Equal-Tempered Scales

with Few Notes

A presentation by John M. Sullivan
at the Bridges conference, 2000

We will start with the assumption that most people perceive musical notes as occuring in pitch classes, which differ by octaves (or factors of 2 in frequency).

The simplest music consists of just a single note, or perhaps unison octaves. Maybe the most interesting example is the One-Note Samba (Samba de uma Nota Só) of Tom Jobim.

Of course, this song really goes on to use two notes in the melody, notes separated by a perfect fifth, or a ratio of 3/2 in frequency. The Richard Strauss quote made famous by the Kubrick film 2001 also starts out using the same two notes.

A fifth is close to half an octave. If we start around the circle of fifths F-C-G, we're almost back where we started. Of course it stretches the imagination a bit to make this exact. If we make a two-note, equal tempered scale, the perfect fifth is approximated by a tritone. These examples really don't sound good.

Continuing around the circle of fifths, we find F-C-G-D-A-E and E is much closer to F than any of the previous notes have been. If we decide that E=F, then we are left with a five-note or pentatonic scale, FGACD. (Often pentatonic music on this scale is considered to be in major, minor or dorian mode if the pitch center is F, D or G, respectively.)

Many common melodies, in music found around the world, use the notes from a pentatonic scale. Examples include Camptown Races (by S. Foster) and Mary Had a Little Lamb (lyrics by S. Hale). We can hear them in a pentatonic scale tuned to have 4 perfect fifths.

If we instead try to switch to an equal-tempered pentatonic scale, the melodies again sound very different. Two-fifths of an octave is relatively close to a perfect fourth, but not at all close to the major third F-A found at one point in the true pentatonic scale.

However, if we play our earlier 2-note song using just two notes of this equal-tempered pentatonic scale, it sounds almost acceptable.

Continuing around the circle of fifths, we find F-C-G-D-A-E-B-F# and again, F# is closer to F than any of the previous notes. If we decide it is close enough to not count as different, then we end up with a seven-note or diatonic scale, CDEFGAB. Depending on the pitch center, we can have the usual major or minor scales, or any of the seven Renaissance modes (Ionian, Dorian, Phrygian, Lydian, Mixolydian, Aeolian, Locrian).

Almost all common melodies use the notes of this diatonic scale. We use Happy Birthday (without permission) as an well-known example that does include all seven notes. Again, we start with the Pythagorean tuning, where all six fifths are perfect.

Now let us try an equal-tempered 7-note scale. When playing Happy Birthday this seems to lose much of the sense of a pitch center, as there are no longer the characteristic whole steps and half steps, but just seven equal steps. The melody seems to have wrong notes as well as notes out of tune. However, this scale does a surprisingly good job of rendering our earlier pentatonic melodies: they are recognizably the right notes, though not quite in tune.

If we continue further, F-C-G-D-A-E-B-F#-C#-G#-D#-A#-E#, we find that E# is much much closer to F than any previous note. In fact it differs from F only by the so-called Pythagorean comma. If we take F=E#, then we get the common twelve-note chromatic scale. It of course can be tuned in many ways, including a Pythagorean tuning with 11 perfect fifths, or an equal-tempered tuning. We will not demonstrate any chromatic melodies, but close with all the previous examples in 12-tone equal-tempered tuning.

We have not discussed equal-tempered scales with 3, 4 or 6 notes because they do not give better approximations to the fifth than the 2-note scale does. These are, of course, subsets of the equal-tempered 12-note scale. The 3-note scale gives augmented triads, and the 4-note scale gives diminished seventh chords, as demonstrated in these arpeggios.

Similarly, no equal-tempered scale between 7 and 12 gives good approximations of fifths.

The MIDI files on this pages were sequenced with Rosegarden, a public domain sequencer running under Linux and Irix. Pitch bends for the different tunings were automatically inserted by Manuel Op de Coul's Scala software, which is designed for exploration of different scales and tunings.