Welcome to the I Can't Sleep podcast,
Where I read random articles from across the web to bore you to sleep with my soothing voice.
I'm your host,
Benjamin Boster,
And today's episode is from a Wikipedia article titled Piano Tuning.
Piano tuning is a process of adjusting the tension of the strings of an acoustic piano so that the musical intervals between strings are in tune.
The meaning of the term in tune in the context of piano tuning is not simply a particular fixed set of pitches.
Fine piano tuning requires an assessment of the vibration interaction among notes,
Which is different for every piano,
Thus in practice requiring slightly different pitches from any theoretical standard.
Pianos are usually tuned to a modified version of a system called equal temperament.
In all systems of tuning,
Every pitch may be derived from its relationship to a chosen fixed pitch,
Which is usually A440 for 440 Hz,
The note A above middle C.
For a classical piano and music theory,
The middle C is usually labeled as C4,
As in scientific pitch notation.
However,
In the MIDI standard definition,
This middle C,
261.
626 Hz,
Is labeled C3.
In practice,
A MIDI software can label middle C as C3-C5,
Which can cause confusion,
Especially for beginners.
Piano tuning is done by a wide range of independent piano technicians,
Piano rebuilders,
Piano store technical personnel,
And hobbyists.
Professional training and certification is available from organizations or guilds,
Such as the Piano Technicians Guild.
Many piano manufacturers recommend that pianos be tuned twice a year.
Many factors cause pianos to go out of tune,
Particularly atmospheric changes.
For instance,
Changes in humidity will affect the pitch of a piano.
High humidity causes the soundboard to swell,
Stretching the strings and causing the pitch to go sharp,
While low humidity has the opposite effect.
Changes in temperature can also affect the overall pitch of a piano.
In newer pianos,
The strings gradually stretch and wooden parts compress,
Causing the piano to go flat,
While in older pianos,
The tuning pins that hold the strings in tune can become loose and not hold the piano in tune as well.
Frequent and hard playing can also cause a piano to go out of tune.
For these reasons,
Many piano manufacturers recommend that new pianos be tuned four times during the first year,
And twice a year thereafter.
An out-of-tune piano can often be identified by the characteristic honky-tonk,
Or beating sound it produces.
This fluctuation in the sound intensity is a result of two or more tones of similar frequencies being played together.
For example,
If a piano string tuned to 440 Hz vibrations per second is played together with a piano string tuned to 442 Hz,
The resulting tone beats at a frequency of 2 Hz,
Due to the constructive and deconstructive interference between the two sound waves.
Likewise,
If a string tuned to 220 Hz,
With a harmonic at 440 Hz,
Is played together with a string tuned at 442 Hz,
The same 2 Hz beat is heard.
Because pianos typically have multiple strings for each piano key,
These strings must be tuned to the same frequency to eliminate beats.
The pitch of a note is determined by the frequency of vibrations.
For a vibrating string,
The frequency is determined by the string's length,
Mass,
And tension.
Piano strings are wrapped around tuning pins,
Which are turned to adjust the tension of the strings.
Piano tuning became a profession around the beginning of the 1800s,
As the pianoforte became mainstream.
Previously,
Musicians owned harpsichords,
Which were much easier to tune,
And which the musicians generally tuned themselves.
Early piano tuners were trained and employed in piano factories,
And often underwent an apprenticeship of about five to seven years.
Early tuners faced challenges related to a large variety of new and changing pianos and non-standardized pitches.
Historically,
Keyboard instruments were tuned using just intonation,
Pythagorean tuning,
And mean-tone temperament,
Meaning that such instruments could sound in-tune in one key,
Or some keys,
But would then have more dissonance in other keys.
The development of well-temperament allowed fixed-pitch instruments to play reasonably well in all of the keys.
The famous well-tempered clavier by Johann Sebastian Bach took advantage of this breakthrough,
With preludes and fugues written for all 24 major and minor keys.
However,
While unpleasant intervals,
Such as the wolf interval,
Were avoided,
The sizes of intervals were still not consistent between keys,
And so each key still had its own distinctive character.
During the 1800s,
This variation led to an increase in the use of quasi-equal temperament,
In which the frequency ratio between each pair of adjacent notes on the keyboard were nearly equal,
Allowing music to be transposed between keys without changing the relationship between notes.
Pianos are generally tuned to an A440 pitch standard that was adopted during the early 20th century in response to widely varying standards.
