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Pythagorean tuning is a system of musical tuning in which the frequency relationships of all intervals are based on the ratio 3:2. In Music, there are two common meanings for tuning: Tuning practice, the act of tuning an instrument or voice Frequency is a measure of the number of occurrences of a repeating event per unit Time. In Music theory, the term interval describes the relationship between the pitches of two Notes Intervals may be described as vertical Superparticular numbers, also called epimoric ratios, are improper Vulgar fractions of the form {n + 1 \over n} = 1 + {1 \over n} Its name comes from medieval texts which attribute its discovery to Pythagoras, but its use has been documented as long ago as 3500 B. "Pythagoras of Samos" redirects here For the Samian statuary of the same name see Pythagoras (sculptor. 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 (although distinct from) many other methods of tuning, such as the common equal temperament. The chromatic scale is a Musical scale with twelve pitches each a Semitone or Half step apart Equal temperament is a Musical temperament, or a system of tuning in which every pair of adjacent notes has an identical Frequency ratio. 53 equal temperament is closely related to Pythagorean tuning because it is extremely similar to a very extended Pythagorean cycle of fifths. 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

Contents

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 perfect fifth ( is the Musical interval between a note and the note seven Semitones above it on the musical scale In Music, an octave ( is the the use of which is "common in most musical systems The two notes G and D, for example, are tuned so that their frequencies are in the ratio 3:2 — if G is tuned to 200 Hz, then the D is tuned to 300 Hz. The hertz (symbol Hz) is a measure of Frequency, informally defined as the number of events occurring per Second. The A a fifth above that D is also tuned in the ratio 3:2 — with the D at 300 Hz, this puts the A at 450 Hz, 9:4 above the original G. When describing tunings, it is usual to speak of all notes as being within an octave of each other, and as this A is over an octave above the original G, it is usual to halve its frequency to move it down an octave. In Music, an octave ( is the the use of which is "common in most musical systems Therefore, the A is tuned to 225 Hz, a 9:8 above the G. The E a 3:2 above that A is tuned to the ratio 27:16 and so on, until the starting note, G, is arrived at again.

In applying this tuning to the chromatic scale, however, a problem arises: no number of 3:2s will fit exactly into an octave. The chromatic scale is a Musical scale with twelve pitches each a Semitone or Half step apart In Music, when ascending from an initial (low pitch by a cycle of justly tuned perfect fifths (ratio 32 ( leapfrogging twelve times one eventually reaches a Because of this, the G arrived at after twelve fifths is about a quarter of a semitone sharper than the G used to begin the process. A semitone, also called a half step or a half tone, is the smallest Musical interval commonly used in Western tonal music and it is considered the The table below (starting at E flat) 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 is a logarithmic unit of measure used for musical intervals. The cent values of the same notes in equal temperament are also given for comparison (marked in the table below as "et-Cents"). Equal temperament is a Musical temperament, or a system of tuning in which every pair of adjacent notes has an identical Frequency ratio.

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 '''circle of fourths''') shows the relationships among the twelve tones of the Chromatic scale, their corresponding

Note Ratio Cents et-Cents Interval
Eb 256:243 90. 22 100 minor second
Bb 128:81 792. 18 800 minor sixth
F 32:27 294. 13 300 minor third
C 16:9  996. 09 1000 minor seventh
G 4:3 498. 04 500 perfect fourth
D 1:1 0 0 unison
A 3:2 701. 96 700 perfect fifth
E 9:8 203. 91 200 major second
B 27:16 905. 87 900 major sixth
F# 81:64 407. 82 400 major third
C# 243:128 1109. 78 1100 major seventh
G# 729:512 611. 73 600 augmented fourth
[D#] [2187:2048] [113. 69] [100] [augmented unison]

The major scale obtained from this tuning is

Note D E F# G A B C# D
Ratio 1/1 9/8 81/64 4/3 3/2 27/16 243/128 2/1
Step 9/8 9/8 256/243 9/8 9/8 9/8 256/243

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. In Music theory, the major scale or Ionian scale is one of the diatonic scales It is made up of seven distinct Notes plus an eighth In modern Music and notation, an enharmonic equivalent is a Note ( enharmonic tone) interval ( enharmonic interval) or This discrepancy, of about 23. 5 cents, or one quarter of a semitone, is known as a Pythagorean comma. In Music, when ascending from an initial (low pitch by a cycle of justly tuned perfect fifths (ratio 32 ( leapfrogging twelve times one eventually reaches a

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. A semitone, also called a half step or a half tone, is the smallest Musical interval commonly used in Western tonal music and it is considered the

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 term key is used in many different and sometimes contradictory ways

Because of the wolf interval, this tuning is rarely used nowadays, although it is thought to have been 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 term key is used in many different and sometimes contradictory ways In Western music, harmony is the use of different pitches simultaneously and 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 western classical music, this usually means music written prior to the 15th century. Classical music is a broad term that usually refers to mainstream music produced in or rooted in the traditions of Western liturgical and Secular music As thirds came 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. Meantone temperament is a Musical temperament, which 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 Superparticular numbers, also called epimoric ratios, are improper Vulgar fractions of the form {n + 1 \over n} = 1 + {1 \over n} However, meantone still has a wolf interval, so is not suitable for all music.

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. The 18th century lasted from 1701 to 1800 in the Gregorian calendar, in accordance with the Anno Domini / Common Era numbering system Well temperament (also circular or circulating temperament is a type of tempered tuning described in twentieth-century Music theory Equal temperament is a Musical temperament, or a system of tuning in which every pair of adjacent notes has an identical Frequency ratio.

Discography

See also

References

Footnotes

  1. ^ West, M. See also Whole-tone scale List of meantone intervals List of intervals in 5-limit just intonation 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 Timaeus ( Greek: Τίμαιος, Timaios) is a theoretical treatise of Plato in the form of a Socratic dialogue, written Background List See also List of meantone intervals In Music, a whole tone scale is a scale in which each Note is separated from its neighbours by the interval of a Whole step. Regular temperament is any tempered system of Musical tuning such that each frequency ratio is obtainable as a product of powers of a finite number of generators The following is a list of intervals of Meantone temperament. L. , "The Babylonian Musical Notation and the Hurrian Melodic Texts", Music & Letters, Vol. 75, no. 2. , May, 1994, pp. 161-179

Notations

External links

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