Quarter tone

Quarter tone

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Encyclopedia
A quarter tone , is a pitch halfway between the usual notes of a chromatic scale, an interval
Interval (music)
In music theory, an interval is a combination of two notes, or the ratio between their frequencies. Two-note combinations are also called dyads...

 about half as wide (aurally, or logarithmically) as a semitone
Semitone
A semitone, also called a half step or a half tone, is the smallest musical interval commonly used in Western tonal music, and it is considered the most dissonant when sounded harmonically....

, which is half a whole tone
Major second
In Western music theory, a major second is a musical interval spanning two semitones, and encompassing two adjacent staff positions . For example, the interval from C to D is a major second, as the note D lies two semitones above C, and the two notes are notated on adjacent staff postions...

.


Many composers are known for having written music including quarter tones or the quarter tone scale, first proposed by 19th-century music theorist Mikha'il Mishaqah, including: Pierre Boulez
Pierre Boulez
Pierre Boulez is a French composer of contemporary classical music, a pianist, and a conductor.-Early years:Boulez was born in Montbrison, Loire, France. As a child he began piano lessons and demonstrated aptitude in both music and mathematics...

, Julián Carrillo
Julián Carrillo
Julián Carrillo Trujillo was a Mexican composer, conductor, violinist and music theorist, famous for developing a theory of microtonal music which he dubbed "The Thirteenth Sound" .-Biography:...

, Mildred Couper
Mildred Couper
Mildred Couper , prominent American composer and pianist, was one of the first musicians to experiment with quarter-tone music. She was based in Santa Barbara, California, but her music and influence were felt around the world...

, Alberto Ginastera
Alberto Ginastera
Alberto Evaristo Ginastera was an Argentine composer of classical music. He is considered one of the most important Latin American classical composers.- Biography :...

, Gérard Grisey
Gérard Grisey
Gérard Grisey was a French composer of contemporary music.-Biography:Gérard Grisey was born in Belfort, France on 17 June 1946. He studied at the Trossingen Conservatory in Germany from 1963 to 1965 before entering the Conservatoire de Paris...

, Alois Hába
Alois Hába
Alois Hába was a Czech composer, musical theorist and teacher. He is primarily known for his microtonal compositions, especially using the quarter tone scale, though he used others such as sixth-tones and twelfth-tones....

, Ljubica Marić
Ljubica Maric
Ljubica Marić was considered to be one of the most original composers to emerge from Yugoslavia. She was a pupil of Josip Štolcer-Slavenski. She was known for being inspired by Byzantine Orthodox church music...

, Charles Ives
Charles Ives
Charles Edward Ives was an American modernist composer. He is one of the first American composers of international renown, though Ives' music was largely ignored during his life, and many of his works went unperformed for many years. Over time, Ives came to be regarded as an "American Original"...

, Tristan Murail
Tristan Murail
Tristan Murail is a French composer. His father, Gérard Murail, is a poet and his mother, Marie-Thérèse Barrois, a journalist. One of his brothers, Lorris Murail, and his younger sister Elvire Murail, aka Moka, also write, and his younger sister Marie-Aude Murail is a French children's writer...

, Krzysztof Penderecki
Krzysztof Penderecki
Krzysztof Penderecki , born November 23, 1933 in Dębica) is a Polish composer and conductor. His 1960 avant-garde Threnody to the Victims of Hiroshima for string orchestra brought him to international attention, and this success was followed by acclaim for his choral St. Luke Passion. Both these...

, Giacinto Scelsi
Giacinto Scelsi
Giacinto Scelsi , Count of Ayala Valva was an Italian composer who also wrote surrealist poetry in French....

, Karlheinz Stockhausen
Karlheinz Stockhausen
Karlheinz Stockhausen was a German composer, widely acknowledged by critics as one of the most important but also controversial composers of the 20th and early 21st centuries. Another critic calls him "one of the great visionaries of 20th-century music"...

, Tui St. George Tucker
Tui St. George Tucker
Tui St. George Tucker was an American composer and recorder player....