Previously,
The pitch standards had gradually risen from about A415 during the late 18th century and early 19th century to A435 during the late 19th century.
Though A440 is generally the standard,
Some orchestras,
Particularly in Europe,
Use a higher pitch standard,
Such as A442.
A stretched string can vibrate in different modes,
Or harmonics,
And when a piano hammer strikes a string,
It excites multiple harmonics at the same time.
The first harmonic,
Or fundamental frequency,
Is usually the loudest and determines the pitch that is perceived.
In theory,
The higher harmonics,
Also called overtones or partials,
Vibrate at integer multiples of the fundamental frequency,
E.
G.
A string with a fundamental frequency of 100 Hz would have overtones at 200 Hz,
300 Hz,
400 Hz,
Etc.
In reality,
The frequencies of the overtones are shifted up slightly due to inharmonicity caused by the stiffness of the strings.
The relationship between two pitches,
Called an interval,
Is the ratio of their absolute frequencies.
The easiest intervals to identify on tune are those where the note frequencies have a simple whole number ratio,
E.
G.
Octave with a 2 to 1 ratio,
Perfect fifth with a 3 to 2,
Etc.
,
Because the harmonics of these intervals coincide and beat when they are out of tune.
For a perfect fifth,
The third harmonic of the lower note coincides with the second harmonic of the top note.
The term temperament refers to a tuning system that allows intervals to beat instead of tuning pure or just intervals.
In equal temperament,
For instance,
A fifth would be tempered by narrowing it slightly,
Achieved by flattening its upper pitch slightly,
Or raising its lower pitch slightly.
Tempering an interval causes it to beat,
Because the actual tone of a vibrating piano string is not just one pitch,
But a complex of tones arranged in a harmonic series.
Two strings that are close to a simple harmonic ratio,
Such as a perfect fifth beat at higher pitches at their coincident harmonics,
Because of the difference in pitch between their coincident harmonics.
Where these frequencies can be calculated,
A temperament may be tuned aurally by timing the beats of tempered intervals.
A common method of tuning the piano begins with tuning all the notes in the temperament octave in the lower middle range of the piano,
Usually F3 to F4.
A tuner starts by using an external reference,
Usually an A440 tuning fork,
More commonly a C523.
23 tuning fork,
To tune a beginning pitch,
And then tunes the other notes in the temperament using tempered interval relationships.
During tuning,
It is common to assess perfect fifths and fourths,
Major and minor thirds,
And major and minor sixths,
Often playing the intervals in an ascending or descending pattern to hear whether an even progression of beat rates has been achieved.
Having established the twelve notes of the chromatic scale,
The technician then replicates the temperament throughout the piano by tuning octaves and cross-checking with other intervals to align each note with others that have already been tuned.
Electronic piano tuning devices are also commonly used.
These are designed to adjust the same tonal complexities that the aural tuner encounters.
The devices use sophisticated algorithms to continuously test the harmonic makeup of each string as it is sounded,
And apply the derived information to determine its optimal pitch within the context of the entire instrument.
There is a table in this article that lists the theoretical beat frequencies between notes in an equal temperament octave.
The top row indicates absolute frequencies of the pitches,
Usually only A440 is determined from an external reference.
Every other number indicates the beat rate between any two tones,
Which shares a row and column with that number in the temperament octave.
Slower beat rates can be carefully timed with a metronome or other such device.
For the thirds in the temperament octave,
It is difficult to tune so many beats per second,
But after setting the temperament and duplicating it one octave below,
All of these beat frequencies are present at half the indicated rate in this lower octave,
Which are excellent for verification that the temperament is correct.
One of the easiest tests of equal temperament is to play a succession of major thirds,
Each one a semitone higher than the last.
If equal temperament has been achieved,
The best rate of these thirds should increase evenly in the temperament region.
The next table in this article indicates the pitch at which the strongest beating should occur for useful intervals.
When tuning a perfect fifth,
For instance,
The beating can be heard not at either of the fundamental pitches of the keys played,
But rather an octave and fifth,
Perfect twelfth,
Above the lower of the two keys,
Which is the lowest pitch at which their harmonic series overlap.
Once the beating can be heard,
The tuner must temper the interval either wide or narrow from a tuning that has no beatings.
The tuning described by the above beating plan provides a good approximation of equal temperament across the range of the temperament octave.
If extended further,
However,
The actual tuning of the instrument becomes increasingly inaccurate because of the deviation of the real partials from the theoretical harmonics.
