1.
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
2.
Isaac Newton
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His book Philosophiæ Naturalis Principia Mathematica, first published in 1687, laid the foundations of classical mechanics. Newton also made contributions to optics, and he shares credit with Gottfried Wilhelm Leibniz for developing the infinitesimal calculus. Newtons Principia formulated the laws of motion and universal gravitation that dominated scientists view of the universe for the next three centuries. Newtons work on light was collected in his influential book Opticks. He also formulated a law of cooling, made the first theoretical calculation of the speed of sound. Newton was a fellow of Trinity College and the second Lucasian Professor of Mathematics at the University of Cambridge, politically and personally tied to the Whig party, Newton served two brief terms as Member of Parliament for the University of Cambridge, in 1689–90 and 1701–02. He was knighted by Queen Anne in 1705 and he spent the last three decades of his life in London, serving as Warden and Master of the Royal Mint and his father, also named Isaac Newton, had died three months before. Born prematurely, he was a child, his mother Hannah Ayscough reportedly said that he could have fit inside a quart mug. When Newton was three, his mother remarried and went to live with her new husband, the Reverend Barnabas Smith, leaving her son in the care of his maternal grandmother, Newtons mother had three children from her second marriage. From the age of twelve until he was seventeen, Newton was educated at The Kings School, Grantham which taught Latin and Greek. He was removed from school, and by October 1659, he was to be found at Woolsthorpe-by-Colsterworth, Henry Stokes, master at the Kings School, persuaded his mother to send him back to school so that he might complete his education. Motivated partly by a desire for revenge against a bully, he became the top-ranked student. In June 1661, he was admitted to Trinity College, Cambridge and he started as a subsizar—paying his way by performing valets duties—until he was awarded a scholarship in 1664, which guaranteed him four more years until he would get his M. A. He set down in his notebook a series of Quaestiones about mechanical philosophy as he found it, in 1665, he discovered the generalised binomial theorem and began to develop a mathematical theory that later became calculus. Soon after Newton had obtained his B. A. degree in August 1665, in April 1667, he returned to Cambridge and in October was elected as a fellow of Trinity. Fellows were required to become ordained priests, although this was not enforced in the restoration years, however, by 1675 the issue could not be avoided and by then his unconventional views stood in the way. Nevertheless, Newton managed to avoid it by means of a special permission from Charles II. A and he was elected a Fellow of the Royal Society in 1672. Newtons work has been said to distinctly advance every branch of mathematics then studied and his work on the subject usually referred to as fluxions or calculus, seen in a manuscript of October 1666, is now published among Newtons mathematical papers
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.
Major second
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In Western music theory, a major second is a second spanning two semitones. A second is an interval encompassing two adjacent staff positions. For example, the interval from C to D is a major second, diminished, minor and augmented seconds are notated on adjacent staff positions as well, but consist of a different number of semitones. The major second is the interval that occurs between the first and second degrees of a scale, the tonic and the supertonic. On a musical keyboard, a second is the interval between two keys separated by one key, counting white and black keys alike. On a guitar string, it is the interval separated by two frets, in moveable-do solfège, it is the interval between do and re. It is considered a step, as opposed to larger intervals called skips. Intervals composed of two semitones, such as the second and the diminished third, are also called tones, whole tones. One source says step is chiefly US. The preferred usage has been argued since the 19th century, Mr. M. in teaching the Diatonic scale calls a tone a step, and a semitone a half step, now, who ever heard of a step in music, or in sound. The largest ones are called major tones or greater tones, the smallest are called minor tones or lesser tones and their size differs by exactly one syntonic comma. Some equal temperaments, such as 15-ET and 22-ET, also distinguish between a greater and a lesser tone, the major second was historically considered one of the most dissonant intervals of the diatonic scale, although much 20th-century music saw it reimagined as a consonance. It is common in different musical systems, including Arabic music, Turkish music and music of the Balkans. It occurs in both diatonic and pentatonic scales, listen to a major second in equal temperament. Here, middle C is followed by D, which is a tone 200 cents sharper than C, the difference in size between a major tone and a minor tone is equal to one syntonic comma. The major tone is the 9,8 interval play, and it is an approximation thereof in other tuning systems, the major tone may be derived from the harmonic series as the interval between the eighth and ninth harmonics. The minor tone may be derived from the series as the interval between the ninth and tenth harmonics. The 10,9 minor tone arises in the C major scale between D and e and G and A, and is a sharper dissonance than 9,8
5.
