FEBRUÄRY 1937 47

OCTAVES AND DECIBELS

By R. VERMEULEN.

Summary. With the aid of practical data, the relationships are discussed between the acoustic magnitudes: Pitch' interval and loudness, and the corresponding physical mag- nitudes:' frequency and sound intensity. The object of this article is to emphasise that the logarithmic units, octaves and decibels, should be given preference in acoustics for denoting physical magnitudes. '

Why is a preference shown in acoustics and In music, where without any close insight into the associated subjects for logarithmic scales and are physics of sound the pitch has always been denoted even new units introduced to permit the employment in musical scores by notes, the concept "musical of such scales?' The use of logarithmic scales in or pitch interval" has been introduced to repr,esent acoustics is justified from a consideration of the a specific difference in pitch. The most important characteristics of the human ear, apart from pitch interval is the 0 c t a v e: Two tones which many other advantages offered in both this and are at an interval of an octave have-such a marked other subjects. Reasons for this are that to a natural similarity that musicians have not consid- marked extent' acoustic measurements, and par- ered it necessary to introduce different names for ticularly technical sound measurements, represent them, but have merely distinguished them by attempts to find a substitute for direct judgement means of indices. In addition to the octave, there by means of the ear. An instructive example of are pitch intervals, such as the fifth, fourth and this is offered in the determination of the charac- several others, which are similarly perfectly natural teristics of a loudspeaker, ill which not only must and self-evident, not only because two tones which the reproduced music arid speech be analysed differ from each other by such a pitch interval critically but inter alia the loudness also deter- harmonise with each other, but also, and this is mined, i.e. the sound output of the loudspeaker the more important for what follows, because such at 'different pitches for an equivalent electrical pitch intervals occur at all pitches with equivalent power input. Yet neither the loudness nor the differences.' pitch is a physical magnitude; these are physiolog- Musicians therefore indicate the pitch in, musical ical and even to some extent psychological factors, notation in such a way that a pitch interval nearly which cannot be evaluated by direct measurement, always corresponds to the same difference in pitch 2), since they are purely subjective. To arrive at a , independent of the absolute pitch value. Thc numerical value it is therefore necessary to make keyboard of the pianoforte is also based on the do with an objective measurement of specific principle that ~qual pitch intervals are reproduced, physical factors, and in the case under discussion, at uniform. intervals throughout the whole compass for instance, the sound output measured as a func- of the instrument. tion of the frequency. To' do this it is very desirable A pitch interval regarded from, the physical to select such units for expressing the results aspect is, however, not representative of a specific arrived at that the numerical values are to some difference, but of a specific ratio of frequen- extent proportional to the corresponding physiolog- cies, an octave corresponding to a doubling of the ical sensations. The same applies to electrical frequency, and the fifth to a ratio of 2 : 3, etc. magnitudes when these are used for sound trans- (see Table I). The logarithm of the frequency mission, as for instance in telephony. shares in common with the pitch the property that on an alteration of the tone by the same Pitch and Frequency pitch interval it always increases by the same amount, irrespective of the absolute frequency It is well-knownthat the pitch is related to the values: frequency of the fundamental tone 1). That this relationship, is, however not a linear function log ft _ log f1 - log f2' will be evident from the following considerations. f2

2) These intervals are not absolutely equal since the pitch 1) That this relationship is not strictly' absolute, but is interval between two successive tones in musical notation determined somewhat by the intensity and timbre, is is usually equal to a whole tone but sometimes also equal left unconsidered here, to a semitone. 48 PHILlPS TECHNICAL REVIEW Vol.' 2, No. 2

Hence in the light of our introductory remarks natural pitch intervals to be obtained with the the logarithm is a suitable measure for the pitch. minimum number of keys per octave in instruments with a fixed keyboard, such as the organ and Octaves pianoforte. The principal natural pitch intervals are sho~ on a logarithmic scale in jig. 1 with a Owing to the fundamental importance of the o ct a veto hearing it appears desirable to select not 10 the Briggsian base, but the number 2 as the base of logarithms. IJl this way we get for the magnitude I of a pitch interval with the octave as unit:

