Generation of musical intervals by a digital method. D. Gossel

Transcription

1 170 PHLlPS TECHNCAL REVEW VOLUME 26 and of an AND block. n order to increase the economy of this system, these two types of block can be supplemented by another active block consisting of two NOR circuits without pulse gate. These can be used with advantage wherever the NOR does not form part of a bistable or polystable circuit. Fig.8 shows a reversible biquinary decade counter with 10 output amplifiers for the digits o The system of circuit blocks described here has been used in electronic weighing installations with digital data encoding and processing. The synchronous biquinary decade counter was developed by P. Muuss of the Hamburg Laboratory. Fig. 8. Construction ofa synchronous reversible biquinary decade counter with output amplifiers. Summary. This paper describes a new system of digital circuit blocks, designed to meet the special needs of industrial measure. ment and control techniques, characterized by the following. a) t contains only two different basic circuits: an active logic circuit in diode-transistor circuit in diode logic. logic (DTL), and a passive logic b) t contains only one type of transistor and two types of diode; there are no emitter-followers. c) The basic circuits of this system can be combined to give not only bistable but also polystable 3, 4 or 5 stable states. circuits with for example d) The bistable circuits can be used for either the synchronous or the asynchronous counting mode. e) The circuit blocks operate reliably under full load in the temperature range from -10 C to +50 C with the most unfavourable values of the resistances and voltages within their tolerances, and with the smallest current amplification and the greatest leakage currents which can occur at the end of life of the transistor. f) The loading table is simple. g) Special circuits, such as multivibrators, monostabe circuits as well as Schmitt triggers, can be realized by simple combi. nation of two active blocks and one or two extra resistors or capacitors. Generation of musical intervals by a digital method D. Gossel : ntroduetion The familiar kinds of musical instrument can be divided into two classes: a) nstruments producing notes whose pitch is not decided upon until the instant of playing: bowed string instruments and certain wind instruments are examples. b) nstruments possessing a store of notes, from which in the course of playing a selection is made in accordance with a programme. All keyboard instruments belong to this class. nstruments in class (b) can only be endowed with a limited store of notes for constructional reasons, and because the technique of execution might otherwise be rendered too difficult; also, the access time for whatever notes are available must be compatible with prae- Dipl.-/g. D. Gosset is a research worker at the Hamburg laboratory of Philips Zentra/laboratorium Gmb H, tical requirements for playing the instrument. This implies the existence of some fixed rule or instruction for selecting individual tones from the continuum of pitch. Several such rules have been laid down at various times in the history of music [1], they find practical expression in the various tonal or tuning systems. The four most important will now be briefly explained and discussed. Tonal systems A tonal system has been defined [2] as a scheme for dividing the octave into a progressive sequence of tones, the principle underlying the division being consistently adhered to and designed to produce musically acceptable intervals. 1. Pythagorean tuning This system dates back to the philosopher who lived during the 6th century BC. t is based upon the fifth,

