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The temperament

Before I delve into the temperament, I should begin by explaining the two principal types of relationships in piano tuning: the unison beating and interval relationships.

A unison is the relationship of the two or three strings that together make up a note for most of the piano. If the strings of a given note aren’t tuned exactly the same, the slightly different pitches produce an ‘out of phase’, or ‘wow-wow-wow’ sound, a regularly repeating beat rate. This unison beating gets slower as the two somewhat different pitches get closer to being the same, then disappears altogether when the two pitches are exactly the same.

When I started in the tuning program at NEC, we learned how to hear and tune unisons by lowering the pitch for one of two strings, then bringing the pitch back up until it matched exactly the other string. We spent a lot of time raising the string too far, then lowering it too much, back and forth. It can be hard to get it perfect right away. This is a good way to get started with practicing tuning.

Interval relationships are how any two notes of an interval interact. In piano tuning, each note is tuned by comparing it to another reference note, using intervals. This is the edge of the cliff right here – if you don’t want to permanently change the way you hear music, STOP READING. Once shown the relationships of intervals from a tuning perspective, it is impossible to un-learn.

Here are some common intervals used in tuning and their relationships:

Thirds – In the temperament section of the piano, thirds have a “beat rate” that gets progressively faster until it gets so rapid that it’s hard to hear. Take the F3A3 third, for example. These two notes together should have a beat rate of seven beats per second. This rate gets progressively slower when moving down the keyboard, faster when moving up.

Fourths – Fourths are almost pure intervals, meaning that both notes interact in a way that produces almost no beat rate. A fourth is a slightly expanded interval, which makes it wider than a pure beatless fourth. Unlike the thirds, fourths remain slightly expanded throughout the entire keyboard.

Fifths – Fifths are almost pure intervals, much like the fourths, except they are slightly contracted. This makes it narrower than a pure (beatless) fifth. Also like the fourths, the fifths remain slightly contracted throughout the keyboard.

Sixths – Sixths have beat rates much like thirds. For example, the A3-C#4 third vs. the A3-F#4 sixth: the sixth will have a beat rate of roughly double that of the third. In the middle section of the piano, where the beat rates are slow enough to be able to hear, thirds and sixths with the same base note will have this same relationship. Like thirds, the beat rate gets progressively slower when moving down the keyboard, faster when moving up.

Octaves – Stretching an octave to address inharmonicity (see stretching the tuning) means making the octave slightly wider than a pure octave. As covered later, just how wide to stretch an octave is really a matter of personal preference and what sounds best to the tuner.

If you are confused about all this talk of interval relationships and beat rates – just relax. This is where one of the major benefits of this website really comes through – not only do I describe what to listen for, you get to hear it for yourself. Knowing how each interval should sound and how they all interact together comes with time and patience, so don’t get too frustrated if it doesn’t come right away.

A notable note about note notation

When I refer to a specific note on the keyboard, I’ll use the standard of note and octave number. Always having to write out ‘F below middle C’ is tedious; F3 (or third F on the keyboard from the bottom) works much better. I should note just to clarify that the lowest several notes on a keyboard are A, Bb and B, and are referred to as A0, Bb0, and B0. The first C from the bottom would then be C1, then up from there.

What is a temperament?

I’ve found trying to explain temperament to someone who knows nothing about piano tuning to be difficult. I’ll give it another try!

Piano tuners need to start somewhere, so we use a tuning fork (usually A440 or C523) as a reference note. All we need is one note that we know is right, then we go from there. I tune the A above middle C to the tuning fork, so that becomes the reference note. It would be possible to tune all the other A’s of the piano from that reference A one octave at a time, but besides not being accurate enough, only tuning the A’s is not very helpful. Tuners use different methods of tuning all the notes inside of an octave in the middle of the piano, typically F3-F4 or A3-A4. Starting with the tuning fork, all of the notes within the octave become the reference notes for the entire tuning. This octave of reference notes is the temperament section.

