Tuning

1. Tuning • Intervals are based on relative pitches • Works fine if you are a solo artist! • Groups of musicians must tune to a common reference pitch • Concert A (440 Hz, maybe) • Middle C (used for pianos) • Concert Bb (used for brass instruments) • All other tunings are taken relative to the agreed upon reference

2. Middle C D E F G A B C D E F G A B C (440 Hz) Assigning Notes to Pitches • We arbitrarily assign note names on a piano using the letters A-G for the white keys • By convention, the A above "Middle" C is fixed at a frequency of 440 Hz

3. Brief History of 440 A • No commonly agreed upon reference pitches before 1600 • Instruments often tuned to organ pipes of local churches • In 1619, composer Michael Praetorius suggested 425 Hz as a standard tuning (the so-called "chamber pitch") • Higher tuning pitches not recommended, due to limited construction techniques for stringed instruments • In 1855, French physicist Jules Lissajous developed a technique for calibrating tuning forks, suggested 435 Hz as the standard pitch • French government (under Napoleon) adopted 435 Hz in 1859 • Adopted internationally in 1885 at a conference in Vienna

4. Lissajous Patterns • Lissajous's apparatus bounced a light beam off mirrors attached to tuning forks • Light produced patterns that could determine relative frequencies of forks, based on standard ratios for intervals • The basic technique is still in use today!

5. History of 440 A, continued • Industrial Age ( late 1800s) led to improvements in metallurgy and construction techniques for instruments • Concert pitch gradually started to creep up • Present day 440 pitch adopted in US in 1939 (later by ANSI) • Modern orchestras (especially in Europe) now use 442 or even 445 as a reference pitch • Note: this "history" is grossly over-simplified (we may never know exactly how standard pitches evolved)

6. Modern Tuning Techniques • Instruments today can be tuned electronically (commercial tuning apparatus - stroboscopes, etc) or acoustically (tuning forks) • Monophonic instruments (i.e. most band instruments) are tuned to a single reference, all other pitches assumed to be "in tune" • Polyphonic instruments (piano, guitar, most orchestra instruments, bagpipes, etc) tune to one reference, all other tunings derived relative to that reference

7. Electronic Tuning Example • An electronic tuner shows exactly what pitch is being played and how far off it is "Sharp" - pitch is too high Just right! "Flat" - pitch is too low

8. Acoustic Tuning • Acoustic tuning is done by comparing the instrument's pitch to a reference • Pitches that are close to each other but out of tune harmonically will "beat" at a frequency equal to the difference between the two frequencies being played • Example: 442 vs 440 beats at 2 Hz • Pitches that are not close will "beat" due to interference in the upper harmonics (good piano tuners use this characteristic)

9. Fourth Fourth Fourth Fourth Third Acoustic Tuning Example • "Standard" tuning on a 6-string guitar is E A D G B E • Tuning by "straight" frets • Fourth == 5 frets, Third == 4 frets • Tuning by harmonics • Fourth == 5th 7th frets, Third == 9th 5th frets • As pitches get close, listen for "beats" • No beats == pitches are in tune

10. Why this Happens • Consider two pitches an octave apart • Coincidental "zero crossings" (shown by arrows) eliminate "beats" • Same effect with a Fifth

11. Out of Tune Pitches • Two pitches a half step apart (no crossings) • Out of tune Fifth (2 cents worth)

12. This all sounds very clinical So how come piano tuners still have jobs?

13. Tuning "for real" Proper tuning of a particular note on a particular instrument is affected by many factors (some we can control, some we cannot) • Psychoacoustics • Physical characteristics of the instrument (i.e. how it is constructed) • Overall temperament of the instrument (i.e. how it is tuned)

14. Psychoacoustics • Our ears process frequencies differently depending on what register the notes are in • Higher frequencies sound "flat" • Lower frequencies sound "sharp" • Professional piano tuners compensate for this by tuning upper registers slightly sharp, and lower registers slightly flat • Differences can be as much as 20-30 cents

15. Intonation • Intonation is how pitches are assigned or determined relative to each other • "Good" intonation means that all notes in all positions are in tune, relatively speaking • "Bad" intonation means that some notes are out of tune • Intonation can be adjusted! • By the manufacturer ("setting up" a guitar) • By the musician (adjusting the embouchure) • Harmonic partials are almost always in tune - problems are often encountered with chords

16. Temperament(Who says scales are boring?) • Temperament is how pitches are adjusted relative to each other when an instrument is tuned • Temperament has a profound effect on intonation • It's impossible to get an instrument to be truly "in tune" • Temperaments have been confounding musicians for almost 5000 years!