, Ivan Alexandrovich Wyschnegradsky, and Iannis Xenakis
Iannis Xenakis
Iannis Xenakis was a Romanian-born Greek ethnic, naturalized French composer, music theorist, and architect-engineer. He is commonly recognized as one of the most important post-war avant-garde composers...

 (see List of quarter tone pieces).

Types of quarter tones


The term quarter tone can refer to a number of different intervals, all very close in size. For example, some 17th- and 18th-century theorists used the term to describe the distance between a sharp and enharmonically distinct flat in mean-tone temperaments (e.g., D–E). In the quarter tone scale, also called 24 tone equal temperament (24-TET), the quarter tone is 50 cents
Cent (music)
The cent is a logarithmic unit of measure used for musical intervals. Twelve-tone equal temperament divides the octave into 12 semitones of 100 cents each...

, or a frequency ratio of 21/24 or approximately 1.0293, and divides the octave into 24 equal steps (equal temperament
Equal temperament
An equal temperament is a musical temperament, or a system of tuning, in which every pair of adjacent notes has an identical frequency ratio. As pitch is perceived roughly as the logarithm of frequency, this means that the perceived "distance" from every note to its nearest neighbor is the same for...

). In this scale the quarter tone is the smallest step. A semitone is thus made of two steps, and three steps make a three-quarter tone or neutral second
Neutral second
A neutral second or medium second is a musical interval wider than a minor second and narrower than a major second. Three distinct intervals may be termed neutral seconds:...

, half of a minor third
Minor third
In classical music from Western culture, a third is a musical interval encompassing three staff positions , and the minor third is one of two commonly occurring thirds. The minor quality specification identifies it as being the smallest of the two: the minor third spans three semitones, the major...

.

In just intonation
Just intonation
In music, just intonation is any musical tuning in which the frequencies of notes are related by ratios of small whole numbers. Any interval tuned in this way is called a just interval. The two notes in any just interval are members of the same harmonic series...

 the quarter tone can be represented as 36:35 or 33:32, approximately half the semitone of 16:15 or 25:24. The ratio of 36:35 is only 1.23 cents narrower than a 24-TET quarter tone. This just ratio is also the difference between a minor third
Minor third
In classical music from Western culture, a third is a musical interval encompassing three staff positions , and the minor third is one of two commonly occurring thirds. The minor quality specification identifies it as being the smallest of the two: the minor third spans three semitones, the major...

 (6:5) and septimal minor third
Septimal minor third
In music, the septimal minor third , also called the subminor third, is the musical interval exactly or approximately equal to a 7/6 ratio of frequencies. In terms of cents, it is 267 cents, a quartertone of size 36/35 flatter than a just minor third of 6/5...

 (7:6).

Quarter tones and intervals close to them also occur in a number of other equally tempered tuning systems. 22-TET
22 equal temperament
In music, 22 equal temperament, called 22-tet, 22-edo, or 22-et, is the tempered scale derived by dividing the octave into 22 equal steps . Each step represents a frequency ratio of 21/22, or 54.55 cents ....

 contains an interval of 54.55 cents, slightly wider than a quarter-tone, whereas 53-TET
53 equal temperament
In music, 53 equal temperament, called 53-TET, 53-EDO, or 53-ET, is the tempered scale derived by dividing the octave into 53 equal steps . Each step represents a frequency ratio of 21/53, or 22.6415 cents , an interval sometimes called the Holdrian comma.- History :Theoretical interest in this...

 has an interval of 45.28 cents, slightly smaller. 72-TET
72 equal temperament
In music, 72 equal temperament, called twelfth-tone, 72-tet, 72-edo, or 72-et, is the tempered scale derived by dividing the octave into twelfth-tones, or in other words 72 equal steps...

 also has equally-tempered quarter-tones, and indeed contains 3 quarter tone scales, since 72 is divisible by 24.

Composer Ben Johnston, to accommodate the just septimal quarter tone
Septimal quarter tone
A septimal quarter-tone is an interval with the ratio of 36:35 , which is the difference between the septimal minor third and the Just minor third , or about 48.77 cents wide. The name derives from the interval being the 7-limit approximation of a quarter tone...

