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N U M B E R A S M U S I C- B U I L D E R : S O N A N C E
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Abstract
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This paper deals with the old question about number and music, centered on pitch phenomena, as interval," Sonance", melody and harmony. It is shown that none of those tools, which constitute the greater part of the last three millennia music, can be understood without the consideration of number and human perception. In this way, psyche, number, thinking and feeling, are united in something related to the old 'logos'. This arrival seems to us very meaningful, giving to our musical conception, roots in the past, and ways of development for the future.
Some useful mathematical tools for interval and harmony representation and dissonance measure, Primal Spaces and Primal Complexity, based on Euler's measure, are presented and applied to actual modal and harmonic music, showing its power to represent the psychological alternation of instability-equilibrium, necessary for all human phenomena, including art and music.
To resume, Music is the Prosody of Universals
Index
)))))))))) Abstract 1. Music, Order & Disorder, Number & Freedom 2. Hypothesis on pitch and consonance perception 3. Pitch, Divisibility, Consonance 4. Representation in PS and Parallelism with musical concepts. 5. Primal Complexity as Dissonance Measure 6. A model of Modal Music 7. Harmonic map in PS3 8. Generalization to the Future. Microtonics? 9. Conclusions 10. Bibliography Appendix I . Absolute and Relative Notation based on Hölder comma Appendix II. Primal Space EP. Appendix III. InstrumentationBack to al Main
1. Music, Order & Disorder, Number & Freedom
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If our aim is to study Music, or at least those aspects that might be quantified , we shall have to pause and consider what is Music and how it differs from sister arts and crafts.
Its sonorous nature sets it apart from the visual arts which remain beyond its borders, (although similar sensations such as repetition, motif, form etc. maintain bridges between these worlds. This aspect is covered more generally by synaethesia.)
Focusing then on sound, we find an area which is formed by Word and Music and within the former but close to the latter, lies poetry, occupying intermediate ground and participating of both. There is however, a marked difference regarding expression: that is, when dealing with words one refers to concepts or ideas linked to the world outside and this in turn, forms the basís of language; in music on the other hand, one is concerned with non conceptual notions. With words we speak of external things, with music we express the world inside.
What then defines Music establishing the frontiers with the sister arts which came after or maybe perhaps gave birth to Music?
Taking into consideration those features of sound which are common to Word and Music, we notice that both resort to tone and intensity and their division into units of various duration and color or timbre, form a chain o series of alternating segments which combined in different ways can express many things.
So if these features essential to Word and Music are common, it must be their respective use that which separates them. It is the concept of Measure that distinguishes Music more than any thing else from other art forms.
In fact, it is Number , the measure of these features which makes Music, Music, bringing to bear Relations, Proportions, Ratios. This creates Harmony in its general sense and its particular musical sense, as will be seen. But what is immediately apparent is the simple relations between the duration of the elements, such as notes , beats; they give rise to a regularity that is the origin of the notions of tempo, bar, beat, rhythm , etc. Equally from the Pitch perspective, simple relations lead to compulsory or preferential intervals (octave, fifth, third); they enable us to build scales and chords. Even traditional musical timbres present combinations of simple elements in simple relations (spectra of harmonic partial). These are the patterns that echo inside us and tell us things.
That is to say that Music, to become Music, renounces the infinite and continuous variety of durations and tones and of its own accord, sets limits in order to build within that limited universe more and more complex forms. This is in no way contradicted by the evolution of forms and styles: These change merely to be substituted by other new ones.
Why this self-imposed limitation? The answer might rest with perception, that mechanism connected to the five senses , which enables us to perceive. Our subjective estimation of these features simplifies and reduces them: only simple relations and measures can be perceived, 1 to 2,3 to 4 and with more difficulty 5 to 4 ( remember usual bar measures). So in music we resort to relations we can follow in a way which all (or almost all) feel but cannot always describe. And in this ineffable perceptions the soul rejoices.
Coming back to number , how does the human ear and mind perceive the phenomenon? We know for instance that octaves are slightly greater than the interval of frequencies represented by the ratio of 2 to 1 (1200 cents) , mainly in the extremes (piano tuning). Any interval analysis must include this "distortion" that the ear imposes on the arithmetical number.
Thus perception plays a part in these measures, therefore in number. It is always "2","3","5" , but in their perceived realization, that may coincide or not with the physical measure. Number, Vibration, Ear are inextricably fused in Music, or better create it. Yet in a scientific approach we must use objective measures: but these methods must include the 'transference function' ( equalizing filter ) of human perception.
However, after this necessary numerical universe is established, as a kind of structure that supports the building, to become Art, Music needs to introduce some freedom, translated in modification which, without attempting against the stability of this building, permits variations or shades which awakens the surprise, the interest and personal expressivity of composer and performer.
We can conclude that in Music ( and for that matter in all Arts) there must coexist, in permanent conflict, these two elements, Order and Freedom (or Freedom and Order following the creed on the moment). Order makes it comprehensible, and Freedom delightful and interesting. Order on its own bores; Freedom left to its own device does not convey anything and bores no less. As we stated in the abstract, Music is Prosody ( expressive freedom ) on Universals (number and order).
