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 PYTHAGORAS |
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MERSENNE |
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BACH |
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HELMHOLTZ |
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JUSTONIC |
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* All quotations on this page are from:
Helmholtz, Hermann; On the Sensations of Tone as a Physiological Basis for the Theory of Music", Fourth German edition, 1877; translated, revised, corrected with notes and additional appendix by Alexander J. Ellis. Reprint: New York, Dover Publications Inc.,1954.
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Hermann Helmholtz
(1821 - 1894)
Discovered how the brain perceives harmony, proved the existence of combinational tones and warned about the disadvantages of equal temperament.
"Music stands in a much closer connection with pure sensation than any other art," Helmholtz noted in 1862, because "... in music, the sensations of tone are the material of the art." p. 2-3
This extraordinary scientist made important contributions to the fields of medicine, physiology, anatomy and physics, while unifying and relating these sciences to the fine arts.
In his book On the Sensations of Tone as a Physiological Basis for the Theory of Music", first published in 1862, Helmholtz connects the boundaries of physical and physiological acoustics on the one side, and of musical science and esthetics on the other. He argues that it is " the theory of the sensations of hearing to which the theory of music has to look for the foundation of its structure." p.4
Physical properties of sound .
Helmholtz begins his analysis of the sensation of hearing by investigating the physical properties of sound. He notes two extremes of sound: noise at one end and musical tones at the other.
He describes a musical tone as "a perfectly undisturbed, uniform sound which remains unaltered as long as it exists" whereas "We perceive that generally a noise is accompanied by a rapid alteration of different kinds of sensations of sound." p.7
A musical tone is made up of its fundamental or prime partial tone.
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"In addition to this, the ear also becomes aware of a whole series of higher musical tones which we shall call harmonic upper partial tones, sometimes simply the upper partials of the whole musical tone or note." p 22
Helmholtz points out that, as we can easily compound noises out of musical tones, we should first study the laws and pecularities of this class of sensations because "musical tones are the simpler and more regular elements of the sensation of hearing." p.8
In this section of the work, Helmholtz describes his analyis of the composition of vibrations, musical tones, sympathetic vibration, and the vowel qualities of tone.
Physiological properties of sound.
Helmholtz next turns his attention to understanding the part played by the ear in the apprehension of the quality of tone.
He discovered that the cochlea contains thousands of elastic appendages, each of which vibrate sympathetically with different frequencies of sounds as they reach the eardrum.
This explains how the brain perceives different frequencies of sound.
Musical science and esthetics.
The third part of his book deals with the construction of musical scales and notes. Some of his more signifigant findings are quoted below:
On scales:
- "Here we come at once upon esthetic ground, and the differences of national and individual tastes begin to appear. Modern music has especially developed the principle of tonality, which connects all the tones in a piece of music by their relationship to one chief tone, called the tonic." p. 5
On keys:
- "There is nothing in the nature of music itself to determine the pitch of the tonic of any composition...In short, the pitch of the tonic must be chosen so as to bring the compass of the tones of the piece within the compass of the executants, vocal or instrumental. " p. 310
Combinational tones:
His expermients confirm the existance of combinational tones heard whenever two musical tones of different pitches are sounded together.
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"The pitch of a combinational tone is generally different from that of either of the generating tones, or of their harmonic upper partials. In experiments, the combinational are readily distinguishable from the upper partial tones, by not being heard when only one generating tone is sounded and by appearing simultaneously with the second tone." p.153
Helmholtz divides them into two classes.
- "Summational tones, having their pitch number equal to the sum of the pitch numbers of the generating tones." p.153
On dssonance and consonance:
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"When two musical tones are sounded at the same time, their united sound is generally disturbed by the beats of the upper partials, so that a greater or less part of the whole mass of sound is broken up into pulses of tone, and the joint effect is rough. This relation is called Dissonance.
