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A More Detailed Explanation of Piano Tuning

Why Does It Matter If The Piano Is In Tune?

Before beginning any detailed article on the subject of tuning, it seems appropriate to try to answer this question. I mean, if you play the keys and they all work, and the music sounds fine to you, why spend the money to have the piano tuned? And what if, after tuning, you don't even hear any difference? How do you justify the expense?

The answer must include a discussion of art itself. Let's face it, life can be difficult, challenging, stressful, and frustrating. It is the moments of beauty found along the way that help make the "journey" feel worthwhile. Beauty is perceived through our senses, and our senses have the ability to become heightened and "educated."

The first time you taste wine with a meal, most people do not "get" what all the fuss is about, or certainly cannot tell why someone would pay hundreds of dollars for a bottle when perfectly good wine can be found at a fraction of that price! However, if you continue the pursuit, especially if someone is there to help you identify the subtleties, over time you begin to appreciate how much a great meal is enhanced by a great wine!

The same experience can be found with music. When a piano is really in tune and well voiced, the sound of the music is balanced and elegant in a way that is not always immediately perceived. However, if you become accustomed to listening to a well tuned piano, it will become your standard. You may not always be able to tell that it is in great tune, until you hear a piano that is NOT, and then you will absolutely hear the difference. In other words, our ears can only be "educated" to the level to which they are exposed.

And as our senses improve, our experiences improve. We perceive music on a deeper level, and the experiences become inspiring in a way that we might not previously have appreciated. If a child is learning to play the piano, the better that piano sounds the higher their standard will become.

No orchestra would consider performing without first making sure that all the instruments were in tune. The last thing you hear before any performance begins is the sound of tuning. The instruments in the orchestra are like strings in a piano. Before great music begins, the elements upon which the harmonies are built must be aligned.

Part 1 - Theory

Entire books have been written that still do not completely cover the complexities of this seemingly simple procedure, but if you take it step by step you can come away with at least a basic sense of understanding, and appreciation for the union of art, craft, and physics that is piano tuning.

Essentially, piano tuning means adjusting the tension on the wires (strings) in the piano to make sure that when they vibrate they produce the frequencies that sound best, whether the note is played by itself or in combinations. Since any note may potentially be played in an number of combinations with other notes, that is not as simple as it sounds. All musical tones have an "ideal" relationship with any other note. The challenge is to tune every note such that it will sound as good as possible in as many combinations as possible.

Naturally, a "system" has evolved for accomplishing this known as "equal temperament" which most tuners use, (though not everyone agrees that this is the best solution.) Add to that the fact that every piano is different and presents a different combination of challenges, and the job of tuning a piano becomes more like a cross between solving a puzzle and shooting at a target. The rules of every puzzle are the same, but the solution is different with each one. Also, the further you are from the target, the harder it is to hit the bulls eye.

The Beat Goes On

Let's start with the unison. You may know, or notice, that the lowest bass notes on a piano have only one rather thick and long string per key. As the notes go higher the strings get shorter and thinner, which you can easily observe in this section, which usually ends with two strings per note. Soon after that, as you travel higher up the keyboard you will find three plain wire strings per note, continuing to the highest "C" at the top.

When a string is tightened it will vibrate when struck. Piano strings are set into motion when struck by a hard felt hammer. A musical tone is nothing more than our ears perception of something vibrating at a regular (rather than fluctuating) frequency. The lowest strings on a piano vibrate at around 27 times per second, while the top notes vibrate around 4000 times per second.

The term "hertz" is abbreviated Hz., and simply means vibrations per second. So if I say that the standard pitch (or frequency) for A above middle C is 440hz., I mean the string at that "A" vibrates 440 times every second.

Let's imagine a string that is tightened until it vibrates at 100 Hz. If I play another string at the same time, and that string is tuned to a different frequency, our ears will simply tell us that we are hearing two different notes. Generally, if there is more than a 10 Hz difference between two strings our ears will hear two separate notes.

However, as the frequencies get closer, it begins to sound more like one note that has a "vibrato," or "wa-wa" type sound. Piano tuners call that "wa-wa-wa" sound a beat. As the two notes get closer and closer in frequency, the beat slows down, but does not go away until the strings are at EXACTLY the same frequency. Here's why.