The practical musician, conscious of the extraordinary complexity of the musical universe usually distrusts these notions of numbers applied to Music, considering as simplistic and, in any case, limiting his creative freedom, a conquest of the western world to which he belongs.
But, without speaking of the oriental world, more traditional in general, which stresses over all a fidelity to some principles (order), it could be said that this western musician make uses, without knowing it, of this numerical simplicity which he might have shunned. When he plays a just fifth, or composes or performs in 3 by 4, he is making numbers without knowing it. Leibnitz once defined Music as an " hidden arithmetical exercise of the soul, unconscious of its counting". The practical musician knows, but he does not knows that he knows, just as the dragonfly knows without knowing of flying but flies. The role of Theory is to disclose this knowledge in conceptual form.
Let me give an example of necessary order in its most ludic, spontaneous and 'innocent' form: when children play, they always play at 'something'; 'this 'something is really the set of rules of the game which the players accept voluntarily to enjoy themselves more, because a game without rules should bore them soon. When in a game , someone says "that's cheating", does it not mean that the collective rules have been broken? Should not even the followers of a ludic conception of music take heed of their masters, children, and admit some degree of regulation and order?.
Here we see another illustration The trees of the figure show by their shape examples of order and number just as in music. The extremes express such rigorous order that it kills the aesthetic emotions, and conversely, extreme disorder that confuses and thereby overrides the aesthetics too. Between them, there lies a group inside which there is probably a balanced combination of order and disorder, number and its violation, and which can be felt. See the figure and decide for yourself.
It is inevitable that in a theoretical exposition such as this one, those elements that represent order, system, organization, should be privileged over those that stand for disorder and freedom, more subtle in nature. Let these lines nevertheless state the case for the ample recognition of this basic duality as the aesthetic foundation of Art. In the following paragraphs, we will first state some psychoacoustics hypotheses on pitch and consonance, . After we define a measure of consonance , the Primal Complexity, which we apply to descriptive model of modal music. Then, in the appendix, the our notation and the Primal Space will be introduced; and a short description of the techniques developed in the LTPM and used in this work.
2. Hypothesis on pitch and consonance perception
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We confront the well-known problem of pitch and consonance perception, setting the following hypothesis:
PH0.
Perception extracts 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 ( as it 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 phenomenon in a musical and artistic way, even violating it by avoiding easy (simpler consonances).
PH1.
We perceive an interval, and its consonance, as the simplest and nearest one 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 sounds that were only varieties in a simpler universe.
It does not mean that we accept as correct these sounds; only we lack of a code for them, and therefore, they simply do 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 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.
PH2.
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" shrouti", shows this character. As we see in [6] with the intervals:
natural third 5:4 do.mi- " Prasârini" is
diffuse, penetrant, shy, sweet, restful.
Pythagoric 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 (up to a certain point) for other neighboring intervals, as in tempered scale; but we believe that loosing an important part of its effect.
We can sum it up by saying that each number has a size and a color: its size is its cardinal; its color is its divisibility (structure).
PH3.
The longer 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. This 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 intervallic size of simple consonances and the variability of more complex ones (moving semitones," pien", variable degree, etc, ).
PH4.
Any judgment of consonance is made in any moment, on the last sounds heard before this moment. It implies that we attribute a measure of consonance to a cluster composed with the last sounds, even those that have actually disappeared: they are held by the short memory, and probably decay in importance as new sounds replace them.
An important corollary is:
PH5.
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 nay (flute) or 'ud (lute).
The choosing of determinate consonances (which usually means depicting others) fixes the main notes of the mode; its hierarchies and structure are built in this way; read M4 in next paragraph.
3. Pitch, Divisibility, Consonance
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Let us see what happen when we sum ( i.e. sound together) two pitched sounds, which have 'combed'the spectra, and the fundamental of one of them is a harmonic of the other. If the fundamental has a frequency F, its harmonics will be 2F, 3F,
{F} = ( F, 2F, 3F,...,nF,..)
{F} = ( F 2F 3F 4F ... nF ....)
{G} = ( G 2G 3G 4G ... nG...)
and if G is el i-harmonic of F, then:
{ F + G } = ( F 2F 3F ...iF...nF...2iF ..... 3iF.....ijF. )
( G 2G 3G ... jG...)
then the j-harmonic of G is equal to the ij-harmonic of F: all the harmonics of G are also harmonics of F. We can say that G reinforces a part of the F spectrum. The primary condition for a perceptive fusion of both spectra are satisfied.
Even when both series are not in harmonic relation, its sum can be a harmonic series: if F/m = G/n = H, then for both series exist a common sound, H, which can be the Fundamental for all the harmonics of F and G, together.
But H will be not the maximum frequency which satisfies this condition (to be the fundamental ) except if m and n are mutually prime: if there is a factor k for which m=k¨m' and n=k¨n', then k¨H will be the fundamental because F/m' = G/n' = k¨H.
