Tuning  creates the scales and chords and blends into harmony the vast sea of overtones of  the  220-odd strings. A good tuning will bring out the piano’s “sweet spot”, finding just the right frequency for the strings to excite and bring out the best in each other. A good tuning, regulation and voicing help the piano take on a life of its own.

Frequency or pitch means how many times a second something vibrates (cycles per second). The A above middle C is normally tuned to a frequency of 440 cycles per second (A=440 c.p.s.), or 440 Hertz. This has been the standard North American pitch for almost 100 years. Most tunings, at least in North America, use A as the starting point. Frequency doubles as the note rises an octave, halves when down an octave, thus (note the lower/higher the note, the lower/higher  the frequency):

  A1 27.5 A2 55 A3 110 A4 220 A5 440 A6 880 A7 1760 A8 3,520

Oddly, many persons tell me they can’t hear the high C on a piano (some have said, “don’t bother tuning the high end, we don’t use it), though they can actually hear much higher frequencies if tested (around 20,000 c.p.s./Hertz). I believe it’s because the sound has a short duration, or sustain.

What is Equal Temperament?

              Pythagoras' demonstration of tension increase raising the frequency of a monochord

To tune a piano, the tuner must set a temperament, where the different notes are reconciled with each other to achieve the most musical compromise, then tune all octaves and all individual notes.

 There are many systems of tuning, all producing different sounding scales and chords. Many are tempered, which means not perfectly in tune (more on this below). The physics of scales have been explored for thousands of years; the Greek Pythagoras made observations on vibrating strings about 2,500 years ago; no doubt the Chinese did, much earlier.

 The basis of tempered tuning is thus:

If, starting on A, one tunes E a perfect fifth (perfectly in tune) above that, the B a perfect fifth above E, F# a perfect fifth above B and so on, tuning through the circle of fifths and arriving back at A, the last A tuned will be much sharper than the first, around 1/8 tone. These two A’s played together will be very out of tune with a pronounced “beating”, a disagreeable throbbing that’s the result of the difference in frequency between the two. The men of old called this difference the “wolf”, and devised a number of systems to sweep it under the rug or smooth it out by distributing this difference through a scale. This is called tempering the scale; equal temperament is the common method used in Western music. In this, all intervals are out of tune by a certain consistent ratio, so all scales, chords and intervals or out the same amount. Earlier temperaments were not so smooth, or the “wolf” was allowed to pile up in keys they simply avoided using, leaving other keys with purer intervals. String players and singers on their own will tend to use pure intervals that are not found on the keyboard.

In equal temperament, all fifths are out just a little. They have a flat, or narrowed, upper note and beat slowly; fourths are sharp, or widened and beat a little faster. Major thirds and sixths are wide and beat much faster yet, minors are narrowed, and so forth. A good tuning will have these beats slowing down descending, and speeding up ascending, nice and smoothly in all intervals. Easier said than done, because there’s an additional bugaboo to bedevil the piano tuner as if setting the temperament isn’t difficult enough, and that is Inharmonicity.

All vibrating bodies, from flicked skipping ropes to earthquakes, generate overtones, or harmonics. That is, they vibrate in fractions, or divisions, that are multiples of a basic frequency, and the shape they take forms waves with up-and-down and side-to-side motion.

                                                                     From phys.unsw.edu.au

These images depict a vibrating string with the resulting waves, in fractions of 1/2, 1/3, 1/4. There are myriad  smaller fractions not depicted here. The top diagram “L”, or fundamental, is what our ears hear (it's really a mirage of the mind. See the article below), namely the sum of all the other vibrations much like the colour spectrum comprises what we see as white light. We’ll call this note A. The next sounds an octave higher than the fundamental, A2, the next sounds an octave and a fifth higher, E2, the next sounds A3, two octaves higher, etc. These are all ratios like 1/2, 1/3/, 1/4. Evoked in succession as sound, they form a major chord: A/A2/E2/A3/C#3 and so forth (I find it surpassingly beautiful that the basis of harmony is folded into any note). The bass strings on a piano have a very complex wave-form, with overtones that extend well into the top range of the piano. In the treble, the wave-form is much simpler, becoming a simple sine-wave a couple of octaves above middle C.

The tricky bit is, because of the stiffness and great tension of the piano strings, their harmonics are out of tune with each other because where the different waves on the string intersect aren’t quite where they ought to be! That is, they don’t correspond to their theoretical values. In effect, some of an individual string’s overtones beat out of tune with each other. This becomes part of the colour of the piano, but in smaller pianos it greatly increases the difficulty of achieving a nicely tempered tuning, where the beating progresses evenly from low to high, across the span of the keyboard.

I routinely tune pianos that have severe problems with inharmonicity, but luckily the ear, with training and practise, can work through this. Sort of….maybe.

The “ideal” piano would use all the same gauge of wire; real pianos do not, because if they did, they’d be about 26 feet long! To produce the low frequencies of the bass (frequency depends on string length, diameter, mass and tension), builders have to wind the bass strings with copper to add mass without too much weight. So, the low A on a small upright is far shorter but much thicker and massive than that on a concert grand. This makes the inharmonicity of the upright far more unruly.

The science of our consciousness and nervous systems and how we perceive and interpret  sound and music is called psychoacoustics. I recommend this excellent Wiki articleif you wish to learn more about this fascinating aspect of mind.

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Piano Tuning:

the Harmony of the Piano

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