This application calculates fret positions according instructions given by John Dowland in his book, "A Varietie of Lute Lessons".
Because 12 tone equal temperament (12TET) divides the octave into 12 equal parts, the only interval that is truly 'pure' (i.e. achieves maximum consonance) is the octave. All other intervals are a bit out. We discuss this problem in detail on another page where we discuss temperaments.
The English lutenist and composer John Dowland (1563-1626) in his book, A Varietie of Lute Lessons, gives instructions for fretting a lute that produces a result that is much sweeter sounding than 12TET. I have used this tuning on a lute and on a Baroque guitar, and I like it very much.
(Note: Much of the information here comes from Jeff Lee's site: https://www.shipbrook.net/jeff/frets.html#fn1)
You can find a facsimile copy of A Varietie of Lute Lessons here: http://onlinebooks.library.upenn.edu/webbin/book/lookupid?key=olbp16050
Dowlands fretting instructions (found on the 15th page) are given below. Dowland describes creating a fret ruler that he then uses to mark the frets on the lute soundboard. If you are making several lutes with the same scale length, then this is still a very good strategy. For a one off construction, I normally measure directly on to the fret board.
I have split Dowland's text into sections to help you follow it, and prefixed each section with the letter name Downland uses for each mark on the ruler. Note that Dowland omits instructions for the 11th fret, and I have used a widely accepted conjectural formula to which I have assigned the letter X.
"Wherefore take a thinne flat ruler of whitish woode, and make it iust as long and straight as from the inward side of the Nut to the inward side of the Bridge, then note that end which you meane to the Bridge with some small marke, and the other end with the letter A, because you may know which belongeth to the one and to the other:
N: then lay the ruler vpon a Table, and take a payre of compasses and seeke out the iust middle of the Ruler : that note with a pricke, and set the letter N. vpon it, which is a Diapason from the A. as appeareth by the striking of the string open.
H: Secondly, part the distances from N. to A.[1] in three parts, then the first part giues you the seauenth fret from the Nut, making a Diapente : in that place also set a pricke, and vpon it the letter H.
B: Thirdly, deuide the distance from the letter H. to the letter A. in eleauen parts : two of which parts from A. giues the first fret, note that with a pricke, and set the letter B. thereon, which maketh a Semitone.
C: Fourthly, diuide the distance from H. to the letter A. in three parts, one of which parts from A. vpward sheweth the second fret, note that with a pricke, and set the letter C. vpon it, which maketh a whole Tone from A.
F: Fiftly, diuide the distance from N. to A. into two parts, there the first part sheweth you the fift fret, sounding a Diatessaron : in that place also set a pricke, and vpon it the letter F.
G: The sixt fret which is a G. must be placed iust in the middest betwixt F. and H. which maketh a Semidiapente.
D: Seuenthly, diuide the distance from the letter B. to A. in three parts, which being done, measure from the B. vpwards foure times and a halfe, and that wil giue you the third fret, sounding a Semiditone : mark that also with a prick, & set thereon the letter D.
J: then set the fourth fret iust in the middle, the which wil be a perfect ditone:
I: then take the one third part from B. to the Bridge, and that third part from B. maketh I. which soundeth Semitonium cum Diapente,
E: then take a third part from the Bridge to C, and that third part maketh E. which soundeth Tonus cum diapente, or an Hexachordo maior.
L: Then take one third part from D. to the Bridge, and that third part from D. maketh L. which soundeth Ditonus cum Diapente.
Now take your LVTE, and lay it vpon a table vpright, and set the Ruler edgewise, betweene the nut and the bridge, and thereby set little marks vpon the necke of the Instrument euen with those on the ruler, because those are the places on which your frets must stand."
Here is a key to some of the terms:
| Term | Interval | Semitones/Fret | Letter | Consonance |
|---|---|---|---|---|
| Unison | Unison | 0 | U | Perfect consonance |
| Semitone | Minor second | 1 | B | Perfect dissonance |
| Tonus | Major second | 2 | C | Median dissonance |
| Semiditone | Minor 3rd | 3 | D | Imperfect consonance |
| Ditone | Major 3rd | 4 | J | Imperfect consonance |
| Diatessaron | Perfect 4th | 5 | F | Median consonance |
| Semidiapente | Diminished 5th | 6 | G | Perfect dissonance |
| Diapente | Perfect 5th | 7 | H | Median consonance |
| Semitonium cum diapente | Minor 6th | 8 | I | Median dissonance |
| Tonus cum diapente | Major 6th | 9 | E | Imperfect dissonance |
| Semiditonus cum diapente | Minor 7th | 10 | X | Imperfect dissonance |
| Ditonus cum diapente | Major 7th | 11 | L | Perfect dissonance |
| Diapeson | Perfect octave | 12 | N | Perfect consonance |
(See https://en.wikipedia.org/wiki/Consonance_and_dissonance the section 'Antiquity and Middle Ages')
This fret calculator simply turns Dowland's instructions for creating a fret ruler into a calculation.
There is precise agreement between Dowland's scheme and 12 tone equal temperament at the 12th fret because this is the octave. There are very significant differences at the 3rd and 4th frets. These differences have the effect of sweetening the sound of the lute.
Here are some useful online resources for lute and baroque guitar players:
String calculator: https://www.niskanenlutes.com/index.php?p=stringcalc
Fret calculator: https://niskanenlutes.com/index.php?p=frets
Lute Society of America fret spreadsheet: http://www.cs.dartmouth.edu/~lsa/download/index.html