbers system is aligned to the diatonic scale at middle C (C4). The origin is specified as a positive or negative offset from middle C, in tetratonic degrees. Increasing the origin by one raises the pitch of the entire numbers system by one tetratonic degree, as the following table shows for group 1 - 5 and Amaj7.

Alignment matters most when switching from a heptatonic function such as Lead to the numbers system or vice versa. If you're playing near the origin when you switch, the transition is smooth, but if you're playing significantly above or below the origin, a jump in pitch occurs. If this jump is undesirable, you can avoid it by adjusting the numbers origin to suit the range of your instrument. You can also change the numbers origin during a performance as a kind of pitch-shifting effect.

Numbers theory

The numbers system was popularized by tenor saxophonist and music educator Jerry Bergonzi in his book "Inside Improvisation, Vol 1: Melodic Structures." The groups can be derived from a tetratonic pattern superimposed onto a heptatonic scale. The pattern consists of three adjacent scale tones followed by a skipped scale tone, as shown in the column heading of the table below. The groups are four-note subsets of this repeating pattern, arranged so that each group starts two scale tones beyond its predecessor.