Un model matematic pentru explicarea armoniei în muzică


Cercetătorii italieni şi ruşi au dezvoltat un model despre care cred că explică de ce oamenii aud unele acorduri ca fiind armonioase şi pe altele ca fiind disonante. Echipa de cercetare susţine că armonia poate fi explicată prin intermediul sistemului neuronal auditiv.

Majoritatea oamenilor pot face diferenţa dintre între armonie şi zgomotul general. De-a lungul timpului au existat mai multe teorii care au sugerat cum şi de ce percepem ca fiind plăcute anumite grupări de note şi pe altele ca fiind neplăcute, disonante. Unii specialişti au formulat ipoteza că creierul nostru este bombardat pur şi simplu cu note pe care le percepem cum vrem noi, însă Bernardo Spagnolo de la Universitatea din Palermi şi colegii săi ruşi nu sunt de acord cu această teorie, spunând că au dezvoltat un model prin care să demonstreze asta.

Echipa de cercetare spune că noi, oamenii, dispunem de diferiţi neuroni în diferte regiuni ale urechii care răspund la frecvenţe diferite. Anumiţi oameni răspund unei anumite note, în timp ce alţii sunt receptivi la altă notă, responsabili pentru această selecţie fiind aşa-numiţii neuroni senzoriali. Dar ce se întâmplă atunci când o persoană percepe ambele note ca fiind plăcute? Pentru a explica asta, cercetătorii au introdus în ecuaţie un al tip de neuron, numit interneuron. În modelul dezvoltat de ei, oamenii de ştiinţă sugerează că neuronii senzoriali transmit semnale interneuronului, care la rândul său transmite semnale creierului în funcţie de ce a auzit de la ceilalţi.

Cerceătorii mai spun că neuronii senzoriali se conformează modelului neuronului integrator cu pierderi (leaky integrate-and-fire), prin care stimulul (în acest caz sunetul) urcă voltajul până la un punct de saturaţie, după care îşi împărştie informaţia prin intermediul neuronilor senzoriali, în acest caz interneuronului, care trimite apoi semnele creierului. Dacă toţi neuronii senzoriali ar bombarda constant cu informaţii interneuronul, acesta din urmă ar fi inundat şi incapabil să proceseze toate informaţiile.

Oamenii de ştiinţă au aplicat apoi o teorie care spune că cu cât un semnal este mai aleatoriu, cu atât este dotat cu mai multă informaţie, propunând un număr pe care l-au numit regularitate. Această regularitate, spun ei, explică de ce ne plac note diferite ascultate laolaltă. Notele “plăcute” cântate împreună generează o regularitate ridicată, pentru că sunt dotate cu mai multe informaţii, în timp ce notele disonante produc o regularitate scăzută.

Cum se explică atunci armonia dintr-o piesă disonantă ca cea de mai jos? :

Sursa: Physorg

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3 comentarii

  1. The harmonic sense of the key (Tonart) in all its ramifications is compre-
    hensible only in relation to the idea of tonality, which should therefore be
    explained before anything else. Tonality is a formal possibility that emerges
    from the nature of the tonal material, a possibility of attaining a certain com-
    pleteness or closure (Geschlossenheit) by means of a certain uniformiry. To
    realize this possibility it is necessary to use in the course of a piece only those
    sounds (Kldnge) and successions of sounds, and these only in a suitable arrange-
    ment, whose relations to the fundamental tone of the key, to the tonic of the
    piece, can be grasped without difficulty. Subsequently, I shall be compelled to
    ake issue with various aspects of tonality and can therefore confine my remarks
    here to just two points: (r) I do not, as apparently all theorists before me have
    done, consider tonality an eternal law, a natural law of music, even though this
    aw is consistent with the simplest conditions of the natural model, that is, of
    the tone and the fundamental chord; all the same, however, (z) it is essential that
    the pupil learn thoroughly the basis of this effect [tonality]
    and how to attain it.

