Transcript Slide 1

THE SECRETS OF VIOL TUNING
A LECTURE / DEMONSTRATION
FREDERICK G. REINAGEL
PAN-PACIFIC GAMBA GATHERING
HONOLULU, HAWAII
AUGUST 2007
WHY DO WE TUNE?
AS HUMAN BEINGS,
WE CONTROL OUR
ENVIRONMENT TO:
AS MUSICIANS, WE
CONTROL OUR ACOUSTIC
ENVIRONMENT (TUNE) TO:
PURSUE THAT WHICH IS
COMFORTABLE OR PLEASURABLE
MAXIMIZE (INTENDED) CONSONANCE
AVOID THAT WHICH IS
UNCOMFORTABLE OR PAINFUL
MINIMIZE (UNINTENDED) DISSONANCE
CREATE ORDER AND STRUCTURE
TO SIMPLIFY OUR MODEL OF REALITY
ACHIEVE UNIFORM INTERVALS AND CHORDS
ENHANCE SOCIAL WELL-BEING
PLAY HARMONIOUSLY WITH OTHERS
TERMINOLOGY
TONE – THE AURALLY PERCEIVED STIMULUS OF A SOUND HAVING A CONSTANT PITCH
PITCH – PROPERTY OF A TONE PERCEIVED AS HIGH(ER) OR LOW(ER), RELATED TO THE
RAPIDITY OF VIBRATION OF THE SOURCE, OR FREQUENCY
PURE TONE – A TONE CONSISTING OF ONLY A FUNDAMENTAL WITH NO HARMONICS
(SINUSOIDAL WAVEFORM)
FREQUENCY – THE NUMBER OF VIBRATIONS OF A TONE IN ONE SECOND
EXPRESSED IN A UNIT CALLED HERTZ
INTERVAL – THE DIFFERENCE IN PITCH BETWEEN TWO TONES
PURE INTERVAL – ONE WHOSE SIMPLE FORM HAS A FREQUENCY RATIO OF TWO
INTEGERS EACH OF WHICH IS NOT GREATER THAN TEN
SIMPLE INTERVAL – ONE WHICH IN NOT GREATER THAN AN OCTAVE
WHOLETONE – THE LARGER INTERVAL OF PITCH IN A DIATONIC SCALE
SEMITONE – A FRACTION OF A WHOLETONE RANGING BETWEEN ONE- AND TWO-THIRDS
HALFTONE – EXACTLY ONE TWELFTH OF AN OCTAVE (EXISTS ONLY IN EQUAL TEMPERAMENT)
MICROTONE – FRACTION OF A WHOLETONE OF LESS THAN ONE-THIRD
NOTE – THE VISUAL SYMBOL OF A MUSICAL TONE
TEMPERAMENT – THE MICROTONAL ALTERATION OF PURE INTERVALS TO SOLVE VARIOUS
TUNING PROBLEMS
THE RELATIONSHIP OF PITCH TO FREQUENCY
g’’
d’’
g’
TREBLE
VIOL
P
I d’
T
C
H
g
MIDDLE C
BASS
VIOL
d
G
100
150
200
250
300
350
400
450
500
FREQUENCY (HERTZ)
550
600
650
700
750
800
THE RELATIONSHIP OF PITCH TO FREQUENCY – LOGARITHMIC SCALE
g’’
d’’
P
I
T
C
H
g’
TREBLE
VIOL
d’
MIDDLE C
g
BASS
VIOL
d
G
50
75
100
150
200
300
400
600
800
FREQUENCY (HERTZ) – LOGARITHMIC SCALE
1200 1600
THE RELATIONSHIP OF PITCH TO FREQUENCY – PITCH COMPRESSION
g’’’
d’’’
g’’
d’’
P
I
T
C
H
g’
TREBLE
VIOL
d’
MIDDLE C
g
7-STRING
BASS
VIOL
d
G
D
GG
50
75
100
150
200
300
400
600
800
FREQUENCY (HERTZ) – LOGARITHMIC SCALE
1200 1600
PITCH MEASUREMENT UNITS
OCTAVE
MACROTONAL
$1
$1
$1
$1
HALFTONE
COMMA
(APPROXIMATE)
25¢
CENT
1¢
MICROTONAL
PYTHAGOREAN – 23.5 ¢
NEWTONIAN (1/53 OCTAVE) – 22.6 ¢
