# Tuplet

In music, a tuplet (also irrational rhythm or groupings, artificial division or groupings, abnormal divisions, irregular rhythm, gruppetto, extra-metric groupings, or, rarely, contrametric rhythm) is "any rhythm that involves dividing the beat into a different number of equal subdivisions from that usually permitted by the time-signature (e.g., triplets, duplets, etc.)" (Humphries 2002, 266). This is indicated by a number (or sometimes two), indicating the fraction involved. The notes involved are also often grouped with a bracket or (in older notation) a slur.

The most common type of tuplet is the triplet.

This rhythm has two tuplets: a triplet on the second beat and a quintuplet on the fourth beat.

## Terminology

The modern term 'tuplet' comes from a rebracketing of compound words like quintu(s)-(u)plet and sextu(s)-(u)plet, and from related mathematical terms such as "tuple", "-uplet" and "-plet", which are used to form terms denoting multiplets (Oxford English Dictionary, entries "multiplet", "-plet, comb. form", "-let, suffix", and "-et, suffix1"). An alternative modern term, "irrational rhythm", was originally borrowed from Greek prosody where it referred to "a syllable having a metrical value not corresponding to its actual time-value, or ... a metrical foot containing such a syllable" (Oxford English Dictionary, entry "irrational"). The term would be incorrect if used in the mathematical sense (because the note-values are rational fractions) or in the more general sense of "unreasonable, utterly illogical, absurd".

Alternative terms found occasionally are "artificial division" (Jones 1974, 19), "abnormal divisions" (Donato 1963, 34), "irregular rhythm" (Read 1964, 181), and "irregular rhythmic groupings" (Kennedy 1994). The term "polyrhythm" (or "polymeter"), sometimes incorrectly used of "tuplets", actually refers to the simultaneous use of opposing time signatures (Read 1964, 167).

Besides "triplet", the terms "duplet", "quadruplet", "quintuplet", "sextuplet", "septuplet", and "octuplet" are used frequently. The terms "nonuplet", "decuplet", "undecuplet", "dodecuplet", and "tredecuplet" had been suggested but up until 1925 had not caught on (Dunstan 1925,). By 1964 the terms "nonuplet" and "decuplet" were usual, while subdivisions by greater numbers were more commonly described as "group of eleven notes", "group of twelve notes", and so on (Read 1964, 189).

## Triplet

The most common tuplet (Schonbrun 2007, 8) is the triplet (Ger. Triole, Fr. triolet, It. terzina or tripletta, Sp. tresillo). Whereas normally two quarter notes (crotchets) are the same duration as a half note (minim), three triplet quarter notes have that same duration, so the duration of a triplet quarter note is ​23 the duration of a standard quarter note.

Similarly, three triplet eighth notes (quavers) are equal in duration to one quarter note. If several note values appear under the triplet bracket, they are all affected the same way, reduced to ​23 their original duration.

The triplet indication may also apply to notes of different values, for example a quarter note followed by one eighth note, in which case the quarter note may be regarded as two triplet eighths tied together (Gherkens 1921, 19).

In some older scores, rhythms like this would be notated as a dotted eighth note and a sixteenth note as a kind of shorthand (Troeger 2003, 172) presumably so that the beaming more clearly shows the beats.

## Tuplet notation

### Notation

Tuplets are typically notated either with a bracket or with a number above or below the beam if the notes are beamed together. Sometimes, the tuplet is notated with a ratio (instead of just a number) — with the first number in the ratio indicating the number of notes in the tuplet and the second number indicating the number of normal notes they have the same duration as — or with a ratio and a note value.

### Rhythm

#### Simple meter

For other tuplets, the number indicates a ratio to the next lower normal value in the prevailing meter (a power of 2 in simple meter). So a quintuplet (quintolet or pentuplet (Cunningham 2007, 111)) indicated with the numeral 5 means that five of the indicated note value total the duration normally occupied by four (or, as a division of a dotted note in compound time, three), equivalent to the second higher note value. For example, five quintuplet eighth notes total the same duration as a half note (or, in 3
8
or compound meters such as 6
8
, 9
8
, etc. time, a dotted quarter note).

