Chapter 16
Keyboard Temperaments and Tuning:
Organ, Harpsichord, Piano
The Just Scale

All intervals are integer ratios in frequency
Major Scale
1
9/8
5/4
4/3
3/2
5/3
15/8
2/1
Unison
Major
2nd
Major
3rd
Perfect
4th
Perfect
5th
Major
6th
Major
7th
Octave
C
D
E
F
G
A
B
C
261.63
294.43
327.04
348.84
392.45
436.05
490.356
523.26
Minor Scale
1
10/9
6/5
4/3
3/2
8/5
9/5
2/1
Unison
Minor
2nd
Minor
3rd
Perfect
4th
Perfect
5th
Minor
6th
Minor
7th
Octave
C
D
Eb
F
G
Ab
Bb
C
261.63
290.70
313.96
348.84
392.45
418.61
470.93
523.26
A Note of Caution
Notes on the Just Scale
Major Scale
The D corresponds to the upper D in the pair found in
Chapter 15. Also, the tones here (except D and B) were the
same found in the beat-free Chromatic scale in Chapter 15.
Minor Scale
Here we use the lower D from chapter 15 and the upper
Ab. In music theory two other minor 7th are recognized,
the grave 7th (16/9) and the harmonic minor 7th (7/4).
Notes on Just Scales


These just scales work (good harmonic
tunings) as long as the piece has no more
than two sharps or flats.
The following notes apply to organ tuning

Organs produce sustained tones and harmonic
relationships are easily heard
The Equal-Tempered Scale

Each octave comprised of twelve equal
frequency intervals

The octave is the only truly harmonic relationship
(frequency is doubled)
12
2 = 1.05946
 Each interval is
 The fifth interval is close to the just fifth



 2  = 1.49831
12
The Difference is
7
whereas the just fifth is 1.5
 1.500 
ln 

1.4983
  1.95  2 cents
1200 
ln(2)
Only fifths and octaves are used for tuning
The Perfect Fifth

Three times the frequency of the tonic down
an octave


3*fo/2
The third harmonic of the tonic equals the
second harmonic of the fifth


3*fo = 2*f5th
The fifth must be tuned down about 2 cents
Tuning Fifths (Organ)

C4 = 261.63 Hz


G4 has a frequency of 1.49831*C4 or 392.00 Hz
Use the second rule of fifths to get the beat
frequency

3(261.63) – 2(392.00) = 0.89 Hz
Notes of Fifth Tuning

The next table above shows the complete
tuning in fifths from C4 through C5.


A trick is to use a metronome set to approximately
the correct beat frequency to get accustomed to
listening for the beats.
The rest of the keyboard is tuned by beat-free
octaves from the notes we have tuned so far.
The Changing Beat Pattern
Tonic
Fifth
3*Tonic
2*Fifth
Difference
261.63 (C4)
392.00 (G4)
784.89
784.00
0.89
392.00 (G4)
587.34 (D5)
1176.00
1174.68
1.32
293.67 (D4)
440.01 (A4)
881.01
880.01
0.99
440.01 (A4)
659.27 (E5)
1320.02
1318.53
1.49
329.63 (E4)
493.89 (B4)
988.90
987.78
1.12
493.89 (B4)
740.00 (F#5)
1481.68
1480.00
1.67
370.00 (F#4)
554.37 (C#5)
1110.00
1108.75
1.25
554.37 (C#5)
830.62 (G#5)
1663.12
1661.25
1.88
415.31 (G#4)
622.26 (D#5)
1245.94
1244.53
1.41
622.26 (D#5)
932.34 (A#5)
1866.79
1864.69
2.11
466.17 (A#4)
698.47 (F5)
1398.51
1396.94
1.58
698.47 (F5)
1046.52 (C6)
2095.40
2093.04
2.36
523.26 (C5)
Just and Equal-Tempered
Interval
Just
EqualTempered
Tonic
1.00
261.63
261.63
Major 2nd
1.13
294.33
293.67
-4
Major 3rd
1.25
327.04
329.63
14
Major 4th
1.33
348.84
349.23
2
Major 5th
1.50
392.45
392.00
-2
Major 6th
1.67
436.05
440.01
16
Major 7th
1.88
490.56
493.89
12
Octave
2.00
523.26
523.26
0
Minor 3rd
1.20
313.96
311.13
-16
Minor 6th
1.60
418.61
415.31
-14
Cent Diff.
Notes on Organ Tuning



Certain intervals sound smoother (or rougher)
than others.
In playing music we seldom dwell on any two
notes long enough to notice precise tuning.
Chords made of three or more notes (equaltempered) create a more “tuned” effect than
the two note intervals would imply.

