A Dictionary of Music and Musicians/Partial Tones

by James Lecky

PARTIAL TONES (Fr. Sons partiels; Ger. Partialtöne, Aliquottöne). A musical sound is in general very complex, consisting of a series of simple sounds called its Partial tones. The lowest tone of the series is called the Prime (Fondamental, Grundton), while the rest are called the Upper partials (Harmoniques; Oberpartialtöne, Obertöne). The prime is usually the loudest, and with it we identify the pitch of the whole compound tone. For each vibration given by the prime the upper partials give respectively 2, 3, 4, 5, 6, 7, etc. vibrations. The number of partial tones is theoretically infinite, but it will be enough here to represent the first 16 partials of C, thus:—

When the notes of this diagram are played on the ordinary Piano, tuned in equal temperament, the Octaves alone agree in pitch with the partial tones. The 3rd, 6th, 9th, and 12th partials are slightly sharper, and the 5th, 7th, 10th, 14th, and 15th much flatter than the notes given above. But even in just intonation the 11th and 13th partials are much flatter than any F♯ and A recognised in music.

When a simple tone is heard, the kind of motion to and fro executed by the sounding body resembles that of the pendulum, and is hence called pendular vibration. [Vibration.] When a compound tone is heard, the form of vibration is more complex, but may be represented as the sum of a series of pendular vibrations of different frequencies. In order that the compound tone shall be musical it is necessary that the vibration should be periodic, and this happens only when the frequencies of the vibrations which sound the upper partials are multiples of that which sounds the prime tone. In the article on Node it has been already explained in what manner a string or the column of air in an organ pipe produces this compound vibration. The real motion, as Helmholtz remarks, is of course one and individual, and our theoretical treatment of it as compound is in a certain sense arbitrary. But we are justified in so treating it, since we find that the ear as well as all bodies which vibrate sympathetically, can only respond to a compound tone by analyzing it into its simple partials.

It may seem difficult to reconcile this with the fact that many ears do not perceive the composite nature of sound. Helmholtz has treated this question at length,^[1] and his explanation may be thus indicated. The different partials really excite different sensations in the ear, but whether they are perceived or not, depends on the amount of attention given to them by the mind. In general we pay attention to our sensations only in so far as they enable us to form correct ideas of external objects. Thus we can distinguish two comparatively simple tones coming from different instruments. On the other hand when a compound tone is produced by one instrument we disregard the several partials because they do not correspond to different portions of the vibrating body; each portion executes the compound motion corresponding to all the partials at once. Moreover it would hinder our musical enjoyment if we were habitually to concentrate our attention on the upper partials, and we have therefore, in general, no interest in doing so. Hence it must not be supposed that when we fail to distinguish the partials of a compound tone they are not really present, or that when we hear them but faintly their intensity is small. Helmholtz gives an experiment which strikingly illustrates this. He obtained two nearly simple tones an Octave apart, and by listening to each tone in succession he was able to distinguish them when sounding together. But he could do so only for a while, for the higher sound was gradually lost in the lower, and a quality of tone different from either was the result. This happened even when the higher was somewhat stronger than the lower sound.

Notwithstanding the difficulty of hearing the upper partial tones, many musicians have been able to do so by their unaided ears. Thus, Mersenne^[2] could distinguish six partials in the tones of strings, and sometimes seven. Rameau^[3] also succeeded in perceiving the partials of the voice, which are much harder to distinguish than those of strings. There are several methods^[4] by which the ear can be trained to recognise the upper partials. It is better to begin with the uneven tones, Twelfth, Seventeenth, etc., which are easier to hear than the Octaves. Touch the note g′ softly on the piano, damp the string, and strike c loudly. Keep the attention directed to the pitch of the g′, and this note will be heard in the compound tone of c. Similarly by sounding e′′ softly and then c loudly, the latter will be observed to contain the former. It must not be supposed that when these partials are heard it is due to an illusion of the ear, for the note e′′ on the piano as ordinarily tuned is appreciably sharper than the 5th partial of c. The difference of pitch between the two sounds proves that one cannot be the echo of the other. There is another and still better method of directing the attention of the ear to any given partial tone. Touch a vibrating string at one of its nodes, for example at ⅛ of its length, and the 5th partial will be heard, faintly accompanied by the 10th, 15th, etc. It will then be easy to hear the 5th partial in the compound tone of the whole string.

The ear is however hardly able to carry out researches of this kind without mechanical assistance. Hence Helmholtz made use of Resonators, which are hollow globes or tubes of glass or metal, having two openings, one to receive the sound, the other to transmit it to the ear. From the mass of compound tone each resonator singles out and responds to that partial which agrees with it in pitch, but is unaffected by a partial of any other pitch. By this means Helmholtz has shown that the number of the partial tones and their relative intensities vary in different instruments, and even in the same instrument, according to the way it is played. These various combinations are perceived by us as different qualities of tone, by which we distinguish the note of a violin from that of a horn, or the note of one violin-player from that of another. The nearest approach to a simple tone is given by tuning-forks of high pitch. Dr. Preyer^[5] was unable to detect any upper partials in forks tuned to g′′ (768 vibrations) or higher. On the other hand, he showed that as many as 10 partials were present in a fork tuned to c (128 vibrations). But these are very weak and can only be heard when great care has been taken to exclude all other sounds. The general effect of such comparatively simple tones is very smooth but somewhat dull, and they seem to be deeper in pitch than they really are. Flutes and wide-stopped organ pipes have few effective partials, and are much inferior in musical effect to open organ pipes and to the piano. The tones of the voice, violin, and horn, are more complex still, and are characterised by fuller and richer qualities. When the partials above the 7th are strong they beat with each other, and the quality becomes harsh and rough as in reed instruments. Mr. Ellis has obtained beats from the 20th partial of a reed and even higher, and Dr. Preyer has proved a reed to possess between 30 and 40 partials.

