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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)

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Tuning


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tuning
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
assigning notes to pitches

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
brief history of 440 a
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
lissajous patterns
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!
history of 440 a continued
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)
modern tuning techniques
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
electronic tuning example
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

acoustic tuning
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)
acoustic tuning example

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
why this happens
Why this Happens
  • Consider two pitches an octave apart
  • Coincidental "zero crossings" (shown by arrows) eliminate "beats"
  • Same effect with a Fifth
out of tune pitches
Out of Tune Pitches
  • Two pitches a half step apart (no crossings)
  • Out of tune Fifth (2 cents worth)
this all sounds very clinical
This all sounds very clinical

So how come piano tuners still have jobs?

tuning for real
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)
psychoacoustics
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
intonation
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
temperament who says scales are boring
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!
review of intervals
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

now assign note names
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

map onto keys
Map onto Keys

C D E F G A B C

taking the fifth

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

a little music history
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

circle of life er fifths

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

back to pythagoras
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

tetrachords

Tone

Tone

Half

Tone

Tone

Tone

Half

Tetrachord

Tetrachord

Diatonic

Tetrachords

C D E F G A B C

pythagorean tuning

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

back to the future
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
many different temperaments
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
so how can we ever tune anything

So how can we ever tune anything?

We get different results by tuning with different intervals!

even tempered tuning
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
even temperament
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
what tuning should i use
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"!