Interval Tuning for Harpsichords

Reflections on Tuning

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2011-08-19 — Original posting (on Blogger)
2014-10-28 — Re-posting as is (WordPress)
2016-06-20 — Brushed up for better readability
2020-09-30 — Correction & clarifications under “Instrument Examples”; extra intro. Thanks to Gary Sarff for suggestions!

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Interval Tuning Pains…

Probably, with this posting I mostly meant to write past pains with interval tuning off my chest …

Before we Even Start…

Some people may wonder why tuning a harpsichord deserves an article of such length and detail on its own. Why is tuning a harpsichord more complex than tuning other string instruments, such as, say, a guitar, or a violin? And: is it much different to tuning a piano?

As for the latter: a harpsichord is a wooden instrument with metal strings. That leads to the principal complication that changes in humidity cause the wood to expand and contract by as much as 1%, whereas metal strings are not affected by humidity changes. Temperature changes also contribute to tuning degradation, but to a far lesser degree.

Instruments such as violins have the same issues. However, their strings are made from material such as gut or plastics such as nylon. These are elastic, which makes tuning mechanically easier. Plus, there are fewer strings to tune.

A modern piano or concert grand, on the other hand, uses metal strings strung onto a cast iron frame. The huge mechanical tension of several hundred steel strings actually mandates a cast iron frame. The latter makes the instrument more rugged and largely avoids the effects of humidity and temperature changes. Thus, the tuning of pianos and concert grands is more stable and persists over weeks, months or longer (most private pianos are tuned once or twice a year). At the same time, such instruments are usually tuned by tuning professionals only, hence is the matter of specialists and beyond the scope of this article.

Basic Challenges

Earlier on I stated that I frequently tune(d) my wife’s harpsichord. To tell the truth: I’m not sure I like the tuning process as such. But I liked the challenge of tuning, at a time when we used recipes based on interval tuning. We did not have electronic tuning aids for a number of years. The basic challenges in interval tuning fall into several categories:

• changing the pitch on a harpsichord (or similar) instrument is inherently non-trivial
• interval tuning is never accurate; if you are lucky, errors average out. And if they don’t, you will simply start over …
• tuning intervals has its own intricacies
• harpsichords with multiple stops may add to the difficulty in tuning.

Basic Terms & Facts

Let’s look at these points one by one, first some principal thoughts on altering the pitch on a string:

Tuning Handle

Piano tuners use a long tuning handle. This gives then the necessary torque (piano strings are under much higher tension than harpsichord strings), but also a fair amount of extra accuracy, permitting to turn the tuning pin by tiny amounts, maybe fractions of a degree.

With historical harpsichords, the tuning pin is essentially a forged nail with a flattened end, and the tuning “handle” typically is a “T”, consisting of a metal tube with a flattened end that is pushed onto the tuning pin, and a short, horizontal, wooden grip that just fits into one’s hand. It takes some exercise to apply the right amount of torque to get the tuning pin to move, but to avoid “overshooting” and broken strings. These tuning “T” handles have the advantage of just turning the pin. A modern tuning handle would apply lateral forces and might loosen the pin.

Tuning Pins

The tuning pins must stick, otherwise the strings would detune rapidly. It therefore requires some exercise to be able to apply the right amount of torque on the short handle to get the pin to move. Careful: don’t to turn it too far, as this might cause the string to break. It is pretty normal that the first change is either not enough or makes too much of a change. In my experience, one usually gets some overshoot, so a second correction will be in the opposite direction. If a tuning is real smooth, that should immediately raise concerns whether the tuning pin is loose!

Strings

Manufacturers of historical instruments prefer not to use industrially manufactured strings, but rather hand-pulled wire (of a special alloy). Such wire naturally has some irregularities; it happens that occasionally a string may initially (after pressing the key) sound “pure”, but may show some slight, often periodic alterations in sound and possibly even the pitch while the sound is fading away. One could exchange such strings. But then, one may also (if it isn’t too bad) regard this a “natural” part of a real harpsichord, as opposed to electronically produced sound;

Interfering Stops

Most harpsichords have at least two stops. One starts tuning a stop at “normal” (8′) pitch; if an instrument has two 8′ stops (such as ours), it is likely better to start with the stop that has the brighter sound, i.e., the stop producing more harmonics, see below. The other stop is initially “turned off”, i.e., the jacks are shifted sideways, such that the plectrum does not reach the string. However, the string is still unmuted when pressing the key: the string is actually not muted – it is just not played, i.e., one of the two strings is played, both are unmuted. This, per se, is OK. However, from a there is some interaction between these strings (through the sound board and the bridge):

In terms of physics, the two strings act as coupled oscillators, i.e., the vibration starts with the string that is played, then gets transferred (partially) to the other string, then back again. This can be heard as periodic swelling or alteration of the sound quality and may be irritating when tuning / listening to interference beats (which are also heard as periodic swelling, albeit not on the main tone, but with the relevant common harmonic, see below). With some training, one learns to ignore this effect; I initially used wedge-shaped rubber pieces (cut from erasers) to mute strings when tuning the first stop. It serves the purpose, but doesn’t look as professional as the rubber comb used by piano tuners;

