Notes on JI
What is Just Intonation? I'd like to explain why the notes we sing
aren't on the piano keyboard.
Let's
start with the basics. Sound is simply vibrations, normally transmitted
through air. A musical sound can be described as having a pitch,
volume, duration, and timbre (or "quality"). Let's look at each aspect
in turn.
- pitch
- The pitch of a musical sound is determined by the frequency
of the vibration. A steady pitch implies a constant frequency.
- volume
- The volume is determined by the amplitude of the vibration.
- duration
- The duration is simply the length of time that the sound is
audible.
- timbre
- The
timbre is what distinguishes the sound of a violin from that of a
trumpet,
for example. It is determined by the relative volume of the sound's
overtones (see below).
Sound is a longitudinal
compression wave, which means that the medium (usually air) undergoes
small variations in pressure that propagate through the medium from the
source to the detector (in this case, your ear). As the pressure
changes impinge on your eardrum, the eardrum vibrates at the same rate,
which causes the small bones in the inner ear to vibrate, and
ultimately causes the fluid in the cochlea to vibrate. Nerve cells in
the cochlea fire in response to the vibrating fluid, and the brain
interprets the nerve impulses as sound.
Pitch is measured in "cycles per second," or "hertz" (abbreviated
"Hz"). A sound with a pitch of middle C has a frequency of 261.626
Hz. The A above middle C is exactly 440 Hz and is the note usually
sounded when an orchestra tunes up.
Pitch
perception is logarithmic, not linear. That means that the perceived
distance between two notes depends on the ratio of their frequencies.
The octave corresponds to a ratio of 2:1 -- the higher note of an
octave interval has exactly twice the frequency of the lower one. Two
octaves corresponds to a ratio of 4:1; three octaves would be 8:1, and
so forth.
The Harmonic Series
When any physical object vibrates, such as a violin string or the air
inside a trumpet, the sound produced is actually a combination of
frequencies, not one single frequency. These frequencies are all
integer multiples of the lowest frequency (called the "fundamental"),
and are referred to as "overtones" or "harmonics." The two systems use
different numbering schemes, such that the fundamental is called the
first harmonic, the first overtone is the same as the second harmonic,
and so forth. The relative amplitude or volume of each harmonic is the
primary component of timbre. A pure sine wave has no overtones.
Here is a frequency analysis of the vowel sound "oo":
Notice the equal spacing in this linear frequency plot. The
fundamental is 220 Hz (A below middle C), the first overtone is at
twice that, 440 Hz (A above middle C), the second overtone is at three
times the fundamental, 660 Hz (E an octave and a third above middle C),
and so on.
Compare the above spectrum to the one below, for the vowel "ee":
Although many of the same frequencies are present, notice that the
relative amplitude of the various overtones are different. Indeed, that
is how we can tell that the sound is "ee" instead of "oo".
Although the overtones are equally spaced in terms of frequency,
because pitch perception is logarithmic, the space between successive
overtones becomes smaller in terms of scale degree as you get
higher in the series. The space between the fundamental and the first
overtone is one octave; from the first to the second is a perfect
fifth; from the second to the third is a perfect fourth; from the third
to the fourth is a major third; from the fourth to the fifth is a minor
third; from the fifth to the sixth is a "sub-minor" third, and so
forth. Here is a grand staff with the first 16 harmonics of low C (note
values are approximate; the numbers above the staff show the actual
cents sharp or flat for the just pitch relative to the corresponding equal-tempered pitch):
The term "cents" in this context means "hundredths of a half-step."
One octave is 1200 cents. A perfect fifth is 702 cents (approximately),
or 2 cents larger than 7 E.T. half-steps. A just major third is
386 cents, or 14 cents smaller than 4 E.T. half-steps, etc. The only
notes in the harmonic series which fall exactly on an E.T. pitch are
the fundamental and its octaves; no other pitch in the harmonic series
exactly matches any key on the keyboard.
What does the harmonic
series have to do with just intonation? Just intonation uses
intervals from the harmonic series, which correspond to small integer
ratios, to produce chords that are perfectly in tune. We call this
condition "lock." Equal-tempered instruments cannot achieve perfect
lock (except for unison and octaves, of course), but fretless
stringed instruments and trombones can and do, just like the human
voice. A cappella singers naturally tune chords using just intonation.
That's what makes a good quartet sound so good.
The barbershop
style uses a certain chord "vocabulary"; some chords are simply not
found in a true barbershop arrangement. The chords which are normally
used in the barbershop style have simple structures, and small
whole-number frequency ratios, which tend to enable the above-mentioned
"lock." I have attempted to enumerate the chords and intervals which
are "allowed" in the barbershop style below.
The
table contains 19 chord types, with the notes indicated relative to a
"C" root. To find the frequency ratios for a just-tuned chord
of any given type, read across the row. For example, in row 2, our
favorite chord, the "barbershop 7" (dominant seventh, indicated here as
simply "7"), spelled C E G Bb and made up of a root, major 3rd, perfect
5th, and flat 7th, has a frequency ratio of 4:5:6:7. That
means that the interval between the 3rd and the root is 5:4, between
the 5th and the root is 6:4 (or 3:2), and between the 7th and the root
is 7:4. The other intervals are 5th to 3rd (6:5), 7th to 5th (7:6),
and 7th to 3rd (7:5).
Note that the table contains two different
columns labeled "Eb" and two columns labeled "Bb". There are two just
intervals which are notated as a "minor 3rd," but they are tuned
differently -- the 7/6 "flat minor third" and the 6/5 plain "minor
3rd." Likewise, two different just intervals are notated as "minor 7th"
-- the "flat minor 7th" (or "barbershop 7th") and the plain "minor
7th." The point is, you can't tell how to tune a given note by looking
at it in isolation, even knowing what the root of the chord is and the
interval between the given note and the root.. There are two pairs of
different intervals that look the same when you look at only the note
and its relationship to the root of the chord. You have to also know
whether this "minor 3rd" is in a chord that's in the "flat m3" family
or "plain m3" family, for example. Fortunately, we don't have to
analyze every chord in order to sing in tune; our ear will tell us
where to put the note, if we'll just let it!
Go on to part 2
