96-EDO
MICROTONAL ACCIDENTALS
AND
INTERVAL NAMING SYSTEM
FOR
PERSIAN MUSIC
(BASED ON VAZIRI'S 24-EDO MICROTONAL NOTATION SYSTEM)
BY : SHAAHIN MOHAJERI
********************
Introduction
Flexibility of intervallic sizes in persian music
Composers of microtonal music have developed a number of systems for accidentals and naming various intervals outside of standard notation of 12-EDO. Although because of different microtonal conseption and taste of composers, there is still limited agreement between them. For example , a system for quarter-tones may not accommodate third-tones, twelfth-tones or some signs may be too similar to others. So while a range of signs may well be appropriate for quarter-tone music, which will be adequate for a majority of composers, others may already be thinking of still smaller intervals. Other issues which may affect the choice of sign are whether or not the microtones are ‘structural’ or ‘ornamental’. Ornamental suggests that microtones are for colour and are inexact; while structural suggests considering each Exact pitches as equal with any other.
If we accept that the interval names such as "major-2nd" etc. really have no meaning outside of a diatonic scale , for scales and systems which has more steps per octave, we need to expand our interval terms based on a knowledge of the traditional interval naming system and accidental.
In this process the known and established signs (the sharp, flat and natural, ….) remain unchanged and new signs are devised which indicate (and usually suggest visually) a deviation from these well established standard accidental.
As a persian microtonalist , I tried to present a method for naming and notating degrees of 96-EDO based on quartertone Accidentals of Vaziri,s 24-EDO system .
But why to extend degrees from 24 to 96? The intervallic flexibility in works of too many Persian traditional music performers and masters Don’t accept any rigid framework like 24-EDO with small number of steps and such EDO is only one of the numerous models to be proposed and a standardized intervallic structure for 17 unequal-divisions of octave can't be practical traditionally. Another factor , affecting fluctuation of intervallic structure , is the modes . for example the interval B-C may vary from 80 to 104 cent according to the mode .
In a research by Jean During in 1979-1980 , he analyzed works of several masters ( Ebadi , Karimi , Musavi , Safvat, …) in each of the dastgahs.
In conclusion , except for fourths , fifths and octaves , intervals fluctuate within a rather broad range beyond which they become no longer acceptable to any musicians .
In table 1 and figure 1, we can see interval sizes of some dastgahs in works of several masters , after being analyzed by jean during . The range of intervals cover a tetrachord or pentachord .
So , we see that 24-EDO can,t show these intervallic flexibility and this is the reason why we see Criticism against it. I think 96-edo is a good framework for those who don’t think about pure harmony and degrees of beating but melodic language. Steps as 12.5 cent is adequate for persian harmono-melodic language and also 96-EDO is sufficient for simulating other systems considering JND (1, 2) of about 5 cent. each degree of 96-EDO is center of a boundary of
±5 cent (7.5 … 12.5 … 17.5) so this boundary have the same accidental and name.(they are realy parts of 960-EDO).
Flexibility of intervallic sizes in persian music
About 96-EDO
96-EDO is a musical intervallic system based on Sixteenth-tones . a Sixteenth-tones is used to refer to microtonal intervals approximately eight as large as the semitone, or approximately 12.5 cents.
Sixteenth-tones is Calculated as the 96th root of 2, or 2 1/96 , an irrational proportion with the approximate ratio of 1.0072464122, and an interval size of exactly 12.5 cents.
