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 OURENVIRONMENT TO:

AS MUSICIANS, WECONTROL OUR ACOUSTICENVIRONMENT (TUNE) TO:

PURSUE THAT WHICH ISCOMFORTABLE OR PLEASURABLE

AVOID THAT WHICH ISUNCOMFORTABLE OR PAINFUL

CREATE ORDER AND STRUCTURETO SIMPLIFY OUR MODEL OF REALITY

ENHANCE SOCIAL WELL-BEING

MAXIMIZE (INTENDED) CONSONANCE

MINIMIZE (UNINTENDED) DISSONANCE

ACHIEVE UNIFORM INTERVALS AND CHORDS

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

100 150 200 250 300 350 400 450 500 550 600 650 700 750 800

FREQUENCY (HERTZ)

MIDDLE C

G

d

g

d’

g’

d’’

g’’

PITCH

TREBLEVIOL

BASSVIOL

THE RELATIONSHIP OF PITCH TO FREQUENCY – LOGARITHMIC SCALE

50 75 100 150 200 300 400 600 800 1200 1600

FREQUENCY (HERTZ) – LOGARITHMIC SCALE

MIDDLE C

G

d

g

d’

g’

d’’

g’’

PITCH

TREBLEVIOL

BASSVIOL

THE RELATIONSHIP OF PITCH TO FREQUENCY – PITCH COMPRESSION

TREBLEVIOL

50 75 100 150 200 300 400 600 800 1200 1600

FREQUENCY (HERTZ) – LOGARITHMIC SCALE

G

d

g

d’

g’

d’’

g’’

PITCH

GG

D

d’’’

g’’’

7-STRINGBASSVIOL

MIDDLE C

$10OCTAVE

HALFTONE

COMMA(APPROXIMATE)

CENT

$1

PITCH MEASUREMENT UNITS

$1

$1

$10$10MACRO-

TONAL

MICRO-TONAL

$1

1¢

PYTHAGOREAN – 23.5 ¢NEWTONIAN (1/53 OCTAVE) – 22.6 ¢SYNTONIC – 21.5 ¢

25¢

$2$2

$2$2

0

5

10

15

20

25

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5

LIMITS OF HUMAN HEARING PITCH DISCRIMINATION

JND(CENTS)

FREQUENCY (HERTZ) – LOGARITHMIC SCALE

250 500 1000 2000 4000 8000

JUST NOTICEABLE DIFFERENCE (JND) AS A FUNCTION OF FREQUENCY

c’ c’’ c’’’ c’’’’ c’’’’’

PITCH

c’’’’’’

HARMONIC MODES OF STRING VIBRATION

1st 2nd 3rd 4th 5th

FUNDA-MENTAL

OCTAVEABOVE

OCTAVEAND A FIFTH

ABOVE

TWOOCTAVES

ABOVE

TWOOCTAVES AND

A MAJORTHIRD ABOVE

6th

TWOOCTAVES

AND A FIFTHABOVE

OCTAVE

FIFTH

FOURTH

MAJOR THIRD

MINOR THIRD

8th

THREEOCTAVES

ABOVE

9th

THREEOCTAVES

AND A MAJORSECOND ABOVE

FOURTH

MAJOR SECOND

THREEOCTAVES AND

A MAJORTHIRD ABOVE

10th

MAJOR SECOND

*

MINORSECOND

4 x 9

5 8

( )15 x 25

16 24

( )8 x 10

9 9

( )4 x 6

5 5

DIFFERENCES OF HARMONIC (PURE) INTERVALS

ASCENDINGPURE INTERVAL

FREQ.RATIO

NAME

OCTAVE

FIFTH

FOURTH

1:2

2:3

3:4

4:5

8:9

DESCENDINGPURE INTERVAL

FREQ.RATIO

NAME

FIFTH

FOURTH

MAJORTHIRD

3:2

4:3

5:4

9:8

10:9

MAJORTHIRD

MAJORSECOND

MAJORSECOND

MAJORSECOND*

4:5 6:5MAJORTHIRD

MINORTHIRD

DIFFERENCE INTERVAL

1 x 3

2 2

( )

( )2 x 4

3 3

3:4

8:9

( )3 x 5

4 4

( )

FREQUENCYRATIO

15:16

9:10

24:25

80:81

NAME

FOURTH

MAJORSECOND

MINORSECOND

MAJORSECOND*

AUGMENTEDUNISON

COMMA

ALTERNATENAME

MAJORWHOLETONE

MINORWHOLETONE

DIATONICSEMITONE

CHROMATICSEMITONE

SYNTONICCOMMA

15:16 25:24AUGMENTED

UNISON 125:128 DIESIS

NEWTONIANCOMMAS (APP)

22 (0.07¢)

9 (-0.14¢)

5 (1.48¢)

8 (-1.27¢)

1 (1.14¢)

3 (-2.75¢)

2 (4.22¢)