The octave can be subdivided in various ways. 2.4 % % ~ "/.1 81.; 5h 91.; '% The most common subdivision has been developed *I' I 1 4------5------*6 8 from the desire to arrive at a scale enabling the I I 3------" I 5 6 Table J. ~----' of 1 2 3 4 5 6 7 '-lr., Frequency 'Milli- I octaves T ~--+% ratio ~ ftJ Pitch ., i:I ~------~------I' 0 Tem- Tem- interval Natural Natural ~ ~ pered perèd ~-~~--~------~~ scale scale I scalc scale 20767 Fig. 1. Pitch intervals and scales. Below the keyboard of the Unison C ~ 1.000 1.000 0 0 pianoforte, the frequency ratios of the tones on the tempered scale are given logarithmically, and below these again the Comma 81/80 1.013 1.000 17.92 0 ratios for tones with natural pitch intervals. The musical Semitone C# 25/24 1.042 1.059 58.89 83.33 notation of the notes is indicated by letters and the frequency Limma 16/15 1.067 1.059 93.U 83.33 ratios of the natural tones based on the fundamental tone C Minor second Dp 27/25 1.080 1.059 ULO 83.33 by vulgar fractions. The tones of the major scale are indicated' by thick lines, and may be regarded as evolved by a threefold 10/9 LUI 1.122 152.0 166.6 application of the major chord whose tones have a frequency Major second D 9/8 1.125 1.122 169.9 166.6 ratio of 4 : 5 : 6, i.e. a major third and a fifth. The figure shows Augmented second D# 75/64 1.172 1.189 228.8 250.0 how with this chord the tones of the scale are obtained. It Minor third Ep 6/5 1.200 1.189 263.0 250.0 follows from the figure that this scale can be represented to Major third E 1.250 1.260 321.9 333.3 a close approximation by those tones on the tempered scale 5/4 which correspond to the white keys on the pianoforte. Diminished fourth Fp 32/25 1.280 1.260 356.1 333.3 The'pitch intervals, which are shown by slightly thinner linés Augmented third E# 125/96 1.302 1.335 380.7 416.5 and which may partly be approximately represented by the Perfect fourth F 4/3 1.333 1.335 414.8 416.5 black keys, have been arrived at by corresponding application of the minor chord. This minor chord may be regarded as a Augmented fourth 25/18 1.389 1.414 473.9 500.0 F# harmonic of the major chord; it is made up from the frequency Diminished fifth Gp 36/25 1.440 1.414 526.1 500.0 ratio (1/4 : 1/5 : 1/6). The tempered scale gives an additional Perf~ct fifth , G 3/2 1.500 1.498 585.0 583.3 two tones between C and D and between F and G, to which Augmented fifth G# 25/16 1.562 1.587 644.0 666.6 no simple pitch intervals of C correspond. It will 'be seen, however, that these are required for arriving at the natural Minor sixth Ap 1.600 1.587 678.1 666.6 8/5 pitch intervals by taking one of the other tones of the scale Major sixth A 5/3 1.667 1.682 737.0 750.0 as roots. Augmented sixth A# 125/72 1.736 1.782 795.8 833.3 Minor seventh Bp . 9/5 1.800 1.782 848.0 833.3 fundamental. tone denoted by "C" as the tone of Major seventh B 15/8 1.875 1.883 906.9 916.6 Diminished octave Cp 48/25 1.920 1.883 941.1 916.6 origin. If the same pitch intervals were furthermore Augmented seventh B# 125/64 1.953 2.000 965.7 1000.0 based on one of the tones obtained in this way, Octave C' 2 2.000 2.000 1000.0 1000.0 new tones would be continually produced, .some of which are shown in the figure by thin lines; it is In addition to the musical notation. (2), the above table gives thus seen that a finite numb~r of scale divisions for the various pitch intervals, which are also indicated in does not suffice. Nevertheless it is found that the fig. 1, the following data: (3) the numerical ratio determining the pitch interval and which for the less simple pitch intervals most important pitch intervals cal!- be obtained is always a simple ratio, e.g. 125/64 = (5/4)3 (i.e. the pitch to a very close approximation by subdividing the interval is composed of three successive 'thirds), The next column (4) gives this fraction in decimals, while (5) gives the octave into twelve equal parts, which are also tempered scale approximating to the natural pitch interval. shown in fig. I and by which additional new tones In the last two colums both the natural (6) and the tempered (7) scales are given in milli-octaves. are .obtained between C and D and between G FEBRUARY 1937 OCTAVES AND DECIBELS 49

"ii

0 '03" 0" ë = o .~ ~E -" ;; Po. ...

