# quarter-tone / 1/4-tone

[Joe Monzo]
###### 1. Approximate musical interval

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.

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###### 2. One degree of 24-edo

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.

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[John Chalmers, Divisions of the Tetrachord]

An interval of about 50 cents (Â¢), 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.

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

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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