Reflections on Tuning
2011-08-29 — Original posting (on Blogger)
2014-10-28 — Re-posting as is (WordPress)
2016-06-19 — Brushed up for better readability
2020-12-11 — Added literature references
2021-11-23 — Minor expansions, corrections
- Introductory Remarks
- The Initial, Traditional Recipe — Werckmeister III
- Calculated Interference Frequencies
- Our First Electronic Aid — Analog
- Switching to a Digital Tuning Meter
- Tuning Beyond the First Octave
- Concluding Anecdote
- Addendum — The Obsolescence of our Korg Tuning Meter
I’m not trying to write a set of instructions on how to tune a harpsichord. Others have done that and are certainly more qualified than I am (see below). In this posting, I merely want to describe my / our experiences with harpsichord tuning over the past (almost) 30 years. People who know how to tune a harpsichord should stop reading here. I don’t want to bore them with trivialities!
Setting the Pitch in the Early 80’s
Some time after taking delivery of the instrument (around 1980), Lea had attended a course on harpsichord tuning. At that time, electronic tuning aides were expensive. So, the course addressed traditional / historic methods for setting up the tuning within the initial octave range. Tuning recipes in baroque times were based upon interval tuning: one tunes a first string using a tuning fork: typically a = 440 Hz or, especially for baroque or pre-baroque tuning, a = 415 Hz. Thereafter, one tunes the initial / central octave range from this one starting point. These two pitch levels are the most common ones. Historically, people used a variety of different standard pitches (even pitches higher than a = 440 Hz). These were often delimited to a specific time at a specific geographical region or location, to a court or an institution.
Switching Between Base Pitch Levels
Our harpsichord is designed to be usable at both a = 440 Hz and a = 415 Hz: essentially a half-tone (99 Cent) difference. The selection is of course not achieved by changing the pitch of all strings by half a tone, but through a transposing keyboard, see below. The former would cause the strings not to have enough tension for the full sound (if tuning down), or, conversely, excessive tension (when tuning up) might cause strings to break.
Apart from this, such a huge change in pitch would make it impossible to tune the instrument at once. Changing the tension of a string so dramatically will inevitably alter the tension (and hence the pitch) of neighboring strings: with such a big change you would constantly alter the pitch of strings already tuned. You could do a very coarse tuning at best, then you will need to restart from scratch. Ideally, you would need to wait for the instrument to adjust itself to the new tension levels, then, after a while (hours), do a second (and later maybe a third) tuning.
Instead, many harpsichords allow for the entire keyboard to be shifted sideways (there is an extra string with every stop). This is a so-called transposing keyboard. One can switch between a = 415 Hz and a = 440 Hz without dramatic retuning. However, as we are not using equal temperament tuning, you should not expect to be able simply to “shift & play”! When shifting up, C major will be using the former C# major keys. When shifting down, it will use the former B major keys. Both these are keys “far away from C major” and will therefore sound much worse than expected — if they are usable at all! Conclusion: after using the transposition mechanism, one still needs to re-tune the instrument.
The transposing keyboard is not exactly historic, but is rather a convenience feature built into many current replicas of old instruments. It permits accommodating continuo / ensemble playing with instruments in either baroque (a = 415 Hz) or modern (a = 440 Hz) pitch settings.
The Initial, Traditional Recipe — Werckmeister III
Initially, we were typically tuning to a = 415 Hz (the bass sounds so full, nice & warm on an Italian harpsichord!!); actually, we used a tuning fork for c’ = 493.5 Hz rather than starting with a’. The reason for this is that a popular / convenient recipe for “Werckmeister III” tuning starts with c’ rather than a’:
- start with tuning C (c’) = 493.5 Hz
- tune the octave above (c”) from c’
- from c” (down towards c’), tune pure fifths and pure fourths C – F – Bb – Eb – Ab – C# – F# – B – E
- Tune the intervals G, D, and A such that the “error” (Pythagorean comma) is distributed evenly between C – G – D – A – E (see the posting “Interval tuning” for details)
- When done, tune E and B for pure intervals A – E – B, hence shifting a quarter of the Pythagorean comma from A – E to B – F#.
