1. THE SECRETS OF VIOL TUNING A LECTURE / DEMONSTRATION FREDERICK G. REINAGEL PAN-PACIFIC GAMBA GATHERING HONOLULU, HAWAII AUGUST 2007

2. 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 AVOID THAT WHICH IS UNCOMFORTABLE OR PAINFUL CREATE ORDER AND STRUCTURE TO SIMPLIFY OUR MODEL OF REALITY ENHANCE SOCIAL WELL-BEING MAXIMIZE (INTENDED) CONSONANCE MINIMIZE (UNINTENDED) DISSONANCE ACHIEVE UNIFORM INTERVALS AND CHORDS PLAY HARMONIOUSLY WITH OTHERS

3. 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

4. THE RELATIONSHIP OF PITCH TO FREQUENCY 100150200250300350400450500550600650700750800 FREQUENCY (HERTZ) MIDDLE C G d g d’ g’ d’’ g’’ PITCHPITCH TREBLE VIOL BASS VIOL

5. THE RELATIONSHIP OF PITCH TO FREQUENCY – LOGARITHMIC SCALE 507510015020030040060080012001600 FREQUENCY (HERTZ) – LOGARITHMIC SCALE MIDDLE C G d g d’ g’ d’’ g’’ PITCHPITCH TREBLE VIOL BASS VIOL

6. THE RELATIONSHIP OF PITCH TO FREQUENCY – PITCH COMPRESSION TREBLE VIOL 507510015020030040060080012001600 FREQUENCY (HERTZ) – LOGARITHMIC SCALE G d g d’ g’ d’’ g’’ PITCHPITCH GG D d’’’ g’’’ 7-STRING BASS VIOL MIDDLE C

7. $10 OCTAVE HALFTONE COMMA (APPROXIMATE) CENT $1 PITCH MEASUREMENT UNITS $1 $10 MACRO- TONAL MICRO- TONAL $1 1¢ PYTHAGOREAN – 23.5 ¢ NEWTONIAN (1/53 OCTAVE) – 22.6 ¢ SYNTONIC – 21.5 ¢ 25¢ $2

8. 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’’’’’’

9. HARMONIC MODES OF STRING VIBRATION 1st 2nd 3rd 4th 5th FUNDA- MENTAL OCTAVE ABOVE OCTAVE AND A FIFTH ABOVE TWO OCTAVES ABOVE TWO OCTAVES AND A MAJOR THIRD ABOVE 6th TWO OCTAVES AND A FIFTH ABOVE OCTAVEOCTAVE FIFTHFIFTH FOURTHFOURTH MAJOR THIRDMAJOR THIRD MINOR THIRDMINOR THIRD 8th THREE OCTAVES ABOVE 9th THREE OCTAVES AND A MAJOR SECOND ABOVE FOURTHFOURTH MAJOR SECONDMAJOR SECOND THREE OCTAVES AND A MAJOR THIRD ABOVE 10th MAJOR SECOND*MAJOR SECOND*

10. MINOR SECOND 4 x 9 5 8 ( ) 15 x 25 16 24 ( ) 8 x 10 9 ( ) 4 x 6 5 DIFFERENCES OF HARMONIC (PURE) INTERVALS ASCENDING PURE INTERVAL FREQ. RATIO NAME OCTAVE FIFTH FOURTH 1:2 2:3 3:4 4:5 8:9 DESCENDING PURE INTERVAL FREQ. RATIO NAME FIFTH FOURTH MAJOR THIRD 3:2 4:3 5:4 9:8 10:9 MAJOR THIRD MAJOR SECOND MAJOR SECOND MAJOR SECOND* 4:56:5 MAJOR THIRD MINOR THIRD DIFFERENCE INTERVAL 1 x 3 2 ( ) 2 x 4 3 3:4 8:9 ( ) 3 x 5 4 ( ) FREQUENCY RATIO 15:16 9:10 24:25 80:81 NAME FOURTH MAJOR SECOND MINOR SECOND MAJOR SECOND* AUGMENTED UNISON COMMA ALTERNATE NAME MAJOR WHOLETONE MINOR WHOLETONE DIATONIC SEMITONE CHROMATIC SEMITONE SYNTONIC COMMA 15:16 25:24 AUGMENTED UNISON 125:128 DIESIS NEWTONIAN COMMAS (APP) 22 (0.07¢) 9 (-0.14¢) 5 (1.48¢) 8 (-1.27¢) 1 (1.14¢) 3 (-2.75¢) 2 (4.22¢) FIFTH2:35:4 ( ) 5:6 MINOR THIRD 14 (1.34¢) 2 x 5 3 4 MAJOR THIRD MINOR DIESIS ONE OCTAVE = THREE MAJOR SECONDS + TWO MAJOR SECONDS* + TWO MINOR SECONDS (3x9 + 2x8 + 2x5 = 53 COMMAS)

