Fret purity

In instrument science , the term fret purity describes the correct placement of the individual frets for stringed instruments with fingerboards (e.g. guitar , electric bass , lute , cister , banjo , mandolin , bouzouki , zither ) .

One speaks of fretless instruments when each fret is placed exactly in such a way that each string running above it assumes exactly the desired pitch when pressed . This pitch depends on the tuning used . Nowadays this is usually the equal temperament (also equal tempered or equal tempered). A special case of fret purity is octave purity .

In the case of the guitar and similar instruments, unlike the viol , the frets cannot be changed after installation, or only with great effort. The consequence of this is that once fret sticks are set incorrectly, they remain wrong and the instrument is therefore not fretless. The fret purity also depends on the type of strings used. For example, an instrument can deviate slightly from the fret purity after it has been drawn up with new, different strings.

Mathematical basis

The following derivation applies to the commonly used equal tuning , in which all semitones have the same frequency ratio:

Denoting the length of the ungegriffenen, "empty" string ( scale ), and , , ..., the string lengths for the higher by 1, 2, ... halftones tones, then for the position of the 1st Federal ${\ displaystyle l_ {0}}$ ${\ displaystyle l_ {1}}$ ${\ displaystyle l_ {2}}$ ${\ displaystyle c}$

{\ displaystyle c = {{l_ {0} -l_ {1}} \ over l_ {0}}}

or ,

{\ displaystyle {l_ {1} \ over l_ {0}} = 1-c}

for the position of the 2nd federal government accordingly

{\ displaystyle {l_ {2} \ over l_ {1}} = 1-c}

or

{\ displaystyle {l_ {2} \ over l_ {0}} = (1-c) ^ {2}}

etc. up to the 12th fret where the octave is fingered and the length of the string that vibrates

{\ displaystyle l_ {12} = {1 \ over 2} \ cdot l_ {0}}

is. This applies

{\ displaystyle {1 \ over 2} = {l_ {12} \ over l_ {0}} = (1-c) ^ {12} \ to c = 0 {,} 056125687 \ ldots}

This means that the 1st fret is 5.61% of the length of the open string from the saddle.

As federal constant is the reciprocal value of signified . ${\ displaystyle c}$ ${\ displaystyle {1 \ over c} = 17 {,} 81715 \ ldots}$

Scale allowance, nut compensation

Vibrato system of an Ibanez Prestige RG 1570 electric guitar. Easily recognizable: the scale allowance set differently on the bridge side.

The mathematical formula only applies if the string tension is assumed to be constant. When the string is pressed down on a fret, the tension increases slightly, which inadvertently increases the tone. The bridge side ensures that every note of a string is lowered by the same amount. All fingered notes are now intoned by extending the length of the scale. The depressed octaved notes, which then sound too low in relation to the open strings, are reduced by shortening, i.e. H. by moving the saddle towards the 1st fret, voiced.

On the saddle side, the amount of compensation required depends on how much the pitch of the individual strings changes with a corresponding increase in tension, which in turn depends on the type of string and how it is played (high or low finger pressure). On the bridge side, it is crucial for the different extent of the scale allowance that strings do not vibrate exactly up to the rest, depending on their rigidity.

literature

Martin Koch: Guitar making. Koch Verlag 1994, ISBN 3-901314-06-7
George Buchanan: String Instruments. Th.Schäfer Verlag 2005, ISBN 3-87870-718-5
Eberhard Meinel: Intonation, tempering and scale compensation for plucked instruments. Lenk & Meinel 2006, ISBN 3-00-018587-9

Web links

Konrad Schwingenstein: Intonation of the guitar (accessed November 28, 2010)
David Schramm / John & William Gilbert: Intonation and Fret Placement (accessed November 28, 2010)