Hysteresis

One positive aspect of harpsichord tuning: hysteresis is a minor worry. Hysteresis is something players of violins and related instruments are often confronted with: they turn a peg, and this loosens or tightens the stretch of string between the peg and the saddle, but not the active part of the string, due to friction between the string and the saddle. The intended pitch change is “delayed”, possibly in either direction. So, after turning the peg a violinist may either pull the middle part of the stringto loosen it, or tighten it by slightly pushing down the string part between the peg and the saddle, in order to make the string move over the saddle. On harpsichords, strings are simple wires, hence friction should be minimal, and any movement of the tuning pin will immediately result in a pitch change.

Fading Sound

Finally, the sound of a harpsichord string (at least in the mid and upper range) rapidly fades away, hence one typically needs to press a key several times in succession in order to be sure that the pitch is correct. In doing so, one should make sure the key is released completely in-between, such that the string is dampened properly: plucking a string that still vibrates may generate additional, unwanted oscillations, harmonics, etc.

Interval Tuning Basics

All this sounds complicated — more complex than it really is: but at least it implies that when one starts tuning harpsichords one should approach the task carefully and with diligence! Let’s turn to interval tuning basics (boring stuff first!):

Tuning in Unison

When you tune a string in unison to another one (e.g., when adjusting the second stop of a harpsichord to have the same pitch as the first stop), the last step in tuning such instruments), they should ideally sound like a single string. If their pitch differs slightly, then you’ll hear periodic variations in the volume of the sound. That is the interference between the two strings. Interferences occur because the two strings at one point vibrate together (“in phase”), so their volume adds up. Then, the slower one (lower frequency) starts lagging behind until the two strings vibrate against each other (“anti phase”). At this point, they partially cancel their sound. This is again followed by an “in phase” period, etc.; every swelling of the sound is called an “interference beat”. The frequency of these beats is the frequency difference between the two strings.

Ideally, when tuning a harpsichord, you should hear less than one beat every 4 – 5 seconds (slower beats are hard to hear because the sound is fading away). This means that in harpsichord tuning, at least in the mid and high range you can tune to an accuracy of maybe 0.2 – 0.25 Hz. At 440 Hz this is about 1 Cent (1/100 of a half-tone).

Harmonics

When tuning the bass register of a harpsichord, 0.25 Hz accuracy is not enough (with the lowest string on our harpsichord — G, — this would be an error of about 5 Cent, or 1/20 of a half-tone). Fortunately, the lower strings sound much longer (so you can listen for slower interferences), plus, with the lower part of the keyboard you will also hear interferences of harmonics of the two strings (octave, octave + fifth, 2 octaves maybe), which should enable you to tune to the same accuracy as with the middle register. Plus, low strings tend to have less tension and are much longer, hence it takes more movement of the tuning pin (per Cent) to make a given adjustment, and these strings are generally easier to tune.

Conversely, at the high end of the keyboard, the tiniest turn of the tuning pin may cause a pitch change of several (5 – 10) Cents. Tuning the top end can be real tricky!

Octaves

Adjusting octaves (the second step in tuning, after tuning the initial octave range) is fairly easy as well: the upper note is at the same time the first harmonic of the lower note, so you can “tune away” interferences at the pitch of the upper of the two notes: if you could separate the two sounds, you would hear that the lower note stays constant, whereas the upper note shows an interference if the octave is not accurate. The accuracy of tuning an octave should be similar to tuning in unison at the pitch of the upper note.

Fifths and Fourths

For the most part, recipes for tuning the first octave are based on adjusting fifths and fourths. Tuning these intervals is different from tuning in unison or tuning an octave, as the upper note is not a harmonic of the lower tone, hence the main interference occurs neither on the lower nor on the upper of the two notes. When tuning a fifth, the interference occurs between the second harmonic of the lower note (octave + fifth) and the first harmonic (octave) of the higher note, e.g.: when adjusting c’ – g’, the interference will occur on g”. The smaller the interval, the farther away the interference: when tuning a fourth — e.g., c’ – f’ — the interference is between the third harmonic (two octaves) of the lower note and the second harmonic (octave + fifth) of the higher note — in the example given: at c”’.

Note that besides this main interference, there are always additional interferences at higher harmonics. However, as these typically have a lower volume, they should be nothing to worry about. If beginners have trouble hearing the main interference, playing the respective key (using the above rules) should help finding it.

Major Thirds

Similarly, when adjusting a major third, the interference is on the fourth harmonic (2 octaves + third) of the lower note, and the third harmonic (2 octaves) of the upper one: c’ – e’ yields a main interference at e”’.

