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Beyond 12-Tone Temperament in Oriental Music.


 (Invited Paper in the 4th Micrötone, Salzburg, may 1991with Late modifications and corrections.)

SUMMARY. In this paper several related topics are reviewed, in order to present coherently oriental music systems. After some physiological hypothesis on tone and consonance perception, a modal music model is proposed. A traditional generative approach to tetrachordal and scale construction is briefly touched, presenting several oriental notations and its problems. Coming to actual playing we offer some analysis which validate and invalidate usual assumptions on that. The examples will be heard and its acoustical analysis shown graphical and numerically, to connect musical impression with objective measurements. In the appendix part we present a definition of dissonance which supports the modal music model, graphics of oriental notations and a short description of techniques used in this work, all developed in the LTPM.
0. Introduction.
1. Nomenclature
2. Theory
        2.1. Hypothesis on consonance perception
        2.2. A model of modal music
        2.3. Tetrachords
        2.4. Scales
        2.5. Notation
3. Practice
        3.1. musical Interpretation of intervals and scales.
        3.2. Experimental Results of tetrachord division
        3.3. An example.
        3.4. Micro melody
        3.5. Pitch control in oriental instruments.
4. Conclusions.
5. Bibliography
        A. Dissonance Measure in LTPM
        B. Techniques employed in our analysis
        C. Graphics of notations of oriental music.

Back al Beginning


I must begin by asking your pardon if I repeat or present as new, ideas already expressed in former symposiums; I could not read their proceedings in German. I will express myself in English, also apologizing to the English speaking persons present today.

I have had been kindly asked by Prof.Hesse to speak you about oriental music and Microtonics. I use the word 'oriental' in a loose way, going from Mauritania to North India; we could better say Islamic Music, not because it all religious but because covers a set of music with some common characteristics to which I will refer myself after some paragraphs (religion however is very important in the classical oriental music).

As this is my first contact with that distinguished audience before going to the main subject of my paper, I must set some assumptions that constitute my philosophy and orient the analysis that I will present to you, of which I will also say some words.

Coming to microtone, we find two main attitudes or philosophies about it: micro intervals as a further equal division of the scale; or, micro intervals as a further development of the harmonic series. However both can join in a similar repertoire of notes, we find the approach essentially different: the first has its own (musical) logic and became an intellectual exercise of limited range; on the contrary, the second has an unlimited way to run and connects with all the tradition from Greece to our days. This is not for comfort but because we refine our feeling of consonance instead of breaking it.

Clearly, we range ourselves in that last option (and we are aware that with that statement we are loosing many potential friends).

0. Introduction


Perhaps it is not absolutely unnecessary to remark that oriental music is firstly, music and after, oriental. The oblivion of that has lead many musicians and músicologist to consider Orient as something exotic, a decorate, behind what we have many picturesque things, certainly agreeable in a holidays journey, but not completely serious, or, in any case, not as serious as our Music (with capital letters). We stress that oriental music IS music; it connects with us and its findings can help us to understand better OUR own music, specially in the domain of tone control, which the West has neglected a little to concentrate in other domains, as harmony and form.

Oriental music is really Oriental Music: many musical systems, some more then 2000 years old, with extensive theory, practice, instruments, etc. According to the main subject of this symposium, I will refer, and briefly, to the intervallic aspects of some of this music, and omit references to the important aspects of rhythm, interpretation, instruments, history, social practice, western influence, mathematics, numerology, ethos theory, religion, philosophy and mysticism, which should need many volumes and more skill and knowledge that I am able to offer.

As the trends of a study determine until a certain point the results and findings of it, I should state what are the basís of my approach to this and other music.

We think that the number is still acting in music perception: rhythm shows it clearly; pitch not so well except in main consonances; but this numerical perception grows with fine education, as Indian music does and ekmelic music needs.

But when speaking of numbers in human perception we must have in mind the findings of psychology, specially psychoacoustics, which shows how the ear-mind perceives the number. For instance we know that in melodic intervals the octaves are a little greater that a factor of 2 in frequency (piano tuning, etc), mainly in treble and bass sounds. Therefore an octave of 1212 cents should be considered like a perfect octave, rather than concluding that 'this culture does not use perfect octaves' as can be read somewhere and sometimes.

We believe that a correct scientific approach must use objective methods of measure; but these methods must include the transference function of human perception in it. We shall present some of our intents in this direction to you.

Therefore, I will first state some psychoacoustics hypothesis, then propose a descriptive model of modal music, and after I will show some measurements to support the model, interpreted through those hypothesis. Then, in the appendix, some definitions on modal consonance measure will be introduced; and a short description of the techniques developed in the LTPM and used in this work.

1. Nomenclature for oriental music


In this paper I will refer myself to some musical areas and cultures, roughly called 'oriental' and covered more o less by Islamic religion. These areas have been and are now in constant mutual influence, but have still special characteristics. Each particular area will be symbolized according to the following table, from west to east.

