 # 11. To what pitches are bass strings normally tuned?

A. Pitch is measured in hertz (hz), which is the rate at which the
string is vibrating back and forth (measured in cycles per
second). The standard definition of pitch is that the A above
middle C is exactly 440 hz. The open A string on a bass is three
octaves below that A, and dropping one octave divides the
frequency by 2. So the A below middle C is 220 hz, the A below
that is 110 hz, and the open A string on the bass is 55 hz.

You can get the pitches for the other two strings in either of two
ways. The first is to use natural tuning, and the second is to use
even-tempered tuning.

Natural tuning is based on the fact that a major chord sounds most
pure if the ratio of the frequencies of the three notes is exactly
4:5:6:8. Thus an A major chord starting on the 440 hz A would be
tuned as follows: A 440 hz, C# 550 hz, E 660 hz, A 880hz. A bass
is tuned in perfect fourths, and as you can see from the E-A
example in the A major chord, the frequencies of two notes in a
perfect fourth are always 6:8, or 3:4. Using this ratio, and
knowing that the open A string on a bass is 55 hz, we can find the
pitches of the other strings just by multiplying or dividing by
4/3, or 1.33333. The problem with natural tuning is that it is
internally inconsistent, because it can produce several different
"correct" pitches for a given note. For example, consider starting
with the 440 hz A, and trying to find the pitch of the A one
octave above it. One way to do that is to say "octaves are in the
ratio 4:8" and conclude that the A one octave above is 880 hz.

However, an equally valid way is to reason as follows. The C# that
is above the A is in the ratio 5:4 with that A, so its pitch must
be 550 hz. Starting on that C#, we can build a C# major chord,
which will have F as its third. The ratio of C# to F must also be
5:4, so that F must have a pitch of 550 * (5/4) = 687.5 hz. Now,
starting on that F, we can build an F major chord with A as the
third. The pitch of that A must be 687.5 * (5/4) = 859 hz, which
is rather different from 880 hz. If you tuned an instrument to
F=687, A=880, and played an F major chord on it, it would sounds
very out of tune.

The solution, which was popularized by JS Bach, is to slightly
fudge the "natural" tuning of each note to average out the errors
so that, while each chord will be a little off, no one chord will
be very wrong and you can play in any key you like. Bach's piece,
"The Well Tempered Clavier", which modulates through all 12 keys,
was written to demonstrate the power of even-tempered tuning.

The formula for even tempering is based on the number of
half-steps between two notes. The ratio of pitch between two notes
that are N half-steps apart is given by

2^(N/12)

This formula was chosen because it makes the octave work out
perfectly; an octave is 12 half steps so the ratio of two notes an
octave apart is just 2 ^ (12/12) or 2^1, or 2. The advantange of
this formula is that it gives the same answer for the pitch of a
note, regardless of what intervals are used to calculate it. In
the above example, the ratio between A and A an octave higher is
2^(12/12) or 2. The ratio of a major third is 2^(4/12) or 1.260.
Starting with A 440, and going up by major thirds, we get C# =
554, F = 698, A = 880, because

1.26^3 = [2^(4/12)]^3 = 2^(12/12) = 2.

For a perfect fourth, which is 5 half-steps, the formula gives a
ratio of 2^(5/12) or 1.33484. Note that this is just slightly
bigger than the ratio of 1.33333 given by the natural tuning, so
it doesn't make a whole lot of difference which one you use in
practice.

Now, to answer the question :) The pitch of an A string is 55hz,
and the other pitches depend on whether you use even-tempered
tuning or natural tuning. The two cases are, for a six-string
bass:

```                B       E       A       D       G       C
Natural         30.938  41.250  55.000  73.333  97.777  130.369
Even-tempered   30.868  41.203  55.000  73.416  97.999  130.812
```

Other tunings are rare but not unknown. Most common is to tune the E
string down to D, giving the tuning D-A-D-G. This has become less
common since 5-string basses became popular but is found on many
older records. Roger Waters of Pink Floyd uses it a lot. Another
common tuning is to tune all strings one half-step flat: Eb, Ag,
Db, Gb (or D#, G#, C#, F# if you like to think of it that way.)
This reduces the tension on the strings, making string bending
easier. Most groups that use this tuning, notably Van Halen,
actually tune down so the guitarist can have the benefits of lower
tension: the bass player just tunes down to match. However, it can
be convenient to have lower string tension on bass as well. Also,
being tuned to E flat instead of E can make things easier if you
are playing with a horn section, since horn music is often written
in such keys as E flat and B flat.

Other artists use even weirder tunings, often setting the string
intervals to fifths, major thirds, tritones, or even unisons.
Michael Manring is probably the most notable artist who does this.
It should be noted that this isn't all that good a thing for the
bass, because the strings are designed so that all four strings
will have the same tension in normal tuning, and thus apply the
same pressure to the neck. If you change the tuning, so that some
strings apply more pressure to the neck than others, the neck can
warp in very odd ways that are not easy to fix. Michael solves
this problem by using a bass with a graphite neck, and if you can
afford to do this, you don't need to worry about the neck warping
(for any reason). But if you have a wooden-necked bass, you might
want to put the bass back into normal tuning after you experiment
with other tunings.

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