1.
Musical temperament
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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. The development of well temperament allowed fixed-pitch instruments to play well in all of the keys. The famous Well-Tempered Clavier by Johann Sebastian Bach takes full advantage of this breakthrough, however, while unpleasant intervals were avoided, the sizes of intervals were still not consistent between keys, and so each key still had its own character. In just intonation, every interval between two pitches corresponds to a whole number ratio between their frequencies, allowing intervals varying from the highest consonance to highly dissonant, for instance,660 Hz /440 Hz constitutes a fifth, and 880 Hz /440 Hz an octave. Such intervals have a stability, or purity to their sound, if, for example, two sound signals with frequencies that vary just by 0. When a musical instrument with harmonic overtones is played, the ear hears a composite waveform that includes a fundamental frequency, the waveform of such a tone is characterized by a shape that is complex compared to a simple waveform, but remains periodic. When two tones depart from exact integer ratios, the shape waveform becomes erratic—a phenomenon that may be described as destabilization, as the composite waveform becomes more erratic, the consonance of the interval also changes. Tempering an interval involves the use of such minor adjustments to enable musical possibilities that are impractical using just intonation. Before Meantone temperament became widely used in the Renaissance, the most commonly used tuning system was Pythagorean tuning, Pythagorean tuning was a system of just intonation that tuned every note in a scale from a progression of pure perfect fifths. This was quite suitable for much of the practice until then. The major third of Pythagorean tuning differed from a just major third by an amount known as syntonic comma, with the correct amount of tempering, the syntonic comma is removed from its major thirds, making them just. This compromise, however, leaves all fifths in this system with a slight beating. Pythagorean tuning also had a problem, which meantone temperament does not solve, which is the problem of modulation. A series of 12 just fifths as in Pythagorean tuning does not return to the pitch, but rather differs by a Pythagorean comma. In meantone temperament, this effect is more pronounced. The use of 53 equal temperament provides a solution for the Pythagorean tuning, when building an instrument, this can be very impractical. Well temperament is the given to a variety of different systems of temperament that were employed to solve this problem, in which some keys are more in tune than others
2.
Limit (music)
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In music theory, limit or harmonic limit is a way of characterizing the harmony found in a piece or genre of music, or the harmonies that can be made using a particular scale. The term limit was introduced by Harry Partch, who used it to give an upper bound on the complexity of harmony, roughly speaking, the larger the limit number, the more harmonically complex and potentially dissonant will the intervals of the tuning be perceived. A scale belonging to a prime limit has a distinctive hue that makes it aurally distinguishable from scales with other limits. Harry Partch, Ivor Darreg, and Ralph David Hill are among the many microtonalists to suggest that music has been slowly evolving to employ higher and higher harmonics in its constructs, in medieval music, only chords made of octaves and perfect fifths were considered consonant. In the West, triadic harmony arose around the time of the Renaissance, the major and minor thirds of these triads invoke relationships among the first 5 harmonics. Around the turn of the 20th century, tetrads debuted as fundamental building blocks in African-American music, in conventional music theory pedagogy, these seventh chords are usually explained as chains of major and minor thirds. However, they can also be explained as coming directly from harmonics greater than 5, for example, the dominant 7th chord in 12-ET approximates 4,5,6,7, while the major 7th chord approximates 8,10,12,15. In just intonation, intervals between pitches are drawn from the rational numbers, since Partch, two distinct formulations of the limit concept have emerged, odd limit and prime limit. Odd limit and prime limit n do not include the same even when n is an odd prime. For a positive odd number n, the n-odd-limit contains all rational numbers such that the largest odd number that divides either the numerator or denominator is not greater than n. In Genesis of a Music, Harry Partch considered just intonation rationals according to the size of their numerators and denominators, since octaves correspond to factors of 2, the complexity of any interval may be measured simply by the largest odd factor in its ratio. Partchs theoretical prediction of the dissonance of intervals are very similar to those of theorists including Hermann von Helmholtz, William Sethares. An identity is each of the odd numbers below and including the limit in a tuning, for example, the identities included in 5-limit tuning are 1,3, and 5. The number 9, though not a prime, is nevertheless an identity in music, simply because it is an odd number. Partch defines identity as one of the correlatives, major or minor, in a tonality, one of the odd-number ingredients, odentity and udentity are, short for Over-Identity, and, Under-Identity, respectively. An udentity is an identity of an utonality, for a prime number n, the n-prime-limit contains all rational numbers that can be factored using primes no greater than n. In other words, it is the set of rationals with numerator and denominator both n-smooth, given a prime number p, the subset of Q + consisting of those rational numbers x whose prime factorization has the form x = p 1 α1 p 2 α2. P r ≤ p forms a subgroup of and we say that a scale or system of tuning uses p-limit tuning if all interval ratios between pitches lie in this subgroup
3.