Some numbers are used inconsistently: for example septuplets (septolets or septimoles) usually indicate 7 notes in the duration of 4—or in compound meter 7 for 6—but may sometimes be used to mean 7 notes in the duration of 8 (Read 1964, 183–84). Thus, a septuplet lasting a whole note can be written with either quarter notes (7:4) or eighth notes (7:8).

To avoid ambiguity, composers sometimes write the ratio explicitly instead of just a single number. This is also done for cases like 7:11, where the validity of this practice is established by the complexity of the figure. A French alternative is to write pour ("for") or de ("of") in place of the colon, or above the bracketed "irregular" number (Read 1964, 219–21). This reflects the French usage of, for example, "six-pour-quatre" as an alternative name for the sextolet (Damour, Burnett, and Elwart 1838, 79; Hubbard 1924, 480).

There are disagreements about the sextuplet (pronounced with stress on the first syllable, according to Baker 1895, 177)—which is also called sestole, sestolet, sextole, or sextolet (Baker 1895, 177; Cooper 1973, 32; Latham 2002; Shedlock 1876, 62, 68, 87, 93; Stainer and Barrett 1876, 395; Taylor & 1879–89; Taylor 2001). This six-part division may be regarded either as a triplet with each note divided in half (2 + 2 + 2)—therefore with an accent on the first, third, and fifth notes—or else as an ordinary duple pattern with each note subdivided into triplets (3 + 3) and accented on both the first and fourth notes. This is indicated by the beaming in the example below.

Some authorities treat both groupings as equally valid forms (Damour, Burnett, and Elwart 1838, 80; Köhler 1858, 2:52–53; Latham 2002; Marx 1853, 114; Read 1964, 215), while others dispute this, holding the first type to be the "true" (or "real") sextuplet, and the second type to be properly a "double triplet", which should always be written and named as such (Kastner 1838, 94; Riemann 1884, 134–35; Taylor & 1879–89, 3:478). Some go so far as to call the latter, when written with a numeral 6, a "false" sextuplet (Baker 1895, 177; Lobe 1881, 36; Shedlock 1876, 62). Still others, on the contrary, define the sextuplet precisely and solely as the double triplet (Stainer and Barrett 1876, 395; Sembos 2006, 86), and a few more, while accepting the distinction, contend that the true sextuplet has no internal subdivisions—only the first note of the group should be accented (Riemann 1884, 134; Taylor & 1879–89, 3:478; Taylor 2001).

#### Compound meter

In compound meter, even-numbered tuplets can indicate that a note value is changed in relation to the dotted version of the next higher note value. Thus, two duplet eighth notes (most often used in 6
8
meter) take the time normally totaled by three eighth notes, equal to a dotted quarter note. Four quadruplet (or quartole) eighth notes would also equal a dotted quarter note. The duplet eighth note is thus exactly the same duration as a dotted eighth note, but the duplet notation is far more common in compound meters (Jones 1974, 20).

A duplet in compound time is more often written as 2:3 (a dotted quarter note split into two duplet eighth notes) than 2:​1 12 (a dotted quarter note split into two duplet quarter notes), even though the former is inconsistent with a quadruplet also being written as 4:3 (a dotted quarter note split into two quadruplet eighth notes) (Anon. 1997–2000).

### Nested tuplets

On occasion, tuplets are used "inside" tuplets. These are referred to as nested tuplets.

## Counting

Tuplets can produce rhythms such as the hemiola or may be used as polyrhythms when played against the regular duration. They are extrametric rhythmic units. The example below shows sextuplets in quintuplet time.