SYNTONIC – 21.5 ¢
LIMITS OF HUMAN HEARING PITCH DISCRIMINATION
JUST NOTICEABLE DIFFERENCE (JND) AS A FUNCTION OF FREQUENCY
PITCH
c’
c’’
c’’’
c’’’’
1000
2
2000
2.5
c’’’’’
c’’’’’’
25
20
15
JND
(CENTS)
10
5
0
0
250
0.5
1 500
1.5
3
4000 3.5
FREQUENCY (HERTZ) – LOGARITHMIC SCALE
8000
4
4.5
HARMONIC MODES OF STRING VIBRATION
O
C
T
A
V
E
F
O
U
R
T
H
F
I
F
T
H
5th
4th
3rd
2nd
1st
8th
6th
M
A
J
O
R
M
I
N
O
R
T
H
I
R
D
T
H
I
R
D
F
O
U
R
T
H
10th
9th
M
A
J
O
R
M
A
J
O
R
S
E
C
O
N
D
S
E
C
O
N
D
*
FUNDAMENTAL
OCTAVE
ABOVE
OCTAVE
AND A FIFTH
ABOVE
TWO
OCTAVES
ABOVE
TWO
OCTAVES AND
A MAJOR
THIRD ABOVE
TWO
OCTAVES
AND A FIFTH
ABOVE
THREE
OCTAVES
ABOVE
THREE
THREE
OCTAVES AND
OCTAVES
A MAJOR
AND A MAJOR
SECOND ABOVE THIRD ABOVE
DIFFERENCES OF HARMONIC (PURE) INTERVALS
ASCENDING
PURE INTERVAL
DESCENDING
PURE INTERVAL
DIFFERENCE INTERVAL
FREQUENCY
RATIO
ALTERNATE
NAME
NEWTONIAN
COMMAS (APP)
FREQ.
RATIO
NAME
FREQ.
RATIO
NAME
1:2
OCTAVE
3:2
FIFTH
3:4
( 12 x 23)
FOURTH
22 (0.07¢)
2:3
FIFTH
5:4
MAJOR
THIRD
5:6
( 23 x 45 )
MINOR
THIRD
14 (1.34¢)
2:3
FIFTH
4:3
FOURTH
8:9
( 23 x 34)
MAJOR
SECOND
MAJOR
WHOLETONE
9 (-0.14¢)
3:4
FOURTH
5:4
MAJOR
THIRD
15:16
( 34 x 45)
MINOR
SECOND
DIATONIC
SEMITONE
5 (1.48¢)
4:5
MAJOR
THIRD
9:8
MAJOR
SECOND
9:10
( 45 x 89)
MAJOR
SECOND*
MINOR
WHOLETONE
8 (-1.27¢)
4:5
MAJOR
THIRD
6:5
MINOR
THIRD
24:25
( 45 x 56)
AUGMENTED
UNISON
CHROMATIC
SEMITONE
3 (-2.75¢)
8:9
MAJOR
SECOND
10:9
MAJOR
SECOND*
80:81
( 98 x 109 )
COMMA
SYNTONIC
COMMA
1 (1.14¢)
15:16
MINOR
SECOND
25:24
AUGMENTED
UNISON
125:128
DIESIS
MINOR
DIESIS
2 (4.22¢)
x 25)
(15
16 24
NAME
ONE OCTAVE = THREE MAJOR SECONDS + TWO MAJOR SECONDS* + TWO MINOR SECONDS (3x9 + 2x8 + 2x5 = 53 COMMAS)
WAVEFORM ANALYSIS/
HARMONIC FUSION
COMPLEX
TONE
H
A
R
M
O
N
I
C
C
O
M
P
O
N
E
N
T
S
6TH
5TH
4TH
3RD
2ND
FUNDAMENTAL
HARMONICALLY-TUNED CHORD
ww
 ww
CHORD
5TH
b’
4TH
g’
3RD
d’
2ND
g
(FUNDAMENTAL)
G
H
A
R
M
O
N
I
C
0.03 SECONDS
COMBINING TWO TONES – FORMATION OF BEATS
10 BEATS
PER
SECOND
COMBINED
TONE
g#
B (4TH FRET
A e-STRING)
S
200 HZ
S
V
I
O
L
a
(OPEN
STRING)
210 HZ
1/5 SECOND
COMBINING TWO TONES – FORMATION OF A SLOW BEAT
1 BEAT
PER
SECOND
COMBINED
TONE
T
R
E
B
L
E
d’’
(OPEN
STRING)
534 HZ
d’’
FRET
a’-STRING
3 CENTS
FLAT)
533 HZ
(5TH
V
I
O
L
2 SECONDS
COMBINING TWO TONES – FORMATION OF A DIFFERENCE TONE
COMBINED
TONES
SHOWING
DIFFERENCE
TONE
d
138 HZ
B
A
S
S
V
I
O
L