13777 1]920 I 1227-4 , 1:!-ID1 - ~ I(l')H , IIO·Ui , IOW IO~!8 , ,J , I 'Hl)) 1)291 ~ , 819! 82;7 ~ C6 ,I 7712 71H:! I r-,? I, , 088. .?fJJ , , "':;"';;', (}IJ7 , 6:?OI I ,...... --:---- , , SiN ~ ,1 ';lbl 521" · i """ = 4,98 d "bH : , · 1 ·flJR r-= CS "". ,""" · 1'- 1 """ 1 }444 · "'" I J{)6. , 1100 I , zu;! · Z7J.! · ,2580 I' : : I' !J!]"'" e., · h-l- -- .- '''''' " 1,1 t'- 2(148 20(01) c" 1933 19;] . . · 11' · I . I IlU 1740 , 1 , 1 . · • I , 1)3-1 , I 'HO ·. · · - :. 1 , 1167 , ORI 1]04 1 1.!'lIl ·. · '0-- ~ , - 1149 d II{lJ :·. · 3 . IOJj c i 1024 .1) 1 %7 ·. ...·. , 1161 8'. · ~--,.._- ,------, t ,., , 77j · "· .. , ,m . - I I " 1 + ."Mi 652 I I I I d ,81 . Hl -' . " " (ill) 111 c2 11 46' ~ 468 " m lill1 ,&I , J ( l\." " i I ". \..: I Hl , H) .-l "; ;.. J2J , J2b J " I " .!'lIl ,_ 5 '" ~. _'._. 2>1} 2;') Cf -, _...!! "_ - -_. -- _'--- -- 24! .!4" I 10 ns 218 1--- ..--.. -- 15 I'),! , I'q , \ '-y--' III III - '" 161 - , '" " I·n • 128 12') c 121 122 / I

'08 109 -: .. % • ., I ss , 56 >. ~- 81 e 81 ~,_., 72 d 1l - '64 ,., r--- C eo ., -

" 48" .. ., r--- I ",. " '-,...-' ,. " 16 rz Cf JO 11" I 27 " , 'c I z" , zz" ~ 10 10

18 d 18 ,. , '6 C2 -.. ____ ------~a07s7 :J ~ ~ .n ... ~ 1:: .... 0 ., oS ., a 0 s le "ii 0" "= " ~ rf ~ ~o..cl "ii s e E-< 2 0 " I -a 0 e "0 g.p:;"= '"~ .....= ~ e ~ = =: 2 ~ ... .§ "0 = :ë en ., ~ ~ ~ =: U ïl E-< :ll UJ" ~ I:Q= Fig. 2. Compass of musical instruments and voices. The frequency range of the fundamental tone for various musical instruments and voices is shown by a continuous line. For a number of tones (heavy dot) the corresponding harmouics have been- included as small dots or a broken line. For speech the characteristic overtones (socalIed harmouics) have been included in the figure. 50 PHILlPS TECHNICAL REVIEW Vol. 2, No. 2 and A. Such uniform subdivision, termed a scale mine the relationship between loudness and sound of just temperament, has the advantage over intensity are far from satisfactory, and their dis- scales based on natural pitch intervals that devia- cussion is beyond the scope of the present article. tions in the pitch intervals on a keyboard from the It is nevertheless, possible to determine with a natural intervals are always of the same magnitude, fair degree of accuracy the equality of loudness of irrespective of the fundamental tone taken as basis.' two sounds, even if they differ considerably in Owing to the general application of the decimal character, At a given frequency and sound intensity, system, a decimal subdivision is favoured in phy- the difference in intensity which produces a- just sical work, and pitch intervals of centi-octaves .and perceptible difference in loudness can moreover be milli-octaves are used; these are also shown in established fairly accurately. It is found that for Table I. a tone of approximately 1000 cycles the ratio I2/I1 The octavo provides a method for ascribing of the sound intensities between which the ear numerical values to differences in pitch. To, be can still just differentiate is constant over a wide able to deduce the absolute pitch from the frequen- range ·and is approximately 1.2. If a logarithmic cy, the frequency of a specific tone must be fixed scale is used two sound intensities just at the as a basis. Various conventions have been adopted threshold of differentia:tion will be roughly the same for this pUJ;pose.The standard of pitch of orchestral distance apart over a greater part of the graph. instruments is based on "A" whose frequency This is very convenient, where, for instance, it is according to the socalled international scale is desirable for the accuracy of a graph to conform 435 cycles, while the socalIed physical scale takes to the acuity or sensitivity of differentiation of the 430.5 cycles for this tone, for it is considered an ear. ¥oreover, with this scale the very wide com- advantage that C3 then has a frequency of lQ24, pass of intensities of 1 : 1013 to which the ear i.e. exactly 10 octaves above 1 cycle. In American reacts (see fig. 3 and 4), can. be covered, which scientific literature a fundamental tone of 1000 cycles is employed in conjunction with decimal db subdivision of the' octave. Fig. 2 shows the compass of a number of common musical 'instruments and brings out the importance of the various frequency bands as well as the. frequency of the tones in the two aforementioned scales.

