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. This was common in the past, but there is less of a tendency to use it in this general sense today.
Used as an exact measurement, the quarter-tone is calculated as the 24th root of 2 -- 24√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-edo (or 24-ET, 24-tET, or 24-eq) scale, also called the "quarter-tone scale" or system.
Several composers began writing music in 24-ET early in the 20th century, including Charles Ives, Richard Stein, Jörg Mager Willi Möllendorf, Ivan Vyschnegradsky, and Alois Hába. Hába had several instruments specially built to be playable in this system, and Schoenberg, Webern, and Berg also experimented with quarter-tones.
24edo approximates the following 13-limit ratios within the usually-accepted human tuning error of about 5 cents:
-------- monzo ------- ~cents error
24edo cents 2 3, 5 7 11, 13 ratio ~cents of 24edo from JI
23 1150 [ -4 4, 1 0 0, -1 > 405 : 208 1153.60605556 -3.60605556
23 1150 [ -1 -2, 1 1 0, 0 > 35 : 18 1151.22961860 -1.22961860
23 1150 [ 6 -1, 0 0 -1, 0 > 64 : 33 1146.72705677 +3.27294323
21 1050 [ -1 -1, 0 0 1, 0 > 11 : 6 1049.36294150 +0.63705850
21 1050 [ 6 0, -1 -1 0, 0 > 64 : 35 1044.86037967 +5.13962033
19 950 [ -2 5, -1 -1 0, 0 > 243 : 140 954.63538399 -4.63538399
19 950 [ 1 -1, -1 0 0, 1 > 26 : 15 952.25894704 -2.25894704
19 950 [ -1 2, 1 0 0, -1 > 45 : 26 949.69605383 +0.30394617
19 950 [ 2 -4, 1 1 0, 0 > 140 : 81 947.31961687 +2.68038313
17 850 [ 1 2, 0 0 -1, 0 > 18 : 11 852.59205937 -2.59205937
17 850 [ 2 -3, 0 0 1, 0 > 44 : 27 845.45293977 +4.54706023
15 750 [ -6 2, 0 0 1, 0 > 99 : 64 755.22794410 -5.22794410
15 750 [ 1 3, -1 -1 0, 0 > 54 : 35 750.72538226 -0.72538226
15 750 [ 4 -3, -1 0 0, 1 > 208 : 135 748.34894531 +1.65105469
15 750 [ 2 0, 1 0 0, -1 > 20 : 13 745.78605210 +4.21394790
13 650 [ -3 -1, 1 1 0, 0 > 35 : 24 653.18461947 -3.18461947
13 650 [ 4 0, 0 0 -1, 0 > 16 : 11 648.68205764 +1.31794236
11 550 [ -3 0, 0 0 1, 0 > 11 : 8 551.31794236 -1.31794236
11 550 [ 4 1, -1 -1 0, 0 > 48 : 35 546.81538053 +3.18461947
9 450 [ -1 0, -1 0 0, 1 > 13 : 10 454.21394790 -4.21394790
9 450 [ -3 3, 1 0 0, -1 > 135 : 104 451.65105469 -1.65105469
9 450 [ 0 -3, 1 1 0, 0 > 35 : 27 449.27461774 +0.72538226
9 450 [ 7 -2, 0 0 -1, 0 > 128 : 99 444.77205590 +5.22794410
7 350 [ -1 3, 0 0 -1, 0 > 27 : 22 354.54706023 -4.54706023
7 350 [ 0 -2, 0 0 1, 0 > 11 : 9 347.40794063 +2.59205937
5 250 [ -1 4, -1 -1 0, 0 > 81 : 70 252.68038313 -2.68038313
5 250 [ 2 -2, -1 0 0, 1 > 52 : 45 250.30394617 -0.30394617
5 250 [ 0 1, 1 0 0, -1 > 15 : 13 247.74105296 +2.25894704
5 250 [ 3 -5, 1 1 0, 0 > 280 : 243 245.36461601 +4.63538399
3 150 [ -5 0, 1 1 0, 0 > 35 : 32 155.13962033 -5.13962033
3 150 [ 2 1, 0 0 -1, 0 > 12 : 11 150.63705850 -0.63705850
1 50 [ -5 1, 0 0 1, 0 > 33 : 32 53.27294323 -3.27294323
1 50 [ 2 2, -1 -1 0, 0 > 36 : 35 48.77038140 +1.22961860
Below is a lattice-diagram showing the ratios in the above table.
It can be seen in general that 24edo gives a good representation of pythagorean chains of ratios whose individual terms (numerator and denominator) contain 11 or (5 and 7), or whose terms compare prime-factors (5 and 13).
Monzo invented a notation based on the 24-edo quarter-tone scale which he calls the quarter-tone staff.
An interval of about 50 cents (¢), one half of a tempered semitone. [note from Monzo: Chalmers here means specifically 12-tone equal temperament.]
Typical quarter tones in Just Intonation have ratios such as 36/35 or 33/34.
It is often stated that quarter-tones are used in music within the Arab/Muslim sphere of cultural influence. The following was submitted by Paul Erlich:
"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."
(For a detailed examination of more ancient Arab tunings, see Monzo, Arab Lute Frettings.)
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