Tuplets may be counted, most often at extremely slow tempos, using the least common multiple (LCM) between the original and tuplet divisions. For example, with a 3-against-2 tuplet (triplets) the LCM is 6. Since 6 ÷ 2 = 3 and 6 ÷ 3 = 2 the quarter notes fall every three counts (overlined) and the triplets every two (underlined):

 1 2 3 4 5 6

This is fairly easily brought up to tempo, and depending on the music may be counted in tempo, while 7-against-4, having an LCM of 28, may be counted at extremely slow tempos but must be played intuitively ("felt out") at tempo:

 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28

To play a half-note (minim) triplet accurately in a bar of 4
4
, count eighth-note triplets and tie them together in groups of four

With a stress on each target note, one would count:

 1 - 2 - 3 1 - 2 - 3 1 - 2 - 3 1 - 2 - 3 1

The same principle can be applied to quintuplets, septuplets, and so on.

In drumming, "quadruplet" refers to one group of three sixteenth-note triplets "with an extra [non-tuplet eighth] note added on to the end", thus filling one beat in 4
4
time (Peckman 2007, 127–28), with four notes of unequal value. Shown below is a "quadruplet" with each note on a different drum in a kit used as a fill (Peckman 2007, 129)

## References

• Anon. 1997–2000. "Music Notation Questions Answered". Graphire Corporation, Graphire.com (Accessed 10 May 2013).
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• Baker, Theodore, Nicolas Slonimsky, and Laura Dine Kuhn. 1995. Schirmer Pronouncing Pocket Manual of Musical Terms. New York: Schirmer Books. ISBN 0-8256-7223-6.
• Cooper, Paul. 1973. Perspectives in Music Theory: An Historical-Analytical Approach. New York: Dodd, Mead. ISBN 0-396-06752-2.
• Cunningham, Michael G. 2007. Technique for Composers. Bloomington, Indiana: AuthorHouse. ISBN 1-4259-9618-3.
• Damour, Antoine, Aimable Burnett, and Élie Elwart. 1838. Études élémentaires de la musique: depuis ses premières notions jusqu'à celles de la composition: divisées en trois parties: Connaissances préliminaires. Méthode de chant. Méthode d’harmonie. Paris: Bureau des Études élémentaires de la musique.
• Donato, Anthony. 1963. Preparing Music Manuscript. Englewood Cliffs, NJ: Prentice-Hall, Inc. Unaltered reprint, Westport, Conn.: Greenwood Press, 1977 ISBN 0-8371-9587-X.
• Dunstan, Ralph. 1925. A Cyclopædic Dictionary of Music. 4th ed. London: J. Curwen & Sons, 1925. Reprint. New York: DaCapo Press, 1973.
• Gehrkens, Karl W. 1921. Music Notation and Terminology. New York and Chicago: The A. S. Barnes Company.
• Hubbard, William Lines. 1924. Musical Dictionary, revised and enlarged edition. Toledo: Squire Cooley Co. Reprinted as The American History and Encyclopedia of Music. Whitefish, Montana: Kessinger Publishing, 2005. ISBN 1-4179-0200-0.
• Humphries, Carl. 2002. The Piano Handbook. San Francisco, CA: Backbeat Books; London: Hi Marketing. ISBN 0-87930-727-7.
• Jones, George Thaddeus. 1974. Music Theory: The Fundamental Concepts of Tonal Music Including Notation, Terminology, and Harmony. New York, Hagerstown, San Francisco, London: Barnes & Noble Books. ISBN 0-06-460137-4.
• Lobe, Johann Christian. 1881. Catechism of Music, new and improved edition, edited and revised from the 20th German edition by John Henry Cornell, translated by Fanny Raymond Ritter. New York: G. Schirmer. (First edition of English translation by Fanny Raymond Ritter. New York: J. Schuberth 1867.)
• Kennedy, Michael. 1994. "Irregular Rhythmic Groupings. (Duplets, Triplets, Quadruplets)". Oxford Dictionary of Music, second edition, associate editor, Joyce Bourne. Oxford and New York: Oxford University Press. ISBN 0-19-869162-9.
• Köhler, Louis. 1858. Systematische Lehrmethode für Clavierspiel und Musik: Theoretisch und praktisch, 2 vols. Leipzig: Breitkopf und Härtel.
• Latham, Alison (ed.). 2002. "Sextuplet [sextolet]". The Oxford Companion to Music. Oxford and New York: Oxford University Press. ISBN 0-19-866212-2.
• Marx, Adolf Bernhard. 1853. Universal School of Music, translated from the fifth edition of the original German by August Heinrich Wehrhan. London.
• Peckman, Jon. 2007. Picture Yourself Drumming: Step-by-Step Instruction for Drum Kit Setup, Reading Music, Learning from the Pros, and More. Boston, MA: Thomson Course Technology. ISBN 1-59863-330-9.
• Read, Gardner. 1964. Music Notation: A Manual of Modern Practice. Boston: Alleyn and Bacon, Inc. Second edition, Boston: Alleyn and Bacon, Inc., 1969., reprinted as A Crescendo Book, New York: Taplinger Pub. Co., 1979. ISBN 0-8008-5459-4 (cloth), ISBN 0-8008-5453-5 (pbk).
• Riemann, Hugo. 1884. Musikalische Dynamik und Agogik: Lehrbuch der musikalischen Phrasirung auf Grund einer Revision der Lehre von der musikalischen Metrik und Rhythmik. Hamburg: D. Rahter; St. Petersburg: A. Büttner; Leipzig: Fr. Kistnet.
• Schonbrun, Marc. 2007. The Everything Music Theory Book: A Complete Guide to Taking Your Understanding of Music to the Next Level. The Everything Series. Avon, Mass.: Adams Media. ISBN 1-59337-652-9.
• Sembos, Evangelos C. 2006. Principles of Music Theory: A Practical Guide, second edition. Morrisville, NC: Lulu Press, Inc. ISBN 1-4303-0955-5.
• Shedlock, Emma L. 1876. A Trip to Music-Land: An Allegorical and Pictorial Exposition of the Elements of Music. London, Glasgow, and Edinburgh: Blackie & Son.
• Stainer, John, and William Alexander Barrett. 1876. A Dictionary of Musical Terms. London: Novello, Ewer and Co.
• Taylor, Franklin. 1879–89. "Sextolet". A Dictionary of Music and Musicians (A.D. 1450–1883) by Eminent Writers, English and Foreign, 4 vols, edited by Sir George Grove, 3:478. London: Macmillan and Co.
• Taylor, Franklin. 2001. "Sextolet, Sextuplet." The New Grove Dictionary of Music and Musicians, second edition, edited by Stanley Sadie and John Tyrrell. London: Macmillan Publishers.
3 (disambiguation)

