Stretched tuning
Encyclopedia
Stretched tuning is a detail of musical tuning
, applied to wire-stringed musical instruments, older, non-digital electric piano
s (such as the Fender Rhodes piano
and Wurlitzer
electric piano), and some sample-based synthesizers
based on these instruments, to accommodate the natural inharmonicity of their vibrating elements. In stretched tuning, two notes an octave
apart, whose fundamental frequencies
theoretically have an exact 2:1 ratio, are tuned slightly farther apart.
or resonant column of air) vibrates at many frequencies
simultaneously: a fundamental frequency
that is usually perceived as the pitch
of the note, and harmonic
s or overtone
s that are multiples of the fundamental frequency and whose wavelength
s therefore divide the tone-generating region into simple fractional segments (1/2, 1/3, 1/4, etc.). (See harmonic series
.) The fundamental note and its harmonics sound together, and the amplitude
relationships among them strongly affect the perceived tone or timbre
of the instrument.
In the acoustic piano
, harpsichord
, and clavichord
, the vibrating element is a metal wire or string
; in many non-digital electric piano
s, it is a tapered metal tine (Rhodes piano
) or reed (Wurlitzer
electric piano) with one end clamped and the other free to vibrate. Each note on the keyboard has its own separate vibrating element whose tension and/or length and weight determines its fundamental frequency
or pitch
. In electric piano
s, the motion of the vibrating element is sensed by an electromagnetic pickup and amplified electronically.
, the relationship between two notes (known musically as an interval
) is determined by evaluating their common harmonic
s. For example, we say two notes are an octave
apart when the fundamental frequency
of the upper note exactly matches the second harmonic of the lower note. Theoretically, this means the fundamental frequency of the upper note is exactly twice that of the lower note, and we would assume that the second harmonic of the upper note will exactly match the fourth harmonic of the lower note.
On instruments strung with metal wire, however, neither of these assumptions is valid, and inharmonicity is the reason.
Inharmonicity
refers to the difference between the theoretical and actual frequencies
of the harmonic
s or overtone
s of a vibrating tine or string
. The theoretical frequency of the second harmonic is twice the fundamental frequency, and of the third harmonic is three times the fundamental frequency, and so on. But on metal string
s, tines, and reeds, the measured frequencies of those harmonics are slightly higher, and proportionately more so in the higher than in the lower harmonics. A digital emulation
of these instruments must recreate this inharmonicity if it is to sound convincing.
The theory of temperaments in musical tuning
do not normally take into account inharmonicity, which varies from instrument to instrument (and from string to string), but in practice the amount of inharmonicity present in a particular instrument will effect a modification to the theoretical temperament which is being applied to it.
is excited into motion by plucking or striking, a complex wave travels outward to the ends of the string. As it travels outward, this initial impulse forces the wire out of its resting position all along its length. After the impulse has passed, each part of the wire immediately begins to return toward (and overshoot) its resting position, which means vibration
has been induced. Meanwhile, the initial impulse is reflected at both ends of the string and travels back toward the center. On the way, it interacts with the various vibrations it induced on the initial pass, and these interactions reduce or cancel some components of the impulse wave and reinforce others. When the reflected impulses encounter each other, their interaction again cancels some components and reinforces others.http://www.speech.kth.se/music/5_lectures/
Within a few transits of the string, all these cancellations and reinforcements sort the vibration into an orderly set of waves that vibrate over 1/1, 1/2, 1/3, 1/4, 1/5, 1/6, etc. of the length of the string. These are the harmonic
s. As a rule, the amplitude
of its vibration is less for higher harmonics than for lower, meaning that higher harmonics are softer—though the details of this differ from instrument to instrument. The exact combination of different harmonics and their amplitudes is a primary factor affecting the timbre
or tone quality of a particular musical tone.
Theoretically, vibration over half the string's length will be twice as fast, and vibration over one-third of the string will be three times as fast, as the fundamental vibration over the whole string's length. In the theoretical string, however, the only force acting to return a part of the string to its rest position is the tension between its ends.
If you try bending a short piece of piano wire or guitar string slightly with your fingers, you can feel the wire's resistance to bending. In a vibrating string, that resistance adds to the effect of string tension in returning a given part of the string toward its rest position. The result is a frequency of vibration higher than the theoretical frequency. And because the wire's resistance to bending increases as its length decreases, its effect is greater in higher harmonics than in lower.
, exactly matching the lowest common harmonic causes a slight amount of stretch, matching the next higher common harmonic causes a greater amount of stretch, and so on. If the interval
is a double octave, exactly matching the upper note to the fourth harmonic of the lower complicates the tuning of that upper note with the one an octave below it.