, uses a small "7" as an accidental to indicate a note is lowered 49 cents, or an upside down "" to indicate a note is raised 49 cents, or a ratio of 36/35. Johnston uses an upward and downward arrow to indicate a note is raised or lowered by a ratio of 33/32, or 53 cents.

Playing quarter tones on musical instruments



Because many musical instruments manufactured today are designed for the 12-tone scale, not all are usable for playing quarter tones. Sometimes special playing techniques must be used.

Conventional musical instruments which cannot play quarter tones (except by using special techniques—see below) include
  • Normally fret
    Fret
    A fret is a raised portion on the neck of a stringed instrument, that extends generally across the full width of the neck. On most modern western instruments, frets are metal strips inserted into the fingerboard...

    ted string instrument
    String instrument
    A string instrument is a musical instrument that produces sound by means of vibrating strings. In the Hornbostel-Sachs scheme of musical instrument classification, used in organology, they are called chordophones...

    s
  • Piano
    Piano
    The piano is a musical instrument played by means of a keyboard. It is one of the most popular instruments in the world. Widely used in classical and jazz music for solo performances, ensemble use, chamber music and accompaniment, the piano is also very popular as an aid to composing and rehearsal...

    s
  • Organs
  • Synthesizers (when design does not permit)
  • Accordion
    Accordion
    The accordion is a box-shaped musical instrument of the bellows-driven free-reed aerophone family, sometimes referred to as a squeezebox. A person who plays the accordion is called an accordionist....

    s
  • Pitched percussion instruments, when tuning does not permit and normal techniques are used


Conventional musical instruments which can play quarter tones include
  • Synthesizers (if design permits)
  • Fretless string instruments (on fretted string instruments it is possible with bending or special tuning)
  • Quarter-tone fretted string instruments
  • Slide brass instrument
    Brass instrument
    A brass instrument is a musical instrument whose sound is produced by sympathetic vibration of air in a tubular resonator in sympathy with the vibration of the player's lips...

    s (trombone
    Trombone
    The trombone is a musical instrument in the brass family. Like all brass instruments, sound is produced when the player’s vibrating lips cause the air column inside the instrument to vibrate...

    )
  • Valved brass instruments (trumpet
    Trumpet
    The trumpet is the musical instrument with the highest register in the brass family. Trumpets are among the oldest musical instruments, dating back to at least 1500 BCE. They are played by blowing air through closed lips, producing a "buzzing" sound which starts a standing wave vibration in the air...

    , horn
    Horn (instrument)
    The horn is a brass instrument consisting of about of tubing wrapped into a coil with a flared bell. A musician who plays the horn is called a horn player ....

    , tuba
    Tuba
    The tuba is the largest and lowest-pitched brass instrument. Sound is produced by vibrating or "buzzing" the lips into a large cupped mouthpiece. It is one of the most recent additions to the modern symphony orchestra, first appearing in the mid-19th century, when it largely replaced the...

    )
  • Woodwind instruments, using special fingering or bending.
    • Flute
      Flute
      The flute is a musical instrument of the woodwind family. Unlike woodwind instruments with reeds, a flute is an aerophone or reedless wind instrument that produces its sound from the flow of air across an opening...

    • Recorder
      Recorder
      The recorder is a woodwind musical instrument of the family known as fipple flutes or internal duct flutes—whistle-like instruments which include the tin whistle. The recorder is end-blown and the mouth of the instrument is constricted by a wooden plug, known as a block or fipple...

    • Clarinet
      Clarinet
      The clarinet is a musical instrument of woodwind type. The name derives from adding the suffix -et to the Italian word clarino , as the first clarinets had a strident tone similar to that of a trumpet. The instrument has an approximately cylindrical bore, and uses a single reed...

    • Oboe
      Oboe
      The oboe is a double reed musical instrument of the woodwind family. In English, prior to 1770, the instrument was called "hautbois" , "hoboy", or "French hoboy". The spelling "oboe" was adopted into English ca...

    • Saxophone
      Saxophone
      The saxophone is a conical-bore transposing musical instrument that is a member of the woodwind family. Saxophones are usually made of brass and played with a single-reed mouthpiece similar to that of the clarinet. The saxophone was invented by the Belgian instrument maker Adolphe Sax in 1846...