Pianos' partials run slightly sharp as increasingly higher orders of the harmonic series are reached.
This problem is mitigated by stretching the octaves as one tunes above and to an extent below the temperament region.
When octaves are stretched,
They are not tuned to the lowest coincidental overtone,
Second partial,
Of the note below,
But instead to a higher overtone,
Often the fourth partial.
This widens all intervals equally,
Thereby maintaining intervallic and tonal consistency.
All Western music,
But Western classical literature in particular,
Requires this deviation from the theoretical equal temperament because the music is rarely played within a single octave.
A pianist constantly plays notes spread over three and four octaves at least,
So it is critical that the mid and upper range of the treble be stretched or widened to better match with the inharmonic overtones of lower registers.
Since the stretch of octaves is perceived and not measured,
The tuner determines which octave needs more or less octave stretching by ear.
Good tuning requires compromise between tonal brilliance,
Accurate intonation,
And an awareness of gradation of timbre through the compass of the instrument.
The name of this modification of the width of the scale is called the piano tuner's octave,
As opposed to the simple two-to-one octave expected from a theoretical harmonic oscillator.
The amount of stretching necessary to achieve the desired compromise is a complicated determination described theoretically as a function of string scaling.
String scaling considers the string's tension,
Length,
Diameter,
Weight per unit length,
And an elasticity in either the string's core wires,
Or any overwinding used to modify the wire's weight.
The overwindings are normally made from a denser,
Heavier,
But less springy metal than the steel used for the core.
Imperfect springiness anywhere in the string wire makes its partials deviate slightly from mathematically pure harmonics,
And no real material used to generating musical tones is perfectly elastic.
The Railsback curve is the result of measuring the fundamental frequencies of stretched tunings and plotting their deviations from unstretched equal temperament.
In small pianos,
The inharmonicity is so extreme that establishing a stretch based on a triple octave makes the single octaves beat noticeably,
And the wide,
Fast-beating intervals in the upper treble beat wildly,
Especially in major 17ths,
Two octaves plus a major third.
By necessity,
The tuner will attempt to limit the stretch.
In large pianos like concert grands,
Less inharmonicity allows for a more complete string stretch without negatively affecting close octaves and other intervals.
So,
While it may be true that the smaller piano receives a greater stretch relative to the fundamental pitch,
Only the concert grand's octaves can be fully widened so that the triple octaves are beatless.
This contributes to the response,
Brilliance,
And singing quality that concert grands offer.
A benefit of stretching octaves is the correction of dissonance that equal temperament imparts to the perfect fifth.
Without octave stretching,
The slow,
Nearly imperceptible beating of fifths in the temperament region,
Ranging from little more than one beat every two seconds to about one per second,
Would double each ascending octave.
At the top of the keyboard,
Then,
The theoretically and ideally pure fifth would be beating more than eight times per second.
Modern Western ears easily tolerate fast beating in non-just intervals,
Seconds and sevenths,
Thirds and sixths,
But not in perfect octaves or fifths.
Happily for pianists,
The string stretch that accommodates inharmonicity on a concert grand also nearly exactly mitigates the accumulation of dissonance in the perfect fifth.
Other factors,
Physical and psychoacoustic,
Affect the tuner's ability to achieve a temperament.
Among physical factors are inharmonic effects due to soundboard resonance in the bass strings,
Poorly manufactured strings,
Or peculiarities that can cause false beats.
False because they're unrelated to the manipulation of beats during tuning.
The principal psychoacoustic factor is that the human ear tends to perceive the higher notes as being flat when compared to those in the midrange.
Stretching the tuning to account for string inharmonicity is often not sufficient to overcome this phenomenon,
So piano tuners may stretch the top octave or so of the piano even more.
Common tools for tuning pianos include the tuning lever or hammer,
A variety of mutes,
And a tuning fork or electronic tuning device.
The tuning lever is used to turn and set the tuning pins,
Increasing or decreasing the tension of the string.
Mutes are used to silence strings that are not being tuned.
While tuning the temperament octave,
A felt strip is typically placed within the temperament middle section of the piano.
It is inserted between each note's trichord,
Muting its outer two strings so that only the middle string is free to vibrate.
A paps mute performs the same function in an upright piano and is placed through the piano action to mute either the two left strings of a trichord or the two right strings similarly.