Picture two strings side by side, going up and down exactly in pace with each other. If you could observe these strings from the side (in VERY slow motion) you might see only one string, because the other would be "hiding" behind it! For all intents and purposes, these two strings together would sound the same as either one alone, just louder. These two strings are vibrating IN UNISON. The musical tone comes to your ear smooth and unwavering.

But if one of the strings goes only one vibration per second faster than the other, what will happen? Half-way through one second, one string will be going up while the other has gotten ahead of it and is already going down. When this happens the energy of one string more or less cancels out the other, and the volume drops. By the end of one second, however, the faster string has caught up and they will be rising together, creating once again a louder tone.

The perception of the tone getting softer and louder in a regular pattern is the source of the beat. So if one string is at 440, and another is at 442, you will hear 2 beats per second. It will not sound like two notes, but one note with a warble in it.

The easiest way to tell if your piano is in tune is to listen to one note at a time. If the tone is "pure" and still, the unison is in tune. If you hear a beat, (often referred to as "twangy") the strings are not in unison.

So now we know where a beat comes from: two frequencies that are too close to sound like different notes, but not exactly the same either, so we hear one tone with a vibrato. (There is a phenomenon known as false beating, whereby a single string does not in fact vibrate purely but has a beat of it's own. That discussion is beyond the scope of this essay, but it is one of the many troubling realities tuners deal with every day...)

Partial Harmonic Overtones

Now for the interesting part. A string under tension will in fact vibrate at a regular frequency, as I said. BUT, that is not ALL it does. When set in motion, a string under tension will do something very surprising. As you might imagine, it will vibrate like a jump rope, along it's entire length. This creates the loudest sound we hear, and we call it the FUNDAMENAL tone.

However, at the same time it will also break itself down into fractional segments and vibrate at a speed inversely proportional to the length. Don't panic. This just means that the string will "break" itself into two halves that each vibrate twice as fast, and also break itself into thirds that vibrate three times as fast, and also break itself down into fourths that vibrate four times as fast, and fifths, and sixths, etc., all at the same time!

Although these faster frequencies are not as easily noticed, once they are pointed out to you they are not at all difficult to hear. These higher frequencies created by a vibrating string are the HARMONICS (also called OVERTONES or PARTIALS) that you have heard about but perhaps never completely understood.

Ok, so now we know that a string produces a spectrum of frequencies, and we know that two frequencies that are close but not the same will produce a beat. The frequencies created by the single string vibrating in halves and quarters produce the notes we call “octaves.” If we start with a given note, we can tune pure octaves by listening to the beat that arises when they are played together, because the fundamental vibration of the higher note is the same as the harmonic from the lower note, so a beat is produced.

The single string also produces a harmonic by vibrating in thirds, producing a note we call a perfect fifth. On a keyboard, the string at C, when vibrating in segments one third it‘s length, will produce a tone that defines the frequency of the G an octave higher. If we try to tune that G to that C, we will hear a beat. When we adjust the tension on the G until there is no beat we will have tuned a “pure” fifth.

Now we find a problem. If we continue to tune pure fifths by continuing this pattern, eventually we will find that it takes us in a circle that comes back around to C. (The Circle, or Cycle, of Fifths.) If we have done a perfect job tuning our fifths “pure” it turns out that the C we end up with will not be the same as the C we would get if we tuned pure octaves from the same starting note.

The Right Temperament For The Job

For this reason tuners and musicians have struggled for centuries to find a compromise. The better job we did in tuning some relationships “pure” the worse other relationships sounded. A system needed to be found that would solve this problem, whereby we could alter or “temper” all 12 notes in one octave range so that they would all be in acceptable relationship with each other.

This dilemma caused the early keyboard makers and piano tuners NO END of grief, frustration, and disagreement. No matter how perfectly a good musician with a good ear tried, he or she could not tune every relationship to be in its pure version; sooner or later SOMETHING would sound wrong, and there was NO way to fix it.