Table I. Nomenclature for oriental music's.
symbol generic area countries
S Spanish flamenco Spain
M maghreb: Lybia,Tunis,Argelia,Maroc
A middle east Arab: Egipt,Irak,Syria,Liban,Arabia,Emirats
T Turk: Turkey
P Persian: Iran,WestAfganistan,Islam.republ.USSR.
I Indian: North India,Pakistan, East Afganistan.
J Javanese: Indonesia, Java, Bali
We will also call some melodic intervals by easy symbols as:
Table II. Nomenclature for melodic intervals.
symbol T t S s M,m A
interval 9:8 10:9 16:15 256:243 11:10,12:11 6:5,7:6
measure 204 182 112 90 150-168 267-316



2.1. Hypothesis on consonance perception


We confront the well-known problem of consonance perception, setting the following hypothesis:


Perception extract from a single sound, when periodic, an impression called pitch: and from combination of pitches an impression called dissonance or consonance. These are natural facts. Culture and education modifies only its valuation (can be found pleasant or not, according those factors and many others).

Consonance has its processing in neural activity. Education can teach to recognize and use this perceptual phenomena in a musical and artistic way, even violating it by avoiding easy (simpler consonances).


We perceive an interval, and its consonance, as the simplest and nearest being compatible with the musical code of the listener.

It means that in a diatonic universe we expect and force the notes we hear as belonging to this diatonic universe. If we shift to a chromatic one, we will have more codes, more categories to classify the sounds we hear. If we shift to a microtonal world we will perceive as meaningful and independent sound that were only varieties in a simpler universe.

It does not means that we accept as correct these sounds; only we lack of a code for them, and therefore, they simply does not exist. It allows us to accept and recognize notes of mistuned instruments.

Another easy proof of that: a perfect temperate 5-degree by octave scale will probably be perceived at first audition as as a pentatonic one. Our usual code, the 12-tone temperate scale, forces these sounds to belong to it. See figure 1 and hear example 1 in the recording: twice the 5 equal-temperate, after the pentatonic using the usual 12-equal temperate.

This effect allows us for instance the enharmony, in which an unique pitch is perceived as forming consonances with two different groups of pitches; it is perceptively modified to fit into both groups.


Any interval has a perceptual SIZE which grows gradually. It has also a COLOR, a character which changes by sudden steps, according to the prime integer numbers imbedded in the consonance.

The names given in western music to the scale degrees (dominant, sensible) point to that color. Oriental perception of these colors have traditionally been more fine and acute. For instance, Indian names for srhuti, shows this character. As we see in [6], the intervals:

natural third             5:4     do.mi-         Prasârini is diffuse, penetrant, shy, sweet, restful.

Pythagorean third   81:64  do.mi           Pritih       is energetic, sensual, joyful, pleasure, love, delight.

Of course this color can be perceived and use after appropriate training: but it is there. By PH1, it can be simulated (until certain point) for other neighboring intervals, as in tempered scale; but we believe that loosing an important part of its effect.

We can resume saying that each number has a size and a colour: its size is its cardinal; its color is its divisibility (structure)


The more time we hear an interval the better we perceive its size, its consonance and its color.

As we increase the density of degrees (number of notes in the octave) our analyzing mechanism needs more time to discriminate between them, the same as any other mechanism, Fourier type or otherwise. It means that, complying with PH1, we can understand quick pentatonic music but we need slow microtonal music. If not we will probable hear something as clusters or glissando. (?)

It seems to be responsible of the fixed intervalic size of simple consonances and the variability of more complex ones (moving semitones, pien, variable degree, etc, ).


Any judgment of consonance is made in any moment, on the last sounds heard before this moment.

It means that we attribute a measure of consonance to a cluster composed with the last sounds, even those that have actually disappeared: they are hold by the near memory, and probably decay in importance as new sounds replace them.

An important corollary is:


Any judgment of consonance must include the reference note (tonic or modal tonic) if the music suggest it.

Consonance is acting during the entire performance: first, in stringed instruments, by the tuning of strings in simple consonances, usually fifths and fourths; secondly, during the actual playing, searching the consonance by repeated essays around it in variable tuning instruments, as naï (flute) or 'ud (luth).

The choosing of determinate consonances (which usually means despising others) fix the main notes of the mode; its hierarchies and structure is build in this way; read M4 in next paragraph.


2.2. A model of Modal Music


Our model of modal music, deduced from many automatic and natural analysis is based on the following characteristics, (several of them are common to other musics). We believe that this model covers the musical systems of the areas named before, and even Gregorian music and many folklores. Lets see its characteristics:

M1. A limited ambitus of pitches are used.

M2. They cover this ambitus in a limited density; usually from five to ten in an octave, mainly seven in many cultures.

This density can be calculated as MI, the mean value of interval: we find 240 cents for any pentonic scale, 171 cents for a heptaphone and 100 cents for a dodecaphonic.

M3. They are not usually perceptually equidistant, which means different sizes of intervals.

We can measure this distribution by defining the Tension of a scale as the mean of differences of each interval with MI ( we could also weight each interval with the frequency of its use in actual music). We will find (without weighting) a tension value of 49 for a Pythagorean scale, 43 for major tempered and 34 for natural, which represent fairly well the subjective feeling.

M4. These pitches are selected between those which form simple consonances with one of them, at least some of them. These ones form, with that one, called modal tonic, an squeletal structure of that music, as we said in PH5. As being simple, they must be octaves, quints, thirds and its complement to the octave, quarts and sixths. As the main notes of the mode, they will appear insistently.

Secondary relations of consonance can be established with the elements of that structure, which is then organized as a tree with different levels and different hierarchy. An structure based on tetrachords (4:3) represents fairly well the central areas considered in our study (P,T,A), but is less clear in the extremes (I,M), where greater consonant intervals ( fifths, sixths, even octaves) are directly used.

