3 - Consonance and Dissonance
[Carter, p 18:] Art in its most primitive state is But it quickly becomes imitation of nature in the wider sense of this idea, that is, imitation of outer of inner nature. In other words, art does not then represent merely but above all ultimately without reference to their What, When, and How. Inference of the original, external object is here perhaps of only secondary importance due to its lack of immediacy. In its most advanced state, art is exclusively concerned with the representation of inner nature. Here its aim is just the imitation of impressions, which have now combined, through association with to form At this stage, inference of the external stimulus is almost certain to be inadequate. At all stages the imitation of the is only relatively faithful. This is true, because of the limits of our abilities; because, whether we are conscious of it or not, the material in which the imitation is presented differs from the material or materials of the stimulus, (so that, for example, visual or tactile sensations might be represented in the material of auditory sensations). If, then, which were the models, then unsurmountable difficulties lie in the way of analysis if the impression on the observing subject is now taken as the point of departure for inquiry. As Schopenhauer shows in his theory of colors, however, a real theory should start with the subject. And, just as he considers the colors one would have to go back to if one would establish a real theory of tones. Now present the harmonic means of music in such a way that they can be directly applied in practice. [Carter, p 19]
[Carter, p 18:]
Art in its most primitive state is
But it quickly becomes
imitation of nature in the wider sense of this idea, that is,
imitation of outer
of inner nature.
In other words,
art does not then represent merely
but above all
ultimately without reference to their What, When, and How.
Inference of the original, external object
is here perhaps of only secondary importance
due to its lack of immediacy.
In its most advanced state,
art is exclusively concerned with the representation of inner nature.
its aim is just the imitation of impressions,
which have now combined,
through association with
At this stage,
inference of the external stimulus is almost certain to be inadequate.
At all stages
the imitation of the
is only relatively faithful.
This is true,
because of the limits of our abilities;
whether we are conscious of it or not,
the material in which the imitation is presented
differs from the material or materials of the stimulus,
(so that, for example,
visual or tactile sensations might be represented
in the material of auditory sensations).
which were the models,
unsurmountable difficulties lie in the way of analysis
if the impression on the observing subject
is now taken as the point of departure for inquiry.
As Schopenhauer shows
in his theory of colors, however,
a real theory should start with the subject.
just as he considers the colors
one would have to go back to
if one would establish a real theory of tones.
present the harmonic means of music
in such a way that they can be directly applied in practice.
[Carter, p 19]
It could happen, nevertheless, that in this way
I achieve more than I am actually striving for,
since my goal is just clarity and comprehensiveness of presentation.
But however little that would displease me, it is still not my purpose.
Therefore, whenever I theorize,
it is less important whether
these theories be right
they be useful as comparisons to
So, for two reasons
I may reject the subject as the basis for my study:
[note from Monzo]
The previous was a preamble. Schoenberg here begins the real discussion of consonance and dissonance.
Thus I may devote my study to the object, the material of music, if I succeed in bringing what I want to show into accord with what is known or surmised concerning this material.
The material of music is the tone; what it affects first, the ear.
The sensory perception releases associations and connects tone, ear, and the world of feeling.
On the cooperation of these three factors depends everything in music that is felt to be art.
Nevertheless, even if a chemical compound does have characteristics other than those of the elements from which it was formed, and if the impression a work of art makes does display characteristics other than those which could be derived from each single component, it is still justifiable for many a purpose, in analyzing the total phenomenon, to bring up for consideration various characteristics of the basic components.
Indeed, the atomic weight and valence of the components permit a conclusion with respect to the molecular weight and valence of the compound.
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 three 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 "science".
In this sense it is of little consequence whether one starts with a correct hypothesis or a false one.
Sooner or later the one as well as the other will certainly be refuted.
Thus, we can only base our thought on such conjectures as will satisfy our formal necessity for sense and coherence without their being considered natural laws.
Should someone succeed in deriving the phenomena solely from the physical properties of tone and explaining them solely on that basis, should the problems be thereby successfully clarified and solved, then it would hardly matter whether our physical knowledge of the nature of tone is correct or not.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; whereas it is not at all a proved fact that more correct or better observation would necessarily yield more correct or better conclusions.
Thus, the alchemists, in spite of their rather poor instruments, recognized the possibility of transmuting the elements, whereas the much better equipped chemists of the 1800s considered the elements irreducible and unalterable, an opinion that has since [Carter, p 20] been disproved.
If this view has now been superseded, we owe this fact, not to better observations, nor to better perception, nor to better conclusions, but to an accidental discovery.
The advance, therefore, did not come as a necessary consequence of anything; it could not have been predicted on the basis of any particular accomplishment; it appeared rather in spite of all efforts, unexpected, undeserved, and perhaps even undesired.