For instance, if F=600 hz and G=400 hz., then there is a frequency, 100hz, of which all the harmonics of F and G are also harmonics; yes, 600/6 = 400/4; but 6 and 4 are not mutually prime, they share the 2 factor. Therefore the fundamental will be 200 hz. the maximum possible.
Therefore to get the fundamental we must obtain the maximum common divisor of both numbers. And, as we can repeat this procedure for more summed harmonic series, we arrive in this way to an important fact:
"The pitch (fundamental frequency) of several mixed pitched sounds is the maximum common divisor of the simple pitches"
In the time domain, as period T is the inverse of frequency F, we will express the same fact as:
"The period of several summed periodic signals is the minimum common multiple (m.c.m.) of the summed periods.
Let see this effect in a graphic translation of pitch and period phenomena : in the next figure we sum periodic webs where we add more and more 'harmonics' as we go down in the graphic; the graphic period can be easily seen. When we sum several, we see again the resulting period which is in fact this m.c.m of the simple periods. And note that this period is the same as we descend in the graphic, which means that this phenomenon does not depend on the harmonic content of summed pitches.
As we are now in the time domain, lets apply this to rhythm: if we play together two rhythms contained in measures of m and n parts of equal duration, i.e. 3 and 4, we will obtain a composed rhythm with a measure of m.c.m parts: 12 parts in this case.
O o o O o o O o o O o o O o o O o o O
X x x x X x x x X x x x X x x x X x x x
I I
We observe now until what point number and perception are united in a so important musical feature as pitch.
But let us consider now the blending or fusion of these simple sounds in one. This fusion will occur only if those periodicities are kept during a temporal interval long enough to be perceived, i.e. several periods. And if they change, they must change together, as 'natural harmonics' do: they remain multiples of the fundamental, even if it changes, pitch phenomenon being maintained.
Even for non exact multiples and divisors there will be periodicities, o better, pseudoperiodicities, provided the differences are not too big, as we perceive pitch even with partials not exactly harmonics; in a sound so inharmonic as the bell, usually a pitch can be ascribed to the sound: as far as we provide perception with a help ( i.e. some partials almost in harmonic relation), it will find by itself a compromise, a explanation, a way to understand, to 'explain' what is offered to it.
The resulting sound will have therefore a 'ground pitch' which will be possible to hear, in case of extreme fusion and a high enough value. But in case of relative blending or/and low value , as 20, 10 o 5 hz. the sensation will be something which we call Consonance. Therefore:
"Consonance is the result of the integrator perceptive mechanism, which tries to unite any combination of sounds in a pitched one"
4. Representation in PS and Parallelism with musical concepts.
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In appendix II we introduce the Primal Space, where every possible rational musical interval can be expressed in these spaces and subspaces: this is the reason for its introduction. All the interval and scale systems (with intervals in rational form) have their representation in these spaces.
We can understand therefore that the Major Pythagoric Scale can be represented in EP2, and the Natural (Zarlino) Scale in EP3, Arquitas System (with its three genera ) in EP4, and so on.
See in fig. a representation of Natural (Zarlino) and Pythagoric Scales. Note there the bigger extension of the last compared with the first.
After adopting a reference frequency, all others form intervals with it, being the interval with itself 1, origin of coordinates (null nuple). Any other frequency related with it by natural (rational) relations will appear in EP as a point. The interval which leads from one to the other will be the vector ( a free vector) which unite both points or frequencies. A new interval will lead to another point and so on.
For instance if we choose a 64 hz. reference for EP2 ( only first two primes, '2' and '3' axis. A jump of an ascending Pythagoric tone 9/8, leads the origin (0,0) to point (-3,2), corresponding to 64 * 9/8 = 72 hz. by interval-vector (-3,2). A new ascending tone, by vector (-3,2) lead us to (-6,4), 81 hz.
(-3,2) + (-3,2) = (-6,4)
or
2 ¨(-3,2) = (-6,4)
Well known vector space concepts remain musically reflected in our EP. The base of space (set of independent vectors which generate all others by linear combination) is one of the most useful, if we choose a fitting base to the scale to be represented and comprehended. For instance, in Pythagoric scale, which uses several (-3,2) intervals it will be easy to take this as one of base vectors. Taking the true fourth ( 4/3) as the new base vector ( we need only two), (2,-1) we can generate and represent the Pythagoric scale:
second 9/8 -- 1¨(-3,2) + 0¨(2,-1) 1 tone third 81/64 -- 2¨(-3,2) + 0¨(2,-1) 2 tones fourth 4/3 -- 0¨(-3,2) + 1¨(2,-1) 1 fourth fifth 3/2 -- 1¨(-3,2) + 1¨(2,-1) 1 tones & 1 fourth sixth 27/16 -- 2¨(-3,2) + 1¨(2,-1) 2 tone & 1 fourth seventh 243/128 -- 3¨(-3,2) + 1¨(2,-1) 3 tones & 1 fourth octave 2/1 -- 1¨(-3,2) + 2¨(2,-1) 1 tone & 2 fourths
As is evident in musical language, if even we choose as base vectors the Pythagoric tone and semitone (9:8 and 256:243) we arrive easily to the familiar definitions " an octave has 5 tones and 2 semitones ": our vector space EP reflects perfectly the traditional language, plus many other musical phenomena that are not traditional.
