A musical temperament is a tuning system that slightly compromises the pure intervals of just intonation to meet other requirements. These pure intervals are based on a perfect fifth that has an exact 3:2 pitch ratio, but tempering narrows or widens them to accommodate a broader range of tuning needs. Temperaments are especially important for fixed-pitch instruments such as keyboards and guitars, which lack any way to easily make fine pitch adjustments during performance. Temperaments can be divided into two categories, regular and irregular. Regular temperaments are generated with a finite number of generator intervals, most commonly a tempered perfect fifth, and a period, usually the octave. These include 12-tone equal temperament, the most commonly used temperament today, and more generally meantone temperaments which were historically used. Irregular temperaments cannot be described with such a generation process, and include the historically used well temperaments, which use an unequal sequence of twelve tempered fifths. 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 takes full advantage of this breakthrough, with pieces written in 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 character. This variation led in the 18th century to an increase in the use of 12-tone equal temperament, in which the frequency ratio between each pair of adjacent notes on the keyboard was made equal. In other words, the ratio between two notes that were one octave apart was kept pure, and the twelve notes in between the octave were equally spaced from one another. This allowed music to be transposed between keys without changing the relationship between notes.
Definition Temperament, in music, the accommodation or adjustment of the imperfect sounds by transferring a part of their defects to the more perfect ones, in order to remedy, in some degree, the false intervals of those instruments, the sounds of which are fixed; as the organ, harpsichord, piano-forte, etc.Temperament is what the Italians call participatione, or system temperato, because it is founded on temperature; that is, on the diminution of some intervals and augmentation of others, by which it partakes of the diatonic and chromatic systems. "Temperament refers to the various tuning systems for the subdivision of the octave," the four principal tuning systems being Pythagorean tuning, just intonation, mean-tone temperament, and equal temperament. In just intonation, every interval between two pitches corresponds to a whole number ratio between their frequencies, allowing intervals varying from the highest consonance to highly dissonant. For instance, 660 Hz / 440 Hz (a ratio of 3 / 2) constitutes a fifth, and 880 Hz / 440 Hz (2 / 1) an octave. Such intervals (termed "just") have a stability, or purity to their sound, when played simultaneously (assuming they are played using timbres with harmonic partials) because pure intervals do not waver or beat regularly. The proportions of their frequencies can be expressed as whole numbers. If one of those pitches is adjusted slightly to deviate from the just interval, a trained ear can detect this change by the presence of beats, which are periodical oscillations in the note's intensity. If, for example, two sound signals with frequencies that vary just by 0.5 Hz are played simultaneously, both signals are out of phase by a very small margin, creating the periodical oscillations in the intensity of the final sound (caused by the superposition of both signals) with a repetition period of 2 seconds (following the equation Tr=1/Δf, Tr being the period of repetition and Δf being the difference in frequencies between both signals), because the amplitude of the signals is only in phase, and therefore has a maximum superposition value, once every period of repetition.
Acoustic physics When a musical instrument with harmonic overtones is played, the ear hears a composite waveform that includes a fundamental frequency (e.g., 440 Hz) and those overtones (880 Hz, 1320 Hz, 1760 Hz, etc.)—a series of just intervals. These just intervals, due to their acoustic nature, are present in many contexts: such as plucked strings and the human voice. The waveform of such a tone (as pictured on an oscilloscope) is characterized by a shape that is complex compared to a simple (sine) waveform, but remains periodic. As the ratios between the frequencies of two tones depart further from simple integer ratios, fewer harmonics coincide, and the shape of the composite waveform becomes less periodic. This 'rougher'-looking waveform may be perceived as having reduced consonance. Every interval created by two sustained tones creates a third tone, called a differential (or resultant) tone. This third tone is equal to the lower pitch subtracted from the higher pitch. This third tone then creates intervals with the original two tones, and the difference between these is called a second differential. Differentials are soft and difficult for the untrained ear to detect. Nevertheless, these relationships between differentials play a large role in determining which tunings create consonant sound. Perceptual studies suggest that maximum roughness is perceived when the frequency of these differentials between harmonics lie at about a quarter of a 'critical bandwith', which varies with overall frequency.
Meantone temperament
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