Tuning (2/5)

How To Tune a Piano:

A Summary of Chapters 6 and 7 from "Piano Servicing, Tuning, and Rebuilding" by Arthur Reblitz

Part II of V

We left off making fun of tuners, so just to clear the air, here's a joke about all of us.
Q:  What's the difference between a musician and a large pizza?
A:  A large pizza can feed a family of four.

In 1925, it was agreed that the international pitch standard, or tuning standard, would be A440, which means the A above middle C would be tuned to 440 hz.  And so it is.  Because octaves have a 2:1 frequency ratio, the A above A440 is A880, and the A below is A220.  To find the twelve theoretical frequencies of the halfsteps within an octave in an equal temperament, the lower frequency is multiplied by some number, x, twelve times in a row, generating these frequencies and ending one octave higher.  Because we know the ratio of octave frequencies is 2:1, it follows that x must be the twelfth root of 2.  Just as the "cubed root of 2" is the name of the number that results in 2 when multiplied by itself 3 times, "the twelfth root of 2" is the name of the number that results in 2 when multiplied by itself twelve times.  Perfect.

In addition to the 1:1 unison and the 2:1 octave, there are other just frequency ratios, some of which are: P5th - 3:2, P4th - 4:3, M3rd - 5:4, and m3rd - 6:5.  For equal temperament, however, a P5th is 2^(7/12):1, a P4th is 2^(5/12):1, etc.

The twelfth root of 2 is about 1.0594631.  These are some theoretical frequencies of notes near middle C, calculated with 1.0594631 and 110/220/440.  Notice that because they are generated through multiplication instead of addition, the difference in frequencies between adjacent halfsteps increases as the pitches get higher.
F 174.614
F# 184.997
G 195.998
G# 207.652
A 220.00
A# 233.082
B 246.942
C 261.626 (Middle C)
C# 277.183
D 293.665
D# 311.127
E 329.628
F 349.228
F# 369.994
G 391.995
G# 415.305
A 440.00

When tuning a piano, one begins with an octave near middle C called the temperament octave, or temperament.  If just intervals (with the exception of the octave) are used in tuning, some intervals sound out of tune and cause fast annoying beats.  The idea behind equal temperament is that every interval except the octave is a little out of tune, with a slow beat between nearly coincident partials.  To find the theoretical number of beats between any two notes, find the partials that are closest and take the difference.  For example, we know that a just P5th has a 3:2 ratio, so the frequency ratio of middle C to the G above it should be close to 3:2, and the third partial of C should be very close to the second partial of G.  The frequency of middle C is 261.626, so its third partial is 784.878.  The frequency of the G above middle C is 391.995, so its second partial is 783.990.  The difference is about .88 beats per second.  Notice that the G came out a tiny bit flat.  In equal temperament, P5ths are narrow, and M3rds and M6ths are wide. It follows then, that P4s are wide, and m6ths and m3rds are narrow. Here are some theoretical beat rates per second in the F-F octave encompassing middle C.
Minor Thirds:
  F-G# 9.42
  F#-A 9.98
  G-A# 10.58
  G#-B 11.20
  A-C 11.87
  A#-C# 12.58
  B-D 13.33
  C-D# 14.12
  C#-E 14.96
  D-F 15.85
Major Thirds:
  F-A 6.93
  F#-A# 7.34
  G-B 7.78
  G#-C 8.24
  A-C# 8.73
  A#-D 9.25
  B-D# 9.80
  C-E 10.38
  C#-F 11.00
Major Sixths:
  F-D 7.93
  F#-D# 8.40
  G-E 8.89
  G#-F 9.42

Remember how wire stiffness creates inharmonicity by generating sharp partials?  Here's a comparison of the frequencies of theoretical harmonics and some actual measurements from the lowest C on the piano.  Interesting.
Theoretical/Actual Measurement
Fundamental 32.703/32.703
2nd Partial 65.406/65.523
3rd Partial 98.109/98.401
4th Partial 130.812/131.434
5th Partial 163.515/164.779
6th Partial 196.218/198.435

As you can see, the difference between theory and reality increases with each partial.  Therefore, when octaves are tuned, they will turn out differently when comparing the 4th partial of the lower note to the 2nd partial of the upper (a 4:2 octave) or the 6th partial of the lower note to the 3rd partial of the upper (a 6:3 octave).  Because the octave is always sharp, all intervals within the temperament should come out a little sharp of their theoretical calculations.  And to make matters worse, every note has its own inharmonicity, and therefore different pianos create different coincident partials.  With the exception of A440, there are no standard frequencies or even beat rates.  How complicated!

Cliffhanger:  In Part III, we will learn something about the Defebaugh F-F Temperament!