Meantone temperament
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Meantone temperament is a musical temperament, which is a system of musical tuning. Quarter-comma meantone is the best known type of meantone temperament, Meantone can receive the following equivalent definitions, The meantone is the mean between the major whole tone and the minor whole tone, i. e. the geometric mean of 9,8 and 10,9. The meantone is the mean of the just major third, i. e. the square root of 5,4, all meantone temperaments are linear temperaments, distinguished by the width of its generator in cents, as shown in the central column of Figure 1. Historically notable meantone temperaments, discussed below, occupy a portion of this tuning continuum. In Figure 1, the tuning ranges of 5-limit, 7-limit, and 11-limit tunings are shown. This last ratio was termed R by American composer, pianist and theoretician Easley Blackwood, 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 relationship between the first two columns is exact, while that between them and the third is closely approximate. Equal temperaments useful as meantone tunings include 19-ET, 50-ET, 31-ET, 43-ET, 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 timbre to match the tuning. A whole number of just perfect fifths will never add up to a number of octaves. If a stacked-up whole number of perfect fifths is to close with the octave, then one of the fifths must have a different width than all of the others. For example, to make the 12-note chromatic scale in Pythagorean tuning close at the octave, one fifth must be out of tune by the Pythagorean comma, wolf intervals are an artifact of keyboard design. This can be shown most easily using a keyboard, such as that shown in Figure 2. On an isomorphic keyboard, any musical interval has the same shape wherever it appears. On the keyboard shown in Figure 2, from any 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 thats a perfect fifth higher than E♯ is B♯, which is not included on the keyboard shown. Because there is no B♯ button, when playing an E♯ power chord, one must choose some other note, such as C, even edge conditions produce wolf intervals only if the isomorphic keyboard has fewer buttons per octave than the tuning has enharmonically-distinct notes. When the perfect fifth is exactly 700 cents wide then the tuning is identical to the familiar 12-tone equal temperament and this appears in the table above when R =2,1. Because of the forced on meantone tunings by the one-dimensional piano-style keyboard, well temperaments
4.
Equal temperament
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An equal temperament is a musical temperament, or a system of tuning, in which the frequency interval between every pair of adjacent notes has the same ratio. In equal temperament tunings, the interval is often found by dividing some larger desired interval, often the octave. That resulting smallest interval, 1⁄12 the width of an octave, is called a semitone or half step, the standard pitch has not always been 440 but has varied and generally risen over the past few hundred years. For example, some music has been written in 19-TET and 31-TET, in Western countries, when people use the term equal temperament without qualification, they usually mean 12-TET. To avoid ambiguity between equal temperaments that divide the octave and ones that divide some other interval, the equal division of the octave. According to this system, 12-TET is called 12-EDO, 31-TET is called 31-EDO. Other instruments, such as wind, keyboard, and fretted instruments, often only approximate equal temperament. Some wind instruments that can easily and spontaneously bend their tone, most notably trombones, use tuning similar to string ensembles, the two figures frequently credited with the achievement of exact calculation of equal temperament are Zhu Zaiyu in 1584 and Simon Stevin in 1585. Kenneth Robinson attributes the invention of equal temperament to Zhu Zaiyu, Zhu Zaiyu is quoted as saying that, in a text dating from 1584, I have founded a new system. I establish one foot as the number from which the others are to be extracted, altogether one has to find the exact figures for the pitch-pipers in twelve operations. Kuttner disagrees and remarks that his claim cannot be considered correct without major qualifications, kuttner proposes that neither Zhu Zaiyu or Simon Stevin achieved equal temperament, and that neither of the two should be treated as inventors. The origin of the Chinese pentatonic scale is traditionally ascribed to the mythical Ling Lun, allegedly his writings discussed the equal division of the scale in the 27th century BC. However, evidence of the origins of writing in this period in China is limited to rudimentary inscriptions on oracle bones, an approximation for equal temperament was described by He Chengtian, a mathematician of Southern and Northern Dynasties around 400 AD. He came out with the earliest recorded approximate numerical sequence in relation to equal temperament in history,900849802758715677638601570536509.5479450, historically, there was a seven-equal temperament or hepta-equal temperament practice in Chinese tradition. Zhu Zaiyu, a prince of the Ming court, spent thirty years on research based on the equal temperament idea originally postulated by his father and he described his new pitch theory in his Fusion of Music and Calendar 乐律融通 published in 1580. An extended account is given by Joseph Needham. Similarly, after 84 divisions the length was divided by a factor of 128,84 =27 =128, according to Gene Cho, Zhu Zaiyu was the first person to solve the equal temperament problem mathematically. Matteo Ricci, a Jesuit in China recorded this work in his personal journal, in 1620, Zhus work was referenced by a European mathematician
5.