Loudness and Sound Intensity

Just as the logarithm of the frequency has been 20760 employed as a measure of the pitch, the logarithm Fig. 3. Range of audibility. The minimum sound intensity of the sound intensity 3) is used to express the just still perceptible to the ear (threshold value) and the maximum sound intensity which can be tolerated by the intensity of an auditory sensation. The justification ear are plotted here as continuous lines as a function of the for this is here however not equally valid since pitch. Thus within this range all tones detectible with the ear are located. The ranges are also shown hatched which cover the auditory sensation 'is not a complete parallel the maximum sound intensities at various frequencies of an to the logarithm of, the sound intensity. The true orchestra of 75 pieces playing forte: below these the cor- responding values for speech are given. (According to relationship between these two magnitudes is J. L. Sivian, H. K. Dunn, S. D. White, Journ. acoust. difficult to estahlish, since the loudness is deter- Soc. Amer., 2, 330, 1931). mined not only by the sound intensity but also " largely by the frequencies composing the sound. would be quite impossible with a linear scale. The I Furthermore, the estimation of equivalent dif- logarithmic scale offers other practical advantages, ferences in loudness is much less accurate and far which will be discussed at the conclusion of this more difficult than is possible with differences in article. pitch. The results of experiments designed to deter- Bel and Neper

3) The sound intensity of a progressive wave is the amount To ascribe a numerical value to the difference in of energy which is conveyed by the sound wave in unit time and per unit of surface normal to the direction of intensity between two sounds we can therefore flow. The unit used to express this intensity is the erg adopt the convention "The logarithm of the ratio per sec per sq. cm, although frequently the intensity is given in watts per sq. cm = 107 erg see+! cm-2 •• of their sound intensities shall be N", although F~BRUARY 1937 OCTAVES AND DECIBELS 51

it is evident that this cumbersome method is useless. For this reason the units "Bel" and "Nep er" have been introduced. V The difference between these two units lies in the 2 log -2 . first place in the choice of the base of logarithms VI used 4), the Briggsian base of 10 being selected Ampl. neper db Int. for the "Bel" and the base of natural logarithms 1,0 1,0 e _: 2.718 for the"N epe r". They also differ in the type of the physical magnitudes to which f,1 they are applied: The bel is always used for express- ing energies or powers, while the neper is used for f,2 1,5

amplitudes, such as sound pressure, velocity of 1,3 sound (sound-particle' velocity), electric currents or voltages. To exemplify the fundamental differ- "4 2,0 ence between these two units and to bring out their t.s 2,5 relationship, the following discussion is instructive: 1,6 The intensity of a plane progressive sound wave 1,7 at a certain point is determined by the sound 3,0 f,8 intensity I (see footnote 3), by the intensity of 1,9 the pressure fluctuations p or by the sound-particle ", 2,0 4,0 velocity v. These factors are related to each

other as follows: 2,2 5,0 p2 2 2.4 I = p' v = - . e c v , . (1) 6,0 ec 2,6 7,0 where e is the density of the medium of propagation 2,8 (for air e is 0.0013 gr per cm3) and c is the velocity 8,0 of propagation through the medium (for air c = 3,0 9,0 approximately 332 m per sec). This equation is fa valid only for plane progressive waves. Similarly, in the transmission of electrical energy, 3,5 either the power W transmitted, the voltage e or the current i can be stated, provided that in the 15 4,0 latter case the resistance R of the consumer is also given, thus: 4,5 20 . e2 W = ei = -= i2R .. . (2) R 5,0 .25