The convenient part of that recipe is that one can tune pure fifths (and fourths) starting at the upper end (c”). The tedious task of distributing the Pythagorean comma is limited to four adjacent fifths / fourths at the other end, i.e., is achieved by tuning “just” three tones g’, d’, and a’. The recipe does have its pitfalls, though (especially for beginners) —
- In the “easy” part with the 8 pure intervals one should keep in mind that small errors may not average out. If for instance you tune every tone by making sure the last change is moving up (in order to avoid the effects of hysteresis, see “Interval tuning”), you may end up with a tendency to leave most tones too high. In this case (starting from the upper end) the fifths may be a little too small, the fourths a little too big. You may then only find out about the error when you see that the residual error is bigger than the Pythagorean comma, i.e., you need to spread too much dissonance over the last four intervals. At this point you start over at square one!
- When tuning the four “impure” intervals at the lower end, keep in mind that the interference in fourths must be 50% faster than in fifths. Also, the higher the pitch position of the interval, the faster the interference. I did the calculus and found that in the case of Werckmeister III tuning at c’ = 493.5 Hz, the interferences should be as follows: c’ – g’: -2.5 Hz (small fifth); g’ – d’: +3.8 Hz (big fourth); d’ – a’: -2.8 Hz; a’ – e’: +4.2 Hz.
- The result can only be checked after shifting the error from a’ – e’ to b’ – f#’. If the outcome is not satisfactory, it is typically faster to start over rather than trying to look for the error(s) and tweak one or several intervals.
One can check the result by comparing the triads for all major keys in the chromatic circle, making sure that at least the keys that one intends to use sound acceptable.
Advantages of the Traditional Method
While from today’s perspective (where most musicians and tuners are using electronic tuning aids) this may sound painful, there are some advantages, too:
- Ideally, all you need is the recipe and a tuning fork (no electronic gear, batteries, etc.).
- What I personally like about this approach is that no two tunings will ever be the same. When you do the final check by going through the entire chromatic circle of keys in order to see which ones are very nice & pure, nice / OK, “a little sharp”, and “hardly usable”, you will easily find out that the results will vary from one tuning to the next. With some training the result will be more and more predictable. However, it always retains some individuality, and if once in a while (hopefully more and more often) the keys that you are going to use are sounding real nice, then you can be proud of your own work and state “this is the result of my own effort — it all paid off in the end!”
Does it Last?
Sadly, harpsichord tunings are always volatile: you may be able to preserve a tuning by making small corrections. Even that will hardly last for more than a couple of days, or 1 – 2 weeks at best, at which point you will typically need to do a fresh tuning. Of course it helps if you keep the instrument at constant humidity and temperature (the former is more critical than the latter, in general).
Of course, we have a humidifier in the room where we have the harpsichord, the virginal and all of Lea’s recorders. The instruments would not survive without that. But this puts us between Skylla and Charybdis: if the regulated humidity level is too low, this is bad for the instrument per se (the wood is shrinking, drying out. It may even develop cracks. A crack in the block holding the tuning pins may be very expensive to repair! At the same time, in warm & humid summer days you would actually need a dehumidifier to keep the humidity at a constant level.
On the other hand, if you turn up the humidity to a high level (55 – 60% would be ideal for harpsichords, recorders and other wooden instruments), not only will you be bothered by the constant or frequent noise of the humidifier (we haven’t found the ideal, noise-less and hassle-free humidifier yet), but also, if the music room is in a house with moderate / limited insulation (such as our house, which will soon be 80 years old), mold may develop on cold corners / walls. When we rearranged the room to accommodate the virginal a couple of years ago, we found that we need to have the walls repainted.
Mold may possibly develop also on music books! That danger of mold can be limited by minimizing the heating in winter (hence reducing the overall water content of the air and minimizing the temperature gradient at the walls). But then, you still want to feel comfortable in such a room — it’s tricky! We lowered the humidity setting from 55% down to 50% — and we try not to alter the setting!