Note that the change of coordinates and base vectors affects the 'position' of figures which represent scales and melodies, but not the relative distances and form. They 'turn ' and deplanes themselves, but keep their structure and form, which is inherent to its constitution by mutual intervals.
We can conclude that EP is a 'musical space' in the sense that musical relations are reflected in it, and conversely, any geometrical property has a musical counterpart. We will center on the metric properties of the space, which will allow us to arrive to Consonance and meet Euler and his beautiful measure for it.
5. Primal Complexity as Dissonance Measure
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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 then: EC(A) = 1 + õ ai¨(pi-1)
where pi are the prime factors of A, ai its exponents, and ´ y õ the signs 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 from Euler omitting the '1' summand because of better geometrical properties of this measure, defining the complexity of A as:
i
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 minimum common multiple by its maximum common divisor.
)))))))))))) ) \ m.c.d (A,B,C,.) // m.c.m (A,B,C,.) \ PC (A,B,C,..) = C (
For instance,
PC(18,16) = C( 2^4¨3^2/2) = C (2^3¨3^2) = 3¨(2-1) + 2¨(3-1) = 7
This is applied to the relative dissonance of simultaneous
(or successive) frequencies integrated in the perceptual system.
We can simplify the formula in an easy algorithmic manner; for the numbers A, B, C,... indexed by j:
PC (A,B,C,..) = õ [max(ai,bi,ci,...)-min(ai,bi,ci,...)] ¨ (pi-1)
j j
where the primal complexity or mutual dissonance of these numbers or frequencies is obtained multiplying for each prime p1, the weight pi-1 for the difference of the major exponent of this prime among all numbers and the minimum, and summing up after all these products.
If we have only a number A, its complexity is null.
As we have seen the measure PC is related to 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, a period in normalized time measure. The factor (pi-1) can be considered as a weight of the prime pi, which we keep from Euler.
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 the vector space PS o Primal Space, where we remember that 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 of it.
Even other generalized Harmonic Distance can be defined, by introducing the r-distances:
*ai-bi* ¨(pi-1) }r 1/r)
PCr(A,B) = { õ
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 infinite , positive, being always a real mathematic distance.
The dissonance between two frequencies appears as an Harmonic Distance, which make a beautiful pair with the Melodic Distance, defined as another distance ( different weight ) in the same space. r is 1 in both.
*a - b * ¨ log p 1 ³ ³ ³ n HD(a,b) = K'¨ õ *a - b * ¨ (p -1) 1 ³ ³ ³n MD(a,b) = K ¨ õ
The arbitrary constants K, K' depends only on the unity chosen 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 frequencies, harmonic content, education, coded scale, etc). These distances are useful to 'describe' the melodic and harmonic processed involved in music.
For instance in the representation in PS of Pythagorean and natural (Zarlino) scales, the harmonic distance must be seen as 'city-block' distance, as walked in a town of blocks (no diagonal).
Now we can measure the dissonance situation and alternatives, the history of a melody in an harmonic (in broad sense) universe. If we take a sound as always 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 dissonance according to its numerical relation with B, its intensity and its duration. All these concepts can be generalized to western harmony. See [42,43] for a more detailed account of these concepts.
Let see a list of the dissonances of some usual intervals, for r=1 and r=2.
)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))) interval ratio cents ED1 ED2 ))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))octave 2:1 1200 1 1 fifth 3:2 702 3 1.7 fourth 4:3 498 4 2.4 major third 5:4 386 6 2.8 minor third 6:5 316 7 2.6 tone 9:8 204 7 4.1 nat.semitone 16:15 112 10 4.7
As for the differences between PC1 and PC2, we note that both grow with the numbers included in the interval ratio, but not in the same way: PC2 more slowly than PC1. And, very interestingly, major and minor thirds are inversed in their order: 5:4 is More consonant than 6:5 in PC1, but Less in PC2.
The reader will judge if these numbers correspond with his own experience, remembering however that its musical practice have surely influenced its primitive valuation.
6. A model of Modal Music
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Our model of modal music, deduced from many automatic and natural analyses 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 ambits of pitches are used.
M2. They cover this ambits 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 pentatonic 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.
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, a squaretail 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. A 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 +))))))))0)))))))0))))0))))))))0)))))))0))))))))0))))),
APPROXIMATE STRUCTURE OF A RAGA SCALE CHARUKESI (India).
