Consonance (music)

Musical consonances

Before even the update of this work in physics and acoustics, the music intuitivemant defined as consonants of the sounds whose fundamental Fréquence S are in an arithmetic report/ratio simple one compared to the other. Thus the intervals of octave (2/1) and of Quinte (3/2) were always regarded as perfect consonances. It is only since the Moyen-âge that third S, major (5/4) and minor (6/5) are also regarded as consonantes, but like imperfect consonances only (see the third article ) . It is as from this time that the theory emerges from the Harmonie, which studies in priority the agreement S obtained by superposition of thirds.

The modern music exploring of the new ways (microtones, sounds inharmonic), of the intervals more distant than the first simple reports/ratios return little by little in the fields of the musical consonances, but like relative consonances.

Accuracy

An acoustic purity could be enough to govern the relationship between the sounds, in the most natural manner which is. It was still employed with the Renaissance for relatively simple parts. Unfortunately, the men want to write and play of the more complex musics, and the obviousness appears bientôt : the intervals are not easy to handle as one wants in all the directions, it is the law of the numbers which dictates it.

We saw higher than the musical intervals could be defined by simple reports/ratios. A simple end of range as C-D-semi installation already a serious problem with this light, to tell the truth the essential problem of the accuracy and systems of accuracy:

to go from C to semi, one can define a third, of a report/ratio 5/4, but one can also wish to pass by a cycle of fifth (which gives us the D in intermediary) C-ground-D-the-semi. Well! but in this case, calculation gives us 3/2 X 3/2 X 3/2 X 3/2 = 5,0625, and not of whole 5 “round whole”! (all this can be checked easily with the ear, without any figure!) Is Which thus the good semi , that of 5,0625 which come from the pure fifths, or that of 5,0000 which come the pure third? There is an irreconcilable difference between the two, almost a quarter of semitone!

This question, the ones will answer one, the others the other, and others, more still, will answer a third compromise solution. It is the variety of these compromises which explains the multiplicity of the systems of accuracy. These compromises do not lose any consonance, which would make them unusable, but their consonances become more or less slightly faded, which forms integral part of their esthetic qualities.

The accuracy of intonation is a concept which can take on various justifications. One will find of them definitions (even constructions and demonstrations) varied in the articles treating of the temperaments (see also Tempérament) and of the intonation. As we saw, it is indeed impossible to give a definition based on an absolute frame of reference, which cannot exist.

Let us not renonçons however to define it, and propose this definition which can be universal:

To play just, it is to use an intonation in conformity with the musical context.

Some consider that “the accuracy is a cultural phenomenon, not an absolute concept”, which returns more or less to the same definition.

Accuracy and consonance, tradition and reality

It goes without saying the accuracy tends naturally towards a maximum of consonance, i.e. the mathematical report/ratio of the interval must be nearest possible to a simple report/ratio such as higher exposed. Nevertheless, it is noted that an ear educated musician will not function in connection with an intangible physical reality such as these simple reports/ratios, but especially by a phenomenon of imitation. Thus, that accentuates the relativity of the concept of accuracy, which can deviate appreciably from the models, was this those of the temperaments or the ranges, worked out and adopted to solve the incompatibilities of intonation as well as possible (e.g.: two Fifth S rising followed from two Quad S downward (report/ratio: 81 /64) do not give a third exactly of report/ratio 5/4, that is to say 80 /64).

It is thus an effect comparable with that of the Mode, which does not look at the intrinsic value of an object, but its conformity with the standard, was it in constant evolution. Thus, a musical medium will be able not to have the same definition of the minor third as another, however similar. One can notice light differences in intonation between two orchestras playing at same time. A large remarkable soloist (standard Pablo Casals) will be able to also introduce (knowingly in its case) of new types of intonation. The fact of evaluating if he plays “just” or “forgery” is not due by no means to objective criteria, but only with the acceptance of his genius, with his established among, could one say. Even if Pablo Casals represents certainemnent an extreme case, of another musicians can have played a considerable part, by the means of the schools of instruments, in the evolution of the practical accuracy, such as it is known at present.

In addition, the non-European or traditional traditions can have references which are due to the acoustic characteristics of traditional instruments. Thus, the Cor of the Alps uses only the series of the natural Harmonique S to play its melodies, and although these intervals are unanimously regarded as “forgery” by traditional musicians (in particular the famous harmonic of row 7, unutilised elsewhere in the Western music), they do not constitute of it less one reference used by the inhabitants of these valleys for the song (perhaps would be necessary it to speak about it with the past?).

Accuracy relativity not of limit?

The case is very different movement of the old Musique, where the spirit of research directed towards the redécouverte of the origins is sharper. There, of many systems of accuracy are employed, sometimes with a seizing result, and thus open new horizons. One could think that the accuracy does not have any more limit. In fact, these old systems are not further away from the average references than an approximate intonation. In another article, one indicates that variations of 10 hundred S are the fact of an orchestra which plays remarkably just. However, in the old temperaments, there are almost no note more moved away of more than 20 hundred (and they are faded notes) values of the equal Tempérament. One remains finally in the same zone of approximation.