    Perhaps it is indefensible to try to
    derive everything that constitutes the physics of harmony from one of the components, say, just from the tone. Some characteristics can be derived from the tone, however, for the very reason that the constitution of the ear, the organ predetermined to receive tone, at least relates to the constitution of the tone somewhat as do well-fitting concave to convex parts. One of the [other] factors,however, the world of our feelings, so completely eludes precisely controlled investigation that it would be folly to place the same confidence in the few conjectures permitted by observation in this sphere that we place in those conjectures that in other matters are called

    [For] it is entirely
    possible that in spite of an observation falsely construed as fundamental we may, by inference or through intuition, arrive at correct results wherereas it is
    not at all a proved fact that more correct or better observation would necessarily yield more correct or better conclusions/

    Once again: the tone is the material of music. It must therefore be regarded,
    with all its properties and effects, as suitable for art. All sensations that it re-
    leases – indeed, these are the effects that make known its properties – bring
    their influence to bear in some sense on the form of which the tone is a com-
    ponent, that is, on the piece of music. In the overtone seriesr* which is one of
    the most remarkable properties of the tone, there appear after some stronger-
    sounding overtones a number of weaker-sounding ones. Without a doubt the
    former are more familiar to the ear, while the latter, hardly perceptible, are
    rather strange. In other words: the overtones closer to the fundamental seem to
    contribute more or more perceptibly to the total phenomenon of the tone –
    tone accepted as euphonious, suitable for art – while the more distant seem to
    contribute less or less perceptibly. But it is quite certain that they all do contri-
    bute more or less, that of the acoustical emanations of the tone nothing is lost.
    And it is just as certain that the world of feeling somehow takes into account
    the entire complex, hence the more distant overtones as well. Even if the ana-
    lyzing ear does not become conscious of them, they are still heard as tone color.
    That is to say, here the musical ear does indeed abandon the attempt at exact
    analysis, but it still takes note of the impression. The more remote overtones
    are recorded by the subconscious, and when they ascend into the conscious
    they are analyzed and their relation to the total sound is determined. But this
    relation is, to repeat, as follows: the more immediate overtones contribute more,
    the more remote contribute /as.r. Hence, the distinction between them is only a
    matter of degree, not of kind. They are no more opposites than two and ten are
    opposites, as the frequency numbers indeed show; and the expressions
    sonance’ and
    which signify an antithesis, are false. It all simply
    depends on the growing ability of the analyzing ear to familiarize itself with the
    remote overtones, thereby expanding the conception of what is euphonious,
    suitable for art, so that it embraces the whole nazural phenomenon.
    What today is remote can tomorrow be close at hand; it is all a matter of
    whether one can get closer. And the evolution of music has followed this course:
    it has drawn into the stock of artistic resources more and more of the harmonic
    possibilities inherent in the tone.

    Now if I continue to use the expressions
    even though they are unwarranted, I do so because there are signs that the
    evolution of harmony will, in a short time, prove the inadequacy of this classi-
    fication. The introduction of another terminology at this stage would have no
    purpose and could hope for little success. Since I still have to operate with these
    notions, I will define consonances as the closer, simpler relations to the funda-
    mental tone, dissonances as those that are more remote, more complicated. The
    consonances are accordingly the 6rst overtones, and they are the more nearly
    perfect the closer they are to the fundamental. That means, the closer they lie to
    the fundamental, the more easily we can grasp their similarity to it, the more
    easily the ear can fit them into the total sound and assimilate them, and the more
    easily we can determine that the sound of these overtones together with the
    fundamental is’restful’and euphonious, needing no resolution. The same should
    hold for the dissonances as well. If it does not, if the ability to assimilate the
    dissonances in use cannot be judged by the same method, if the distance from
    the fundamental is no measure of the degree of dissonance, this is even so no
    evidence against the view presented here. For it is harder to gauge these differ-
    ences precisely, since they are relatively small. They are expressed by fractions
    with large denominatorsl and as it requires some thought to say whether 8/234
    is larger or smaller than 4f
    68o, because a mere estimate can lead one astray, the
    mere estimate made by the ear is just as undependable. Efforts to make use of
    the more remote consonances (today called
    as artistic means thus
    led necessarily to many an error, to many a detour. The vay of history, as we
    can see it in that which has acrually been selected by practice from the practic-
    able dissonances, hardly leads here to a correct judgment of the real relations.
    That assertion is proved by the incomplete or unusual scales of many other
    peoples, who have, nevertheless, as much right as we to explain them by appeal
    to nature. Perhaps their tones are often even more natural than ours (that is,
    more exact, more correct, better); for the tempered system, which is only an
    expedient for overcoming the difficulties of the material, has indeed only a
    limited similarity to nature. That is perhaps an advantage, but hardly a mark of

