Acoustics

Period of a tone. Siren

How do the oscillations of a high tone differ from those of a low tone? Answer: By their velocity; for a high tone, the sounding body must vibrate more frequently per second than for a low tone. This and related questions are answered experimentally by the siren of Charles Cagniard de la Tour 1777-1859 1819, an instrument with which you can

1. generate tones of a given frequency,
2.
count directly the oscillations of a sounding body.

The most important component of a siren (there are different lay-outs) is a circular disk out of foil, cardboard, etc., which can be rotated like a wheel and has holes, which are uniformly distributed around concentric circles. As a rule, the disk (Fig. 312) has several circles of holes, which differ by the number of holes (A.Seebeck). If a strong jet of air is blown through a narrow tube at a circle of holes and the disk rotates fast and uniformly, a tone arises at a sufficiently large rate of rotation. From where does the tone come? As long as the disk is at rest, the stream of air passes uniformly through the tube. If the disk rotates, the air flow becomes stronger as a hole passes the mouth of the little tube. It becomes weaker in between as the disk passes the tube. In other words: The air exits in bursts and becomes at equal time intervals a sequence of generating impulses of longitudinal waves and, if they are frequent enough, a tone is generated. If you direct the air stream at the outer circle of the disk (with two circles of holes Fig. 312), the stream of air is more often interrupted than if it is directed at the inner circle, that is, a higher tone results than for the inner circle. The number of turns of the disk per second and the number of holes of a circle of holes yields the number of impacts per second, that is the period of the tone. If the inner circle has 8 holes and the disk rotates 30 times per second, the tone has 240 cycles per second, that is, the tone has 240 Hertz. Hence you can generate with a single row of holes the period of any tone: You let the siren turn so fast that it gives when blown at an equally high tone, the period of which is equal to the number of holes multiplied by the number of turns per second, its Hertz number.

Interval

If the disk has several rows of holes (the siren of H.N.Dove 1803-1879 has, as a rule, four with 16, 12, 10 and 8 holes), you can produce at the same rate of rotation tones with different Hertz. You can thus examine the change of the height of a tone with its Hertz number. You find out in this way, that two tones, t1 and t2, have for your ear the same difference in height as the tones T1 and T2 (one says: "lie apart by the same interval"), if the Hertz numbers of t1 and t2 are in the same ratio as those of T1 and T2. The row with 8 holes, for example, at 30 cycles per second, yields a tone of 8·30 Hertz, the row with 16 holes at the same rate of rotation 16·30 Hertz, whence it is higher than the first. The interval by which it sounds higher is called an octave and we say: "The second tone lies by one octave higher than the first" and call the lower tone the base tone, the higher one its octave. If the siren runs more slowly, say at 25 cycles, each of the tones changes its height, each becomes lower, but the difference in height remains the same - an octave. If the siren runs faster, say at 35 cycles, both tones rise, but their height difference remains unchanged - an octave. What has remained unchanged in spite of the different rates of rotation (25 - 30 - 35) is the ratio 1:2, the ratio of Hertz numbers of the lower tones (8·25, 8·30, 8·35) to those of the higher tones (16·25, 16·30, 16·35). The lower tone has always half as many cycles as the one which is higher by one octave. Only when the Hertz number of two tones are in the ratio 1 : 2, their height difference is sensed as that of one octave. The equality of the physiologically sensed tones manifests itself in the equality of their Hertz ratios.

Tone scale

The height difference between the base tone and the octave is rather large. But as the distance between 1 and 2 can be filled by the fractions (11/3, 11/2,2/3, etc.), that between the base tone and the octave can be filled by an infinite sequence of smaller intervals. You can think of an infinitely large number of intervals, but we are only interested in those used in music - a comparatively small number.

It subdivides to start with - and this is arbitrariness1 - the distance between the base tone and octave into seven steps, that is, introduces between them six tones. This sequence of tones is called a tone scale. The individual tones are called: Base tone, Second, Third, Fourth, Fifth, Sixth, Seventh, Octave.

Fig. 313 displays the ratios of the Hertz numbers of these tones. That of the base tone corresponds to the length of the first of the eight lines. The fractions entered beside the other lines indicate the ratio of the Hertz number of the corresponding tone to that of the base tone. The Fifth has 3/2, the Fourth 4/3 times as many cycles as the base tone! Those numbers state the intervals between the base tone and the corresponding tone of the scale.

