# Paper on quarter tones



## xmetalhead69 (Nov 30, 2010)

Heres the story, for the final in my Intro to World Music class, I'm writing my paper about quarter tones. I'm pretty sure some of you have some knowledge on their usage etc. Any information you guys could share with me, particularly citable sources, would be AWESOME. thanks


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## Necris (Dec 1, 2010)

What exactly are you looking for, composers who utilized them or what?


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## Holy Katana (Dec 1, 2010)

You should do it on extended just intonation instead. Much more interesting than quartertones in terms of microtonal music.


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## Varcolac (Dec 1, 2010)

Arabian oud music is rife with quarter tones. It's performed on a fretless instrument so anything's technically possible, but Arab music uses 24-TET.

Arab tone system - Wikipedia, the free encyclopedia


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## xmetalhead69 (Dec 1, 2010)

After talkng with my prof, I'm gonna start with an explanation of whole/semi tones, then go into microtonal music


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## bostjan (Dec 1, 2010)

Microtonal music is so much more than just quarter tones.

Quarter tone music has long been a fundamental part of Persian and Arabic music. The traditional Islamic prayer scales all contain intervals that involve quarter tones, three-quarter tones or one-and-a-quarter tones.

Quarter tones are primarily used for two purposes: 1) to create neutral intervals that are neither major nor minor 2) to create diminished scale intervals.

I don't have the citation, but there was a group that studied the tones that babies make when first attempting musical singing, and the group founf that other than major thirds and fifths, neutral seconds were very common among early vocalization attempts.

using quarter tones, you can make a new microchromatic scale (in equal division tuning):
C Ddb Db Dd D Dt/Edb Eb Ed E Fb F Ft F# Gd G Gt Ab Ad A Bdb Bb Bd B B#
In rough interval notation: r d2 m2 n2 M2 A2/d3 m3 n3 M3 d4 p4 ? A4 d5 p5 A5 m6 n6 M6 d7 m7 n7 M7 A7.

The above example is for 24 notes, as is used in Arabic music, but it's easy to come up with 29 notes or even more with quarter tones.

A 24-EDO quarter tone is 50 cents. A just quarter tone is 41 cents. A 22-EDO quarter tone is 55 cents. A 27-EDO quarter tone is 44 cents. A 29-EDO quarter tone is 41 cents. Many other examples of quarter tone tunings exist, particularly those used in antiquity and in very experimental music.


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## bostjan (Dec 1, 2010)

Also, speaking of step tones generated by differing tunigs-

In just tuning, there are typically two different takes on the same step.

For instance, a whole step can be 204 cents or 182 cents, depending on the context. A half step can be 112 cents or 133 cents.

A neutral step, which is a fundamental step in arabic music, also refered to as a three-quarter step (as a single quarter step is very rarely used in world music) has been described by Arabic music theorists as either 150 cents or 165 cents.

Stepping up from a perfect harmonic seventh (based upon the seventh harmonic), used in barbershop quartets and other highly chordal music, to a just minor sixth is 155 cents, which is very close to a neutral step or three-quarter step.

A quarter step used in just tuning under a diminished context is not related to a three-quarter step used in a neutral context, as a correction step to get out of neutral must be used, that is not equivalent to a just quarter-tone of 41 cents.

Moving from a n2 to a m3 is 151 cents.
n2 to n3 is, in fact 182 cents, a lesser whole step.
n2 to M3 is 221 cents, far narrow of an augmented tone of 275 cents.

Incidentally, going from a M3 to the perfect eleventh harmonic is 165 cents, the same as going from root to n2. And the distance from m6 to h7 is exactly the same as the distance from a diminished fourth to a perfect tritone (155 cents).


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## Mr. Big Noodles (Dec 1, 2010)

Indian music uses an identical scale to the Western diatonic scale, called "Bilawal". They have a solfege system, too, called "Sargam", with the swara (syllables) Sa Re Ga ma Pa Dha Ni (Sa), corresponding to Do Re Mi Fa Sol La Ti (Do). Notes are either natural, sharp (tivra), or flat (komal). Sa is always natural, as is Pa, so five out of the seven pitches can be moved up or down a semitone, which, fully realized, results in a Western chromatic scale. Then, there are ten other pitches between those, for a full twenty-two pitched octave. A lot of books concerning Indian music cover this in some form, and I think that "The Life of Music In North India" by Daniel Neuman covers this decently at some point. If not, check another book. Because of the close relation to Western scales, you're probably best off looking at Indian and Arabic music in the beginning.



bostjan said:


> Incidentally, going from a M3 to the perfect eleventh harmonic is 165 cents, the same as going from root to n2. And the distance from m6 to h7 is exactly the same as the distance from a diminished fourth to a perfect tritone (155 cents).