3 is a number, numeral, and glyph.

3, three, or III may also refer to:

3 BC, the third year before the AD era

March, the third month

Computable real function

In mathematical logic, specifically computability theory, a function ${\displaystyle f\colon \mathbb {R} \to \mathbb {R} }$ is sequentially computable if, for every computable sequence ${\displaystyle \{x_{i}\}_{i=1}^{\infty }}$ of real numbers, the sequence ${\displaystyle \{f(x_{i})\}_{i=1}^{\infty }}$ is also computable.

A function ${\displaystyle f\colon \mathbb {R} \to \mathbb {R} }$ is effectively uniformly continuous if there exists a recursive function ${\displaystyle d\colon \mathbb {N} \to \mathbb {N} }$ such that, if

${\displaystyle |x-y|<{1 \over d(n)}}$

then

${\displaystyle |f(x)-f(y)|<{1 \over n}}$

A real function is computable if it is both sequentially computable and effectively uniformly continuous,

These definitions can be generalized to functions of more than one variable or functions only defined on a subset of ${\displaystyle \mathbb {R} ^{n}.}$ The generalizations of the latter two need not be restated. A suitable generalization of the first definition is:

Let ${\displaystyle D}$ be a subset of ${\displaystyle \mathbb {R} ^{n}.}$ A function ${\displaystyle f\colon D\to \mathbb {R} }$ is sequentially computable if, for every ${\displaystyle n}$-tuplet ${\displaystyle \left(\{x_{i\,1}\}_{i=1}^{\infty },\ldots \{x_{i\,n}\}_{i=1}^{\infty }\right)}$ of computable sequences of real numbers such that

${\displaystyle (\forall i)\quad (x_{i\,1},\ldots x_{i\,n})\in D\qquad ,}$

the sequence ${\displaystyle \{f(x_{i})\}_{i=1}^{\infty }}$ is also computable.