DO 2:1 do 03))))))))))))))))))))))))))))))0)))))))))))))))))))))), * 3:2 sol 4:3 * 1 /))))))))))))))))0)))))))))))))3)))))0))))))))))))))))1 * 5:4 mi- 6:5 *16:15 la- 5:4 * 2 /))))))))0)))))))3)))))0)))))))3)))))3))))))))))))))))1 * 9:8 d 10:9 *16:15 fa * *10:9 sib= 9:8 * 3 /))))))))3)))))))3)))))3)))))))3)))))3)))))))3))))))))1 * * * * * * * * do re mi fa sol lab- sib= do' +))))))))0)))))))0)))))0)))))))0)))))0)))))))0)))))))),
T t S T S t T
APPROXIMATE STRUCTURE of Modus TRITUS PLAGAL (Gregorian). do FA do 0+))))))))))))))))))))))3)))))))))))))))))))))))))))))), * 4:3 fa 3:2 * 1 /))))))))0)))))))))))))3)))))))))))))))0))))))))))))))1 * 9:8 re 6:5 * 5:4 la- 6:5 * 2 /))))))))3)))))))0)))))3))))))))0))))))3))))))))))))))1 * * 10:9 *16:15* 9:8 * 10:9 *16/15sib- 9:8 * 3 /))))))))3)))))))3)))))3))))))))3))))))3)))))3))))))))1 * * * * * * * * do re mi fa sol la- sib- do' +))))))))0)))))))0)))))0))))))))0))))))0)))))0)))))))), T t S T t S T
M5. This structure can be shifted in frequency in a continuous way according to particular conditions (instruments, voice, even mood) without changing its significance. There is no absolute pitch in that music. But after it is chosen in a performance, it does not change, it will sound during all the performance, always in the conscience of performer and listeners (M,S,A,T,P), even actually sounded(I,P).
M6. The use of those 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 pentacord, and is called genus or ajnas in Arabic music. The interpreter presents slowly and carefully these notes one after another, appearing gradually to the listener. With that fixed reference, each pitch acquires a very particular and strong function. The consonance is not the only expressive mean: the motivic development, like a prosody, is essential to the mode. But what is more melody makes use also 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 notes in an 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 makes the melody move and give life to the music, in an alternative arsis-thesis.
M9. This arsis-thesis pair is repeated in all the levels. The highest is the whole melody and its end, which represent to the hearer coming back to the reference, felt as peace and resolution of dissonance problems proposed in the melody.
M10. The particular consonances (intervals) chosen 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 language (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 example 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 constructs 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 effectivity!) rules the performer 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 these 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 these principles, see the score of a Rast Taqsim from Liban, played by Buyun in 1920 on tanbur, long necked luth, and note the sections which correspond to different genus.
In the figure 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 frequency 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 cadencies.
7. Harmonic map in PS3
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By choosing Fifth an Major Third as axis of PS3 ( third is Octave, normal to the represented plane) we have a true map of harmonic relationship between notes and between tonalities and chords. A chord is a polygonal and its dissonance with the tonic is represented by its global distance to that tonic.
Harmony can be seen then as a movement of figures from a chord on tonic to far places and coming back.
This map in very similar to the schema of parental tonalities which Schönberg offers in its book "Structural Functions of Harmony". This shows how our Primal Space, build only from number theory, converges with an schema deduced from music and harmonic comprehension only:.
Schönberg :
3)))))))))))3 * * MI M Mi m * Sol M * Mi m Si b M * * 3)))))))))))3)))))))))))3)))))))))))3 * * * * La M * La m * Do M * Do m * Mi b M * * * * 3)))))))))))3)))))))))))3)))))))))))3 * * Re M Fa m * Fa M * Fa m La b M * * 3)))))))))))3
and ours:
23)))))))))))3)))))))))))3)))))))))))3)))))))))))3)))))))))))3 * * * * * * * * * * * * *si *mib *Sol M *si *re# * 1 3)))))))))))3)))))))))))3)))))))))))3)))))))))))3)))))))))))3 * * * * * * * * * * * * *mi *lab *Do M *mi *sol# * 0 3)))))))))))3)))))))))))3)))))))))))3)))))))))))3)))))))))))3 * * * * * * * * * * * * *la *reb *Fa M *la *do# * -13)))))))))))3)))))))))))3)))))))))))3)))))))))))3)))))))))))3
