page_title: Equal temperament
section_title: Various Western equal temperaments
hierarchical_section_title: Equal temperament / Other equal temperaments / Various Western equal temperaments
caption_attribution_description: English: 16-tet scale on C (75 cent steps). Easley Blackwood, Jr.'s notation for 16 equal temperament. Intervals are notated similarly to those they approximate and there are fewer enharmonic equivalents.
context_page_description: An equal temperament is a musical temperament or tuning system, which approximates just intervals by dividing an octave into equal steps. This means the ratio of the frequencies of any adjacent pair of notes is the same, which gives an equal perceived step size as pitch is perceived roughly as the logarithm of frequency.
  In classical music and Western music in general, the most common tuning system since the 18th century has been twelve-tone equal temperament, which divides the octave into 12 parts, all of which are equal on a logarithmic scale, with a ratio equal to the 12th root of 2. That resulting smallest interval, ​¹⁄₁₂ the width of an octave, is called a semitone or half step.
  In Western countries the term equal temperament, without qualification, generally means 12-TET.
  In modern times, 12-TET is usually tuned relative to a standard pitch of 440 Hz, called A440, meaning one note, A, is tuned to 440 hertz and all other notes are defined as some multiple of semitones apart from it, either higher or lower in frequency. The standard pitch has not always been 440 Hz. It has varied and generally risen over the past few hundred years.
context_section_description: 24 EDO, the quarter tone scale (or 24-TET), was a popular microtonal tuning in the 20th century probably because it represented a convenient access point for composers conditioned on standard Western 12 EDO pitch and notation practices who were also interested in microtonality. Because 24 EDO contains all of the pitches of 12 EDO, plus new pitches halfway between each adjacent pair of 12 EDO pitches, they could employ the additional colors without losing any tactics available in 12-tone harmony. The fact that 24 is a multiple of 12 also made 24 EDO easy to achieve instrumentally by employing two traditional 12 EDO instruments purposely tuned a quarter-tone apart, such as two pianos, which also allowed each performer (or one performer playing a different piano with each hand) to read familiar 12-tone notation. Various composers including Charles Ives experimented with music for quarter-tone pianos. 24 EDO approximates the 11th harmonic very well, unlike 12 EDO.
  19 EDO
  is famous and some instruments are tuned in 19 EDO. It has slightly flatter perfect fifth (at 694 cents), but its major sixth are less than a single cent away from just intonation's major sixth (at 884 cents). Its minor third is also less than a cent from just intonation's. Its perfect fourth (at 503 cents), is only 5 cents sharp than just intonation's and 3 cents sharp from 12-tet's.
  23 EDO is the largest EDO that fails to approximate the 3rd, 5th, 7th, and 11th harmonics (3:2, 5:4, 7:4, 11:8) within 20 cents, making it attractive to microtonalists looking for unusual microtonal harmonic territory.
  27 EDO is the smallest EDO that uniquely represents all intervals involving the first eight harmonics. It tempers out the septimal comma but not the syntonic comma.
  29 EDO is the lowest number of equal divisions of the octave that produces a better perfect fifth than 12 EDO. Its major third is roughly as inaccurate as 12-TET; however, it is tuned 14 cents flat rather than 14 cents sharp. It tunes the 7th, 11th, and 13th harmonics flat as well, by roughly the same amount. This means intervals such as 7:5, 11:7, 13:11, etc., are all matched extremely well in 29-TET.
  31 EDO was advocated by Christiaan Huygens and Adriaan Fokker. 31 EDO has a slightly less accurate fifth than 12 EDO, but provides near-just major thirds, and provides decent matches for harmonics up to at least 13, of which the seventh harmonic is particularly accurate.
  34 EDO gives slightly less total combined errors of approximation to the 5-limit just ratios 3:2, 5:4, 6:5, and their inversions than 31 EDO does, although the approximation of 5:4 is worse. 34 EDO doesn't approximate ratios involving prime 7 well. It contains a 600-cent tritone, since it is an even-numbered EDO.
  41 EDO is the second lowest number of equal divisions that produces a better perfect fifth than 12 EDO. Its major third is more accurate than 12 EDO and 29 EDO, about 6 cents flat. It is not meantone, so it distinguishes 10:9 and 9:8, unlike 31edo. It is more accurate in 13-limit than 31edo.
  46 EDO provides slightly sharp major thirds and perfects fifth, giving triads a characteristic bright sound. The harmonics up to 11 are approximated within 5 cents of accuracy, with 10:9 and 9:5 being a fifth of a cent away from pure. As it's not a meantone system, it distinguishes 10:9 and 9:8.
  53 EDO is better at approximating the traditional just consonances than 12, 19 or 31 EDO, but has had only occasional use. Its extremely good perfect fifths make it interchangeable with an extended Pythagorean tuning, but it also accommodates schismatic temperament, and is sometimes used in Turkish music theory. It does not, however, fit the requirements of meantone temperaments, which put good thirds within easy reach via the cycle of fifths. In 53 EDO, the very consonant thirds would be reached instead by using a Pythagorean diminished fourth (C-F♭), as it is an example of schismatic temperament, just like 41 EDO.
  72 EDO approximates many just intonation intervals well, even into the 7-limit and 11-limit, such as 7:4, 9:7, 11:5, 11:6 and 11:7. 72 
caption: Easley Blackwood's notation system for 16 equal temperament: intervals are notated similarly to those they approximate and there are fewer enharmonic equivalents.[27] About this soundPlay (help·info)
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