Tritone
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In music theory, the tritone is strictly defined as a musical interval composed of three adjacent whole tones. For instance, the interval from F up to the B above it is a tritone as it can be decomposed into the three adjacent whole tones F–G, G–A, and A–B, according to this definition, within a diatonic scale there is only one tritone for each octave. For instance, the above-mentioned interval F–B is the only tritone formed from the notes of the C major scale, a tritone is also commonly defined as an interval spanning six semitones. According to this definition, a diatonic scale contains two tritones for each octave, for instance, the above-mentioned C major scale contains the tritones F–B and B–F. In twelve-equal temperament, the tritone divides the octave exactly in half, in classical music, the tritone is a harmonic and melodic dissonance and is important in the study of musical harmony. The tritone can be used to avoid traditional tonality, Any tendency for a tonality to emerge may be avoided by introducing a note three whole tones distant from the key note of that tonality. Contrarily, the found in the dominant seventh chord helps establish the tonality of a composition. These contrasting uses exhibit the flexibility, ubiquity, and distinctness of the tritone in music, the condition of having tritones is called tritonia, that of having no tritones is atritonia. A musical scale or chord containing tritones is called tritonic, one without tritones is atritonic, since a chromatic scale is formed by 12 pitches, it contains 12 distinct tritones, each starting from a different pitch and spanning six semitones. According to a complex but widely used naming convention, six of them are classified as augmented fourths, under that convention, a fourth is an interval encompassing four staff positions, while a fifth encompasses five staff positions. The augmented fourth and diminished fifth are defined as the produced by widening the perfect fourth. They both span six semitones, and they are the inverse of each other, meaning that their sum is equal to one perfect octave. In twelve-tone equal temperament, the most commonly used tuning system, in most other tuning systems, they are not equivalent, and neither is exactly equal to half an octave. Any augmented fourth can be decomposed into three whole tones, for instance, the interval F–B is an augmented fourth and can be decomposed into the three adjacent whole tones F–G, G–A, and A–B. It is not possible to decompose a diminished fifth into three adjacent whole tones, the reason is that a whole tone is a major second, and according to a rule explained elsewhere, the composition of three seconds is always a fourth. To obtain a fifth, it is necessary to add another second, for instance, using the notes of the C major scale, the diminished fifth B–F can be decomposed into the four adjacent intervals B–C, C–D, D–E, and E–F. Using the notes of a scale, B–F may be also decomposed into the four adjacent intervals B–C♯, C♯–D♯, D♯–E♯. Notice that the diminished second is formed by two enharmonically equivalent notes
6.
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
7.
Perfect fifth
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In music theory, a perfect fifth is the musical interval corresponding to a pair of pitches with a frequency ratio of 3,2, or very nearly so. In classical music from Western culture, a fifth is the interval from the first to the last of five notes in a diatonic scale. The perfect fifth spans seven semitones, while the diminished fifth spans six, for example, the interval from C to G is a perfect fifth, as the note G lies seven semitones above C. Play The perfect fifth may be derived from the series as the interval between the second and third harmonics. In a diatonic scale, the dominant note is a fifth above the tonic note. The perfect fifth is more consonant, or stable, than any other interval except the unison and it occurs above the root of all major and minor chords and their extensions. Until the late 19th century, it was referred to by one of its Greek names. Its inversion is the perfect fourth, the octave of the fifth is the twelfth. The term perfect identifies the perfect fifth as belonging to the group of perfect intervals, so called because of their simple pitch relationships and their high degree of consonance. However, when using correct enharmonic spelling, the fifth in Pythagorean tuning or meantone temperament is actually not a perfect fifth. The perfect unison has a pitch ratio 1,1, the perfect octave 2,1, the perfect fourth 4,3, within this definition, other intervals may also be called perfect, for example a perfect third or a perfect major sixth. In terms of semitones, these are equivalent to the tritone, the justly tuned pitch ratio of a perfect fifth is 3,2, meaning that the upper note makes three vibrations in the same amount of time that the lower note makes two. The just perfect fifth can be heard when a violin is tuned, if adjacent strings are adjusted to the ratio of 3,2, the result is a smooth and consonant sound. Keyboard instruments such as the piano normally use a version of the perfect fifth. In 12-tone equal temperament, the frequencies of the perfect fifth are in the ratio 7 or approximately 1.498307. An equally tempered fifth, defined as 700 cents, is about two cents narrower than a just perfect fifth, which is approximately 701.955 cents. Kepler explored musical tuning in terms of ratios, and defined a lower imperfect fifth as a 40,27 pitch ratio. His lower perfect fifth ratio of 1.4815 is much more imperfect than the equal temperament tuning of 1.498, the perfect fifth is a basic element in the construction of major and minor triads, and their extensions
8.