f#’
(4TH FRET
d’’-STRING)
345 HZ
a
(OPEN
STRING)
207 HZ
STRETCHING THE CHAIN OF FOURTHS - REGULAR TEMPERAMENTS
DIMINISHED SECOND (AUGMENTED SEVENTH)
DIMINISHED SIXTH (AUGMENTED THIRD)
DIMINISHED THIRD (AUGMENTED SIXTH)
DIMINISHED SEVENTH (AUGMENTED SECOND)
DIMINISHED FOURTH (AUGMENTED FIFTH)
SIMPLE
INTERVAL
NAMES
(INVERSION
INTERVAL
NAMES)
DIMINISHED OCTAVE (AUGMENTED UNISON)
DIMINISHED FIFTH (AUGMENTED FOURTH)
MINOR SECOND (MAJOR SEVENTH)
MINOR SIXTH (MAJOR THIRD)
MINOR THIRD
(MAJOR SIXTH)
PURE
FOURTH =
22 COMMAS
A#
D#
G#
C#
F#
B
E
MINOR
SEVENTH
(MAJOR
SECOND)
A
D
G
C
ONE COMMA
SHORT OF 5:16
ONE COMMA SHORT OF 5 OCTAVES
F
Bb
Eb
Ab
Db
Gb
4 X 22 = 88
2 X 53 – 17 = 89
12 X 22 = 264
5 X 53 = 265
SCALE PROPERTIES
SCALE TYPE
SCALE
CHARACTERISTICS
MUSIC NOTATION AND
INTERVAL TERMINOLOGY
HARMONIC
(JUST INTONATION)
UNEQUAL WHOLETONES
UNEQUAL SEMITONES
NON-COINCIDENT ENHARMONICS
UNDERDETERMINED
REGULAR TEMPERAMENTS
(EXCEPT EQUAL)
EQUAL WHOLETONES
UNEQUAL SEMITONES
NON-COINCIDENT ENHARMONICS
UNIQUELY
DETERMINED
CIRCULAR (WELL)
TEMPERAMENTS
UNEQUAL WHOLETONES
UNEQUAL SEMITONES
COINCIDENT ENHARMONICS
OVERDETERMINED
EQUAL TEMPERAMENT
EQUAL WHOLETONES
EQUAL SEMITONES
COINCIDENT ENHARMONICS
OVERDETERMINED
THE CASE FOR NON-EQUAL TEMPERAMENT FOR VIOL (AND LUTE) TUNING
16TH AND 17TH CENTURY KEYBOARD TUNINGS
MOVABLE FRETS
SYLVESTRO GANASSI – IN LETTIONE SECONDA, “ …..OTHER MEANS OF ADJUSTMENT OF FRETS
UP AND DOWN, ACCORDING TO THE REQUIREMENTS OF YOUR EAR. I COULD PROVIDE
A LENGTHY DISCOURSE ON THE REASONS BEHIND THE NECESSARY ADJUSTMENTS,
RESULTING FROM DIFFERENCES BETWEEN MAJOR AND MINOR SEMI-TONES …….”*
HANS GERLE – TUNING BASED ON 1/6-COMMA MEANTONE
VINCENZO GALILEI – DISCUSSED THE USE OF TASTINI
JOHN DOWLAND – BASED HIS TUNING ON GERLE
MODERN PRACTICE OF DIVIDING THE FIRST FRET
MODERN PRACTICE OF DIVIDING OTHER FRETS
*TRANSLATION BY RICHARD BODIG
DIVIDING THE COMMA
¼-COMMA
MEANTONE
d’
ONE OCTAVE
53 COMMAS
PURE
MAJOR THIRD
17 COMMAS
PURE FOURTH
22 COMMAS
18 COMMAS
d
c’
g
e’
a’
d’’
PYTHAGOREAN
ONE COMMA
22.6 CENTS
g
c’
e’
CHECKING OPEN-STRING TUNING BY COMPARING HARMONICS
+1
6
-1
+2
-2+3
-3
+4
-4
+5
-5,6
5TH
4TH
HARMONIC HARMONIC
3RD
HARMONIC
1
PLAY HARMONICS ONE AT A TIME BY LIGHTLY TOUCHING STRINGS AT THE INDICATED POINTS AND
APPLYING A QUICK, FIRM BOW STROKE NEAR THE BRIDGE. LIFT THE BOW AT THE END OF THE STROKE
TO ALLOW THE STRING TO RING.
