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!

Tuning (1/5)

How To Tune a Piano:

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

Part I of V

All good mommies have to take breaks from kissing their babies to do other things, like get manicures, continue promising careers, or learn something about piano tuning.  Because I don't have nails or a career, I decided to start opening my tuning book during Drakeson's afternoon naps.  After 2 pages of "Tuning Theory and Terminology," I was smitten.  I'm probably not the only mommy without nails or a career, so for the rest of us out there, this is for you.

When something vibrates, it causes the surrounding air vibrate, and our eardrums process these vibrations as sound.  When the vibrations are irregular, we hear noise.  When the vibrations are regular and within the range of human hearing, we hear a musical tone.  The vibration speed, called frequency, is measured in cycles per second called hertz or hz.  The higher the hz, the higher the pitch.  When all other factors are equal, wires that are shorter, thinner, or under more tension produce higher pitches.  Also, stiffness is measured by ratio of shortness and thickness.  Stiffer wires are also higher.

When a wire vibrates, it simultaneously divides itself into 2 vibrating halves, 3 thirds, 4 fourths, and so on.  It's difficult to imagine, so here's a picture.  A vibrating wire does all these things at once.

The imaginary pitch on the left of the picture is called the fundamental or the first partial.  While we are listening to this imaginary fundamental, we can also more softly hear the second pitch in the diagram, called the second partial or first overtone.  The frequency of either half is twice that of the fundamental.  For the third partial or second overtone, the sound is softer than the second partial, and the frequency of each third is three times as fast.  Etc.  A harmonic is a frequency that is an exact multiple of the fundamental.  Theoretically this is every partial, but in reality, wire stiffness causes partials to be inexact, and the difference between the two is called inharmonicity.

Aside: The first diagram creates a unison with the fundamental frequency.  Because the next diagram has twice the frequency, the pitch sounds one octave higher.  The third sounds a fifth and an octave above, the fourth two octaves above, and the fifth a major third and two octaves above.

If two wires that are tuned to the same pitch are struck at the same time, the resulting sound is louder due to constructive interference.  If they are not synchronized, however, they will produce a softer tone due to destructive interference.  Let's assume one wire is tuned at x hz, and a second wire is tuned at x+4 hz.  The second string will vibrate 4 more times than the first in one second, and the vibrations of both strings will be synchronized 4 times.  Those times, which are louder than the rest of the second, create periodic shifts in volume called beats.  Beats occurring more than 15 times a second are difficult to hear, and the sound is instead interpreted as two distinct pitches.  Partials and fundamentals can each cause beats if the frequencies are close.  The goal of tuning a piano is to create the fewest beats when any combination of pitches is played.

Reblitz goes on to teach note names and define intervals, but all my friends know that part.  If you don't and you would like to, just give me a call.  What we didn't know, however, is that tuners like unisons, M2nds, m3rds, M3rds, P4ths, P5ths, M6ths, 8ves, M10ths, P12ths, P15ths, and M17ths.  Adjacent or successive intervals are the same interval starting a half step apart (C-E, C#-F).  Contiguous intervals are the same interval with one overlapping pitch (C-E, E-G#).  When there are three or more contiguous intervals, they are called stacked (C-E, E-G#, G#-C).  This brings me to the final point of our first lesson.  Tuners call all the black notes sharps and don't believe in flats.  They also do things like call diminished fourths major thirds.  They keep us in tune, though, so let's forgive them.  See you in Part II.