Chapter 16

Keyboard Temperaments and Tuning: Organ, Harpsichord, Piano

The Just Scale

All intervals are integer ratios in frequencyMajor Scale

1 9/8 5/4 4/3 3/2 5/3 15/8 2/1

UnisonMajor

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

UnisonMinor 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.

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).

Minor Scale

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) Each interval is = 1.05946

12 2

712 2

cents 2 1.95 ln(2)1.49831.500

ln1200

The fifth interval is close to the just fifth = 1.49831 whereas the just fifth is 1.5

The Difference is

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

Equal-Tempered

Cent Diff.

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

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 (equal-tempered) 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#. C and B# are not in tune.

A perfect 5th is 702 cents. 702+702+702+702+702+702+702+702+702+702+70

2+702= 8424 cents An octave is 1200 cents.

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 Freq. Note Freq. Note Freq. Note Freq. Note

261.63 C4                

174.42 F3                

116.28 Bb2 232.56 Bb3            

155.04 Eb3 310.08 Eb4        

        206.72 Ab3 413.44 Ab4    

            275.63 Db4 551.25 Db5

                367.50 Gb4

Column jumps indicate octave changes

Gathering Results into One Octave

Note FrequencyCent 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

IntervalJust

Interval CCent 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

L

r

Density = d

Stretched between rigid supports, the frequency of harmonic n is… 4

1

d

T

Lr

1n f n

Clearly, fn = nf1

Some Dependencies

L

1 fn

r

1 f n

T f n

as r , f (larger strings lower tones)

as T , f (more tension higher tones)

as L , f (longer strings lower tones)

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…

2d

Y

L

rn bar) (hingedf

22

n

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

double the length and the frequency is up two octaves 2n L

1 f

the opposite behavior of the string, as r , f

r f n

Real Strings We need to combine the string and bar dependencies Felix Savart found…

(bar)f (flexible)f ion)under tens string (stifff 2n

2nn

The stiffness (bar) contribution is rather small compared to the tension contribution in real strings. We can approximate the above work as…

)Jn (1 ion)under tens string (flexiblenf ion)under tens string (stiff f 21n

3

2

4

2TL

Yr J

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) 1 2 3 4 5 6

Piano String 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.

)J - (1 L Lc

The clamp has the effect of shortening the string length to Lc. Lc is related to J. The effect of the termination is small.

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. These differ by 2 cents.

Tonic Fifth ET Fifth Cent Diff

261.63 392.445 392.002 2.0

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).

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 FifthCentDiff

"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.