1.
Musical temperament
–
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.
Scale (music)
–
In music theory, a scale is any set of musical notes ordered by fundamental frequency or pitch. A scale ordered by increasing pitch is a scale. Some scales contain different pitches when ascending than when descending, for example, the Melodic minor scale. Due to the principle of equivalence, scales are generally considered to span a single octave. A musical scale represents a division of the space into a certain number of scale steps. A measure of the width of each scale step provides a method to classify scales, based on their interval patterns, scales are put into categories including diatonic, chromatic, major, minor, and others. A specific scale is defined by its interval pattern and by a special note. The tonic of a scale is the selected as the beginning of the octave. Typically, the name of the scale specifies both its tonic and its interval pattern, for example, C major indicates a major scale with a C tonic. Scales are typically listed from low to high, most scales are octave-repeating, meaning their pattern of notes is the same in every octave. An octave-repeating scale can be represented as an arrangement of pitch classes. The distance between two notes in a scale is called a scale step. The notes of a scale are numbered by their steps from the root of the scale, for example, in a C major scale the first note is C, the second D, the third E and so on. Two notes can also be numbered in relation to other, C and E create an interval of a third. A single scale can be manifested at many different pitch levels, for example, a C major scale can be started at C4 and ascending an octave to C5, or it could be started at C6, ascending an octave to C7. As long as all the notes can be played, the octave they take on can be altered, the pitch distances or intervals among the notes of a scale tell us more about the sound of the music than does the mere number of tones. The notes of a scale form intervals with each of the notes of the chord in combination. A 5-note scale has 10 of these intervals, a 6-note scale has 15, a 7-note scale has 21
3.
Octave
–
In music, an octave or perfect octave is the interval between one musical pitch and another with half or double its frequency. It is defined by ANSI as the unit of level when the base of the logarithm is two. The octave relationship is a phenomenon that has been referred to as the basic miracle of music. The most important musical scales are written using eight notes. For example, the C major scale is typically written C D E F G A B C, two notes separated by an octave have the same letter name and are of the same pitch class. Three commonly cited examples of melodies featuring the perfect octave as their opening interval are Singin in the Rain, Somewhere Over the Rainbow, the interval between the first and second harmonics of the harmonic series is an octave. The octave has occasionally referred to as a diapason. To emphasize that it is one of the intervals, the octave is designated P8. The octave above or below a note is sometimes abbreviated 8a or 8va, 8va bassa. For example, if one note has a frequency of 440 Hz, the note one octave above is at 880 Hz, the ratio of frequencies of two notes an octave apart is therefore 2,1. Further octaves of a note occur at 2n times the frequency of that note, such as 2,4,8,16, etc. and the reciprocal of that series. For example,55 Hz and 440 Hz are one and two away from 110 Hz because they are 1⁄2 and 4 times the frequency, respectively. After the unison, the octave is the simplest interval in music, the human ear tends to hear both notes as being essentially the same, due to closely related harmonics. Notes separated by a ring together, adding a pleasing sound to music. For this reason, notes an octave apart are given the note name in the Western system of music notation—the name of a note an octave above A is also A. The conceptualization of pitch as having two dimensions, pitch height and pitch class, inherently include octave circularity, thus all C♯s, or all 1s, in any octave are part of the same pitch class. Octave equivalency is a part of most advanced cultures, but is far from universal in primitive. The languages in which the oldest extant written documents on tuning are written, leon Crickmore recently proposed that The octave may not have been thought of as a unit in its own right, but rather by analogy like the first day of a new seven-day week
4.
Cent (music)
–
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
5.
