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## The Hoffmann fret calculator

### Introduction

#### Is it possibile to obtain pythagorean and meantone temperaments on fretted instruments (gambas, lutes, theorboes etc.) without tuning problems?

This fret calculator is the answer. It shows the exact position where to place the frets on the fretboard and where the necessary additional frets (tastini) are to be placed

#### Why is it necessary to add tastini?

On different strings, certain frets determine two types of intervals, a smaller one and a larger one. For example: on the first fret of the C or F string one finds a chromatic semitone (C-C#, F-F#); on the same fret of the A or E string one finds a diatonic semitone (A-Bb, E-F). In other words: on the very same fret one can find both sharp and flat notes. Both Mi and Fa, as a theorethician of the Renaissance period would have said. In the meantone and in the pythagorean temperaments there are large differences between the chromatic and the diatonic semitones; depending on the temperament and on the lenght of the string, this difference may be too large to be manageable on the same fret. On an instrument with a vibrating lenght of 70 cm. tuned with the 1/4 comma meantone temperament, the resulting difference would be of 1,5 cm. It would be useless to try to adjust the tuning moving up or down the finger on the fret: in such a case we do need a tastino.

#### How to fix a tastino?

Solution number 1: tie a second fret and place it at the correct distance from the main fret, using a slighty thicker piece of gut for the lower fret, as usual. It is a "clean" and final solution although a little complicated.
Solution number 2: glue on the fretboard a piece of fret (or a section of a wooden match, or whatever you wish), only for the strings that need it.
Solution number 3, for the gamba: split the fret, in other words divide the two parts of the double fret. This solution is not very "clean" but very easy.

#### Which frets need the tastino?

It dipends on the temperament that you use, but also on the key you play in: do you need a G# or an Ab? Do you need a D# or an Eb? Or even an A#? Moreover, it depends on the technical requirements of the pieces you want to play: on the gamba, if you play polyphonic music, an adjustment on the sixth fret of the lower strings is never necessary: we are not interested the C# position on the fifth string.

#### What should I do, then?

Step 1: With the help of the Fret Calculator find out where exactly on the fretboard you should place your frets, according to the temperament you want to use and for all the possibile intervals. The Calculator Basic Version only determines the most important intervals on the most common strings. With the Calculator Advanced Version you can determine any interval on any possibile string.
Step 2: Decide which are the notes that you really need for the music you are going to play.
Step 3: Place the frets according to the positions determined by the Calculator and add tastini where they are necessary.

#### Example

Bass viol in D, 1/4 comma meantone temperament, d minor and a minor keys:

#### Why does the Fret Calculator distinguish between meantone, pythagorean and other temperaments?

The meantone and pythagorean temperaments are regular temperaments: all the chromatic semitones are mathematically equal. The same applies to diatonic semitones. For gambists and lutenists this means that, with well placed tastini, one can play perfectly in tune. Obviously, even the equal temperament is regular, since in it all the semitones are the same.
Differently, the more "modern" temperaments of the 1600s and 1700s (Werckmeister, Vallotti, Kirnberger etc.) are irregular. Each interval, according to its position inside the circe of fifths, can therefore have a different width. In theory, it could be necessary to have not only one tastino but several tastini for each fret. But in practice the difference is usually so small that it is enough to adjust the tuning by ear, either applying a slightly lighter pressure with the finger o simply moving the finger a little away from the fret. See Calculator Werckmeister, Vallotti.

#### A few words on the history of temperaments on fretted stringed instruments

Several theoreticians of the Renaissance and Baroque periods have dealt with the problems related to the temperaments on fretted stringed instruments: Martin Agricola, Silvestro Ganassi, Nicola Vicentino, Juan Bermudo, Gioseffo Zarlino, Francisco Salinas, Ettore Bottrigari, Vincenzo Galilei, Giovanni Maria Artusi, Marin Mersenne, Thomas Mace, Thomas Salmon, Danoville, etc., but an universally accepted solution to these problems was not found. Many theoreticians simply preferred the equal temperament (Salinas, Bottrigari, Galilei). But this meant, according to several opinions, that lutes and viols could never play together with organs and harpsichords, since the resulting cacophony would have offended the ear (Artusi). Others tried to explain with long and complicated logical and mathematical discussions how to place frets according to the meantone and pythagorean temperaments (Agricola, Ganassi, Mersenne), but they did not realize (or pretended not to realize) that there were several conflicts with certain frets on certain strings. Many theoreticians (Ganassi, Mace) instead, relied on the practical experience of the performing musician who should have been able to adjust the tuning by ear, either by moving the fret(s) or by adjusting the fingerÃ­s position or pressure on the fretboard. "Bad Christians who are secretly sinners" they were called by Bermudo, and he was right because it is certainly impossibile to cope with the difference between the chromatic and the diatonic semitone with one fret only. Someone, finally, suggested that double frets should be used (Bermudo, Salomon), but others totally disagreed and strongly defended the equal temperament (Galilei). The reason for this preference had probably technical and instrumental motivations, since, in fast runs double frets are extremely unconfortable.

#### To learn more:

Mark Lindley, Lutes, Viols and Temperaments, Cambridge, Cambridge University Press, 1984.
Bettina Hoffmann, Di tasti e temperamenti. Questioni di compatibilità, in: La viola da gamba, Palermo, L´Epos, 2010.
Bibliography about temperaments: http://www.huygens-fokker.org/docs/bibliography.html

© Bettina & Friedrich Hoffmann