    Our major scale, the series of tones do,re, mi, fa, sol, re, si, the tones that also pro-
    vided the basis for the Greek and church modes, we can explain as having been
    found through imitation of nature. Intuition and inference (Kombination)
    assisted in translating the most important characteristic of the tone, the over-
    tone series, from the vertical (as we imagine the position of all simultaneous
    sounds) into the horizontal, into separate, successive tones. The natural model,
    the tone, exhibits the following characteristics:
    r. A musical sound (Klang) is a composite, made up of a series of tones
    sounding together, the overtones; hence, it forms a chord. From a fundamental,
    C, these overtones are:
    do ,sol,do1,mi1,sol1,sibemol,do2,re2,mi2,fa1,sol2 Etc.
    z. In this series the c is the strongest sound because it occurs the greatest
    number of times, and because it is actually played or sung itself as a funda-
    3. After the do the next strongest tone is sol, because it occurs earlier in the
    series, therefore more often than the other tones.
    If we think of this g as a real tone (as indeed occurs when the overtone series
    is realized horizontally, when, for example, the fifth of a horn tuned in do is
    played), it then has overtones itself (as a tone actually played); these are:
    sol,re,sol,si,re etc.l
    and at the same time this & together with its overtones, presupposes the do
    (fundamental of the horn). Thus it happens that the overtones of the overtones
    also contribute to the total sound.
    4. An actual tone (the sol) appears as dependent upon a tone a 5th below, the
    The conclusion that follows from the foregoing:
    This tone, do, is likewise dependent upon the tone a fifth below it, .Fa.
    Now if the C is taken as the midpoint, then its siruation can be described by
    reference to two forces, one of which pulls downward, toward fa, the other
    upward, toward sol:

    Adding up the overtones (omitting repetitions) we get the seven tones of
    our scale. Here they are not yet arranged consecutively. But even the scalar
    order can be obtained if we assume that the further overtones are also in effect.
    And that assumption is in fact not optional; v/e must assume the presence of the
    other overtones. The ear could also have defined the relative pitch of the tones
    discovered by comparing them with taut strings, which of course become
    longer or shorter as the tone is lowered or raised. But the more distant overtones
    were also a dependable guide. Adding these s/e get the following:

    The discovery of our scale was a stroke of luck in the development of our
    music, not only with regard to its success, but also in the sense that we could
    just as well have found a different scale, as did for example the Arabs, the
    Chinese and
    Japanese, or the gypsies. That their music has not evolved to such
    heights as ours does not necessarily follow from their imperfect scales, but can
    also have to do with their imperfect instruments or with some other circum-
    stance which cannot be investigated here. Moreover, it is not to our scale alone
    that we owe the evolution of our music. And above all; this scale is not the last
    word, the ultimate goal of music, but rather a provisional stopping place. The
    overtone series, which led the ear to it, still contains many problems that will
    have to be faced. And if for the time being we still manage to escape those
    problems, it is due to liitle else than a compromise between the natural inter-
    vals and our inability to use them – that compromise which we call the tem-
    pered system, which amounts to an indefinitely extended truce. This reduction
    of the natural relations to manageable ones cannot permanently impede the
    evolution of music; and the ear will have to attack the problems, because it is so
    disposed. Then our scale will be transformed into a higher order, as the church
    modes were transformed into major and minor modes. Whether there will then
    be guarter tones, eighth, third, or (as Busonir thinks) sixth tones, or whether
    we will move directly to a tJ-tone scale that Dr. Robert Neumann has calcu-
    lated,2 we cannot foretell. Perhaps this new division of the octave will even be
    untempered and will not have much left over in common with our scale. How-
    ever that may be, attempts to compose in quarter or third tones, as are being
    undertaken here and there, seem senseless, as long as there are too few instru-
    ments available that can play them. Probably, whenever the ear and imagination

    have matured enough for such music, dre scale and the instruments will all at
    once be available. It is certain that this movement is now afoot, certain that it
    will lead to something. It may be that here again many digressions and errors
    will have to be overcome; perhaps these, too, will lead to exaggerations or to
    the delusion that now the ultimate, the immutable has been found. Perhaps here,
    once again, Iaws and scales will be erected and accorded an aesthetic timeless-
    ness. To the man of vision, even that will not be the end. He recognizes that
    any material can be suitable for art – if it is well enough defined that one can
    shape it in accordance with its supposed nature, yet not so well defined that the
    imagination has no unexplored territory left in which to roam, in which to
    establish mystical connection with the universe. And since we can sdll hope
    that the world will long continue to be a riddle to our intelligence (Verstand),
    we can say in spite of all Beckmessers that the end of art is not yet at hand.