Here is an example of this structure for the natural (Zarlino) scale on c:

c 2:1 c
0 +))))))))))))))))))))))))))))))0)))))))))))))))))))))),
  *     3:2                       sol 4:3                    *
1 /))))))))))))))))0)))))))))))))3)))))))0))))))))))))))1
  *     5:4         e-     6:5      * 10:9  a-     6:5        *
2 /))))))))0)))))))3))))0))))))))3)))))))3))))))))0)))))1
  * 9:8     d  10:9  * *  f 9:8     *        * 9:8   b- *    *
3 /))))))))3)))))))3))))3))))))))3)))))))3))))))))3)))))1
  *        *       *     *        *        *        *     *
  c        d        e-   f         g        a-        b-     c

M5. This structure can be shift in frequency in a continuous way according to particular conditions (instruments, voice, even mood) whitout changing its significance. There is no absolute pitch in that music. But after it is choused in a performance, it does not change, it will sound during all the performance, always in the conscience of interpret and listeners (M,S,A,T,P), even actually sounded (I,P).

M6. The use of that notes is mainly melodic, moving by conjoint degrees, and covering a limited part of the total ambits. This part is usually a tetrachord or a pent cord, and is called genus or ajnas in Arabic music. The interpret present slowly and carefully these notes one after another, appearing gradually to the listener. With that fixed reference, each pitch acquire a very particular and strong function. The consonance is not the only expresive mean: the motivic development, like a prosody, is essential to the mode. But also melody makes use of (more subtle) consonances.

M7. The structure described in M4. can be partially changed during the piece, making a kind of modulation (modal modulation) which modifies some of the consonances and its intervals, but not the modal tonic. It is as if a part of the tree changes its branches and hierarchy, even some of the pitches in the maqam. It means that we can find more than seven notas by octave, by joining the ones used in different moments of the performance. This music can be seen as moving on that structure, through branches (intervals) and knots (notes).

M8. This ideal structure is realized (given reality in sound world) by making audible the consonances, which establish references, and the dissonances, which give instable moments which must be resolved: that make the melody to move and give life to the music, in an alternative arsis-tesis.

M9. This arsis-thesis pair is repeated in all the levels. The highest is the whole melody and its end, which represent to the perceptor the coming back to the reference, felt as peace and resolution of dissonance problems proposed in the melody.

M10. The particular consonances (intervals) choosed determine a particular musical climate, a character, an emotive mood. This is called maqam(A,T), raga(I), tub(M) or any word according to the epoch, country and languaje (as tonoi, in Old Greece). We think of that as a global harmonic timbre, specific of each maqam, which has an effect on audience.

M11. The most pure exemple of this melodic use is the so-called taqsim(A), alap(I) istahbar(M) or muhtasari(P). In this improvised form (note the apparent paradox), the player construct the maqam o raga by establishing the scale by dividing the strong consonances in smaller units which became melodic intervals, by sections. See in figure 2 the analysis of the music heard in example 2, an alap(I) of raga ahir bhairav on santur (see paragraph 3.5 and see slide of this instrument).

M12. Following traditional (tested affectivity!) rules he develops the scales and intervals with almost prescribed notes of beginning and end for each section, and also prescribed order of each genus. But out of this rules he will stay longer or not in a section according to his mood. The taqsim can take from two or five minutes to half an hour.

M13. A long duration of a section is achieved by delaying sagely the resolution which this section demands, i.e., going to a partial consonance.

To illustrate this principles, see the score (fig 3) and hear the recording (exemple 3), of a Rast Taqsim from Liban, played by Buyun in 1920 on tanbur, long necked lute, and note the sections which correspond to different genus.

In fig.4 we present the double table of frequency of melodic intervals between pairs of notes, a list of the melody, two cumulative histograms: the first of notes, showing the relative predominance of C (modal tonic), and E and G after,i,e, the hierarchy. The second histogram shows the frecuency of intervals measured in semitones: semitones and tones are dominant, which reflects the melodic style of fragment.

Down in the figure we can se a rough score of the melody and a measure of the modal dissonance (which include tonic, see appendix A) of the melody, viewed as a function of time: low values are rests on consonance, and high, peaks of dissonance, instable moments which needs resolution, as western cadencia.


2.3. Tetrachordal divisions and Consonant intervals.


There is an extense literature on theoretical intervals and genus (ajnas) in classical greek [3, 6, 7, 8, 24, 25], mediaeval ecclesiastic [4, 8, 25], Arabic [5, 6, 7, 12, 16, 18, 28, 36, 48, 49], Indian [6, 45, 46], Persian [23, 27], maghrbi [5, 7, 9, 28,] and Turkish [11, 22, 23, 30, 51] music. But we dispose of less information on how the musician, the priests of this cults, play this music [18, 27, 38, 39, 40, 45].

The acoustical analysis of the music of several non western modal musics, specially the Arabic, has shown the existence of of tetrachordal divisions which differ of the traditionally accepted. As an example, we center our attention on a ton greater than the diatonic one, 9/8, which can be represented by the 8/7 ratio.

In the following paragraphs we review the theoretical divisions of tetrachord, ajnas, which include this great tone, and contrast these ajnas with the results of our research, which allow us to recognize the trend of some interpreted music toward these interval and ajnas.

We consider modal music as based on consonant intervals, as described in our model. These intervals are present in all maqams, the strings of instruments, the harmonics of wind instruments. Even maqams and ajnas with non simple tonic, as sikah, use consonants intervals as reference and support (i.e. rast-naua).