19 equal temperament
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In music,19 equal temperament, called 19-TET, 19-EDO, or 19-ET, is the tempered scale derived by dividing the octave into 19 equal steps. Each step represents a ratio of 19√2, or 63.16 cents. 19-edo is the tuning of the temperament in which the tempered perfect fifth is equal to 694.737 cents. Division of the octave into 19 equal-width steps arose naturally out of Renaissance music theory, the greater diesis, the ratio of four minor thirds to an octave was almost exactly a nineteenth of an octave. Interest in such a system goes back to the 16th century. Costeley understood and desired the circulating aspect of this tuning, in 1577, music theorist Francisco de Salinas in effect proposed it. Salinas discussed 1⁄3-comma meantone, in which the fifth is of size 694.786 cents, the fifth of 19-edo is 694.737, which is less than a twentieth of a cent narrower, imperceptible and less than tuning error. Salinas suggested tuning nineteen tones to the octave to this tuning, in the 19th century, mathematician and music theorist Wesley Woolhouse proposed it as a more practical alternative to meantone temperaments he regarded as better, such as 50-edo. The composer Joel Mandelbaum wrote his Ph. D. thesis on the properties of the 19-edo tuning, Mandelbaum and Joseph Yasser have written music with 19-edo. Easley Blackwood has stated that 19-edo makes possible a substantial enrichment of the tonal repertoire, here are the sizes of some common intervals and comparison with the ratios arising in the harmonic series, the difference column measures in cents the distance from an exact fit to these ratios. For reference, the difference from the fifth in the widely used 12 equal temperament is 1.955 cents. Beta scale Elaine Walker Levy, Kenneth J. Costeleys Chromatic Chanson, Annales Musicologues, Moyen-Age et Renaissance, Tome III, howe, Hubert S. Jr. 19-Tone Theory and Applications, Aaron Copland School of Music at Queens College. Sethares, William A. Tunings for 19 Tone Equal Tempered Guitar, Experimental Musical Instruments, Vol. VI, hair, Bailey, Morrison, Pearson and Parncutt, Rehearsing Microtonal Music, Grappling with Performance and Intonational Problems, Microtonalism. ZiaSpace. com - 19tet downloadable mp3s by Elaine Walker of Zia, the Music of Jeff Harrington, Parnasse. com. Jeff Harrington is a composer who has several pieces for piano in the 19-TET tuning. Chris Vaisvil, GR-20 Hexaphonic 19-ET Guitar Improvisation Arto Juhani Heino, Artone 19 Guitar Design, naming the 19 note scale Parvatic
6.
31 equal temperament
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In music,31 equal temperament, 31-ET, which can also be abbreviated 31-TET, 31-EDO, also known as tricesimoprimal, is the tempered scale derived by dividing the octave into 31 equal-sized steps. Play Each step represents a ratio of 31√2, or 38.71 cents. 31-ET is a good approximation of quarter-comma meantone temperament. More generally, it is a diatonic tuning in which the tempered perfect fifth is equal to 696.77 cents. In 1666, Lemme Rossi first proposed an equal temperament of this order, shortly thereafter, having discovered it independently, scientist Christiaan Huygens wrote about it also. Huygens not only realized this, he went farther and noted that 31-ET provides an excellent approximation of septimal, the composer Joel Mandelbaum used this tuning system specifically because of its good matches to the 7th and 11th partials in the harmonic series. The tuning has poor matches to both the 9,8 and 10,9 intervals, however, it has a match for the average of the two. Practically it is close to quarter-comma meantone. This tuning can be considered a meantone temperament, many chords of 31-ET are discussed in the article on septimal meantone temperament. Chords not discussed there include the neutral thirds triad, which might be written C–E–G, C–D–G or C–F–G, and the Orwell tetrad, usual chords like the major chord is rendered nicely in 31-ET because the third and the fifth are very well approximated. Also, it is possible to play chords and supermajor chords. It is also possible to render nicely the harmonic seventh chord, for example on C with C–E–G–A♯. The seventh here is different from stacking a fifth and a minor third and this difference cannot be made in 12-ET
7.
