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A schismatic temperament is a musical tuning system that results from tempering the schisma of 32805:32768 (1.9537 cents) to a unison. It is also called the schismic temperament, Helmholtz temperament, or quasi-Pythagorean temperament.
In Pythagorean tuning all notes are tuned as a number of perfect fifths (701.96 cents 
play (help·info)). The major third above C, E, is considered four fifths above C. This causes the Pythagorean major third, E+ (407.82 cents play (help·info)), to differ from the just major third, E♮ (386.31 cents play (help·info)): the Pythagorean third is sharper than the just third by 21.51 cents (a syntonic comma play (help·info)).
Ellis's "skhismic temperament" [1] instead uses the note eight fifths below C, F♭-- (384.36 cents play (help·info)), the Pythagorean diminished fourth or schismatic major third. Though spelled "incorrectly" for a major third, this note is only 1.95 cents (a schisma) flat of E♮, and thus more in tune than the Pythagorean major third. As Ellis puts it, "the Fifths should be perfect and the Skhisma should be disregarded [accepted/ignored]."
In his eighth-schisma "Helmholtzian temperament" [1] the note eight fifths below C is also used as the major third above C. However, in the "skhismic temperament" pure perfect fifths are used to construct an approximate major third, while in the "Helmholtzian temperament" approximate perfect fifths are used to construct a pure major third. To raise the Pythagorean diminished fourth 1.95 cents to a just major third each fifth must be narrowed, or tempered, by 1.95/8 = 0.24 cents. Thus the fifth becomes 701.71 cents instead of 701.96 cents. As Ellis puts it, "the major Thirds are taken perfect, and the Skhisma is disregarded [tempered out]."
Whereas schismatic temperaments achieve a ratio with a number of fifths, each tempered by a fraction of the schisma; meantone temperaments achieve a ratio with fifths, each tempered by a fraction of the syntonic comma (81:80, 21.51 cents). As meantone temperaments are often described by what fraction of the syntonic comma is used to alter the perfect fifths, schismatic temperaments are often described by what fraction of the schisma is used to alter the perfect fifths (thus quarter-comma meantone temperament, eighth-schisma temperament, etc.).
In both eighth-schisma tuning and quarter-comma meantone the octave and major third are just, but eighth-schisma has much much more accurate perfect fifths and minor thirds (less than a quarter of a cent off from just intonation). However, quarter-comma meantone has a large advantage in that the major third and minor third are spelled as such, whereas in schismatic tunings, they're represented by the diminished fourth and augmented second (if spelled according to their construction in the tuning). This places them well outside the span of a single diatonic scale, and requires both a larger number of pitches and more microtonal pitch-shifting when attempting common-practice Western music.
Various equal temperaments lead to schismatic tunings which can be described in the same terms. Dividing the octave by 53 provides an approximately 1/29-schisma temperament; by 65 a 1/5-schisma temperament, by 118 a 2/15-schisma temperament, and by 171 a 1/10-schisma temperament. The last named, 171, produces very accurate septimal intervals, but they are hard to reach, as to get to a 7/4 requires 39 fifths. The −1/11-schisma temperament of 94, with sharp rather than flat fifths, gets to a less accurate but more available 7:4 by means of 14 fourths. Eduardo Sabat-Garibaldi also had an approximation of 7:4 by means of 14 fourths in mind when he derived his 1/9-schisma tuning.
