The Kirnberger temperaments are three irregular temperaments developed in the second half of the 18th century by Johann Kirnberger. Kirnberger was a student of Johann Sebastian Bach who held great admiration for his teacher and was one of his principal proponents.[1]
The first Kirnberger temperament, "Kirnberger I", had similarities to Pythagorean temperament, which stressed the importance of perfect fifths all throughout the circle of fifths. A complete circle of perfect fifths is not possible, because when the circle comes to an end at the tone it began, it will have overshot its original pitch. Thus, if one tunes C–G, G–D, D–A, A–E, E–B, B–F♯, F♯–C♯, C♯–G♯ (A♭), A♭–E♭, E♭–B♭, B♭–F, F–C… the new "C" will not be the same frequency as the first. The two "C"s will have a discrepancy of about 23 cents (a comma), which would be unacceptable. This difference between the initial "c" and final "c" that is derived from performing a series of perfect tunings is generally referred to as the Pythagorean comma. In Kirnberger I, the D–A fifth is reduced by a syntonic comma, making the major thirds F–A, C–E, G–B, and D–F# pure, though the fifth based on D is a ratio of 40/27 instead of 3/2. Many tuning systems have been developed to "spread around" that comma, that is, to divide that anomalous musical space among the other intervals of the scale.
Practical Temperaments: Kirnberger II
Kirnberger II temperament
Kirnberger's first method of compensating for and closing the circle of fifths was to split the "wolf" interval, known to those who have used meantone temperaments, in half between two different fifths. That is, to compensate for the one extra comma, he removed half a comma from two of the formerly perfect fifths in order to complete the circle. In so doing, he allowed the remaining fifths to stay pure. At the time, however, pure thirds were more valued than pure fifths (meantone temperament had eight pure thirds and sacrificed four entire chords to achieve this end.) So, Kirnberger allowed for three pure thirds, the rest being slightly wide and the worst being three Pythagorean thirds (22 cents wider than pure) on the opposite end of the circle from the pure thirds. To put it graphically:
C-----G-----D------A-----E-----B-----F♯-----C♯-----Ab(G♯)-----Eb-----Bb-----F-----C p p −½ −½ p p p p p p p p |__________pure 3rd______| |__________pure 3rd______| |_______pure 3rd________| |__________Pythag. 3rd_________| |_________Pythag. 3rd___________| |________Pythag. 3rd___________|
The above table represents KirnbergerII temperament. The first row under the intervals shows either a "p" for pure, or "−½" for those intervals narrowed to close the circle of fifths (D–A), (A–E). Below these are shown the pure 3rds (between C–E, G–B, D–F♯), and Pythagorean (very wide) 3rds (B–D♯, F♯–A♯(B♭), D♭–F.)
Temperament, however, is a give-and-take situation: None is a perfect solution. It must also be remembered that temperament applies only to instruments with fixed pitch: Any keyboard instrument, lutes, viols, and so forth. Wind musicians, singers, and string players all have a degree of control over the exact pitch and intonation of what they play, and may therefore be free of such restrictive systems. When the two classes of players come together, it is important when evaluating a temperament to consider the tendencies of the instruments vs. those of the temperament. KirnbergerII would only have been applicable to harps and keyboard instruments such as the harpsichord and organ. The advantages of this system are its three pure major thirds and its ten pure fifths; the disadvantages are, of course, the two narrow "half-wolf" fifths and the three Pythagorean, super-wide thirds. The chords are not entirely unusable but certainly must not be used frequently nor in close succession within the course of a piece.
Kirnberger III
KirnbergerIII temperament
After some disappointment with his sour, narrow fifths, Kirnberger experimented further and developed another possibility, later named the KirnbergerIII.
This temperament splits the Syntonic comma between four fifths instead of two; ¼-comma tempered fifths are used extensively in meantone and are much easier to tune and to listen to. This also eliminates two of the three pure thirds found in KirnbergerII. Therefore, only one third remains pure (between C and E), and there are fewer Pythagorean thirds. A greater middle ground is reached in this improvement, and each key is closer to being equal to the next. The drawback is an aesthetic one: Fewer chords have pure thirds and fifths. But every temperament system is a mix of give-and-take compromises; each finds a way of dealing with the comma.