Therefore we shall confine our list to the tetrachordal divisions which uses consonant intervals. And we consider the more consonant the more simpler is the ratio of frequencies, and over all, those of the form (m+1)/m, being m an positiv integer. This is the traditional approach, which we adopt.

However these limitations the possibilities are very numerous. Usually, when dealing with European modal music aditional restrictions are imposed: only quintal and tertian harmony are admitted, that is, only 2,3,5 are the prime numbers which are accepted as factors in the numbers which appear in the interval ratio. Thus 9/8 are accepted because 9/8 = 3*3/2*2, but not 11/10, because 11 is a prime number.

But, for Arabic and other related musics these restrictions are too narrow, as they cannot explain the 'quartertone intervals'. We admit therefore also the numbers 7, 11 and 13, which allow more flexible divisions. Indeed, the neutral tierce is well represented by the 11/10, 12/11, 13/12 or 14/13 intervals.

Here we shall see only those genus which includes the great ton or 'tonus maximus', 8/7, as one at least of its three intervals. As the complement to the quart, 4/3, is 7/6 ( = (4/3)/(8/7) ), we look for two ratio which product is 7/6: the possibilities are


Table III. Tetrachordal divisions including Great Tone (8:7)
1. 8/7 . 8/7 . 49/48 = 4/3 231 + 231 + 38 = 498 2. 8/7 . 9/8 . 28/27 = 4/3 231 + 204 + 63 = 498
3. 8/7 . 10/9 . 21/20 = 4/3 231 + 182 + 85 = 498
4. 8/7 . 13/12 . 14/13 = 4/3 231 + 139 + 128 = 498

These genus are well-known from the antiquity. The greeks described and Ptolomeus (180 B.C.), called them 'middle diatonic' for our second division and 'smoothed diatonic' for the third [8]. Also the Arabic theoretics treated them. Al-Farabi quote and describe the firts 3 ajnas, and even recommend the first and second for the execution on the Tanbur of Bagdad; in fig.5 can be seen the possibility of playing these intervals with this instrument, according to the theretic tuning of its frets (appendix C); and see in [7], vol.2, notes 4,5, comments to Al-Farabi book. Ibn-Sina too include these ajnas, as Farabi does, within the class 'strong'.

If we change now the order of these intervals we find 6 combinations for each genus. Following arab nomenclature, we will name those combinations the 'rast type', the 'hiyas type' and the 'kurdi type' according the position of the bigger interval in first, second or third positions, respectively. The position of the middle interval provide us with three other types. We have thus six main forms:


Table IV. Division of Just Tetrachords.




1. rast greater middle smaller     *.........*......*....*
2. nahauand greater smaller middle *.........*....*......*
3. hiyasA middle greater smaller   *......*.........*....*
4. baiati middle smaller greater   *......*....*.........*
5. hiyasB smaller greater middle   *....*.........*......*
6. kurdi smaller middle greater    *....*......*.........*

According to the actual used tetrachord division we will name rast-2 the 231-204-63 division or baiati-3 the 182-85-231 division.

But the evolution of music brought other division, probably derived from the formers, which cannot be reflected by just tetrachords. In fact we find modal genus with ambitus of diminished and augmented fourths, better included in perfects fifths. We find also disminished fifths ( not in I). Even sometimes, the modal tonic in not a main consonance: mode sikah (A) and segah (T,P) has the tonic in the third degree (segah means third) of a perfect natural pentachord. For instance:

Table V. Irregular Genus
area mode ambitus tonic ASPECT
A saba just pent. 1 Ã..M.    .*..m.*..S.*.....A........*
T segah just pent. 3          *....T... *.t.....Ã..S.,,*....T...*
I todi just pent. 1 Ã.S.     .*..m....*....A.......*.S.*
A karjigar dim. pent. 1 Ã..M..*..m.*..T.....*.S...*

2.4. Scales of modes


By joining different tetrachord we can construct thousands of scales, but only some exist in the musical world or have existed.

The joining can be made by conjoint union, disjoint union or even by superposition. The joining is made according the characteristics of the tetrachords involved, in a way which forms greater consonances, as the octave, in a first and fundamental place. The process is the same that the one used in clasícal greek music, but extended to this defective tetrachords.

The notes of joining, as being consonants with the other two extremes of the tetrachords, became important notes of the structure, as tonic, dominant and other more subtle functions not stated in western theory.

The joining in not only a theoretic generative way of form scales; it is also a musical joining, as the modal use of it, described in our model, shows.

The number of modes in these cultures is very big. Numbers of this modes change but some typical numbers (only approximate) could be:

Table VI. Typical number of modes in oriental musics
area basic known in present bibliography
T 12 30 110
A 12 30 120
I 10 40 70-100
M 9 18 26

If we accept only deep undestanding only a little number can be mastered ( 1 or 2, as a great indian musician answered when he was dying).

2.5. Notation for Oriental Scales


The notation of oriental scales is so variable as the cultures that we try to cover. We can find, as we have here, the temperated approach and the not temperated approach. But we must have in mind that in both, the aim is to have as many sounds as necesary to cover all the scales of the modes of this musical world. As additional sources of confusion, some systems take the Pythagorean major scale as reference, others the natural-zarlino major scale, others the western tempered major scale.

Moreover, sometimes the names and alterations try to represent with little error the interval in the scale; others only to point to an area (i.e, like the position of a finger on the instrument).

