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Topic: Equal tempered

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	[No title]
		Schematic of Equal Temperament In Figure 5, the Pythagorean comma is indicated by the upper-case "P" and 1/12 of the Pythagorean comma by the lower-case "p." The direction of the arrow indicates the ascending harmonic interval.
		One equally tempered semitone equals the twelfth root of 2, an irrational number whose value is normally given in a decimal approximation as 1.05946 (100 cents.) The tuning of the chain of equal tempered fifths in six place decimals is given in Figure 7.
		Equal Temperament Using the diagram in Figure 5, the 700-cent logarithmic tuning of an equally tempered fifth is narrower than a 702 cent perfect fifth by 2 cents, 1/12 of the 24-cent Pythagorean comma.
	www.societymusictheory.org /mto/issues/mto.98.4.4/mto.98.4.4.scholtz.art (7652 words)

	Equal Temperament (Site not responding. Last check: 2007-11-07)
		The equal tempered scale is the common musical scale used at present, used for the tuning of pianos and other instruments of relatively fixed scale.
		One of the advantages of the equal tempered scale is that it is the same in any musical "key", so that compositions may be freely transposed up or down without changing the musical intervals.
		The equal tempered intervals may be compared with Just and Pythagorean temperaments which maintain the exact-integer-ratio rule for the main intervals, while equal temperament departs from that standard.
	hyperphysics.phy-astr.gsu.edu /hbase/music/et.html (311 words)

	Mathematics of musical scales - Wikipedia, the free encyclopedia
		The most important scale in the Western tradition is the diatonic scale, but the scales used and proposed in various historical eras and parts of the world have been many and varied.
		The tempering can involve either the irregularities of well temperament or be constructed as a regular temperament, either some form of equal temperament or some other regular meantone, but in all cases will involve the fundamental features of meantone temperament.
		Equal tempered scales have been built using 19 equally spaced tones, and also 24 equally spaced tones.
	en.wikipedia.org /wiki/Mathematics_of_musical_scales (1119 words)

	Equal tempered (Site not responding. Last check: 2007-11-07)
		Equal temperament is a scheme of musical tuning inwhich the octave is divided into a series of equal steps (equal frequency ratios).
		Twelve tone equal temperament was designed to permit the playing of music in all keyswith an equal amount of mis- tuning in each, while still approximating just intonation.
		The equal tempered version of the Bohlen-Pierce scale consists of the ratio 3:1, 1902 cents, conventionally an octave and a just fifth, usedas a tritave, and split into a thirteen tone equal temperament where each step is
	www.therfcc.org /equal-tempered-19446.html (1019 words)

	Guitar Lessons at Guitar Tricks: Todays Tuning...
		At the beginning of the 18th century "well tempered tuning" was developed in order to avoid this problem, and this was replaced by equal tempered tuning in the 19th century.
		EQUAL TEMPERED TUNING (chromatic tuning): In 1559 prince Chu Tsai Yü (China) was doing calculations on frequency ratios that allowed him to play an instrument chromatically (actually he worked out the 12th root of 2:1 precisely to nine decimal places).
		The resulting equal tempered tuning is therefore also called "Chromatic tuning." Basically this tuning is a controlled detuning of the partial tones.
	www.guitartricks.com /trick.php?trick_id=5648&s_id=39 (680 words)

	Equal temperament
		Other equal temperaments do exist (some music has been written in 31-TET for example), but they are so rare that when people use the term equal temperament it is usually understood that they are talking about the twelve tone variety.
		The distance between each step and the next is aurally the same for any two adjacent steps, though, because steps form an geometric sequence, the difference in frequency increases from one to the next.
		This may be used for atonal music, such as that written with the twelve tone technique or serialism, or tonal music.
	www.fact-index.com /e/eq/equal_temperament.html (684 words)

	Scales: Just vs Equal Temperament
		The "equal tempered scale" was developed for keyboard instruments, such as the piano, so that they could be played equally well (or badly) in any key.
		The equal tempered system uses a constant frequency multiple between the notes of the chromatic scale.
		For the equal temperament scale, the frequency of each note in the chromatic scale is related to the frequency of the notes next to it by a factor of the twelfth root of 2 (1.0594630944....).
	www.phy.mtu.edu /~suits/scales.html (626 words)

	Articles - Equal temperament (Site not responding. Last check: 2007-11-07)
		The first person known to introduce a mathematically accurate specification for equal temperament is probably Chu Tsai-Yu in the Ming Dynasty, who published a theory of the temperament in 1584.
		Twelve tone equal temperament was introduced in the West to permit the playing of music in all keys with an equal amount of mis-tuning in each, without having to provide more than 12 pitches per octave on instruments, while still roughly approximating just intonation intervals.
		At the time equal temperament was beginning to take hold in the West, many people perceived the much-increased mis-tuning of the music, relative to meantone temperament, as a disgrace.
	www.lastring.com /articles/Equal_temperament?mySession=a9e439fce5ce654178188b21df809232 (1690 words)