1 We are not aware of this arbitrariness, because the sound of the scale has become so much part of ourselves that we consider it to be a matter of course. However, you should realize that within the regime of the same octave a major and a minor scale (and eventually yet the chromatic scale) have arisen.

What about the intervals between neighbouring tones? As has been said above, the interval between the base tone and the Octave has been subdivided into seven intervals (steps). But the steps have not the same height. The Second oscillates 9/8 times as often as its neighbour below, but by no means every tone of the scale oscillates 9/8 times its lower neighbour's fwequency. If you convert all the fractions to the same denominator, you find that the Hertz numbers of the scale are

 24 27 30 32 36 40 45 48

In order to find the interval between two neighbours, you must divide the Hertz number of the higher tone by that of the lower tone, when you will find the intervals

 9/8 10/9 16/15 9/8 10/9 9/8 16/15

Correspondingly, your ear senses differently the interval between neighbours.

You can envisage the difference of the intervals graphically. For example, the Second has 9/8 as many cycles at its lower neighbour. Hence there correspond to 100 cycles of the base sound 112.5 cycles of the Second: The interval 9/8 means an increase by 12.5 % over the deeper tone. The interval 10/9 is an increase by 11.1 %, the interval 16/15 an increase of 6.67 % over the lower tone. If you measure this increase like the incline of a road, which is stated by so many m per 100 m of the horizontal length, and substitute cycles instead of metres, you arrive at Fig. 314, The angle between the lines leading upwards and the horizontal yields the individual intervals. The interval 9/8 or 10/9 is called a full tone, the interval 16/15 a half tone.

Diatonic, major- and minor-, chromatic, temperate Scales

If you step beyond the Octave and consider it to be the base scale, and proceed from there in the same intervals of the sequence of the first scale, you obtain a similar scale. Every tone of the new scale is related to its base tone in the same ratio, in which the corresponding tone of the previous scale is related to its base tone. And since the base tone of the new scale has twice as many oscillations as the previous one, every individual tone of this new scale has twice as many cycles as the corresponding tone of the previous scale. and therefore sounds one octave higher. In the same manner, you can continue the new Octave - as one calls a sequence of eight successive tones - upwards to the upper bound of perceptible sounds. However, you can also continue the scale, first described, downwards and thus align Octave after Octave to the lower bound of the perceptible sounds. This complete scale is called the diatonic scale (Greek: diateinein = extend).

If you employ as starting base tone the tone with 261 Hertz - designated traditionally by c1 - and align Octaves upwards and downwards, you obtain the C-major scale. It corresponds - with an at present unimportant restriction, which will arise later on - to the sequence of tones, given by a piano, as you strike one white key after another. They are denoted by c d e f g a h c in countries, where German is spoken, and by ut re mi fa sol la si ut elsewhere. The musician Guido v. Arezzo 992-1050 is supposed to have taken these first syllables from the poem

 ut queant laxis resonare fibris mira gestorum famuli tuorum solve polluti labii reatum Sancte Ioannes

It is called the Major scale in contrast to the Minor scale, which differs from the former physically by another sequence of intervals. The intervals of the Major scale are (Fig. 313 bottom):

 9/8 10/9 16/15 9/8 10/9 9/8 16/15

Those of the Minor scale:

 9/8 16/15 10/9 9/8 16/15 9/8 10/9

Hence the sound also differs essentially from that of the Major scale. In the Minor scale, the Hertz numbers of neighbouring tones within an Octave are related to that of the base tone like 1 : 9/8 : 6/5 : 4/3 : 3/2 : 8/5 : 9/5 : 2. A Three Chord is the tone sequence Base, Third, Fifth, in the Major scale with the Hertz number proportions 1 : 5/4 : 3/2 or 4 : 5 : 6, in the Minor scale with 1 : 6/5 : 3/2 or 10 : 12 : 15. We will not give further details of the Minor scale,