    • Bassoon
      Bassoon
      The bassoon is a woodwind instrument in the double reed family that typically plays music written in the bass and tenor registers, and occasionally higher. Appearing in its modern form in the 19th century, the bassoon figures prominently in orchestral, concert band and chamber music literature...

  • Harmonica
    Harmonica
    The harmonica, also called harp, French harp, blues harp, and mouth organ, is a free reed wind instrument used primarily in blues and American folk music, jazz, country, and rock and roll. It is played by blowing air into it or drawing air out by placing lips over individual holes or multiple holes...

  • Harp
    Harp
    The harp is a multi-stringed instrument which has the plane of its strings positioned perpendicularly to the soundboard. Organologically, it is in the general category of chordophones and has its own sub category . All harps have a neck, resonator and strings...

  • Pianos, if specially tuned
  • Pitched percussion instruments, when tuning permits, or using special techniques


Experimental instruments have been built to play in quarter tones, for example a quarter tone clarinet
Quarter tone clarinet
A quarter tone clarinet is an experimental clarinet designed to play music using quarter tone intervals. Using special fingerings, quarter tones may be produced by a skilled player on a conventional clarinet. However, such fingerings are awkward in rapid passages, and results tend to vary from one...

 by Fritz Schüller (1883–1977) of Markneukirchen
Markneukirchen
Markneukirchen is a town in the Vogtlandkreis district, in the Free State of Saxony, Germany. It lies in between the Erzgebirge and the Fichtelgebirge in the Elstergebirge, southeast of Plauen, and northeast of Asch ....

.

Other instruments can be used to play quarter tones when using audio signal processing
Audio signal processing
Audio signal processing, sometimes referred to as audio processing, is the intentional alteration of auditory signals, or sound. As audio signals may be electronically represented in either digital or analog format, signal processing may occur in either domain...

 effects such as pitch shifting.

Pairs of conventional instruments tuned a quarter tone apart can be used to play some quarter tone music. Indeed, "quarter tone pianos" have been built which consist essentially of two pianos stacked one above the other in a single case, one tuned a quarter tone higher than the other.

Music of the Middle East


While the use of quarter tones in modern Western music is a more recent and experimental phenomenon, these and other microtonal intervals have been an important part of the music of the Iran
Iran
Iran , officially the Islamic Republic of Iran , is a country in Southern and Western Asia. The name "Iran" has been in use natively since the Sassanian era and came into use internationally in 1935, before which the country was known to the Western world as Persia...

 (Persia), Arab world
Arab world
The Arab world refers to Arabic-speaking states, territories and populations in North Africa, Western Asia and elsewhere.The standard definition of the Arab world comprises the 22 states and territories of the Arab League stretching from the Atlantic Ocean in the west to the Arabian Sea in the...

, Armenia
Armenia
Armenia , officially the Republic of Armenia , is a landlocked mountainous country in the Caucasus region of Eurasia...

, Turkey
Turkey
Turkey , known officially as the Republic of Turkey , is a Eurasian country located in Western Asia and in East Thrace in Southeastern Europe...

, Assyria
Assyria
Assyria was a Semitic Akkadian kingdom, extant as a nation state from the mid–23rd century BC to 608 BC centred on the Upper Tigris river, in northern Mesopotamia , that came to rule regional empires a number of times through history. It was named for its original capital, the ancient city of Assur...

, Kurdistan and neighboring lands and areas for many centuries.


Many Arabic maqamat
Arabic maqam
Arabic maqām is the system of melodic modes used in traditional Arabic music, which is mainly melodic. The word maqam in Arabic means place, location or rank. The Arabic maqam is a melody type...

 contain intervals of three-quarter tone size; a short list of these follows.
  1. Shoor (Bayati)
    شور (بیاتی)
    D E F G A B C D
  2. Hussayni
  3. Siga
    سيكاه
    E F G A B C D E
  4. Rast
    راست
    C D E F G A B C
    with a B replacing the B in the descending scale
  5. ‘Ajam
    • Sabba
      صبا
      D E F G A B C D


The persian philosopher and scientist Al-Farabi
Al-Farabi
' known in the West as Alpharabius , was a scientist and philosopher of the Islamic world...

 described a number of intervals in his work in music, including a number of quarter tones.

