Pythagorean tuning

Pythagorean tuning is based on a stack of perfect fifths, each tuned in the ratio 3:2, the next simplest ratio after 2:1, which is the ratio of an octave. The two notes A and D, for example, are tuned so that their frequencies are in the ratio 3:2 — if D is tuned to 200 Hz, then the A is tuned to 300 Hz. The E a fifth above that A is also tuned in the ratio 3:2 — with the A at 300 Hz, this puts the E at 450 Hz, 9:4 above the original D. When describing tunings, it is usual to speak of all notes as being within an octave of each other, an ...

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Pythagorean tuning, Pythagorean tuning - Method, Pythagorean tuning - Discography, Pythagorean tuning - Source

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Pythagoreanism is a term used for the esoteric and metaphysical beliefs held by Pythagoras and his followers, the Pythagoreans, who were much influenced by mathematics and probably a main inspiration source to Plato and platonism. One main subject that is part of pythagoreanism is musica universalis, the music of the spheres. Some Surat Shabda Yoga, Satgurus considered the music of the spheres to be a term synonymous with the Shabda or the Audible Life Stream in that tradition, becaus ...

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• Pythagoreanism - Influence
• Pythagoreanism - Pythagorean cosmology
• Pythagoreanism - Influences
• Pythagoreanism - Reference

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Some fixed just intonation scales and systems, such as the diatonic scale above, produce wolf intervals. The above scale allows a minor tone to occur next to a semitone which produces the awkward ratio 32:27 for C:A, and still worse, a minor tone next to a fourth giving 40:27 for E:A. Moving A down to 10/9 alleviates these difficulties but creates new ones: D:A becomes 27:20, and A:F# becomes 32:27. You can have more frets on a guitar to handle both A's, 9/8 with G and 10/9 with G so that C:A can be played as 6:5 while D:A can still b ...

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Just intonation, Just intonation - The diatonic scale in just intonation, Just intonation - Why isn't just intonation used much?, Just intonation - Singing in just intonation, Just intonation - Bagpipe tuning, Just intonation - Non-western tuning, Just intonation - Western composers who specified just intonation

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Pythagorean thought was dominated by mathematics, but it was also profoundly mystical. In the area of cosmology there is less agreement about what Pythagoras himself actually taught, but most scholars believe that the Pythagorean idea of the transmigration of the soul is too central to have been added by a later follower of Pythagoras. On the other hand it is impossible to determine the origin of the Pythagorean account of substance. It seems that the Pythagorean account begins with Anaximander's account of the ultimate substance of things a ...

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Pythagoreanism, Pythagoreanism - Influence, Pythagoreanism - Pythagorean cosmology, Pythagoreanism - Influences, Pythagoreanism - Reference

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Most composers don't specify how instruments are to be tuned, although historically most have assumed one tuning system which was common in their time; in the 20th century most composers assumed equal temperament would be used. However, a few have specified just intonation systems for some or all of their compositions, including Glenn Branca, Wendy Carlos, Stuart Dempster, Arnold Dreyblatt, Kyle Gann, Kraig Grady, Lou Harrison, Ben Johnston, Elodie Lauten, Pauline Oliveros, Harry Partch, Robert Rich, Terry Riley, James Tenney, Ernesto Rodrig ...

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Just intonation, Just intonation - The diatonic scale in just intonation, Just intonation - Why isn't just intonation used much?, Just intonation - Singing in just intonation, Just intonation - Bagpipe tuning, Just intonation - Non-western tuning, Just intonation - Western composers who specified just intonation

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Harmony is the use and study of pitch simultaneity and chords, actual or implied, in music. It is sometimes referred to as the "vertical" aspect of music, with melody being the "horizontal" aspect. Very often, harmony is a result of counterpoint or polyphony, several melodic lines or motifs being played at once, though harmony may control the counterpoint. The word harmony comes from the Greek ἁρμονία harmonía meaning "a fastening or join". The conce ...

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In Indian music, the basic unaltered diatonic scale is considered to be 1/1, 9/8, 5/4, 4/3, 3/2, 27/16, 15/8, 2/1. This would appear problematic, since (27/16):(5/4) = 27:20 (a wolf interval), not 4:3. But Indian music uses melodies over a drone dyad (usually 1/1 and 3/2), so these two pitches (27/16 and 5/4) would seldom be heard sounding together. See sargam and swara. [The just scale with the ratios 1/1, 9/8, 5/4, 4/3, 3/2, *5/3*, 15/8, 2/1 gives (5/3):(5/4) = 4:3 (a perfect fourth), and allows these notes to sound together in a co ...