2
COMPARE THE PITCHES OF THE HARMONIC PAIRS INDICATED BY CIRCLES WITH THE SAME NUMBER.
3
THE PITCH CORRESPONDING THE “+” SIGN SHOULD BE VERY SLIGHTLY HIGHER THAN THAT OF THE “-”.
(EXCEPTION: FOR 1/4-COMMA, -3 SHOULD EXACTLY MATCH +3)
4
THE PITCH OF THE HARMONIC INDICATED BY “6” ON THE LOWEST STRING SHOULD EXACTLY MATCH THE
OPEN STRING PITCH OF THE HIGHEST STRING.
CHECKING FRET TUNING BY SYMPATHETIC RESONANCE
5
2
1
4
1,6
3
2
5
2
3
1,6
2
3
3
3
5
4
4
1,6
5
5
2
6
1
MAKE SURE ALL OPEN STRINGS ARE TUNED AS ACCURATELY AS POSSIBLE.
2
PLAY FRETTED NOTE BY FIRMLY STOPPING STRINGS AT THE INDICATED POINTS AND APPLYING A LONG,
FIRM BOW STROKE NEAR THE BRIDGE. MAINTAIN PRESSURE ON STOPPED STRING AND LIFT THE BOW AT
THE END OF THE STROKE TO ALLOW THE BOWED STRING TO RING.
3
CHECK THAT OPEN STRING(S) INDICATED BY THE NUMBER IN THE STOPPED POINT CIRCLE ARE
RESONATING SYMPATHETICALLY (RESONANCE TONES MAY BE AN OCTAVE ABOVE THE BOWED TONE). THIS
MAY DONE VISUALLY, OR BY LIGHTLY TOUCHING THE RESONATING STRING TO FEEL ITS VIBRATION.
4
IF SYMPATHETIC RESONANCE IS NOT EXHIBITED, COMPARE PITCHES OF THE STOPPED AND OPEN
STRINGS AND ADJUST FRET TO EXACTLY MATCH PITCHES.
5
CHECK THAT FRETTED TONE PRODUCES SYMPATHETIC RESONANCE WITH NO BEATING.
WHERE DO YOU PUT THE FIRST FRET?
BASS OR TREBLE VIOL
NUT
1ST
FRET
D#
Eb
G#
Ab
C#
Db
E#
F
A#
Bb
D#
Eb
D#
G#
C#
E#
A#
D#
Eb
Ab
Db
F
Bb
Eb
D#
G#
C#
E#
A#
D#
Eb
Ab
Db
F
Bb
Eb
2ND
FRET
EQUAL TEMPERAMENT
(1/12 COMMA)
1/6 COMMA
1/4 COMMA
WHERE DO YOU PUT THE FOURTH FRET?
BASS OR TREBLE VIOL
3RD
FRET
4TH
FRET
F#
Gb
B
Cb
E
Fb
G#
Ab
C#
Db
F#
Gb
F#
B
Gb
Cb
E
G#
C#
F#
Fb
Ab
Db
Gb
F#
B
E
G#
C#
F#
Gb
Cb
Fb
Ab
Db
Gb
5TH
FRET
EQUAL TEMPERAMENT
(1/12 COMMA)
1/6 COMMA
1/4 COMMA
WHERE DO YOU PUT THE THIRD FRET?
