Pythagorean tuning is a system of musical tuning in which the frequency relationships of all intervals are based on the ratio 3:2. Its name comes from medieval texts which attribute its discovery to Pythagoras, but its use has been documented as long ago as 3500 B.C. in Babylonian texts^[1]. It is the oldest way of tuning the 12-note chromatic scale and, as such, it is the basis for many other methods of tuning, such as meantone temperament and the common equal temperament. 53 equal temperament is also closely related to Pythagorean tuning because it is extremely similar to a very extended Pythagorean cycle of fifths. In music, there are two common meanings for tuning: Tuning practice The act of tuning an instrument or voice. ... FreQuency is a music video game developed by Harmonix and published by SCEI. It was released in November 2001. ... In music theory, the term interval describes the difference in pitch between two notes. ... Sesquialterum (plural: sesquialtera) or sesquialter refers to the improper rational fraction . It is a superparticular number. ... Pythagoras of Samos (Greek: ; circa 580 BC – circa 500 BC) was an Ionian (Greek) philosopher[1] and founder of the religious movement called Pythagoreanism. ... The chromatic scale is the scale that contains all twelve pitches of the Western tempered scale. ... Meantone temperament is a system of musical tuning. ... An equal temperament is a musical temperament -- that is, a system of tuning intended to approximate some form of just intonation -- in which an interval, usually the octave, is divided into a series of equal steps (equal frequency ratios). ... In music, 53 equal temperament, called 53-TET, 53-EDO, or 53-ET, is the tempered scale derived by dividing the octave into fifty-three equally large steps. ...

Method

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 D and A, 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, and as this E is over an octave above the original D, it is usual to halve its frequency to move it down an octave. Therefore, the E is tuned to 225 Hz, a 9:8 above the D. The B a 3:2 above that E is tuned to the ratio 27:16 and so on, until the starting note, D, is arrived at again. The perfect fifth or diapente is one of three musical intervals that span five diatonic scale degrees; the others being the diminished fifth, which is one semitone smaller, and the augmented fifth, which is one semitone larger. ... In music, an octave (sometimes abbreviated 8ve) is the interval between one musical note and another with half or double its frequency. ... The hertz (symbol: Hz) is the SI unit of frequency. ... In music, an octave (sometimes abbreviated 8ve) is the interval between one musical note and another with half or double its frequency. ...

In applying this tuning to the chromatic scale, however, a problem arises: no number of 3:2s will fit exactly into an octave. Because of this, the D arrived at after twelve fifths have been tuned up is about a quarter of a semitone sharper than the D used to begin the process. The table below (starting at E flat rather than D) illustrates this, showing the note name, the ratio above D, and the value in cents above the D for each note in the chromatic scale. The cent values of the same notes in equal temperament are also given for comparison (marked in the table below as "et-Cents"). The chromatic scale is the scale that contains all twelve pitches of the Western tempered scale. ... When one ascends by a cycle of justly tuned perfect fifths (ratio 3:2), leapfrogging 12 times, one eventually reaches a note around seven octaves above the note one started on, which, when lowered to the same octave as the starting point, is 23. ... A semitone (also known in the USA as a half step) is a musical interval. ... The cent is a logarithmic unit of measure used for musical intervals. ... An equal temperament is a musical temperament -- that is, a system of tuning intended to approximate some form of just intonation -- in which an interval, usually the octave, is divided into a series of equal steps (equal frequency ratios). ...

In order to keep the ratios in this table relatively simple, fifths are tuned down from D as well as up. The first note in the circle of fifths given here is E flat (equivalent to D#), from which five perfect fifths are tuned before arriving at D, the nominal unison note. In music theory, the circle of fifths (or cycle of fifths) is an imaginary geometrical space that depicts relationships among the 12 equal-tempered pitch classes comprising the familiar chromatic scale. ...

In equal temperament, and most other modern tunings of the chromatic scale, pairs of enharmonic notes such as E flat and D sharp are thought of as being the same note — however, as the above table indicates, in Pythagorean tuning, they theoretically have different ratios, and are at a different frequency. This discrepancy, of about 23.5 cents, or one quarter of a semitone, is known as a Pythagorean comma. In music, an enharmonic is a note which is the equivalent of some other note, but spelled differently. ... When one ascends by a cycle of justly tuned perfect fifths (ratio 3:2), leapfrogging 12 times, one eventually reaches a note around seven octaves above the note one started on, which, when lowered to the same octave as the starting point, is 23. ...

