You
can tune a piano, but you can’t tuna fish.


Or
so goes the title of the REO
Speedwagon album. I have always disagreed. Both parts are wrong. You can tune a fish by going up and down
the scales. And, contrary to popular belief, you can’t tune a piano. Read on,
and I’ll explain why you can’t.


The circle of fifths


As
a first example, we consider the circle of fifths. No, I don’t mean a circle
of my buddies Jack Daniels, José Cuervo, Johnny Walker and Jim Beam.




I
mean the circle of fifths on the piano. From C to G is an interval known as a
fifth. From a G to a D is another fifth. This eventually winds up back at C,
so the whole thing forms a circle with twelve notes. I suspect that many of
those who have read this far are well aware of this.


C
→ G → D → A → E → B → F# → D♭ → A♭ → E♭
→ B♭
→ F → C


Now
for two facts putting these to facts to their logical conclusion will lead to
a contradiction that will prove that you can’t tune a piano. First fact: The
ratio of frequencies from a note to the note that is one octave higher is
two. If the A below middle C is tuned to 440 Hertz (440 cycles per second),
then the A above middle C is 880 Hertz.


If
you count the octaves as you go through the circle of fifths, you will find
that they span seven octaves. Put this together with the first fact, and you
will see that the frequency of the last entry in the circle is exactly 128
times the frequency of the first C in the circle. 128. Remember that number.


Now
for the second fact: The ratio of frequencies from a note to the note which
is one fifth above that note (as in from C to the next G up), is 3/2. Thus,
if we tune to A 440, the E just above middle C is 660 Hertz.


We’re
coming up on the contradiction, so pay attention.


There
are twelve steps in the circle of fifths^{1}. If
each of those steps is 3/2, then the ratio of frequencies from the C at the
end of the circle to the C at the start should be (3/2)^{12}, which
is 531,441 divided by 4,096, which is about 129.746. The astute student of
mathematics will note that 129.746 is not equal to 128.


The
conclusion is that this set of three statements is not consistent: An octave
is a ratio of two. A fifth is a ratio of three halves. The circle of fifths
winds up back where it started.


A Hint from Pythagoras


As
the story goes, Pythagoras happened to be walking by the blacksmith’s shop
one day when he noticed the musical tones that were created by the different
blacksmiths pounding on the iron with their hammers. As the story goes,
Pythagoras was intrigued, and started experimenting to try to understand what
caused the tones to be different. As the story goes, he found that the weight
of the hammers was the important factor, and that the ratio of weights told
what chord the hammers would make when they were pounded together.


He
determined a very simple, yet powerful fact. Simple ratios make for consonant
notes. The octave, with a ratio of two to one, is a simple ratio, and the two
notes sound consonant. The fifth, with the simple ratio of three to two, is
also consonant. The major third has a ratio of five to four, and this is also
consonant.


This
story comes to us by way of Iamblichus, who was born in the third century AD,
about eight centuries after Pythagoras. The story was wrong^{2}, but the conclusion was correct. The note that is
produced when you hammer on a piece of iron depends on the piece of iron, and
not the weight of the hammer. Pythagoras probably came to this conclusion by some
other means, perhaps listening to the tone produced by a string held taught
by differing amounts of weight, or perhaps by listening to the tone produced
by a column of air of different lengths.


Defining a C major scale


Regardless
of how the “simple ratios” rule was determined, it can be used to define all
the notes in a C major scale. Much like early Rock and Roll, we just need
three chords: the tonic, the dominant, and the subdominant, that is, C, G,
and F.




Let
us start by assuming that we have tuned our instrument so that middle C is
equal to 240 Hz. This is a bit off, but it makes for easier calculations.
With this definition of C, then G must be three halves of that (360 Hz), and
E must be five fourths (300 Hz).


The
G chord allows us to build from the definition of G (360 HZ) to ascertain
that the fifth above that, D, is 540 Hz. This is in the next octave, so we
can divide by two to bring it into this octave. Thus, the D above middle C is
270 Hz. From the G chord, we can also ascertain B, the third above the G.
This makes B out to be 450 Hz.


The
F below middle C is a fifth below C. That means that the frequency of F, when
multiplied by three halves, gives you 240 Hz. Thus, F must be two thirds of
the frequency of C, or 160 Hz. This is in the octave below middle C, so we
multiply this by two to get 320 as the frequency of F above middle C.


The
note A is the final note in the key of C major. This is five fourths the
frequency of F, which gives us 400 Hz. Here is the full table for a C major
scale, with C tuned to 240 Hz.




Trouble brewing


This
was all pretty simple. Given that the circle of fifths ran into trouble, it
should come as no surprise that there might be problems underlying our simple
C major scale as well. The trouble can be exemplified by asking a seemingly
simple question, “what is the ratio for a minor third?” The C major scale
gives us three opportunities to figure this out. D to F is a minor third, as
is E to G, and A to C.


The
ratio from E to G is six fifths (1.200), as is the ratio from A to C. But the
other example of a minor third is D to F, with a ratio of 32 / 27, or 1.185. We
have two problems here. The first is that we don’t know for sure what a minor
third is. The second is that sometimes a minor third has a ratio that is
simple (6/5) and sometimes it is not (32/27). What went wrong?


