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 (making them narrower than a perfect fifth), 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.
Figure 1. Comparison between Pythagorean tuning (blue), equal-tempered (black), quarter-comma meantone (red) and third-comma meantone (green). For each, the common origin is arbitrarily chosen as C. The values indicated by the scale at the left are deviations in cents with respect to equal temperament.
Notable meantone temperaments
Twelve-tone equal temperament, obtained by making all semitones the same size, each equal to one-twelfth of an octave (with ratio the 12throot of 2 to one ( 12√2 : 1 ), narrows the fifths by about 2cents or 1/ 12 of a Pythagorean comma, and produces out-of-tune thirds that are only slightly better than in Pythagorean tuning. Twelve-tone equal temperament is roughly the same as 1/ 11 comma meantone tuning.
Quarter-comma meantone, which tempers each of the twelve fifths by 1 / 4 of a syntonic comma, is the best known type of meantone temperament, and the term meantone temperament is often used to refer to it specifically. Four ascending fifths (as C G D A E) tempered by 1 / 4 comma produce a perfect major third (C E), one syntonic comma narrower than the Pythagorean third that would result from four perfect fifths. Quarter-comma meantone has been practiced from the early 16thcentury to the end of the 19th. Nowadays it is standardised and extended by a division of the octave into 31equal steps.
This proceeds in the same way as Pythagorean tuning; i.e., it takes the fundamental (say, C) and goes up by six successive fifths (always adjusting by dividing by powers of 2 to remain within the octave above the fundamental), and similarly down, by six successive fifths (adjusting back to the octave by multiplying by powers of 2 ). However, instead of using the 3 / 2 ratio, which gives "perfect" fifths, this must be multiplied by the fourth root of 81 / 80 . ( 81 / 80 is the "syntonic comma": the ratio of a just major third ( 5 / 4 ) to a Pythagorean third ( 81 / 64 ).) Equivalently, one can use 4√5 instead of 3 / 2 , to produce slightly reduced fifths. This results in the interval C E being a "perfect third" ( 5 / 4 ), and the intermediate seconds (C D, D E) dividing C E uniformly, so D C and E D are equal ratios, whose square is 5 / 4 . The same is true of the major second sequences F G A and G A B. However, there is still a "comma" in meantone tuning (i.e. the F♯ and the G♭ have different pitches; they are not the same as in 12 TET). The meantone comma is actually larger than the Pythagorean one, and in the opposite pitch direction (sharp vs. flat).
In third-comma meantone, the fifths are tempered by 1 / 3 comma, and three descending fifths (such as A D G C) produce a perfect minor third (A C) one syntonic comma wider than the Pythagorean comma that would result from three perfect fifths. Third-comma meantone can be approximated extremely well by a division of the octave in 19equal steps.
The tone as a mean
The name "meantone temperament" derives from the fact that all such temperaments have only one size of the tone, between the major tone (9:8) and minor tone (10:9) of just intonation, which differ by a syntonic comma. In any regular system (i.e. with all but one of the fifths the same size)[1] the whole tone (as C D) is reached after two fifths (as C G D), while the major third is reached after four fifths (C G D A E): The mean tone therefore is exactly half of the meantone temperament's major third (in cents, or equivalently the square root in frequency).
This is one sense in which the tone is a mean; it is a median or intermediate value between 10 / 9 and 9 / 8 . Specifically, it is their geometric mean: = 1.1180340 as a frequency frequency ratio, equivalent to 193.156cents – the quarter-comma whole-tone size. However, any intermediate tone qualifies as a "mean" in the sense of being intermediate, and hence a valid choice for some meantone system.