Now if the correct explanation must be the goal of all inquiry, even though the explanation does almost always turn out to be wrong, we still do not have to allow our pleasure in searching for explanation to be spoiled.
On the contrary, we should be satisfied with this pleasure as perhaps the only positive result of all our trouble.
From this point of view it is thus of little importance for the explanation of harmonic problems, whether science has already refuted the function of overtones or only raised some doubts.
Should one succeed, as said, in defining the problems sensibly and presenting them intelligibly, even though the overtone theory be false, then the goal could still be reached - even if it turned out after some time that both, overtone theory and explanation, were false (but this outcome is by no means inevitable).
I can make the attempt with so much the more confidence since, as far as I know, no one has yet refuted the theory beyond all doubt; and since no man is able to examine and prove everything himself, I, too, have to get along with the existing knowledge as long as I may and can believe in it.
Therefore, I will proceed in my study from the possibly uncertain overtone theory because what I can deduce from it seems to agree with the evolution of the harmonic means.
[note from Monzo]
From this point to the end of the chapter, Schoenberg draws heavily on Bellermann 1862/1901, Der Contrapunkt, chapter 2.
Once again: the tone is the material of music.
It must therefore be regarded, with all its properties and effedts, as suitable for art.
All sensations that it releases - indeed, these are the effedts that make known its properties - bring their influence to bear in some sense on the form of which the tone is a component, that is, on the piece of music.
In the overtone series, 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 contribute 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 analyzing 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 [Carter, p 21:] are recorded by the subconscious, and when they ascend into the conscious they are analyzed and their relation to the toatl sound is determined.
But this relation is, to repeat, as follows: the more immediate overtones contribute more, the more remote contribute less.
Hence, the distinction between them is only a matter of degree, not of kind.
They are no more opposites than 2 and 10 are opposites, as the frequency numbers indeed show; and the expressions "consonance" and "dissonance", 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, htereby expanding the conception of what is euphonious, suitable for art, so that it embraces the whole natural phenomenon.
What tody is remote can tomorrow be clsoe 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 "consonance" and "dissonance", 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 classification.
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 fundamental tone, dissonances as those that are more remote, more complicated.
The consonances are accordingly the first 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 differences precisely, since they are relatively small.
They are expressed by fractions with large denominators; and as it requires some thought to say whether 8/234 is larger or smaller than 23/680, because a mere estimate can lead one astray, the mere estimate made by the ear is just as undependable.
[note from Monzo]
Below is a an analysis of the prime-factor monzos and decimal sizes of the fractions used by Schoenberg in his illustration:
2 3 5 7 11 13 17 8/234 = 3 -2, 0 0 0, -1 0 = 0.034188034 23/680 = -3 0, -1 0 0, 0 -1 = 0.033823529
compare Bellermann 1862/1901, p. 15:
Wie der menschliche Verstand einfache Verhältnisse leichter als verwickeltere zu übersehen im Stande ist, so faßt auch unser Ohr solche Intervalle mit größerer Sicherheit auf, denen ein einfaches Zahlenverhältnis zu Grunde liegt. Diese einfachen Verhältnisse sind daher die wohltönenderen und bilden die Konsonanzen in der Musik; jene komplicierteren dagegen die Dissonanzen ... As the human understanding of the conditions of simple ratios is more easily surveyed than those of the more complicated ratios, so also our ear understands such intervals with greater security, for which a simple numerical ratio is appropriate to reason. These simple ratios are the more probable-sounding therefore and form the consonances in the music; those more complicated against it the dissonances ...
Schoenberg presumably got the idea to use these large denominators from the table of pythagorean ratios presented by Bellermann, altho as is obvious from the table of monzos, Schoenberg's denominators do not match the terms of any pythagorean ratios.
Efforts to make use of the more remote consonances (today called "dissonances") as artistic means thus led necessarily to many an error, to many a detour.
The way of history, as we can see it in that which has actually been selected by practice from the practicable 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 superiority.
[Carter, p 22:]
The most nearly perfect consonance (after the unison) is the first tone of the overtone series, thus the one that occurs most frequently, consequently the strongest: the octave.
The next most nearly perfect are the 5th and the Major 3rd.
The Minor 3rd and the Major and Minor 6ths are, in part, not relations of the fundamental, in part, not relations in the ascending direction.
That explains why it used to be questioned whether they are consonances at all.
On the other hand, the 4th, known as an imperfect consonance, is a relation of the fundamental, but in the opposite direction; it could, therefore, be counted with the perfect consonances, in the same sense as the Minor 3rd and the Major and Minor 6ths, or simply with the consonances, as indeed often happens.
but the evolution of music has gone a different way here and has given the 4th a peculiar position.
Only the following are designated as dissonances:
thus Diminished and Augmented 8ves, 4ths, 5ths, etc.
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2005.11.05 - added comments about Bellermann 1862/1901.
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