Notes on JI
More about just intonation.
If we take the notes in a major scale and tune each one so that it
makes a just interval relative to the key note (tonic), we get a JI
major scale. This is the scale that a melody singer should use in a
barbershop song, so that the song stays in key and the harmony singers
can more easily tune their parts to the melody. In the key of C, it
looks like this:
The numbers above the staff give the tuning of each note (in cents)
relative to the equal tempered note with the same name. The numbers
below the staff give the size of each interval ("step"). Note that
there are three different step sizes in a JI scale, not two. The "half
steps" are 112 cents, the "large whole steps" are 204 cents, and the
"small whole steps" are 182 cents. To hear this scale sung against a
drone on the tonic C, click here.
For comparison, here is
an equal-tempered scale. Notice the noisy quality of the third and
sixth, in particular.
The ratios used in the JI major scale are 1:1, 9:8, 5:4, 4:3, 3:2, 5:3,
15:8, and 2:1. Using only these pitches, we can form major triads
I, IV, and V (C, F, and G in the key of C) and minor triads vi and iii
(Am and Em) that are perfectly in tune, but the other triads (ii and
vii°)
sound out of tune. In particular, the perfect fifth that is
in the ii chord (Dm) sounds "perfectly" awful. Looking at
our scale, we see that D is +4 cents and A is -16 cents, which means
that the D-A interval is 20 cents flatter than an equal-tempered fifth,
when it needs to be 2 cents sharper.
To hear the progression C F Dm G C performed using the JI scale above,
click here. Notice
the harsh sound of the third chord, the Dm. To make it sound in tune,
the A (in the baritone part) should be raised by 22 cents. If
we do so, we get this
progression. Much sweeter! The notes with annotated pitch adjustments
relative to ET:
The third note in the baritone part does not lie in the JI scale.
To play this short passage perfectly in tune on a keyboard, you would
need two differently tuned A's. This is the reason you can't play
barbershop songs on a piano or other
keyboard instrument. Even if you re-tune your keyboard to the just
scale for a given key, (possible, but difficult), you will
still have chords that require tweaking to be in tune. Of course, if
you only use the I-vi-IV-V-I progression (and there are a lot of songs
that do!), you can actually make tuned chords with
a (specially tuned)
keyboard, so long as you don't modulate to another key!