In the case of quarter-comma meantone, in addition, where the major third is made narrower by a syntonic comma, the whole tone is consequently made half a comma narrower than the major tone of just intonation (9:8), or half a comma wider than the minor tone (10:9). This is another sense in which the whole tone in quarter-tone temperament may be considered "the" mean tone; it explains why quarter-comma meantone is often considered the exemplary meantone temperament, since it lies midway (in cents) between its possible extremes.[2]
Meantone temperaments
For a tuning to be meantone, its fifth must be between 685+5/7 and 700¢ in size. Note that 7 TET is on the flatmost extreme, 12 TET is on the sharpmost extreme, and 19 TET forms the midpoint of the spectrum.
"Meantone" can receive the following equivalent definitions:
The meantone is the geometric mean between the major whole tone (9:8 in just intonation) and the minor whole tone (10:9 in just intonation).
The meantone is the mean of its major third (for instance the square root of 5:4 in quarter-comma meantone).
The family of meantone temperaments share the common characteristic that they form a stack of identical fifths, the whole tone (major second) being the result of two fifths minus one octave, the major third of four fifths minus two octaves. Meantone temperaments are often described by the fraction of the syntonic comma by which the fifths are tempered: quarter-comma meantone, the most common type, tempers the fifths by 1 / 4 of a syntonic comma, with the result that four fifths produce a just major third, a syntonic comma lower than a Pythagorean major third; third-comma meantone tempers by 1 / 3 of a syntonic comma, three fifths producing a just major sixth (and hence a just minor 3rd), a syntonic comma lower than a Pythagorean one.
A meantone temperament is a linear temperament, distinguished by the width of its generator (the fifth, often measured in cents). Historically notable meantone temperaments, discussed below, occupy a narrow portion of this tuning continuum, with fifths ranging from approximately 695to 699cents.
Meantone temperaments can be specified in various ways: By what fraction (logarithmically) of a syntonic comma the fifth is being flattened (as above), what equal temperament has the meantone fifth in question, the width of the tempered perfect fifth in cents, or the ratio of the whole tone to the diatonic semitone. This last ratio was termed " R " by American composer, pianist and theoretician Easley Blackwood, but in effect has been in use for much longer than that.[citation needed] The ratio is useful because it gives an idea of the melodic qualities of the tuning, and if R happens to be a rational number N / D , then so is 3 R + 1 / 5 R + 2 or 3 N + D/ 5 N + 2 D, which gives an idea of the size of fifth, in terms of logarithms base2, and which immediately tells us what division of the octave we will have.[clarification needed]
If we multiply by 1200¢, we have the size of fifth in cents.
In these terms, some historically notable meantone tunings are listed below. The second and fourth column are corresponding approximations to the first column. The third column shows how close the second column's approximation is to the actual size of the fifth interval in the given meantone tuning from the first column.
Neither the just fifth nor the quarter-comma meantone fifth is a rational fraction of the octave, but several tunings exist which approximate the fifth by such an interval; these are a subset of the equal temperaments( "NTET" ), in which the octave is divided into some number (N) of equally wide intervals.
Equal temperaments useful as meantone tunings include (in order of increasing generator width) 19 TET(~ + 1 / 3 comma),50 TET(~ + 2 / 7 comma),31 TET(~ + 1 / 4 comma),43 TET(~ + 1 / 5 comma), and 55 TET(~ + 1 / 6 comma). The farther the tuning gets away from quarter-comma meantone, however, the less related the tuning is to harmonic timbres, which can be overcome by tempering the partials to match the tuning – which is possible, however, only on electronic synthesizers.[3]
Comparison of perfect fifths, major thirds, and minor thirds in various meantone tunings with just intonation
Approximation of just intervals in meantone temperaments
12-ET
19-ET
31-ET
43-ET
50-ET
55-ET
Wolf intervals
A whole number of just perfect fifths will never add up to a whole number of octaves, because log2 3 is an irrational number. If a stacked-up whole number of perfect fifths is to close with the octave, then one of the intervals that is enharmonically equivalent to a fifth must have a different width than the other fifths. For example, to make a 12note chromatic scale in Pythagorean tuning close at the octave, one of the fifth intervals must be lowered ("out-of-tune") by the Pythagorean comma; this altered fifth is called a "wolf fifth" because it sounds similar to a fifth in its interval size and seems like an out-of-tune fifth, but is actually a diminished sixth (e.g. between G♯ and E♭). Likewise, 11 of the 12perfect fourths are also in tune, but the remaining fourth is an augmented third (rather than a true fourth).