It is seen that the product e c in equation (1) 5,5 30 bears a close analogy to the resistance R in equation (2). e c is termed the "acoustic resistance". For air 6,0 e c = 41 gr cm-2 sec-I. FigA gives the relationship 40 6,5 between the ratio of the amplitudes, the number . of nep ers or decibels and the ratio of the intensities 7,0 50 or energy values. 7,5 It follows from equations (1) and (2) that the 60 8,0 logarithm of the ratio of two powers' or intensities 8,5 ---+- 70 is double the logarithm of the ratio of the corre- 9,0 -1--f9 ---L__ 80 sponding voltage, current; pressure,. or velocity 90 amplitudes. 9,5 fa 100 207(;9 4 ) We shall write loglOa as 19 a and log. a as ln a, the simple relationships between these logarithms being: Fig. 4. Scale subdivision in amplitude ratios, nepers, decibels 19 a = 0.434 ln a and ln a = 2.3 19 a. and intensity ratios. 52 PHILIPS TECHNICAL REVIEW Vol. 2, No. 2

In the following we shall term the logarithm To determine the ratio of the amplitudes the number of the ratio of two magnitudes a difference in level, is sought corresponding to half the number of deci- thus arriving at the following definitions for the bels. bel and néper: Bel: The difference in level between two power s Sound Intensity Level and Loudness Level or in tensitie s is N b els, when: Decibels and nepers are units for expressing differences in level. With pure tones the loudness or itself can be established by taking a specific sound intensity as zero level. The difference in decibels Decibel: The decibel is the tenth part of a bel, compared with zero level is termed the sound so that' the difference in level of two power s intensity level. or intensities is n decibels when: Tabel n. Relationships between different acoustic magnitudes or for plane progressive waves in air.

Nep er: The difference in level between two Sound- Amplitude Decibels Phons Intensity Sound of a m p litud e s (of voltage, current, pressure, veloc- pressure particle vibrations velocity in air at ity, etc.) is M nep ers when: Above db above 9 in air, 1000 10-16 2.5'10-16 10- Dynes per 10-3 cm cycles, . or wattsper watts per watts per sq. cm 6 sq. cm sq. cm sq. cm per sec. 10- cm

In the light of these definitions the difference in 64 60 0.25 0.32 8 1.8 level of two powers or intensities can 0 n I y be 65 61 0.32 0.36 9 2.0 expressed in bels or decibels, while the difference 66 62 0.40 0.40 10 2.2 in level between two voltages or veliocities must 67 63 0.50 0.45 U 2.5 68 64 0.63 0.50 2.8 not be expressed in bels but in nepers. To-calculate 12 69 65 0.8 0.56 14 3.2 the difference in level between two powers the 70 66 1.0 0.63 16 3.6 measured voltages (e), currents (i), velocities (v) 71 67 1.25 0.71 18 4.0 or pressures (p) are naturally useful, provided the 72 68 1.6 0.80 20 4.5 resistance of the consumer is the same in both 73 69 2.0 0.89 22 5.0 74 70 2.5 1.0 25 5.6 instances. The following equation can then be used: 75 71 3.2 1.1 28 6.3 72 W2, e2 76 4.0 1.25 32 7 n= 10lg -= 20 19- 77 73 5.0 1.4 36 8 W el I 78 74 6.3 1.6 40 9 If it is remembered that log 2 = 0.30 and that 79 75 8 ·1.8 45 10 1 decibel corresponds to an energy increase of 80 76 10 2.0 50 11 81 77 12.5 2.2 approximately 25 per 'cent, the intensity ratio 56 12 82 78 16 2.5 63 14 corresponding to a specific number of decibels can 83 79 20 2.8 71 16 readily be calculated mentally. The following series 84 80 25 3.2 80 J8 of corresponding values is then found:

Decibels It is a matter of serious concern that not only is more than one zero level used but that moreover Ratio reference is frequently omitted to the actual zero level used as a basis of comparison 6). In America 5) The following equivalents are employed for converting nep ers into decibels, provided that the resistances are equivalent: ' G) The whole statement of data may as a result become 1 decibel = 0.U5 nep er; 1 neper = 8.7 decibels. meaningless. "The intensity of a loudspeaker is 80 decibels" But it is incorrect to calculate the amplification of an has no meaning, unless it is stated which intensity is taken amplifier with an input impedance of Rl = 1 megohm and as zero level and at what distance this intensity is measured, a resistance load of R2 = 1000 ohms, in terms of decibels or, if the total power output is meant, which power value as n = 20 19 e2/cl' The amplification is moreover given has been selected as zero level. The sentence above can by the expression: therefore mean: "The sound intensity in the axis of the loudspeaker at a distance of 2 m is 80 decibels above . (e 22 R) e R e2 16 n = 10 19 R • --i = 20 19..J. + 10 19 Rl = 201g - + 30. 10- watts per sq."cm", or also e.g.: "The total output ,2cl el,' 2 el' of the loudspeaker is 80 decibels above 1 microwatt." FEBRUARY 1937 OCTAVES AND DECIBELS 53 db - - t20 t20 L/. tt;----r--.... 1/1' ~ I'...... 1111 tOO too r- /, ---r--.... -V Ill-- - 90--- i"- ~ j ~ E;:: t-t- ~ .-' 11 § ~ - l- j 80 ~ i"-o- 80 V ~ :'. r- ~..... ~ ::--..F:: :::i'--r- I---r-- 70 - J /, .....1' <, ~ ..... -- - ~ r-... ~ I'--ro- 60 / I--f- L/, 60 " r-, 1'1' r--...... - r- .... ~ ~ r--....r-. 50, I V .... I' I' i' <,~ r-.... """ 40 ~ r--.. 40 IJ _/...... ""'" -..", r-, r- /11 I"-r-, t'-- 30 - ./ <, <, ...... "'- ..." r-, Î't'-- r-- 11 20 20 .I / <, r..... -...... r-- 11"" "'''' r-, t'--l'- 10 """ J I' ---..... 1.1 .... t...... " o ~" 0 r-- 11"" -....--; V cycles 20 . 100 500 tOOO 5000 10000 206/9 Fig. 5. Sound intensity level in decibels above 10-16 watts per sq.cm as a' function of the frequency for constant loudness. Thenumbers next to the lines indicate-the loudness level, i.e. the sound intensity level of the corresponding loudness for. a 1000-cycle tone. The bottom line represents the threshold of audibility of the human ear. It is striking that the corresponding sound intensity increases considerably with diminishing frequency, while the curves for higher loudness values are much :flatter. ' The irregularity which occurs in all curves in the band between 2000 and 6000 cycles is probably due to the diffraction of the sound waves around the head. In measurements of the sound pressure in the ear canals the curves are smooth. (Accordingto H; FIe t cher and W. A. Munson, Journ. acoust. Soc. Amer., 5, 82, 1933)..

16 an intensity of 10- watts per sq.cm which lies For sounds differing in character the loudness approximately at the threshold of audibility of may differ considerably at the same intensity level. the ear is used as zero level; in Germany the In particular pure tones with different frequencies standard intensity level corresponding to 1 dyne and at the same level exhibit very marked differen- per sq.cm sound pressure in a plane wave-is put ces in loudness. Infig. 5 the intensity level at con- equal to 70 decibels. Zero level then has a pressure . stant loudness is plotted as a function of the of 0.32.10-3 dyne per sq. cm and a sound intensity frequency. It is seen that a m..ich higher sound 16 of 2.5.10- watts per sq. cm. In general the use intensity corresponds to an equivalent 100~dness of the decibel is limited to differences in level at low frequencies than at medium frequencies. only and the level itself is expressed in 'phons, The difference may be as great as 60 decibels, 70 phons corresponding to a sound pressure of 1 _i.e. by a factor of 1 million. . dyne per sq. cm. In Table 11, the comparison is The sound intensity which is just at the' limit given for a plane progressive wave between the of audibility is termed the threshóld value. The intensity level (decibels above 10-16 watts ,per line of constant loudness corresponding to, this sq. cm and phons) and the intensity, as well as the value is marked with the numeral 0 in the figure. corresponding . effective' ~alue .of the pressure The other lines are marked with the sound intensity fluctuations and the sound-particle velocity. As level which is required to obtain the loudness in a guide the amplitudes of the air particles are given question at 1000 cycles. This numerical value is in the last column for a frequency of 1000 cycles 7). termed the loudness level. Also for compound 7) The amplitude a is calculated from the effective value sounds,' such as street noises, the loudness level . of the sound-particle velocity for a specific frequency f can be determined by comparison 'with a tone of. using the expression: 1000 cycles. A selection of loudness levels of various a = 2:f.V2. common sounds and noises is reproduced infig. 6. 54 PHILlPS TECHNICAL REVIEW Vol. 2, No. 2