*F++6 *A+6 *C#7 *E#-7 *G##=7 * 7 3)))))))))))3)))))))))))3)))))))))))3)))))))))))3)))))))))))3 * * * * * * *Bb++5 *D+6 *F#6 *A#-6 *C##=7 * 6 >44444444444344444444444>)))))))))))>44444444444344444444444> 5 * 5 5 * 5 5Eb++5 *G+5 5B5 5D#-6 *F##=6 5 5 >)))))))))))3)))))))))))>)))))))))))>)))))))))))3)))))))))))> 5 * 5 5 * 5 5Ab++4 *C+5 5E5 5G#-5 *B#=5 5 4 >)))))))))))>)))))))))))>)))))))))))>)))))))))))3)))))))))))> 5 * 5 5 * 5 5Db++4 *F+4 5A4 5C#-5 *E#=5 5 3 >44444444444344444444444>)))))))))))>44444444444344444444444> * * * DD * * * *Gb++3 *Bb+3 *D4 *F#-4 *A#=4 * 2 3)))))))))))>44444444444344444444444>)))))))))))>444444444443 * 5 . * . 5 5 * *Cb++3 5Eb+3 . *G3 . 5B-3 5D#=4 * 1 3)))))))))))>)))))))))))3)))))))))))>)))))))))))>)))))))))))3 * 5 . * . 5 . 5 * *Fb++2 5Ab+2 . *C3 . 5E-3 . 5G#-3 * 0 3)))))))))))>)))))))))))>)))))))))))>)))))))))))>)))))))))))3 * 5 . * . 5 . 5 * *Bbb++1 5Db+2 . *F2 . 5A-2 . 5C#=3 * -13)))))))))))>44444444444344444444444>)))))))))))>444444444443 * * * * * * *Ebb++1 *Gb+1 *Bb1 *D-2 *F#=2 * -2344444444444>)))))))))))>44444444444344444444444>)))))))))))3 * 5 5 * 5 * *Abb++0 5Cb+1 5Eb1 *G-1 5B=1 * -33)))))))))))>)))))))))))>)))))))))))3)))))))))))>)))))))))))3 * 5 5 * 5 * *Dbb++0 5Fb+0 5Ab0 *C-1 5E=1 * -43)))))))))))>)))))))))))>)))))))))))>)))))))))))>)))))))))))3 * 5 5 * 5 * *Gbb++(-1) 5Bbb+(-1) 5Db0 *F-0 5A=0 * -5344444444444>)))))))))))>44444444444344444444444>)))))))))))3 Fig. PARENTAL MAP of NO TEMPERATED NOTES The Primal Subspace PS3 (only primes 2,3,5 ) when refered to two axis, Fifth (ordinate) and Major Third (abscisa). The third axis is Octave, perpendicular to this plane (not represented). Numbers mean relative octave. 1 octave= 53 hc (hölder comma) # and b mean 5 hc. + and - mean 1 hc. Neighbor points means consonante notes. Far points dissonant. Different names mean Enharmonic notes when equal in temperate Temperament means to warp this map equalizing these enharmonic. (i.e: d and d-, fa#- and gb+, ... ): The plane became a sphere. Axial Movement Interval Relation 1- Dison- 2(nor) -------------------------------------------------------------- Perpendicular: Octave 2 : 1 0 0 Vertical: Just Quint 3 : 2 3 3 Horizontal: Major Third 5 : 4 6 6 Oblique (\): Minor Third 6 : 5 7 6.15
8. Generalization to the Future. Microtonics?
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Are these numbers and measures valid only for rhythmic and tonal music ?. Without rejecting non tonal and non rhythmic music, as concrete music , and a part of electronic as well as a lot of experimental music, we believe them valid in possible generalizations of these past tools, by means of an increasing complexity which avoids complacency in easy consonance.
For instance, let us construct a scale based on harmonics, 7 to 14, one or our Regular Scales. As for any one, each degree has a relative dissonance with tonic which allows the play of tension and distension. Also chords can find its function, and so a harmony can be developed. Thinking in all possible scales, this generalized harmony would take centuries to develop (search, perceive, regulate).
The chords in this 'tonality' have the following PC.
chord (degrees) PC1
7 9 11 20
8 10 12 8
9 11 13 26
10 12 14 13
11 13 15 28
It should be possible, then, to make harmonic liaisons in this different universe.
9. Conclusions
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0. In music there coexists necessarily order and freedom in a harmonic struggle. The main order component is number, the main freedom component is pragmatic prosody.
1. Number is acting in Music meanwhile pitch, octaves, fifths, and thirds, parts, bars, rhythms are used in Music. The numbers implied are rather Relations, Ratio between the acoustic feature measures used in Music, as Duration, Pitch, Intensity.
2. The perception impose its own 'rounding' on these numbers: they are perceptive numbers. Perception accomplishes an active role, simplifying, trying to understand and comprehend. It will perceive numbers as simple as possible.
3. The Primal Space EP, is a good tool to represent the mechanics of music and the perception of it. It is a Perceptive Map.
4. Pitch and Consonance are similar phenomena (as Harmony and Timbre are for themselves): they are related with the period of repetition of composed sounds.
5. The Primal Complexity is a good measure of Consonance and Dissonance, and has an illuminating representation in EP, as Distance.
6. Consonance and Dissonance constitute, both, the main motif of Music, in dialectical opposition. Our soul looks for consonance after dissonance; as for food after hunger.
7. Modal Music can be considered as a journey from home (modal tonic) to home, after going to far (dissonant) places. The dissonance of these places is determined for the intervallic structure of scale, which fixes points of tension (more dissonance ) and relaxations ( dissonance), all centered on tonic.
8. Harmonic music can be considered the same. Only more complex aggregations of tones are used, and therefore higher levels of dissonance (complexity) are reached.
9. These principles can be extended to even more complex tone aggregations, as Microtony can provide in an infinite manner. And in Time, to complex polyrythms.