7-limit tuning
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7-limit or septimal tunings and intervals are musical instrument tunings that have a limit of seven, the largest prime factor contained in the interval ratios between pitches is seven. Thus, for example,50,49 is a 7-limit interval, for example, the greater just minor seventh,9,5 Play is a 5-limit ratio, the harmonic seventh has the ratio 7,4 and is thus a septimal interval. Similarly, the chromatic semitone,21,20, is a septimal interval as 21÷7=3. The harmonic seventh is used in the seventh chord and music. Compositions with septimal tunings include La Monte Youngs The Well-Tuned Piano,4, and Lou Harrisons Incidental Music for Corneilles Cinna. The Great Highland Bagpipe is tuned to a ten-note seven-limit scale,1,1,9,8,5,4,4,3,27,20,3,2,5,3,7,4,16,9,9,5. In the 2nd century Ptolemy described the septimal intervals, 7/4, 8/7, 7/6, 12/7, 7/5, and 10/7. Those considering 7 to be dissonant include Gioseffo Zarlino, René Descartes, Jean-Philippe Rameau, Hermann von Helmholtz, A. J. von Öttingen, Hugo Riemann, Colin Brown, the 7-limit tonality diamond, This diamond contains four identities. Similarly, the 2,3,5,7 pitch lattice contains four identities and thus 3-4 axes, laMonte Young created a lattice containing only identities 3 and 7, thus requiring only two axes, for The Well-Tuned Piano. It is possible to approximate 7-limit music using equal temperament, for example 31-ET
8.
Major third
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In classical music from Western culture, a third is a musical interval encompassing three staff positions, and the major third is a third spanning four semitones. Along with the third, the major third is one of two commonly occurring thirds. It is qualified as major because it is the larger of the two, the major third spans four semitones, the third three. The major third may be derived from the series as the interval between the fourth and fifth harmonics. The major scale is so named because of the presence of this interval between its tonic and mediant scale degrees, the major chord also takes its name from the presence of this interval built on the chords root. The older concept of a made a dissonantly wide major third with the ratio 81,64. The septimal major third is 9,7, the major third is 14,11. A helpful way to recognize a third is to hum the first two notes of Kumbaya or of When the Saints Go Marching In. A descending major third is heard at the starts of Goodnight, Ladies and Swing Low, in equal temperament three major thirds in a row are equal to an octave. This is sometimes called the circle of thirds, in just intonation, however, three 5,4 major thirds are less than an octave. For example, three 5,4 major thirds from C is B♯, the difference between this just-tuned B♯ and C, like that between G♯ and A♭, is called a diesis, about 41 cents. The major third is classed as an imperfect consonance and is considered one of the most consonant intervals after the unison, octave, perfect fifth, a diminished fourth is enharmonically equivalent to a major third. For example, B–D♯ is a third, but if the same pitches are spelled B and E♭. B–E♭ occurs in the C harmonic minor scale, the major third is used in guitar tunings. For the standard tuning, only the interval between the 3rd and 2nd strings is a third, each of the intervals between the other pairs of consecutive strings is a perfect fourth. In an alternative tuning, the tuning, each of the intervals are major thirds. Decade, compound just major third Ear training List of meantone intervals Doubling the cube, 21/3 = 3√2
9.