Wolf intervals are an artifact of keyboard design, and keyboard players using a key that is actually in-tune with a different pitch than intended.[4] This can be shown most easily using an isomorphic keyboard, such as that shown in Figure2.
On an isomorphic keyboard, any given musical interval has the same shape wherever it appears, except at the edges. Here's an example. On the keyboard shown in figure2, from any given note, the note that's a perfect fifth higher is always upward-and-rightward adjacent to the given note. There are no wolf intervals within the note-span of this keyboard. The problem is at the edge, on the note E♯. The note that's a perfect fifth higher than E♯ is B♯, which is not included on the keyboard shown (although it could be included in a larger keyboard, placed just to the right of A♯, hence maintaining the keyboard's consistent note-pattern). Because there is no B♯ button, when playing an E♯power chord (open fifth chord), one must choose some other note, such as C, to play instead of the missing B♯.
Even edge conditions produce wolf intervals only if the isomorphic keyboard has fewer buttons per octave than the tuning has enharmonically-distinct notes (Milne 2007 harvnb error: no target: CITEREFMilne2007 (help)). For example, the isomorphic keyboard in figure2 has 19buttons per octave, so the above-cited edge-condition, from E♯ to C, is not a wolf interval in 12tone equal temperament (TET), 17TET, or 19TET; however, it is a wolf interval in 26TET, 31TET, and 50ET. In these latter tunings, using electronic transposition could keep the current key's notes on the isomorphic keyboard's white buttons, such that these wolf intervals would very rarely be encountered in tonal music, despite modulation to exotic keys.[5]
Isomorphic keyboards expose the invariant properties of the meantone tunings of the syntonic temperament isomorphically (that is, for example, by exposing a given interval with a single consistent inter-button shape in every octave, key, and tuning) because both the isomorphic keyboard and temperament are two-dimensional (i.e., rank2) entities (Milne 2007 harvnb error: no target: CITEREFMilne2007 (help)). One-dimensional N key keyboards (where N is some number) can expose accurately the invariant properties of only a single one-dimensional tuning in NTET; hence, the one-dimensional piano-style keyboard, with 12keys per octave, can expose the invariant properties of only one tuning: 12TET.
When the perfect fifth is exactly 700cents wide (that is, tempered by approximately 1/11 of a syntonic comma, or exactly 1/12 of a Pythagorean comma) then the tuning is identical to the familiar 12tone equal temperament. This appears in the table above when R = 2:1 .
Because of the compromises (and wolf intervals) forced on meantone tunings by the one-dimensional piano-style keyboard, well temperaments and eventually equal temperament became more popular.
Using standard interval names, twelve fifths equal six octaves plus one augmented seventh; seven octaves are equal to eleven fifths plus one diminished sixth. Given this, three "minor thirds" are actually augmented seconds (for example, B♭ to C♯), and four "major thirds" are actually diminished fourths (for example, B to E♭). Several triads (like BE♭F♯ and B♭C♯F) contain both these intervals and have normal fifths.
Extended meantones
All meantone tunings fall into the valid tuning range of the syntonic temperament, so all meantone tunings are syntonic tunings. All syntonic tunings, including the meantones and the various just intonations, conceivably have an infinite number of notes in each octave, that is, seven natural notes, seven sharp notes (F♯ to B♯), seven flat notes (B♭ to F♭) (which is the limit of the orchestral harp, which allows 21pitches in an octve); then double sharp notes, double flat notes, triple sharps and flats, and so on. In fact, double sharps and flats are uncommon, but still needed; triple sharps and flats are almost never seen. In any syntonic tuning that happens to divide the octave into a small number of equally wide smallest intervals (such as 12, 19, or 31), this infinity of notes still exists, although some notes will be equivalent. For example, in 19ET, E♯ and F♭ are the same pitch; and in just intonation for C major, C♯D are within 8.1¢, and so can be tempered to be identical.
