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METHOD Pythagorean tuning is based on a stack of 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. In applying this tuning to the . 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 below table (starting at E flat rather than D) illustrates this, showing the note name, the ratio above D, and the value in Cent s 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"). 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 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 ''. 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 '' flatter. (38.2KB), which is the same three fifths tuned in Equal Temperament , each of them tolerably well in tune. 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. 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. 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 , became more popular. However, meantone still has a wolf interval, so is not suitable for all music. From around the 18th Century , the need grew for instruments to change key, and therefore to avoid a wolf interval, this led to the widespread use of Well Temperament s and eventually Equal Temperament . DISCOGRAPHY
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