An absolute limit (and it is still relative !) is the variation from which the doubt starts to be installed on the interval which is actually played. It any more is not recognized, as by a lack of Intelligibilité… Nevertheless, one can easily find people who sing in a kind of Parlando, and their listeners are still delighted. Another limit there is reached: that of the total loss of consonance.

Acoustic consonance of the intervals

Physical study

The theory of the consonance was studied at the 19th century by the physicist Hermann Ludwig von Helmholtz starting from the phenomenon of Résonance. Hemholtz used hollow spheres (called since resonators of Helmholtz) provided with two short collars tubular diametrically opposite. When the sound contained a Harmonique Fréquence equal to the frequency of resonance of the cavity of the resonator, or close to that Ci, this harmonic was amplified what made it possible to insulate it. Thanks to a series of resonators of this type, Helmholtz could determine the intensity of the harmonics of a natural sound. In its physiological Théorie of the music , Helmholtz developed the idea that the consonance of an interval was all the more large as the Battement S between harmonics close one to the other were not very fast.

Consonance and acoustics

One can define the consonance by the state in which the sonority of a musical interval watch the least disorder, or minimum of sound effect - acoustic state of purity . It is easy to note that this state can be reached only when the two sounds are in a simple report/ratio of frequencies. For example, if the relationship between the vibrations of two sounds is from 3 to 2 (either 3/2), one will hear a Quinte, if the report/ratio is from 5 to 4 (or 5/4), it will be a third , etc If this report/ratio is not very exactly precise, of the disturbances will occur in sonority, and the feeling to lose this acoustic purity, which is a remarkable acoustic phenomenon, will be sharp.

Purity and beats

The purity of a interval is defined by the audible absence of Battement (or by the weakest possible beat, to see the third ) (concept of minimum of sound effect ). That can occur only if the two notes are in a simple report/ratio of frequency:

the simplest report/ratio is the octave (2/1), and the consonance is so perfect that one can often doubt the presence of two notes. Indeed, all the harmonics of the note top are already present in the note of bottom.

the Quinte (3/2) is the distinct interval more consonant, this is why it is at the base of the music.

the Quarte is the inversion of the fifth (4/3), it is slightly less consonante.

the third major ''' pure ''' is of a report/ratio 5/4, while the third pythagorician of a report/ratio 81/64, is not pure, its " consonance" being very far away from the natural report/ratio 5/4. The major third of the equal Tempérament is of a report/ratio $\ sqrt {2} /1$ , which is a little less far away from the natural report/ratio 5/4.

for the following intervals, it becomes difficult to speak about purity, a relatively audible beat always remaining, even for rigorous simple reports/ratios such as 6/5 (minor third) and 9/8 (your major).

Nevertheless, another consideration enters in account: proximity of the relationship of the interval with a simple report/ratio. One can recognize that the strict consonance (= pure) is in fact present, but deteriorated, and that gives rise to the Battement, which is almost unperceivable if the report/ratio is close.

The simplest harmonics and intervals

The simplest Harmonique of a sound of frequency F is 2xF is the octave, whose frequency is double the fundamental one (it is called harmonic " of row 2").

For the intervals which follow, the principle of equivalence of the octaves enables us to consider only the harmonics whose frequency lies between the fundamental frequency F (often noted F ₀) and that of the higher octave 2xF.

The frequency 3xF “will be thus brought back” in interval 1 to 2, by dividing it by 2, i.e. while going down from an octave to obtain the Fx3/2 frequency. This interval, corresponding to a report/ratio of frequencies 3/2 or 1,5, is simplest after the octave, and has a paramount importance in the Western music. It is called the interval of “Quinte”.

The musicians of the Antiquity and the Moyen-âge considered that only were perfectly harmonious “consonants” i.e., the intervals of octave and fifth.

The octave being the interval between two notes whose report/ratio is 2/1, and the fifth, the interval between two notes whose report/ratio is 3/2, the interval which separates them corresponds to a ratio of 4/3 named “Quarte”:

(2/1)/(3/2) = 4/3 ---> Octave - Fifth = Quad

Thus the quad is the " Inversion " fifth, because it is the complement of this one compared to the octave: Fifth + Quad = Octave ((3/2) * (4/3) = 2/1)

The interval between the fifth and the quad corresponds to the report/ratio of frequencies 9/8 named “your major” or “second major”:

(3/2)/(4/3) = 9/8 ---> Fifth - Quad = Your major

Consonance affected by the inharmonicity

The purity of a its musical (or more precisely of sound stamp) is it also defined by a consonance, that of the harmonics which constitute it between them. There too, a beat can appear if the sound is not pure. This deterioration of the purity of the stamp is measured by the Inharmonicité. The musical instruments are generally far from inharmonic.

The Piano is known for its inharmonicity, especially in the case of the upright pianos, which leads to installations of the system of agreement (spacing of the octaves), that Serge Cordier theorized in his Tempérament equal to perfect fifths.

The Cloche S are very strongly inharmonic, but in such a proportion that makes it possible to find other consonances, which constitutes all the art of the Fondeur of bells.

See too

Internal bonds

Beat
Consonance (tonal harmony)
Inharmonicité
Consonance and dissonance
theoretical and technical Glossary of the Western music
Temperament

External bonds

IRCAM-Assayag See framed: a short history of the consonance , in the middle of the article.

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