After the center strings are all tuned,
Or right if a paps mute is used,
The felt strip can be removed note by note,
Tuning the outer strings to the center strings.
Wedge-shaped mutes are inserted between two strings to mute them,
And the paps mute is commonly used for tuning the high notes in upright pianos because it slides more easily between hammer shanks.
In an aural tuning,
A tuning fork is used to tune the first note,
Generally A4,
Of the key of intervals and checks until the tuner is satisfied that all the notes in the octave are correctly tuned.
The rest of the piano is then tuned to the temperament octave,
Using octaves and other intervals as checks.
If an electronic tuning device is used,
The temperament step might be skipped,
As it is possible for the tuner to adjust notes directly with the tuning device in any reasonable order.
Piano acoustics is the set of physical properties of the piano that affect its sound.
It is an area of study within musical acoustics.
The strings of a piano vary in diameter,
And therefore in mass per length,
With lower strings thicker than upper.
A typical range is from 0.
240 inches from the lowest bass strings to 0.
031 inches,
String size 13,
For the highest treble strings.
These differences in string thickness follow from well-understood acoustic properties of strings.
Given two strings equally taut and heavy,
One twice as long as the other,
The longer will vibrate with a pitch one octave lower than the shorter.
However,
If one were to use this principle to design a piano,
I.
E.
If one began with the highest notes and then doubled the length of the strings again and again for each lower octave,
It would be impossible to fit the bass strings onto a frame of any reasonable size.
Furthermore,
When strings vibrate,
The width of the vibrations is related to the string strength.
In such a hypothetical ultra-long piano,
The lowest strings would strike one another when played.
Instead,
Piano makers take advantage of the fact that a heavy string vibrates more slowly than a light string of identical length and tension.
Thus,
The bass strings on the piano are shorter than the double-with-each-octave rule would predict,
And are much thicker than the others.
The other factor that affects pitch,
Other than length,
Density,
And mass,
Is tension.
Individual string tension in a concert grand piano may average 200 pounds,
And have a cumulative tension exceeding 20 tons each.
Any vibrating thing produces vibrations at a number of frequencies above the fundamental pitch.
These are called overtones.
When the overtones are integer multiples,
E.
G.
2x,
3x,
6x of the fundamental frequency called harmonics,
Then neglecting damping the oscillation is periodic,
I.
E.
It vibrates exactly the same way over and over.
Many enjoy the sound of periodic oscillations.
For this reason,
Many musical instruments,
Including pianos,
Are designed to produce nearly periodic oscillations.
That is,
To have overtones as close as possible to the harmonics of the fundamental tone.
In an ideal vibrating string,
When the wavelength of a wave on a stretched string is much greater than the thickness of the string,
The theoretical idea being a string of zero thickness and zero resistance to bending,
The wave velocity on the string is constant,
And the overtones are at the harmonics.
That is why so many instruments are constructed of skinny strings or thin columns of air.
However,
For high overtones with short wavelengths that approach the diameter of the string,
The string behaves more like a thick metal bar.
Its mechanical resistance to bending becomes an additional force to the tension,
Which raises the pitch of the overtones.
Only when the bending force is much smaller than the tension of the string are its wave speed and the overtones pitched as harmonics unchanged.
The frequency raised overtones above the harmonics,
Called partials,
Can produce an unpleasant effect called inharmonicity.
Basic strategies to reduce inharmonicity include decreasing the thickness of the string or increasing its length,
Choosing a flexible material with a low bending force,
And increasing the tension force so that it stays much bigger than the bending force.
Winding a string allows an effective decrease in the thickness of the string.
In a wound string,
Only the inner core resists bending,
While the windings function only to increase the linear density of the string.
The thickness of the inner core is limited by its strength and by its tension.
Stronger materials allow for thinner cores at higher tensions,
Reducing inharmonicity.
Hence,
Piano designers choose high quality steel for their strings,
As its strength and durability help them minimize string diameters.
If string diameter,
Tension,
Mass,
Uniformity,
And length compromises were the only factors,
All pianos could be small,
Spinet-sized instruments.
Piano builders,
However,
Have found that longer strings increase instrument power,
Harmonicity,
And reverberation,
And help produce a properly tempered tuning scale.
With longer strings,
Larger pianos achieve the longer wavelengths and tonal characteristics desired.
Piano designers strive to fit the longest strings possible within the case.
Moreover,
All else being equal,
The sensible piano buyer tries to obtain the largest instrument compatible with budget and space.