Historically significant is the eighth-schisma tuning of Hermann von Helmholtz and Norwegian composer Eivind Groven. Helmholtz had a special Physharmonica (a harmonium by Schiedmayer) with 24 tones to the octave.[ citation needed ] Groven built an organ internally equipped with 36 tones to the octave which had the ability to adjust its tuning automatically during performances; the performer plays a familiar 12-key (per octave) keyboard and in most cases the mechanism will choose from among the three tunings for each key so that the chords played sound virtually in just intonation.[ citation needed ] A 1/9-schisma tuning has also been proposed by Eduardo Sabat-Garibaldi, who together with his students uses a 53-tone to the octave guitar with this tuning.[ citation needed ]
Mark Lindley and Ronald Turner-Smith argue that schismatic tuning was briefly in use during the late medieval period. [2] [ need quotation to verify ] This was not temperament but merely 12-tone Pythagorean tuning. Justly tuned fifths and fourths generate a reasonable schismatic tuning and therefore schismatic is in some respects an easier way to introduce approach justly tuned thirds into a Pythagorean harmonic fabric than meantone. However, the result suffers from the same difficulties as just intonation – for example, the wolf B-G♭ here arises all too easily when availing oneself of the concordant schismatic substitutions just outlined – so it is not surprising that meantone temperament became the dominant tuning system by the early Renaissance. Helmholtz's and Groven's systems get around some, but not all, of these difficulties by including multiple tunings for each key on the keyboard, so that a particular note can be tuned as G♭ in some contexts and F♯ in others, for example.
In music, just intonation or pure intonation is the tuning of musical intervals as whole number ratios of frequencies. An interval tuned in this way is said to be pure, and is called a just interval. Just intervals consist of tones from a single harmonic series of an implied fundamental. For example, in the diagram, if the notes G3 and C4 are tuned as members of the harmonic series of the lowest C, their frequencies will be 3 and 4 times the fundamental frequency. The interval ratio between C4 and G3 is therefore 4:3, a just fourth.
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 temperament is a musical temperament, that is a tuning system, 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 the same way as Pythagorean tuning, as a stack of equal fifths, but it is a temperament in that the fifths are not pure.
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. 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♯ (Play (help·info)), 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 refer to tempering is the Pythagorean comma.
In classical music from Western culture, a diesis is either an accidental, or a very small musical interval, usually defined as the difference between an octave and three justly tuned major thirds, equal to 128:125 or about 41.06 cents. In 12-tone equal temperament three major thirds in a row equal an octave, but three justly-tuned major thirds fall quite a bit narrow of an octave, and the diesis describes the amount by which they are short. For instance, an octave (2:1) spans from C to C', and three justly tuned major thirds (5:4) span from C to B♯. The difference between C-C' (2:1) and C-B♯ (125:64) is the diesis (128:125). Notice that this coincides with the interval between B♯ and C', also called a diminished second.
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. For example, C is adjacent to C♯; the interval between them is a semitone.
In music theory, a minor third is a musical interval that encompasses three half steps, or semitones. Staff notation represents the minor third as encompassing three staff positions. The minor third is one of two commonly occurring thirds. It is called minor because it is the smaller of the two: the major third spans an additional semitone. For example, the interval from A to C is a minor third, as the note C lies three semitones above A. Coincidentally, there are three staff positions from A to C. Diminished and augmented thirds span the same number of staff positions, but consist of a different number of semitones. The minor third is a skip melodically.
In music, the schisma (also spelled skhisma) is the interval between a Pythagorean comma (531441:524288) and a syntonic comma (81:80) and equals 5(38)⁄215 or 32805:32768 = 1.00113, which is 1.9537 cents (Play (help·info)). It may also be defined as:
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.
The diaschisma is a small musical interval defined as the difference between three octaves and four perfect fifths plus two major thirds. It can be represented by the ratio 2048:2025 and is about 19.5 cents. The use of the name diaschisma for this interval is due to Helmholtz; earlier Rameau had called that interval a "diminished comma" or comma minor.
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 × 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. Play (help·info) Each step represents a frequency ratio of 21⁄53, or 22.6415 cents, an interval sometimes called the Holdrian comma.
In musical tuning theory, a Pythagorean interval is a musical interval with frequency ratio equal to a power of two divided by a power of three, or vice versa. For instance, the perfect fifth with ratio 3/2 (equivalent to 31/ 21) and the perfect fourth with ratio 4/3 (equivalent to 22/ 31) are Pythagorean intervals.
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.
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.
Five-limit tuning, 5-limit tuning, or 5-prime-limit tuning (not to be confused with 5-odd-limit tuning), is any system for tuning a musical instrument that obtains the frequency of each note by multiplying the frequency of a given reference note (the base note) by products of integer powers of 2, 3, or 5 (prime numbers limited to 5 or lower), such as 2−3·31·51 = 15/8.