TENOR VIOL
2ND
FRET
3RD
FRET
A#
Bb
D#
Eb
G#
Ab
B#
C
E#
F
A#
Bb
A#
D#
G#
B#
E#
A#
Bb
Eb
Ab
C
F
Bb
A#
D#
G#
B#
E#
A#
Bb
Eb
Ab
C
F
Bb
4TH
FRET
EQUAL TEMPERAMENT
(1/12 COMMA)
1/6 COMMA
1/4 COMMA
FIFTH
DIMINISHED
FIFTH
AUGMENTED
FOURTH
4:5
FOURTH
MAJOR
THIRD
5:6
DIMINISHED
FOURTH
MINOR
THIRD
AUGMENTED
SECOND
MAJOR
SECOND
MINOR
SECOND
AUGMENTED
UNISON
HOW FRETS MIGRATE AS A FUNCTION OF TEMPERAMENT
1/4 COMMA
NUT
1/6 COMMA
1/12 COMMA
(EQUAL)
UNTEMPERED
(PYTHAGOREAN)
24:25 15:16
9:10
8:9
HARMONIC INTERVALS
3:4
2:3
TUNING TECHNIQUES
BOWED TONE – USE LIGHT BOW PRESSURE NEAR THE BRIDGE
MINIMIZES PITCH BENDING, ENHANCES UPPER HARMONICS
FREE RINGING TONE – PLUCKED OR AFTER BOW STROKE (BOW LIFTED)
PLUCKING NEARER THE NUT ENHANCES UPPER HARMONICS
AXIAL FORCE ON PEG – AVOID BENDING FORCE ON NECK OF VIOL
USE OF A PEG WRENCH
PEG BOX SQUEEZING OR STRING TUGGING – NOT RECOMMENDED
USE OF ELECTRONIC AIDS
VISUAL DISPLAY – LEARN ¢ DEVIATIONS FOR NON-EQUAL TEMPERAMENTS
AUDIO REFERENCE TONE
SHOULD BE MULTI-TEMPERAMENT
ONE MUST DEVELOPE A GOOD EAR FOR BEATS
CONSORT TUNING IN NON-EQUAL TEMPERAMENTS
MULTI-TEMPERAMENT AUDIO REFERENCE IS BEST
ELSE, USE A SINGLE VIOL (PREFERABLY A BASS) THAT IS MOST LIKELY TO BE
IN BEST TUNE (INCLUDING FRETS) AS AN AUDIO REFERENCE. OTHER VIOLS
TUNE UNISONS OR OCTAVES TO IT (FIFTHS AND FOURTHS ARE NOT PURE)
DEVIATIONS FROM EQUAL TEMPERAMENT IN CENTS
SIXTH COMMA
FIFTH COMMA
QUARTER COMMA
B#
-14.7
-21.1
-30.8
E#
-13
-18.8
-27.4
A#
-11.4
-16.4
-24
D#
-9.8
-14.1
-20.5
G#
-8.1
-11.7
-17.1
C#
-6.5
-9.4
-13.7
F#
-4.9
-7
-10.3
B
-3.3
-4.7
-6.8
E
-1.6
-2.3
-3.4
A
0
0
0
D
1.6
2.3
3.4
G
3.3
4.7
6.8
C
4.9
7
10.3
F
6.5
9.4
13.7
Bb
8.1
11.7
17.1
Eb
9.8
14.1
20.5
Ab
11.4
16.4
24
Db
13
18.8
27.4
Gb
14.7
21.1
30.8
Cb
16.3
23.5
34.2
Fb
17.9
25.8
37.6
VIOL MAINTENANCE ISSUES
THE MAJOR CAUSE OF POOR TUNING IS POOR VIOL MAINTENANCE
PEGS
FRETS
STRINGS
BRIDGE
NUT
PROPERLY
DOPED
(NO CREAKING)
LEARN
TO TIE
YOUR OWN
(VdGSA
WEBSITE)
NOT FALSE
FEET FLAT
TO BELLY
TIGHTLY
GLUED
TO NECK
PROPERLY
FIT TO
PEGBOX
ORIENTED
TO A
COMFORTABLE
ANGLE
REPLACED
WITH
MACHINE PEGS
(PEGHEDS.COM)
NOT TOO
LOOSE
NOT TOO
TIGHT
USE
MONOFILIAMENT
NYLON
PROPERLY
WOUND
AROUND
PEGS
NOT
CURVED
OR
WARPED
STRING
NOTCHES
LUBED
WITH
GRAPHITE
(NO. 2 PENCIL)
STRING
NOTCHES
LUBED
WITH
GRAPHITE
WHEN ARE WE PERFECTLY IN TUNE – OR WHEN IS IT GOOD ENOUGH?
