Whilst in the Services I played the piano in an Army dance band for a short period. Arriving at a Garrison Hall in India, we found the piano to be far from concert pitch. There was no alternative but to re-tune the instrument as quickly as possible, using a strip of bamboo between adjacent strings, a tuning lever and a great deal of patience. That experience taught me that piano tuning is very much the domain of an expert: at that time I little realised that pulling up tuning throughout can be very dangerous if the frame has hair cracks. Anyway, the tuning was completed in time for the concert and the piano passed muster!
Thank goodness that the modern electronic keyboard instrument only requires the adjustment of a single oscillator — or twelve tunings at the most — with the widely used divider system. Free phase generator systems, on the other hand, still require adjustment of each note generated and thus multiple tunings similar to the piano.
Despite the fact that electronic music facilities include glide, slalom and portamento, steady tones need accurate tuning. For this purpose a standard of pitch is necessary — an absolute reference to which all other notes can be related. Until the advent of broadcasting and recording there was no real standard. By international agreement, the pitch standard is that A above middle C is 440.00Hz (at 8' pitch). The BBC uses this frequency before the start of programme material as a tuning signal.
Although A = 440Hz is the ideal, other factors may have to be taken into account. For example, if an organ is to be played in duet with a piano which has deviated slightly from concert pitch, the organ can be easily brought into line. 'Sharp pitch' instruments used in military bands may be encountered occasionally so it will be necessary to re-tune to these if accompanying.
Having considered the international pitch standard, the question of intervals arises. Certain orchestral instruments are under total control of the player in this respect, such as the trombone and stringed instruments without frets. A certain amount of pitch bending is possible with brass and woodwind instruments but the intervals between notes are basically fixed by the position of wind holes and operation of valves. Excluding the synthesiser, keyboard instruments have fixed tonal intervals.
Whilst it is a simple matter to tune octaves or perfect fifths by listening to the beats produced, musicians always had tuning problems until equal temperament tuning was adopted. The pure diatonic scale and its frequency ratios are shown in Table 1.
Due to the oddities of these frequency ratios, difficulties arise. For example, the first two notes in the key of C (C and D) have a different ratio from the first two notes in the key of D major (D and E), so transposition is impossible. For a while it was thought that making all the whole tone intervals equal would solve the problem, but cumulative errors arose when attempting to tune in perfect fifths.
Equal temperament tuning was adopted during the eighteenth century, where the frequency ratio between each semitone was made identical — based on the twelfth root of two. This method is a compromise: and has been very successful but it does have the disadvantage that perfect intervals involve a slight beat because they are no longer perfect in the true sense!
When the T.O.S. (Top Octave Synthesiser) came to our rescue some years ago, its critics imagined it would not be sufficiently accurate to satisfy the ear of a trained musician: for tuning purposes, each semitone is divided into 100 cents — and a trained ear can detect errors of a few cents. The twelfth root of two is a complex number (1.0594631) for a digital divider system to cope with but the manufacturers have done their homework well.
To set the record straight, bearing in mind the problems the T.O.S. overcomes for amateur and professional builders alike, I have looked closely at a typical T.O.S., the AY-1-0212A from General Instrument Microelectronics. This device will accept an input frequency of 2.5MHz and so will produce 2' pitch without breaking back (the AY-1-0212 accepts up to 1.5MHz and so will only provide 4' pitch).
Table 2 shows the results obtained when supplying the AY-1-0212A with just over 2MHz to provide top C at 8372.016Hz.
Column one shows the divisors that are applied to the incoming oscillator signal, the next two columns the note of the scale and its frequency. For comparison, the precise E.T. figures are shown and are found by using the twelfth root of two as a constant divisor. The deviations are tiny indeed and in the worst case (A#) amount to one part in a thousand. In musical terms, this represents an error of less than 2 cents which a trained musician might be hard put to detect. With reverberation added to the instrument, this small deficiency would probably be covered.
The music industry must be thankful to GIM and other manufacturers of these devices for solving the problem of accurate tuning intervals. Additionally, the ability to re-tune the whole compass simultaneously is indeed a boon. Incidentally, I should mention that any constructor using TDA 1008 divider-keyers should use a TOS capable of top C = 16744Hz (such as AY-3-0214) if the full capabilities of the keyers are to be realised.
Some instruments still employ twelve master oscillators or an individual oscillator per note of the compass, in which case tuning is more complex. The middle octave of a well-tuned piano could be used to set the scale of a divider instrument, tuning each note until beats are eliminated. The various octaves of a free phase instrument can be tuned from these, again eliminating beats. The piano is intentionally mistuned progressively in each direction from the middle of the keyboard, so only its centre should be used as a tuning aid.
Alternatively, using a pocket calculator 440.00Hz can be multiplied or divided by 1.0594631 repetitively (using it as a constant) to give the E.T. semitone frequencies above and below the tuning standard. Having tabulated the figures, a frequency meter will indicate whether the oscillator is tuned correctly.
If neither a good piano nor frequency meter are to hand, tuning has to be tackled the hard way.
Tuning by introducing audible beats relies on the presence of harmonics, so I would suggest 8', 4' and 2' Diapason tabs are used simultaneously. The table shows A = 440Hz as the starting point with E tuned against it. Our E.T. calculations will show that E ought to be 329.6275Hz and its fourth harmonic therefore becomes 1318.51Hz. The third harmonic of A is 1320Hz, so the two harmonics will beat at 1.49Hz (or 89 beats per minute) when the fundamentals are correctly tuned to each other. The rest of the table is based on a similar calculation for each note.
Before embarking on the sequence of Table 3, allow the oscillators to settle down, then use a tuning fork or the BBC signal to set 440Hz to zero beat. When arriving at the last step, do not adjust A unless it has drifted from the standard. There should be 60 beats per minute between D and A and, if this is not the case, the process must be repeated and repeated... while trying to keep your temperament!
Feature by Ken Lenton-Smith
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