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Aural Piano tuning is the art of making adjustments to the tensions in the strings of the piano, so that the instrument is “in tune”. INTRODUCTION The meaning of the term “in tune” in the context of piano tuning is not as straightforward as it might seem, as it does not refer to the assignment of consistently fixed pitches. Fine piano tuning requires an assessment of the interaction between notes, which can be different for each piano, thus requiring slightly different pitches from a theoretical, mathematical standard. Pianos are tuned to a system called Equal Temperament , which determines the basic size of the musical intervals. However, aural fine tuning is largely a process of producing good tone, and the tuning of precise equal temperament is only the start. The relationship between tone and string tensions is complex; perceived tone, like perceived pitch, is in part dependent on psycho-acoustic factors. SCOPE Piano tuning is distinct from repairs or other maintenance that may be carried out - for example, regulation of the action. There are typically around 220 strings in a full sized piano, which may have a combined tension of up to approximately 20 tonnes. Fine tuning involves making “micro-adjustments” to the tensions of the strings, in order to make corresponding “micro-adjustments” to the tonal properties of notes and intervals. It is interesting to note that pitch is not a fixed, mathematical standard. Since the 1500s, the frequency used for the standard A pitch has ranged from 403 to 567 Hz . The U.S. standard has been A=440 since around 1920. THEORY The tuning of the piano is most often “explained” in terms of “standard” theory that was developed in the 19th century, which in the context of modern acoustics, constitutes a simplified and idealised acoustical model. The theory provides a useful guide that is still widely used. As an acoustical theory, however, it is not recognised today as a definitive description either for the acoustical behaviour of the piano, or for the art of aural piano tuning. The property of piano tone with which "traditional" theory deals, can be approximately modelled in a relatively straightforward way. This is a property that is (somewhat loosely) termed “ Beating ”. Piano tone consists of a complex recipe of ingredients, and “beating” is (with training) an audible form of “movement” - most often heard as a “pulsating” - in some of the ingredients that make up the tone. A large part of all the beating that occurs in piano tone can be adjusted by the tuner, because it is very sensitive to changes in string tensions. Thus it is often said that piano tuners listen to “beats”, within the piano tone. However, not all “beating” that occurs in piano tone is sensitive to changes of string tension, in the way the standard theory assumes. “Beating” is not the only property of piano tone that is adjustable through fine tuning. All “beating” contributes to tonal quality. TEMPERAMENT AND “BEATING” Modern tuning requires all musical intervals of any one kind to be the same “size”. The “size” of a musical interval is defined by Temperament theory (which is not the same as tuning theory) in a precise, straightforward and idealised way. Applied to piano tones, the temperament theory definition for the “size” of any musical interval has only approximate meaning. This is because in temperament theory, musical intervals are defined by simple, whole number ratios, or occasionally by a single decimal number. Piano tone itself is acoustically far too complex to be properly described using only simple, whole number ratios, or single decimal numbers. However, if we simply accept the use of the term “size” applied to a musical interval, in an intuitive way, we can get an intuitive idea of the tuning system used on pianos. By dividing the octave into 12 “equally sized” semitones, all intervals of any one kind (e.g. major thirds or “perfect fifths”) will also be the same size, since all intervals of any given kind contain the same number of semitones. Tuning to this principle is called Equal Temperament tuning. The amount of "beating" in a musical interval will affect its tone quality. Generally speaking, too much beating corresponds to a "poor" tone. An important consequence of equal temperament tuning is that the tone quality of every interval except the octave (and compound octaves), will not, according to the theory, contain the minimum possible amount of “beating”. (In standard theory, the "amount" of beating is the speed of the beat - no other beat parameters are represented). Each interval that must contain beating is called a “tempered interval”. An “untempered”, “pure” or “just” interval, would, in the context of the theory, be one containing no “beating”. According to the theory, equal temperament is achieved with a precise amount of beating in each interval, which is different for every interval of any given kind. The business of standard tuning theory is to specify the amount (speed) of beating necessary in each interval. “Beating”, however, is an inherent part of piano tone, and it is not only a consequence of the applied tuning system. In fine aural tuning, the relationships between the amounts of “beating” in the various tempered intervals must be finely controlled using empiric principles. In “standard” theory, the application of beat rates is treated as a technique for achieving a pre-specified set of frequencies for the notes. This necessarily differs from aural tuning practice, in which beats are treated as a property in their own right, according to the empiric principles. Tuners often tune the middle octave of the piano by beating, and tune the remaining pitches by octaves - that is, they match all the Ds with the tuned middle D, all the Cs with the tuned middle C, etc. The relationship between beats and frequencies is much more complicated than as described by standard theory, due to the presence of Inharmonicity , and because of bridge and soundboard effects. The effects of inharmonicity (mode frequency dispersion) are relatively well known, whilst relatively little is currently known about bridge and soundboard effects. THE “STANDARD” THEORY OF BEATS The basic premiss of the “traditional” theory for piano tuning “by beats”, is that the Equal Temperament tuning condition of the piano can be defined as a specific set of frequencies, one frequency for each note of the 12-note chromatic scale. This set is treated as an “axiomatic frequency set”. From this principle, the theory is then used to calculate “beat rates” (beat frequencies) that must, according to the theory, be present in each of the tempered intervals in the scale. The tuning scenario modelled by the “standard” theory is effectively one in which there is only one string per note, and the string motions are assumed to be acoustically isolated from each other. The “standard” theory considers only real frequency values. It is assumed that the frequency components present in each note fall in the harmonic series , where is the fundamental frequency. It is then assumed that the size of the equally tempered semitone can be computed such that 12 adjacent equally tempered semitones will together constitute an octave, whose upper and lower notes have the frequency ratio 2:1 (as in temperament theory). The frequency ratio between the fundamentals of two notes separated by an equally tempered semitone will then be given by . The fundamental frequencies for all the notes of an equally tempered chromatic scale whose lowest note has the fundamental frequency , are then given by (Eqn. 1) where the notes are numbered consecutively upwards from the lowest note of the scale, for which ''m''=1. Each musical interval type (e.g. major thirds, perfect fifths, etc.) has associated with it a fundamental ''harmonic ratio'' (some intervals also have secondary and tertiary ratios) as ascribed by temperament theory, which in tuning theory translates as the frequency ratio between the fundamentals of any two notes separated by that interval, when the interval is tuned without tempering. In the untempered tuning condition, some harmonic frequency components of the notes of any interval will coincide in value according to , for any integer (Eqn. 2) where is the numerator and is the denominator of the fundamental harmonic ratio. In equal temperament tuning, in which and are determined by Eqn. (1), this equality will not occur for any interval (except octaves). Thus in the equally tempered intervals (Eqn. 3) The inequality indicates the presence of two close-frequency components. Standard textbooks in acoustics show that two (real) close-frequency components of similar amplitude will exhibit a beat whose beat rate is equal to the frequency difference between the components. Thus with knowledge of the fundamental frequency set calculated from Eqn. (1), and knowledge of the harmonic ratios, the expected beat rates for all the intervals can be easily computed. The complete set of beat rates (in beats per second) for a scale F to F spanning across middle C, when the A has a fundamental frequency of 220 Hz (which it would have for tuning at International Concert Pitch), is given in the table below. Each column heading gives the name of the lowest note of the interval, for each interval type specified on the rows. The table specifies beat rates for intervals extending beyond a 13 note scale itself, and covers two octaves of the compass. CERTIFIED PIANO TUNERS The Piano Technicians Guild certifies individuals who pass the guild's technicals exams as a " Registered Piano Technician ". PIANO TUNING SCHOOLS/SUPPLIES
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