Perfect tritone? Not augmented fourth or diminished fifth? Tritones are hardly "perfect" intervals; the harmonic ratio of the tritone in many tuning systems is rarely an even ratio, nor is it anywhere near the fundamental in terms of the harmonic series.


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## ixlramp (Dec 1, 2010)

List of musical modes

Scroll down to '24 tone modes'. These are modes in the system of '24 equal divisions of the octave' (24EDO) where each division is a 50 cent quartertone.

The string of numbers represent the number of 50 cent quartertones between consecutive tones of the mode, starting from the tonic: 3 = 150 cents, 5 = 250 cents.

Note that the Arabic modes (Maqam) are only rough approximations of the traditional pitches which are not based on a system of 24 equal divisions. The pitches of a Maqam may have originally been derived from Just Intonation, but over time they have been transferred aurally from master to student and artistically tweaked in very subtle ways. The pitches do not conform to any mathematical theory and there are regional variations.

The above is also the case for Indian Classical music modes (Raga).


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## ixlramp (Dec 1, 2010)

Two examples of 24EDO metal:








Seal of Graphiel, myspace







M.A.N., myspace


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## ixlramp (Dec 1, 2010)

There was an attempt to modernise Arabic music at the Cairo Congress of Arab Music 1932:

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

So Egypt modernised and Turkish music now continues with the traditional pitches.


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## bostjan (Dec 1, 2010)

SchecterWhore said:


> Perfect tritone? Not augmented fourth or diminished fifth?



Correct.



> Tritones are hardly "perfect" intervals; the harmonic ratio of the tritone in many tuning systems is rarely an even ratio, nor is it anywhere near the fundamental in terms of the harmonic series.



You mean in equal temperaments? In just tuning, the tritone is a unique ratio, neither a fourth nor a fifth. It is an interval originating in traditional African music, which does not follow the numbering system we use in post-renaissance western music.

But as soon as you mention quarter tones, you aren't talking about traditional western music.


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## Mr. Big Noodles (Dec 2, 2010)

bostjan said:


> You mean in equal temperaments? In just tuning, the tritone is a unique ratio, neither a fourth nor a fifth. It is an interval originating in traditional African music, which does not follow the numbering system we use in post-renaissance western music.


In well-tempered European tonal and modal music, the tritone is not accepted as a perfect or pure interval. Whatever the case, I was just wondering about your usage of the term, "perfect tritone". I'll look into it, though. Got any sources on the naming? Thanks.



> But as soon as you mention quarter tones, you aren't talking about traditional western music.


Truth. However, "quarter tone" assumes a derivative of the Western chromatic scale in which the symmetry of the chromatic scale is further replicated on the level of the quarter tone in 24 EDO. Microtones, on the other hand, are a different story, and the very term is much more general.



ixlramp said:


> Note that the Arabic modes (Maqam) are only rough approximations of the traditional pitches which are not based on a system of 24 equal divisions. The pitches of a Maqam may have originally been derived from Just Intonation, but over time they have been transferred aurally from master to student and artistically tweaked in very subtle ways. The pitches do not conform to any mathematical theory and there are regional variations.





ixlramp said:


> So Egypt modernised and Turkish music now continues with the traditional pitches.



Thanks for bringing this up. Ottoman (Turkish) modal theory conceives of microtones in terms of ninths of a tone, as illustrated in this diagram from Wikipedia:






I took a workshop on Baltic music, and the instructor said that there isn't a whole lot of agreement on how sharp or flat a note will be in any given maqam (or at least it's divided by region), although I have no way of verifying this.


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## xmetalhead69 (Dec 2, 2010)

I love you guys! Also glad this only has to be 3 pages


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## bostjan (Dec 2, 2010)

SchecterWhore said:


> In well-tempered European tonal and modal music, the tritone is not accepted as a perfect or pure interval. Whatever the case, I was just wondering about your usage of the term, "perfect tritone". I'll look into it, though. Got any sources on the naming? Thanks.
> 
> Truth. However, "quarter tone" assumes a derivative of the Western chromatic scale in which the symmetry of the chromatic scale is further replicated on the level of the quarter tone in 24 EDO. Microtones, on the other hand, are a different story, and the very term is much more general.