Coordinate system

In geometry, a coordinate system is a system that uses one or more numbers, or coordinates, to uniquely determine the position of the points or other geometric elements on a manifold such as Euclidean space. The order of the coordinates is significant, and they are sometimes identified by their position in an ordered tuple and sometimes by a letter, as in "the x-coordinate". The coordinates are taken to be real numbers in elementary mathematics, but may be complex numbers or elements of a more abstract system such as a commutative ring. The use of a coordinate system allows problems in geometry to be translated into problems about numbers and vice versa; this is the basis of analytic geometry.

Dotted note

In Western musical notation, a dotted note is a note with a small dot written after it. In modern practice, the first dot increases the duration of the basic note by half of its original value. This means that a dotted note is equivalent to writing the basic note tied to a note of half the value – for instance, a dotted half note is equivalent to a half note tied to a quarter note. Subsequent dots add progressively halved value, as shown in the example to the right. Though theoretically possible, a note with more than three dots is highly uncommon; only quadruple dots have been attested.The use of a dot for augmentation of a note dates back at least to the 10th century, although the exact amount of augmentation is disputed; see Neume.

A rhythm using longer notes alternating with shorter notes (whether notated with dots or not) is sometimes called a dotted rhythm. Historical examples of music performance styles using dotted rhythms include notes inégales and swing. The precise performance of dotted rhythms can be a complex issue. Even in notation that includes dots, their performed values may be longer than the dot mathematically indicates, a practice known as over-dotting.

Duration (music)

In music, duration is an amount of time or how long or short a note, phrase, section, or composition lasts. "Duration is the length of time a pitch, or tone, is sounded." A note may last less than a second, while a symphony may last more than an hour. One of the fundamental features of rhythm, or encompassing rhythm, duration is also central to meter and musical form. Release plays an important part in determining the timbre of a musical instrument and is affected by articulation.

The concept of duration can be further broken down into those of beat and meter, where beat is seen as (usually, but certainly not always) a 'constant', and rhythm being longer, shorter or the same length as the beat. Pitch may even be considered a part of duration. In serial music the beginning of a note may be considered, or its duration may be (for example, is a 6 the note which begins at the sixth beat, or which lasts six beats?).

Durations, and their beginnings and endings, may be described as long, short, or taking a specific amount of time. Often duration is described according to terms borrowed from descriptions of pitch. As such, the duration complement is the amount of different durations used, the duration scale is an ordering (scale) of those durations from shortest to longest, the duration range is the difference in length between the shortest and longest, and the duration hierarchy is an ordering of those durations based on frequency of use.Durational patterns are the foreground details projected against a background metric structure, which includes meter, tempo, and all rhythmic aspects which produce temporal regularity or structure. Duration patterns may be divided into rhythmic units and rhythmic gestures (Winold, 1975, chap. 3). But they may also be described using terms borrowed from the metrical feet of poetry: iamb (weak–strong), anapest (weak–weak–strong), trochee (strong–weak), dactyl (strong–weak–weak), and amphibrach (weak–strong–weak), which may overlap to explain ambiguity.

Giovanni Battista Pescetti

Giovanni Battista Pescetti (c. 1704 – 20 March 1766) was an organist and composer. Born in Venice around 1704, he studied under Antonio Lotti for some time. Having spent some time writing operas in and around Venice, he left for London in 1736, becoming director of the Opera of the Nobility in 1737.

After having to leave London when hostility arose against Catholic Italians, he returned to Venice in 1745 and became Second Organist at St Mark's Basilica. He died in Venice.

