Dowland Lute Fretting

The scale given by Manuel Op de Coul is correct as far as it goes.
However, Dowland's lute fretboard had only ten frets. The first ten
pitches in the scale given by de Coul give the correct fretting for the
lowest string, and the 11th note does appear on another string. However,
his scale is only a subset of the complete fretting, which is given below
in an excerpt from my book (from the highest pitch down).

(I hope the ASCII text format doesn't garble it too much in everyone's
viewer)

On each line I give the Letter-name, my JustMusic notation (^ = exponent,
* = multiply), Ratio, Semitones (which are cents divided by 100), and
Fret and String (with Dowland's letter tablature):

HIGHEST OCTAVE
F 3^2*7^-1*11^1 99/56 9.86 10th fret, 6th string
(6l)
E 3^3 27/16 9.06 9th fret, 6th
string (6k)
D#/Eb 3^2*11^1*31^-1 99/62 8.10 8th fret, 6th string (6i)
D 3^1 3/2 7.02 7th fret, 6th
string (6h)
C#/Db 3^1*17^-1 24/17 5.97 6th fret, 6th string
(6g)
C 3^-1 4/3 4.98 5th fret, 6th
string (6f)
C 3^3*7^-1*11^1 297/224 4.88 10th fret, 5th string (5l)
B 3^4 81/64 4.08 9th fret, 5th
string (5k)
B 3^1*11^1*211^-1 264/211 3.88 4th fret, 6th string (6e)
A#/Bb 3^3*11^1*31^-1 297/248 3.12 8th fret, 5th string (5i)
A#/Bb 3^1*7^-1*11^1 33/28 2.84 3rd fret, 6th string (6d)
A 3^2 9/8 2.04 2nd fret, 6th
string (6c)
& 7th fret,
5th string (5h)
G#/Ab 3^1*11^1*31^-1 33/31 1.08 1st fret, 6th string (6b)
G#/Ab 3^2*17^-1 18/17 0.99 6th fret, 5th string
(5g)
G n^0 1/1 0.00 open 6th string
(6a)
& 5th fret,
5th string (5f)

MIDDLE OCTAVE
G 3^4*7^-1*11^1 891/448 11.90 10th fret, 4th string
(4l)
F# 3^5 243/128 11.10 9th fret, 4th
string (4k)
F# 3^2*11^1*211^-1 396/211 10.90 4th fret, 5th string (5e)
F 3^4*11^1*31^-1 891/496 10.14 8th fret, 4th string
(4i)
F 3^2*7^-1*11^1 99/56 9.86 3rd fret, 5th string
(5d)
E 3^3 27/16 9.06 2nd fret, 5th
string (5c)
& 7th fret,
4th string (4h)
D#/Eb 3^2*11^1*31^-1 99/62 8.10 1st fret, 5th string
(5b)
D#/Eb 3^3*17^-1 27/17 8.01 6th fret, 4th string
(4g)
D#/Eb 7^-1*11^1 11/7 7.82 10th fret, 3rd string
(3l)
D 3^1 3/2 7.02 open 5th string
(5a)
& 5th fret,
4th string (4f)
& 9th fret,
3rd string (3k)
C#/Db 11^1*31^-1 44/31 6.06 8th fret, 3rd string
(3i)
C#/Db 3^3*11^1*211^-1 297/211 5.92 4th fret, 4th string (4e)
C 3^-1 4/3 4.98 7th fret, 3rd
string (3h)
C 3^3*7^-1*11^1 297/224 4.88 3rd fret, 4th string
(4d)
B 3^4 81/64 4.08 2nd fret, 4th
string (4c)
B 3^-1*17^-1 64/51 3.93 6th fret, 3rd
string (3g)
A#/Bb 3^3*11^1*31^-1 297/248 3.12 1st fret, 4th string (4b)
A#/Bb 3^-3 32/27 2.94 5th fret, 3rd
string (3f)
A#/Bb 3^1*7^-1*11^1 33/28 2.84 10th fret, 2nd string
(2l)
A 3^2 9/8 2.04 open 4th string
(4a)
& 9th fret,
2nd string (2k)
A 3^-1*11^1*211^-1 704/633 1.84 4th fret, 3rd string
(3e)
G#/Ab 3^1*11^1*31^-1 33/31 1.08 8th fret, 2nd string (2i)
G#/Ab 3^-1*7^-1*11^1 22/21 0.81 3rd fret, 3rd string (3d)
G n^0 1/1 0.00 2nd fret, 3rd
string (3c)
& 7th fret,
2nd string (2h)

