What does “setting a temperament” mean?
Before the advent of the Electronic Tuning Device, (ETD), the only way a piano tuner could tune a piano was to first “set the temperament”. But what does setting the temperament mean, and why is it called that?
The meaning of the word temperament comes from the verb “temper”. For modern Western music, the most common tuning system divides the octave into 12 equal parts, referred to as equal temperament. A piano tuner has to “temper”, or adjust the strings, so that all the coincidental partials and harmonics line up. But it gets more complicated than that.
Piano strings have harmonics that are out of tune, referred to as inharmonicity. Inharmonicity is the degree to which the frequencies of partials depart from whole multiples of the fundamental frequency. When a piano string is struck by a hammer it is displaced, and it is forced back towards it’s straight state by a combination of its tension and its stiffness. Inharmonicity will be greatest in strings where the stiffness is large in comparison to the tension. In the plain wire sections, length is the primary influence on stiffness. The less elastic the strings are (shorter, thicker, and stiffer), the more inharmonicity they exhibit. For instance, a stiff string under low tension (such as those found in the bass notes of small upright pianos) exhibits a high degree of inharmonicity, while a thinner string under higher tension (such as a treble string) will exhibit less inharmonicity.
When tuning coincidental partials (partials that are shared by the two notes being tuned), they are tempered so that selected partials of the intervals note’s will be not have beats. Because of inharmonicity, only one pair of coincident partials in any given interval can be tuned beat less. The inharmonicity will determine the distance in cycles per second (cps) between intervals, sometimes referred to as the piano’s stretch.
This inharmonicity causes a deviation of the actual fundamental pitch from being a perfect match to the mathematical formula for equal temperament. As part of the process of spacing all 12 notes of an octave, a piano tuner has to make minor adjustments to the spacing of each note within the octave. It’s this process of making those adjustments that is referred to as setting the temperament. Once the temperament has been set, it is lays a solid foundation for tuning of the remainder of the piano.
A good analogy to setting a temperament is by comparing it to the revolutions of the earth around the sun. It would be mathematically convenient if there were exactly 12 months per solar year, but it is not so — therefore we have developed a way to adapt the length of each month so that the year comes out right. The calendar system is like the temperament in that they try to make the octave (the “year”) come true.
On a side note, the tuning process consists of tuning the A above middle C, referred to as A4, to a tuning fork which vibrates at 440 Hz (or beats per second), and then setting the temperament from that note. The American music industry reached an informal standard of 440 Hz in 1926, and some began using it in instrument manufacturing. In 1936 the American Standards Association recommended that the A above middle C be tuned to 440 Hz. This standard was taken up by the International Organization for Standardization in 1955 (reaffirmed by them in 1975) All pianos, wind, and string instruments, are manufactured to this standard, so that they can all be played together, regardless of in which country the instrument is made.