TO ACHIEVE ZERO TUNING ERROR
TIME SPENT TUNING
100%
0%
TO ACHIEVE ACCEPTABLE TUNING WITH TYPICAL VIOL MAINTENANCE (TVM)
TIME SPENT PLAYING
TO ACHIEVE ACCEPTABLE TUNING WITH GOOD VIOL MAINTENANCE (GVM)
TIME AVAILABLE FOR SESSION
DEGREE
OF TUNING
SATISFACTION
0
GVM
DEGREE
OF SESSION
ENJOYMENT
100%
ACCEPTABLE
TUNING ERROR
MARGINALLY
TOLERABLE
TVM
SWEET
SPOT
0
TUNING ERROR
THE SECRETS OF GOOD VIOL TUNING
MAINTAIN YOUR VIOL IN TIP-TOP CONDITION
BE WILLING AND ABLE TO TUNE YOUR FRETS AS READILY AS YOUR PEGS
IF FRETS ARE NOT DOUBLED, TUNE FRETS APPROPRIATE TO THE PIECE BEING PLAYED
LEARN TO RECOGNIZE AND LISTEN FOR BEATS – ESPECIALLY SLOW BEATS
LEARN THE “FEELING” OF THE SOUND WHEN BEATS DISAPPEAR
ALWAYS TUNE ALL FOURTHS SLIGHTLY WIDER THAN PURE
(ALL FIFTHS SLIGHTLY NARROWER THAN PURE)
n
ALWAYS TUNE FLATTED NOTES HIGH ( n S ARE “FLATS” IN SHARPED KEY SIGNATURES)
ALWAYS TUNE SHARPED NOTES LOW ( S ARE “SHARPS” IN FLATTED KEY SIGNATURES)
ALWAYS TUNE AND PLAY YOUR VIOL(S) WITH THE APPROPRIATE A-FREQUENCY
THE DAY BEFORE A CONSORT SESSION
CAREFULLY LISTEN TO INTONATION AS YOU PLAY AND QUICKLY CORRECT TUNING
PROBLEMS AT THE NEXT BREAK IN THE MUSIC
WHEN CHECKING AN OPEN STRING, LISTEN BEFORE TURNING THE PEG
IT MAY ALREADY BE GOOD ENOUGH (NO PERCEPTIBLE BEATING)
CONSIDER DOUBLING THE FRET IN THE FIRST FRET POSITION (DON’T SPLIT THE EXISTING FRET)
IF YOU ENJOY THE SONORITY OF PURE HARMONIES, DOUBLE FRETS IN OTHER FRET POSITIONS,
PARTICULARLY THE FOURTH AND SIXTH FOR BASSES, AND ALSO THE THIRD FOR TENORS
BIBLIOGRAPHY
Barbour, J. Murray. Tuning and Temperament: A Historical Survey. East Lansing: Michigan State
College Press, 1951; Reprint, New York: Dover Publications, 2004
Duffin, Ross W. How Equal Temperament Ruined Harmony. New York: W. W. Norton & Co., 2007
Ganassi, Sylvestro. Regola Rubertina and Lettione Seconda. Venice 1542-1543. English translation
by Richard Bodig. Artarmon NSW Australia: Saraband Music, 1999
Gouk, Penelope. Music, Science, and Natural Magic in Seventeenth-Century England.
London: Yale University Press, 1999
Helmholtz, Herman von. On the Sensations of Tone; translated by A. J. Ellis, 4th edition 1885
Reprint, New York: Dover Publication, 1954
Lindley, Mark: ‘Temperaments, 9. Fretted Instruments’
Grove Music Online ed. L. Macy (Accessed 24 January 2007), http://www.grovemusic.com
Meyer, Christian: ‘Observations pour une analyse des tempéraments des instruments à cordes pincées:
le luth de Hans Gerle (1532)’, Revue de musicologie © 1985 Société Française de Musicologie
Moore, Brian C. J.: Introduction to the Psychology of Hearing. New York: Academic Press, 2003
Shepherd, Martin: ‘Tuning and Temperament”, http://luteshop.co.uk/tuning.htm
Wikla, Arto: ‘Vincenzo Galileo [sic] about glued frets etc.’, http://www.cs.helsinki.fi/u/wikla/mus/fronimo.html