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Just intonation, Just intonation - The diatonic scale in just intonation, Just intonation - Why isn't just intonation used much?, Just intonation - Singing in just intonation, Just intonation - Bagpipe tuning, Just intonation - Non-western tuning, Just intonation - Western composers who specified just intonation

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The word 'vegetarian' was coined in 1847 when the British Vegetarian Society was formed. Before this, vegetarians were known as Pythagoreans. The pentagram (five-pointed star) was an important religious symbol used by the Pythagoreans. It was called "health". ...

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Pythagoreanism, Pythagoreanism - Influence, Pythagoreanism - Pythagorean cosmology, Pythagoreanism - Influences, Pythagoreanism - Reference

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It is possible to tune the familiar diatonic scale or chromatic scale in just intonation, in many ways, all of which make certain chords purely tuned and as consonant as possible, and others considerably more dissonant and indeed seeming out-of-tune to modern ears (see below for more on this). The prominent notes of a given scale are tuned so that the ratios of their frequencies are comprised of relatively small integers. For example, in the key of G major, the ratio of the frequencies of the not ...

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Just intonation, Just intonation - The diatonic scale in just intonation, Just intonation - Why isn't just intonation used much?, Just intonation - Singing in just intonation, Just intonation - Bagpipe tuning, Just intonation - Non-western tuning, Just intonation - Western composers who specified just intonation

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Western common practice music usually cannot be played in just intonation, even when it is confined to a single key. This is because the supertonic chord, or ii-chord, which is the most important of the minor triads in a major key, serves to bridge between the dominant and subdominant, having a root at once a minor third below the root of the subdominant triad, and hence sharing two of its notes, and a fifth above the root of the dominant triad or dominant seventh chord. The problem becomes still worse when modulation, the key changes so imp ...

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Mathematics of musical scales, Mathematics of musical scales - Pythagorean tuning, Mathematics of musical scales - Just intonation, Mathematics of musical scales - Temperament, Mathematics of musical scales - Equal temperament, Mathematics of musical scales - Sound samples, Mathematics of musical scales - Source

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All musical tuning have advantages and disadvantages. Twelve tone equal temperament is the standard and most usual tuning system used in western music today because it gives the advantage of modulation to any key without dramatically going out of tune, as all keys are equally and slightly out of tune. However, just intonation provides the advantage of being entirely in tune, with at least some, and possible a great deal, loss in ease of modulation. Referring to 12-tet the composer Terry Riley, who has written music for both tuning systems, h ...

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Musical tuning, Musical tuning - Subjects in general, Musical tuning - Ways of tuning the twelve-note chromatic scale, Musical tuning - Tunings of other scale systems, Musical tuning - Comparisons and controversies between tunings

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The traditional Pythagorean tuning of the diatonic, also known as Ptolemy's "ditonic diatonic", has two identical 9/8 tones in succesion, making the other interval 256/243: hypate parhypate lichanos mese | 256/243 | 9/8 | 9/8 | -498 -408 -204 0 cents However, the most common tuning in practice from about the 4th century BC to the 2nd century AD appears to have been Archytas's diatonic, or Ptolemy's "tonic diatonic", which has the superparticular 28/ ...

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Diatonic genus, Diatonic genus - Tunings of the diatonic

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Vincenzo Galilei (father of Galileo Galilei) may have been the first person to advocate equal temperament (in a 1581 treatise). The first person known to introduce a mathematically accurate specification for equal temperament is probably Chu Tsai-Yu (朱載堉) in the Ming Dynasty, who published a theory of the temperament in 1584. Soon after, European mathematicians Simon Stevin (1585, inspired by V. Galilei) and Marin Mersenne ( ...

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Equal temperament, Equal temperament - Explanation, Equal temperament - History, Equal temperament - Twelve-tone equal temperament, Equal temperament - Cent values of equal temperament, Equal temperament - Non-12 TET, Equal temperament - Sources

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In diatonic or tonal theory intervals are labelled according to their diatonic function and according to the number of members or degrees they span in a diatonic scale. The interval number of a note from a given tonic note is the number of staff positions enclosed within the interval, as shown at right. Intervals larger than an octave are called compound intervals; for example, a tenth is known as a compound third. Intervals larger than a thirteenth are rarely spoken of, since going above this by stacking thirds would re ...