Solving such dilemmas is at the heart of precise tuning
by ear, and all solutions involve some stretching of the higher notes upward and the lower notes downward from their theoretical frequencies. In shorter pianos the wire stiffness in the bass register is proportionately high and therefore causes greater stretch; on larger concert grand pianos this effect is reduced. Online sourceshttp://www.postpiano.com/support/updates/tech/Tuning.htm suggest that the total amount of "stretch" over the full range of a concert grand piano may be on the order of ±35 cents
.
Musical tuning
In music, there are two common meanings for tuning:* Tuning practice, the act of tuning an instrument or voice.* Tuning systems, the various systems of pitches used to tune an instrument, and their theoretical bases.-Tuning practice:...
, applied to wire-stringed musical instruments, older, non-digital electric piano
Electric piano
An electric piano is an electric musical instrument.Electric pianos produce sounds mechanically and the sounds are turned into electrical signals by pickups. Unlike a synthesizer, the electric piano is not an electronic instrument, but electro-mechanical. The earliest electric pianos were invented...
s (such as the Fender Rhodes piano
Rhodes piano
The Rhodes piano is an electro-mechanical piano, invented by Harold Rhodes during the fifties and later manufactured in a number of models, first in collaboration with Fender and after 1965 by CBS....
and Wurlitzer
Wurlitzer
The Rudolph Wurlitzer Company, usually referred to simply as Wurlitzer, was an American company that produced stringed instruments, woodwinds, brass instruments, theatre organs, band organs, orchestrions, electronic organs, electric pianos and jukeboxes....
electric piano), and some sample-based synthesizers
Sample-based synthesis
Sample-based synthesis is a form of audio synthesis that can be contrasted to either subtractive synthesis or additive synthesis. The principal difference with sample-based synthesis is that the seed waveforms are sampled sounds or instruments instead of fundamental waveforms such as the saw waves...
based on these instruments, to accommodate the natural inharmonicity of their vibrating elements. In stretched tuning, two notes an octave
Octave
In music, an octave is the interval between one musical pitch and another with half or double its frequency. The octave relationship is a natural phenomenon that has been referred to as the "basic miracle of music", the use of which is "common in most musical systems"...
apart, whose fundamental frequencies
Frequency
Frequency is the number of occurrences of a repeating event per unit time. It is also referred to as temporal frequency.The period is the duration of one cycle in a repeating event, so the period is the reciprocal of the frequency...
theoretically have an exact 2:1 ratio, are tuned slightly farther apart.
Fundamentals and harmonics
In most musical instruments, the tone-generating component (a stringStrings (music)
A string is the vibrating element that produces sound in string instruments, such as the guitar, harp, piano, and members of the violin family. Strings are lengths of a flexible material kept under tension so that they may vibrate freely, but controllably. Strings may be "plain"...
or resonant column of air) vibrates at many frequencies
Frequency
Frequency is the number of occurrences of a repeating event per unit time. It is also referred to as temporal frequency.The period is the duration of one cycle in a repeating event, so the period is the reciprocal of the frequency...
simultaneously: a fundamental frequency
Fundamental frequency
The fundamental frequency, often referred to simply as the fundamental and abbreviated f0, is defined as the lowest frequency of a periodic waveform. In terms of a superposition of sinusoids The fundamental frequency, often referred to simply as the fundamental and abbreviated f0, is defined as the...
that is usually perceived as the pitch
Pitch (music)
Pitch is an auditory perceptual property that allows the ordering of sounds on a frequency-related scale.Pitches are compared as "higher" and "lower" in the sense associated with musical melodies,...
of the note, and harmonic
Harmonic
A harmonic of a wave is a component frequency of the signal that is an integer multiple of the fundamental frequency, i.e. if the fundamental frequency is f, the harmonics have frequencies 2f, 3f, 4f, . . . etc. The harmonics have the property that they are all periodic at the fundamental...
s or overtone
Overtone
An overtone is any frequency higher than the fundamental frequency of a sound. The fundamental and the overtones together are called partials. Harmonics are partials whose frequencies are whole number multiples of the fundamental These overlapping terms are variously used when discussing the...
s that are multiples of the fundamental frequency and whose wavelength
Wavelength
In physics, the wavelength of a sinusoidal wave is the spatial period of the wave—the distance over which the wave's shape repeats.It is usually determined by considering the distance between consecutive corresponding points of the same phase, such as crests, troughs, or zero crossings, and is a...
s therefore divide the tone-generating region into simple fractional segments (1/2, 1/3, 1/4, etc.). (See harmonic series
Harmonic series (music)
Pitched musical instruments are often based on an approximate harmonic oscillator such as a string or a column of air, which oscillates at numerous frequencies simultaneously. At these resonant frequencies, waves travel in both directions along the string or air column, reinforcing and canceling...