Alexander John Ellis
–
Alexander John Ellis, FRS was an English mathematician and philologist, who also influenced the field of musicology. He changed his name from his fathers name Sharpe to his mothers maiden name Ellis in 1825 and he is buried in Kensal Green Cemetery, London. He was born Alexander John Sharpe in Hoxton, Middlesex to a wealthy family and his father James Birch Sharpe was a notable artist and physician, who was later appointed Esquire of Windlesham. His mother Ann Ellis was from a background, but it is not known how her family made its fortune. Alexanders brother James Birch Sharpe junior, died at the Battle of Inkerman and his other brother William Henry Sharpe served with the Lancashire Fusiliers after moving north with his family to Cumberland, due to military work. Alexander was educated at Shrewsbury School, Eton College and Trinity College, Cambridge, initially trained in mathematics and the classics, he became a well-known phonetician of his time. Through his work in phonetics, he became interested in vocal pitch and by extension, in musical pitch as well as speech. Ellis is noted for translating and extensively annotating Hermann von Helmholtzs On the Sensations of Tone, the second edition of this translation, published in 1885, contains an appendix which summarises Ellis own work on related matters. In his writings on musical pitch and scales, Ellis elaborates his notion and notation of cents for musical intervals and this concept became especially influential in Comparative musicology, a predecessor of ethnomusicology. Analyzing the scales of various European musical traditions, Ellis also showed that the diversity of systems cannot be explained by a single physical law. In part V of his work On Early English Pronunciation, he applied the Dialect Test across Britain and he distinguished forty-two different dialects in England and the Scottish Lowlands. This was one of the first works to apply phonetics to English speech and he was acknowledged by George Bernard Shaw as the prototype of Professor Henry Higgins of Pygmalion. He was elected in June 1864 as a Fellow of the Royal Society, elliss son Tristram James Ellis trained as an engineer, but later became a noted painter of the Middle East. Ellis developed two phonetic alphabets, phonotype, which used many new letters, and palæotype, which replaced many of these with turned letters, small caps, and italics. Two of his novel letters survived, ⟨ʃ⟩ and ⟨ʒ⟩ were passed on to Sweets Romic alphabet,1885, On the Musical Scales of Various Nations, Journal of the Society of Arts 33, p.485. (Link is to a HTML transcription 1890, English Dialects – Their Sounds and Homes Chisholm, Hugh, ed. Ellis, dictionary of National Biography,1901 supplement. London, Smith, Elder & Co. M. K
6.
Perfect fourth
–
In classical music from Western culture, a fourth is a musical interval encompassing four staff positions, and the perfect fourth is a fourth spanning five semitones. For example, the interval from C to the next F is a perfect fourth, as the note F lies five semitones above C. Diminished and augmented fourths span the same number of staff positions, the perfect fourth may be derived from the harmonic series as the interval between the third and fourth harmonics. The term perfect identifies this interval as belonging to the group of perfect intervals, so called because they are neither major nor minor, up until the late 19th century, the perfect fourth was often called by its Greek name, diatessaron. Its most common occurrence is between the fifth and upper root of all major and minor triads and their extensions. A perfect fourth in just intonation corresponds to a ratio of 4,3, or about 498 cents, while in equal temperament a perfect fourth is equal to five semitones. A helpful way to recognize a fourth is to hum the starting of the Bridal Chorus from Wagners Lohengrin. Other examples are the first two notes of the Christmas carol Hark, the Herald Angels Sing or El Cóndor Pasa, and, for a descending perfect fourth, the second and third notes of O Come All Ye Faithful. The perfect fourth is a perfect interval like the unison, octave, and perfect fifth, in common practice harmony, however, it is considered a stylistic dissonance in certain contexts, namely in two-voice textures and whenever it appears above the bass. Conventionally, adjacent strings of the bass and of the bass guitar are a perfect fourth apart when unstopped, as are all pairs. Sets of tom-tom drums are also tuned in perfect fourths. The 4,3 just perfect fourth arises in the C major scale between G and C, play The use of perfect fourths and fifths to sound in parallel with and to thicken the melodic line was prevalent in music prior to the European polyphonic music of the Middle Ages. In the 13th century, the fourth and fifth together were the concordantiae mediae after the unison and octave, in the 15th century the fourth came to be regarded as dissonant on its own, and was first classed as a dissonance by Johannes Tinctoris in his Terminorum musicae diffinitorium. In practice, however, it continued to be used as a consonance when supported by the interval of a third or fifth in a lower voice. Modern acoustic theory supports the medieval interpretation insofar as the intervals of unison, octave, the octave has the ratio of 2,1, for example the interval between a at A440 and a at 880 Hz, giving the ratio 880,440, or 2,1. The fifth has a ratio of 3,2, and its complement has the ratio of 3,4, ancient and medieval music theorists appear to have been familiar with these ratios, see for example their experiments on the Monochord. In early western polyphony, these simpler intervals were generally preferred, however, in its development between the 12th and 16th centuries, In the earliest stages, these simple intervals occur so frequently that they appear to be the favourite sound of composers. Later, the more complex intervals move gradually from the margins to the centre of musical interest
7.
