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24-EDO is a musical intervallic system based on quarter-tones . a quarter-tone is used  to  refer to microtonal intervals approximately half as large as the semitone, or approximately 50 cents.

The earliest confirmed written reference to an interval in this size range is by Philolaus, who called it a "diaschisma". There is less of a tendency to use "quarter-tone" in this general sense today.

Used as an exact measurement, the quarter-tone is calculated as the 24th root of 2, or 2 1/24 , an irrational proportion with the approximate ratio of 1.02930223664, and an interval size of exactly 50 cents.

The quarter-tone is the size of one division in the 24-edo which is also called the "quarter-tone scale" or system .

Approximating 24-EDO by EDL system

We can approximate 24-EDO by N-EDL (Equal divisions of length) intervallic system. This system is constructed by dividing an assumed string length to N equal divisions . So, we have rational intervals .To approximate 24-EDO, 640-EDL (with 320 equal divisions of length in octave) is selected.  Maximum error in this approximation is 2.06 cent and minimum is -1.98 cent , which are below the usually-accepted human tuning error of about 5 cents or pitch JND (1, 2) .EDL systems are the best way to fret string instruments with a good approximation.So , for example if we want to fret a setar , with string length of 640 mm (between bridge and nut) in 640-EDL ver 24-EDO (640-EDL version of  24-EDO), first we must fret octave considering compensation (Pressing string behind a fret stretches it and the intervals will sound sharper...so we must replace of frets. It is done by changing position of octave fret and then calculating repalcement of other frets due to change in position of octave fret)and then moving toward nut and fretting with accuracy of 640/640 or

1 mm .

Comparision between 640-EDL and 24-EDO (Many thanks to SCALA)

Difference  between 640-EDL ver 24-EDO and 24-EDO

Sizes of 24 division in 640 -EDL ver 24-EDO

24-EDO and limits

24-EDO does not provide more accurate approximations to the basic intervals (3/2, 5/4, 6/5, and their octave-complements) than 12-edo In the 5-limit.

The big improvements of 24-edo over 12-edo are the approximations to prime-factors 11, 13, 31and 37(http://www.tonalsoft.com/enc/q/quarter-tone.aspx) .24-edo approximates the following 13-limit ratios within the usually-accepted human tuning error of about 8-5 cents :

1

2

1&2 -Approximation of 24-EDO to 13-limit ratio

(From:

24 Equal Divisions of Octave and Ali Naqi Vaziri

24 Equal Divisions of Octave or 24-EDO and new accidentals for music notation,was firstly suggested for Persian music by Ali Naqi Vaziri .

Vaziri was born in Tehran, Persia in 1887. He had his first lessons on the tar (a traditional Persian instrument) by his uncle at the age of 15, and a few years later studied Western music theory under Yavar Agha-Khan (a music instructor at Tehran's Polytechnique, "Dar-ol-Fonoun") and under a priest at the St. Louis School in Tehran. At the same time he continued practicing the tar intensively and finally succeeded in transcribing the radif (the repertoire of Persian classical music) according to the performance of Mirza-Abdollah and Agha Hossein-Gholi (prominent tar soloists in that period).
In several books and articles it has been mentioned that this transcribed radif had been lost and that only the "chahargaah" mode had survived, which Vaziri presented to the Persian National Music Conservatory Library. However, many years later Moussa Ma'rufi (tar soloist, professor of the National Conservatory and a pupil of Vaziri) announced in "Iran's Music Magazine" (Vol.11, No. 10, Jan. 1964) that this radif had been made available to him by Vaziri, based on which Ma'rufi had prepared a complete version of the radif.
After working with Mirza-Abdollah and Agha Hossein-Gholi, Vaziri practiced the radif orally with Montazem-ol-Hokama. In 1918 he went to Europe, where he continued studying the piano, composition and voice in Paris and Berlin.

On his return to Persia (Iran) in 1923, Vaziri established a private music school. Because of Vaziri's new methods in teaching, after a short time many students left his school. Alongside teaching, he gave lectures and Persian music concerts in the new style, established a "Musical Club" and attempted to revive traditional Persian music with the so-called scientific method.

At that time women were not allowed to attend such programs, but Vaziri succeeded in receiving the officials' permission to form two classes for young girls in his school. His own daughter, Badri, was a tar player, ballet dancer and writer, educated in Switzerland and Belgium.
In 1928 Ali-Naghi Vaziri became the director of the Tehran Conservatory of Music, and in 1934, due to conflict with the Persian (Iranian) court, he was discharged. A few years later, in view of his research in the fields of art history, concord of poetry with music and the prosody of Persian poetry, he was selected as professor at the University of Tehran.

After the September 1941 events, when Reza Shah was exiled to Johannesburg, Vaziri again became director of the Conservatory for five years and, in collaboration with Rouhollah Khaleghi, established the "Novin Orchestra" at Radio Tehran.
With a change in the Cabinet, Vaziri was dismissed again in 1946 and until the end of his life (1979) he no longer appear on the music scene and worked merely as a university professor.

