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EXPLANATION The distance between each step and the next is ''aurally'' the same for any two adjacent steps; though, because steps form a Geometric Sequence , the difference in Frequency increases from one to the next. A Linear sequence of one frequency difference would create ever smaller intervals ( Ratio s), such as the Harmonic Series . (''See also Logarithmic Scale .'') In theoretical writings, material belonging to an equal temperament is often given an Integer Notation , meaning a single integer is used to represent each pitch. This simplifies and generalizes discussion of pitch material within the temperament. Furthermore, by applying the Modular Arithmetic where the modulo is the number of divisions of the octave (usually 12), these integers can be reduced to Pitch Class es, which removes the distinction (or acknowledges the similarity) between pitches of the same name, e.g. 'C' is 0 regardless of octave register. HISTORY Vincenzo Galilei (father of Galileo Galilei ) may have been the first person to advocate equal temperament (in a 1581 treatise). The first person known to have attempted a numerical specification for equal temperament is probably Chu Tsai-Yu (朱載堉) in the Ming Dynasty , who published a theory of the temperament in 1584. It is likely that this idea was spread to Europe by way of trade which had been intensified just at the moment when Chu Tsai-Yu went into print with his new theory. Within fifty-two years of Chu's publication, the same ideas had been published by Marin Mersenne and Simon Stevin . From 1450 to about 1800 there is evidence that musicians expected much less mistuning (than that of Equal Temperament) in the most common keys, such as C major. Instead, they used approximations that emphasized the tuning of Third s or Fifth s in these keys, such as Meantone Temperament . Some theorists, such as Giuseppe Tartini , were opposed to the adoption of Equal Temperament; they felt that degrading the purity of each chord degraded the aesthetic appeal of music. Some listeners claim that the equal-tempered difference is especially troubling in the lower register, and had somewhat constrained composers in the classical and romantic eras from writing chords narrower than octave for the left hand in keyboard music, while such examples in cello parts of string quartets are more common. Others take issue with dissonance in the higher register, where Beating between harmonics of mistuned consonances is faster, and where combinational tones, often an entire semitone out-of-tune in equal temperament, are louder. String ensembles and vocal groups, who have no mechanical tuning limiations, often use a tuning much closer to Just Intonation , as it is naturally more Consonant . Other instruments, such as Wind , Keyboard , and Fretted -instruments, often only approximate equal temperament, where technical limitations prevent exact tunings. J. S. Bach wrote The Well-Tempered Clavier to demonstrate the musical possibilities of Well Temperament , where in some keys the consonances are even more degraded than in equal temperament. It is reasonable to believe that when composers and theoreticians of earlier times wrote of the moods and "colors" of the keys, they described the subtly different dissonances of particular tuning methods, though it is difficult to determine with any exactness the actual tunings used in different places at different times by any composer. (Correspondingly, there is a great deal of variety in the particular opinions of composers about the moods and colors of particular keys.) Twelve tone equal temperament took hold for a variety of reasons. It conveniently fit the existing keyboard design, and was a better approximation to Just Intonation than the nearby alternative equal temperaments. It permitted total harmonic freedom at the expense of just a little purity in every interval. This allowed greater expression through Modulation , which became extremely important in the 19th century music of composers such as Chopin , Schumann , Liszt , and others. True equal temperament was not theoretically possible until 1863 when Hermann Helmholtz published the first rigorous scientific study of tone and acoustics. Even so, it was not routinely applied as a tuning system until piano manufacturers adopted it in the early 20th century. It is in the environment of equal temperament that the new styles of symmetrical tonality and Polytonality , Atonal Music such as that written with the Twelve Tone Technique or Serialism , and Jazz (at least its piano component) developed and flourished. TWELVE-TONE EQUAL TEMPERAMENT The ratio between two adjacent semitones can be found with a few steps: :1. Let ''a''''n'' be the frequency of a tone ''n'', with ''a''12 an Octave above ''a''0. This creates twelve tones for each octave. :2. Since the frequency ratio of a tone from one octave to the next is 2:1, the ratio of the frequency of one tone (''a''12) to the frequency of a tone an octave lower (''a''0) is 2:1 as well, so :: :3. Since the frequencies of the tones are in a geometric sequence, the frequency for a tone ''k'' (relative to the tone designated zero) will be equal to ''s''''k''''a''0 where ''s'' is the constant ratio between adjacent frequencies. This gives for ''k'' = 12, ::$a\_\{12\}\; =\; s^\{12\}\; a\_0$ :: :4. Since ''a''12 / ''a''0 was found to be two, the formula with constant ratio ''s'' is ::$2\; =\; s^\{12\}$ ::$s\; =\; \backslash sqrt${Link without Title} {2} Therefore, the ratio between two adjacent frequencies is equal to the Twelfth Root Of Two or approximately 1.05946309 to one. :$s\; =\; \backslash sqrt${Link without Title} {2} \approx 1.05946309 The half tone interval: :$1\; :\; 2^\{1/12\}$ is also known as 100 '' Cent s''. 1 cent is therefore the ratio between two tone frequencies with an interval of one hundredth of an equal-tempered semitone. The distance between two notes whose frequencies are ''f''1 and ''f''2 is 12 log2(''f''1/''f''2) half tones, that is 1200 log2(''f''1/''f''2) cents. Cent values of equal temperament The following table shows the values of the intervals of 12 TET, along with one interval from Just Intonation that each approximates, and the percentage by which they differ:   16/9 1777778
	15/8	1875000

  2/1 2000000

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