Assyrian/Syriac Church Music Scale:
  • 1 - Qadmoyo (Bayati)
  • 2 - Trayono (Hussayni)
  • 3 - Tlithoyo (Segah)
  • 4 - Rbi‘oyo (Rast)
  • 5 - Hmishoyo
  • 6 - Shtithoyo (‘Ajam)
  • 7 - Shbi‘oyo
  • 8 - Tminoyo

Quarter tone scale



Known as gadwal in Arabic, the quarter tone scale was developed in the Middle East in the eighteenth century and many of the first detailed writings in the nineteenth century Syria describe the scale as being of 24 equal tones. The invention of the scale is attributed to Mikhail Mishaqa
Mikhail Mishaqa
Mikhail Mishaqa born in Rashmayyā, Lebanon is "the first historian of modern Ottoman Syria" as well as the "virtual founder of the twenty-four equal quarter tone scale".Mikhail's great-grandfather converted to Catholicism...

 whose work Essay on the Art of Music for the Emir Shihāb (al-Risāla al-shihābiyya fi 'l-ṣinā‘a al-mūsīqiyya) is devoted to the topic but also makes clear his teacher Sheikh Muhammad al-‘Attār (1764-1828) was one of many already familiar with the concept.

The quarter tone scale may be primarily considered a theoretical
Music theory
Music theory is the study of how music works. It examines the language and notation of music. It seeks to identify patterns and structures in composers' techniques across or within genres, styles, or historical periods...

 construct in Arabic music. The quarter tone gives musicians a "conceptual map" with which to discuss and compare intervals by number of quarter tones and this may be one of the reasons it accompanies a renewed interest in theory, with instruction in music theory being a mainstream requirement since that period.

Previously, pitches of a mode were chosen from a scale consisting of seventeen tones, developed by Safi 'I-Din al-Urmawi in the thirteenth century.

In popular music


The Japanese multi-instrumentalist and experimental musical instrument builder Yuichi Onoue developed a 24-TET quarter tone tuning on his guitar.
Norwegian guitarist Ronni Le Tekro
Ronni Le Tekro
Ronni Le Tekrø is a Norwegian guitarist best known for playing with the Norwegian hard rock band TNT and as a solo guitarist....

 of the band TNT (band)
TNT (band)
TNT is a Norwegian hard rock/glam metal band from Trondheim, formed in 1982. The band has released twelve studio albums, three EPs and two live albums while going through numerous line-up changes since its formation...

 used a quarter-step guitar on the band's third studio album, Intuition
Intuition
Intuition may refer to:*Intuition , immediate a priori knowledge or experiential belief*Intuition , acquiring beliefs in ways that bypass ordinary justificationIn music:*Intuition...

.

Ancient Greek tetrachords


The enharmonic genus
Enharmonic genus
The enharmonic genus has historically been the most mysterious and controversial of the three Greek genera of tetrachords. Its characteristic interval is a major third, leaving the remainder of the tetrachord to be divided by two intervals smaller than a semitone...

 of the Greek
Ancient Greece
Ancient Greece is a civilization belonging to a period of Greek history that lasted from the Archaic period of the 8th to 6th centuries BC to the end of antiquity. Immediately following this period was the beginning of the Early Middle Ages and the Byzantine era. Included in Ancient Greece is the...

 tetrachord
Tetrachord
Traditionally, a tetrachord is a series of three intervals filling in the interval of a perfect fourth, a 4:3 frequency proportion. In modern usage a tetrachord is any four-note segment of a scale or tone row. The term tetrachord derives from ancient Greek music theory...

 consisted of a ditone
Ditone
In music, a ditone is the interval 81:64, which is 407.82 cents. It is the major third in Pythagorean tuning. The ditone is evenly divisible by two major tones and is wider than a just major third by a syntonic comma...

 or an approximate major third
Major third
In classical music from Western culture, a third is a musical interval encompassing three staff positions , and the major third is one of two commonly occurring thirds. It is qualified as major because it is the largest of the two: the major third spans four semitones, the minor third three...