Tuning the temperament is all about relationships of intervals. It might be helpful to use an analogy of a family reunion. Imagine you are the host of the annual reunion, and none of the family members are getting along. As host, your job is to help everyone find common ground and to end the…dischord. Easier said than done! As soon as you get Aunt B and Uncle C to stop arguing, that sets off cousin G complaining even worse! Patiently and methodically, you patch up all the relationships until there is …harmony. Tuners can think of themselves as relationship counselors, from a musical point of view.


Critical foundation

If you take away nothing else from this tutorial, let it be this: an accurate and stable temperament is the critical foundation to any good tuning. It simply will not work without the accuracy and stability. The temperament becomes the blueprint, the reference for which the rest of the tuning is based.

Temperament Accuracy

Learning to tune a temperament accurately is very challenging. As the temperament progresses, notes which were already tuned can be used as checks, to make sure it’s moving in the right direction.

In some cases, the easiest and best thing to do is to just memorize the sound specific intervals should make. My first teacher told us when we were just starting, to memorize the sound of the F3A3 third, which should be seven beats per second. There are no checks for that since it is at the beginning of the temperament - just memorize how it sounds. Having a reference temperament to listen to and compare, such as the temperament audio file of this tutorial, is extremely helpful. The more I tuned, the more I memorized how each interval relationship should sound, and tuning the temperament became much easier and faster.

Not to be discouraging, but on many pianos it will be impossible to duplicate a perfect temperament. Most pianos have their inaccuracies and idiosyncrasies, and getting all the intervals to fall into place can be frustrating. One of the most successful strategies I employ when a temperament just won’t work out on a particular piano (usually it’s a shorter upright) is to change the beat rates of the thirds. I have some Baldwin Hamilton studios that I have to start by slowing down the F3A3 third from seven beats per second to five, and consistently slow down the progression on the thirds throughout the temperament. It will make the rest of the tuning smoother to slightly modify the speeds throughout the temperament instead of leaving in one jarring interval that won’t fit into place.

Accuracy is so important in the temperament because whatever errors are there initially get magnified throughout the rest of the piano. This leaves a tuning sounding uneven and disjointed. I’ll get more into this later, but confidence in my temperament accuracy allows me to tune faster with fewer checks as I go along. With a solid temperament in place, tuning with octaves and fifths gives me such a narrow window for error that I usually don’t end up having to use most other checks as I tune.

Temperament stability

The absolute need for accuracy and the fact that the temperament becomes the reference for tuning the entire piano makes tuning stability in this section ESSENTIAL. Since the temperament is the foundation upon which the rest of the tuning is built, it has to be rock-solid.

I had frustrating temperament stability issues earlier on. I would tune the piano, take out the temperament strip and tune the unisons, and I’d be in trouble. The temperament wouldn’t end up the same as when I’d left it, and I’d have to spend time doing reconstructive surgery to my poor foundation. Sometimes I could get it to fit back into place with the rest of the piano, and sometimes it was almost better to just tune it all over again. Until I got better at hammer control and setting the tuning pins, I had to come up with a way to compensate for my temperament tuning instability.

One of the solutions I found to keeping the temperament rock-solid was to tune the temperament section first with just the rubber mutes, then insert the temperament strip to “fine tune” the section again. Unless the tuning just needs a touch-up, I always use the rubber mutes first. There are other techniques I employ to keep the temperament rock-solid, but I’ll get to those a little later.

A word about equal temperament

Okay, a number of words, actually. Historical temperaments used to favor some notes over others, depending on which key the music was written in. Today there are a number of equal temperaments. Which one to use is largely a matter of personal preference (most tuners probably use whichever temperament they were originally shown). Some temperaments are, by design, more accurate than others. On an individual level, some temperaments are easier for a person to use than others. Temperaments are all really just different solutions to the same puzzle.