Calculated Interference Frequencies
Back to tuning. I calculated the above Hz-numbers for the frequency of the interference beats (and I did that for a whole series of tuning temperaments). Soon, I realized that estimating Hz values is difficult for the human ear / brain, so I then calculated beats per minute (metronome numbers). I would dial these into an electronic metronome that could be switched to silent mode, so I could adjust the frequency of the interference to a blinking LED.
This actually has been described by others, too, so it is not my invention. Here is the frequency table that I used for a couple of years, for tuning starting with c’ = 493.5 Hz (i.e., a = 415.3 Hz — we have a second table for a = 440 Hz). Note that this was before we learned about Bradley Lehman’s interpretation of Bach’s tuning recipe (Lehman, 2005a and Lehman, 2005b), see also my posting “What tunings did Bach use?“.
Our Initial Set of Temperaments
In this table (click on the image to enlarge), the various tuning labels mean
- Meantone temperament tuning (see my posting “Progress in tuning?”)
- organ tuning by Arnolt Schlick
- original tuning of the third organ in the abbey of Muri / Freiamt, Switzerland
- alleged organ tuning by Gottfried Silbermann (two versions)
- Werckmeister III tuning
- Bernhard Billeter’s proposal for a “Bach harpsichord tuning” (Bernhard Billeter, 1979)
- Bernhard Billeter’s proposal for a (more versatile) organ tuning (Bernhard Billeter, 1979)
- equal temperament tuning
How to Use the Metronome Numbers
Negative numbers indicate intervals that are tuned too small (compared to “pure” intervals), typical for fifths. The red numbers indicate large fifths or small fourths — a full or partial wolf fifth. The yellow rows are those which we typically used on our harpsichord. Tuning with a metronome makes the process a bit more “controlled”, i.e., it minimizes the chances for major errors. However, it still is sequential tuning. Therefore, errors may or may not cancel out. Also, beginners may still make “sign errors”, i.e., tune a “short fourth” or a “wide fifth” by mistake, which definitely means at least a partial re-start.
With most tunings, the last number ideally is for checking only: c” is tuned as octave from c’. If after reaching F, the fifth F – C is far off, you need to look for errors or start over. Overall, as it is sequential and hence susceptible to (partially) cumulative errors, the above “metronome method” still produces unique, individual results. If all goes well, it also offers some “adventure / achievement feeling”!
Limitations of the Metronome Method
One typically cannot use numbers above 300 with metronomes. In tunings with a wolf fifth you would not tune that interval, but just all others, from either end of the scale. Note that because you can’t really check the wolf fifth for its degree of impurity, the original recipes for such tunings are better than purely sequential methods (only shown here for comparison). Typical Meantone temperament tuning recipes also use thirds for tuning and a more “branched scheme”, i.e., they involve shorter interval sequences / tuning chains, making major cumulative errors less likely.
Our First Electronic Aid — Analog
A new era in tuning started for us in November 1989 (in Germany): my birthday present for Lea was an analog tuning meter (Korg); this ended the above sessions with “chained / sequential tuning”, the use of a silent metronome for tuning, as well as the use of a tuning fork.
In 1989, tuning meters were either analog (with a needle indicating the amount of deviation, ideally in Cent units and with a range of +/- 50 Cent), or they indicated the pitch by a series of red & green LEDs giving a rough idea about the deviation from equal temperament tuning. The latter are unusable for tuning a harpsichord, or for any serious, accurate tuning (they may be OK for beginners tuning a guitar). You need a meter that yields an accuracy of around 1 Cent, i.e., 1/100th of a half-tone (equal temperament). You also need the ability to set the base pitch with an accuracy of 1 Hz or better. At that time, analog meters were rather expensive: I paid around DM 450, if I remember correctly.
How is this Different from Older Methods?