10. Bibliography.
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[ 1] BENADE,A.H. Fundamentals of musical Acoustics. Oxford N.York,1976
[ 2] BLACKWOOD,E. The Struct.of Recognizable Diatonic Scales Princ.Pr. Prince.1985
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[ 5] CHERKI, S. Al-Moustadraf dans le regles.. Rabat, 1972
[ 6] DANIELOU,A. Traité de Musicologie Comparée. Hermann Paris, 1959
[ 7] ERLANGER,R.D' La Musique Arabe. Vol.1-6 Geuthner Paris 49-64
[ 8] GEVAERT,F.A. Histoire&Théorie d.l.Musique d.l'Antiquité G.Olms Hildes 1965
[ 9] GUETTAT, M. La Musique Clasíque du Maghreb Sindbad Paris, 1980
[10] HABA,A. Nuevo Tratado de Armonía R.musical Madrid,1979
[11] HAKKI,I,Ö. Türk Mûsikîsi Nazariyati ve Usûlleri. Ötüken Ne.Istanb,1984
[12] HELMHOLTZ,H. On the Sensation of Tone. Dover N.York,1954
[13] HERF,F.R(com) Mikrotöne I Helbling Innsbr.1985
[14] HERF,F.R(com) Mikrotöne II Helbling Innsbr.1987
[15] HESSE,H.P. Grundlagen der Harmonik in mikrotaler MusiK Helbling Innsbr.1989
[16] JARGY,Simon La Musique Arabe (Que sais je?) P.U.F. Paris, 1971
[17] JEANNETEAU,J. Los Modos Gregorianos Abadía de Silos, 1985
[18] Al-KATIB Kitab Kamal Adab Al-Gina Siria,s.IX
(SHILOAH,A.) La Perfection des Conn.Music. Geuthner Paris, 1972
[19] Al-Fasí Kitab Al Jumu Fi Al Musiqi ...
(FARMER,H.G.)An Old Moorish Lute Tutor Civil Pr Glasgo,1933
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[21] KASZIM, A Terminology of Oriental Music AlJamhuri Bagdad,1964
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[23] LACHMANN,R. Música de Oriente. Labor Barcel.1931
[24] LALOY,L. Aristoxéne de Tarente A.Forni Paris 1904
[25] LEON TELLO,F. Estudios de Historia de la Teoría musical C.S.I.C. Madrid.1962
[26] LLOYD,H.B Intervals, Scales and Temperament S.Martin N.York,1963
[27] MALM, W.P. Culturas Mus.Pacifico,Cercano Oriente&asía Alianza Madrid 1985
[28] MAHDI,Sal.El La Musique Arabe Leduc Paris, 1972
[29] MESSIAEN,O. Technique de mon Langage musical Leduc Paris 1944
[30] ÖZCAN, I.H. Türk Mûsikîsi Nazariyati ve Usulleri Ötüken Istamb.1984
[31] ROEDERER,J.G. Intr.Physics and Psychoacoustics of Music Springer N.York,1979
[32] Al-RAJAB,H Al Maqam Al-Iraqui AlMa'arif Bagdad,1961
[33] REINHART,K&U. Turquie. Les Traditions musicales. Buchet/Ch.Paris, 1969
[34] SACHS,K. La Música en la Antiguedad Labor Barcel.1934
[35] SACHS,K. Musicología Comparada Eudeba B.Aires1966
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Apendix I . Absolute and Relative Notation based on Hölder comma
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To represent intervals more exactly than by number of semitones, we need finer divisions of octave. These intervals appears in both theoretical and practical scales. We adopt a notation based on Hölder Comma, the 53th part of an octave: with an error smaller than two cents the traditional simple intervals which compose Pythagoric and Zarlino Scales. Even intervals which include 7 as factor, as 8/7 or 28/27 (Arquitas genus) are represented with less of 5 cents error.
We name notes or sounds according the Pythagoric Major Scale. The usual names, A, B, C, D, E, F, G, appear withoutalteration , as the tone 9/8 measures 9 comma, and the semitone 256/243, 4 comma. Temperate intervals will apper without alterations too, ans Zarlino will have its 3th, 6th and 7th degrees lowered a comma: C D E- F G A- B- C
The deviations on this base scale will be expressed by the number of comma lowered or raised according the following synbols:
comas up down
1 + -
2 ++ =
5 b
Will so cover all the 53 'notes': between tones:
notes: C C+ C++ Db- Db Db+ Db++ D= D- D
comma: 1 2 3 4 5 6 7 8 9
and between semitones:
notes: E E+ F= F- F
comma: 1 2 3 4
and similarly the others. This notation is similar, but not identical to that used by Danielou [10], and, also to that used in the traditional Turkish music [17],[30]. The first, very precise is something complex, as using a Zarlino base scale, with great and small tones and '1/4' interval. The second uses a Pythagoric base scale tone, but up to 8 types of sharp and flat.