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"But there are certain determinate ratios between pitch numbers, for which this rule suffers an exception, and either no beats at all are formed, or at least only such as have so little intensity that they produce no unpleasant disturbance of the united sound. These exceptional cases are called Consonances." p.194
Pythagorean problem solved by Helmholtz:
- "The enigma which, about 2,500 years ago, Pythagoras proposed to science, which investigates the reasons of things, 'Why is consonance determined by the ratios of small whole numbers?' has been solved by the discovery that the ear resolves all complex sounds into pendular oscillations, according to the laws of sympathetic vibration, and it regards as harmonious only such excitements of nerves as continue without disturbance." p. 279
On equal tempered intonation:
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"As regards musical effect, the difference between the just and the equally tempered, or the just and the Pythagorean intonation, is very remarkable. The justly intoned chords, in favourable positions, not withstanding the rather piercing quality of the tone of the vibrators, possess a full and as it were saturated harmoniousness; they flow on, with a full stream, calm and smooth, without tremor or beat." p.319
- "The difference between natural and tempered intonation is greatest and most unpleasant in the higher Octaves of the scale, because here the false combinational tones of the tempered intonation are more observable, and the number of beats for equal differences of pitch becomes larger, and hence the roughness greater." p. 319
- "It should be observed that the disturbances due to beats in the tempered scale, are the less observable the swifter the motion, and the shorter the duration of the single notes." p. 322
- In 1851, at the Great Exhibition, no English organ was tuned in equal temperament, but the only German organ exhibited (Schultze's) was so tuned.." p.549
On forced rapidity:
"Hence in rapid passages, with the soft quality and moderate intensity of tone, the evils of tempered intonation are but little apparent. Now, almost all instrumental music is designed for rapid movement, and this forms its essential advantage over vocal music. We might, indeed, raise the question whether instrumental music had not rather been forced into rapidity of movement by this very tempered intonation, which did not allow us to feel the full harmoniousness of slow chords to the same extent as is possible from well-trained singers, and that instruments had consequently been forced to renounce this branch of music." p. 323
On music and space:
"Upon this reposes also the characteristic resemblance between the relations of the musical scale and of space, the resemblance which appears to me of vital importance for the peculiar effects of music. It is an essential character of space that every position within its like bodies can be placed, and like motions can occur. Everything that is possible to happen in one part of space is equally possible in every part of space and is perceived by us in precisely the same way. This is also the case with the musical scale. Every melodic phrase, every chord, which can be executed at any pitch, can also be executed at any other pitch in such a way that we immediately perceive the marks of their similarity." p. 370
Don't knock it until you've tried it:
"Musicians have contested, in a very dogmatic manner, the correctness of the propositions here advanced. I do not doubt for a moment that many of these antagonists of mine really perform very good music, because their ear forces them to play better than they intended, better than would really be the case if they actually carried out the regulations of the school and played exactly in Pythagorean or tempered intonation. On the other hand, it is generally possible to convince oneself from their very writings, that these writers have never taken the trouble to make methodical comparison of just and tempered intonation. I can only once more invite them to hear, before uttering judgments, founded on an imperfect school-theory, concerning matters which are not within their own personal experience. Those who have no time for such observations, should at any rate glance over the literature of the period during which equal temperament was introduced. When the organ took the lead among musical instruments, it was not yet tempered. And the pianoforte is doubtless a very useful instrument for making the acquaintance of musical literature, or for domestic amusement, or for accompanying singers. But for artistic purposes its importance is not such as to require its mechanism to be made the basis of the whole system of music." p. 428
The difference is really striking:
- "It must not be imagined that that the difference between tempered and just intonation is a mere mathematical subtilty without any practical value. That this difference is really striking even to unmusical ears, is shewn immediately by actual experiments with properly tuned instruments." p.320
Final word from Helmholtz:
- "And after all, I do not know that it was so neccessary to sacrifice correctness of intonation to the convienience of musical instruments." p. 327
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