11. FUNDA- MENTAL 2ND 4TH 6TH 5TH COMPLEX TONE HARMONICCOMPONENTSHARMONICCOMPONENTS 3RD WAVEFORM ANALYSIS/ HARMONIC FUSION

12.  HARMONICALLY-TUNED CHORD (FUNDA- MENTAL) 2ND 3RD 5TH 4TH     CHORD HARMONICHARMONIC g d’ b’ g’ G 0.03 SECONDS

13. g# (4 TH FRET e-STRING) 200 HZ COMBINING TWO TONES – FORMATION OF BEATS a (OPEN STRING) 210 HZ BASS VIOLBASS VIOL 10 BEATS PER SECOND 1/5 SECOND COMBINED TONE

14. d’’ (5 TH FRET a’-STRING 3 CENTS FLAT) 533 HZ d’’ (OPEN STRING) 534 HZ TREBLE VIOLTREBLE VIOL 1 BEAT PER SECOND COMBINED TONE COMBINING TWO TONES – FORMATION OF A SLOW BEAT 2 SECONDS

15. BASS VIOLBASS VIOL a (OPEN STRING) 207 HZ COMBINED TONES SHOWING DIFFERENCE TONE d 138 HZ f  ’ (4TH FRET d’’-STRING) 345 HZ COMBINING TWO TONES – FORMATION OF A DIFFERENCE TONE

16. MINOR SEVENTH (MAJOR SECOND) ONE COMMA SHORT OF 5 OCTAVES ONE COMMA SHORT 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A DD GG FF BEADGCF BB EE AA DD GG CC STRETCHING THE CHAIN OF FOURTHS - REGULAR TEMPERAMENTS DIMINISHED SECOND (AUGMENTED SEVENTH) 12 X 22 = 264 5 X 53 = 265 4 X 22 = 88 2 X 53 – 17 = 89 PURE FOURTH = 22 COMMAS SIMPLE INTERVAL NAMES (INVERSION INTERVAL NAMES)

17. SCALE PROPERTIES SCALE TYPE HARMONIC (JUST INTONATION) REGULAR TEMPERAMENTS (EXCEPT EQUAL) CIRCULAR (WELL) TEMPERAMENTS EQUAL TEMPERAMENT SCALE CHARACTERISTICS UNEQUAL WHOLETONES UNEQUAL SEMITONES NON-COINCIDENT ENHARMONICS EQUAL WHOLETONES UNEQUAL SEMITONES NON-COINCIDENT ENHARMONICS UNEQUAL WHOLETONES UNEQUAL SEMITONES COINCIDENT ENHARMONICS EQUAL WHOLETONES EQUAL SEMITONES COINCIDENT ENHARMONICS MUSIC NOTATION AND INTERVAL TERMINOLOGY UNDERDETERMINED UNIQUELY DETERMINED OVERDETERMINED

18. THE CASE FOR NON-EQUAL TEMPERAMENT FOR VIOL (AND LUTE) TUNING 16 TH AND 17 TH 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

19. PURE MAJOR THIRD DIVIDING THE COMMA ONE COMMA 22.6 CENTS PYTHAGOREAN ¼ -COMMA MEANTONE ONE OCTAVE 53 COMMAS PURE FOURTH 22 COMMAS d’’a’e’c’gd g e’ 18 COMMAS 17 COMMAS d’

20. CHECKING OPEN-STRING TUNING BY COMPARING HARMONICS 1PLAY 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. 2COMPARE THE PITCHES OF THE HARMONIC PAIRS INDICATED BY CIRCLES WITH THE SAME NUMBER. 3THE PITCH CORRESPONDING THE “+” SIGN SHOULD BE VERY SLIGHTLY HIGHER THAN THAT OF THE “-”. (EXCEPTION: FOR 1/4-COMMA, -3 SHOULD EXACTLY MATCH +3) 4THE PITCH OF THE HARMONIC INDICATED BY “6” ON THE LOWEST STRING SHOULD EXACTLY MATCH THE OPEN STRING PITCH OF THE HIGHEST STRING. -2+3 +2 -3 +4 - 4 +5 -5,6 +1 6 3 RD HARMONIC 4 TH HARMONIC 5 TH HARMONIC

