53 notes in an octave - go nut, but make sense

Back to the Pythagorean time, when he found notes based on the pure perfect fifth ascending, 3/2 ratio of the original frequency. After conducting it 12 times and bringing it down with 7 octaves. The 13th note is closed to the original notes, it is closed but not off about 23.46 cents (1 cent = 1/1200 an octave), about a quarter-tone. This is known as the Pythagorean comma [(1.5^12)/(2^7)]. (If you really want to see the maths, click here.)

People all over the world notice this problem, there are 3 choices for musicians.

  1. Completely ignore it, just limit the use of the 5-7 notes in the music, simple harmony and texture, like the music before the renaissance period.
  2. Adjust 12 notes we have and the ratios of each interval, with more complex harmony.
    1. Meantone temperament - sound pure but some keys are not useable
    2. Equal temperament - all keys can be used, but sound the same
    3. Unequal (Well) temperament - in between meantone and equal temp.
  3. Keep finding new notes with pure 5th or other simple ratios intervals. Let's talk about this today.
Back to the title, 53-note tuning, was first developed by Jiao Gong (?-?). Jing Fang (78–37 BCE) promoted it to the public. Originally, Jiao would like to expand the temperament to 12x5=60 notes, echoing the idea of the hexagram of I Ching, 64 (2^6) sequences in the table. However, they calculated with pure fifth and found the 54th note very close to the original note.  [(1.5^53)/(2^31)] about 1.002090314 or 3.615046 cents, call Jing Fang Schisma (click here to see the maths). Way smaller than the Pythagorean comma (23.46 cents). The slight out-of-tune is hard to be noticeable. Combining the 2nd and 3rd choices above, since the schisma is so narrow and if we divide it equally into 53 steps, we can have a perfect 2/1 octave and other notes are moved very slightly, this is the 53-TET (equal) temperament.
A piece of mine using 53-TET.

Using this scale in music can guarantee the majority of intervals are in pure ratios, sound pure. But there are tons of notes in an octave. Although the Chinese developed the temperament, the culture that adopt it is Turkisk music (Yarman,2007). Like the Chinese first invented the equal temperament, Western musicians employed it later.


Yarman, O. (2007). A Comparative Evaluation of Pitch Notations in Turkish Makam Music:Abjad Scale & 24-Tone Pythagorean Tuning – 53 Equal Division of the Octave as a Common Grid. journal of interdisciplinary music studies. fall 2007, volume 1, issue 2, art. #071203, pp. 43-61

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