The Imperfect Perfection of the Tempered Tuning
History
About 550 BC, Pythagoras discovered that when he tuned the strings of a harp in twelve clean fifths (7 octaves), the high note was quite higher than seven octaves.
From that time through the Middle Ages, a lot of experimenting took place, mostly based on trial and error. It was not until Andreas Werckmeister (1645-1706), an organ builder who experimented with different tuning, introduced in 1697 the well tempered tuning in a publication called Hypomnemata Musica. In 1702, Johann Gottfried Walther(1685-1748), a distant cousin of Johann Sebastian Bach (1685- 1750), became an apprentice of Werckmeister. At a later time, he introduced Werckmeister's Theory to Johann Sebastian Bach. Most of Bach's compositions before 1720 recommend a tuning called Werckmeister 3 which is still used today for the authentic reproduction of Bach's earlier works. Bach completed Book One with the Preludes and the Fugues of the Well-tempered Clavier in 1722 which require the well-tempered tuning - the only tuning that allows us to play in all twelve keys.
The tempered tuning became the standard in Germany around 1800, and about 50 years later, in England. Until the tempered tuning became the standard, bridge, nut and fret compensation was a common practice.
After the tempered tuning was established, the next problem came up when steel strings replaced gut strings on fretted instruments. Steel strings are much stiffer than gut strings, and due to their stiffness, don't start to vibrate directly at the bridge, the nut or frets, which causes an intonation problem. We compensate for this with adjustable bridge saddles. It's not perfect but as close as we can get. Another problem is the nut. If the slot in the nut is not cut deep enough and you play a note on the first fret, you will bend the string, and the note will be sharp. The slot in the nut should be cut precisely, as deep as the height of a fret and not by a hair deeper or the string will buzz on the first fret. The result is that the guitar plays easier, and the notes are right.
Why Do We Need The Tempered Tuning?
When we tune a piano in twelve clean fifths, starting with an A to E, E to B, B to F#, etc. till we reach, after twelve steps, the note A again, we recognize that this A is almost 1.365% too high. This arrives out of the following formula:
| The Octave | = Fundamental x 2.0 |
|
|
= Fundamental x 1.5 |
| The Factor for 7 Octaves |
|
= 128.000 |
| The Factor for 12 Clean 5ᵗʰs |
|
= 129.746 |
Low A 27.5Hz x 2^7
=
3520Hz High A
Low A 27.5Hz x 1.5^12
=
3568Hz High A
This shows that we cannot use clean fifths to calculate a scale or the fret spacings of a guitar because we compound the difference by a factor of twelve.
=
Fundamental X = Fundamental X
The Tempered Tuning
We divide the octave into twelve equally tempered intervals:
The Octave A Halftone
The Fifth
=
Fundamental X
2^(12/12) 2 2^(1/12) 2^(7/12)
=
=
1.05946309436
=
1.49830707688
The Factor for Seven Octaves
=
2^7
=
128
The Factor for 12 Tempered Fifths
=
2 (7/12) 12
= 128
Here you can see that the tempered fifth is only a very small fraction lower than a clean fifth.
Calculating The Tempered Fret Spacing
Let me show you an easy way how to calculate the fret spacing of any desired scale in less than one minute:
Because the octave is precisely half the speaking length of a string, we have to use the reciprocal of a tempered halftone step as a factor. The factor is: .5^¹/¹² = .94387431268
Example for a 25.5" scale:
Press on a scientific calculator - .5 XY 12 1/X X X 25.5 =
Press on a small pocket calculator - .943875 X X 25.5 =
(Each time you press = there is the next number)
I use a calculator with twelve digits, but all you need are six digits to be precise.