They are other differences: some of the systems use symbols only for note the sounds which appears in the scales when played in the right, traditional position in the instrument; there is no transposition, except for the traditional scales which are (never completely) transposed of others. These systems are very centered in his culture. Some other systems try to cover any traditional scale: they are general, usually made by (western) músicologues, and cover the whole octave by minute intervals more or less equally spaced. We call 'international' this systems.

About the names of sounds, we must say that, in traditional systems, each one have a different name for each sound for a two octave ambitus, as in old Greek music. These names point sometimes to the character of the interval with tonic, its order in the scale. We have as much names as sounds in the traditional system. For instance, 22 different names for each shruti in indian scale, as chandovatî, dayavatî, rajanî, etc. In Turkish, Persian and arab music we will have rast, nim zirguleh, zirguleh, tik zirguleg, dugah, etc.

As we said before, each name represent a sound which form with the modal tonic a particular interval. They are relative names. Each instrument or singer will choose the reference pitch which suits him and construct the desired scale on that. For instance, the same name (rast) is applied to the sound which utters a naï (flute) with all the holes closed. As these flutes are constructed in about nine o ten sizes (between almost 1 meter to 30 cm) we have the same number of different sounds for the same name.

The absolute pitch (in herzs) is therefore variable. But it is being more and more being fixed, as instrumental groups are created and even western instruments are adopted. Bur even here we find difficulties. Each musical system has taken western notation in its own way, so an score has to be transported to be understood in our system, or read changing the clef. We find this particular use in this cultures.

Table VII. Absolute pitch of base notes in oriental systems.
area base note written in G-clef sounds preferently(A=440hz)
A rast C C, B
T rast G D, C
P rast C Bb
I sa C C#,D,A
M dhil C C

That lead us to the problem of original notation of this music and transcription to ours.

We found more problems dealing with names and written oriental music. Sometimes similar names conveys diferent relative sounds and sometimes, different names conveys similar sound (but not necessarily similar modal use of it!).

For instance the note segah in always a note which forms with tonic an interval greater than minor third (6:5) and smaller than Pythagorean major third (5:4). But it varies greatly in this ambitus.

Table VIII. Different value of interval of same name.
area name measure(cents)
T rast-segah 386
iraq rast-segah 380
egypt rast-segah 350
P rast-segah 330

It should also be remembered the concept of interval and its size in this section: they are not melodic, scale, modal intervals, only minute intervals which, when joined, will form the former.

We will try to present some of these notations, together with western 12 tones scale, the 72-degree notation used in Salzburg, and mine, in order to be compared. In table IX we can see a summary of those systems and in appendix C a schema that represent each system in actual relative sizes. For each system we list its main user or creator, the musical area of use, the number of minute interevals for octave, their sizes in cents, the base scale for notation and the symbols used to alter this basic degrees.




3.1. musical Interpretation of intervals and scales


How this intervals and scales are interpreted on the

instruments?. There are two main approaches: the first (A,P,T,M) chhoses the situation of the scale to have as many fix sounds as possible: a basic tetrachord of the mode will be sounded in two free strings if the instruments is tuned in this way, as in 'ud(A,T,P); any other use could be possible but should not preserve the purity of main consonances and would be difficult to play.

In fact, probably the inverse process have actually happened: on a basic instrument, new modes and scales have been developped profitint the main consonances which it offers in his fixed pitches. In this way a limited number of sounds (notes) suffice to play many different scales.

In an instrument as described, with free strings s1 and s2, the scale types in Table III would be played:

s1-- tetrachord----- s2 ------------------s3
rast     *.........*......*....*
baiati   *......*....*.........*
hiyasB   *....*.........*......*
rast     *.........*......*....*

(The scales played in actual arab music would be Rast, Bayati, Sikah and Chahargah)

Of course it means that for a same instrument and tuning, each scale will repose on a different fundamental, as in gregorian music. If this tessitura does not fits the melodies to be played and/or the ambitus of the voice, the instrument can be retuned in limited ambitus (1,2 semitones) or transport the entire scale to another situation, but almost allways to a similar (easy consonance) situation.

A differet approach is used in indian music (I): as fixed drone sounds are used for each instrument, each scale is allways played in the same place of this instrument: it is the duty of the player to retune it each time he changes the mode, which for some instruments as the saranghi, meant to retune about forty strings(melody's, support's, resonance's); however, this changes are rare during a performance. In fretted string instruments as sitar and north veena, also these frets must be shifted ( the process can take one hour when done very carefully). The same scale type of last exemple would be:

rast *.........*......*....*
baiati *......*....*.........*
hiyasB *....*.........*......*
rast *.........*......*....*

Both processes are similar to play diatonic scales on the different white keys or to transport every scale to c as fundamental.


3.2. Experimental Results of tetrachord division


Applying our method of statistical analysis of intervals, described in Sánchez [38], we have found melodic intervals which are functionally whole tones, that is, they are perceptively whole tones for opposition to semitones, neutral tones and augmented tones.

But our measurements shows without any doubt, that, in many occasíons, their size is larger: we find often values of 224, 232 and 240 cents, nearer of the 8/7 interval (231 cents), than of the diatonic whole tone, 9/8 (204 cents).

And it is surprising to find them in instrumental and vocal music, in indian, andalusi, arabian, turkish and spanish (cante hondo) musics. And that in the same places of scale, the first interval of a rast type tetrachord, and the second interval of a hiyas1. Lets see some results:

In orden to know how musicians plays the music, we performed extensive measurements on music from oriental cultures. Here are some results on the 8:7 interval [40] , with information about country, mame of player(s), musical instrument, maqam or similar concept of fragment, tetrachord inside of the scale of the maqam, intervals found in the analysis, type of the tetrachord according our proposal, and order of division of tetrachord as listed before.