Vaziri's techniques in the playing of the tar caused the next generation of tar players to develop their style in performance. According to music critic Parviz Mansouri, "He was the first tar player in the history of Persian art music to add "firmness" and "power" to the sweet style of previous players of this instrument."
On the one hand, Vaziri loved Persian music so much that he was unable to accept having the extremist advocates of modernism put it to rest on the pretext of its age; on the other hand, since he loved to modernize Persian music, he was persistent in adapting his knowledge of Western music to Persian music.
Training such musicians as Rouhollah Khaleghi, innovation of symbols in Persian melody notation, translation of Western musical terms into the Persian language, writing several methods, books and articles about classical music of Persia have been some of his services.Vaziri died on September 9, 1979, at the age of 91.

In western culture , several composers began writing music in 24-EDO 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

In middle east we can see the impact of westernization on music of some countries such as Egypt , as testified  to in the reports issued by the Congress of Arab Music held in Cairo in 1932.The scale of 24 quarter-tones was the subject of fierce discussion at this Congress, 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 .

In Egypt , theorists divide the octave scale into small microtones comparable to those discussed earlier by al-Farabi and Safi ad-Din. Several types of micro-intervals have been advocated,  including the comma division (roughly one-ninth of a whole step), which is found in some Syrian theories. Yet, it is generally conceived that the Maqamat are based on a referential octave scale consisting of twenty-four equal quarter-tones. Despite the essentially aural nature of Arab music, Western notation has become fully established, and extra symbols are widely used. In addition to the regular flat and sharp signs, new symbol lowers a note by approximately a quarter tone while another symbol raises a note by roughly a quarter tone .

Parallel to some middle eastern theorists who proposed 24-EDO for arabic music (1 , 2 , 3 , 4 , 5) , vaziri , as the father of modern persian music notation , was persistent in adapting his knowledge of Western music to Persian music And suggested this reformulation to fascilitate the composition of polyphonic pieces in a system which was traditionally monophonic. His efforts also brought about the notation of microtonal raising and lowering of pitches . Although his efforts was severly excoriated , but accepted as a framework for intervallic structure of Persian music.

Accidental proposed by Ali Naqi Vaziri

Vaziri proposed 2 accidentals for indicating the quarter tone pitches outside of standard notation. These accidental were named as KORON and SORI  which alter the pitch -1/4 and +1/4 tone :

Accidentals of quarter tones proposed by Vaziri

Criticism against 24-EDO

Although accepted as a framework for intervallic structure of Persian music , Vaziri's model  is severly criticized by Players , composers and scholars  who belive in intervallic flexibility  .

One of critics is Hormoz Farhat, composer and musicologist. He writes, "In the course of the twentieth century, three separate theories on intervals and scales of Persian music have been proposed. The first of these, put forward in the 1920s by Vaziri, identifies a 24-quarter-tone scale as the basis for Persian music... Vaziri's quarter-tone theory, which is arrived at by way of a further division of the western equidistant 12-note chromatic scale, is entirely irrelevant to Persian music. It is an artificial creation devised to make possible the adoption of a kind of harmonic practice, based on western tonal harmony. It would be difficult to accept Vaziri was not aware of the fact that Persian music makes no use of the quarter-tone and that intervals other than the semi-tone and the whole-tone are not achieved through multiples of the quartertone. He must simply have believed in the desirability of their being adjusted to correspond to an equidistant quarter-tone scale so that a kind of harmony may be imposed upon the music. Clearly, he did not propose to do this in order to destroy the music, but, as he saw it, to advance its possibilities into the realm of polyphony. He and many other Middle Eastern musicians of the early twentieth century regarded a monophonic musical tradition as intrinsically inferior. Their aim was to make the necessary adjustments so that polyphonic writing could be admitted into their music, and understandably they took western music as their model .

Buno Nettl in his book "The Radif of Persian Music: Studies of Structure And CulturalContext10" writes about inefficiency of  24-EDO and all the rigid models for intervallic structure and says that the intervals are not the first problem of masters and performes in Persian music but melodic sense.

Mohammad Rezâ Darvishi (Persian Composer and Scholar) in Fourth chapter of  his book '' A View on the West (An account of the influence of the western music on Iran’s regional music , P. 212)'' after umerating some intervallic models like farhat's model , criticize Vaziri's opinions and attitudes.Thus, Darvishi argues that Persian music is not based on a precise tuning definition like 24-EDO.

Jean during in his book writes that The koron ( ) which lowers the note by approximately a 1/4 tone, corresponds in fact to intervals varying more or less from one performer to an other. Thusthe interval D -C fluctuates between 34 and 38 savarts (136 and 152 cents)In the same way the pitch of certain tones fluctuates according to the modes. For instance the interval B-C may vary in practice from 20 to 26 s. (80 to 104 c.), according to the musicians and/or the  modes .  The master of tar and radif repertoire of Persian art music , in " ketab-e-sal-e-sheyda ,vol.1 " , believes that system of vaziri ( 24-EDO) has changed the real taste of persian music. the real size of quarter tones is related to size of diatonic semitones. Diatonic semitone in traditional music is smaller than 12-EDO and so the size of quarter tones which are related to them are different from 24-EDO. Lotfi believe that although vaziri was a honourable musician , tar player and researcher but his theory of 24-EDO and his other western minds were historic mistakes in Persian music. Morteza Hannaneh , Persian composer and Hornist , in his book "The lost scale "

After a long discussion about Quarter tones and "Sori – Koron" of  vaziri and works of Dr.Mehdi Barkeshli , Persian acoustician on quarter tones , concludes that 24-Edo can't explain the intervallic structure of Persian music . According  to his researches , each tone is divided into 2 semitone and then the second semitone  is divided into two parts or the tone is divided into 2 unequal 3/4 and 1/4  tone.