.) The fundamental note and its harmonics sound together, and the amplitude
Amplitude
Amplitude is the magnitude of change in the oscillating variable with each oscillation within an oscillating system. For example, sound waves in air are oscillations in atmospheric pressure and their amplitudes are proportional to the change in pressure during one oscillation...
relationships among them strongly affect the perceived tone or timbre
Timbre
In music, timbre is the quality of a musical note or sound or tone that distinguishes different types of sound production, such as voices and musical instruments, such as string instruments, wind instruments, and percussion instruments. The physical characteristics of sound that determine the...
of the instrument.
In the acoustic piano
Piano
The piano is a musical instrument played by means of a keyboard. It is one of the most popular instruments in the world. Widely used in classical and jazz music for solo performances, ensemble use, chamber music and accompaniment, the piano is also very popular as an aid to composing and rehearsal...
, harpsichord
Harpsichord
A harpsichord is a musical instrument played by means of a keyboard. It produces sound by plucking a string when a key is pressed.In the narrow sense, "harpsichord" designates only the large wing-shaped instruments in which the strings are perpendicular to the keyboard...
, and clavichord
Clavichord
The clavichord is a European stringed keyboard instrument known from the late Medieval, through the Renaissance, Baroque and Classical eras. Historically, it was widely used as a practice instrument and as an aid to composition, not being loud enough for larger performances. The clavichord produces...
, the vibrating element is a metal wire or string
Strings (music)
A string is the vibrating element that produces sound in string instruments, such as the guitar, harp, piano, and members of the violin family. Strings are lengths of a flexible material kept under tension so that they may vibrate freely, but controllably. Strings may be "plain"...
; in many non-digital electric piano
Electric piano
An electric piano is an electric musical instrument.Electric pianos produce sounds mechanically and the sounds are turned into electrical signals by pickups. Unlike a synthesizer, the electric piano is not an electronic instrument, but electro-mechanical. The earliest electric pianos were invented...
s, it is a tapered metal tine (Rhodes piano
Rhodes piano
The Rhodes piano is an electro-mechanical piano, invented by Harold Rhodes during the fifties and later manufactured in a number of models, first in collaboration with Fender and after 1965 by CBS....
) or reed (Wurlitzer
Wurlitzer
The Rudolph Wurlitzer Company, usually referred to simply as Wurlitzer, was an American company that produced stringed instruments, woodwinds, brass instruments, theatre organs, band organs, orchestrions, electronic organs, electric pianos and jukeboxes....
electric piano) with one end clamped and the other free to vibrate. Each note on the keyboard has its own separate vibrating element whose tension and/or length and weight determines its fundamental frequency
Fundamental frequency
The fundamental frequency, often referred to simply as the fundamental and abbreviated f0, is defined as the lowest frequency of a periodic waveform. In terms of a superposition of sinusoids The fundamental frequency, often referred to simply as the fundamental and abbreviated f0, is defined as the...
or pitch
Pitch (music)
Pitch is an auditory perceptual property that allows the ordering of sounds on a frequency-related scale.Pitches are compared as "higher" and "lower" in the sense associated with musical melodies,...
. In electric piano
Electric piano
An electric piano is an electric musical instrument.Electric pianos produce sounds mechanically and the sounds are turned into electrical signals by pickups. Unlike a synthesizer, the electric piano is not an electronic instrument, but electro-mechanical. The earliest electric pianos were invented...
s, the motion of the vibrating element is sensed by an electromagnetic pickup and amplified electronically.
Intervals and inharmonicity
In tuningMusical tuning
In music, there are two common meanings for tuning:* Tuning practice, the act of tuning an instrument or voice.* Tuning systems, the various systems of pitches used to tune an instrument, and their theoretical bases.-Tuning practice:...
, the relationship between two notes (known musically as an interval
Interval (music)
In music theory, an interval is a combination of two notes, or the ratio between their frequencies. Two-note combinations are also called dyads...