Table X. Actual Tetrachordal Divisions including Great Ton (8:7)
Country Musician Instrument Maqam Tetrachord Intervals Type Div.
A Liban Ba'yun tanbur rast 1 236 147 141 rast -4
A Liban Ba'yun tanbur rast 2 222 141 128 rast -3
S Spain Menese voice.. martinete 1 243 165 98 rast -3
I India Banerjee sitar charukesi 1 233 164 102 rast -3
A Iraq M.Bachir 'ud nahauand 2 242 182 75 rast -3
M Maroc Rais orq.voice&orq istihlal 1 223 164 120 rast -3
T Turkey Öczimi nai rast 2 243 101 157 nahau-3 A Liban Ba'yun tanbur rast 2 133 235 142 hiyaB-4
A Iraq S.Chukur 'ud baiati 1 134 146 224 kurdi-4

We found also these intervals in Javanese music, but out of the tetrachord boundaries.

The analysis have been made on long fragments, in order to detect the 'trend' of the ear and finger of player. We have selected, naturally, only the results that illustrate this special point (interval 8:7); but others can be made looking to other characteristic.

From this table we do not conclude that these scales uses necessarily this interval; but it proofs that diatonic model is not enough to explain oriental music in general. And it is specially interesting that great tone actually played, because it is usually considered a very old interval, discarded even by medieval theorists. It also shows the utility of objective measurements for musicologists.

How can we explain these intervals ?. In some cases they came out of the instrument: it is the case of the tanbur, which does not allow other "whole tones" that 182 cents (10/9) and 231 (8/7), due to the fixed frets [12],[39]. We must remark that those frets and intervals are not arbitrary, they came from the tradition, and therefore agree with the musical system. See fig.

In the case of the 'ud (short necked luth) we can explain that for the positions of the fingers and/or for the tuning of strings. If we tune the main strings asíran-dukah-naua-kirdan of the 'ud in the usual Pythagorean way, we have the 8/7 intervals between dukah, free string and rast on asíran string, by raising the dukah about one coma, until its seventh harmonics harmonizes with kirdan. (we tried this tuning and the result was far of being shocking).

But in vocal music we have no instrumental raisons: we must suppose that the ear and the spirit, in order to perceive the ajnas and maqamat differences, reacts in the same apparently crude way shown in the list of 'tetrachord types', by comparing the relatives sizes of intervals to order them as 'greater than' or 'smaller than'.

Does it means that any interval produces the same effect than another belonging to the same category otr type?. Not at all, but we believe that each type represents a 'family'.

These families can be recognized in practical music: for instance, Baiati presents a fist interval which varies greatly, between about 133 cents in the Persian 'schur' until about 180 cents in Turkish 'uchak', that is, between a 'big semitone' and a 'small minor tone'. And even within the Arabic music, the range is no less of 135-169 cents [7][48][49].

Inside of the family, subfamilies can be found according to the country, region and even individuals. But a link can be traced between these variations: the common type, described before.

3.3. An example.


Lets see and hear the first part of a pesrev, in maqam pengigah. In this maqam, which roughly reposes on a major scale (lets be C major), the pesrev begins with a natural major consonance on D, rast type, which give us D,E,F#-,G, and after shifts to a major consonance on C, forming the C,D,E-,F tetrachord, also rast type (remember point 8 of our model).

We find therefore the notes C,D,E-,E,F,F#-,G in the first fifth of the scale and G,A,Bb,B,C. It makes 10 main notes in the octave, but all these notes are used in a diatonic way, thus having only 4 notes in a tetrachord, even if we use them in an almost chromatic way (12 bar).

Lets see in some detail this process:

turkish maqam Pengigah:
* 3 : 4 *
* 4 : 5 *
do re mi- fa sol
* * * * *
* * * * *
do re mi fa# sol
* 4 : 5 *
* 3 : 4 *
. 204 . 182 .23. 80 . 112 . 114 .
* * * * * * *
do re mi- mi fa fa# sol
sol la <si si do do# rel
. 231 . 182 .23. 80 . 112 . 114 .

Lets see the score and analysis of the first bars in fig.6: the music can be heard in exemple.4. As told before, the base scale of turkish music reposes on rast note which is written as G. However it sounds usually C now, D before and, in this particular exemple, Bb. This music will sound like beginning in C minor and going to Sib major.D, but will be read as beginning in A minor and going to G major. In our analysis we have called DO (C) the tonic, to clarify the exemple.

3.4. Micromelody


But intervals in tetrachords as heve been measured does not represent the richness of the subtle tone use in oriental modal music. What is for the western listener a short motiv of two or three notes is really for musician and oriental listener a subtle melodie that goes beyond the rigid mark of the scale. And, as it is not usuaklly coded, each interpretation is different, reacting to the audience in a lively way. In exemple 5 in clear this pitch use with the indian shenai, a double reed wind instrument, paired with a violin. In fig.7 is shown the scale analysis of it.

We find also in this music different pitch for the same note when going up or down, as can be seen in the figure 8 and heard in the musical exemple 6.

3.5. Pitch control in oriental instruments.


Here is a short description of pitch control in oriental instruments which allows this subtle effects.