Perhaps one reason for the complex chords of
music since Beethoven.
The Circle of Fifths
Key Signature Derived from
the Circle of Fifths
C
0#
G
1#
F#
D
2#
F#
C#
A
3#
F#
C#
G#
E
4#
F#
C#
G#
D#
B
5#
F#
C#
G#
D#
A#
F#
6#
F#
C#
G#
D#
A#
E#
C#
7#
F#
C#
G#
D#
A#
E#
B#
C
0b
F
1b
Bb
Bb
2b
Bb
Eb
Eb
3b
Bb
Eb
Ab
Ab
4b
Bb
Eb
Ab
Db
Db
5b
Bb
Eb
Ab
Db
Gb
Gb
6b
Bb
Eb
Ab
Db
Gb
Cb
Cb
7b
Bb
Eb
Ab
Db
Gb
Cb
Fb
Pythagorean Comma

Start from C and tune perfect 5ths all the way
around to B#.


A perfect 5th is 702 cents.


702+702+702+702+702+702+702+702+702+702+702
+702= 8424 cents
An octave is 1200 cents.


C and B# are not in tune.
1200+1200+1200+1200+1200+1200+1200= 8400
cents
8424 - 8400 = 24 cents = Pythagorean Comma
Pythagorean Comma
More Precisely
Note
Circle of Fifths
Seven Octaves
C
261.63
G
392.45
D
588.67
A
883.00
E
1324.50
B
1986.75
F#
2980.13
Db
4470.19
Ab
6705.29
Eb
10057.94
Bb
15086.90
F
22630.36
C
33945.53
33488.64
Cent. Dif.
23.46
A Well-Tempered Tuning

Werckmeister III


Created by Andreas Werckmeister in 1691 –
useful for baroque organ, harpsichord, etc.
The system contains eight pure fifths, the
remaining fifths being flattened by ¼ the
Pythagorean Comma.
Werckmeister III
Start with a reference note (C4)
Tune a beat free major third above (E4).
Construct a series of shrunken fifths so that we end
up back at E6, which will tune with E4.




The interval is found by dividing the Pythagorean Comma
into four equal parts (23.46/4 = 5.865). So instead of the
perfect fifths being 702 cents, they are 696.1 cents.
The Shrunken Fifth


Using the cents calculator, the fifth interval will be
1.49492696.
The just interval of the perfect fifth is 1.5, so each fifth is
about 5.9 cents short.


The first six steps in the tuning are…
Note
Frequency
Beats
C4
261.63
E4
327.04
G4
391.12
1.33
D5
584.69
3.98
A5
874.07
8.93
E6
1306.67
17.83
The perfect fifth above C4 would have a frequency of 392.45 Hz, so
we tune for a beat frequency of (392.45 – 391.12) 1.33 Hz. Other
entries in the final column above are calculated in a similar fashion.
Werckmeister III (the perfect fifths)
Recall that the newly tuned fifths produced an E6
in tune with E4. The next step is to retune E6 to be
a perfect fifth above the A5 already determined.





That would put it at a frequency of 1311.05 Hz.
This also retunes the E4 to 327.76 Hz.
The new E6 can now be used to tune B6 a perfect
fifth above it at 1966.58 Hz.
Starting from C4 again tune perfect fifths
downward to Gb. We raise the pitch an octave
periodically to remain in the center of the
keyboard.
Downward Perfect Fifths
Freq.
Note
261.63
C4
174.42
F3
116.28
Bb2
Freq.
Note
232.56
Bb3
155.04
Eb3
Freq.
Note
310.08
Eb4
206.72
Ab3
Column jumps indicate octave changes
Freq.
Note
413.44
Ab4
275.63
Db4
Freq.
Note
551.25
Db5
367.50
Gb4
Gathering Results into One
Octave
Note
Frequency
Cent Difference
from C
C
261.63
0
Db
275.63
90
D
292.35
192
Eb
310.08
294
E
327.76
390
F
348.84
498
Gb
367.50
588
G
391.12
696
Ab
413.44
792
A
437.04
888
Bb
465.12
996
B
491.65
1092
C
523.26
1200
Werckmeister Circle of Fifths

Numbers in the intervals refer to differences
from the perfect interval.