 and a semitone
Semitone
A semitone, also called a half step or a half tone, is the smallest musical interval commonly used in Western tonal music, and it is considered the most dissonant when sounded harmonically....

 which was divided into two microtones. Aristoxenos, Didymos
Didymos
Didymos was an ancient Greek music theorist in the last century before Christ. He was a predecessor of Ptolemy at the library at Alexandria.- Theory :We know of his theory only indirectly from the works of Porphyry and Ptolemy...

 and others presented the semitone as being divided into two approximate quarter tone intervals of about the same size, while other ancient Greek theorists described the microtones resulting from dividing the semitone of the enharmonic genus as unequal in size (i.e., one smaller than a quarter tone and one larger) .

Interval size in equal temperament


Here are the sizes of some common intervals in a 24-note equally tempered scale, with the interval names proposed by Alois Hába
Alois Hába
Alois Hába was a Czech composer, musical theorist and teacher. He is primarily known for his microtonal compositions, especially using the quarter tone scale, though he used others such as sixth-tones and twelfth-tones....

 (neutral third, etc.) and Ivan Wyschnegradsky (major fourth, etc.):
interval name size (steps) size (cents) midi just ratio just (cents) midi error
octave
Octave
In music, an octave is the interval between one musical pitch and another with half or double its frequency. The octave relationship is a natural phenomenon that has been referred to as the "basic miracle of music", the use of which is "common in most musical systems"...

24 1200 2:1 1200.00 0.00
semidiminished octave 23 1150 2:1 1200.00 −50.00
supermajor seventh 23 1150 35:18 1151.23 −1.23
major seventh
Major seventh
In classical music from Western culture, a seventh is a musical interval encompassing seven staff positions , and the major seventh is one of two commonly occurring sevenths. It is qualified as major because it is the larger of the two...

22 1100 15:8 1088.27 +11.73
neutral seventh
Neutral seventh
A neutral seventh is a musical interval wider than a minor seventh but narrower than a major seventh . Four distinct intervals may be termed neutral sevenths:* A septimal neutral seventh has a ratio of 64:35 or about 1045 cents....

21 1050 11:6 1049.36 +0.64
minor seventh
Minor seventh
In classical music from Western culture, a seventh is a musical interval encompassing seven staff positions , and the minor seventh is one of two commonly occurring sevenths. The minor quality specification identifies it as being the smallest of the two: the minor seventh spans ten semitones, the...

20 1000 16:9 996.09 +3.91
supermajor sixth
Supermajor sixth
In music and tuning, a supermajor sixth is a supermajor interval that is noticeably wider than a major sixth but noticeably narrower than an augmented sixth, and may be a just interval of 12:7....

/subminor seventh
19 950 7:4 968.83 −18.83
major sixth
Major sixth
In classical music from Western culture, a sixth is a musical interval encompassing six staff positions , and the major sixth is one of two commonly occurring sixths. It is qualified as major because it is the largest of the two...

18 900 5:3 884.36 +15.64
neutral sixth
Neutral sixth
A neutral sixth is a musical interval wider than a minor sixth but narrower than a major sixth . Three distinct intervals may be termed neutral sixths:...

17 850 18:11 852.59 −2.59
minor sixth
Minor sixth
-Subminor sixth:In music, a subminor sixth or septimal sixth is an interval that is noticeably narrower than a minor sixth but noticeably wider than a diminished sixth.The sub-minor sixth is an interval of a 14:9 ratio or alternately 11:7....

16 800 8:5 813.69 −13.69
subminor sixth 15 750 14:9 764.92 −14.92
perfect fifth
Perfect fifth
In classical music from Western culture, a fifth is a musical interval encompassing five staff positions , and the perfect fifth is a fifth spanning seven semitones, or in meantone, four diatonic semitones and three chromatic semitones...

14 700 3:2 701.95 −1.95
lesser septimal tritone
Septimal tritone
The lesser septimal tritone is the interval with ratio 7:5 . The inverse of that interval, the greater septimal tritone, is an interval with ratio 10:7...