Tuning Pure Intervals

Now, let’s look at actual tuning of pure intervals:

• When adjusting pure intervals (fourths, fifths), you “simply” tune until the interference disappears, i.e., until its frequency is (almost) zero. When the tuning is far off, the interference may be too fast to be heard. However, then the ear should have no trouble indicating an error. The closer you get to the pure interval, the slower the interference, see above: you will never be accurate to 0.000 Hz!
• There is a fair chance that such residual errors cancel out when adjusting a chain of pure intervals, e.g.; C – F – Bb – Eb – Ab – C# – F# – B – E, as for “Werckmeister III” tuning. However, occasionally they may add up rather than cancel each other out. In such cases, you may find that you need to start over again!

Tuning non-pure Intervals

Tuning non-pure intervals has additional pitfalls:

• In order to compensate (distribute) the Pythagorean comma, fifths need to be tuned too small, but fourths too big, ignoring the wolf fifth in pre-baroque tuning.
• Unfortunately, the interference gives no indication whether an interval is too big or too small, and the required deviations from pure intervals are often too small for the ear to make this decision.
• One common mishap in interval tuning is that you intent to make a small correction, but the tuning pin sticks; when it finally moves, the correction is too big, and by mistake you tune an interval too big in lieu of too small, or vice versa. If this goes unnoticed, you will find that the “ends don’t match up” (i.e., the last, remaining interval has a large error), and you need to start over again!
• The interference frequency depends on the interval. Assume you want to tune c’ – g’ (fifth) and d’ – g’ (fourth) to have the same error (say, 1/4 of the Pythagorean comma, or 6 Cent). The interference for the fourth will be substantially faster than for the fifth, because with c’ – g’ the interference is on g” whereas for d’ – g’ the interference is at d”’, i.e., a fifth higher than the interference of the fifth. Hence you can expect the interference to be 50% faster for the fourth.
• For the very same reason, the frequency of the interference on a given interval depends on the pitch of the two notes involved. The interference for a given error with c” – g” will be twice as fast as with c’ – g’.

Example for Pitch-Dependency

As an illustration for the last two points: assume you want to set up equal temperament tuning for a = 440 Hz (starting with c’). In this case, the interference frequency for the interference frequencies for C – G, G – D, D – A, etc. will be -0.9 Hz (fifth), +1.3 Hz (fourth), -1.0 Hz, +1.5 Hz, -1.1 Hz, +1.7 Hz, …

Conclusions

Conclusion: if the tuning recipe requires “distributing the error between C – G, G – D, D – A, and A – E”, that is tricky for beginners, doable for experienced musicians. However, setting up equal temperament tuning using the above methods is impossible for people with little experience, difficult even for tuning professionals. There are aids that help in such tasks:

• One can calculate the frequency of the interference for a given pitch and deviation from a pure interval.
• One can use the help of a silent metronome to set the frequency of the interference beats more accurately. In the above example of equal temperament tuning one would adjust the interference frequency to metronome numbers of 53, 80, 60, 89, 67, 100 beats per minute, and so on. Of course, you still need to deal with the inevitable tuning errors!

This all sounds like a complex, difficult task. Keep in mind that up till a few decades ago (recipe-based) interval tuning was the only option for tuning. Luckily, with the advent of electronic tuning aids, this burden has gone away!

Interval Tuning and Instrument Type

A question at last: is the adjustment dependent on the instrument type? Strictly speaking: no, as for a given temperament and absolute pitch, the interference frequencies will be the same for a harpsichord, a virginal, a grand piano, an organ, etc. — however,

• the volume of the interference depends on the volume of the relevant harmonics, and those depend on the instrument type
• instruments with a sound that rapidly fades away make it more difficult to adjust the interference frequency
• bad strings may change pitch as the tone fades away
• the ear’s perception of a sound pitch is somewhat dependent on the volume of the tone(s). This has no influence on the frequency of the interference, but it may still be irritating, e.g., when looking down an open organ pipe as opposed to looking at the pipe (the labium of a flue pipe) from the front while tuning.

Instrument Examples

Some examples for the above:

• Harpsichords typically have a bright tone that is rich in harmonics. This should make it easy to hear harmonic interferences. However, the tone fades away quickly, and there may be other limitations, as discussed above.
• Virginals and spinets are very similar to a harpsichord in their sound characteristics, though they typically have only one stop. However, the sound of a virginal is less rich in harmonics, particularly with muselar-type instruments, where the string is plucked near the center. This means that the base frequencies dominate the sound. That again may make it somewhat harder to hear the interferences.
• The sound of a piano string takes longer to fade away, which should help in tuning. The sound may contain a lower volume in harmonics, though, which could make it harder to identify the interferences.
• Finally, organ pipes have the advantage of producing constant sound, so the fading sound is not an issue at all. One would expect organs to be the easiest to tune (ignoring the fact that there may be thousands of pipes … ).

Sorry for the length of this posting. I meant this to be part of the next posting, which again was to be a part of the posting “Progress in tuning?“. It looks like whatever I start in this area grows to a book; I split that book, the parts grow to books again, etc. — reminds me of the Sorcerer’s Apprentice … 😉

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