The ¼ refers to ¼ of the Pythagorean Comma.
Notes



Bach’s tuning was similar (he divided the
Pythagorean Comma into five parts).
Either one of these well-tempered tunings
admits to all 24 keys (major and minor).
This is the basis of Bach’s Das
Wohltemperirte Clavier.
Comparison Table




The next slide is similar to Table 16.1
I show the just intervals and the
Werckmeister frequencies that are generated
with a variety of tonics.
Cent differences in these two tunings are also
given
Notice that the minor intervals are all flat in
the Werckmeister III
Just - Werckmeister III
Interval
Just
Interval
C
Cent
Diff
G
Cent
Diff
F
Cent
Diff
D
Cent
Diff
Bb
Cent
Diff
A
Cent
Diff
Eb
Cent
Diff
Major 2nd
1.125
294.43
-12
440.01
-12
392.45
-6
328.89
-6
523.26
0
491.67
0
348.84
0
Major 3rd
1.250
327.04
4
488.90
10
436.05
4
365.43
10
581.40
10
546.3
16
387.60
16
Major 4th
1.333
348.84
0
521.49
6
465.12
0
389.79
6
620.16
0
582.72
6
413.44
0
Major 5 th
1.500
392.45
-6
586.68
-6
523.26
0
438.52
-6
697.68
0
655.55
0
465.12
0
Major 6 th
1.667
436.05
4
651.86
10
581.40
10
487.24
16
775.20
16
728.39
16
516.80
22
Major 7 th
1.875
490.36
4
733.35
4
654.08
4
548.15
10
872.10
4
819.44
16
581.40
10
Minor 3 rd
1.200
313.96
-22
469.34
-16
418.61
-22
350.82
-10
558.14
-22
524.44
-4
372.10
-22
Minor 6 th
1.600
418.61
-22
625.79
-16
558.14
-22
467.75
-10
744.19
-22
699.26
-4
496.13
-16
Musical Implications



The table clearly shows that transposing
yields different flavor or mood
Modulating to another key also produces
different moods depending on the key that
was just left.
Equal temperament loses these changes.
Physics of Vibrating Strings
Flexible Strings
Density = d
r
L
Stretched between rigid supports, the
frequency of harmonic n is…
Clearly,
 1  T 1
f n  n 
 Lr  d 4
fn = nf1
Some Dependencies
fn 
1
L
as L , f  (longer strings  lower tones)
fn 
1
r
as r , f 
(larger strings  lower tones)
as T , f 
(more tension  higher tones)
fn  T
Physics of Vibrating Strings
Hinged Bars

Because strings are under tension, they are
stiff and take on some of the properties on
thin bars. The frequencies of the harmonics
are…
 r  Y  
f n (hinged bar)  n 2  2 


L  d  2

All the symbols have their same meaning and Y =
Young’s Modulus, is a measure of the elasticity of
the string.

Clearly, for a bar fn = n2f1
Some Dependencies
1
fn  2
L
double the length and the frequency is up
two octaves
fn  r
the opposite behavior of the string,
as r , f 
Real Strings


We need to combine the string and bar dependencies
Felix Savart found…
f n (stiff string under tens ion)  f n2 (flexible)  f n2 (bar)

The stiffness (bar) contribution is rather small
compared to the tension contribution in real
strings. We can approximate the above work
as…
f n (stiff string under tens ion)  nf 1 (flexible string under tens ion)  (1  Jn 2 )
 r 4 Y   
J   2  
 TL  2 
3
And is small (about 0.00016)
Departures from Harmonic
Series


The perfect Harmonic Series is nf1
We can make sure departures from the
perfect series are small if we make J small



Using long strings (increase L)
Make the strings taut (increase T, the tension)
Make the strings slender (decrease r)
Sample Series
Component (n)
Piano String
1
2
3
4
5
6
261.63 523.51
785.91
1049.23 1313.23 1578.68
Pipe Organ (nf1) 261.63 523.26
784.89
1046.52 1308.15 1569.78
Difference
0.00
0.25
1.02
2.71
5.08
8.90
Physics of Vibrating Strings
The Termination

Strings act more like
clamped connections to
the end points rather
than hinged
connections.