    If the scale is imitation of the tone on the horizontal plane, that is, note after
    note, then chords are imitation on the vertical, notes sounded together. If the
    scale is analysis, then the chord is synthesis ofthe tone. It is required ofa chord
    that it consist of three different tones. The simplest of such chords is, obviously,
    that one which most closely resembles the simplest and most evident aspects of
    the tone, that one which consists of fundamental, major third, and perfect
    fifth – the major triad. It imitates the euphony of the single tone by omitting
    the more distant overtones and reinforcing the more immediate. The triad is
    without doubt similar to the tone, but it is no more similar to its model than,
    say, Assyrian reliefs are to their human models. Such triads might have come
    into use harmonically, as it were, when someone discovered it was possible to
    sing, at 6rst, the 6fth with the fundamental, later, the third as well; or it might
    have happened by the singing of parts in such a way that they came together on
    none other than such chords as these. Neither way, however, can today be
    proved with any certainty. It is probable that simultaneous sounds of this kind
    were already felt to be euphonious before the polyphonic manner of writing
    was able to use them. Yet the possibiliry is not to be excluded that, conversely,
    monophonic melody and scales existed before the chords, and that the step
    from monophonic to polyphonic music did not occur through dre setting of
    chords as accompaniment to tones or melodic progressions, but rather through
    the singing of two or three melodies at the same time, one of which
    “rr.ntrlully became the principal melody. However it may have been in the earliest days of
    music, both methods, harmonic and polyphonic, have cooperated equally for at
    Ieast four hundred years in promoting the evolution of our present-day music.
    It is therefore hardly appropriate to formulate chords on only one of the rwo
    principles. It is hardly appropriate, on the one hand, to present chords as if
    they had germinated and developed spontaneously, as they are usually presented
    in the teaching of harmony; nor is it appropriate, on the other hand, to explain
    polyphony as nothing else but voice leading that merely follows certain con-
    ventional rules and does not consider the chords resulting from the coincidence
    of parts, as usually happens in the teaching of counterpoint. It is much more
    correct to say that the development of harmony was not only essentially influ-
    enced by melodic principles, that the development of the possibility of voice

  2. Applied to our present concern, that means: Let the pupil learn the laws and
    effects of tonality just as if they still prevailed, but let him know of the tenden-
    cies that are leading toward their annulment. Let him know that the conditions
    leading to the dissolution of the system are inherent in the conditions upon
    which it is established. Let him know that every living thing has within it that
    which changes, develops, and destroys it. Life and death are both equally
    present in the embryo. What lies between is time. Nothing intrinsic, that is;
    merely a dimension, which is, however, necessarily consummated. Let the pupil
    learn by this example to recognize what is eternal: change, and what is temporal:
    being(das Bestehen). Thus he will come to the conclusion that much of what
    has been considered aesthedcally fundamental, that is, necessary to beauty, is by
    no means always rooted in the nature of things, that the imperfection of our
    senses drives us to those compromises through which we achieve order. For
    order is not demanded by the object, but by the subject.l

  3. Am studiat și eu problema consonanței dpdv matematic. De obicei sună consonant frecvențele care au raport un număr întreg mic (sol/do = 3/2, mi/sol=5/4). Acestea au proprietarea să fie armonice ale unei note cu 1-2 octave mai jos. Fiind armonice ale acestei note, combinația lor va genera alte armonice ale aceeași note, deci un sunet organizat. Două note cu raport ne-simplu (fa#/do = radical_din_2) generează prin combinație un spectru împrăștiat de sunete, o cacofonie.

    Aproximări ale gamei temperate Do major (frecvențe):
    Do = 1 * Do (prima)
    Re = 9/8 * Do (secunda)
    Mi = 5/4 * Do (terta mare)
    Fa = 4/3 * Do (cvarta)
    Sol = 3/2 * Do (cvinta)
    La = 5/3 * Do (sexta mare)
    Si = 15/8 * Do (~ septima)
    DO = 2 * Do (cvarta)

    Mai multe la http://mihvoi.intelsoft.ro/work/Alte/Notele%20si%20gamele%20muzicale.doc
    (dar nu știu să cânt!)

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