2 1965, No. 4/5/6 MUSCALNTERVALS BY DGTAL METHOD 171 which represents a frequency ratio of 3 : 2 and is. the simplest musical interval of all with the exception of the octave, with its frequency ratio of 2 : 1. A succession of musically new notes is produced when the gamut is transversed at intervals of a fifth, and in this respect the fifth.differs from the octave. The notes of the Pythagorean scale are arrived at by superimposing n fifths in this way and then returning through m octaves to end up in the starting octave; the Pythagorean-type interval r, is thus the result of multiplying by (3/2)n and dividing by 2m. Accordingly, the underlying law is (3/2)n Pv = --; 1 ~ Pv < 2. (l) 2 m ln the Pythagorean system all fifths are true 3 : 2 intervals. The thirds, having a frequency ratio of 81 : 64, do not represent a straightforward harmonic interval and are ciassed as dissonant. A property shared by Pythagorean tuning with all scales having "just" intonation, and some tempered scales, is that it does not form a closed system. t is impossible in principle to arrive exactly at an octave (say) by superimposing a "pure" interval like a true fifth upon itself, because the relevant frequency ratio C represents a simple fraction that cannot yield a whole Fig. 1. Natural harmonic tuning: all the intervals arise out of the octave, fifth, fouth and third. of the scale number when raised to a higher power (which is what the stacking process amounts to) [2]. n fact the octave is not one of the intervals that can be derived from eq. (), the relation underlying the Pythagorean system. Nowadays Pythagorean tuning possesses only historical interest; on account of its dissonant thirds, it can only serve as vehicle for a single melodic line. 2. Natural harmonic tuning octave 2 mojor seventh if minor seventh t major sixth f > minor sixth s. major -third 5 perfect fifth f major third minor diminished fifth H- perfect fourth f major third major third -1 minor third f major minor full tone f r fourth This system is the result of introducing a new interval of the simple harmonic type, the major third with its frequency ratio of 5: 4, to supplement the fifth (3 : 2) and the fourth (4 : 3); the fourth is that interval which, added to a fifth, is needed to complete the octave. Fig. 1shows how all the other harmonic tones are engendered by the octave, fifth, fourth, major third and minor third (6 : 5), this last being the interval which, added to a major third, forms a fifth. The natural harmonic system is quite a practical propositionforinstruments in class(a) as defined above, but is unsuitable for tuning those in class (b) because too few of the resulting fifths are true to the harmonic series, and this fact limits the possibilities of modulation. semifone T 5 third third major third '_'. " third ~,~.,; T > D~ D G? G é 8 cr F for organs. The resulting scale contains harmonically true thirds and tempered fifths. The stacking of four perfect fifths engenders the dissonant Pythagorean third (81 : 64), which exceeds the true major third (5 : 4) by an interval known as the syntonic comma (81 : 80). As compared with true perfect fifths, the tempered fifths ofthe mean-tone system are too flat by a quarter of a syntonic comma,i.e. by4-y81/80 ~ 1.003; the difference is too small to be disturbing, and four of these mean-tone fifths engender a true major third. The mean-tone system provides as many as eleven musically acceptable fifths, but the twelfth fifth needed to close the circle is far too wide and scarcely tolerable to the ear (it used to be known as a "wolf"). 4. Scale of equal temperament Practically all kinds of keyboard instrument nowadays are tuned to the scale of equal temperament (or "well tempered" tuning system). Essentially, the system is due to Andreas Werckmeister and Georg Neidhard (c. 1700) although earlier mention of it is to be found in the works of Ramis de Pareja ( ). The system closes back on itself via twelve equally -e, 3. Mean-tone tuning The system, proposed by Arnold Schlick in 1511, was quite commonly employed in the past, especially [1] W. Dupont, Geschichte der musikalischen Temperatur, Bärenreiter-Verlag, Kassel [2] Adapted from W. H. Westphal, Physikalisches Wörterbuch, Part, Springer, Berlin 1952, pp. 546 and 547.