1 1 1 1 1 1 0 110 2 0 3 0 4 0 510 6 0 7 0 SlO 9 0 1110 do 13 0 I 1 I I I 1 1+ I I I 1 I 1 I Tearing Whispering Conversation ' Voice calling paper Ticking of clocks -- Radio music -- Dwelling houses and alarm bells Vacuum cleaner

o doI Typing Printing Boiler Average --:-~ Office ---,,--- dwell, house private busy room wor k s forge T{xrb 0- Other Rooms - Factories - generator ---- Restaurant ---~ Large orchestra ------

-- Railway train -- -- Aircraft cabin-- with windows with windows In Vehicles closed open - Motor car-

Omnibus and tramcar - Underground- railway o

Aircraft at Street noises - Motor horn- distance of3 m -' -- Traffic-- --Thunder'-- t t Sounds in the open G a r den t London, quiet street Berlin, Potsdamer Platz Berlin-Westend t + Berlin, Tauentzienstr. New-York, + min. at night Chicago business street M 0 to r Lorry car -- Motor cycle --

I 20 so 1

Fig. 6. Range of loudness level in phons for various common sounds and noises.

Comparison of Diagrams with Logarithmic and- linear scale the band below 1000 cycles, which Linear Scales is the most important as regards audibility, has b~en' compressed to an insignificant width. To, illustrate clearly the differences between the ,On the other hand the band above 5000 cycles, linear and logarithmic scales used for graphical which 'has a compass of only 11/ octaves, takes representation, the same arbitrary characteristic 2 up a ~isproportionately large part of the figure. showing, e.g. the variation of sound intensity of a Ioudspeaker, has been plotted in various ways in 2) Secondly (cf. á and b) the widths of the various fig. 7 as a function of the frequency. . peaks and troughs of the curves are equal 1) The first point which becomes apparent, for on the logarithmic ,scitle where they are the instance, on a comparison of a' and b, is 'that same percentage fractions of the frequency. with the logarithmic frequency scale all octaves This gives a better insight into the damping are shown to an equal extent, while with the of the resonant frequencies, which in certain

-, FEBRUARY 1937 OCTAVES AND DECIBELS 55

10 a 9 9~~~-r-+--r-+-+-~-+~~+-+-4--+~ 8 , . 7

.6

5 , 4

3

2 I'\. 1 l I\. J I1 o 25 50 lOO 200 500 f(XX) 2000 5000 fO(}()()Hz äb +20 dbr.r-rTTrn~-r.--r-rTTrn~-r.--r-.rrrn.-, +16 C +40r'rr~++H+--~--+-~++H+~~--+-~++H+~ +12 +20rr-r~~+--r~-+-r++tHlt--r~-+-r++tH~~ +8 - +4 \. _j_ , ...I o ...I I 50 toO 200 500 1000 2000 5000 fOCXXJHz -4 \ - 8 -12 - 16 I -20 V1- 25 50 toO 200 500 1000 2000 5000 10000 Hz -5 -4 -3 -2 -I 0 +1 +2 +3 +4 l.UIII~(.11111'25,. 50 toO 200 500 fOOO 2000 5000 1)000 Hz

octave 20770

Fig. 7. Representation of one and the same frequency charac- b abscissa and ordinate both with linear scales; teristic on linear and logarithmic scales along the abscissa c abscissa and ordinate both with logarithmic scales. and ordinate. d as for c but with reduced scale along the ordinate. a abscissa with logarithmic 'scale, ordinate with linear scale; e as for a, but with reduced scale along the ordinate.