Tuning with a tuning meter is different from all of the above in several ways:
- Primarily, there is no “chained” tuning. One tunes every tone individually and independently. The pitch of any string (within the first octave) does not depend on the tuning of any other string. So, the (absolute) accuracy of the tuning of all individual strings (and the accuracy of the tuning meter) is the only factor determining the final tuning quality.
- One can tune the common (and convenient) range a .. a’ for all tuning schemes — or in any other range, as required by the tuning scheme / table;
- Equal temperament tuning simply means tuning all strings to “0” (0 Cent deviation);
- Adjusting the initial octave range does not involve interval tuning (see my posting “Interval tuning”). You don’t need to train your ear to listen for the frequency of interference beats. The only challenge is the ability to tune individual strings to the desired accuracy, see again my posting “Interval tuning”;
- All tuning temperaments involve the same effort, the same difficulty, from Meantone to equal temperament tuning;
- One can switch to a different base pitch (e.g., between a = 440 Hz and a = 415.3 Hz) using the very same scheme / recipe (see the table below). It just requires changing the base pitch at the tuning meter.
A Table for Tuning Meters
We are now tuning in the range a .. a’, and we use the following table for setting up the initial octave:
The table (click on the image to enlarge) indicates deviations from equal temperament tuning, in cents (1 cent = 1/100 half-tone), starting on “A”. The numbers in this table are rounded to half-cent values. The various temperament labels mean
- Meantone: meantone temperament tuning (see above)
- Muri—Organ: original tuning of the third organ in the Abbey of Muri / Freiamt, Switzerland
- Schlick—Organ: organ tuning by Arnolt Schlick (c.1455 – 1521)
- Werckmeister III: one of the tuning schemes described by Andreas Werckmeister (1645 – 1796)
- Silbermann 1 / Silbermann 2: alleged tuning schemes by the German organ builder Gottfried Silbermann (1683 – 1753)
- Billeter — Bach/WTC: Bernhard Billeter’s original proposal for a “Bach harpsichord tuning” (Bernhard Billeter, 1979)
- Billeter — Organ: Bernhard Billeter’s original proposal for a (more versatile) organ tuning (Bernhard Billeter, 1979)
- Bach WTC / Lehman: Bradley Lehman’s interpretation of Bach’s header decoration (Lehman, 2005a and Lehman, 2005b)
- Bach WTC / Lehman, mod. Billeter: Bernhard Billeter’s slightly moderated version of Bradley Lehman’s proposal (Billeter, 2007)
- Bach WTC / Jobin & Billeter: Bernhard Billeter’s adaptation of Émile Jobin’s reading of Bach’s header decoration, as published by Pierre-Alain Clerc (Clerc, 2007).
- Equal Temperament: equal temperament tuning, just for reference.
Since we know about them, we tune our harpsichord following one of the recipes from the “WTC” family of temperaments. My first selection would be “Bach WTC / Jobin & Billeter” from the table above. We are happy with this. Additional information on the tuning aspects of Bach’s Well-tempered Clavier are also available in Wikipedia.
Switching to a Digital Tuning Meter
2010, after having used it for 21 years, we trashed the old Korg tuning meter (it started falling apart) and bought a new, fully electronic model (Korg LCA-120), for about a third of the price (inflation included) of the previous one (CHF 165). The new meter simulated an analog meter by LCD. The analog version may have been marginally more accurate in the display / reading. However, with harpsichord tuning, 1 Cent accuracy is about what one can achieve, considering the rapid fading, and natural oscillations / variations in the reading, due to string irregularities, etc.
There are also digital tuning meters with built-in historic tunings. In our case I think these would have been a waste of money, because
- the choice of pre-programmed tunings is typically small, and
- little known or newer tuning schemes such as the one that Bach may have used for his Well-Tempered Clavier (see “What tunings did Bach use?“) will be missing anyway.
- Some tuning meters may permit defining one’s own tuning. This is wasted time if you have a table such as the one above (or others found in literature).