3 172 PHlLPS TECHNCAL REVEW VOLUME 26 tempered fifths. The tempering consists in the division between these twelve fifths of the ditonic comma ( : ), the interval by which the octave is exceeded when twelve harmonically true fifths are su- tain: perimposed. This means that the equally tempered fifth is only a twelfth of a ditonic comma (or about 1.001) flatter than a true perfect fifth. To put it another way, the scale of equal temperament divides the octave into twelve exactly equal semitones, or intervals having a magnitude ofl2y2. The advantage of having a closed tuning system is that one can modulate successively from one key to another ad libitum, but this is only at the price of major thirds that are a little too sharp and minor thirds that are a little too flat. However, as experience has shown, the ear becomes accustomed to this accentuation of the major and minor character of the relevant modes. Fig. 2 is a representation of the four tuning systems just discussed. The octave covers an angle of 217 in this diagram. The magnitude of any interval can be expressed in "cents", a logarithmic unit divised by H. Bellerman and H. J. Ellis round about The multiplication of frequency ratios is thereby reduced to the addition of the corresponding cent values; the formula is cent 1200 J ~- J og. Thus the octave, an interval of = 2, has a cent value Fig. 2. The four most important tuning systems compared. The octave, having a cent value of 1200, extends over the full circumference of the circle (360 ); accordingly, an angle of.1 0 in the above diagram represents a difference in pitch amounting to 3 l(a cents. ET = scale of equal temperament NH = natural harmonic tuning MT = mean-tone tuning PY = Pythagorean tuning (2) of i = 1200 and the equally tempered semitone, for which = 12/2, is equivalent to 100 cents. On converting Eq. 2 to Naperian logarithms we ob- i -= 1730 n, cent (2a) and the following approximate expression for small intervals such that = 1 + s : i - R::! 1730 e. cent (2b) t will be seen from fig. 2 that the major third in the scale of equal temperament lies intermediate between the Pythagorean and the natural harmonic third. Tone production and the tuning of musical instruments Of particular interest within the framework of the present article are instruments belonging to class (b). With the exception of certain electromechanical organs (the Hammond Organ for example) they all have an independent oscillator for each note, or at least one for each of the twelve notes of the chromatic scale, these oscillators being switched on and off if required [3]. For reasons that are well known the oscillators (which may take the form of strings, wires, air columns, reeds or electronic LF generators) have to be "tuned", i.e. retuned, from time to time, and for instruments employing the system of equal temperament, the operation usually consists in correcting the pitch of a chain of fifths [4]. The tuner's professional skilllies in an ability to find the right tempering for the fifths; only then will the circle close in twelve steps. n principle, tuning could alternatively be done in a sequence of fourths, major sevenths or semitones. n fact the use of the semitone as a tuning interval would do away with the need to work back continually to the starting octave; this must invariably be done if any other interval is employed. But it is almost impossible to judge the tempering of a dissonant interval byear, and so the consonant fifth and fourth are preferred. The successful tuning of a musical instrument, in fact, calls for concentration, time and a trained musical ear. Specialists able to do the work are nowadays becoming fewer and fewer. n the next section a small electronic device is described which permits of exact tuning to the scale of equal temperament in the shortest possible time. Adjustment of a note to the right pitch is done with a visual aid, the needle of a measuring instrument for example, and does not in any way necessitate a trained musical ear. Operation of the device is so simple that it might be worthwhile considering the desirability of

4 1965, No. 4/5/6 MUSCAL NTERVALS BY DGTAL METHOD 173 simplifying it together with certain keyboard instruments, occasional tuning of the instrument thus being left to the user. The device is also likely to be widely adopted for tuning church organs, ajob that has to be done every so often on account of seasonal temperature changes. This routine is particularly laborious and time-consuming in the case of the larger instruments with their multiplicity of pipes and registers. frequency-divider Yzy whh switching Zt~196) ~2=185 Generation of intervals by digital means As is well known it is possible in digital technique to divide a given frequency /0 by any desired whole number z. Where the value of z is on the large side the usual practice is to feed /0 into a scaler which has been adjusted to recognize a preselected z value; having counted this number of input pulses, the scaler emits a zeroing pulse and starts to count anew. The zeroing pulses form a train with a recurrence frequency off = Jo/zo t is