If you have seen a Windsor zither-banjo you may have noticed that the portion of the second fret which lies underneath the first string is set slightly further up the neck than is the rest of the fret. Why did Windsor decide to do this?

The explanation is at the same time complicated and simple. Many people with sensitive ears experience difficulty in tuning stringed instruments – and the difficulty experienced only increases with the ability to hear fine gradations between pitches. The modern system of tones used in Western music is a fascinating compromise which took centuries to evolve. This system, in which the octave is split into twelve equal semitones, enables music to be played in all twelve keys. In his book ‘Temperament’, Stuart Isacoff describes how this tuning system, known as ‘equal temperament’, gradually became accepted in spite of fierce opposition in musical, philosophical and even religious circles.

In earlier times other musical scales were made use of. These were based on the pure intervals between tones which occur in the natural harmonic series.

Pythagorus worked out that if a string is divided into half then this subdivision will vibrate at twice the frequency as and sound exactly one octave above the original string. If the string is divided into thirds then the portion of the string which is two thirds the length of the original string will vibrate 3/2 times as fast as and will sound a perfect fifth above the original string. Examples of the interval of the fifth are C up to G, F up to C and G up to D. Other natural intervals are similarly found: thus, if divided into quarters, the portion which is three quarters the original length will vibrate 4/3 times faster than and sound a perfect fourth (e.g. C up to F, G up to C, etc.) above the original string. If divided into fifths, the portion which is four fifths the original length will vibrate 5/4 times faster than and sound a major third, (e.g. C up to E, F up to A, G up to B, etc.) above the original vibrating string.

The natural intervals given by these simple mathematical ratios are experienced as "pure" or "in tune" by the inner ear. However, beautiful sounding though they are, these natural intervals are not easily contained within the Western diatonic scale. To be more precise, the problem arises when the music played involves keys with several flat or sharp notes, i.e. involving the five black notes located at their familiar positions in between the seven diatonic notes of the C major scale.

If a scale beginning on C contains the naturally "in tune" minor third from C to Eb, the naturally "in tune" major third from C to E and the naturally "in tune" perfect fourth from C to F, then it is found that the semitone from Eb to E is significantly smaller than the semitone from e.g. E to F. The mathematical explanation for this is not so complicated. The ratios of these intervals are: minor third C-Eb 6/5; major third C-E 5/4, perfect fourth C-F 4/3. The gap, i.e. the semitone, between Eb and E is given by 6/5 divided by 5/4. This works out at 25/24. The gap between E and F is given by 5/4 divided by 4/3. This equals 16/15. This is also a semitone but as we see quite a lot bigger – in fact, one and a half times bigger than that between Eb an E! This difference is clearly audible. Corresponding discrepancies appear with other intervals in the scale and the result is that a piece of music in, say, F# major will sound quite excruciating. For a thorough study of the problems kicked up by Pythagorus’s pure ratios see ‘Treatise on Harmony’ by Jean-Philippe Rameau.

To enable music to be played in all keys the purity of the natural intervals had to be sacrificed and were replaced by the not-quite-in-tune intervals contained within the scale of equal temperament. This scale is constructed thus. The octave is the interval in which the upper tone vibrates at twice the frequency of the lower tone. The mathematical formula which gives the ratio of successive tones in a twelve-note scale of equal temperament turns out to be the twelfth root of two. This provides us with a scale in which each semitone step is proportionally identical to the next. This allows us to build banjos with frets at fixed positions. As mentioned, however, the intervals are no longer pure, i.e. no longer in tune. They are nothing like as badly out of tune as those which crop up trying to play F# major music on an instrument tuned with natural intervals based around the C major scale but, nevertheless, to a keen ear they are not quite in tune. Thus, for example, the tempered perfect fifth (e.g. C to G) is ever so slightly flat when compared with the natural perfect fifth. The tempered major third (e.g. C to E) is slightly sharp of the natural major third while the tempered minor third (e.g. C to Eb) is slightly flat in comparison with its natural counterpart.

Back to our Windsor banjo! Concentrating solely on getting the open strings in the best possible tune, we would try to tune our 1^st, 2^nd and 3^rd strings (D, B, G) so as to attain a natural perfect 5^th from G to D, natural major third from G to B and natural minor third from B to D. This makes a lovely, in-tune open G major chord. Thus tuned, all the barré positions would also be in perfect tune (that is, in tune with themselves - other things permitting, such as correct bridge position, low string action, evenly gauged strings etc.) However, thus far we have only played chords in which all three strings are stopped at the same fret. Play a C major chord, i.e. 3^rd string open G, 2^nd string C (1^st fret) and 1^st string E (2^nd fret), and we are now at the mercy of the fret positioning in order to attain chords in tune with themselves. Played on an instrument fretted according to the scale of equal temperament this C major chord will not be in tune. Why not?

The major third between the C and the E now consists of the nicely in tune natural minor third between the open strings B to D PLUS two tempered semitones (D to E on the first string) MINUS one tempered semitone (B to C on the second string). In other words, we start with a natural minor third - which is, as mentioned earlier, a wider interval than the tempered minor third - and we add a tempered semitone to it. The result is a major third which is a slightly wider interval than the tempered major third. However, we remember that our sweet sounding in-tune natural major third is actually a NARROWER interval than the tempered version. So when we play the 1^st and 2^nd strings of the C major chord described above on a banjo which doesn’t have Windsor’s idiosyncratic second fret we notice that the upper note, i.e. the E on the first string, is sharp.

This is the fault which Windsor tried to correct by setting the 1^st string portion of the 2^nd fret back up the neck. Unfortunately, in getting this C major chord in better tune he then put other chords out of tune. For example, in the 2^nd fret barré chord of A major the 1^st string E is no longer a sweet sounding in-tune perfect fifth above the 3^rd string A but is slightly flat and sounds rather deflated. The other obvious point is that, had Windsor wanted all his major thirds between first and second string tones to be in better tune, then he would have had to set the first string portion of EVERY fret slightly back from the remainder of the fret. This would in turn have put every barré position chord out of tune. It seems that Windsor felt that the C major chord, being used so often, deserved special attention and that getting it in tune was worth any consequent out-of-tuneness in the other chords which involved the 1^st string 2^nd fret.