To get around this problem, Pythagorean tuning uses the above 12 notes from E flat to G sharp shown above, and then places above the G sharp another E flat, starting the sequence again. This leaves the interval G#—Eb sounding badly out of tune, meaning that any music which combines those two notes is unplayable in this tuning. A very out of tune interval such as this one is known as a wolf interval. In the case of Pythagorean tuning, all the fifths are 701.96 cents wide, in the exact ratio 3:2, except the wolf fifth, which is only 678.49 cents wide, nearly a quarter of a semitone flatter. When the twelve notes within the octave are tuned using meantone temperament, one of the fifths will be much sharper than the rest. ... A semitone (also known in the USA as a half step) is a musical interval. ...

Wolf fifth.ogg (33.1KB) is a sound file demonstrating this out of tune fifth. The first two fifths are perfectly tuned in the ratio 3:2, the third is the G#—Eb wolf fifth. It may be useful to compare this to Et fifths.ogg (38.2KB), which is the same three fifths tuned in equal temperament, each of them tolerably well in tune. An equal temperament is a musical temperament -- that is, a system of tuning intended to approximate some form of just intonation -- in which an interval, usually the octave, is divided into a series of equal steps (equal frequency ratios). ...

If the notes G# and Eb need to be sounded together, the position of the wolf fifth can be changed (for example, the above table could run from A to E, making that the wolf interval instead of Eb to G#). However, there will always be one wolf fifth in Pythagorean tuning, making it impossible to play in all keys in tune. In music theory, the key identifies the tonic triad, the chord, major or minor, which represents the final point of rest for a piece, or the focal point of a section. ...

Because of the wolf interval, this tuning is rarely used nowadays, although it is thought it was once widespread. In music which does not change key very often, or which is not very harmonically adventurous, the wolf interval is unlikely to be a problem, as not all the possible fifths will be heard in such pieces. In music theory, the key identifies the tonic triad, the chord, major or minor, which represents the final point of rest for a piece, or the focal point of a section. ... Harmony is the use and study of pitch simultaneity, and therefore chords, actual or implied, in music. ...

Because fifths in Pythagorean tuning are in the simple ratio of 3:2, they sound very "smooth" and consonant. The thirds, by contrast, which are in the relatively complex ratios of 81:64 (for major thirds) and 32:27 (for minor thirds), sound less smooth. For this reason, Pythagorean tuning is particularly well suited to music which treats fifths as consonances, and thirds as dissonances. In classical music, this usually means music written prior to the 16th century. As thirds became to be treated as consonances, so meantone temperament, and particularly quarter comma meantone, which tunes thirds to the relatively simple ratio of 5:4, became more popular. However, meantone still has a wolf interval, so is not suitable for all music. Classical music is a broad, somewhat imprecise term, referring to music produced in, or rooted in the traditions of, European art, ecclesiastical and concert music, encompassing a broad period from roughly 1000 to the present day. ... (15th century - 16th century - 17th century - more centuries) As a means of recording the passage of time, the 16th century was that century which lasted from 1501 to 1600. ... Meantone temperament is a system of musical tuning. ... Meantone temperament is a system of musical tuning. ... Sesquiquartum refers to the improper vulgar fraction The sesquiquartum in musical harmony is the ratio corresponding to the interval ditonus. ...

From around the 18th century, as the need grew for instruments to change key, and therefore to avoid a wolf interval, this led to the widespread use of well temperaments and eventually equal temperament. (17th century - 18th century - 19th century - more centuries) As a means of recording the passage of time, the 18th century refers to the century that lasted from 1701 through 1800. ... Well temperament (also circular or circulating temperament) is a type of tempered tuning described in twentieth-century music theory. ... An equal temperament is a musical temperament -- that is, a system of tuning intended to approximate some form of just intonation -- in which an interval, usually the octave, is divided into a series of equal steps (equal frequency ratios). ...

Discography

• Gothic Voices - Music fo the Lion-Hearted King (Hyperion, CDA66336, 1989), directed by Christopher Page (Leech-Wilkinson)
• Lou Harrison performed by John Schneider and the Cal Arts Percussion Ensemble conducted by John Bergamo - Guitar & Percussion (Etceter Records, KTC1071, 1990): Suite No. 1 for guitar and percussion and Plaint & Variations on "Song of Palestine"

Gothic Voices is a United Kingdom based vocal ensemble specialising in repertoire from the 11th to the 15th century. ... Christopher Page is an expert on medieval music, instruments and performance practice, and the author of four books: Voices and Instruments of the Middle Ages (1987), The Owl and the Nightingale: Musical Life and Ideas in France 1100-1300 (1989), The Summa Musice: A Thirteenth-Century Manual for Singers (1991... Lou Silver Harrison (May 14, 1917 - February 2, 2003) was an American composer. ... John Schneider is the name of the following unrelated people: John Schneider is an actor best known for his television roles, particularly The Dukes of Hazzard. ... John Bergamo (b. ...