Klop, G.C. (1974). Harpsichord Tuning. Raleigh, NC: The Sunbury Press. — called "Kirnberger2" in Lieberman & Miller (2006). Lou Harrison. p.80. ISBN0-252-03120-2; presumably other Kirnberger temperaments as well.
In music, there are two common meanings for tuning:
Pythagorean tuning is a system of musical tuning in which the frequency ratios of all intervals are based on the ratio 3:2. This ratio, also known as the "pure" perfect fifth, is chosen because it is one of the most consonant and easiest to tune by ear and because of importance attributed to the integer 3. As Novalis put it, "The musical proportions seem to me to be particularly correct natural proportions." Alternatively, it can be described as the tuning of the syntonic temperament in which the generator is the ratio 3:2, which is ≈ 702 cents wide.
Meantone temperaments are musical temperaments, that is a variety of tuning systems, obtained by narrowing the fifths so that their ratio is slightly less than 3:2, in order to push the thirds closer to pure. Meantone temperaments are constructed similarly to Pythagorean tuning, as a stack of equal fifths, but they are temperaments in that the fifths are not pure.
In music theory, an interval is a difference in pitch between two sounds. An interval may be described as horizontal, linear, or melodic if it refers to successively sounding tones, such as two adjacent pitches in a melody, and vertical or harmonic if it pertains to simultaneously sounding tones, such as in a chord.
Well temperament is a type of tempered tuning described in 20th-century music theory. The term is modeled on the German word wohltemperiert. This word also appears in the title of J. S. Bach's famous composition "Das wohltemperierte Klavier", The Well-Tempered Clavier.
In music theory, the syntonic comma, also known as the chromatic diesis, the Didymean comma, the Ptolemaic comma, or the diatonic comma is a small comma type interval between two musical notes, equal to the frequency ratio 81:80 (= 1.0125). Two notes that differ by this interval would sound different from each other even to untrained ears, but would be close enough that they would be more likely interpreted as out-of-tune versions of the same note than as different notes. The comma is also referred to as a Didymean comma because it is the amount by which Didymus corrected the Pythagorean major third to a just major third.
In music theory, the wolf fifth is a particularly dissonant musical interval spanning seven semitones. Strictly, the term refers to an interval produced by a specific tuning system, widely used in the sixteenth and seventeenth centuries: the quarter-comma meantone temperament. More broadly, it is also used to refer to similar intervals produced by other tuning systems, including Pythagorean and most meantone temperaments.
In musical tuning, the Pythagorean comma (or ditonic comma), named after the ancient mathematician and philosopher Pythagoras, is the small interval (or comma) existing in Pythagorean tuning between two enharmonically equivalent notes such as C and B♯, or D♭ and C♯. It is equal to the frequency ratio (1.5)12⁄27 = 531441⁄524288 ≈ 1.01364, or about 23.46 cents, roughly a quarter of a semitone (in between 75:74 and 74:73). The comma that musical temperaments often "temper" is the Pythagorean comma.
A semitone, also called a half step or a half tone, is the smallest musical interval commonly used in Western tonal music, and it is considered the most dissonant when sounded harmonically. It is defined as the interval between two adjacent notes in a 12-tone scale, visually seen on a keyboard as the distance between two keys that are adjacent to each other. For example, C is adjacent to C♯; the interval between them is a semitone.
A schismatic temperament is a musical tuning system that results from tempering the schisma of 32805:32768 to a unison. It is also called the schismic temperament, Helmholtz temperament, or quasi-Pythagorean temperament.
In music theory, a comma is a very small interval, the difference resulting from tuning one note two different ways. Strictly speaking, there are only two kinds of comma, the syntonic comma, "the difference between a just major 3rd and four just perfect 5ths less two octaves", and the Pythagorean comma, "the difference between twelve 5ths and seven octaves". The word comma used without qualification refers to the syntonic comma, which can be defined, for instance, as the difference between an F♯ tuned using the D-based Pythagorean tuning system, and another F♯ tuned using the D-based quarter-comma meantone tuning system. Intervals separated by the ratio 81:80 are considered the same note because the 12-note Western chromatic scale does not distinguish Pythagorean intervals from 5-limit intervals in its notation. Other intervals are considered commas because of the enharmonic equivalences of a tuning system. For example, in 53TET, B♭ and A♯ are both approximated by the same interval although they are a septimal kleisma apart.