Pescetti was active as a teacher of composition in Venice; his most famous students being Josef Mysliveček (1737–1781) and Antonio Salieri (1750–1825). Although his output consists mainly of operatic works, a considerable amount of Pescetti's compositions were written for harpsichord, some intended to be performed on pipe organ, including his Six Sonatas, composed around 1756. Today, his keyboard sonatas are generally performed on a modern piano, though various recordings exist that use the intended organ.

Groove (music)

In music, groove is the sense of propulsive rhythmic "feel" or sense of "swing". In jazz, it can be felt as a persistently repeated pattern. It can be created by the interaction of the music played by a band's rhythm section (e.g. drums, electric bass or double bass, guitar, and keyboards). Groove is a significant feature of popular music, and can be found in many genres, including salsa, funk, rock, fusion, and soul.

From a broader ethnomusicological perspective, groove has been described as "an unspecifiable but ordered sense of something that is sustained in a distinctive, regular and attractive way, working to draw the listener in." Musicologists and other scholars have analyzed the concept of "groove" since around the 1990s. They have argued that a "groove" is an "understanding of rhythmic patterning" or "feel" and "an intuitive sense" of "a cycle in motion" that emerges from "carefully aligned concurrent rhythmic patterns" that stimulates dancing or foot-tapping on the part of listeners. The concept can be linked to the sorts of ostinatos that generally accompany fusions and dance musics of African derivation (e.g. African-American, Afro-Cuban, Afro-Brazilian, etc.).The term is often applied to musical performances that make one want to move or dance, and enjoyably "groove" (a word that also has sexual connotations). The expression "in the groove" (as in the jazz standard) was widely used from around 1936 to 1945, at the height of the swing era, to describe top-notch jazz performances. In the 1940s and 1950s, groove commonly came to denote musical "routine, preference, style, [or] source of pleasure."

Index of music articles

Metric modulation

In music, metric modulation is a change in pulse rate (tempo) and/or pulse grouping (subdivision) which is derived from a note value or grouping heard before the change. Examples of metric modulation may include changes in time signature across an unchanging tempo, but the concept applies more specifically to shifts from one time signature/tempo (metre) to another, wherein a note value from the first is made equivalent to a note value in the second, like a pivot or bridge. The term "modulation" invokes the analogous and more familiar term in analyses of tonal harmony, wherein a pitch or pitch interval serves as a bridge between two keys. In both terms, the pivoting value functions differently before and after the change, but sounds the same, and acts as an audible common element between them. Metric modulation was first described by Richard Franko Goldman (1951) while reviewing the Cello Sonata of Elliott Carter, who prefers to call it tempo modulation (Schiff 1998, 23). Another synonymous term is proportional tempi (Mead 2007, 65).

A technique in which a rhythmic pattern is superposed on another, heterometrically, and then supersedes it and becomes the basic metre. Usually, such time signatures are mutually prime, e.g., 44 and 38, and so have no common divisors. Thus the change of the basic metre decisively alters the numerical content of the beat, but the minimal denominator (18 when 44 changes to 38; 116 when, e.g., 58 changes to 716, etc.) remains constant in duration. (Nicolas Slonimsky 2000)

Multiple birth

A multiple birth is the culmination of one multiple pregnancy, wherein the mother delivers two or more offspring. A term most applicable to placental species, multiple births occur in most kinds of mammals, with varying frequencies. Such births are often named according to the number of offspring, as in twins and triplets. In non-humans, the whole group may also be referred to as a litter, and multiple births may be more common than single births. Multiple births in humans are the exception and can be exceptionally rare in the largest mammals.

Each fertilized egg (zygote) may produce a single embryo, or it may split into two or more embryos, each carrying the same genetic material. Fetuses resulting from different zygotes are called fraternal and share only 50% of their genetic material, as ordinary full siblings from separate births do. Fetuses resulting from the same zygote share 100% of their genetic material and hence are called identical. Identical twins are always the same sex, except in cases of Klinefelter syndrome (also known as XXY syndrome and 47,XXY syndrome).