Inharmonicity increases continuously as notes get further from the middle of the piano,
And is one of the practical limits on the total range of the instrument.
The lowest strings,
Which are necessarily the longest,
Are most limited by the size of the piano.
The designer of a short piano is forced to use thick strings to increase mass density,
And is thus driven into accepting greater inharmonicity.
The highest strings must be under the greatest tension,
Yet must also be thin enough to allow for a low mass density.
The limited length of steel,
I.
E.
A too-thin string will break under the tension,
Forces the piano designer to use very short and slightly thicker strings,
Whose short wavelengths thus generate inharmonicity.
The natural inharmonicity of a piano is used by the tuner to make slight adjustments in the tuning of a piano.
The tuner stretches the notes,
Slightly sharpening the high notes,
And flatting the low notes to make overtones of lower notes have the same frequency as the fundamentals of higher notes.
The Railsback curve,
First measured in the 1930s by O.
L.
Railsback,
A U.
S.
College physics teacher,
Expressed the difference between inharmonicity-aware stretched piano tuning and theoretically correct equal-tempered tuning,
In which the frequencies of successive notes are related by a constant ratio,
Equal to the 12th root of 2.
For any given note on the piano,
The deviation between the actual pitch of that note and its theoretical equal-tempered pitch is given in cents,
Hundreds of a semitone.
The curve is derived empirically from actual pianos tuned to be pleasing to the ear,
Not from an exact mathematical equation.
As the Railsback curve shows,
Octaves are normally stretched on a well-tuned piano.
That is,
The high notes are tuned higher and the low notes tuned lower than they are in mathematically idealized equal-tempered scale.
Railsback discovered that pianos were typically tuned in this manner not because of a lack of precision,
But because of inharmonicity in the strings.
For a string vibrating like an ideal harmonic oscillator,
The overtone series of a single-played note includes many additional higher frequencies,
Each of which is an integer multiple of the fundamental frequency.
But in fact,
Inharmonicity caused by piano strings being slightly inflexible makes the overtones actually produced successively higher than they would be if the string were perfectly harmonic.
Inharmonicity in a string is caused primarily by stiffness.
The stiffness is the result of piano wires' inherent hardness and ductility,
Together with string tension,
Thickness,
And length.
When tuners adjust the tension of the wire during tuning,
They establish pitches relative to notes that have already been tuned.
Those previously tuned notes have overtones that are sharpened by inharmonicity,
Which causes the newly established pitch to conform to the sharpened overtone.
As the tuning progresses up and down the scale,
The inharmonicity,
Hence the stretch,
Accumulates.
It is a common misconception that the Railsback curve demonstrates that the middle of the piano is less inharmonic than the upper and lower regions.
It only appears that way because that is where the tuning starts.
Stretch is a comparative term.
By definition,
No matter what pitch the tuning begins with,
There can be no stretch.
Further,
It is often construed that the upper notes of the piano are especially inharmonic because they appear to be stretched dramatically.
In fact,
Their stretch is a reflection of the inharmonicity of strings in the middle of the piano.
Moreover,
The inharmonicity of the upper notes can have no bearing on tuning because their upper partials are beyond the range of human hearing.
As expected,
The graph of the actual tuning is not a smooth curve,
But a jagged line with peaks and troughs.
This might be the result of imprecise tuning,
Inexact measurement,
Or the piano's innate variability in string scaling.
It has also been suggested with Monte Carlo simulation that such a shape comes from the way humans match pitch intervals.
All but the lowest notes of a piano have multiple strings tuned to the same frequency.
The notes with two strings are called bichords,
And those with three strings are called trichords.
These allow the piano to have a loud attack with a fast decay,
But a long sustain in the Attack-Decay-Sustain-Release,
ADSR system.
The trichords create a coupled oscillator with three normal modes,
With two polarizations each.
Since the strings are only weakly coupled,
The normal modes have imperceptibly different frequencies,
But they transfer their vibration energy to the sounding board at significantly different rates.
The normal mode in which the three strings oscillate together is most efficient at transferring energy,
Since all three strings pull in the same direction at the same time.
It sounds loud,
But decays quickly.
This normal mode is responsible for the rapid staccato attack part of the note.
In the other two normal modes,
Strings do not all pull together,
E.
G.
One pulls up while the other two pull down.
There is a slow transfer of energy to the sounding board,
Generating a soft but near constant sustain.
Thank you for watching.
