This article is from the Rec.music.makers.bass FAQ, by Kalle Kivimaa Kalle.Kivimaa@hut.fi with numerous contributions by others.

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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