Many musical instruments are capable of very fine distinctions of pitch, such as the human voice, the trombone, unfretted strings such as the violin, and lutes with tied frets. These instruments are well-suited to the use of meantone tunings.
On the other hand, the piano keyboard has only twelve physical note-controlling devices per octave, making it poorly suited to any tunings other than 12ET. Almost all of the historic problems with the meantone temperament are caused by the attempt to map meantone's infinite number of notes per octave to a finite number of piano keys. This is, for example, the source of the "wolf fifth" discussed above. When choosing which notes to map to the piano's black keys, it is convenient to choose those notes that are common to a small number of closely related keys, but this will only work up to the edge of the octave; when wrapping around to the next octave, one must use a "wolf fifth" that is not as wide as the others, as discussed above.
The existence of the "wolf fifth" is one of the reasons why, before the introduction of well temperament, instrumental music generally stayed in a number of "safe" tonalities that did not involve the "wolf fifth" (which was generally put between G♯ and E♭).
Throughout the Renaissance and Enlightenment, theorists as varied as Nicola Vicentino, Francisco de Salinas, Fabio Colonna, Marin Mersenne, Christiaan Huygens, and Isaac Newton advocated the use of meantone tunings that were extended beyond the keyboard's twelve notes,[6][7][8] and hence have come to be called "extended" meantone tunings. These efforts required a corresponding extension of keyboard instruments to offer means of controlling more than 12notes per octave, including Vincento's Archicembalo, Mersenne's 19ET harpsichord, Colonna's 31ET sambuca, and Huygens's 31ET harpsichord.[9] Other instruments extended the keyboard by only a few notes. Some period harpsichords and organs have split D♯ / E♭ keys, such that both E major / C♯ minor (4sharps) and E♭ major / C minor (3flats) can be played with no wolf fifths. Many of those instruments also have split G♯ / A♭ keys, and a few have all the five accidental keys split.
All of these alternative instruments were "complicated" and "cumbersome" (Isacoff 2009), due to
(a) not being isomorphic, and
(b) not having a transposing mechanism,
which can significantly reduce the number of note-controlling buttons needed on an isomorphic keyboard (Plamondon 2009 harvnb error: no target: CITEREFPlamondon2009 (help)). Both of these criticisms could be addressed by electronic isomorphic keyboard instruments (such as the open-source hardware jammer keyboard), which could be simpler, less cumbersome, and more expressive than existing keyboard instruments.[10]
Use of meantone temperament
References to tuning systems that could possibly refer to meantone were published as early as 1496 (Gaffurius).[11]Aron (1523) was unmistakably discussing quarter-comma meantone,[12] however, the first mathematically precise meantone tuning descriptions are found in late 16th century treatises by Zarlino[13] and deSalinas.[14] These authors both described the 1 / 4 comma, 1 / 3 comma, and 2 / 7 comma meantone systems. Fogliano mentioned the quarter-comma system, but offered no discussion of it.[citation needed]
Of course, the quarter comma meantone system (or any other meantone system) could not have been implemented with complete accuracy until much later, since devices that could accurately measure pitch frequencies didn't exist until the mid-19th century. But tuners could use precisely the same method that "by ear" tuners have used until recently: Go up by fifths, and down by octaves, or down by fifths, and up by octaves, and "temper" the fifths so they are "slightly" smaller than just 3/ 2 ratios.
For 12tone equally-tempered tuning, they would have to be tempered by a little less than a "1/4 comma", since they must form a perfect cycle, with no comma at the end, whereas the meantone tuning still has a residual comma.
