Equal temperament
The equal temperament is a tuning system in which all twelve semitones in an octave are the same size (100 cents). Other names are: equal tempered / equal temperament or equal temperament; to distinguish it from the same stage systems with different number of stages ( eg 19 or 24 ) the name of 12 - EDO (Equal Division of the Octave) is in use. The term often colloquially used temperament is too imprecise, since the equal temperament is only one possible way to control the temperature intervals.
Just intonation in keyboard instruments suffers from the problem that there is always only a limited number of clean -sounding chords available. This problem solves the equal temperament by evenly distributes the inevitable impurities on all pitches, so that all the chords and keys are the same useful. In earlier systems, this tuning is not the case, which was limited to other keys, the possibility of modulation. Critics of equal temperament tuning regret, however, that in it the individual character of individual keys of the earlier mean- or well tempered tunings perish.
Moods with tempered intervals
The dominant in Western music equal temperament system is particularly important for instruments whose pitch and key number or the number of notes per octave is determined by design parameters, eg keyboard instruments such as organ, harpsichord, piano, or mallets and vielsaitige Plucked, where a conditional tonart retuning or adjust the pitch during the match is not possible.
Other instruments, such as string or wind instruments, on the other hand may well intone pure, where the player can compensate from case to case, the system-related impurities by slight adjustment of the pitch.
Depending on in which harmonic context, a sound is played, but this would actually have a slightly different pitch to be heard in a chord pure ( beatless free). For example, the sound does not match the tone Gis As, and this problem is ultimately in all tones of a scale, depending on the harmonic context in which they are used. For keyboard instruments, therefore, a scale was needed, which was first realized in the Mean moods and then into the well-tempered tunings. Feature of all these tunings is that they were developed on the basis of musical aspects. The exact location of all twelve semitones is in meantone or well-tempered tuning so determined that some scales or chords sound pure, other, mostly the sound of rare common, impure.
Alone in the mathematically determined equal temperament tuning keys all sound the same ( impure).
Intervals in equal temperament tuning
In equal temperament, the octave is divided into twelve identical halftone steps:
Thus, the Pythagorean comma is balanced, the twelve consecutively executed pure fifths (for example CGDAEH - Fis - Cis - Gis Dis -Ais - ice -His ) occurs. These fifths are now all voted by 1/ 12 of commas deeper, so that the open bottom spiral closes the circle of fifths. Compared to the Pythagorean ( the quint pure ) atmosphere with the perfect fifth of 702 cents equal temperament has a slightly scaled- fifth of 700 cents; according to the Quarte (in: 498 cents, equally-: 500 cents) of equal temperament tuning - which complements the fifth to the octave - by about 2 cents more than a perfect fourth. The major third ( 386 cents) in just intonation is at the same stage tuning ( 400 cents) to at least about 14 cents enlarged ( " sharpened "); the minor sixth (in: 814 cents, equally-: 800 cents) is the same amount too small. The minor third (in: 316 cents, equally-: 300 cents), in turn, is as much as about 16 cents too tight, the major sixth (in: 884 cents, equally-: 900 cents) too far voted by the same amount.
Such a tuned instrument except the octave contains not a single "ideal", that is, in a simple integer frequency ratio purely tuned interval longer, and the deviations are also quite audible. In today's music perception but which is generally perceived as acceptable ( habituation ).
History
The equal temperament was in 1584 by Chu Tsai -yu (朱 载 堉) in China nine-digit numbers can be calculated fairly accurately using a system for the first time. In Europe, these calculations were, however, known until 1799 without Chu Tsai -yii was named. 1588 offered Gioseffo Zarlino an exact geometric representation. Simon Stevin first described in Europeans Vande Spiegheling the Singconst ( manuscript at or prior to 1600) a close approximation using a method developed by him to the square root calculation, but said mistakenly, while ensuring natural major thirds.
As equally- designated According moods of the 16th century were based, as practiced by Vincenzo Galilei, mostly on the halftone with the ratio 18:17 ( about 99 cents).
Especially in the 17th century, the equal temperament not only of theorists such as Pietro Mengoli and Marin Mersenne, but also by composers, instrument makers and practicing musicians was discussed. This is evidenced, for example, a dispute over moods between Giovanni Artusi and Monteverdi shortly after in 1600. Girolamo Frescobaldi recommended the equal temperament of the organ in the Basilica of S. Lorenzo in Damaso.