We prefer our notation because we find it simple, intuitive, reducible to the usual temperate notation by supressing the comma alterations and surprisinly efficient to represent the traditional scales. However it 'rounds' the intervals and notes, as any temperament, only more finely than the usual one. Apendix II. Primal Space EP.
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It is well known that every positive integer number can be expressed in a unique way as a product of prime factors:
n=Ý
0)0 aA =
* * p ³n=1 ³
where p are the primes 2,3,5,7,..., an, the exponents of them in the decomposition and Ý is infinity .
2 2 1
For instance, 180 = 2 ¨ 3 ¨ 5 ; all other primes have
the 0 power for this number 180.
We can also write only these exponents as their bases are constants (the primes series):
2 3 5 7 11 13 17 19 ...
p1 p2 p3 p4 p5 p6 p7 p8 ...
)))))))))))))))))))))))))))))))))))))a1 a2 a3 a4 a5 a6 a7 a8 ...
The n-uple whisch represents this number 180 is therefore:
180 = { 2 2 1 0 0 0 ... }
( 1 is a trivial prime, not considered ).
In general:
a = { a1 a2 a3 a4 a5 }
where we understand that every exponent after the 5th is null.
We call 'A' the number, 'a' its associated n-uple.
Every integer has its n-uple, and every n-uple its integer:
1 = { 0 0 0 0 0 0 ... }
2 = { 1 0 0 0 0 0 ... }
3 = { 0 1 0 0 0 0 ... }
180 = { 2 2 1 0 0 0 ... }
Even every rational number (quotients of integers) can also
be expressed as n-uples: 9/8 = 3^2 / 2^3 = 2^(-3)¨ 3^2. And so:
9/8 = { -3 2 0 ... }.
3/2 = { -1 +1 0 ... }
4/3 = { +2 -1 0 ... }
32/27= { +5 -3 0 ... }
It is very easy to show that all sum of n-uples are n-uples,
that every product of integer numbers for n-uples are also n- uples, and that these operations correspond to products of integers and powers of integers:
(10/9)*(9/8) = { 1 -2 1..} + { -3 2 0 .} = { -2 0 1..} = 5/4
(3/2)^3 = { -1 1 0 ..} + { -1 1 0 ..} + { -1 1 0 ..} =
= 3 ¨ { -1 1 0 ... } = { -3 3 0 ... } = 27/8
Generally:
a + b = { a1 a2 a3 a4 a5 } + { b1 b2 b3 }
= { a1+b1 a2+b2 a3+b3 a4+0 a5+0 }
Ý
0)0 ( a + b )=
* * p ³ ³ = A.B³
1
with the sufficient propieties of the sum: (comnmutative,
associative, distributive).
And the product Ó.a = B , to multiply the elements of n-
uple by an integer corresponds to the power of the relative number:
B = Ó.a = Ó ¨ { a1 a2 a3 a4 a5 } =
= { Ó¨a1 Ó¨a2 Ó¨a3 Ó¨a4 Ó¨a5 } =
Ý Ý
0)0 Ó¨ a / 0)0 a \ Ó Ó=
* * p ³ = ( * * p ³ ) = A1 ³ \ 1 ³ /
with every property (distributive, associative) satisfied.
There exists also a Null Element ( number 1, n-uple null ) and an Inverse Element for each element (inverse numbe r, negated n- upla).
Therefore the set of n-uples, a well known Vector Space, is isomorphic with the set of rational numbers (Q) with the operations of product (*) and integer power (^). Every necessary and sufficient condition is satisfied. So:
{ n-uples, + , ¨ ) isomorphic { Q, x, ^ )
We can therefore consider both vector spaces as identical.
We will call this Vector Space the Primal Space, PS. It is also easy to apply all the properties of the Vector Space of n.uple, and to show that the ensemble of Primes is a Base of this Space, a natural base. Many other bases can be found, with the condition that none of the base vectors can be found as linear combination of the others.
Moreover, the sets of rational numbers that use only some of
the primes, not all, are also Vector Spaces, Subspaces of EP. We will call them according to the maxime prime present in them. Then, EP1 only uses 2 as factors ( the sets of powers of 2 ), EP2 includes 2 and 3, EP3, 2,3,5 and so on.
Apendix III. Instrumentation
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The examples, measurements and graphics shown in this presentation have been made with the following systems, all developped in this Laboratory, the LTPM.
ANALMELO.
Based on ATARI.
Connected with any MIDI keyboard can record, store,
edit, and output, play, any music in this standard.
Every kind of analysis can be made on the stored music,
as:
Consonance analysis of harmonies or melodies.
Histograms of notes, durations and intensities in a
performance.
Motive analysis and synthesis, in the melodic field.
(see in fig.4 this interval and consonance
analysis by Analmelo).
SETS
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
Symthesis of speech
Coding-recognition of speech.
(see in fig.10 the melodic (pitch) analysis of
example 7 by SETS)
EULERCO6
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).
ESCA9T
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 example 8, with
some tones and semitones constructed with these
little intervals. All the analyses in this paper
have been done also by it.)
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