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Interval music, Interval music - Frequency ratios, Interval music - Interval number and quality, Interval music - Shorthand notation, Interval music - Enharmonic intervals, Interval music - Steps and skips, Interval music - Pitch class intervals, Interval music - Ordered and unordered pitch and pitch class intervals, Interval music - Generic and specific intervals, Interval music - Cents, Interval music - Comparison of different interval naming systems, Interval music - Consonant and dissonant intervals, Interval music - Inversion, Interval music - Interval roots, Interval music - Interval cycles, Interval music - Other intervals, Interval music - Sources

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The Classical Orders are largely known through the writings of Vitruvius, particularly De Archetura (The Ten Books of Architecture) and studies of classical architecture by Renaissance architects and historians. Within a classical order elements from the positioning of triglyphs to the overall height and width of the building were controlled by principles of proportionality. See also:

Proportion architecture, Proportion architecture - Sacred Proportions, Proportion architecture - Feng Shui, Proportion architecture - Classical Orders, Proportion architecture - Vitruvian Proportion, Proportion architecture - Renaissance Orders, Proportion architecture - Le Modulor, Proportion architecture - The Plastic Number

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In diatonic or tonal theory intervals are labelled according to their diatonic function and according to the number of members or degrees they span in a diatonic scale. The interval number of a note from a given tonic note is the number of staff positions enclosed within the interval, as shown at right. Intervals larger than an octave are called compound intervals; for example, a tenth is known as a compound third. Intervals larger than a thirteenth are rarely spoken of (but see 8va for use of 15ma). ...

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Interval music, Interval music - Frequency ratios, Interval music - Interval number and quality, Interval music - Shorthand notation, Interval music - Enharmonic intervals, Interval music - Steps and skips, Interval music - Pitch class intervals, Interval music - Ordered and unordered pitch and pitch class intervals, Interval music - Generic and specific intervals, Interval music - Cents, Interval music - Comparison of different interval naming systems, Interval music - Consonant and dissonant intervals, Interval music - Inversion, Interval music - Interval roots, Interval music - Interval cycles, Interval music - Other intervals, Interval music - Sources

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The average value of the twelve fifths must equal the 700 cents of equal temperament. If eleven of them have a flattened meantone value of 700-ε cents, the wolf will equal 700+11ε cents. In terms of frequency ratios, the product of the fifths must be 128, and if f is the size of the meantone fifths, 128/f11 will be the size of the wolf. In 1/4-comma meantone, the meantone fifth is of size 51/4, 3.422 cents flatter than 700 cents, and so the wolf is 37.637 cents sharper than 700 cents, which is 35.683 cen ...

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Wolf interval, Wolf interval - Temperament and the wolf, Wolf interval - Sound files demonstrating the wolf

Read more here: » Wolf interval: Encyclopedia II - Wolf interval - Temperament and the wolf

The term meantone temperament is sometimes used to refer specifically to quarter-comma meantone. However, systems which flatten the fifth by differing amounts but which still equate the major whole tone, which in just intonation is 9/8, with the minor whole tone, tuned justly to 10/9, are also called meantone systems. Since (9/8) / (10/9) = (81/80), the syntonic comma, the fundamental character of a meantone tuning is that all intervals are generated from fifths, and the syntonic comma is tempered to a unison. While the term meanto ...

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Meantone temperament, Meantone temperament - Meantone temperaments, Meantone temperament - Wolf intervals and extended meantones

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"Well tempered" means that the 12 notes per octave of the standard keyboard are tuned in such a way that it is possible to play music in any major or minor key and it will not sound perceptibly out of tune. In most tuning systems used before 1700, one or more intervals on the 12-note keyboard were so far from any pure interval that they were unusable in harmony and were called a "wolf". The most used system immediately before Werckmeister was meantone in which one fifth (usually E♭–G♯) was nearly two commas wider than pure ...

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Well temperament, Well temperament - Origins, Well temperament - Forms of well temperament

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The distance between each step and the next is aurally the same for any two adjacent steps; though, because steps form a geometric sequence, the difference in frequency increases from one to the next. A linear sequence of one frequency difference would create ever smaller intervals (ratios), such as the harmonic series. See also logarithmic scale. Equal temperaments allow the use of integer notation; a single integer can be used to represent the pitch. The pitch classes can then be expressed in terms of modular arithmetic modulo the number of divisions of the octave, and this expedites mathematic ...

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Equal temperament, Equal temperament - Explanation, Equal temperament - History, Equal temperament - Twelve-tone equal temperament, Equal temperament - Cent values of equal temperament, Equal temperament - Non-12 TET, Equal temperament - Sources

Read more here: » Equal temperament: Encyclopedia II - Equal temperament - Explanation