| |
Tempered Tuning for Acoustic Guitar
By Kevin Ryan
I learned this method of
temper-tuning a guitar while I was a professional piano
tuner in Ohio about 24 years ago. This concept is used by
the best piano tuners. If you don’t care a farthing about
the theory and want to begin using the method, dispensing
with these arcane facts, skip to
"The Method".
Terms & Theory
Fundamental: A vibrating
guitar string produces a series of tones called the
fundamental and it's harmonics (also called
partials or overtones). The fundamental is the
lowest note--the pitch produced by the entire string length
in motion.
Harmonic: A vibrating string moves
in a complex but predictable fashion. In addition to the
entire string moving in its parabola (arc), the string can
be divided conceptually into halves. Each half of the
string also produces its own (almost independent) parabola
and, thus, a pitch called a partial. Since the length of
this part of the string is exactly half, it vibrates at
twice the speed of the whole string and thus produces a
pitch that is an octave higher. (E.g., 440=A; 880=A an
octave higher). Divide the string into 3 equal lengths (the
7th fret is 1/3 the length of the string!) and you have
another section of string with its own vibrating pattern and
pitch (i.e., another partial). Divide the string into four
equal lengths (at--you guessed it--the 5th fret) and you
have yet another partial. These partials are always
there, but when you just touch the string at these
frets without actually fretting the string, you dampen the
fundamental and isolate the partial at that fret,
giving the illusion that you have "produced" the harmonic.
Pluck an open string and listen for the partials along with
the fundamental.
Inherent Inharmonicity: This is the
phenomenon that necessitates tempered tuning in a stringed
instrument. As we saw above, the fundamental of a string
produces a tone along with, say, the 1st partial of that
string (harmonic at the 12th fret). Let us assume the note
is A (oscillating at 440 cycles). In theory, the 1st
partial will be oscillating at 880 cycles per second.
But it is not! It oscillates slightly faster-- i.e.,
sharper. Why? The answer lies in the physical properties
of a vibrating string. The proportion of stiffness to
diameter in a half-section of a string is greater than
the proportion in the whole string. And the shorter the
section, the greater the difference in the proportion. In
any string length (whether fretted or open), the partials
are all progressively sharper than the fundamental. This is
Inherent Inharmonicity. Gosh, this all happens when you
play that note? You bet it does!
Mathematical Intervals Adding to the
problem of properly tuning a guitar is this quirk concerning
intervals of mathematical purity. You see, each given chord
wants intervals of mathematical purity for that
particular chord! The mathematically pure intervals of
a "C" chord, for example, are different that for a "D"
chord. The differences are slight but they are real. That
is why when you tune your guitar to "D" while fretting a
"D" chord, the "D" will sound great but the other chords
will sound slightly wrong. (This has probably driven you to
distraction--now you know why!)
Compensation When a string is fretted, the
string is slightly stretched as it is pushed down onto the
fret. This, of course, minimally sharps the string. To
compensate for this slight sharping during fretting and
to compensate for some of the Inherent Inharmonicity, the
bridge and saddle are traditionally moved away from the nut
by a carefully calculated distance (on the Ryan Mission,
this amount is .140"). But this cannot fully compensate for
all of the above hurdles to excellent intonation. For that,
we need Tempered Tuning.
Tempered Tuning This is a method
of tuning that addresses all of the above factors. In
essence, this method takes the inharmonicity of all six
strings and the slight mathematical discrepancy
between the whole scales and divides the variation
equally among each string. This means that while no
one chord or interval is perfect (and it is physically
impossible for them all to be perfect), they are all
only slightly off. But off by such a small,
consistent amount that no ordinary ear can detect any
dissonance. What follow are the steps to achieve this
tempered tuning. You can learn it quickly. Master it and
you will tune your guitar quicker and slicker than the
other kids on the block! Fail to master it, and studies
show you will spend 7.52 years of your life tuning your
guitar.
|
|