FIFTH2:3 5:4 ( )5:6MINORTHIRD 14 (1.34¢)2 x 5

3 4

MAJORTHIRD

MINORDIESIS

ONE OCTAVE = THREE MAJOR SECONDS + TWO MAJOR SECONDS* + TWO MINOR SECONDS (3x9 + 2x8 + 2x5 = 53 COMMAS)

FUNDA-MENTAL

2ND

4TH

6TH

5TH

COMPLEXTONE

HARMONIC

COMPONENTS

3RD

WAVEFORM ANALYSIS/HARMONIC FUSION

HARMONICALLY-TUNED CHORD

(FUNDA-MENTAL)

2ND

3RD

5TH

4TH

CHORD

HARMONIC

g

d’

b’

g’

G

0.03 SECONDS

g#(4TH FRETe-STRING)

200 HZ

COMBINING TWO TONES – FORMATION OF BEATS

a(OPEN

STRING)210 HZ

BASS VIOL

10 BEATSPER

SECOND

1/5 SECOND

COMBINEDTONE

d’’(5TH FRETa’-STRING3 CENTS

FLAT)533 HZ

d’’(OPEN

STRING)534 HZ

TREBLE VIOL

1 BEATPER

SECOND

COMBINEDTONE

COMBINING TWO TONES – FORMATION OF A SLOW BEAT

2 SECONDS

BASS VIOL

a(OPEN

STRING)207 HZ

COMBINEDTONES

SHOWINGDIFFERENCE

TONEd

138 HZ

f’(4TH FRETd’’-STRING)

345 HZ

COMBINING TWO TONES – FORMATION OF A DIFFERENCE TONE

MINORSEVENTH(MAJOR

SECOND)

ONE COMMA SHORT OF 5 OCTAVES

ONE COMMASHORT OF 5:16

MINOR SIXTH (MAJOR THIRD)

DIMINISHED FIFTH (AUGMENTED FOURTH)

DIMINISHED OCTAVE (AUGMENTED UNISON)

MINOR THIRD(MAJOR SIXTH)

MINOR SECOND (MAJOR SEVENTH)

DIMINISHED FOURTH (AUGMENTED FIFTH)

DIMINISHED SEVENTH (AUGMENTED SECOND)

DIMINISHED THIRD (AUGMENTED SIXTH)

DIMINISHED SIXTH (AUGMENTED THIRD)

A D G F B E A D G C F B E A D GC

STRETCHING THE CHAIN OF FOURTHS - REGULAR TEMPERAMENTS

DIMINISHED SECOND (AUGMENTED SEVENTH)

12 X 22 = 2645 X 53 = 265

4 X 22 = 882 X 53 – 17 = 89

PUREFOURTH =

22 COMMAS

SIMPLEINTERVAL

NAMES(INVERSIONINTERVALNAMES)

SCALE PROPERTIES

SCALE TYPE

HARMONIC(JUST INTONATION)

REGULAR TEMPERAMENTS(EXCEPT EQUAL)

CIRCULAR (WELL)TEMPERAMENTS

EQUAL TEMPERAMENT

SCALECHARACTERISTICS

UNEQUAL WHOLETONESUNEQUAL SEMITONES

NON-COINCIDENT ENHARMONICS

EQUAL WHOLETONESUNEQUAL SEMITONES

NON-COINCIDENT ENHARMONICS

UNEQUAL WHOLETONES UNEQUAL SEMITONES

COINCIDENT ENHARMONICS

EQUAL WHOLETONESEQUAL SEMITONES

COINCIDENT ENHARMONICS

MUSIC NOTATION ANDINTERVAL TERMINOLOGY

UNDERDETERMINED

UNIQUELYDETERMINED

OVERDETERMINED

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 FRETSUP AND DOWN, ACCORDING TO THE REQUIREMENTS OF YOUR EAR. I COULD PROVIDEA 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

PUREMAJOR THIRD

DIVIDING THE COMMA

ONE COMMA22.6 CENTS

PYTHAGOREAN

¼-COMMAMEANTONE

ONE OCTAVE53 COMMAS

PURE FOURTH22 COMMAS

d’’a’e’c’gd

g c’ e’

18 COMMAS

17 COMMAS

d’

CHECKING OPEN-STRING TUNING BY COMPARING HARMONICS

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.

-1

-2+3

+2

-3 +4

- 4 +5

-5,6

+1

6

3RD

HARMONIC

4TH

HARMONIC

5TH

HARMONIC

CHECKING FRET TUNING BY SYMPATHETIC RESONANCE

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 OPENSTRINGS AND ADJUST FRET TO EXACTLY MATCH PITCHES.

5 CHECK THAT FRETTED TONE PRODUCES SYMPATHETIC RESONANCE WITH NO BEATING.

32

4 3

5 2

3 5

2 4

35 2

1

2

3

4

5

6

1,6

1,6

1,6

5

WHERE DO YOU PUT THE FIRST FRET?