The C-major scale is not sufficient for Music as it contains too few tones between the Base and the Octave. A musician demands that he can use every tone as Base and can continue from it in the intervals Second, Third, Fourth, etc. In the scale, described so far, this is not possible. Let there be given a piano, which has only white keys, that is, only yields the C-scale and a musician, who wants to play on it a tune, which starts with the intervals Second, Third, Fourth. Then with whatever tone the tune starts, the first tone is followed by one in the interval 9/8, followed by the interval 10/9 and then 16/15, if the tune is to sound right; only then he has at his disposal the sequence Second,Third, Fourth. However, for example, if you do not want to start from c1, but from the next higher tone d1 as Base (that is, to transpose a tune by a complete tone upwards), you will recognize the insufficiency of the Major scale, because in the given scale d1 is followed by the interval 10/9 (Fig. 313), then 16/15 and then 9/8 - which are quite different tones from the ones you want. The available tones have the Hertz numbers 261. 294. 326, 348,3911/2. However, if you do not want to start from c1 with 261 Hertz, but from d1 with 294 Hertz, then it must be followed by the Second with 9/8·294 = 331 Hertz, the Third with 10/9·331 = 368 Hertz, the Fourth with 16/15·368 = 392 Hertz. Not a single one of these three tones is available in the C-major scale. Hence an introduction of 6 tones between the Base and the Octave is not sufficient.

If it were desired to meet completely a musician's demand to employ each tone as Base and to proceed from it in pure (mathematically exact) intervals, you would have to introduce between the Base and the Octave 29 tones (for the combined Major and Minor scales), whence the piano would require 30 keys, including the Base and Octave (Helmholtz ). However, several of these tones lie so close together, that their intervals may be neglected, whence certain groups, the terms of which differ very little from each other, are replaced by a single tone for introduction into the incomplete scale (Fig. 313). This is how five more keys were fitted in between the Base and Octave, in fact, with the large intervals 9/8 and 20/9, that is, in between the Base and the Second, Second and Third, Fifth and Sixth, Sixth and Seventh (the piano's black keys).

In this way, the Octave was subdivided into 12 steps and the Chromatic scale created (Greek: crwma = colour, because initially the inserted tones were written and printed in colour). However, one went further and made the 12 steps equally large, that is, one created a scale, in which the interval between all neighbouring keys was the same (Andreas Werckmeister, organ builder, 1645-1706). However, one did not change the interval of 1 : 2 between the Base and the Octave, whence follows the interval between any two neighbouring sounds in the twelve step scale: We denote the Hertz number of the base by b and look for the number, by which we must multiply, in order to find the Hertz number of the upper neighbour x. Then the Hertz number of the first tone beyond the Base is b ·.x, that of the second, which is to execute x times as many oscillation, is b ·.x2, that of the third b ·.x3, that of the twelfth b ·.x12. The twelfth tone after the Base is the Octave, which has 2b Hertz, since the Base has b Hertz, whence

b ·.x12 = 2 · b , that is, x12 = 2 and x = = 1.0595.

This is the interval between two neighbouring tones of the new scale - the uniformly tempered scale, also called the scale with "well-tempered temperature". Temperature denotes here the equalization of the (hitherto different) intervals between them. (Bach 1685-1750 must have written his Well Tempered Piano composition around the time, when this scale was introduced! (Rainer Radok)

The interval is still a little smaller than the interval 16/15 = 1.067, the increase from one tome of the scale to the next higher tone, that is, yet a bit softer that even the smallest interval of the former scale (Fig. 314); it is only 5.95% of the Hertz number of the neighbouring tone. However, since this new interval does not coincide with any of the former one the scale becomes totally different, Only the ratio between the Base and the Octave remains. If we present the Hertz numbers of the pure scale by

 240 270 300 320 360 400 450 480

the corresponding tones of the tempered scale have the numbers

 254.27 285.42 302.38 320.37 359.6 403.63 453.05 480

They are joined in this scale by the tones with the Hertz numbers

 254.27 285.42 339.4 380.97 427.63

The tempered scale dominates in Music; however, it is only perfectly pure in its Octaves. Fig. 315 presents the names of the tones for the Major and Minor scales as well as for the tempered scale. Horizontal lines correspond to the individual tones, their vertical distances are determined by the acoustic intervals.

Concert-pitch

Hitherto we have been concerned with the intervals, that is, the height intervals and the ratios of the Hertz numbers, forming intervals. The absolute heights of individual tones and the absolute values of their Hertz numbers have hardly been mentioned. (If we compare two tones with mountain peaks, we have only considered the relative height of one to the other and the inclination of the path from one to another, but not the absolute heights of the peaks above the mean sea level.).

We can only fix the Hertz number of a single tone arbitrarily; by this number (and height of the tone), also those of every other tone of the scale are fixed, for all intervals, related to this tone, are determined by multiplication (or division) of the Hertz number of the tone under consideration with the numbers, which specify the intervals. If one were only concerned with a single instrument, for example, one piano or one singing voice, the absolute height of the Base would be unimportant; it would just be a bit higher or lower. However, this is no longer sufficient when several instruments sound in unison. Apart from the fact that each instrument must naturally be right by itself, also the base tone of one instrument must be at the same height as that of the other, because otherwise dissonance would occur, that is, an instrument would be too high or too low compared with another instrument.

