1. Keyboard Temperaments and Tuning: Organ, Harpsichord, Piano
Chapter 16
Keyboard Temperaments and Tuning: Organ, Harpsichord, Piano

2. 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
261.63
294.43
327.04
348.84
392.45
436.05
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
261.63
290.70
313.96
348.84
392.45
418.61
470.93
523.26

3. A Note of Caution

4. 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).

5. 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

6. 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 =
The fifth interval is close to the just fifth
= whereas the just fifth is 1.5
The Difference is
Only fifths and octaves are used for tuning

7. 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

8. Tuning Fifths (Organ) C4 = 261.63 Hz
G4 has a frequency of *C4 or Hz
Use the second rule of fifths to get the beat frequency
3(261.63) – 2(392.00) = 0.89 Hz

9. 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.

10. The Changing Beat Pattern
Tonic
Fifth
3*Tonic
2*Fifth
Difference
(C4)
(G4)
784.89
784.00
0.89
(D5)
1.32
(D4)
(A4)
881.01
880.01
0.99
(E5)
1.49
(E4)
(B4)
988.90
987.78
1.12
(F#5)
1.67
(F#4)
(C#5)
1.25
(G#5)
1.88
(G#4)
(D#5)
1.41
(A#5)
2.11
(A#4)
(F5)
1.58
(C6)
2.36
(C5)

11. Just and Equal-Tempered
Interval
Just
Equal-
Tempered
Cent Diff.
Tonic
1.00
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 0 
Minor 3rd
1.20
313.96
311.13
-16
Minor 6th
1.60
418.61
415.31
-14

12. 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.

13. The Circle of Fifths

14. Key Signature Derived from the Circle of Fifths0# G 1# F# D 2# C# A 3# G# E 4# D# B 5# A# 6# E# 7# B# C 0b F 1b Bb 2b Eb 3b Ab 4b Db 5b Gb 6b Cb 7b Fb

15. 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.
= 8424 cents
An octave is 1200 cents.
= 8400 cents
= 24 cents = Pythagorean Comma

16. Pythagorean Comma More Precisely
Note
Circle of Fifths
Seven Octaves C
261.63 G
392.45 D
588.67 A
883.00 E B F# Db Ab Eb Bb F
Cent. Dif.
23.46

17. 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.

18. 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 cents.

19. The Shrunken Fifth
Using the cents calculator, the fifth interval will be
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
17.83
The perfect fifth above C4 would have a frequency of Hz, so we tune for a beat frequency of ( – ) 1.33 Hz. Other entries in the final column above are calculated in a similar fashion.

20. 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 Hz.
This also retunes the E4 to Hz.
The new E6 can now be used to tune B6 a perfect fifth above it at 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.

21. Downward Perfect Fifths
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

22. Gathering Results into One Octave
Note
Frequency
Cent Difference
from C C
261.63 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
523.26
1200

23. Werckmeister Circle of Fifths
Numbers in the intervals refer to differences from the perfect interval.
The ¼ refers to ¼ of the Pythagorean Comma.

24. 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.

25. 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

26. Just - Werckmeister III
Interval
Just Interval C
Cent Diff G F D Bb A Eb
Major 2nd
1.125
294.43
-12
440.01
392.45 -6
328.89
523.26
491.67
348.84
Major 3rd
1.250
327.04 4
488.90 10
436.05
365.43
581.40
546.3 16
387.60
Major 4th
1.333
521.49 6
465.12
389.79
620.16
582.72
413.44
Major 5 th
1.500
586.68
438.52
697.68
655.55
Major 6 th
1.667
651.86
487.24
775.20
728.39
516.80 22
Major 7 th
1.875
490.36
733.35
654.08
548.15
872.10
819.44
Minor 3 rd
1.200
313.96
-22
469.34
-16
418.61
350.82
-10
558.14
524.44 -4
372.10
Minor 6 th
1.600
625.79
467.75
744.19
699.26
496.13

27. 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.

28. Physics of Vibrating Strings Flexible Strings
Density = d r L
Stretched between rigid supports, the frequency of harmonic n is…
Clearly, fn = nf1

29. Some Dependencies as L , f  (longer strings  lower tones)
as r , f  (larger strings  lower tones)
as T , f  (more tension  higher tones)

30. 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…
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

31. Some Dependencies
double the length and the frequency is up two octaves
the opposite behavior of the string, as r , f 

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

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

34. 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)

35. Sample Series Component (n) 1 2 3 4 5 6 Piano String 261.63 523.51
785.91
Pipe Organ (nf1)
523.26
784.89
Difference
0.00
0.25
1.02
2.71
5.08
8.90

36. 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.

37. 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.

38. 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.

39. Change the Frequency

40. 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.

41. 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.

42. 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.

43. 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.

44. 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.

45. 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
C4 (harmonic)
523.26
784.89
Cent difference
0.00
0.83
2.25
4.48
6.71
9.79
C5 (Benade’s piano)
523.70
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.

46. “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 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
2.0
The one cent difference suggested in the text would give Hz. Two times this is and three times the middle C is (a beat frequency of 0.45).

47. 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“
261.63
-13.7
327.7

48. 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.