A. String:

1. change of string

all instruments except those of only one string (folkloric).

2. longitudinal moving of finger ON string

continuous: violin(A,I), sarod,

úd (A,M,T,P)

rabab(M), djose(A,Iraq)

3. by frets;

3.1. moving chromatic:

north vina,sitar, dilruba (I)

3.2. microtonal:

tanbur(T), tarhn(P),

4. longitudinal moving of finger BY string


saranghi(I), kamanja rumi(T)

5. lateral pulling of string (meend)

north vina,sitar (I)

6. combination of plaques

qanun (A,T,M)

7. displacement of bridge

santur (A,P,I)

8. pressing of string

santur (A,P,I)

B. Wind

1. total opening or closing of holes

nai(A,T,P), flutes (I)

2. partial opening or closing of holes

nai(A,T,P), flutes (I)

3. change of head-instrument iangle

nai(A,T,P), flutes (I)

C. Percussion 1. pressing of membranes

tabal (I)

2. Change of membrane (drum)

3. Change of place of strike.


4. Conclusions.


We resume some considerations on oriental modal musics:

1. They do not use in general microtonal intervals as distinct codes in its systems.

2. They have more than seven sounds in an octave, from 17 to 22, divided in seven groups of which only one group (scale) is used at a moment: each memeber of a group represent zones or melodic functions.

3. Combination of this sounds give many varieties of scales: thousands theoretics, one hundred in the books, thirty to fifty known by expert musicians, ten basic and popular.

4. The apparition of intervals in the surrounding area of the 8/7 interval, or great tone, suggests that pitagoric scale cannot explain all the actual intervals played in this musics. It seems to prove that other divisions of tetrachord are also in use nowadays in the ear of the musician. And lets remember once more that he is the door through music comes out.

5. A global model for modal music is valid in an extense area. This model reflects the hyerarchy of scale degrees in a structure.


Apendix A. Disonance Measure in LTPM


A natural number can be expressed as a product of whole prime numbers, equal or different: we define with Euler the complexity of the number A as:

i ai i

A = ´ pi entonces: EC(A) = 1 + õ ai¨(pi-1)

where pi are the prime factors of A, ai its exponents, and ´ y õ the signes for generalized sum and product. For instance, the complexity of 8 es 1+3(2-1)=4 but, the one of 9, 1+2(3-1)=5; therefore 8 is simpler than 9.

We depart fron Euler omitting the '1' summand because of better geometrical properties of this measure, defining the complexity of A as


C(A) = õ ai¨(pi-1)

For several natural numbers A,B,C.. we define the relative Primal Complexity as the complexity of a number N, the quotient of its minimun common multiple by its maximum common divisor,, which is again the Euler measure (EC) minus 1.


/ m.c.m (A,B,C,.) \

PC (A,B,C,..) = C ( )))))))))))))))) )

\ m.c.d (A,B,C,.) /


For instance, PC(18,16) = C( 2^4¨3^2 / 2 ) = C ( 2^3¨3^2)

= 3¨(2-1) + 2¨(3-1) = 7 = EC -1

This is applied to the relative dissonance of simultaneous (or successive) frequencies integrated in the perceptual system. This is easily seen that the measure PC is the period of the sum of frequencies A,B,C,.. divided by the maximum common divisor of the periods of components. It is then, essentially, a relative time measure.

For two frequencies, A,B, this measure can be seen as a distance:

PC(A,B) = õ *ai-bi*¨(pi-1)


And it is really a distance in a vector space, which we call PS, Primal Space, where a point is a (positive) integer, the axis represent the primes, the vector components the exponents of the primes in its decomposition; a vector is an interval, a segment which goes from a point to another, and its modulus the distance between them. The distance is a weighted distance, with a weight pi-1, the prime minus 1 (module of base vectors). This vector space is not only an analogy, it a true vector normed space which verifies all the necessary and sufficient conditions.

Even other Aharmonic Distance can be defined, by introducing the r-distances:

r 1/r)

PCr(A,B) = { õ *ai-bi* ¨(pi-1) }


which coincides with our former definition for r=1 and with the euclidean (usual geometric) for r=2. The exponent r can vary between 1 and infinitus, positive, to be a real mathematic distance.

The disonance betwwen two frecuencies appears as an Harmonic Distance, which make a beautifull pair with the Melodic Distance, defined as another distance ( diferent weight ) in the same space. r is 1 in both.



MD(a,b) = K ¨ õ *a - b * ¨ log p

1 ³ ³ ³



HD(a,b) = K'¨ õ *a - b * ¨ (p -1)

1 ³ ³ ³


The arbitrary constants K,K' depends only on the unity choosed to measure them: K=1731 for neperian logarithm and the cent as unity; K'=1 for Euler-like unity.

Both distances are perceptual, which means that they approach the sensation felt by a listener; however, many other factors are involved in the perceptions of the melodic and harmonic distance or interval (range of frecuencies, harmonic content, education, coded scale, etc). This distances are useful to 'describe' the melodic and harmonic processed involved in music.

For instance lets show the representation in our space PS of Pythagorean and natural (Zarlino) scales. The harmonic distances must be seen as 'city-block' distance, as walked in a town in block (no diagonal) (fig.8).