The selection of a generally valid Base for the conformity of the individual instruments is the concern of technical music performances (say: opera) and thus became an international concern: Since the Concert-pitch Conference in Vienna 1885, the Normal Tone is that of a tuning fork, which has 435 complete cycles (that is, 870 simple) oscillations per second (you say now: It has 435 Hertz). This tone is the pitch tone a1. If you accept it and compute the C-major scale in pure pitch (with the intervals 10/9, 9/8. 16/15), you obtain for the two consecutive Octaves from c'1 to h'1 the following set of numbers I and for the same two Octaves in pitch with well tempered temperature (with the interval ) the numbers II:

You can continue this table arbitrarily in both directions. But to such an unlimited computable sequence of numbers does not correspond to an unlimited sequence of audible tones.

Audibility bounds of tone heights and their musical employment

The bounds of audibility are wider apart in youth than at an old age. On an average, the lower bound lies at 16 - 20, the upper bound at 20 000 - 25000 Hertz. The tones employed in Music lie in the range 40 - 5000; Helmholtz refers to the E of the Double-bass at 41 and the d5 of the Piccolo-flute at 4702 Hertz. Since we know the Hertz number n, which belongs to a given tone, we can also compute the corresponding wave lengths. We know that l = v/n, where v. the velocity of propagation of sound in air for all Heats numbers, is equal to 342 m (at 15º C), whence the wave, corresponding to the concert-pitch, is 342/435 = 0.786 m long. The waves of the bounds of the musically employed tones with 41 and 4702 Hertz have the lengths of 8.4 m and 7.3 cm.

Doppler Principle

The height of a tone depends on its Hertz number n, that is, on the number n of waves, whichenter the ear in one second. If these waves cover the segment AB (= ab = aA = b'a) (Fig. 316), your ear accepts every second the segment of the wave AB, provided its distance from the source of the sound does not change. This situation changes, if you approach the source or travel away from it very fast! If you approach, that is, move to meet the waves, for example, during 1 sec, naturally your ear receives more waves than it receives, when you stay in place. Assume that you have moved in 1 sec to the point c. Instead of letting your ear move during this second at uniform velocity to c, imagine that it were transposed suddenly to c only at the last instant of this second as the end of the wave distance ab just arrives, that is in the direction towards the sound source of the waves, which fill the distance e; it then absorbs in addition to the n waves, which came to it, yet all the waves , which cover the distance between a and c and which, if it had waited for them at rest, only would have reached it during the next second.

What we have assumed to be compressed into the last instant, in fact is spread uniformly over each whole second. During this second, the ear has received more waves, that is, it has heard a tone with a larger Hertz number than it would have heard at rest. In other words: The tone rises for the ear, as long as it moves very quickly towards the source of the sound.

Analogous reasoning shows that fast motion away from the source of sound leads to reception of a lower tone. Let the ear as it moves for one second uniformly away from the source of sound have reached c' and be transplanted suddenly at the first instant of the second, as the start of the wave train AB arrives, to c'. The waves which arrive at a (where the ear was to start with) during this second extends to b', whence at c' (where it is now) only the waves lying between c' and b'; however, the waves between c' and a have not yet caught up with the ear at the end of this second. During this second - the events, which we have compressed into one instant, have indeed taken place during the course of a second - the ear indeed received fewer waves, that is, a tone of smaller Hertz number than it would have received when at rest. While the ear moved, it heard a lower tone.

You obtain the same result when the source of sound moves. The calculation yields: If the source approaches the ear at 21 m/sec, the tone increases by half a tone. Experiments agree completely with this theory - referred to as Doppler Principle. You can detect this change in height of the tone as fast trains move towards you or away from you.

The elementary considerations above yield: If n is the Hertz number of the source of sound (at rest), l the wave length, v the velocity of sound, vb that of the observer, then he receives, when at rest, n = v/l waves; however, when he moves towards the source of sound, vb/l more waves. He receives n' = v + vb/l = n(1 + vb/v) waves. Hence the tone increases in the ratio 1:(1 + vb/v). If the observer travels away from the source of sound, he hears a tone which is lower in the ratio 1:(1 - vb/v).