12 600 7:5 582.51 +17.49
undecimal tritone or semi-augmented fourth
Major fourth
In music, a major fourth is the interval from the quarter tone scale, named by Ivan Wyschnegradsky, between the perfect fourth and the augmented fourth and thus 550 cents...

11 550 11:8 551.32 −1.32
perfect fourth
Perfect fourth
In classical music from Western culture, a fourth is a musical interval encompassing four staff positions , and the perfect fourth is a fourth spanning five semitones. For example, the ascending interval from C to the next F is a perfect fourth, as the note F lies five semitones above C, and there...

10 500 4:3 498.05 +1.95
tridecimal major third
Major third
In classical music from Western culture, a third is a musical interval encompassing three staff positions , and the major third is one of two commonly occurring thirds. It is qualified as major because it is the largest of the two: the major third spans four semitones, the minor third three...

9 450 13:10 454.21 −4.21
septimal major third
Septimal major third
In music, the septimal major third , also called the supermajor third and sometimes Bohlen–Pierce third is the musical interval exactly or approximately equal to a just 9:7 ratio of frequencies, or alternately 14:11. It is equal to 435 cents, sharper than a just major third by the septimal...

9 450 9:7 435.08 +14.92
major third
Major third
In classical music from Western culture, a third is a musical interval encompassing three staff positions , and the major third is one of two commonly occurring thirds. It is qualified as major because it is the largest of the two: the major third spans four semitones, the minor third three...

8 400 5:4 386.31 +13.69
undecimal neutral third
Neutral third
A neutral third is a musical interval wider than a minor third but narrower than a major third . Three distinct intervals may be termed neutral thirds:...

7 350 11:9 347.41 +2.59
minor third
Minor third
In classical music from Western culture, a third is a musical interval encompassing three staff positions , and the minor third is one of two commonly occurring thirds. The minor quality specification identifies it as being the smallest of the two: the minor third spans three semitones, the major...

6 300 6:5 315.64 −15.64
septimal minor third
Septimal minor third
In music, the septimal minor third , also called the subminor third, is the musical interval exactly or approximately equal to a 7/6 ratio of frequencies. In terms of cents, it is 267 cents, a quartertone of size 36/35 flatter than a just minor third of 6/5...

5 250 7:6 266.88 −16.88
tridecimal minor third 5 250 15:13 247.74 +2.26
septimal whole tone
Septimal whole tone
In music, the septimal whole tone, septimal major second, or supermajor second is the musical interval exactly or approximately equal to a 8/7 ratio of frequencies. It is about 231 cents wide in just intonation. Although 24 equal temperament does not match this interval particularly well, its...

5 250 8:7 231.17 +18.83
whole tone, major tone 4 200 9:8 203.91 −3.91
whole tone, minor tone 4 200 10:9 182.40 +17.60
neutral second
Neutral second
A neutral second or medium second is a musical interval wider than a minor second and narrower than a major second. Three distinct intervals may be termed neutral seconds:...

, greater undecimal
3 150 11:10 165.00 −15.00
neutral second, lesser undecimal 3 150 12:11 150.64 −0.64
15:14 semitone 2 100 15:14 119.44 −19.44
diatonic semitone, just
Just intonation
In music, just intonation is any musical tuning in which the frequencies of notes are related by ratios of small whole numbers. Any interval tuned in this way is called a just interval. The two notes in any just interval are members of the same harmonic series...

2 100 16:15 111.73 −11.73
21:20 semitone 2 100 21:20 84.47 +15.53
28:27 semitone 1 50 28:27 62.96 −8.42
septimal quarter tone
Septimal quarter tone
A septimal quarter-tone is an interval with the ratio of 36:35 , which is the difference between the septimal minor third and the Just minor third , or about 48.77 cents wide. The name derives from the interval being the 7-limit approximation of a quarter tone...

1 50 36:35 48.77 +1.23


Moving from 12-TET
Equal temperament
An equal temperament is a musical temperament, or a system of tuning, in which every pair of adjacent notes has an identical frequency ratio. As pitch is perceived roughly as the logarithm of frequency, this means that the perceived "distance" from every note to its nearest neighbor is the same for...

 to 24-TET allows the better approximation of a number of intervals. Intervals matched particularly closely include the neutral second
Neutral second
A neutral second or medium second is a musical interval wider than a minor second and narrower than a major second. Three distinct intervals may be termed neutral seconds:...