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
9.
Pythagorean comma
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It is equal to the frequency ratio 531441,524288, or approximately 23.46 cents, roughly a quarter of a semitone. The comma which musical temperaments often refer to tempering is the Pythagorean comma, the diminished second, in Pythagorean tuning, is defined as the difference between limma and apotome. It coincides therefore with the opposite of a Pythagorean comma, and can be viewed as a descending Pythagorean comma, equal to about −23.46 cents. As described in the introduction, the Pythagorean comma may be derived in multiple ways, Difference between two enharmonically equivalent notes in a Pythagorean scale, such as C and B♯, or D♭, Difference between Pythagorean apotome and Pythagorean limma. Difference between twelve just perfect fifths and seven octaves, Difference between three Pythagorean ditones and one octave. A just perfect fifth has a ratio of 3/2. It is used in Pythagorean tuning, together with the octave, as a yardstick to define, with respect to an initial note. Apotome and limma are the two kinds of semitones defined in Pythagorean tuning, namely, the apotome is the chromatic semitone, or augmented unison, while the limma is the diatonic semitone, or minor second. A ditone is a formed by two major tones. In Pythagorean tuning, a major tone has a size of about 203.9 cents, the 6♭ and the 6♯ scales* are not identical - even though they are on the piano keyboard - but the ♭ scales are one Pythagorean comma lower. Disregarding this difference leads to enharmonic change, * The 7♭ and 5♯, respectively 5♭ and 7♯ scales differ in the same way by one Pythagorean comma. Scales with seven accidentals are used, because the enharmonic scales with five accidentals are treated as equivalent. This interval has serious implications for the various tuning schemes of the scale, because in Western music,12 perfect fifths. Equal temperament, today the most common tuning used in the West, reconciled this by flattening each fifth by a twelfth of a Pythagorean comma. Take the just fifth to the power, then subtract seven octaves. He concludes that raising this number by six whole tones yields a value G which is larger than that yielded by raising it by an octave and this much smaller interval was later named Mercators comma
10.
Five-limit tuning
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Powers of 2 represent intervallic movements by octaves. Powers of 3 represent movements by intervals of perfect fifths, powers of 5 represent intervals of major thirds. Thus, 5-limit tunings are constructed entirely from extensions of three basic purely-tuned intervals, hence, 5-limit tuning is considered a method for obtaining just intonation. If octaves are ignored, it can be seen as a 2-dimensional lattice of pitch classes extending indefinitely in two directions, however, most tuning systems designed for acoustic instruments restrict the total number of pitches for practical reasons. It is also typical to have the number of pitches in each octave. In that case, the system can also be thought of as an octave-repeating scale of a certain number of pitches per octave. For example, if we have a 5-limit tuning system where the note is C256, then fC =256 Hz. There are several ways to define E above this C. Using thirds, one may go up one factor 5 and down two factors 2, reaching a frequency ratio of 5/4, or using fifths one may go up four factors of 3 and down six factors of 2, reaching 81/64. The prominent notes of a scale are tuned so that their frequencies form ratios of relatively small integers. Here the row headed Natural expresses all these ratios using a common list of natural numbers. In other words, the lowest occurrence of this one-octave scale shape within the series is as a subset of 8 of the 25 harmonics found in the octave from harmonics 24 to 48 inclusive. The three major thirds are correct, and three of the thirds are as expected, but D to F is a semiditone or Pythagorean minor third. As a consequence, we obtain a scale in which EGB and ACE are just minor triads, but the DFA triad doesnt have the minor shape or sound we might expect, being. Furthermore, the BDF triad is not the diminished triad that we would get by stacking two 6,5 minor thirds, being instead, Another way to do it is as follows. The three major thirds are still 5,4, and three of the thirds are still 6,5 with the fourth being 32,27. FAC and CEG still form just major triads, but GBD is now, there are other possibilities such as raising A instead of lowering D, but each adjustment breaks something else. It is evidently not possible to get all seven diatonic triads in the configuration for major, for minor and that demonstrates the need for increasing the numbers of pitches to execute the desired harmonies in tune