an easy matter to select divisors Zt such that the corresponding recurrence frequencies fi represent musically acceptable intervals. f for example we choose Zl obtain a true perfect fifth: = 2 and Z2 = 3, we shall inserted ~ at A or B -=--=-=- 3 Zl 2 All the other intervals are obtainable in a similar way. The great advantage of the digital method of generating musical intervals is that the intervals produced are independent of/o. The absolute position of an interval in the gamut can thus be changed by varying /0, and this fact can be exploited for the purpose of transposition. Digital equipment and methods could conceivably be used for carrying out investigations into musical aesthetics, enabling concords with various degrees of tempering to be quickly and conveniently produced and compared, and appraised as a function of absolute pitch. A digital tuner The intervals between notes in the scale of equal temperament are given by the law underlying the system: Now, v is always a positive integer, so the above relation yields irrational numbers which do not correspond exactly to any interval that can be produced by digital means (the only exception is the octave, for which v = 12). However, W. Schott of this laboratory found that the quotient of 196 : 185..approximates very closely to the equal-tempered semitone, whose value is 12 Vl, the error being only 5 X 10-6 t is on this, and on the fact that the absolute pitch of intervals generated by frequency division can be varied at will, (3) (4) Fig. 3. Arrangement for tuning a musical instrument to the scale of equal temperament. The sensing element can take the form of a microphone, for example, or a magnetic pick up; none is needed for tuning electronic musical instruments. A moving coil measuring instrument, an electric lamp, a magic eye or the like can serve as a visual display device. that the tuner represented sche.natically in fig. 3 is based. The output waveform from an LF generator whose frequency /0 is continuously variable over the range between 40 kc/s and 90 kc/s, which covers little more than an octave, is fed to a frequency-dividing stage which has facilities for division by Zl = 196 or by Z2 = 185, as desired. Thus two frequencies.j'i = /0/196, and /2 = /0/185, are alternatively available from the output of frequency-divider A; and provided fo is constant, the separation between them is almost exactly equivalent to an equally-tempered semitone. The frequency fpu of the note to be tuned is picked up by a sensing element whose nature depends on the musical instrument being dealt with, and after amplification and conversion into a pulse train it is applied to a discriminating circuit in which it is compared with either /1 or f2. one of these serving as a standard or test frequency fst.. The difference-indicating arrangements are ph asesensitive; they should preferably take the form of a visual display. As the picked-up frequency is gradually [3] D. Wolkov, Electronic organ tone generators, Audio 46, No. 2, 34-44, and No. 3, 30, 32, 65, [4] o. Funke, Theorie und Praxis des Klavierstimmens, published by Das Musikinstrument, Frankfurt a.m, 1958.

5 174 PHLlPS TECHNCAL REVEW' VOLUME26 adjusted to exact equality with the standard frequency a fluctuation in the visual display becomes slower and slower and. finally stops altogether; the fluctuation may appear in the movement of a needle against a scale, ceasing when the 'needle finally comes to rest, or in the dimming and brightening of a small electric lamp, which finally acquires a constant brightness level. The tuning procedure is as follows. a) Adjust the LF generator to give its standard frequency of cis, which can if desired be controlled by a built-in quartz crystal. n switch position Z2 = 185 a standard test frequency of 1st = 440 cis will be obtained, and can be used to correct the A above middle' C on the musical instrument being tuned. t is scarcely necessary. to point out that this standard A can be adjusted to any other desired pitch cis for example. b) Switch now to Zl = 196, with the result that the standard frequency is lowered by a semitone. This new standard can be used to tune A flat above middle C on the instrument. c) Switch back to Z2 = 185 and decrease fo untillst is in unison with A flat on the instrument, as just corrected. d) Switch to Zl = 196 and tune G above middle C. e) Switch back to Z2 = 185 and decrease fo untillst is in unison with G on the instrument, as just corrected. f) Switch to ai= 196 and tune F sharp above middle C, and so on. n the course of twelve downward semitonal shifts, carried out in the manner described above, all the required intervals can be found and one finishes up an octave below the note first tuned. Taking account of the systematic error involved in each semitonal shift, which is 5x 10-6, the octave thus arrived at is true to within about 6 X 10-5 This may be compared with the smallest deviation from unison that the ear is capable of perceiving in the most sensitive range of its response curve, around 1 kc/s; this smallest detectable difference is 4x 10-3, or two orders of magnitude greater