Harmonic seventh
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The harmonic seventh interval play, also known as the septimal minor seventh, or subminor seventh, is one with an exact 7,4 ratio. This is somewhat narrower than and is, particularly sweet, sweeter in quality than a minor seventh. The harmonic seventh may be derived from the series as the interval between the seventh harmonic and the fourth harmonic. Composer Ben Johnston uses a small 7 as an accidental to indicate a note is lowered 49 cents, thus, in C major, the seventh partial, or harmonic seventh, is notated as ♭ B with 7 written above the flat. Instead, Hagerman and Sundberg found that tuning of major and minor third intervals in barbershop lies between just and equal temperament, the harmonic seventh differs from the augmented sixth by 224/225, or about one-third of a comma. The harmonic seventh note is one third of a semitone flatter than an equal tempered minor seventh. When this flatter seventh is used, the dominant seventh chords need to resolve down a fifth is weak or non-existent and this chord is often used on the tonic and functions as a fully resolved final chord. The twenty-first harmonic is the seventh of the dominant, and would then arise in chains of secondary dominants in styles using harmonic sevenths. The Tonal Phoenix, A Study of Tonal Progression Through the Prime Numbers Three, Five, & Sundberg, J. Fundamental frequency adjustment in barbershop singing. Journal of Research in Singing,4, 3-17
10.
Quarter-comma meantone
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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, 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, later theorists Gioseffo Zarlino and Francisco de Salinas described the tuning with mathematical exactitude. In a meantone tuning, we have diatonic and chromatic semitones, in Pythagorean tuning, these correspond to the Pythagorean limma and the Pythagorean apotome, only now the apotome is larger. Two fifths up and an octave down make up a whole tone consisting of one diatonic, four fifths up and two octaves down make up a major third, consisting of two diatonic and two chromatic semitones, or in other words two whole tones. This large interval of a seventeenth contains 5 + + + =20 −3 =17 staff positions. In Pythagorean tuning, the size of a seventeenth is defined using a stack of four justly tuned fifths,4 =8116 =8016 ⋅8180 =5 ⋅8180. In quarter-comma meantone temperament, where a just major third is required, by definition, however, a seventeenth of the same size must be obtained, even in quarter-comma meantone, by stacking four fifths. Since justly tuned fifths, such as used in Pythagorean tuning, produce a slightly wider seventeenth. Notice that, in quarter-comma meantone, the seventeenth is 81/80 times narrower than in Pythagorean tuning and this difference in size, equal to about 21.506 cents, is called the syntonic comma. This implies that the fifth is a quarter of a syntonic comma narrower than the justly tuned Pythagorean fifth. The difference between two sizes is a quarter of a syntonic comma, ≈701.955 −696.578 ≈5.377 ≈21.5064 cents. In sum, this system tunes the major thirds to the just ratio of 5,4, most of the tones in the ratio √5,2. This is achieved by tuning the seventeenth a syntonic comma flatter than the Pythagorean seventeenth and it is this that gives the system its name of quarter-comma meantone. The whole chromatic scale, can be constructed by starting from a base note. This method is identical to Pythagorean tuning, except for the size of the fifth, the construction table below illustrates how the pitches of the notes are obtained with respect to D, in a D-based scale. As in Pythagorean tuning, this method generates 13 pitches, but A♭ and G♯ have almost the same frequency, the table above shows a D-based stack of fifths. Some authors prefer showing a C-based stack of fifths, ranging from A♭ to G♯, the only difference is that the construction table shows intervals from C, rather than from D
11.
Cent (music)
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The cent is a logarithmic unit of measure used for musical intervals. Twelve-tone equal temperament divides the octave into 12 semitones of 100 cents each, alexander J. Ellis based the measure on the acoustic logarithms decimal semitone system developed by Gaspard de Prony in the 1830s, at Robert Holford Macdowell Bosanquets suggestion. It has become the method of representing and comparing musical pitches. Like a decibels relation to intensity, a cent is a ratio between two close frequencies, for the ratio to remain constant over the frequency spectrum, the frequency range encompassed by a cent must be proportional to the two frequencies. An equally tempered semitone spans 100 cents by definition, an octave—two notes that have a frequency ratio of 2, 1—spans twelve semitones and therefore 1200 cents.0005777895. For example, in just intonation the major third is represented by the frequency ratio 5,4, applying the formula at the top shows that this is about 386 cents. The equivalent interval on the piano would be 400 cents. The difference,14 cents, is about a seventh of a half step, as x increases from 0 to 1⁄12, the function 2x increases almost linearly from 1.00000 to 1.05946. The exponential cent scale can therefore be accurately approximated as a linear function that is numerically correct at semitones. That is, n cents for n from 0 to 100 may be approximated as 1 +0. 0005946n instead of 2 n⁄1200. The rounded error is zero when n is 0 or 100, and is about 0.72 cents high when n is 50 and this error is well below anything humanly audible, making this piecewise linear approximation adequate for most practical purposes. It is difficult to establish how many cents are perceptible to humans, one author stated that humans can distinguish a difference in pitch of about 5–6 cents. The threshold of what is perceptible, technically known as the just noticeable difference, also varies as a function of the frequency, the amplitude and the timbre. In one study, changes in tone quality reduced student musicians ability to recognize, as out-of-tune and it has also been established that increased tonal context enables listeners to judge pitch more accurately. Free, online web sites for self-testing are available, while intervals of less than a few cents are imperceptible to the human ear in a melodic context, in harmony very small changes can cause large changes in beats and roughness of chords. When listening to pitches with vibrato, there is evidence that humans perceive the mean frequency as the center of the pitch, normal adults are able to recognize pitch differences of as small as 25 cents very reliably. Adults with amusia, however, have trouble recognizing differences of less than 100 cents, iring noticed that the Grad/Werckmeister and the schisma are nearly the same and both may be approximated by 600 steps per octave. Yasser promoted the decitone, centitone, and millitone, for example, Equal tempered perfect fifth =700 cents =175.6 savarts =583.3 millioctaves =350 centitones
12.