	Temperaments and the circle of fifths
		The well tempered tunings are circulating (no wolf) temperaments, some of the modified mean tone temperaments such as 18th century English, contain very small "wolves" where the comma has been slightly exceeded, and some 5ths have been widened by 1/12th or 1/6th of a comma.
		Placing equal temperament in the miscellaneous section may seem strange, as it is a circulating temperament with no wolf, and permits unlimited modulation.
		Equal temperament is the standard piano tuning in the western world and is likely to remain so.
	www.kirnberger.fsnet.co.uk /TEMPS4.html (2428 words)

	Well v.s. equal temperament
		Equal temperament is appropriate for some music of the 20th century, especially atonal music, and music based on the whole tone scale, but not for the works of the 18th and 19th centuries.
		Equal temperament, the modern and usually inappropriate system of tuning used in western music, is based on the twelfth root of 2.
		One main drawback to equal temperament is that all major thirds are quite a bit off from where they ought to be, roughly fourteen percent of a semitone.
	www.math.uwaterloo.ca /~mrubinst/tuning/tuning.html (1354 words)

	History of Tuning and Temperament
		Tempering is an adjustment of the intervals between notes away from pure.
		In equal temperament, every 5th is narrow by 2 cents, (702 - 2 = 700), and every 4th is wide by 2 cents, (498 + 2 - 500).
		Just intonation, meantone and equal temperament are classified as regular systems because all, or all but one of the 5ths, are tempered equally.
	www.terryblackburn.us /music/temperament/stoess.htm (2797 words)

	The Equal Tempered Scale and Peculiarities of Piano Tuning
		Understanding the equal tempered scale and the rational for the stretching of the upper octaves in piano tuning involves an understanding of something about both the physics of sound, and the history of music.
		With the evolution of composition technique in the 18th century increasing the use of harmonic modulation a change was advocated to the equal tempered scale.
		In the equal tempered scale, the twelve intervals are spread evenly between the octaves.
	www.precisionstrobe.com /apps/pianotemp/temper.html (4260 words)

	JSyn Docs: Class EqualTemperedTuning
		Equal Tempered Tuning Calculates frequencies based on pitch indices.
		Every interval between successive notes in the tuning is equal to the next, thus the name.
		A typical Western piano is tuned to twelve tone equal temperament, or "12 TET".
	www.softsynth.com /jsyn/docs/autodocs/com/softsynth/jsyn/EqualTemperedTuning.html (239 words)

	The RealTime Tuner: Background
		In Equal Temperament all intervals except octaves are slightly out of tune, but the out-of-tuneness is distributed in such a way that music played in one key makes exactly the same tuning compromises as when played in any other key.
		Equal Temperament had already been discovered in Bach's time, but he and other composers of his era generally preferred not to use it.
		The current near-universal adoption of Equal Temperament has been based on the assumption that it is essential for free modulation, there being no practical way to retune a keyboard instrument as the internal harmonies of a piece change.
	socrates.berkeley.edu /~wcooper/background.html (1257 words)

	John Greschak - Tempo Scales in Polytempo Music: A Survey
		The equal tempered scale to which Pople refers is as follows (rounded to two decimal places): 40.00, 41.77, 43.62, 45.55, 47.57, 49.67, 51.87, 54.17, 56.57, 59.07, 61.69, 64.42, 67.27, 70.25, 73.36, 76.61 and 80.00.
		Here, he is motivated by a desire to have a scale of durations that has the same equal tempered structure as that of a scale of pitches, for use in the composition of serial music.
		In the standard equal tempered tuning system that is used today, for which the frequency of the A above middle C is 440 Hz, the frequency of this low C and the B below it is approximately 32.70 Hz and 30.87 Hz, respectively.
	www.greschak.com /polytempo/ptts.htm (3001 words)

	Bingo-card Lattices, Equal-tempered Musical Scales
		The utility of the bingo-card lattices lies in their ability to show the "vanishing commas", i.e., the small intervals which are tempered out in a particular EDO.
		Below is a lattice using triangular geometry, resulting in a hexagonal cell representing each lattice-point, showing many of these intervals, with most of the more important ones labeled with the names in common use among tuning theorists c.
		16edo tempers out the magic comma [-1 5] and also the rather large ratio 128/135 [3 1] (= ~92 cents) called "larger limma" by Ellis and "mean semitone" by Rameau, and their derivatives.
	www.tonalsoft.com /enc/b/bingo.aspx (2244 words)

	Just Intervals?
		The equal tempered value is 1.12462048, indicating that 9/8 is a better simple ratio than 7/6 which is 1.11666667.
		Whether these intervals are just or not, or tempered equally or unequally to some degree or the other, whether through habituation or inadvertency, the just values are, speaking both acoustically and psychologically, the fundamental, "special relationship", to use Benande's expression, of frequencies which have a psychoacoustic value to the human mind.
		As people have an unconscious, conditioned, memory of pitch, they similarly have a conditioned sense of tempered intervals, particularly thirds and sixths, while at the same time another impetus exists, and that is toward the sense of just values for these intervals.
	www.ptg.org /pipermail/pianotech/2002-May/107940.html (714 words)