Quarter-comma meantone, or 1 / 4 -comma meantone, was the most common meantone temperament in the sixteenth and seventeenth centuries, and was sometimes used later. In this system the perfect fifth is flattened by one quarter of a syntonic comma ( 81 : 80 ), with respect to its just intonation used in Pythagorean tuning ; the result is 3 / 2 × [ 80 / 81 ] 1 / 4 = 4√5 ≈ 1.49535, or a fifth of 696.578 cents. This fifth is then iterated to generate the diatonic scale and other notes of the temperament. The purpose is to obtain justly intoned major thirds. It was described by Pietro Aron in his Toscanello de la Musica of 1523, by saying the major thirds should be tuned to be "sonorous and just, as united as possible." Later theorists Gioseffo Zarlino and Francisco de Salinas described the tuning with mathematical exactitude.
In music, 53 equal temperament, called 53 TET, 53 EDO, or 53 ET, is the tempered scale derived by dividing the octave into 53 equal steps. Each step represents a frequency ratio of 21⁄53, or 22.6415 cents, an interval sometimes called the Holdrian comma.
In music, septimal meantone temperament, also called standard septimal meantone or simply septimal meantone, refers to the tempering of 7-limit musical intervals by a meantone temperament tuning in the range from fifths flattened by the amount of fifths for 12 equal temperament to those as flat as 19 equal temperament, with 31 equal temperament being a more or less optimal tuning for both the 5- and 7-limits. Meantone temperament represents a frequency ratio of approximately 5 by means of four fifths, so that the major third, for instance C–E, is obtained from two tones in succession. Septimal meantone represents the frequency ratio of 56 (7 × 23) by ten fifths, so that the interval 7:4 is reached by five successive tones. Hence C–A♯, not C–B♭, represents a 7:4 interval in septimal meantone.
In music, 22 equal temperament, called 22-TET, 22-EDO, or 22-ET, is the tempered scale derived by dividing the octave into 22 equal steps. Each step represents a frequency ratio of 22√2, or 54.55 cents.
In musical tuning, a temperament is a tuning system that slightly compromises the pure intervals of just intonation to meet other requirements. Most modern Western musical instruments are tuned in the equal temperament system. Tempering is the process of altering the size of an interval by making it narrower or wider than pure. "Any plan that describes the adjustments to the sizes of some or all of the twelve fifth intervals in the circle of fifths so that they accommodate pure octaves and produce certain sizes of major thirds is called a temperament." Temperament is especially important for keyboard instruments, which typically allow a player to play only the pitches assigned to the various keys, and lack any way to alter pitch of a note in performance. Historically, the use of just intonation, Pythagorean tuning and meantone temperament meant that such instruments could sound "in tune" in one key, or some keys, but would then have more dissonance in other keys.
Werckmeister temperaments are the tuning systems described by Andreas Werckmeister in his writings. The tuning systems are numbered in two different ways: the first refers to the order in which they were presented as "good temperaments" in Werckmeister's 1691 treatise, the second to their labelling on his monochord. The monochord labels start from III since just intonation is labelled I and quarter-comma meantone is labelled II. The temperament commonly known as "Werckmeister III" is referred to in this article as "Werckmeister I (III)".
"Young temperament" may refer to either of a pair of circulating temperaments described by Thomas Young in a letter dated July 9, 1799, to the Royal Society of London. The letter was read at the Society's meeting of January 16, 1800, and included in its Philosophical Transactions for that year. The temperaments are referred to individually as "Young's first temperament" and "Young's second temperament", more briefly as "Young's No. 1" and "Young's No. 2", or with some other variations of these expressions.
A regular diatonic tuning is any musical scale consisting of "tones" (T) and "semitones" (S) arranged in any rotation of the sequence TTSTTTS which adds up to the octave with all the T's being the same size and all the S's the being the same size, with the 'S's being smaller than the 'T's. In such a tuning, then the notes are connected together in a chain of seven fifths, all the same size which makes it a Linear temperament with the tempered fifth as a generator.
The circulating temperament today referred to as Vallotti temperament is a shifted version of Young's second temperament. Its attribution to the 18th-century organist, composer, and music theorist, Francesco Vallotti is a mistake, since there is no evidence that he ever suggested it. It is however audibly indistinguishable from a slightly different temperament that was in fact devised by Vallotti.
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