In the German -speaking area was used for equally- the concept of equal temperament, according to Andreas Werckmeister 1707 in Musical Paradoxal - Discourse: " ... if the temperature is so set / that all fifths 1/12 commat: ... float, and a Accurates ear the same to also to the registry install and tune white / so then certainly a wohltemperirte Harmonia, is found by the gantzen Circul and through all Clavis itself. " Werckmeister expressly says that's not that the beat frequencies are equal. The question raised by him difficulty, equally- to vote, such as a piano tuner can master just the fact that he knows the different beat frequencies of the different fifths in the high altitudes of the own piano and uses the voices.
The practical significance, however, remained low until the 18th century. It multiplied the proponents of equal temperament tuning, which included, for example, Jean -Philippe Rameau and Friedrich Wilhelm Marpurg. Until the late 18th century, the equal temperament gained the upper hand against unequal stage moods and sat down definitively in the 19th century.
This, however, lost the keys of characters for new compositions in importance because different keys no longer sounded different in this regard. When listing older works on equally- tuned instruments are often lost essential artistic aspects of composition for the same reason, so translated, for example, older composers in their time like bad sounding " impossible" keys in order to make negative situations such as pain or sin sound experience.
Today, instruments with fixed pitches, like the piano or the guitar, tuned by default equally-. Many organs and harpsichords but be historicized provided with other, non- stage tunings.
Quantitative aspects of the equal temperament tuning
Frequency calculation
The mathematical rule to determine the shades on the entire gamut of equal temperament tuning is
Where f0 is the frequency of a desired output sound ( for example, the standard frequency of a ' at 440 Hz). i is the half-step - distance to the selected tone at the frequency f0. Such a mathematical result is called geometric sequence. If you want to pay off the frequencies above equidistant note names on a straight line, so you have to use einfachlogarithmisches paper. It is tempting to label not the tens, but to use the logarithm.
If you want, for example, the frequency of the tone g ' determine, as you counts his semitone distance from the pitch a' from (i = minus 2, as you counts down ), and sets the values into the equation:
For the sound g'' is obtained according to a semitone to f0 of i = 10:
As you can see, g'' has twice the frequency of g '. The octave purity remains so preserved, whereas all other intervals are slightly impure.
Frequencies and cents values
When comparing intervals using the unit cents. Where: 1 octave = 1200 cents.
- If the difference is negative, the interval is narrower than the same temperature-controlled pure.
- * Tritone ( Excessive fourth), defined as: Major third ( frequency ratio 5/4) plus Large second ( frequency ratio 9/8) = fifth ( frequency ratio 3/2) minus diatonic semitone ( frequency ratio of 16/15 ). The augmented fourth ( for example, C - F sharp or G flat C, frequency ratio 45/32 according to 590 cents) is less than the diminished fifth (for example, F # -C or C- Ges, frequency ratio 64/45 according to 610 cents) in just intonation. In gleichstufiger mood but both are equal to half an octave (600 cents).
- Note on major second and minor seventh: In just intonation, there are two whole tones with the frequency ratios 9/8 and 10 / 9th Accordingly, there are two minor seventh with the frequency ratios 2:9 / 8 = 16 /9 and 2:10 / 9 = 9 / 5th
Special shapes
The division of the octave into twelve tones with the same frequency relative to its neighboring tones, although the most common, but not the only way to approach the pure intervals. With more tones per octave to better approximations can be achieved. Gleichstufige organizations that have actually been used are, for example:
- Neunzehnstufiges sound system
- Einunddreißigstufiges sound system of 1606
- Quarter-tone system
Was in the new music of the 20th and 21st century and experimented with numerous equal temperament (and others) sound systems, with the octave as in 17, 19, 31, 53, 72 equal steps is divided.
Occasionally, other intervals are divided as the octave. So, for example, creates Karlheinz Stockhausen for his electronic Study II from 1952 a sound system that is based on the division of an interval with the frequency ratio 5/1 in 25 equal steps. Since the stages distances are slightly larger than the traditional tempered semitone, creating a sound system which is capable of producing ( discordant ) clay mixtures suitable.