The clarinet and the stopped organ pipe are exceptions to the general rule, for they give only the unevenly numbered partials 1, 3, 5, 7, 9, etc. Neither of these instruments will set into vibration a resonator an Octave or two Octaves above it in pitch, proving that the 2nd and 4th partials are absent. The resulting quality of tone is hollow and nasal, and may be obtained from a string, by plucking or bowing it in the middle. The effect is to make a Loop there, and hence to prevent the vibrations of the halves, quarters, etc. of the string, which require a Node at that point. [See Node.]

Helmholtz has also discovered that the different vowel sounds are due to various combinations of simple tones, and he verified his theory by reproducing several vowels from a series of tuning-forks set in motion by electricity. Each fork had a resonator the mouth of which could be opened or closed in order to obtain any required degree of intensity.

Bells, gongs, and drums have a variety of secondary tones generally inharmonic with the prime, and the result is that their vibration is not periodic. Hence the sounds they produce are felt to be more of the nature of noise than musical tone, and this explains why they are so much less used than other instruments. Tuning-forks also produce very weak inharmonic tones, not only when struck, but, as Dr. Preyer has shown, when bowed strongly.

The use of upper partials is, then, to produce different qualities of tone, for without them, all instruments would seem alike. Thus Dr. Preyer found that for the Octave c^iv–c^v (2048 to 4096 vibrations) many good observers were unable to distinguish the tones of forks from those of reeds, unless both were very loud. Moreover organ-builders have long been accustomed to obtain artificial qualities of tone by combining the Octave, Twelfth, Fifteenth, Seventeenth, etc. in the so-called compound stops (Sesquialtera, Mixture, Cornet). This was done not from any knowledge of the theory, but from a feeling that the quality of the single pipe was too poor for musical effect.

A still more important use of the upper partials is in distinguishing between consonance and dissonance. It was formerly supposed that the dissonance of two musical sounds depended solely on the complexity of the ratio between their prime tones. According to this view c′–f′♯ being as 45:32, would be dissonant even if there were no upper partials. Helmholtz has however shown that when c′ and f′♯ are struck together on any instrument whose tones are compound, the dissonance arises from the 3rd and 4th partials of c′ beating with the 2nd and 3rd of f′♯, thus (1):

and that the prime tones continue sounding without interruption. Hence when c′ and f′♯ are simple tones they give no beats, and in fact form as smooth a combination as c′ and f′. This theory has been carefully verified by Dr. Preyer. He used tuning-forks having from 1000 to 2000 vibrations per second; and by bowing them in such a manner as to get practically simple tones, he found that 5:7, 10:13, 14:17, and many like intervals were pronounced by musicians to be consonant. By stronger bowing the upper partial and resultant tones were brought out, and then these intervals were immediately felt to be dissonant. In the consonant intervals, on the other hand, the upper partials either coincide and give no beats, or are too far apart to beat roughly. Thus in the Fourth c′–f′ the affinity between the two notes depends on their possessing the same partial c′′′, and this relation is but slightly disturbed by the dissonance of g′′ and f′′ (see (2) above).

This theory also explains why such intervals as 11:13 are excluded from music. They are not consonant, for though they have a common partial it is high and feeble, and to get to it we have to pass over a mass of beating intervals. Nor are 11:13 connected by a series of consonant intervals as is the case with the dissonances in ordinary use. For example, C and F♯ are linked together thus, C–G–D–F♯, or thus, C–E–B–F♯.

Though the partial tones are generally heard simultaneously, they are sometimes separated by being made to traverse a considerable distance before reaching the ear. Regnault^[6] found that when a compound tone is sent through a long tube, the prime is heard first, then the 2nd partial, then the 3rd, and so on. He also noted that the velocity of sound increases or diminishes with its intensity. Hence, as the lower partials are usually the louder, they arrive before the higher.

The word 'harmonics' was formerly (and is sometimes even now) used to mean partial tones. But a harmonic produced by touching a string at one of its nodes, or by increasing the force of wind in an organ pipe, is not a simple tone. If we touch the string at ⅓ of its length we quench the 1st, 2nd, 4th, 5th, 7th, etc. tones, but leave the 3rd, 6th, pth, I2th, etc. unchecked. Hence it is proposed by Mr. Ellis to limit the word 'harmonics' to its primary sense of a series of compound tones whose primes are as 1, 2, 3, 4, 5, etc., and to use the words 'partial tones' to mean the simple tones of which even a harmonic is composed.

[ J. L. ]

1. 'Sensations of Tone,' pp. 93–105.
2. 'Harmonie Universelle,' Paris, 1636, pp. 208, 9, and 221 of the 4th book on Instruments. He gives's false ratio for the 7th partial, viz. 20:3 instead of 7:1.
3. 'Nouveau Système de Musique théorique.' Paris, 1726, Preface.
4. Helmholtz, 'Sensations of Tone,' pp. 79–82.
5. 'Akustische Untersuchungen.' Jena, Guslar Fischer, 1879.
6. Helmholtz, 'Sensations of Tone,' p. 721.