It is the size of one degree, and thus the basic "step" size, in the 96-EDO, also called the "sixtheenth-tone scale" or system. 96-EDO is capable of giving good representations of Persian music intervals by many tuning theorists. It is also a superset which contains several subset EDOs like :
48 32 24 16 12 8 6
96-EDO and limits
It should be emphasized that 96-EDO approximates several just-intonation tuning systems for primes up to for example 23.According to:
http://launch.groups.yahoo.com/group/tuning/message/21486 and
http://tonalsoft.com/enc/number/72edo.aspx
we can say that :
- 5-Limit : Every factor of 5 we put in the numerator we have to use the 12- EDO system 1/16octave lower (or 1/16 tone lower) than standard 12- EDO. And the reverse for every factor of 5 you put in the denominator. For an interval of 5/4 (386.31 major third ) we must lower 12-EDO major third of 400 cent by (1/16) to get 387.5 cent which has a difference of -1.186 cent . for an interval of 25/16 (772.627 classic augmented fifth) which has 52 in numerator , we must lower 12-EDO augmented fifth of 800 cent by 2*(1/16) to get 775 cent which has a difference of 2.373 cent.Or for an interval of 32/25 (427.373 classic diminished fourth) which has 5-2 in denominator , we must increase 12-EDO diminished fourth of 400 cent by 2*(1/16) to get 425 cent which has a difference of -2.373 cent.
A spreadsheet about approximating 96-EDO with 5-limit.
- 7-Limit : For an interval of 7/4 (968.826 harmonic seventh) which has 71 in numerator , we must lower 12-EDO minor seventh of 1000 cent by 3*(1/16) to get 962.5 cent which has a difference of -6.326 cent. Or for an interval of 8/7 (231.174 septimal whole tone) which has 7-1in denominator , we must increase 12-EDO major tone of 200 cent by 3*(1/16) to get 237.5 cent which has a difference of 6.326 cent.So , we havn't a good approximation.
A spreadsheet about approximating 96-EDO with 7-limit.
- 11-Limit For an interval of 11/8 (551.318 undecimal semi-augmented fourth) which has 111 in numerator , we must lower 12-EDO augmented fourth of 600 cent by 4*(1/16) to get 550 cent which has a difference of -1.318 cent. Or for an interval of 16/11 (648.682 undecimal semi-diminished fifth) which has 11-1 in denominator , we must increase 12-EDO diminished fifth of 600 cent by 4*(1/16) to get 550 cent which has a difference of 1.318 cent.
A spreadsheet about approximating 96-EDO with 11-limit
- 13-Limit : For an interval of 13/8 (840.528 tridecimal neutral sixth) which has 131 in numerator , we must lower 12-EDO major sixth of 900 cent by 5*(1/16) to get 837.5 cent which has a difference of -3.028 cent. Or for an interval of 16/13 (359.472 tridecimal neutral third) which has 13-1 in denominator , we must increase 12-EDO major third of 300 cent by 5*(1/16) to get 362.5 cent which has a difference of 3.028 cent.
13-Limit interval of 13/12 (138.573 tridecimal 2/3-tone) is a good approximation for the 11th step of 96-EDO (137.5 cent) with difference of 1.0727 cent.
13-Limit interval of 39/35 (187.343) is a good approximation for the 15th step of 96-EDO (187.5 cent) with difference of 0.1570 cent.
- 17-Limit : 17/16 (104.955 17th harmonic) which has 171 in numerator , has a difference of +4.955 cent with 12-EDO first step. 32/17 (1095.045 17th subharmonic) which has 17-1 in numerator, has a difference of -4.955 cent with 12-EDO 11th step.
17-Limit interval of 18/17 (98.955 Arabic lute index finger) is a good approximation for the first step of 12-EDO (100 cent) with difference of 1.0454 cent.
17-Limit interval of 136/135 (12.7767 cent) is a good approximation for the first step of 96-EDO (12.5 cent) with difference of 0.2767 cent.
17-Limit interval of 35/34 (50.184 cent) is a good approximation for 24-EDO quartertone or the 4th step of 96-EDO (50 cent) with difference of 0.1842 cent.
- 19-Limit :
- 23-Limit :23/16 ( 628.2743 23rd harmonic) is a good approximation for 625 cent with difference of 3.2743 cent. 32/23 ( 571.7257 23th subharmonic) has a difference of -3.2743 cent with 575 cent.
23-Limit interval of 23/15 ( 740.0056 cent) is a good approximation for 737.5 cent with difference of 2.5056 cent.
23-Limit interval of 30/23 ( 459.9944 cent) is a good approximation for 462.5 cent with difference of - 2.5056 cent.
23-Limit interval of 23/17 (523.3189 cent) is a good approximation for 525 cent with difference of - 1.6811 cent.