circumstances are responsible for' abnormal- 6) The objection is frequently advanced against ities. a logarithmic scale that the irregularities of a 3) Thirdly (cf. a and c) it is evident that the linear characteristic are smoothed out. In general intensity scale brings out more clearly the peak this is not the case, for although the peaks are at 60 cycles than the trough at 1000 cycles,' reduced in size as compared with the troughs while these are of the same size on the logari- (cf. a and c) or regarded from the acoustical thmic scale. aspect they are reduced to their true propor- tions, the magnitude of the deviations yet 4) The character of the curve above 5000 cycles depends entirelyon the choice of unit for the (cf. band c) is almost flat with the purely linear scale (cf. c with e, where the logarithmic scale scale, but on the logarithmic scale still reveals brings out greater irregularities, and a with d marked irregularities, which. certainly, affect where this is the case with the linear scale). the quality of sound reproduction. 7) The frèquency and intensity range which can -5) It is interesting to note that on measurements be conveniently covered in a single diagram is with e.g_ a quarter of the power the shape of practically unlimited when using a logarithmic the curve is not changed at all if a logarithmic' scale and with a iin~ar scal~ is limited to a 'ratio scale is used. The only difference is a dis- of 1 : 10. placement of the whole curve downward .hy .6 decibels. When using a linear intensity scale 8) A disadvantage of the logarithmic scale for (cf.. a and e) the curve is however considerably many applications is that the z~ro is located distorted· and in case e appears to be much at -- oo and also that negative values cannot smoother. be plotted. But since the ear itself has a specific 56 PHILlPS TECHNICAL REVIEW Vol. 2, No. 2

threshold value no objection can be raised to although on this scale the total amplification or this from acoustical considerations. damping (attenuation) can be :found directly 9) With a linear scale the numerical value of the by expressing the corresponding components intensity of a compound sound is equal to the in decibels and adding them. In many instances , sum of the numerical values of its components. the ability to do this is alone sufficiently im- With a logarithmic scale this is not the case, portant to justify the use of decibels or nep'ers.,

THE PLAYING SPEED'OF GRAMOPHONE RECORDS

By J. DE BO~R.

'If a gramophone record is not played at the speed for this record is 76 revolutions per minute, correct speed 1), both the pitch and time of the music instead of 78 revs. per min. as printed on the will be reproduced incorrectly. At Philips Labor- record. atory a series of ,experiments have been carried This divergence was confirmed by an observer out with the object of establishing the reaction with a perfect ear. According to this observer also, of listeners to this effect. A disc of Bee t ho ven' s . the pitch (C minor) of the music reproduced was Fifth Symphony was played at different speeds, correct when the playing speed was 76 rev. per and each member of the audience of about 30 per- minute. In addition, the frequencies of two tones on sons was then asked whether the speed was correct, the record, e' and g~/, were also determined too fast, or too slow. The results of these tests are objectively. With an equal tempered scale with sho~ graphically in fig. 1,/ where the percentage a' = 435 cycles, these frequencies should be 308' of the listeners who judged' that the record had and 77~ cycles respectively. These tones we,re been played at the right speed is given,' fo~ dif- recorded on a strip of film by the Phi I i ps -Mil I'é r , ferent playing speeds. system of sound recording. The film strip was passed t:h~~~gh the reproducing machine in the form' of an endless band, so that the pitch could be readily r. t ~.. ~ compared by the beat method with a calibrated' o tone generator. The frequencies were found to have the specified values when the playing speed of the records was 76 (± 0.3) revolutions per min. This value is ,in agr~ement with the results obtained , in the tests with an audience and with those made by an observer ~th a perfect ear. 40r---~-----+----~----~----~--~ The difference between the' optimum playing o speed and the specified speed is probably not due ' ~r----+----~----~----+-----4----~ to the speed of the recording apparatus, making the discs, being incorrect. It is more feasible that the observers took as a basis an equal tempered ~'V~----"--;;7.=:2----'/I:f.I1:---.~75:------::7.l,-'8---BO.L--82 Reutmin. scale with a' = 435 cycles, while' the orchestral

20400 instruments were tuned to a higher pitch. , Fig. 1. Percentage of listeners who judge reproduetion to be The rèsult of this investigation demonstrates good at different playing speeds. ' that the ,public is able' to distinguish whether the pitch and time of the music reproduced are' cor- Owing to the small number of listeners entailed rect. Tlie' curve in fig. 1 again shows an increase at the accuracy of the points plotted is somewhat a speed of approximately 80.5 revolutions per min., uncertain, although the shape of the curve was which may be associated with the fact that at this later confirmed in a second series of tests. it may speed the notes are just transposed by half a tone be deduced from this curve that t~e correct playing and that the music is then reproduced at a pitch more familiar to, listeners than at intermediate 1) This speed is defined as the speed at which the record was originally recorded. playing speeds.