17. Review of Intervals Ratio Interval f0 Start f0x9/8 Second f0x5/4 Third f0x4/3 Fourth Ration Interval f0x3/2 Fifth f0x5/3 Sixth f0x15/8 Seventh f0x2 Octave

18. Now Assign Note Names Name Interval C 1/1 Start D 9/8 Second E 5/4 Third F 4/3 Fourth Name Interval G 3/2 Fifth A 5/3 Sixth B 15/8 Seventh C 2/1 Octave

19. Map onto Keys C D E F G A B C

20. This one doesn't work! Taking the Fifth Name Interval C 1/1 Start D 9/8 Second E 5/4 Third F 4/3 Fourth Name Interval G 3/2 Fifth A 5/3 Sixth B 15/8 Seventh C 2/1 Octave Corresponding notes in each row are perfect Fifths (C-G, D-A, E-B, F-C), and should be separated by a ratio of 3/2

21. A Little Music History • Much of what we understand today about tuning and temperament was discovered by the ancient Greeks (specifically, Pythagoras and his followers) • Harmonic Series, Intervals, etc • One of the oldest tunings is the Pythagorean tuning, which is based on the interval of the Fifth • Tuning Factoid: the notes of any diatonic scale can be rearranged in sequence such that the interval between each consecutive note is a Fifth: C D E F G A B becomes: F C G D A E B

22. You can even buy a wristwatch whose face is a Circle of Fifths! Circle of Life, er, Fifths By extending this idea (and utilizing both black and white keys on a piano), it is possible to start at any note, go up twelve perfect Fifths, and end up at the same note from whence you started (just in a different octave) We call this the Circle of Fifths; it is an important fundamental concept that is the basis for much of modern music theory

23. Back to Pythagoras • The Pythagoreans based their tuning on Fourths and Fifths, which were considered harmonically "pure": C F G C • The Fourth was subdivided into two tones (whole step interval), and a half tone (half step interval) • This arrangement of intervals is called a tetrachord • Two tetrachords can be concatenated together (separated by a whole step) to create a diatonic scale Fourth Fifth Fourth Fifth

24. Tone Tone Half Tone Tone Tone Half Tetrachord Tetrachord Diatonic Tetrachords C D E F G A B C

25. All whole step intervals are equal at 9/8 (204 cents) All half step intervals are equal at 256/243 (90 cents) Pythagorean Tuning Name Interval C 1/1 Start D 9/8 Second E 81/64 Third F 4/3 Fourth Name Interval G 3/2 Fifth A 27/16 Sixth B 243/128Seventh C 2/1 Octave

26. Back to the Future • Using the Circle of Fifths, we can start at any arbitrary note at the "bottom" of the circle, and reach this note again at the "top" of the circle (in a different octave) by adding twelve perfect Fifths • The "top" note will be 6 octaves above the bottom "note" • We can then try to return to the original note by halving the frequency of the "top" note six times • Mathematically: (3/2)12 ÷ 26 == 531441/5524188 == 1.0136/1 • But this should be 1/1 because it's the same note! • This difference between a note's frequency as calculated via the Circle of Fifths versus its frequency calculated via octaves is called a comma

27. Many Different Temperaments • Pythagorean Tuning • "Just" Tuning (four different modes!) • Mean-tone Tuning • Well-tempered Tuning • J S Bach's Well-Tempered Clavier • And of course … • P D Q Bach's Short-Tempered Clavier

28. So how can we ever tune anything? We get different results by tuning with different intervals!

29. Even Tempered Tuning • Historically, different tunings and temperaments have been used to improve the intonation of an instrument • Instruments sound "best" in only one "key" • This is a problem if you want to transpose, or use inharmonic intervals • Starting in the 1850s, musicians began to use "even" temperaments • Much Classical and Romantic music required this, as composers began to experiment with fuller, more textured sounds and different key changes • Makes it easier to tune pianos, harps, and organs

30. Even Temperament • Even temperament divides an octave into 12 equally spaced half steps • Every half step is always 100 cents • Every whole step is always 200 cents • Intervals are calculated based on multiples of 21/12 • All intervals of like size will have the same multiplier • Some intervals may not "sound" in tune, but we live with it to get more flexibility

31. What tuning should I use? • In general, Even/Equal Temperaments are easiest to deal with • Some "period" pieces may sound better in their original tunings • Experiment with it and see what sounds "best"!