96-EDO

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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 intervalnaming 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 Duringin 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 (or1/16 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 maximu m difference of about -6 and +6 cent:

                                                                                               

                                                                                     Advocators of 96-EDO

 

Pascale Criton

 

www.pascalecriton.org/en/biography

 

Julián Carrillo

 

- Julian Carrillo and the 13th Sound

- Sonido 13

-PreludioNostálgico-JuliánCarrillo-JorgeMacielNegrete-Piano

- Ahualulco del Sonido 13 (pt.1) (pt. 2) (pt. 3) (pt. 4)

-... The Mexican who tried to revolutionize the world of classical music was once nominated for the Nobel Prize in Physics?

- Julián Carrillo y el Sonido 13

- A film in Carrillo's hometown of Ahualulco, San Luis Potosí, Mexico. Directed by Mario Mendoza and edited by Patricio Hinojosa, in
Spanish with English subtitles.The video discusses basic theory about 12 equal temperament and how the composer/author divided semitones to produce 24, 48 and 96 equal. Excerpts from his microtonal piano and string works are also included, along with various scenes from Ahualulco and his life.
pt. 1: http://www.youtube. com/watch? v=K16MJjfQIiQ
pt. 2: http://www.youtube. com/watch? v=7DaCJ0A8m1w
pt. 3: http://www.youtube. com/watch? v=mLWJWNIr31I
pt. 4: http://www.youtube. com/watch? v=mRkcnHsEj04

Carrillo Ives Partch Harrison Xenakis Scelsi

  

                                                                                        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

 

Jose Antonio Martin Salinas

 

He was born in Spain in 1968, studied a BA (hons.) degree in Contemporary Arts at Nottingham Trent University.He is currently doing research in music technology and composition at the London Metropolitan University.

Bellophones in 96-EDO

Salinas notation for 96-EDO

Conic Bellophone Demo

 

Name of intervals in 96-EDO

 

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

minor second major second

minor third major third

minor sixth major sixth

minor seventh major seventh

·  Major seconds are two semitones, also called a whole step,minor seconds are one semitone, also called a half step.

·  Major thirds are four semitones, minor thirds are three semitones.

·  Major sixths are nine semitones, minor sixths are eight semitones.

·  Major sevenths are eleven semitones, minor sevenths are ten semitones.

2- Perfect : The prefix perfect identifies intervals as belonging to the group of perfect intervalsbecause of their extremely simple pitch relationships resulting in a high degree of consonance and also because when they are inverted they remain perfect (a perfect fourth inverts to a perfect fifth and vice versa).

·  A perfect fourth is five semitones.

·  A perfect fifth is seven semitones.

·  A perfect octave is twelve semitones.

A perfect unison occurs between notes of the same pitch, so it is zero semitones.

3- Neutral or Median : neutral or median interval is a musical interval between a minor and a major major. neutral is a quarter tone sharp from minor and a quarter tone flat from major.

I have seen "median" in http://home.earthlink.net/~kgann/Octave.html for these 2 intervals :

18/11 852.592 Undecimal "median" sixth or Undecimal neutral sixth

11/9 347.408 Undecimal "median" thirdorUndecimal neutral third

4-Augmentation: In music and music theory augmentation is the lengthening or widening of rhythms, melodies, intervals, chords. The opposite is diminution (as in "a diminished triad").An interval is augmented if it is widened by a chromatic semitone ( or diatonic semitone) . Thus an augmented fifth, for example, is a chromatic semitone wider than the perfect fifth.Historically, augmented means increased from the perfect or the major, but the actual change in width could be:

(a) a semitone

(b) a quartertone

(c) a 3/4 tone (Sesquisemitone)

(d) a Tone

5- Semi-augmented : An interval is Semi-augmented if it is widened by a Quartertone. Thus an semi- augmented fifth is a quatertone wider than the perfect fifth.

6- Sesqui-augmented : An interval is Sesqui-augmented if it is widened by 3 Quartertoneor 3/2 Semitone.Thus a Sesqui-augmented fifth is 3 quatertone wider than the perfect fifth

7- Double-augmented : An interval is Double-augmented if it is widened by 2 Semitones.Thus C to D ## is a doubly augmented second (C to D is a major second, C to D sharp is an augmented second).

8- Diminution : from Italian diminuimento , An interval is diminished if it is narrowed by a chromatic semitone ( or diatonic semitone) .Thus a diminished fifth, for example, is a chromatic semitone narrower than the perfect fifth. The opposite is augmented. Historically, Diminished has always meant reduced in width from the perfect or the minor (whichever exists), but the actual change in width could be:

(a) a semitone

(b) a quartertone

(c) a 3/4 tone (Sesquisemitone)

(d) a Tone

9- Semi-diminished: An interval is Semi-diminished if it is narrowed by a Quartertone. Thus an semi- diminished fifth, for example, is a quatertone narrower than the perfect fifth.