A multiple pregnancy may be the result of the fertilization of a single egg that then splits to create identical fetuses, or it may be the result of the fertilization of multiple eggs that create fraternal fetuses, or it may be a combination of these factors. A multiple pregnancy from a single zygote is called monozygotic, from two zygotes is called dizygotic, or from three or more zygotes is called polyzygotic.

Similarly, the siblings themselves from a multiple birth may be referred to as monozygotic if they are identical or as polyzygotic if they are fraternal.

Petrus de Cruce

Petrus de Cruce (also Pierre de la Croix) was active as a cleric, composer and theorist in the late part of the 13th century. His main contribution was to the notational system.

Point (geometry)

In modern mathematics, a point refers usually to an element of some set called a space.

More specifically, in Euclidean geometry, a point is a primitive notion upon which the geometry is built, meaning that a point cannot be defined in terms of previously defined objects. That is, a point is defined only by some properties, called axioms, that it must satisfy. In particular, the geometric points do not have any length, area, volume or any other dimensional attribute. A common interpretation is that the concept of a point is meant to capture the notion of a unique location in Euclidean space.

Polytempo

The term polytempo or polytempic is used to describe music in which two or more tempi occur simultaneously.In the Western world, the practice of polytempic music has its roots in the music theory of Henry Cowell, and the early practices of Charles Ives. Later on, composer Elliott Carter, in the fifties, began polymetric experiments in his string quartets that inevitably amounted to polytempic behavior by nature of several competing lines at different surface speeds. At around the same time, composer Henry Brant expanded on Ives's The Unanswered Question to create a spatial music in which entire ensembles, separated by vast distances, play in distinct simultaneous tempi.

Some types of African drumming exhibit this phenomenon.Today's composers are employing polytempi as a compositional strategy to create total and complete independence of line in polyphonic music. Composers such as Conlon Nancarrow, David A. Jaffe, Evgeni Kostitsyn, Kyle Gann, Kenneth Jonsson, John Arrigo-Nelson, Brian Ferneyhough, Karlheinz Stockhausen, Frank Zappa, and Peter Thoegersen have used various methods in achieving polytempic effects in their music.

Polytempic music also harkens to the rhythmic practices of some Renaissance and medieval composers (see hemiola).

Prime k-tuple

In number theory, a prime k-tuple is a finite collection of values representing a repeatable pattern of differences between prime numbers. For a k-tuple (a, b, ...), the positions where the k-tuple matches a pattern in the prime numbers are given by the set of integers n such that all of the values (n + a, n + b, ...) are prime. Typically the first value in the k-tuple is 0 and the rest are distinct positive even numbers.

Region of interest

A region of interest (often abbreviated ROI), are samples within a data set identified for a particular purpose. The concept of a ROI is commonly used in many application areas. For example, in medical imaging, the boundaries of a tumor may be defined on an image or in a volume, for the purpose of measuring its size. The endocardial border may be defined on an image, perhaps during different phases of the cardiac cycle, for example, end-systole and end-diastole, for the purpose of assessing cardiac function. In geographical information systems (GIS), a ROI can be taken literally as a polygonal selection from a 2D map. In computer vision and optical character recognition, the ROI defines the borders of an object under consideration. In many applications, symbolic (textual) labels are added to a ROI, to describe its content in a compact manner. Within a ROI may lie individual points of interest (POIs).

Tenuto

Tenuto (Italian, past participle of tenere, "to hold") is a direction used in musical notation. The precise meaning of tenuto is contextual: it can mean either hold the note in question its full length (or longer, with slight rubato), or play the note slightly louder. In other words, the tenuto mark may alter either the dynamic or the duration of a note. Either way, the marking indicates that a note should receive emphasis.Tenuto is one of the earliest directions to appear in music notation. Notker of St. Gall (c. 840–912) discusses the use of the letter t in plainsong notation as meaning trahere vel tenere debere in one of his letters.

The mark's meaning may be affected when it appears in conjunction with other articulations. When it appears with a staccato dot, it means non legato or detached. When it appears with an accent mark, because the accent indicates dynamics, the tenuto means full or extra duration.