Perfect fifth
–
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.
Safi al-Din al-Urmawi
–
Safi al-Din al-Urmawi al-Baghdadi or Safi al-Din Abd al-Mumin ibn Yusuf ibn al-Fakhir al-Urmawi al-Baghdadi was a renowned musician and writer on the theory of music, possibly of Persian origin. Safi al-Din Abd al-Muʾmin ibn Yusof ibn Fakhir al-Ormawi al-Baghdadi, renowned musician and writer on the theory of music, was born c.613 AH and he died in Baghdad on 28 Ṣafar 693 AH, at the age of about 80. According to the Encyclopedia of Islam The sources are silent about the origin of his family. He may have been of Persian descent Qutb al-Din al-Shirazi calls him afdal-i Īrān, based on its terminology, Al-Urmawis international modal system was intended to represent the predominant Arab and Persian local traditions. In his youth, he went to Baghdad and was educated in the Arabic language, literature, history and he made a name for himself as an excellent calligrapher and was appointed copyist at the new library built by the Abbassid caliph al-Mustaṣim. He had also studied Shafii law and comparative law at the Mustansiriyya Madrasa which opened in 631 AH. This qualified him to assume a post in al-Mustaʿsims juridical administration and, after 656 AH, to head the supervision of the foundations in Iraq until 665 AH and his musical talent made him survive the fall of Baghdad, by generously accommodating one of Hulagu’s officer. Hulagu the Mongol ruler was impressed by al-Urmawi and doubled his income relative to the Abbassid era. His musical career, however, seems to have been supported mainly by the Juwayni family, especially by Shams al-Din Muḥammad, after the demise of his patrons, he fell into oblivion and poverty. He was placed under arrest on account of a debt of 300 dinars and he died in the Shafii Madrasat al-Khalil in Baghdad. As a composer, al-Urmawi cultivated the vocal forms of ṣawt, qawl, in the anonymous Persian Kanz al-tuḥaf of the 8th century AH, he is also credited with the invention of two stringed musical instruments, the nuzha and the mughnī. Al-Urmawis most important work are two books in Arabic Language on music theory, the Kitab al-Adwār and Risālah al-Sharafiyyah fi l-nisab al-taʾlifiyyah, the first book was written while he still worked in the library of al-Mustasim. The Abbassid caliph was well known for his addiction to music, the second book was dedicated to Sharaf al-Din Harun Juvayni who ordered him to compile it. The Kitab al-Adwār is the first extant work on music theory after the writings on music of Avicenna. It also contains depictions of contemporary musical metres, and the use of letters and numbers for the notation of melodies. It is the first time that occurs in history, making it a unique work of greatest value. Al-Urmawis international modal system was intended to represent the predominant Arab, by its conciseness it became the most popular and influential book on music for centuries. No other Arabic music treatise was so often copied, commented upon, the Kitab al-Adwār was conceived as a compendium of the standard musical knowledge of its time
9.