than the error in the octave. Where facilities are required for tuning the parallel octaves in bass or treble along with the twelve notes of the middle register, it is an easy matter to incorporate a chain of octave-dividers at A or E, i.e. on the standard-frequency or pick-up side of the tuner, the lower registers being covered in the former case and the upper ones in the latter, These octave-dividers are straightforward bistable circuits ("flip-flops") that divide the incoming frequency by two. One type of frequency comparator circuit is shown in jig. 4a; its mode of functioning is explained in, figs. 4b and c. t is assumed that both Ust and Upu, the standard and picked-up voltages, can each assume either of two U st Usto--C:=J---, u"u!o--c::::::::r-""'_+-r: _t t U ~t U U U U U U U tc p U ~t======:===~~== t Usf _t t.1 U Ol] Lf,u t -hp --'-0-'---'-0-'--'-0-'-"- U-'-"-O-r-".-'-0--'----'-0--'---'--[ tc 10 o o o o o o o ft==================== Fig. 4. a) Discriminator circuit. b) Pulse trains operative in the casefst=l=fpu. c) Shapes assumed by the same voltages whenfst = fpu. values only, namely zero or the negative maximum. Further, the ratio between the pulse duration and the recurrance period is assumed to be constant. The circuit functions as a NOR gate: the presence of either of the two negative voltages on the transistor input suffices to switch it on. The transistor only switches to the non-conducting state when both circuit inputs carry zero potential. f the standard and picked-up frequencies differ (fig. 4b) there will arise at the collector ofthé transistor a train of pulses Uc whose breadth fluctuates cyclically. The corresponding mean voltage ti; also fluctuates cyclically, but is a continuous function of time; it is this quantity that is visualized. f on the other handlst is equal to fpu, voltage U c will be a train of pulses whose breadth is uniform, though dependent on the purely fortuitous phase relationship between the two incoming signals. The result will be a steady reading on the measuring instrument. The visual display method makes it possible to detect very slow fluctuations (down to a frequency of 0.01 cis) that cannot be perceived by the ear.

6 1965, No. 4/5/6 MUSCAL NTERVALS, BY_DGTAL METHOD 175 The type of sensing element depends on the kind of musical instrument to be tuned. A microphone is to be preferred for picking up organ tones; for piano tuning a special magnetic pick-up is proposed, fitted with permanent magnet pole pieces which allow it to be attached to the wires on either side of the one being tuned, in such a way that it has a damping effect on vibrations in these neighbouring wires. The pick-up also embodies a small transistor amplifier coupled to a feedback circuit, which serves to keep the wire being tuned in a state of sustained vibration. For correct frequency discrimination it is necessary that the resonator under test should supply a continuous train of oscillations. Experiments have shown that damped vibrations, such as are produced by percussion of a piano wire, are useless for this purpose. An electronic musical instrument like the "Philicorda" is the least demanding as regards the ancillary equipment required for tuning. No electro-acoustic transducer is required, because alternating voltages at the frequencies under test are available from the instrument anyway. n experiments on a "Philicorda" which had first been completely detuned in a random manner, it was possible with the aid of the digital tuner to bring the instrument back to exact conformity with the equal-temperament scale in a matter of barely ten minutes. frequency divider Yz. with switching Y Fig. 5. Purely melodic musical instrument using digital interval generation. Musical instruments based on digital interval generation Musical instruments of "the purely melodic type (fig.5) or ofthe type equipped for harmony (fig. 6) can be devised on the basis of this digital method of generating musical intervals; the "polyphonic" models are naturally dearer, the price depending on the number of melodic lines required. Both types have much the same fundamental design as the digital tuner. One essential difference is that the purely melodic type incorporates one frequency divider with facilities for selecting 12 different divisors Zl Z12, whereas the harmonizing type is equipped with n frequency dividers up to a maximum of 12, for each of which at least one divisor zv, and possibly a set of m such divisors, is available. A feature common to all such instruments is that it is impossible in principle for themto go out of tune, since all the notes are produced by numerical division of a single master frequency fc; and all can be transposed at will, simply by altering fo. The all-important factor governing the choice of divisors is the type of tonal system desired and, in some cases, the exactness or truth the individual intervals are required to have. To finish up, we shall briefly describe a possible choice for the natural harmonic tuning. + + t t further octave (+2) dividers Fig. 6. "Polyphonic" musical instrument using digital interval generation.