21. CHECKING FRET TUNING BY SYMPATHETIC RESONANCE 1MAKE SURE ALL OPEN STRINGS ARE TUNED AS ACCURATELY AS POSSIBLE. 2PLAY 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. 3CHECK 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. 4IF SYMPATHETIC RESONANCE IS NOT EXHIBITED, COMPARE PITCHES OF THE STOPPED AND OPEN STRINGS AND ADJUST FRET TO EXACTLY MATCH PITCHES. 5CHECK THAT FRETTED TONE PRODUCES SYMPATHETIC RESONANCE WITH NO BEATING. 3 2 4 3 5 2 3 5 2 4 3 5 2 123456123456 1,6 5

22. WHERE DO YOU PUT THE FIRST FRET? EQUAL TEMPERAMENT (1/12 COMMA) 1/6 COMMA 1/4 COMMA 1 ST FRET 2 ND FRET NUT DEDE DEDE ABAB EFEF CDCD DD AA EE CC GG DD EE BB F DD AA EE GAGA BASS OR TREBLE VIOL DD AA EE CC GG DD EE BB F DD AA EE

23. WHERE DO YOU PUT THE FOURTH FRET? EQUAL TEMPERAMENT (1/12 COMMA) 1/6 COMMA 1/4 COMMA FF CC GG E B FF GG DD AA FF CC GG 4 TH FRET 5 TH FRET 3 RD FRET FGFG FGFG CDCD GAGA EFEF BCBC FF CC GG E B FF GG DD AA FF CC GG BASS OR TREBLE VIOL

24. WHERE DO YOU PUT THE THIRD FRET? EQUAL TEMPERAMENT (1/12 COMMA) 1/6 COMMA 1/4 COMMA AA EE BB GG DD AA BB FC AA EE BB 3 RD FRET 4 TH FRET 2 ND FRET ABAB ABAB EFEF BCBC GAGA DEDE AA EE BB GG DD AA BB F C AA EE BB TENOR VIOL

25. UNTEMPERED (PYTHAGOREAN) 1/12 COMMA (EQUAL) 1/6 COMMA 1/4 COMMA AUGMENTED UNISON HOW FRETS MIGRATE AS A FUNCTION OF TEMPERAMENT FIFTH FOURTH MAJOR THIRD MINOR THIRD MAJOR SECOND MINOR SECOND AUGMENTED FOURTH DIMINISHED FIFTH DIMINISHED FOURTH AUGMENTED SECOND 15:169:108:95:64:53:42:3 HARMONIC INTERVALS NUT 24:25

26. 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)

27. SIXTH COMMAFIFTH COMMAQUARTER 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 A000 D1.62.33.4 G3.34.76.8 C4.9710.3 F6.59.413.7 Bb8.111.717.1 Eb9.814.120.5 Ab11.416.424 Db1318.827.4 Gb14.721.130.8 Cb16.323.534.2 Fb17.925.837.6 DEVIATIONS FROM EQUAL TEMPERAMENT IN CENTS

28. VIOL MAINTENANCE ISSUES THE MAJOR CAUSE OF POOR TUNING IS POOR VIOL MAINTENANCE PEGS PROPERLY DOPED (NO CREAKING) PROPERLY FIT TO PEGBOX ORIENTED TO A COMFORTABLE ANGLE REPLACED WITH MACHINE PEGS (PEGHEDS.COM) FRETS LEARN TO TIE YOUR OWN (VdGSA WEBSITE) NOT TOO LOOSE NOT TOO TIGHT USE MONO- FILIAMENT NYLON STRINGS NOT FALSE PROPERLY WOUND AROUND PEGS BRIDGE FEET FLAT TO BELLY NOT CURVED OR WARPED STRING NOTCHES LUBED WITH GRAPHITE (NO. 2 PENCIL) NUT TIGHTLY GLUED TO NECK STRING NOTCHES LUBED WITH GRAPHITE

29. 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 DEGREE OF TUNING SATIS- FACTION 100% MARGINALLY TOLER- ABLE 0 TUNING ERROR DEGREE OF SESSION ENJOYMENT 0 GVM TVM ACCEPTABLE SWEET SPOT

30. 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) 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-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

31. 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, 4 th 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.comhttp://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 MusicologieSociété 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.htmhttp://luteshop.co.uk/tuning.htm Wikla, Arto: ‘Vincenzo Galileo [sic] about glued frets etc.’, http://www.cs.helsinki.fi/u/wikla/mus/fronimo.html