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
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
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
• f0 x 9/8 Second
• f0 x 5/4 Third
• f0 x 4/3 Fourth

Ration Interval f0 x 3/2 Fifth f0 x 5/3
Sixth f0 x 15/8 Seventh f0 x 2 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
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
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
Tetrachords
C D E F G A B C
25
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
243/128 Seventh 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"!