quarter-tone

• 1. Used in a general sense by many people to refer to microtonal intervals approximately half as large as the semitone, or thus measuring approximately 50 cents.

  There is less of a tendency to use it in this general sense today.
  
  

• 2. Used as an exact measurement, it is calculated as the 24th root of 2, or 2^(1/24), an irrational proportion with the approximate ratio of 1:1.029302236643, and an interval size of exactly 50 cents.

  It is the size of one degree, and thus the basic "step" size, in the 24-eq (or 24-ET) scale, also called the "quarter-tone scale" or system.

Several composers began composing in 24-ET early in the 20th century, including Charles Ives, Ivan Vyschnegradsky, and Alois Hába. Hába had several instruments specially built to be playable in this system.

In addition to the rational approximations to the actual "quarter-tone" as given below by John Chalmers, it should be noted that several larger intervals of 24edo approximate 11- and 13-limit ratios very closely:

24edo   cents   ~ratio   ~cents   ~cents error from ratio

2(3/24)   150    12:11  150.6370585   - 0.637058501
2(7/24)   350    11:9   347.4079406   + 2.592059366
2(11/24)  550    11:8   551.3179424   - 1.317942365
2(13/24)  650    16:11  648.6820576   + 1.317942365
2(17/24)  850    18:11  852.5920594   - 2.592059366
2(21/24) 1050    11:6  1049.362941    + 0.637058501

2(5/24)   250    15:13  247.741053    + 2.258947039
2(9/24)   450    13:10  454.2139479   - 4.213947904
2(15/24)  750    20:13  745.7860521   + 4.213947904
2(19/24)  950    26:15  952.258947    - 2.258947039

I have invented a notation based on the quarter-tone scale which I call the quarter-tone staff.

[from Joe Monzo, JustMusic: A New Harmony]

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An interval of about 50 cents (¢), one half of a tempered semitone. [Chalmers here means specifically 12-tone equal temperament. -Monzo]

Typical quarter tones in Just Intonation have ratios such as 36/35 or 33/34.

[from John Chalmers, Divisions of the Tetrachord]

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From Paul Erlich, private communication:

"In 1905-6 the Kitab al-musiqa al-sharqi ('The book of eastern music') by Kamil al-Khula'i (1879-1938) established the equidistance of quartertones in the octave. This scale of 24 quarter-tones was the subject of fierce discussion at the Congress of Cairo in 1932, where the participants divided into two opposing camps; the Egyptians supported the division of the octave into 24 equal quarters, while the Turks (represented by Yekta Bey) and the Syro-Lebanese (Sabra and Tawfiq al-Sabbagh) rejected the system of equal division.

"In 1959 and 1964 the Egyptians organized two symposia to settle the differences of opinion arising from the controversy at the 1932 Congress over the equidistance of quarter-tones. The aim of these symposia was to establish the principle of equal temperament on the basis of the quarter-tone and give official sanction to its teaching.

[from: New Grove's Dictionary of Music and Musicians, entry on "Arabic Music", 6(ii) Theory p.812]

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Updated:

2000.02.14
2002.01.25
2003.06.08 -- added table of 11- and 13-limit ratios

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