	Tunings and Cognition
		Tempered intervals, then, are fine deviations (usually below 15-20 cents) from the "pure" consonance of small integer ratios.
		In tuning keyboard instruments, tempering of some or all of the intervals is inevitable.
		In equal temperament the octave is divided precisely into 12 equal intervals.
	eamusic.dartmouth.edu /~kov/lerdahl/tuningPaper.html (925 words)

	Meantone
		Those tempered fifths are less nasty than you evidently believe they are; and the major thirds are much, MUCH better (pure, or nearly so) than they are in E.T. The quality of any given triad--its perceptible character--comes almost completely from the major and minor thirds in it, not the fifths.
		Equal temperament has no interval in it that can be expressed as a rational number (because it's all based on an irrational number, the twelfth root of 2).
		Equal temperament is based on a concession that all intervals are imperfect.
	www.bach-cantatas.com /Topics/Meantone.htm (2919 words)

	Guitar Notes: TEMPERAMENT
		The main compromise in equal temperament is in the third and sixth intervals.
		There are people for whom the compromise in the major third that is caused by use of equal temperament is actually uncomfortable.
		If tuning to harmonics this way worked, then the number ((4/3)^4)(5/4) would have to be equal* to 4 (which is the frequency ratio between the low E on the 6th string and the E on the open 1st string - a double octave).
	www.guitarnotes.com /notes/noteget.cgi?temperament (836 words)

	Graphic Design (Site not responding. Last check: 2007-11-07)
		So equal-tempered scales (ones in which the smallest interval is always the same frequency ratio) compromise by mistuning the fifths slightly while keeping the octaves perfect.
		Most western instruments today are tuned to equal temperament, though continuous instruments such as violin, trombone and voice can produce other scales.
		Equal temperament became all the rage during the industrial revolution, when musical instruments became mass produced.
	www.scottkim.com /a5 (500 words)

	Equal temperament : Equal tempered
		Equal temperament was designed to permit the playing of music in all keys with an equal amount of mis-tuning in each.
		There is some reason to believe that when composers and theoreticians of this era wrote of the "colors" of the keys, they described the subtly different dissonances of particular tuning methods.
		However, equal temperament also made possible far more harmonically complex music, as it enabled musicians to move rapidly between keys without re-tuning: J.
	www.fastload.org /eq/Equal_tempered.html (459 words)

	Harmonic errors in equal tempered musical scales (Site not responding. Last check: 2007-11-07)
		The equal tempered scale usually refers to the musical scale with 12 equal divisions of the octave, but here I take an equal-tempered scale to be any scale where the frequency of each note is related to the next by a constant multiplier.
		A first pass at finding equal tempered scales that work is to compare powers of 2 with powers of the other small primes and look for near matches.
		For similar evaluations of equal tempered scales, using some different criteria, including those important to melody, see Paul Erlich's article Tuning, Tonality, and Twenty-Two-Tone Equal Temperament and Georg Hajdu's Low Energy and Equal Spacing; the Multifactorial Evolution of Tuning Systems, Interface, 22, (1993), 319-333.
	www.uq.net.au /~zzdkeena/Music/EqualTemperedMusicalScales.htm (2765 words)

	What are the frequencies of musical notes like G in k-hertz?
		Both of these pitch standards define what are called 'equal tempered chromatic scales.' Mathematically, this means that each successive pitch is related to the previous pitch by a factor of the twelfth root of 2.
		Note when counting steps that there is a single half-tone (step) between B and C, and E and F. These pitch scales are referred to as 'equal tempered' or 'well tempered.' This refers to a compromise built into the use of the 12th root of 2 as the factor separating each successive pitch.
		In this equal temperament system of tuning, the frequencies of notes on a keyboard are related by a fairly simple mathematical relationship involving the number of keys (half-steps) between the notes.
	www.physlink.com /Education/AskExperts/ae165.cfm (1027 words)

	[No title]
		In any case, it is all very well for the equal tempered lobby to make such claims, but this overlooks the music played on equally tempered pianos that was written for well-tempered or mean tone tunings, which is certainly historically incorrect.
		One of the more amusing claims in favour of equal temperament is that J.P. Kirnberger tried to persuade musicians to adopt it, when in fact his tuning systems, which were published near the end of his life are very different from equal temperament, favour the popular keys, and contain at least one pure major 3rd.
		Already in use in Germany in Bach’s time, was a term for equal temperament, or what was generally thought to reflect the manner of tuning this system: Gleichschwebende temperatur (equal beating temperament); it was considered that equal temperament was obtained by tuning all twelve 5ths flat to beat equally.
	www.kirnberger.fsnet.co.uk /Temps3.htm (1972 words)

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