) is determined by evaluating their common harmonic
Harmonic
A harmonic of a wave is a component frequency of the signal that is an integer multiple of the fundamental frequency, i.e. if the fundamental frequency is f, the harmonics have frequencies 2f, 3f, 4f, . . . etc. The harmonics have the property that they are all periodic at the fundamental...
s. For example, we say two notes are an octave
Octave
In music, an octave is the interval between one musical pitch and another with half or double its frequency. The octave relationship is a natural phenomenon that has been referred to as the "basic miracle of music", the use of which is "common in most musical systems"...
apart when the fundamental frequency
Fundamental frequency
The fundamental frequency, often referred to simply as the fundamental and abbreviated f0, is defined as the lowest frequency of a periodic waveform. In terms of a superposition of sinusoids The fundamental frequency, often referred to simply as the fundamental and abbreviated f0, is defined as the...
of the upper note exactly matches the second harmonic of the lower note. Theoretically, this means the fundamental frequency of the upper note is exactly twice that of the lower note, and we would assume that the second harmonic of the upper note will exactly match the fourth harmonic of the lower note.
On instruments strung with metal wire, however, neither of these assumptions is valid, and inharmonicity is the reason.
Inharmonicity
Inharmonicity
In music, inharmonicity is the degree to which the frequencies of overtones depart from whole multiples of the fundamental frequency....
refers to the difference between the theoretical and actual frequencies
Frequency
Frequency is the number of occurrences of a repeating event per unit time. It is also referred to as temporal frequency.The period is the duration of one cycle in a repeating event, so the period is the reciprocal of the frequency...
of the harmonic
Harmonic
A harmonic of a wave is a component frequency of the signal that is an integer multiple of the fundamental frequency, i.e. if the fundamental frequency is f, the harmonics have frequencies 2f, 3f, 4f, . . . etc. The harmonics have the property that they are all periodic at the fundamental...
s or overtone
Overtone
An overtone is any frequency higher than the fundamental frequency of a sound. The fundamental and the overtones together are called partials. Harmonics are partials whose frequencies are whole number multiples of the fundamental These overlapping terms are variously used when discussing the...
s of a vibrating tine or string
Strings (music)
A string is the vibrating element that produces sound in string instruments, such as the guitar, harp, piano, and members of the violin family. Strings are lengths of a flexible material kept under tension so that they may vibrate freely, but controllably. Strings may be "plain"...
. The theoretical frequency of the second harmonic is twice the fundamental frequency, and of the third harmonic is three times the fundamental frequency, and so on. But on metal string
Strings (music)
A string is the vibrating element that produces sound in string instruments, such as the guitar, harp, piano, and members of the violin family. Strings are lengths of a flexible material kept under tension so that they may vibrate freely, but controllably. Strings may be "plain"...
s, tines, and reeds, the measured frequencies of those harmonics are slightly higher, and proportionately more so in the higher than in the lower harmonics. A digital emulation
Digital piano
A digital piano is a modern electronic musical instrument, different from the electronic keyboard, designed to serve primarily as an alternative to a traditional piano, both in the way it feels to play and in the sound produced. It is meant to provide an accurate simulation of a real piano. Some...
of these instruments must recreate this inharmonicity if it is to sound convincing.
The theory of temperaments in musical tuning
Musical tuning
In music, there are two common meanings for tuning:* Tuning practice, the act of tuning an instrument or voice.* Tuning systems, the various systems of pitches used to tune an instrument, and their theoretical bases.-Tuning practice:...
do not normally take into account inharmonicity, which varies from instrument to instrument (and from string to string), but in practice the amount of inharmonicity present in a particular instrument will effect a modification to the theoretical temperament which is being applied to it.
Vibration of wire strings
When a stretched wire stringStrings (music)
A string is the vibrating element that produces sound in string instruments, such as the guitar, harp, piano, and members of the violin family. Strings are lengths of a flexible material kept under tension so that they may vibrate freely, but controllably. Strings may be "plain"...
is excited into motion by plucking or striking, a complex wave travels outward to the ends of the string. As it travels outward, this initial impulse forces the wire out of its resting position all along its length. After the impulse has passed, each part of the wire immediately begins to return toward (and overshoot) its resting position, which means vibration
Vibration
Vibration refers to mechanical oscillations about an equilibrium point. The oscillations may be periodic such as the motion of a pendulum or random such as the movement of a tire on a gravel road.Vibration is occasionally "desirable"...
has been induced. Meanwhile, the initial impulse is reflected at both ends of the string and travels back toward the center. On the way, it interacts with the various vibrations it induced on the initial pass, and these interactions reduce or cancel some components of the impulse wave and reinforce others. When the reflected impulses encounter each other, their interaction again cancels some components and reinforces others.http://www.speech.kth.se/music/5_lectures/
Within a few transits of the string, all these cancellations and reinforcements sort the vibration into an orderly set of waves that vibrate over 1/1, 1/2, 1/3, 1/4, 1/5, 1/6, etc. of the length of the string. These are the harmonic
Harmonic
A harmonic of a wave is a component frequency of the signal that is an integer multiple of the fundamental frequency, i.e. if the fundamental frequency is f, the harmonics have frequencies 2f, 3f, 4f, . . . etc. The harmonics have the property that they are all periodic at the fundamental...