See also

• Pythagorean intervals
• Enharmonic scale
• Temperament
• Timaeus (dialogue), in which Plato discusses Pythagorean tuning
• List of musical intervals
• Whole-tone scale
• Regular temperament
• List of meantone intervals

The intervals of Pythagorean tuning are just intervals involving only powers of two and three. ... An enharmonic scale is a musical scale in which there is no exact equivalence between a sharpened note and the flattened note it is enharmonically related to. ... In musical tuning, a temperament is a system of tuning which slightly compromises the pure intervals of just intonation in order to meet other requirements of the system. ... Timaeus is a theoretical treatise of Plato in the form of a Socratic dialogue, written circa 360 BC. The work puts forward speculation on the nature of the physical world. ... Equal-tempered refers to 12-tone equal temperament. ... In music, a whole tone scale is a scale in which each note is separated from its neighbors by the interval of a whole step. ... Regular temperament is a system of musical tuning such that each frequency ratio is obtainable as a product of powers of a finite number of generators, or generating frequency ratios. ... The following is a list of intervals of meantone temperament. ...

References

1. ^ West, M.L., "The Babylonian Musical Notation and the Hurrian Melodic Texts", Music & Letters, Vol. 75, no. 2., May, 1994, pp. 161-179

• Daniel Leech-Wilkinson (1997), "The good, the bad and the boring", Companion to Medieval & Renaissance Music. Oxford University Press. ISBN 0-19-816540-4.

External links

• Tonalsoft Encyclopaedia of Tuning

In music, there are two common meanings for tuning: Tuning practice The act of tuning an instrument or voice. ... In music, just intonation, also called rational intonation, is any musical tuning in which the frequencies of notes are related by ratios of whole numbers. ... The 43-tone scale is a just intonation scale with 43 pitches in each octave invented and used by Harry Partch. ... Regular temperament is a system of musical tuning such that each frequency ratio is obtainable as a product of powers of a finite number of generators, or generating frequency ratios. ... An equal temperament is a musical temperament -- that is, a system of tuning intended to approximate some form of just intonation -- in which an interval, usually the octave, is divided into a series of equal steps (equal frequency ratios). ... An equal temperament is a musical temperament -- that is, a system of tuning intended to approximate some form of just intonation -- in which an interval, usually the octave, is divided into a series of equal steps (equal frequency ratios). ... In music, 19 equal temperament, called 19-tet, 19-edo, or 19-et, is the scale derived by dividing the octave into 19 equally large steps. ... In music, 22 equal temperament, called 22-tet, 22-edo, or 22-et, is the scale derived by dividing the octave into 22 equally large steps. ... A quarter tone is an interval half as wide (aurally, or logarithmically) as a semitone, which is half a whole tone. ... In music, 31 equal temperament, called 31-tet, 31-edo, or 31-et, is the scale derived by dividing the octave into 31 equally large steps. ... In music, 53 equal temperament, called 53-TET, 53-EDO, or 53-ET, is the tempered scale derived by dividing the octave into fifty-three equally large steps. ... In music, 72 equal temperament, called 72-tet, 72-edo, or 72-et, is the scale derived by dividing the octave into twelfth-tones, or in other words 72 equally large steps. ... Meantone temperament is a system of musical tuning. ... Quarter-comma meantone was the most common meantone temperament in the sixteenth and seventeenth centuries, and was sometimes used later. ... LucyTuning is a form of meantone temperament, in which the fifth is of size 600+300/π (= approximately 695. ... In music, septimal meantone temperament, also called standard septimal meantone or simply septimal meantone, refers to the tempering of 7-limit musical intervals by a meantone temperament tuning in the range from fifths flattened by the amount of fifths for 12 equal temperament to those as flat as 19 equal... In music, schismatic temperament is the temperament which results from tempering the schisma of 32805:32768 to a unison. ... In music, miracle temperament is a regular temperament invented by George Secor which has as a generator an interval, called the secor, which serves as both the 15/14 and 16/15 semitones. ... Well temperament (also circular or circulating temperament) is a type of tempered tuning described in twentieth-century music theory. ...