The Black Page

"The Black Page #1" is a piece by American composer Frank Zappa known for being extraordinarily difficult to play. Originally written for the drum kit and melodic percussion (as "The Black Page Drum Solo"), the piece was later rearranged in several versions, including the disco "easy teenage New York version" (commonly referred to as "The Black Page #2") and a so-called "new-age version", among others.

Drummer Terry Bozzio said of the piece:

He wrote it, because we had done this 40-piece orchestra gig together and he was always hearing the studio musicians in LA, that he was musing on that, talking about the fear of going into sessions some morning and being faced with "the black page". So he decided to write his "Black Page". Then he gave it to me, and I could play parts of it right away. But it wasn't a pressure thing, it just sat on my music stand and for about 15 minutes every day for 2 weeks, before we would rehearse, I would work on it. And after 2 weeks I had it together and I played it for him. And he said, "Great!", took it home, wrote the melody and the chord changes, brought it back in. And we all started playing it.

On the double live album Zappa in New York (recorded 12/1976, released 3/1978), Zappa noted the "statistical density" of the piece. It is written in common time with extensive use of tuplets, including tuplets inside tuplets. At several points there is a crotchet triplet (sixth notes) in which each beat is counted with its own tuplet of 5, 5 and 6; at another is a minim triplet (third notes) in which the second beat is a quintuplet (actually a tuplet of 7), and the third beat is divided into tuplets of 4 and 5. The song ends with a crotchet triplet composed of tuplets of 5, 5, and 6, followed by two tuplets of 11 in the space of one. Australian drummer Chris Quinlan explained and demonstrated the polyrhythms, nested polyrhythms (quintuplets and sextuplets played within a 3:2 polyrhythm), polymeters (5/4 phrases played over 4/4 time signature) and structure of "The Black Page" on "Melbourne Musos" TV Show, episode 122, aired in 1999 on Australian Community TV station, C31 Melbourne.

Zappa would re-arrange the song into "The Black Page #2" shortly after his band's mastery of the piece. This second version has a disco beat, but nevertheless retains nearly every metric complexity from #1. One notable difference in this version is that the final set of tuplets feature a rhythmic change and are repeated three times to conclude the song. The 1991 live album Make a Jazz Noise Here includes a so-called "new age version", which incorporates lounge and reggae music. The 1994 album You Can't Do That on Stage Anymore, Vol. 4 featured a version from 1984 that had a ska motif. Both of these versions included guitar solos from Zappa.

In 2001, Terry Bozzio and Chad Wackerman released the video "Solos and Duets" which features "The Black Page" played as a duet between the two ex-Zappa drummers with a transcription of the piece scrolling along the bottom of the screen as it is being played .

In 2006, "The Black Page" was featured on Zappa Plays Zappa - Tour de Frank, an ambitious effort by Dweezil Zappa to bring Zappa music to the stage again, played by himself and a new band. The 2006 tour also included, as special guests, Zappa alumni singer & woodwind player Napoleon Murphy Brock, drummer Bozzio, and guitarist Steve Vai. In the 2006 shows, "The Black Page" was played first as a drum solo by Bozzio and then a second time as a guitar duet with Steve Vai.

In 2010, "The Black Page" was performed by a young British percussionist Lucy Landymore in the BBC Young Musician of the Year percussion final.

In 2014 "The Black Page" was immortalized by Terry Bozzio in the form of art he calls Rhythm & Sketch. On canvas Terry's sketch of Zappa with "The Black Page" is layered with a rhythmic pattern of light traces from Terry's drumsticks. This was a limited run of 25 canvases and sold out quickly.

The Desert Music

The Desert Music is a work of music for voices and orchestra composed by the minimalist composer Steve Reich. It is based on texts by William Carlos Williams and takes its title from his poetry anthology The Desert Music and Other Poems. The composition consists of five movements, and in both its tempi and arrangement of thematic material, the piece is in a characteristic arch form (ABCBA). It was composed in 1983.

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