23-Limit interval of 34/23 ( 676.6811 cent) is a good approximation for 675 cent with difference of 1.6811 cent.
23-Limit interval of 27/23 ( 277.5907 cent) is a good approximation for 275 cent with difference of 2.2776 cent.
23-Limit interval of 46/27 ( 922.4093 cent) is a good approximation for 925 cent with difference of - 2.2776 cent.
23-Limit interval of 23/19 ( 330.7613 cent) is an approximation for 325 cent with difference of 5.7613 cent.
96-EDO and 5-limit lattice
Lattice is a visual representation of the mathematical relationships of musical intervals in 2-, 3-, or multi-dimensional space, consisting of points which represent the interval as positions calculated according to the Fundamental Theorem of Arithmetic. Lattices may be based upon two types of factoring: either odd or prime - similar to the two types of limit. Here is a 2-dimensional example of a 5-limit lattice.It shows the ratios which correspond to exponents of only the prime-factors 3 and 5, thus it is a 2-dimensional or planar system. The lattice theoretically continues infinitely in all four directions, the 3-axis radiating outward from the central 1:1 to the south-west in the positive direction and to the north-east in the negative, and the 5-axis radiating outward from 1:1 to the south-east in the positive direction and to the north-west in the negative.
(From : http://www.tonalsoft.com/enc/j/just.aspx)
Another presentation for 5-limit ratios:
(From :http://www.csufresno.edu/folklore/Olson/CHRDARAY.GIF)
As mentioned above every factor of 5 we put in the numerator you have to use the 12- EDO system 1/16 tone lower .We can see 96-EDO and 5-limit just intervals and their difference in arbitary boundaries of 3(-6..7) 5(-10..10) in the following lattice diagrams:
Considering any interval of 5-limit as (x,y) which x is power of 3 and y is power of 5 , we can have the lower lattice in arbitary boundaries of 3(-6..7) 5(-10..10) :
For more information about lattice go to
UNISON VECTORS AND PERIODICITY BLOCKS IN THE THREE-DIMENSIONAL (3-5-7-)HARMONIC LATTICE OF NOTES
Approximating 5-Limit just intonation by 96-EDO
Degrees of 96-EDO approximating 5-limit lattice
Difference of 96-EDO with 3,5 and 7 limit in arbitary boundaries of
3 (-4….4) 5 (-2….2) 7 (-5….5)
You can see minimum and maximum difference of about -6 and +6 cent:
Advocators of 96-EDO
Pascale Criton
www.pascalecriton.org/en/biography
- Julian Carrillo and the 13th Sound
- Julián Carrillo y el Sonido 13
Sauter's 1/16 tone / microtone piano
Vincent-Olivier Gagnon
Born in Nicolet in 1975, Vincent-Olivier Gagnon graduated with highest distinction in instrumental composition at the Conservatoire de Musique de Montréal under the direction of Serge Provost. He is presently pursuing an Artist’s Diploma at the same institution under the tutelage of Michel Gonneville. He has also briefly studied electro-acoustic music with Yves Daoust, as well as counterpoint and fugal writing with Jacques Faubert. His years of training naturally led to different avenues of musical research, most notably the study of micro-intervals and temperaments. A desire to share the unique poetry of this extraordinary soundscape marks all of his recent productions. He’s pursuing creative research in microtonality with Isabelle Panneton and Caroline Traube at the music faculty of the University of Montreal, where he’s writing an opera whose libretto by Mickaël Bouffard is inspired by the emperor Hardrian’s love for Antinous.
-http://pages.interlog.com/~nmc/2004-2005.pdf
Name of intervals in 96-EDO system
Basic concepts
Diatonic and chromatic intervals
All perfect, major and minor intervals are diatonic.
Additionally, the tritone and the diminished 5th are diatonic.
All other intervals are chromatic.
1-Major / Minor : In music, the adjectives major and minor just mean large and small, so a major third is a relatively wider interval, and a minor third a relatively narrow one. A minor interval is one less semitone than its equivalent major interval.The intervals of the second, third, sixth, and seventh (and compound intervals based on them) may be major or minor:
Minor and Major intervals