How tuners could identify a "quarter comma" reliably by ear is a bit more subtle. Since this amounts to about 0.3% of the frequency which, near middle C (~264Hz), is about one hertz, they could do it by using perfect fifths as a reference and adjusting the tempered note to produce beats at this rate. However, the frequency of the beats would have to be slightly adjusted, proportionately to the frequency of the note.
In the past, meantone temperaments were sometimes used or referred to under other names or descriptions. For example, in 1691 Huygens[15] introduced what he believed to be a new division of the octave. He referred several times, in a comparative way, to a conventional tuning arrangement, which he indicated variously as "temperament ordinaire", or "the one that everyone uses".[15] But Huygens' description of that conventional arrangement was quite precise, and is clearly what is now called quarter-comma meantone temperament.[16]
Although meantone is best known as a tuning environment associated with earlier music of the Renaissance and Baroque, there is evidence of continuous usage of meantone as a keyboard temperament well into the middle of the 19thcentury.
"The mode of tuning which prevailed before the introduction of equal temperament, is called the Meantone System. It has hardly yet died out in England, for it may still be heard on a few organs in country churches. According to Don B. Yñiguez, organist of Seville Cathedral, the meantone system is generally maintained on Spanish organs, even at the present day." — G. Grove (1890)[17]
Meantone temperament has had considerable revival for early music performance in the late 20thcentury and in newly composed works specifically demanding meantone by composers including Adams, Ligeti, and Leedy.
An equal temperament is a musical temperament or tuning system that approximates just intervals by dividing an octave into steps such that the ratio of the frequencies of any adjacent pair of notes is the same. This system yields pitch steps perceived as equal in size, due to the logarithmic changes in pitch frequency.
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.
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.
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.
In music, two written notes have enharmonic equivalence if they produce the same pitch but are notated differently. Similarly, written intervals, chords, or key signatures are considered enharmonic if they represent identical pitches that are notated differently. The term derives from Latin enharmonicus, in turn from Late Latin enarmonius, from Ancient Greek ἐναρμόνιος, from ἐν ('in') and ἁρμονία ('harmony').
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.
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, 31 equal temperament, 31-ET, which can also be abbreviated 31-TET or 31-EDO, also known as tricesimoprimal, is the tempered scale derived by dividing the octave into 31 equal-sized steps. Each step represents a frequency ratio of 31√2, or 38.71 cents.
In music, 19 equal temperament, called 19 TET, 19 EDO, 19-ED2 or 19 ET, is the tempered scale derived by dividing the octave into 19 equal steps. Each step represents a frequency ratio of 19√2, or 63.16 cents.
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.
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 greatly admired his teacher; he was one of Bach's principal proponents.
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)".
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.
References
↑ Barbour, J. Murray (1951). Tuning and Temperament: A historical survey. East Lansing, MI. p.xi.{{cite book}}: CS1 maint: location missing publisher (link)
↑ Barbour, James Murray (2004). Tuning and Temperament: A historical survey. Courier Corporation. ISBN978-0-486-43406-3.[pageneeded]
↑ Duffin, Ross W. (2007). How Equal Temperament Ruined Harmony (and why you should care). W. W. Norton & Company. ISBN978-0-393-06227-4.[pageneeded]
↑ Isacoff, Stuart (2009). Temperament: How music became a battleground for the great minds of western civilization. Knopf Doubleday Publishing Group. ISBN978-0-307-56051-3.[pageneeded]
↑ Paine, G.; Stevenson, I.; Pearce, A. (2007). The Thummer mapping project (ThuMP)(PDF). 7th International Conference on New Interfaces for Musical Expression (NIME 07). pp.70–77.
This page is based on this Wikipedia article Text is available under the CC BY-SA 4.0 license; additional terms may apply. Images, videos and audio are available under their respective licenses.
