Piano Tuning Tactics: Harmonics, Temperament, and Electronic Tuners

March 7, 2014
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As a sound engineer, I’ve known and worked with many accomplished piano players, and I’m an avid pianist myself. But until I started down the path of tuning pianos, I had no clue that the concept of a piano being perfectly “in tune” does not actually exist. Most of us know when unisons and intervals are way off. But what constitutes “in tune” in terms of being musically pleasing? The answer lies in understanding two key factors: temperament and inharmonicity.
 

It’s a String Thing

Piano strings exhibit inherent inharmonicity. This means that the fundamental pitch, let’s say the note A vibrating at precisely 440Hz, will exhibit harmonic content (also called overtones or partials) that is not evenly distributed as in say, a pipe organ, a Hammond organ, some synths, or many other instruments. The second harmonic, which in theory should be 880Hz, will in fact be slightly higher in pitch than twice the fundamental. The third, fourth, fifth and other ascending harmonics will rarely, if ever, be exact arithmetic multiples of the fundamental.  

Moreover, when a piano string is struck by the hammer, it becomes physically displaced, causing an initial and temporary pitch change in its fundamental and harmonic frequencies. This is difficult to hear because the pitch displacement in the fundamental is small, yet magnified in the harmonics that are multiples of the fundamental frequency. The result of this inharmonicity is usually perceived as harshness, but not always. In addition to what you can hear, seeing the pitch visually wandering on an Electronic Tuning Device (ETD) readily shows what’s happening.

Shorter upright pianos, spinets, and baby grands all have short strings that exacerbate inharmonicity—so will low-quality string material. Conversely, the longer strings on medium- and full-sized grand pianos, along with their typically higher quality metal alloys, are much more forgiving; partials are more closely in-tune with each string’s fundamental pitch. Less inharmonicity equates to purer sound quality, but that’s not always the goal of a piano manufacturer. Harmonic content can be part of a piano’s unique tonality.

The “scale” of a piano—that is, the string length in relation to string tension, intended pitch, and resultant harmonic content—is a huge element that distinguishes one piano from another. Variations on scaling include the duplex scale, invented by Theodore Steinway. This is a short “non-speaking” segment of the string length, isolated from the string’s “speaking” length by a bridge and roughly tuned to an octave or fifth above the note. In some pianos the non-speaking strings can be tuned; in others they can’t. 

Exactly how partials present themselves in a given string is not readily predictable, and it’s only one of many factors in a piano’s character. Were it not for the highly complex interaction of the strings, the bridge, the soundboard, the hammers, and the structure of the piano case itself, then all pianos of a given size would sound pretty much the same. But they don’t, and we revel in that fact.


Temperament

Temperament is the frequency relationship of each interval to another within the “temperament octave” that dictates the pitch of all upper and lower octaves, and one size does not fit all! Numerous forms of temperament range from equal to ancient. To name just a few: 14th-century Pythagorean (1373), DeMorgan unequal (1843), Moscow equal-beating, William Hawkes Improved (1807), and the list goes on.

Temperament can be (and often is) set by ear by skilled piano tuners who count beats—audible oscillations between not-quite-in-tune notes—though the modern-day preference of many professional tuners is to hand this task over to an ETD. Fifths can be made perfect (no beating) at the expense of fourths and thirds, or the opposite can be true. One aural tuning guide, authored by W. Dean Howell, specifies 6.93 beats per second (bps) between F33 and A37. Under this same guidebook, A37 to the higher E44 should beat at 0.74 bps to achieve a perfect fifth. Point being, there is no such thing as beat-free intervals throughout the scale. 


Electronic Tuning Devices

Modern electronic tools give professional tuners a whole new way to work. Pianists who can’t afford to pay a tuner to show up every week might also take advantage of these, but they’ll need to work through the rather steep learning curve of handling a tuning lever (also called a tuning hammer) to properly set the tuning pins. It’s not easy, but it’s not rocket science, either. If you just move the lever until you hear the desired pitch without “setting” the tune, stability will be elusive. This involves torquing the tuning pin upward in pitch, then relieving the torque, and then carefully adjusting each string’s tension for stability. 

ETDs range from dedicated hardware such as Peterson’s industry-standard strobe tuners to modern apps that run on laptops, iOS, and Android. I was privileged to try out two leading apps while repeatedly tuning a year-old Yamaha upright piano in a small church—an application that many professional tuners might encounter. Both were easy to learn, yet sophisticated. 

  
 Fig. 1. TuneLab’s primary display. The black bars move left or right to indicate fine pitch, while the graph below indicates coarse pitch.Fig. 2. One of Verituner’s main displays showing the inharmonicity table (left), pitch in cents (center), current note, settings, and the inharmonicity I-beam (right).
 

TuneLab. I started with TuneLab (tunelab-world.com), an app I ran on an iPhone 4S. The display shows an FFT-style coarse pitch graph, while simultaneously providing a series of black bars that move either left (for flat) or right (for sharp), to establish fine pitch.. The coarse pitch graph lets you see two or more unisons at the same time that are not in-tune, due to its excellent resolution. This is a very helpful feature, especially in the upper octaves. TuneLab can also sample strings in order to identify their inharmonicity and thus calculate the projected inharmonicity of the entire piano’s scale.

 
Verituner. Next was Verituner (veritune.com), a product with significant design depth. It runs on iOS or Windows PCs, and is also available as a standalone device called Verituner Pocket. I loved Verituner’s relative initial simplicity, combined with its ability to make complex adjustments to the prescribed starting points, thus giving the professional tuner a broad palette of possibilities. The display depicts a rotating icon including several additional elements. Coarse pitch is shown much like a speedometer needle; fine pitch is depicted by a spinning rotor. Inharmonicity is displayed as a table for each string, providing valuable information about a given piano’s harmonic quirks. Verituner looks at string inharmonicity in the background as each note is tuned. It doesn’t extrapolate, but rather, makes a direct calculation for each string. 
 

Stretch Tuning

Partials (ascending harmonics) tend to skew sharp in piano strings. So if each fundamental frequency were the only basis for a tuning strategy, most pianos would sound lifeless. For this reason, most by-ear tuners and most ETDs will “stretch” the upper and the lower octaves in relation to the reference “temperament tuning octave.” Both the TuneLab and Verituner ETDs do the work for you. You simply specify stretch tuning and you’ll get it, without having to count beats per second. Both apps allow adjustments to the stretch parameters, which is important for making a piano sound the best it can.

 

Parting Thoughts

Piano tuning is an art form backed by decades of scientific research. But science and the mathematical analysis of harmonics can never tell us what to enjoy when we listen to music. While we can analyze frequency content, we have yet to bridge the gap between objective analysis and why one subtle difference in tuning a piano might make all the difference in how a performer responds to the instrument. ETDs bring us much closer to this understanding, though none should ever be taken as a substitute for listening. 


Ken DeLoria has had a 45-year relationship with music in general and pianos in particular. He works in the audio industry as an engineer, most notably as the founder and CEO of loudspeaker maker Apogee Sound Inc. He can be reached at kdeloria@gmail.com.

 

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