So, we had to find a way to CHANGE the relationships SIGHTLY from their pure state. To change is to TEMPER; we had to TEMPER the notes so that we could play any two notes together and have them sound, if not PERFECT, at least acceptable.

This is why tuners centuries ago invented what was called the TEMPERMENT. It is a system for slightly altering the notes from the mathematically perfect relationships to one where they would still sound acceptable in all possible combinations.

In the early years, there were many differing ideas as to how to best accomplish this. The one thing they all had in common, however, was the fact that some relationships were made pure or closer to pure at the expense of others.

These systems each had their own merit, but they all severely restricted the composer. Music needed to be written in a specific key so that the music would fit the tuning. The most famous example of this is Bach's Well Tempered Clavier. In this case, the word “Well” does not refer to the quality of temperament, but a specific system of temperament that Bach preferred. The music was specifically written to maximize and utilize the qualities of that tuning system.

As composers began writing music that moved through different keys it became clear that a better solution had to be found. That solution was called "equal temperament," and is the system used almost universally today. We simply decided that of all musical relationships, the one that could NOT be compromised was the octave. If we simply placed the 11 notes that fell between them at a precisely equal, mathematically proportional distance from each other, we find that although no other musical intervals would be pure, ALL similar intervals would be off by the same amount.

For example, the notes "C" and "E" are in a musical relationship known as a third, (because "E" is the third note away from "C" in the major scale.) Similarly, "F" and "A" are also thirds. When you play either C/E or F/A, with equal temperament neither sounds any better or worse than the other; they are both equally "off." This gives our ears a chance to adjust to an expected degree of imperfection that we can learn to accept because it is consistent.

By carefully listening to the beats created by certain musical intervals and adjusting the notes until all the similar intervals had consistent beats, the early tuners were able to evolve a series of aural checks that would guide them through the process of setting one of each of the twelve notes so that all twelve were in the best possible "equally" tempered relationship.

Although mathematically you might think that these notes would have to be the same on all pianos, that turns out not to be true, because the beats are created between the HARMONICS of the notes, and so we have to adjust the fundamental frequency so that the harmonic we are listening to ends up in the right place. (Because two different pianos may have different size wires with differing lengths for the same note, the harmonics one piano will produce at middle C will not be the same as another.)

The use of electronic tuning devices (ETD’s) has greatly improved the overall consistency of tuning especially in the temperament range. However, even the most sophisticated device will fail to do the best possible job. These devices listen to a sampling of harmonics and make predictions based upon those measurements. Some devices claim to measure ALL the possible harmonics.

Still, if you use three or four of the top brands available on the same piano, they will all offer slightly differing results. Ultimately, it is the trained ear that must listen to the "suggested" tuning and decide if it is in fact the ideal sound.

So now at least we can tune one full octave of notes, from "C" to shining "C," as it were. (Sorry, bad piano tuner humor...) Assuming we could tune all the strings in those unisons perfectly, we could indeed be confident that at least those thirteen notes were now “in tune.”

Always Stretch Your Octaves Well Before Any Strenuous Exercise

Yet, the story continues. Now that we have the notes within one octave in tune, do we simply tune each higher note from the octave below? That is, do we just take our next note and tune a pure octave from its octave below, and so on?

Well, yes and no. Let's go back to our single vibrating string. It clearly and audibly vibrates in it's full length, its half, quarter, and eighth. If we are listening to middle "C" then, it produces a frequency at one, two and even three octaves higher, at the same time. Now I know I said earlier that the vibration is inversely proportional to the length, and this is true. BUT, the length is unfortunately not exactly what we would expect.

Because the string is an actual, physical, vibrating steel wire, with thickness and under tension, we find that as it divides itself into vibrating fractions, each fraction is slightly SMALLER than 1/2, 1/3, 1/4, etc., because the point where the string pivots (the node) is not a mathematical point, but actually a measurable length of wire that for all intents and purposes holds still.

Each harmonic is somewhat sharper than expected , and the higher we go the sharper it gets. This quality is known as inharmonicity. So now we have ANOTHER problem. If I want to tune the highest "C" on a piano, what should I compare it with? If I listen to middle "C", I find that I have to bring the top "C" SIGNIFICANTLY sharper in order to have it "line up" with middle "C's" harmonic.

