1. Why Not Tune Every Piano Note to Its Exact Ideal Frequency?
Using a tuner that shows pitch with a needle, you might think setting all 88 keys to their exact target frequencies would make a piano sound perfect. However, tuning guides state that the goal is not simply to match each note to an ideal value. Instead, the aim is to spread out the distortion that appears as "beats" when multiple notes are played together, creating a sound that is not unpleasant to the ear [1]. There is no single correct answer for how many beats to leave; tuners decide this by listening.
In Western countries and Japan, tuning with a tuning fork and the tuner's ear is the mainstream method [2]. Even if you set each note to its ideal value, the perceived pitch changes with volume, harmonics shift, and strings resonate with each other, which can make the overall sound poor. Tuners and apps exist, but the ear plays the major role. So, what are they listening for? The clue is "beats."
2. What Are Beats? How Slightly Different Pitches Make the Volume Rise and Fall
When two notes with slightly different pitches sound together, the volume rises and falls repeatedly. This is called a "beat." The number of beats per second equals the difference in frequency (the number of vibrations per second) between the two notes. For example, 440 Hz and 444 Hz produce 4 beats per second. The smaller the difference, the slower the beats; if they match exactly, the beats disappear.
In contrast, notes in integer multiples of each other, like 220 Hz and 440 Hz, do not produce noticeable beats and sound harmonious [3]. As the relationship moves away from integer multiples, beats become more audible. In pianos, beats often come from mismatches between harmonics, not just the fundamental notes.
In the "equal temperament" system commonly used for pianos, some combinations naturally produce beats. For instance, the major third interval of F and A has a slight mismatch between F's 5th harmonic and A's 4th harmonic, creating audible beats [3]. In "just intonation," another tuning system, these would match and the beats would vanish. Thus, equal temperament is a system that assumes beats will exist. How do tuners handle these unavoidable beats?
3. Why Piano Tuners Count Beats Instead of Matching Frequencies
Tuners focus on the number of beats rather than the exact frequency of each note. If they only matched the fundamental frequencies, the harmonics would not align, and the overall sound would be chaotic [5]. A piano has about 230 strings, each under about 80 kg of tension [4]. To manage this, tuners follow a set procedure.
First, they strike a tuning fork to produce 440 Hz. If 440 Hz is the target, they tune the piano's A note so that the beats disappear [5]. Then, they assign specific beat rates to intervals like perfect fifths, fourths, and major thirds. For example, D and G might have about 0.7 beats per second, while F and A (a major third) might have about 7 beats per second [5]. These are rough guides, and practices vary among tuners. They set these rates in the middle register and extend them by octaves to the whole range.
Pianos have 3 strings for each note in the middle and high registers (1-2 thick strings in the low register). Once the middle string is set, the tuner silences one side string with felt and tunes the other to match the middle string, removing beats [6]. Only when all 3 strings are tuned to the same pitch is the note complete. This process assumes strings vibrate ideally. Do real strings do that?
4. Why Stiff Piano Strings Mean High Notes Are Tuned Higher
Real strings are stiff, unlike ideal strings. This stiffness causes higher harmonics to be slightly higher than integer multiples of the fundamental frequency (a phenomenon called inharmonicity). The deviation increases with higher harmonics [7]. Because of this, if you tune to the calculated frequency, the beats will not match the calculation.
Therefore, tuners use "stretched tuning," making octaves slightly wider than a 2:1 ratio. High notes are tuned higher than theoretical values, and low notes lower. The amount of stretch varies by instrument and range. An English-language guide suggests a range of about ±35 cents [8]. A cent is 1/100 of a semitone, so 35 cents is about 1/3 of a semitone. Raising 440 Hz by 35 cents results in about 449 Hz. This ±35 figure is a rough guide, not a fixed rule.
Is this just because of the strings? A study by Aalto University in Finland had 15 tuners and 18 orchestra musicians tune based on "piano-like" sounds with shifted harmonics. The resulting tuning curve resembled a Railsback curve and was significantly different from the tuning produced when listening to sounds with clean harmonics [9]. The prevailing theory is that stretched scales are explained by string inharmonicity. In this experiment, the result followed the string's natural quirks.
5. When Was the Standard A Pitch Decided? A=440 Hz and the 1939 Meeting
Tuning starts with the tuning fork's A note. The widespread adoption of A=440 Hz is relatively recent. In May 1939, representatives from France, Germany, the Netherlands, Italy, and the UK agreed in London to set A=440 Hz as the standard. This led to ISO Recommendation R16 in 1955 and ISO 16 in 1975 [10].
Before that, standards varied by country and era. In 1860s France, A was 435 Hz; in 1899, British piano tuners used A=439 Hz [10]. If you play 440 Hz and 435 Hz together, the 5 Hz difference creates 5 beats per second. Even small differences in Hz are audible as beats.
April 4th is "Piano Tuning Day." The standard A note corresponds to the initial of April, and the frequency is 440 Hz. The International Association of Piano Builders and Technicians (IAPBT) established this day in 1993 [11].
6. How to Hear Beats and Calculate Them Yourself
Let's calculate. If you play A at 440 Hz and another note at 444 Hz simultaneously, how many beats per second are there? Subtracting the frequencies gives 4 beats. For 7 beats per second (the guide for F and A), you hear a beat about every 0.14 seconds. Tapping a desk 7 times per second helps you feel that speed.
To hear it, try with a friend using recorders. Play the same fingering simultaneously. If one player adjusts to be slightly higher, you may hear beats. If you don't hear them, it is not a failure; checking the calculation is sufficient. To protect your ears, use a low volume and do not play for long.
Research pages on tuning knowledge are available in Japanese [1][5][10]. They include examples of counting beats and procedures for tuning the middle register. Note that the ±35 cents guide comes from English-language sources. Adults should read these English materials together with Japanese tuning association or manufacturer documents to compare them.