A440 vs. C523 tuning forks

Different temperaments call for using different reference notes to begin, usually either A440 or C523 tuning forks. I’ve always used an A440 fork, so I have some bias here. Since A440 is the concert standard for pitch (or possibly A441 or A442) it makes sense to me to start with that as the tuning reference. Using C523 as the reference to tune the piano only to try to end up at A440 just doesn’t seem very direct to me. However, there are a lot of tuners who use a C fork instead of A, so it must work for other tuners.

Why I created my own equal temperament

First, a little background:

The first temperament I learned was an A3-A4 temperament at the tuning program of New England Conservatory. This temperament is tuned with thirds and sixths, using fourths and fifths as checks. I’ve long since forgotten the name of the temperament, but first all the naturals are tuned, then all the sharps.

The second temperament I learned was an F3-F4 temperament when I apprenticed at Steinway Hall. This temperament (the Steinway temperament) is tuned with emphasis on fourths and fifths, using thirds and sixths more as checks. Talk about turning someone’s tuning world on its head! I’ll be honest: I had a very hard time learning this temperament. I struggled for many weeks with limited success trying to get the temperament to work. I knew my time was running out and if I didn’t work things out fast, my career at Steinway would be very short-lived.

Desperate times call for desperate measures! Now knowing two different temperaments intimately, I could see what I liked about each one and what didn’t work for me. I took what I saw as the strengths of each and used these to create a new temperament. It took some time to work out, but when I was finished I was able to tune a temperament far more accurately than I had ever been able to do before.

There were several major factors I incorporated into this new temperament. I liked the use of thirds and sixths to tune the temperament, but also fourths and fifths. I didn’t want to favor one group over the other; for either group to be any less important. Notes within my temperament are tuned using thirds, fourths, fifths and sixths, and each of these intervals have equal importance.

I like to use as many checks as possible using whatever notes have been tuned already. The A temperament I learned had many checks. As far as I can remember from ten years ago, the F temperament had fewer immediate checks and several “bridging” checks - several notes had to be tuned before going back to use the checks. This felt to me like driving blind, so I tried to incorporate as many immediate checks as possible.

Since I had learned both an A-A and F-F temperament and was familiar with all the interval relationships in between, it was natural for me to expand to an F-A temperament. You never can have too many checks!

Important notes

I recently noticed that Randy Potter (of correspondence course fame) teaches an F-A temperament as well. I guess I can’t claim to be the only one to have thought of the idea! As far as our two temperaments go, any casual comparison will show that the similarity ends there.

I stated earlier that I created the temperament out of necessity, and that I used what I thought were the best aspects of both temperaments I had learned. I did what worked best for me – I certainly don’t claim that my temperament is better or an improvement over the first two I learned. The last impression I want to make is that neither were “good enough” for me!

The McCullough temperament

Is it fair to give the temperament I created my last name? Please don’t think my ego to be too large! Still, it’s not every day that someone comes up with a new temperament, so I’ll give myself a little credit. A German name might instantly give it more credibility, but alas all my ancestors came from the UK. So without further ado, here is a chart of the McCullough temperament:

Tuning intervals and checks (♫)

A4 – A3 OCTAVE BELOW ♫ A4, F3 (after the next…)

Rough temperament “with wings”

Another technique I use to improve temperament tuning stability is to extend the rough tuning of the temperament section by several notes in either direction. This goes back to the critical need for tuning stability in this section especially. I find that changes in string tension of a note have the greatest effect on the neighboring several notes. If I rough-tune only the notes of the temperament, tuning the immediately neighboring notes would probably affect the notes of the edges of the temperament, including A440. Rough tuning several notes outside the temperament provides me with some assurance that the critical foundation that I started with will still be in place when I am finished.

Using a temperament strip

I use a temperament strip to fine-tune the temperament section from F3-A4. After I use rubber mutes to do a rough tuning of the temperament, I use the end of a wire-handled mute to insert the strip between the strings of each note, which mutes off the left and right strings. I keep the strip in place until the entire piano has been tuned, then I remove it to tune the unisons in the temperament section.

Audio files of temperament

Audio of temperament fine-tuned with temperament strip

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