Now it will sound perfect when played with middle C, but too sharp when played with the "C" just one octave below it. If, however, I decide to tune high "C" to the "C" one octave below, it will sound flat when played with middle "C".

The art of tuning notes sharper than expected (as you go up, and flatter than expected as you go down) so that they line up with the harmonics generated by the middle notes is called "stretching the octaves" and no two tuners (or electronic aids) do it exactly the same. Here is where one of the real artistic elements of piano tuning happens. On one hand we want consistency, and we want our top and bottom notes to sound in tune, yet we have no idea which notes they will be used with by the player; probably ALL at one time or another.

Once again, most tuners choose to find a middle ground, setting their octaves somewhere in the middle of the possible range, but this is not always the best solution. The right solution must be found by understanding the piano, the artist, and the type of music most usually played. This takes years of practice and experience.

Do the Math

For those who have managed to make it this far and STILL want to fully understand why we have twelve notes in western music and why we have to employ a temperament when tuning so called “fixed pitch” instruments, I offer this mathematical exercise. (Otherwise, skip this part and go to Part II below.)

The twelve notes we use in western music are not randomly chosen. They represent frequencies discovered by expanding the naturally occurring relationship of one note to another of 2 to 1, which occurs naturally in a single vibrating string vibrating in halves, combined with the notes discovered by the relationship of 3 to 1 which occurs when a single vibrating string breaks into three vibrating segments.

As explained above, by simply touching a vibrating string at the half-way point we can increase the volume of the harmonic generated by that segment. Pythagoras discovered the simple mathematical relationship whereby the halves are found to vibrate twice as fast, the thirds three times as fast, etc.

If you compare the results of the coexistence of the ratio 2:1 with 3:1 you can discover the mathematical root of the problem.

Pick any frequency you want to be your starting note. Multiply that number by two repeatedly to get the theoretical frequencies of all the octaves that might fall within a pianos range. ( For Example a starting frequency of 25 Hz. Would yield a note at 50Hz., 100Hz., 200 Hz., 400 Hz., etc.)

Now if you start with that same note and multiply by three you will get a frequency different from the ones discovered in the octaves sequence above. Remember that once you have discovered a new frequency, you can divide that by two as often as needed to bring that new note down an octave or two. Otherwise the numbers will too quickly fall above the pianos range.

This process of multiplying by three and then dividing that answer by two as needed will ultimately create twelve distinct frequencies (including the starting note) that are different from the frequencies we found in our octave multiplication.

However, the thirteenth time we multiply by three we get a frequency that is just slightly larger than one we got to with our octave sequence. It is not exactly the same, but too close to be considered a new note. This small mathematical difference is known as the “Comma of Didymus.”

Thus, pure fifths cannot coexist with pure octaves. In equal temperament we slightly narrow each fifth so that the mathematical difference is factored equally into each successive interval, and will allow the circle of fifths to end up at the same place as the succession of octaves. Once we have 12 notes, we can deduce the theoretical frequencies of their octaves, and discover the beat rates that must result when different intervals (which share harmonics and thus generate beats) are played. Using this beat information, we can proceed to temper the piano according to the beats observed when various notes are played together.

Part 2 - Technique

At this point you have a general idea of some of the challenges tuners face when deciding where to place each note. Now we can begin to discuss the physical process by which the strings are adjusted.

The art of manipulating the tuning pins so that the strings will actually stay where we put them is extremely difficult to master. Even if there was a device that could give us with absolute certainty the best frequency for each string, and it was so affordable that any pianist could own one, we would not be in danger of being put out of business just yet!

After 4 or 5 hours of wrestling with the strings, the inexperienced tuner would find their piano in worse shape than when they started. Learning to master the aural part of tuning, at least theoretically, can be done in a few months, but learning the "hand/ear" coordination necessary to produce a professional result takes decades of daily practice.

In the article that follows ( Why Do Pianos Go Out of Tune?) I explain the ways in which a tuner can "set" the pin and "set" the string to achieve maximum stability. The challenge is to turn the pin as little as possible, using tiny, controlled "jerks" to achieve the desired result.