10- Sesqui-diminished: An interval is Sesqui-diminished if it is narrowed by 3 Quartertoneor 3/2 Semitones.Thus a Sesqui-diminished fifth is 3 quatertones narrower than the perfect fifth .

11- Double- diminished: An interval is Double- diminished if it is narrowed by 2 Semitones. Thus C to G bb is a doubly diminished fifth (C to G is a perfect fifth, C to G flat is a diminished fifth).

(Semi-augmented ,Semi-diminished , Sesqui-augmentedandSesqui-diminished were recommended by Ivor Darreg for 24-EDO in his article " The place of quatertones in Today's Xenharmonics ")

12- Enharmonic :An enharmonic is a note which is the equivalent of some other note but spelled differently. For example, in equal temperaments , the notes C♯ (C sharp) and D♭ (D flat) are enharmonically equivalent - that is, they are represented by the same key (on a musical keyboard, for example), and thus are identical in pitch, although they have different names and function :

                

step

1

2

3

4

Major third

F♯

 

A♯

 

Minor third

 

G♭

 

B♭

Diminished fourth

F♯

 

 

B♭

Double augmented Second 

 

G♭

A♯

 

              

How to name an interval in 96-EDO?

1- Name or the label of an interval is firstly determined by counting the number of degrees between the two notes beginning with one for the lower note. The number of degrees between F and B for example is 4, therefore the interval is a fourth:

Interval Number of Degrees between unison and the note

Octave 12

Seventh 11

Sixth 9

Fifth 7

Fourth 5

Third 4

Second 2

Unison(Prime) 0

2-The name of any interval is secondly qualified using the terms of below tables.These are called interval quality .We have 2 level for Interval quality:

-First level which consist of these qualifiers:

Major

Minor

Neutral( 50 cent Sharper than minor , 50 cent flatter than major)

Perfect

***********

Double-augmented  (+200 cent from perfect or major)

Sesqui-augmented  (+150 cent from perfect or major)

Augmented  (+100 cent from perfect or major)

Semiaugmented ,Super (+50 cent from perfect or major)

Semidiminished ,Sub (- 50 cent from perfect or minor)

Diminished  (- 100 cent from perfect or minor)

Sesquidiminished  (- 150 cent from perfect or minor)

Doublediminished  (- 200 cent from perfect or minor)

- An interval of a 2nd, 3rd, 6th or 7th must be : major, minor, augmented or diminished

- An interval of a 4th or 5th must be one : perfect, augmented or diminished

- Second level which consist of these qualifiers to show ranges of shading for intervals:

In music, acute is applied to a tone which is sharp, or high; opposed to grave( Not acute or sharp).Here,these terms n addition to upper and lower are used to denote 4 shades of an interval .  

Acute (+25 Cent higher in pitch )

Upper (+12.5 Cent higher in pitch )

Lower (-12.5 Cent lower in pitch )

Grave (- 25 Cent lower in pitch )

                                                                                            *********** Examples ***********

637.5 cent = Lower semidiminished fifth = lower sesquiaugmented fourth

325 cent = Acute minor third

 712.5 cent = Upper fifth

687.5 cent = Lower fifth

875 cent = Grave sixth

250 cent = Semiaugmented second = Sesquidiminished third = Super major second = Sub minor third

362.5 cent = Upper neutral third

262.5 cent = Upper semiaugmented second = Upper sesquidiminished third

Grave and acute are also used in a List of intervals found in :www.xs4all.nl/~huygensf/doc/intervals.html

In the table below, we see that any interval is related to :

1- perfect , major ,minor and neutral qualifiers

2-Different shade of augmentation or Diminution

For example an interval with size of 612.5 cent is named as upper augmented fourth.

 

                                                                                  Enharmonics in 96-EDO      

In modern music, an enharmonic is a note (or key signature) which is the equivalent of some other note (or key signature), but spelled differently. It is the same exact note or key with either of two possible names. For example, in 12-EDO , the notes C♯ and D♭ are enharmonically equivalent - that is, they are identical in pitch, although they have different names and diatonic functionality.

In a given diatonic scale, an individual note name may only occur once. In the key of F for example, the major scale is: 'F, G, A, B♭, C, D, E, (F)'. Thus, the 'B♭' is called 'B♭' rather than 'A♯' as we already have a note named 'A' in the scale. The scale of F♯ major is: 'F♯, G♯, A♯, B, C♯, D♯, E♯, (F♯)'; thus we use the term 'A♯' instead of 'B♭' as we need the name 'B' to represent the 'B' note in the scale, and 'E♯' instead of 'F' as we need the name 'F' to represent the 'F♯' note in the scale. Some examples in 96-EDO from the above table:

637.5 cent = Lower semidiminished fifth = lower sesquiaugmented fourth fourth

250 cent = Semiaugmented second = Sesquidiminished third tird

262.5 cent = Upper semiaugmented second = Upper sesquidiminished third

Inversion of intervals

An interval is inverted by raising or lowering either of the notes the necessary number of octaves, so that both retain their names (pitch class) and the one which was higher is now lower and vice versa, changing the perspective or relation between the pitch classes. For example, the inversion of an interval consisting of a C with an E above it is an E with a C above it - to work this out, the C may be moved up, the E may be lowered, or both may be moved.Under inversion, perfect intervals remain perfect, major intervals become minor and the reverse, augmented intervals become diminished and the reverse and so on :

Grave

Acute

Neutral

Neutral

Diminished

Augmented

Semi-diminished

Semi-augmented

Perfect

Perfect

Major

Minor

Upper

Lower

Sesqui-diminished

Sesqui-augmented

Double-diminished

Double-augmented

Some examples :

875 cent = Grave sixth ↔ 325 cent = Acute minor third (1200 - 875 = 325)

712.5 cent = Upper fifth ↔ 487.5 cent = Lower fourth (1200 - 712.5 = 487.5)

Traditional interval names sum to nine: seconds become sevenths and the reverse, thirds become sixes and the reverse and so on . In traditional music theory a complement is the interval which, when added to the original interval, spans an octave in total. For example, a major 3rd is the complement of a minor 6th. The complement of any interval is its inverse (or inversion), except for the octave and the unison which are each other's complements.

Interval

Its

inversion

Some of compliments

8

1

9

7

2

9

6

3

9

5

4

9

4

5

9

3

6

9

2

7

9

1

8

9

96-EDO NOTATION SYSTEM

I have proposed a new notation system for notating musical pitches in 96-EDO.This is based on 24-EDO notation system of Ali Naqi Vaziri . He 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 :< xml="true" ns="urn:schemas-microsoft-com:office:office" prefix="o" namespace="">

All of the new accidentals in 96-EDO are those used in 24-EDO with Flags pointing up / down or right / left to indicate the direction of pitch alteration. Each flag represents ± 12.5 cent or1 degree of 96-EDO (1º96) . Accidentals that mirror each other vertically indicate equal-but-opposite amounts of pitch alteration. So , size of each accidentals depends on the number and direction of flags.

 

Sagittal notation system and 96-EDO

In http://users.bigpond.net.au/d.keenan/sagittal/Sagittal.pdf by: George D. Secor and David C. Keenan , we can see a set of symbols for musicians (who use microtonal subdivisions of 12-equal and have little or no desire to be involved with ratios) to notate tunings relative to 12-EDO. We call this the 12R (12-relative) or trojan symbol set.Here we see symbols related to 96-EDO:

- During , Jean , 1991 , Le Repertoire-Model de la Musique Iranienne , Teheran , Editions Soroush.

- http://tonalsoft.com/enc/q/quarter-tone.aspx

- http://www.duke.edu/~azomorod/history.html

- http://www.iran-heritage.org/interestgroups/AliNaghi.htm

- Nettl , Bruno, , 1993,The Radif of Persian Music: Studies of Structure And CulturalContext, < xml="true" ns="urn:schemas-microsoft-com:office:smarttags" prefix="st1" namespace=""> Champaign: Elephant and Cat.

- http://tonalsoft.com/enc/l/limit.aspx

- http://www.darvishi.com/pages/books.htm A View on the West (An account of the influence of the western music on Iran’s regional music) Tehran, Mâhoor 1994.

- The Radif as a Basis for a Computer Music Model: http://crca.ucsd.edu/~syadegar/

- Morteza Hannaneh , 2004, The lost scale , Tehran , Soroush press

- http://ourworld.compuserve.com/homepages/Neil_Hawes/theory45.htm

- http://www.aboutmusictheory.com/music-intervals.html

- http://www.kuke.com/TextData/list_4.html

- http://hyperphysics.phy-astr.gsu.edu/hbase/music/cents.html

- www.ee.kth.se/php/modules/publications/reports/2006/XR-EE-SIP_2006_002.pdf

- http://launch.groups.yahoo.com/group/tuning/message/21486

- http://tonalsoft.com/enc/l/lattice.aspx

- http://www.o-art.org/history/Computer/composers/Scholz/Lattice_notes.html

- http://www.geocities.com/jamsalinas/96etnotationmay21st.html

- http://www.tonalsoft.com/enc/e/equal-temperament.aspx

- http://ourworld.compuserve.com/homepages/Neil_Hawes/theory45.htm

- http://www.aboutmusictheory.com/music-intervals.html

- http://www.kuke.com/TextData/list_4.html

- http://www.tonalsoft.com/enc/number/72edo.aspx

- http://www.tonalsoft.com/enc/h/hewm.aspx

- http://users.bigpond.net.au/d.keenan/sagittal/Sagittal.pdf