LOWEST OCTAVE
F#/Gb 3^-1*11^1*31^-1 176/93 11.04 1st fret, 3rd string (3b)
F#/Gb 17^-1 32/17 10.95 6th fret, 2nd string
(2g)
F 3^-2 16/9 9.96 open 3rd string
(3a)
& 5th fret,
2nd string (2f)
F 3^2*7^-1*11^1 99/56 9.86 10th fret, 1st string
(1l)
E 3^3 27/16 9.06 9th fret, 1st
string (1k)
E 11^1*211^-1 352/211 8.86 4th fret, 2nd string
(2e)
D#/Eb 3^2*11^1*31^-1 99/62 8.10 8th fret, 1st string (1i)
D#/Eb 7^-1*11^1 11/7 7.82 3rd fret, 2nd string
(2d)
D 3^1 3/2 7.02 2nd fret, 2nd
string (2c)
& 7th fret,
1st string (1h)
C#/Db 11^1*31^-1 44/31 6.06 1st fret, 2nd string (2b)
C#/Db 3^1*17^-1 24/17 5.97 6th fret, 1st string
(1g)
C 3^-1 4/3 4.98 open 2nd string
(2a)
& 5th fret,
1st string (1f)
B 3^1*11^1*211^-1 264/211 3.88 4th fret, 1st string (1e)
A#/Bb 3^1*7^-1*11^1 33/28 2.84 3rd fret, 1st string (1d)
A 3^2 9/8 2.04 2nd fret, 1st
string (1c)
G#/Ab 3^1*11^1*31^-1 33/31 1.08 1st fret, 1st string (1b)
G n^0 1/1 0.00 open 1st string
(1a)


As can be seen from the table, some pitches are represented on two
strings, and one is represented on three strings.

The ratios derived from this fretting utilize the prime numbers 3, 7, 11,
17, 31, and 211. The strings are tuned (from the bottom up) 1/1 - 4/3 -
16/9 - 9/8 - 3/2 - 1/1. The 1st, 3rd, 4th, 8th and 10th frets all
incorporate 11^1. The 6th fret uses 17^-1, the 1st and 8th frets both
use 31^-1, and the 4th fret uses 211^-1. Prime number 5 (giving the
standard just-intonation 3rds and 6ths) is notably absent.

Dowland wrote his pieces in tablature, as did most other composers of
lute music, so there was no question as to which ratio would be played
for a pitch where there was a choice available (for example, there are
three different D#/Eb's within a range of 28 cents in the middle octave).

The exact string measurements which produce this fretting are given by
Dowland in his son Robert's book "Varietie of Lute Lessons", published in
London in 1610. They are also available in Appendix 2 of Diana Poulton's
"John Dowland -- His Life and Works" [University of California Press,
1972], page 458. A drawing of the fretboard with cents values was
published on the cover of the first issue of Johnny Reinhard's journal
"Pitch, for the International Microtonalist" [Autumn 1986].

This fretting can be considered a temperament, and it is eminently
suitable for Elizabethan-era lute music.

Hoogewerf's playing of the Dowland pieces at last year's AFMM was
outstanding. He performed them on a guitar whose fretboard had six
grooves running the length of it, one under each string, with 24 (at
least, possibly more) slideable frets in each groove, each fret long
enough for just one string. This guitar is capable of retuning to any
possible scale within the space limitations of the frets and fingers.

Joseph L. Monzo
monz@juno.com
4940 Rubicam St., Philadelphia, PA 19144-1809, USA
phone 215 849 6723

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