s. As a rule, the amplitude
Amplitude
Amplitude is the magnitude of change in the oscillating variable with each oscillation within an oscillating system. For example, sound waves in air are oscillations in atmospheric pressure and their amplitudes are proportional to the change in pressure during one oscillation...
of its vibration is less for higher harmonics than for lower, meaning that higher harmonics are softer—though the details of this differ from instrument to instrument. The exact combination of different harmonics and their amplitudes is a primary factor affecting the timbre
Timbre
In music, timbre is the quality of a musical note or sound or tone that distinguishes different types of sound production, such as voices and musical instruments, such as string instruments, wind instruments, and percussion instruments. The physical characteristics of sound that determine the...
or tone quality of a particular musical tone.
Theoretically, vibration over half the string's length will be twice as fast, and vibration over one-third of the string will be three times as fast, as the fundamental vibration over the whole string's length. In the theoretical string, however, the only force acting to return a part of the string to its rest position is the tension between its ends.
If you try bending a short piece of piano wire or guitar string slightly with your fingers, you can feel the wire's resistance to bending. In a vibrating string, that resistance adds to the effect of string tension in returning a given part of the string toward its rest position. The result is a frequency of vibration higher than the theoretical frequency. And because the wire's resistance to bending increases as its length decreases, its effect is greater in higher harmonics than in lower.
Tines and reeds
Tines and reeds differ from strings in that they are held at one end and free to vibrate at the other. The frequencies of their fundamental and harmonic vibrations are subject to the same inharmonicity as strings. However, because of the comparative thickness of the bars that terminate the tines in an electric piano, the larger (and stronger) vibrations tend to "see" termination points slightly deeper in the bar than do smaller, weaker vibrations. This enhances inharmonicity in tines.Effects on tuning
Inharmonicity "stretches" harmonics beyond their theoretical frequencies, and higher harmonics are stretched proportionally more than lower. Thus, in our example of an octaveOctave
In music, an octave is the interval between one musical pitch and another with half or double its frequency. The octave relationship is a natural phenomenon that has been referred to as the "basic miracle of music", the use of which is "common in most musical systems"...
, exactly matching the lowest common harmonic causes a slight amount of stretch, matching the next higher common harmonic causes a greater amount of stretch, and so on. If the interval
Interval (music)
In music theory, an interval is a combination of two notes, or the ratio between their frequencies. Two-note combinations are also called dyads...
is a double octave, exactly matching the upper note to the fourth harmonic of the lower complicates the tuning of that upper note with the one an octave below it.
Solving such dilemmas is at the heart of precise tuning
Musical tuning
In music, there are two common meanings for tuning:* Tuning practice, the act of tuning an instrument or voice.* Tuning systems, the various systems of pitches used to tune an instrument, and their theoretical bases.-Tuning practice:...
by ear, and all solutions involve some stretching of the higher notes upward and the lower notes downward from their theoretical frequencies. In shorter pianos the wire stiffness in the bass register is proportionately high and therefore causes greater stretch; on larger concert grand pianos this effect is reduced. Online sourceshttp://www.postpiano.com/support/updates/tech/Tuning.htm suggest that the total amount of "stretch" over the full range of a concert grand piano may be on the order of ±35 cents
Cent (music)
The cent is a logarithmic unit of measure used for musical intervals. Twelve-tone equal temperament divides the octave into 12 semitones of 100 cents each...
.
See also
- Piano acousticsPiano acousticsPiano acoustics are those physical properties of the piano which affect its acoustics.-String length and mass:The strings of a piano vary in thickness, and therefore in mass per length, with bass strings thicker than treble. A typical range is from 1/30 inch for the highest treble strings to 1/3...
- Piano tuningPiano tuningPiano tuning is the act of making minute adjustments to the tensions of the strings of a piano to properly align the intervals between their tones so that the instrument is in tune. The meaning of the term in tune in the context of piano tuning is not simply a particular fixed set of pitches...
- Stretched octave
- Electronic tunerElectronic tunerThe term electronic tuner can refer to a number of different things, depending which discipline you wish to study.In the Discipline of radio frequency electronics an electronic tuner is a device which tunes across a part of the radio frequency spectrum by the application of a voltage or appropriate...