, neutral third
Neutral third
A neutral third is a musical interval wider than a minor third but narrower than a major third . Three distinct intervals may be termed neutral thirds:...

, and (11:8) ratio, or the 11th harmonic. The septimal minor third
Septimal minor third
In music, the septimal minor third , also called the subminor third, is the musical interval exactly or approximately equal to a 7/6 ratio of frequencies. In terms of cents, it is 267 cents, a quartertone of size 36/35 flatter than a just minor third of 6/5...

 and septimal major third
Septimal major third
In music, the septimal major third , also called the supermajor third and sometimes Bohlen–Pierce third is the musical interval exactly or approximately equal to a just 9:7 ratio of frequencies, or alternately 14:11. It is equal to 435 cents, sharper than a just major third by the septimal...

 are approximated rather poorly; the (13:10) and (15:13) ratios, involving the 13th harmonic, are matched very closely. Overall, 24-TET can be viewed as matching the 11th harmonic more closely than the 7th.

See also

  • Temperament
  • Microtonal music
    Microtonal music
    Microtonal music is music using microtones—intervals of less than an equally spaced semitone. Microtonal music can also refer to music which uses intervals not found in the Western system of 12 equal intervals to the octave.-Terminology:...

  • List of quarter tone pieces
  • List of meantone intervals
  • Tonsilabo
    Tonsilabo
    A tonsilabo is a syllable describing a tone of a certain relative frequency. Tonsilabos are used for writing or typing music in the form of text, as an alternative to music notation by design .-History:...

    s

Further reading

  • Bartolozzi, Bruno (1967). New Sounds for Woodwind. London, New York: Oxford University Press.
  • Bousted, Donald (2002). "Microtonality, the Recorder and the Quarter-Tone Recorder Manual". The Recorder Magazine 22, no. 3 (Fall): 99–102.
  • Bousted, Donald (2005). "Next Step Quarter-Tone Resources: Melody". The Recorder Magazine 25, no. 3 (Fall): 88–91.
  • Caravan, Ronald R. (1979). Preliminary Exercises and Etudes in Contemporary Techniques for Clarinet: Introductory Material for the Study of Multiphonics, Quarter Tones, and Timbre Variation. [Oswego, N.Y.]: Ethos Publications.
  • Ellis, Don (1975). Quarter Tones: A Text with Musical Examples, Exercises and Etudes. Plainview, N.Y.: Harold Branch Pub. Co.
  • MacDonald, John (1822). A Treatise on the Harmonic System Arising from the Vibrations of the Aliquot Divisions of Strings According to the Gradual Progress of the Notes from the Middle, to the Remote Extremes: Explaining Simply, by Curved Delineations, the Manner in Which the Harmonic Tones, Half and Quarter Notes, Are Generated and Produced on Every Corresponding Part of the String; and under a Copious Explanatory Description Illustrated by Musical and Appropriate Plates, Giving an Easy and Familiar Adaptation of the Whole to the Purposes of Composition and Instrumental Music, and More Particularly, to the Practice of the Violin, Tenor, Violoncello and Double Bass, on All the Strings, and in Every Compass of These Instruments, by Every Practical Mode of Execution; with Some Musical Animadversions Introductory of the General Subject, Briefly Alluding to the Rise and Progress of Music, and to the Corrections of Temperament: and Stating Various Improvements of Instruments, Experimentally Ascertained: Concluding with an Application or Two of the Principle of Musical Notes, to Purposes of Utility, and a Reference to Terms Less Generally Noticed. London: Printed for the Author, and Sold by T. Preston.
  • Möllendorff, Willi, and Joe Monzo (2001). Music with Quarter-Tones: Experiences at the Bichromatic Harmonium. [United States]: J. Monzo.
  • Rees, Carla (2007). "Eva Kingma and the Quarter-Tone Flute". Pan: The Flute Magazine 26, no. 4:23-29.
  • Rewoldt, Todd (2000). "Altissimo Quarter-Tones for the Alto Saxophone". Saxophone Symposium 25:56–69.

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