7 PHlLPS TECHNlCAL REVEW VOLUME26. The lowest note to be produced is associated with the highest divisor Z: All the other tones are given by where <Z: 10 Je= -. This means that the divisors must be 11 Z b; z; = Z- = --, v a; where a; represents the numerator and b, the denominator of any of the fractions appearing in fig. 1; these fractions stand for cancelled-out. frequency ratios. Since all Z values must be integers and no b; value is a factor of its a~,a sensible move will be to make Z the lowest common multiple of all the values assumed by c.: Z = 2x2x2x2x3x3xS = 720 All the other smallest whole-number values of z; can be found on inserting Zl = 2x2x2x2x3x3x5 = 720in eq. (7) (theyare set out in Table ). The choice of Table J. Divisors required for producing the intervals of a natural harmonic scale by the digital method, shown against the notes of the scale. Note C D flat 16/ D 9/8 640 E flat 6/5 600 E 5/4 576 F 4/3 540 G flat 36/ G 3/2 480 A flat 8/5 450 A 5/3 432 B flat 9/5 400 B 15/8 384 C' as highest divisor entails a certain outlay of circuit elements - 10 bistables, say, and the associated decoding circuits. This outlay can be greatly reduced if the designer confines himself to producing only the consonant intervals at their true values, accepting slight deviations in the dissonant ones. For example, let us suppose that only the octave, fifth fourth and the major and minor thirds need be taken into account in fixing Z. The highest divisor then becomes Z = 2 X 2 X 3 X 5 = 60. The new set of divisors and the errors they involve are displayed in Table l; a positive sign means that the tone in question is too sharp. The quantity ~f (5) (6) (7) Table n. Divisors required for, and errors involved by, an approximation to natural harmonic tuning. Note Error percentage in cents C D flat D E flat 50 E 48 F 45, - - G flat G A flat A B flat B 32 C( components required is now greatly reduced. Six bistables will be ample, and even so, seven of the intervals produced are true as against five which are not. Some of the deviations are rather large; for example, that in the minor sixth amounts to almost the fourth part of a semitone. However, a better approximation can be obtained by choosing Zl three times as large. The result of so doing is that the minor seventh and diminished fifth all become true intervals. With a highest divisor of ai = 2x2x3x3xS = 180 the number of bistables required is eight, and the tuning that results is as shown in Table ll. The scale arrived at in this way would be quite acceptable musically. Table m. Divisors and errors entailed by a closer approximation,;, than that in Table l. ;; Note C 180 D flat 169 D 160 E flat 150 E 144 F 135 G flat 125 G 120 A flat 112 A 108 B flat 100 B 96 C' Error.d' Percentage in cents (,'" ~. ( ,. :i;:-;h Summary. A method familiar from digital techniques, enabling a.' given frequency to be divided by any desired whole number, can be exploited for producing musically acceptable intervals. The more important tuning systems are briefly reviewed. A description follows of a device suitable for quick and accurate tuning of keyboard instruments; it produces an equally-tempered semitone, which can be transposed through the gamut at will. The' true ',', semitonal interval, whose value is 12Vz, can be closely approxi-'. mated by performing the division 196 : 185, the error being only' 5 X 10-6 Also described is a procedure for arriving at a complete scale of notes for a melodic instrument, or one affording facilities 'for harmony. By way of example, numerical values are given for the intervals of natural harmonic scales obtainable by the digital. method.. Volume 26, 1965, No. 4/5/6 pages Published 21st December 1965