A quarter tone certainly can derive from a western half tone. In the last 250 years or so, however, western music has become reliant on the chromatic scale. 

{To be fair, equal divided octave tuning was not a western idea, but was borrowed from eastern cultures. China, Indonesia, Persia and Mesopotamia all had the idea of equal temperaments hundreds, if not thousands, of years before the idea caught on in the west.}

Prior to the late 18th century, almost all western music was based on just tuning interval theory. Basic intervals like perfect fifth and major third predominated western compositional approaches. 

In African music, the scale tones didn't stick to such a numbering scheme. Although the name "tritone" is taken from the western idea of three whole tones, the term as I use it, and as it was explained to me, refers to a just interval of 7:5 borrowed from traditional African music~ which is the context in which it came into use in American blues music and subsequently into rock music.

In a purely western context, a tritone would consist of three whole tones: root to major second, major second to major third, and major third to augmented fourth. A quadratone then would be an augmented fifth and a pentatone an augmented sixth. In this case a hexatone would be an augmented seventh, or an octave if a comma is introduced, or if tones are limited through equal temperament to exclude an augmented seventh from existing in the tuning system.

The tritone really doesn't have that much to do with quarter tones, though. It has a lot more to do with seven-limit just tuning. I think I only mentioned it because it was approximately 3/4 of a tone from the diminished fourth.

As a guitarist, I like to think of quarter tones in a 24-EDO sense, but in a broader sense there is a lot more to it. Certainly, discussing quarter-tones with a specific context in mind is much easier than discussing them in a completely open context. -and even easier to discuss with a specific tuning system and context in mind, such as their use in modern arabic music with 24-EDO tuning.


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## ixlramp (Dec 2, 2010)

Here's how to re-string a conventionally fretted guitar to play in 24EDO ...

Restring with 3 identical pairs of strings, the pairs tuned in unison, and with fifth intervals between the pairs. You'll need to buy single strings to put this set together and the intonation will need to be reset. Check that the nut slots are large enough for the strings you intend to use.

For example:

D ..............high
D
G
G
C
C ..............low

Then tune up one of each pair by 50 cents, easy to do with a normal guitar tuner. When the tuner flips back and forth between one chromatic note and the next non-stop, that is 50 cents.

D +50 cents ........high
D
G +50 cents
G
C +50 cents
C .........................low

So you then have the conventional notes on the C G D strings. Directly above each conventional note on the same fret you have a quartertone sharp version of that note. This enables quartertone / 24EDO scales to be played and lots of undecimal just intervals.

The reason for tuning the pairs in fifths is to obtain a reasonable pitch range, since this method does reduce the range of a guitar.


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## ixlramp (Dec 2, 2010)

Maqam World, a good site explaining the Arabic Maqamat. Useful page here.


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## Mr. Big Noodles (Dec 2, 2010)

Excellent posts, ixlramp. I'd like to try out that quarter tone conversion for standard guitar frettings; the alterations to left hand technique would be interesting.


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## ixlramp (Dec 2, 2010)

SchecterWhore, I call it 'Bichromatic tuning', I came up with the idea independently but of course it had been done before. The tuning can be tweaked to play several simple Just Intonation scales too.

Lots of detail here at Not Only Music.

The tuning reduces the range of the guitar so an 8 or 10 string would be perfect.


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## Mr. Big Noodles (Dec 3, 2010)

ixlramp said:


> I call it 'Bichromatic tuning'



Do you know how official that sounds? If I had money, I'd give you a research grant right now.


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## GJaunz (Dec 3, 2010)

I found this dude on youtube who added frets to his guitar to get quartertones in there. pretty cool stuff, good luck on the paper!


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## ixlramp (Dec 3, 2010)

SchecterWhore said:


> Do you know how official that sounds?


Yeah! It does


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## ixlramp (Dec 3, 2010)

Recently I thought up a new version of bichromatic tuning that is more suited to chords, and perhaps a little less suited to melody. This is the tuning I briefly mentioned to bostjan a while back. I'll go into detail in another thread, but briefly, for 24EDO the tuning is all-neutral-thirds ...

2400+50c
1200+900c
1200+550c
1200+200c
1050c
700c
350c
0c

The tuning follows the classical chord structure of Root 3rd 5th 7th 9th 11th etc.

Playing across a single fret gives you a neutral triad / neutral seventh / neutral ninth chord. The 11th + 3rd harmonic 0-550-700c chord is easily played.

I need more strings ...


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