24-EDO in Middle East

The twentieth century is marked by an increase in the role of Western theory, notation, instruments, and overall musical attitudes. In his Kitab al-Musiqa al-Sharqi, written around 1904, the Egyptian theorist and composer Kamil al-Khula'i mentioned that the piano, the accordion, and the mouth organ were becoming common household instruments in Egypt. The twentieth century also marked the continuation and growth of a medium that had begun in the nineteenth century and flourished in Egypt: the musical theater. Dramas mainly by European authors were Arabized and presented as combinations of acting, singing, and sometimes dancing. Among the theatrical artists were the Syrian-born Abu Khalil al-Qabbani (1841-1902), who also performed at the Columbian World Fair in Chicago in 1893, and the Egyptian Shaykh Salamah Hijazi (1852-1917), a Sufi-trained singer and stage actor whose theatrical songs were heard on early recordings throughout the entire Arab world.

Between World War I and the late 1920s, Cairo witnessed the rise of a new theatrical form, a type of musical play that typically combined comedy and vaudeville and was comparable to the European operetta. Among the prime contributors to this form was the celebrated composer Shaykh Sayyid Darwish (died 1923), who is now considered the father of modern Egyptian music. By the early 1930s, the impact of Westernization on Egyptian music was considerable, as testified to in the reports issued by the Congress of Arab Music held in Cairo in 1932.

In Egypt and the Levant, theorists divide the octave scale into small microtones comparable to those discussed earlier by al-Farabi and Safi ad-Din. Several types of micro-intervals have been advocated, including the comma division (roughly one-ninth of a whole step), which is found in some Syrian theories. Yet, it is generally conceived that the Maqamat are based on a referential octave scale consisting of twenty-four equal quarter-tones. Despite the essentially aural nature of Arab music, Western notation has become fully established, and extra symbols are widely used. In addition to the regular flat and sharp signs, the symbol lowers a note by approximately a quarter tone while the symbol raises a note by roughly a quarter tone.

Hosam adeeb nashed pages on quarter-tone

Scales of Persian Music

Playing and displaying Persian music with free softwares

Arab tone system

Music Notation in arab and turkish music

Ajnas  اجناس

24-EDO in western world

Writings of Ivor Darreg on quarter-tone

24-EDO and 48-EDO notation chart from Microtonal trumpet

Micro 3 font for quarter-tone

The Extended Helmholtz-Ellis JI Pitch Notation for quarter-tone

The clarinet of the 21 century + Quarter-tones

Quarter tone clarinet

Fingering for quarter-tone flute

List of quarter tone pieces

ON THE NATIVE SONGS OF NEW ZEALAND

Polynesian Mythology and Ancient Traditional History of the New Zealand

24-EDO in Newmusicbox

List of quarter tone pieces

Schönberg's notation of a portion of a quarter-tone scale

Music with quarter-tones - Willi Möllendorf

Quarter tone signs

Quarter tone signs

Quarter tone signs

Quarter-tone marimba and ......

Some quarter tone scales

Micro 3 - SYMBOLS for MICROTONAL (19-, 21-, 24-, 36-, 72-NOTE) MUSIC

Sim's accidentals for 24-EDO and 72-EDO

Quartertone marimba

Skinner, Myles. Quarter Tones. PhD diss.

Tui St. George Tucker , quartertone composer

Below is a list showing advocates of 24-EDO with approximate dates:

1906 -- Richard H. Stein (first published 24-tET scores)

1908 -- Arnold Schönberg (schematic sketch, no extant compositions)

1908 -- Anton Webern (early drafts of two songs)

1916 -- Charles Ives

1917 -- Willi von Möllendorff

1918 -- Jörg Mager

1920 -- Alois Hába (and subsequently many of his students)

1924 -- Julián Carrillo

1932 -- Adopted as standard tuning in Egypt and elsewhere in Arabic world and persian music

1933 -- Ivan Wyschnegradsky (Treatise on Quartertone Harmony)

1940s -- Julián Carrillo, 1/4th-tone piano

1941 -- Mildred Couper

1948 -- Pierre Boulez (original version of Le Soleil des Eaux)

1950s -- Giacinto Scelsi (very loosely-conceived intonation)

1967 -- Tui St. George Tucker

1969 -- Györgi Ligeti (Ramifications)

1960s-2000s -- John Eaton

1980s -- Brian Ferneyhough (very loosely-conceived intonation)

1983 -- Leo de Vries

1994 -- Joe Monzo