But in order of approach more the dissonance situations in actual music, we must introduce the influence of the relative presence of each sound, because it is clear that in a given tonal reference, the actual dissonance of a weak and short note cannot be the same as that of a strong and durable one. Therefore we must introduce the perceptual influence of each note weighting it, i.e, weighting its contribution, its exponents ai. We introduce the Real Harmonic Distance, RHD, as:


1 n

RHD(A,B) = K'¨ ))) ¨ õ *w a - v b * ¨ (p -1)

w+v 1 ³ ³ ³

Of course the sums of weights must be 1, as in any relative weighting policy; we divide each one by its sum.

The weights w, of A, and v, of B, must represent the perceptual relative influence of them; we can take as a primary approximation of it, the area of the note represented in a intensity-duration space, both in a logarithm-like scale; the product of the intensity in Db and its duration in 'figure-scale' (whole note=8, half=4, quarter=2, crochet=1...), also a logarithm scale.

Now we can measure the disonance situation and alternatives, the history of a melody in an harmonic (in broad sense) univers. If we take a sound as allways present (actual o perceptually), as modal reference, we can fix B and measure the RAD of each note A in the melody with B. A note A will have a disonance according to its numerical relation with B, its intensity and its duration.

All this concepts can be generalized to western harmony. See [42,43] for a more detailed account of this concepts.


Apendix B. Instrumentation


The examples, measurements and graphics shown in this presentation have been made with the following sistems, all developped in this Laboratory, the LTPM.


Based on ATARI.

Connected with any MIDI keyboard can record, store, edit, and output, play, any music in this standard.

Every kind of analisis can be made on the stored music,


Consonance analysis of harmonies or melodies.

Histograms of notes, durations and intensities in a performance.

Motiv analysis and synthesis, in the melodic field.

(see in fig.4 this interval and consonance analysis by Analmelo).


Based on PC-AT, connected with analog-digital converters.

Support input, output, storing in memory and disk

of analogic signals.

Extensive analysis:

Pitch, fourier, LPC

Synthesis of speech

Coding-recognition of speech.

(see in fig.10 the melodic (pitch) analysis of example 7 by SETS)


Based on PC-AT

Consonance Measure of:

any combination of tones.

any combination of composed tones (timbre) whole scales

Draws scales or melodies in a EP, (vector space of consonance). (fig.9).


Based on PC-AT, connected with Bruel&Kjaer frequency analyzer.

Tune the DX-7-II in any arbitrary scale.

Gets spectra from analyzer and performs extensive processing as:

Interpolation, smoothing, peak estimation,

Scale analysis.

Formant estimation.

Graphic printing of processed spectra

Sonagraph style of presentation.

(See in fig.11. the pitch analysis of a DX7-II keyboard tuned by ESCA9T in a 72-equal-degree scale, which can be heard in exemple 8, with some tones and semitones constructed with these little intervals. All the analysis in this paper have been done also by it.)

figuras microtónicas

medidas microtónicas

afinacion microtónica (escala de 72 por octava, grabada)


motivos generacion

analisis estadistico.


escalas regulares.

diagramas de consonancia de e.a.

espacios vectoriales (artículo) distancia

escalas árabes

tetracordios y conexion( tono, junto y solapado)

medidas de escalas aarbes (artículo escalas)


notacion microtónica ascii


tanbur e intervalos. sonido.


percepcion intervalos

fracción contínua libros=


figuras anal9t

figuras ada1


fórmulas de disonancia




alusiones a flamenco, india, turca, árabe, indonesia




microtonía en matiz (gamaka, vibrato, mordente, melisma)

umm kulthum


alusiones a todos los o --------------------------

microtonía de duraciones, ha sido tocada


motivos arbes, qaflah


codigos orientales más finos:



notaciones india, turca, arabe, persa


notas: repertorio mínimo de sonidos que da cuenta de todos los intervalos que salen en una música

: no es una escala


each number has a size and a colour

its size is its cardinal

its color is its divisibility (structure)


hipotesis psicoacústica numero 1

each interval is comprehended inside of the conventión where it happens.(an equal tempered scale of 5 or 7 is inderstood as



recordings: cassette and DAT

graphics: transparencies

written material


psychoestetics in LTPM.



Appendice 1


The players and musical recordings analysed and described here are:

Mohiuddin Ba'yun, tanbur. Liban.

Baidaphon Company B 082 703, Cairo, 1920. Reedited in CD by Institut du Monde Arabe, vol 1, Ocora, Paris,1987. Rast taqsim extensively analysed in Sánchez&Odeimi [7]

Menese, singing martinete, Spain.

Private cassette with selections of non-instrumental flamenco (hondo) music, made by D.Antonio Espinós, expert in this field. First quatrain. Martinete is of Rast type in first notes.

Nikhil Banerjee, sitar, India.

Recordings of Ragas Bhairavi and Charukesi. Raga Charukesi has first tetrachord of rast type.

Munir Bachir, 'ud, Iraq.

Le 'Ud Classique Arabe. Harmonia Mundi, Paris 1984. Taqsim nahauand.

Maroc. Musique clasíque andalou-mahgrebine.

Orquestre de Fez, dirigée par Hajj Abdelkrim Raïs. Face B. Baytain (vocal solo) in nouba Istihlal (similar to Rast) sung by Abdel Rahim Suiri. Ocora, HM 57.588 588. Paris, 1986

Ustad Öczimi, turkish ney. Professor in Islamic University of Konia, Turkey. Private recording of main maqamat specially made for the author. Maqam rast.

Salman Chukur, 'ud. Iraq.

Iraq, craddle of Civilization, Ministry of Culture, Baghdad, 1982. First record (of three). Taqsim Baiati.


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