Syntonic comma
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The comma is referred to as a comma of Didymus because it is the amount by which Didymus corrected the Pythagorean major third to a just major third. Namely,81,64 ÷5,4 =81,80, the difference between four justly tuned perfect fifths, and two octaves plus a justly tuned major third. A just perfect fifth has a size of 3,2, a just major third has a size of 5,4, and one of them plus two octaves is equal to 5,1. The difference between these is the syntonic comma, namely,81,16 ÷5,1 =81,80. The difference between one octave plus a justly tuned minor third, and three justly tuned perfect fourths, namely,12,5 ÷64,27 =81,80. The difference between the two kinds of major second which occur in 5-limit tuning, major tone and minor tone, namely,9,8 ÷10,9 =81,80. The difference between a Pythagorean major sixth and a justly tuned or pure major sixth, namely,27,16 ÷5,3 =81,80. On a piano keyboard a stack of four fifths is exactly equal to two octaves plus a major third, in other words, starting from a C, both combinations of intervals will end up at E. Using justly tuned octaves, fifths, and thirds, however, the ratio between their frequencies, as explained above, is a syntonic comma. Pythagorean tuning uses justly tuned fifths as well, but uses the complex ratio of 81,64 for major thirds. Quarter-comma meantone uses justly tuned major thirds, but flattens each of the fifths by a quarter of a syntonic comma and this is one of the reasons why 12-tone equal temperament is currently the preferred system for tuning most musical instruments. Mathematically, by Størmers theorem,81,80 is the closest superparticular ratio possible with regular numbers as numerator and denominator. A superparticular ratio is one whose numerator is 1 greater than its denominator, such as 5,4, and a number is one whose prime factors are limited to 2,3. Thus, although smaller intervals can be described within 5-limit tunings, the syntonic comma has a crucial role in the history of music. It is the amount by which some of the notes produced in Pythagorean tuning were flattened or sharpened to produce just minor and major thirds, in Pythagorean tuning, the only highly consonant intervals were the perfect fifth and its inversion, the perfect fourth. The Pythagorean major third and minor third were dissonant, and this prevented musicians from using triads and chords, in late Middle Ages, musicians realized that by slightly tempering the pitch of some notes, the Pythagorean thirds could be made consonant. For instance, if the frequency of E is decreased by a comma, C-E. But the fifth C-G stays consonant, since only E has been flattened, since then, other tuning systems were developed, and the syntonic comma was used as a reference value to temper the perfect fifths in an entire family of them
♭
Play (help·info). Similarly, the tuning with pure 7:4 intervals has a fifth of size 696.88 cents 
, C-E♭-G♭-A, C-E♭-F♯-A or C-D♯-F♯-A. It also has a septimal augmented triad, which in various inversions can be C-E-G♯, C-E-A♭ or C-F♭-A♭. It has both a dominant seventh chord, C-E-G-B♭, and an otonal tetrad, C-E-G-A♯; the latter is familiar in common practice harmony under the name German sixth. It likewise has utonal tetrads, C-E♭-G-B
). The 11 is pure using this method if the fifth is of size 697.30 cents, very close to the fifth of 74 equal temperament. On the other hand, 13 meantone fourths up and two octaves down (e.g. C-G