Each string is wrapped around a finely threaded metal tuning pin, which in turn is driven into a block of wood constructed with layers of hard maple. The grain of each layer runs in a different direction, creating an incredible hard and stable block of wood, which cannot swell or shrink with humidity changes.

Tuning a piano is a lot like shooting at a target. You would like to hit the center of the bulls eye, but the further you are from the target the more difficult that is. The farther the strings are in pitch from where they need to be, the more we have to turn the pins. That in turn increases the likelihood that the string will not stay exactly where we placed it, because the more you move the string the more difficult it becomes to maintain even tension throughout it's entire length.

Further, although the vibrating strings produce the tone, by themselves they would be so quiet as to be barely audible. The strings pass over and are pressed down against a wooden bridge, which in turn is attached to a large wooded diaphragm know as the sound board or sounding board. As we bring the strings "up to pitch" we continually increase the downward pressure on the soundboard, which will in turn change the tension on the strings we first tuned. That is why we have to do a "pitch-raise," or rough tuning first. (See FAQ's: What is a pitch Raise? What is A-440?)

Part 3 - Electronic Tuning Devices

Prior to the introduction of electronic tuning aids, we all tuned "by ear." The process began by striking a tuning fork to sound the standard pitch of A-440hz., and then one string from the "A" above middle "C" was set to be in unison with that tone.

Then, using that "A" as our starting point, we proceeded to "set the temperament." By understanding how fast the beats created between any two notes should be, we could use various patterns to tune one string from another, continually checking to see if the beats were correct, and if they were "progressing" smoothly.

Once we had all twelve notes "tempered" to our liking, we would tune (and "stretch") the octaves so that each higher note made the least audible beats when played with the octaves below. Since most of the notes had two or three strings, we would mute all but one string until we had that one right, then take out the mute and tune the unison, as described earlier. Sounds pretty simple.

However, even the most skilled aural tuner can make mistakes. There were many attempts "along the way" to develop a device that could tell the tuner where to set each string, and a complete discussion of the history of these devices is beyond the scope of this article. Suffice to say that they continued to develop and improve. Today there are a number of choices. Some are "self-contained" units, and others are software programs that run on a lap-top or hand held computer.

Each of these systems has merit, and it is important for a good tuner to try them all, to determine which one best supports his or her style.

Basically, they each attempt in different ways to "read" and measure the harmonics produced by the strings, so that the best possible tuning solution can be found mathematically. These devices provide many benefits. They greatly speed up the process and accuracy of pitch-raising, allowing for a more stable result. Also, they help maintain consistency as we go up or down the keyboard. Finally, once we have tuned the piano and are satisfied with the result, we can "save" the tuning in the internal memory, so that in the future we do not have to go through as many steps, checking the results along the way.

There are problems, however. First, I have never seen a device provide information that could not be improved by a well trained "aural" tuner, so although the electronic "feedback" is extremely valuable, it is not by any means perfect. Second, in the hands of an inexperienced tuner who has not taken the time to develop the physical skills needed, the devices can become a crutch, and the piano is not likely to stay in tune. Finally, there is always user error. Tuners misread the data, tuning a note other than the one indicated, for example. Sometimes, especially in the bass, a tuner can be so far off that the device is reading the wrong harmonic.

Thus, like any other tool, these devices must be used correctly and are never a substitute for skill and experience, and cannot be utilized to their full potential unless the technician has a thorough understanding of all the fundamental principles which this article has introduced.

I hope that this has helped give the reader a more profound understanding of this most extraordinary instrument, and the challenges it presents to the technician attempting to bring it to its full potential.

It is this fascinating union of art, science and craft that keeps us looking forward to each tuning, year after year. When a piano is well tuned, the music it produces washes over us like a great sigh. It somehow resonates in unison with the frequencies of our spirits, and in fact helps put US back in tune, which is the magic of music. The greatest physicists of our time now believe that the universe itself is, at its core, a complex interaction of subatomic, vibrating strings... So this is where the matter of tuning ends, and philosophy, physics, and metaphysics begins.

I'll save that for a future article!

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