The clamp has the effect
of shortening the string
length to Lc. Lc is
related to J. The effect
of the termination is
small.
L c  L  (1 - J )
Physics of Vibrating Strings
The Bridge and Sounding Board
We use a model where the string is firmly
anchored at one end and can move freely on a
vertical rod at the other end between springs
FS is the string natural frequency
FM is the natural frequency of the block and spring to which the
string is connected.
Results


The string + mass acts as a simple string
would that is elongated by a length C.
The slightly longer length of the string gives a
slightly lower frequency compared to what we
would have gotten if the string were firmly
anchored.
Change the Frequency
Results



For the case of FS > FM, chapter 10 suggests
that the mass on the spring lags the string by
up to one-half cycle.
As the string pulls up the mass is moving
down and vice versa.
The string acts as though it were shortened
by a length C. The shortened length raises
the pitch over a firmly anchored string.
Example of a Guitar String



Consider the D string of
a guitar and its first
several.
Harmonic
Frequency
Ratio
1
146.83
1.000
2
293.95
2.002
3
440.49
3.000
4
589.52
4.015
5
738.16
5.027
The ratios show a tendency to grow larger
because of the effects of string stiffness.
Irregularities in the sequence occur when one of
the guitar body resonant frequencies happens to
be near one of the partials.
Bigger is Better


Larger sounding boards have overlapping
resonances, which tend to dilute the
irregularities.
Thus grand pianos have a better harmonic
sequence than studio pianos.
Pitch of a Single String Sound

Because the piano string has a slightly
inharmonic series, the perceived pitch of a
key may vary from an instrument with strictly
harmonic sequences.
The Piano Tuner’s Octave


When a tuner tunes an octave to “sound
right” we find there are still beats, but there is
a reduction in the “tonal garbage.”
For the C4 – C5 octave this is achieved when
the fundamental of C5 is 3 cents higher than
2*C4.
A Real Piano Tuning

Below I list the first few partials of C4 and the
resulting C5 and its partials.
C4 (Benade’s piano)
261.63
523.51
785.91
1049.23
1313.23
1578.68
C4 (harmonic)
261.63
523.26
784.89
1046.52
1308.15
1569.78
Cent difference
0.00
0.83
2.25
4.48
6.71
9.79
C5 (Benade’s piano)
523.70
1048.81
1571.11
Cent difference (C5: C4)
0.63
-0.69
-8.32
Beat Frequency (C5- C4)
0.19
-0.42
-7.57
Note: The values used here for the C4 partials are the same as
were used previously to compare piano to organ tuning and
introduced the inharmonic factor J. Also notice that the C5 is not
3 cents sharp of the second harmonic of C4.
“Perfect” Fifths on the Piano

A condition of least roughness for the fifth is
obtained when the tuning is about 1 cent
higher than 3/2 * fundamental.



Below I have calculated the fifth interval based on
the middle C of 261.63 Hz.
The first fifth is 1.5*C4 and the second one is the
equal tempered fifth interval. Tonic
Fifth
ET Fifth
These differ by 2 cents.
261.63
392.445
392.002
The one cent difference suggested in the text would give 392.67
Hz. Two times this is 785.34 and three times the middle C is
784.89 (a beat frequency of 0.45).
Cent Diff
2.0
The “Perfect” Third

Similarly, the “perfect” third (condition of least
roughness) is found for a setting 3.5 cents
sharp of the 5/4 interval. Numerically,
Tonic
Third
ET Fifth
Cent
Diff
"Perfect“
Third
261.63
327.038
329.633
-13.7
327.7
Concluding Comments


Piano and harpsichord tuning is not marked
by beat-free relationships, but rather
minimum roughness relationships.
The intervals not longer are simple numerical
values.