EQUAL TEMPERAMENT(1/12 COMMA)

1/6 COMMA 1/4 COMMA

1ST

FRET

2ND

FRET

NUT

DE

DE

AB

EF

CD

DAECGD

EBFDAE

GA

BASS OR TREBLE VIOL

DAECGD

EBFDAE

WHERE DO YOU PUT THE FOURTH FRET?

EQUAL TEMPERAMENT(1/12 COMMA)

1/6 COMMA 1/4 COMMA

FCGEBF

GDAFCG4TH

FRET

5TH

FRET

3RD

FRET

FG

FG

CD

GA

EF

BC

FCGEBF

GDAFCG

BASS OR TREBLE VIOL

WHERE DO YOU PUT THE THIRD FRET?

EQUAL TEMPERAMENT(1/12 COMMA)

1/6 COMMA 1/4 COMMA

AEBGDA

BFCAEB3RD

FRET

4TH

FRET

2ND

FRET

AB

AB

EF

BC

GA

DE

AEBGDA

BFCAEB

TENOR VIOL

UNTEMPERED(PYTHAGOREAN)

1/12 COMMA(EQUAL)

1/6 COMMA

1/4 COMMA

AU

GM

EN

TE

DU

NIS

ON

HOW FRETS MIGRATE AS A FUNCTION OF TEMPERAMENT

FIF

TH

FO

UR

TH

MA

JOR

TH

IRD

MIN

OR

TH

IRD

MA

JOR

SE

CO

ND

MIN

OR

SE

CO

ND

AU

GM

EN

TE

DF

OU

RT

H

DIM

INIS

HE

DF

IFT

H

DIM

INIS

HE

DF

OU

RT

H

AU

GM

EN

TE

DS

EC

ON

D

15:16 9:10 8:9 5:6 4:5 3:4 2:3

HARMONIC INTERVALS

NUT

24:25

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 TONESHOULD BE MULTI-TEMPERAMENTONE 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 BEIN BEST TUNE (INCLUDING FRETS) AS AN AUDIO REFERENCE. OTHER VIOLSTUNE UNISONS OR OCTAVES TO IT (FIFTHS AND FOURTHS ARE NOT PURE)

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

DEVIATIONS FROM EQUAL TEMPERAMENT IN CENTS

VIOL MAINTENANCE ISSUES

THE MAJOR CAUSE OF POOR TUNING IS POOR VIOL MAINTENANCE

PEGS

PROPERLYDOPED

(NO CREAKING)

PROPERLYFIT TO

PEGBOX

ORIENTEDTO A

COMFORTABLEANGLE

REPLACEDWITH

MACHINE PEGS(PEGHEDS.COM)

FRETS

LEARNTO TIE

YOUR OWN(VdGSA

WEBSITE)

NOT TOOLOOSE

NOT TOOTIGHT

USEMONO-

FILIAMENTNYLON

STRINGS

NOT FALSE

PROPERLYWOUNDAROUND

PEGS

BRIDGE

FEET FLATTO BELLY

NOTCURVED

ORWARPED

STRINGNOTCHES

LUBEDWITH

GRAPHITE(NO. 2 PENCIL)

NUT

TIGHTLYGLUED

TO NECK

STRINGNOTCHES

LUBEDWITH

GRAPHITE

WHEN ARE WE PERFECTLY IN TUNE – OR WHEN IS IT GOOD ENOUGH?

100%

TIME AVAILABLE FOR SESSION

0%

TIME SPENT TUNING

TIME SPENT PLAYING

TO ACHIEVE ZERO TUNING ERROR

TO ACHIEVE ACCEPTABLE TUNING WITH TYPICAL VIOL MAINTENANCE (TVM)

TO ACHIEVE ACCEPTABLE TUNING WITH GOOD VIOL MAINTENANCE (GVM)

TUNING ERROR

DEGREEOF TUNING

SATIS-FACTION

100%

MARGINALLYTOLER-ABLE

0 TUNING ERROR

DEGREEOF SESSIONENJOYMENT

0

GVM

TVM

ACCEPTABLESWEETSPOT

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 BEATSLEARN THE “FEELING” OF THE SOUND WHEN BEATS DISAPPEAR

ALWAYS TUNE ALL FOURTHS SLIGHTLY WIDER THAN PURE(ALL FIFTHS SLIGHTLY NARROWER THAN PURE)

ALWAYS TUNE SHARPED NOTES LOW ( S ARE “SHARPS” IN FLATTED KEY SIGNATURES)

ALWAYS TUNE FLATTED NOTES HIGH ( S ARE “FLATS” IN SHARPED KEY SIGNATURES)

ALWAYS TUNE AND PLAY YOUR VIOL(S) WITH THE APPROPRIATE A-FREQUENCYTHE DAY BEFORE A CONSORT SESSION

CAREFULLY LISTEN TO INTONATION AS YOU PLAY AND QUICKLY CORRECT TUNINGPROBLEMS AT THE NEXT BREAK IN THE MUSIC

WHEN CHECKING AN OPEN STRING, LISTEN BEFORE TURNING THE PEGIT 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 StateCollege 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 translationby 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 1885Reprint, 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