The
difficulty is that we have implicitly accepted two different notes as being
the same note. First, there is the F that is a fifth down from C. Second,
there is the F that is a minor third up from D. The piano forces these two
different notes into the same key. The keyboard of the piano is a lie. The
two different notes are only 1.25% different, but they are different notes.


All
we need to do to fix this up is to add a second F to the keyboard, defining
it to be a minor third above D. And of course, we need to officially define a
minor third to be the simple ratio 6 / 5. And all is well with the world.


Well,
at least until someone wants to play a chord that has a sharp or a flat. Or
until someone wants to play anything by Cole Porter. Then Debussy comes along
and asks you to play in sixteen sharps. Pretty soon, your piano has exactly 4.76
gazillion keys. Every chord is perfectly in tune. Now, If your name is Ruben, then you could probably
learn to play such a monster, or at least a toy version of it. All the rest
of us mere mortals have learned to live with a piano tuning that is a
compromise, where each key is tuned to be a compromise between all the frequencies
that it is supposed to play.


Two halves don’t make a whole


Every
choir director I have ever known has played a trick on us during warm ups.
Maybe it’s been on purpose, or maybe they don’t realize that it’s just a
cruel joke. A joke like asking a third grader to find an integer fraction
that, when squared, gives you 2. Choir directors ask you to sing a chord, and
then move down by a half step and then another. Then they ask you to go up a
full step. The cruel task master will then chastise the choir for going flat.
Cruel trick. There is a perfectly reasonable explanation for going flat that
has absolutely nothing to do with the indolence or skill of the choir.


First,
let’s consider what “going down a half step” means. If I am the lead voice,
and I am singing a C, then a half step down is to B. That is, a half step is
from do to ti. From the chart above, we can see that going down a half step
means multiplying the frequency by 15 / 16. And the next half step, you just
repeat that^{3}. So, after two half steps, the
frequency is now 225 / 256 times the original frequency.


Then
the choir director commands to go up by a whole step. If you are singing the
root of the chord, then you are going to be going from do to re. Looking at
the chart above, do to re is the interval from C to D, which means
multiplying the frequency by 9 / 8. This means that the final landing point
is (225 / 256) X (9 / 8) = 2025 / 2048. This is about 0.98877, just over 1%
short of returning back home.


The
choir director, in tyrannical ignorance, will demand that two half steps be
equal to a whole step, and accuse the choir of (God forbid) going flat. But
the choir is just doing what it was told.


I
hope this section has caused some choir directors to have a bit more
humility. Maybe some of them will even find contrition in their hearts, and
will be willing to apologize for their misguided despotism.


If
any of my previous choir directors feel such a need, rest assured that
apologies are not necessary. I have been blessed through the years with many,
many choir directors who were inspirational magicians, and I want to thank
them for the glorious moments of ethereal chords.


That
is why I say that you can’t tune a piano.


1) Any
alcoholic will note a certain irony in that.
2)
This of course was not the only thing that history
has gotten wrong about Pythagoras. See a previous
blog for another example.

Wednesday, July 11, 2012
You can tune a piano...
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Thanks for the namecheck! And I will also take this opportunity to stand contritely in penance for my despotic choir director ways :)
ReplyDeleteUnwittingly or not, you've hit on the concept of musical "temperament," or different modes of tuning based on mathematical ratios.
The modern method of tuning keyboard instruments is known as equal temperament. The piano is not tuned with regard to the circle of fifths, but, rather, octave by octave with each interval therein tuned proportionally within the octave. In this way, every musical key sounds the same; in other historic tunings, such as ones based on the circle of fifths (called Pythagorean tuning), certain keys will sound consonant while others will sound dissonant to our modern ears.
One of my friends, who has a DMA in historical musicology, refers to equal temperament as "equally out of tune." When I tune my harpsichord, I don't use equal temperament  rather, I experiment with a number of temperaments that would have been familiar to earlier composers. I have an app on my tablet that allows me to select one of about 20 temperaments and then tune it accordingly. Very cool.
Do some research on "musical temperament"  you will find it a fascinating topic.
 Ruben
Please rest assured that your cruel despotism has left only superficial scars, Ruben. :) Therapy, brandy, and time will heal all.
DeleteThe post was somewhat unwitting, and somewhat witting. I based it all on the rule of simple ratios. I have read a bit on different temperaments. Maybe I need to read a book I have: "Science & Music" by James Jeans?
Hey maybe this is silly me commenting on such an old article but I thought I'd give some insight. A choir has the lovely ability to make chords that are impossible on most instruments. In equal temperament the interval of every note is the 12th root of 2. In just scales the ratio is more simple and is set to nice easy ratios like 3/2 and 4/3 but when a choir sings they can harmonise to any ratio and this opens a huge window of opportunity to be creative. A skilled choir can give a certain feeling to each chord that instruments cannot by harmonising to the feeling of the music rather than intervals. I have heard many Tchaikovsky pieces that do this and it really has a nice sound. Our ears really are better than any calculator when it comes to music and each song will work best in a different tuning.
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