Quarter tone
–
A quarter tone play, is a pitch halfway between the usual notes of a chromatic scale or an interval about half as wide as a semitone, which is half a whole tone. Many composers are known for having written music including quarter tones or the quarter-tone scale —proposed by 19th-century music theorists Heinrich Richter in 1823, george Tucker, Ivan Alexandrovich Wyschnegradsky, and Iannis Xenakis. The term quarter tone can refer to a number of different intervals, for example, some 17th- and 18th-century theorists used the term to describe the distance between a sharp and enharmonically distinct flat in mean-tone temperaments. In the quarter tone scale, also called 24 tone equal temperament, the tone is 50 cents, or a frequency ratio of 24√2 or approximately 1.0293. In this scale the quarter tone is the smallest step, a semitone is thus made of two steps, and three steps make a three-quarter tone play or neutral second, half of a minor third. The 8-TET scale is composed of three-quarter tones, in just intonation the quarter tone can be represented by the septimal quarter tone,36,35, or by the undecimal quarter tone,33,32, approximately half the semitone of 16,15 or 25,24. The ratio of 36,35 is only 1.23 cents narrower than a 24-TET quarter tone and this just ratio is also the difference between a minor third and septimal minor third. Quarter tones and intervals close to also occur in a number of other equally tempered tuning systems. 22-TET contains an interval of 54.55 cents, slightly wider than a quarter-tone, whereas 53-TET has an interval of 45.28 cents, 72-TET also has equally tempered quarter-tones, and indeed contains 3 quarter tone scales, since 72 is divisible by 24. Johnston uses an upward and downward arrow to indicate a note is raised or lowered by a ratio of 33,32, because many musical instruments manufactured today are designed for the 12-tone scale, not all are usable for playing quarter tones. Sometimes special playing techniques must be used, other instruments can be used to play quarter tones when using audio signal processing effects such as pitch shifting. Pairs of conventional instruments tuned a tone apart can be used to play some quarter tone music. Indeed, quarter-tone pianos have been built, which consist essentially of two stacked one above the other in a single case, one tuned a quarter tone higher than the other. Many Persian dastgah and Arabic maqamat contain intervals of three-quarter tone size, the quarter tone scale may be primarily a theoretical construct in Arabic music. Previously, pitches of a mode were chosen from a scale consisting of seventeen tones, the Japanese multi-instrumentalist and experimental musical instrument builder Yuichi Onoue developed a 24-TET quarter tone tuning on his guitar. Several quarter-tone albums have been recorded by Jute Gyte, a one-man avantgarde black metal band from Missouri, the enharmonic genus of the Greek tetrachord consisted of a ditone or an approximate major third, and a semitone, which was divided into two microtones. Intervals matched particularly closely include the second, neutral third. The septimal minor third and septimal major third are approximated rather poorly, overall, 24-TET can be viewed as matching the 11th and 13th harmonics more closely than the 7th
10.
Regular diatonic tuning
–
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. In the ordinary diatonic scales the Ts here are tones and the Ss are semitones which are half, but in the more general regular diatonic tunings, the two steps can be of any relation within the range between T=171.43 and T=240 cents. Note that regular diatonic tunings are not limited to the notes of the scale which defines them. As the semitones get larger, eventually the steps are all the size. Regular here is understood in the sense of a mapping from Pythagorean diatonic such that all the relationships are preserved. These scales however are not included as regular diatonic tunings, all regular diatonic tunings are also Linear temperaments, i. e. Regular temperaments with two generators, the octave and the tempered fifth. One can use the fourth as an alternative generator. Another moment of symmetry with two interval sizes. g, from E to F between notes five steps apart in the cycle. Here, the seven equal system is the limit as the chromatic semitone tends to zero, however, his range of recognizability is more restrictive than regular diatonic tuning. For instance, he requires the diatonic semitone to be at least 25 cents in size and this includes 1/3 comma meantone - achieves pure minor thirds 6/5, fifth is 694.786 cents. So for instance, a 1/8 schizma temperament will achieve a pure 8/5 in a chain of eight fifths. 53 equal temperament achieves an approximation to Schismatic temperament. At around 703. 4-705.0 cents, with fifths mildly tempered in the wide direction, at 705.882 cents, or tempered in the wide direction by 3.929 cents, the result is the diatonic scale in 17 tone equal. Beyond this point, the major and minor thirds approximate simple ratios of numbers with prime factors 2-3-7, such as the 9/7 or septimal major third. At the same time, the regular tones more and more closely approximate a large 8/7 tone and this septimal range extends out to around 711.111 cents or 27-ed2, or a bit further. This combination is necessary and sufficient to define a set of relationships among tonal intervals that is constant across the syntonic temperaments tuning range. Hence, it defines a constant mapping -- all across the tuning continuum -- between the notes at these tonal intervals, and the corresponding partials of a pseudo-harmonic timbre. Hence, the relationship between the temperament and its note-aligned timbres can be seen as a generalization of the special relationship between Just Intonation and the Harmonic Series