- Also, I find it confusing to dial “Vallotti” (or whatever pre-programmed scheme there is) and then tune all notes to zero: it would feel like I have no idea what I’m doing!
Make Sure You are Not Off by a Half-Tone!
Both our analog as well as our new, digital tuning meter also detects and indicates the currently played tone. Normally, we hardly look at that. However, I think it is a good idea to check at least for the first string (a) that you tune. Imagine that you have been on vacation for a couple of weeks, and back home it has been raining after a dry period. It is easily possible that the harpsichord went up in tuning, possibly by more than 50 Cent. Once that is the case, it can easily happen that by mistake you tune the instrument half a tone (100 Cent) too high (e.g., a = 466 Hz in lieu of a = 440 Hz). This may cause strings to break etc.!
The opposite (low tuning by half a tone, a = 415 in lieu of a=440) may happen as well. This will at least avoid danger to the strings, but the sound of the instrument will not be very good. People with absolute pitch don’t have that problem, of course!
Tuning Beyond the First Octave
All of the above is about tuning the first, central octave, a .. a’. The rest is the same for all methods. I first tune from the initial octave to the upper end of the keyboard (the tricky part first: higher pitch strings are harder to tune, see “Interval tuning”). Then I tune down from (a) to the bottom. This happens by tuning octaves from the initial range, a .. a’. I would never consider using an ordinary tuning meter to tune octaves. Tuning octaves is easier than other intervals. Nevertheless it pays to do this as carefully as the central octave range. Errors will propagate from one octave to the next one! At the end, I will also shift the transposing keyboard to tune the unused string, just to make sure it does not “wander off” to extreme pitch levels.
Once one has tuned the entire stop, on instruments with multiple stops one then needs to tune the second stop in unison with the first one (I do this top-down, again the doing trickier part first). Tuning strings in unison is relatively easy, but it still pays to do this carefully as well, as interferences would be very audible even when playing one single key with both stops engaged.
Enough said! I think (or hope) this concludes my long posts. But let me finish off this one with a little tuning anecdote / mishap:
In our last year in Germany, back in summer 1992, I offered to tune the harpsichord for a concert in which Lea played recorder in a small baroque music ensemble (in the church in Seeheim-Jugenheim). I carefully tuned the instrument and quite liked the result (others confirmed this judgement). I made one mistake, though: the base pitch was a tad (maybe 2 Hz) high. With the harpsichord this was not a problem at all. However, it was a little too high for the recorder. The player can just fully push the body into the headpiece, thereafter it is not possible to rise the pitch further, other than by playing louder! So, Lea (sadly, of course, and to her dismay) was unable to play softly; she was a little stressed after the concert. Luckily, this went unnoticed otherwise …
Addendum — The Obsolescence of our Korg Tuning Meter
Once had the digital Korg LCA-120 tuning meter, we realized that it had limitations similar to those of its analog predecessors. The two damping positions proved hardly adequate for harpsichord tuning: strings are often enough inherently a tad impure, causing the needle / indication to jump around nervously. However, there are new tools: we now use an iPhone or iPad to tune our harpsichord.
Tuning Apps for iPhone / iPad
One simple iOS app for tuning that we have mostly used so far is “Tuner!” by Peter Deelstra: simple, but efficient, and accurate enough. A more sophisticated app with a limited selection of pre-programmed temperament (e.g., Werckmeister III) is “Tuner by plusadd”, and finally, there is “PitchLab Guitar Tuner” (free, with in-app purchases), which comes with tons of features that you will not use with a harpsichord, but still an app usable for the latter. At least with the “unlock everything” option it allows you to program your desired temperament(s). I have also downloaded the “n-Track Tuner” app (Pro version). So far could not make much sense out of it: a sophisticated gimmick without much practical use for tuning a harpsichord, in my opinion.
I recently found another, new iOS tuning app: “Well-Tempered“, by Niklas Saers. This comes with a mind-boggling set of around 270 different pre-set tunings. One can set all of these to start with any half-tone (and the base pitch is adjustable, too). So far, I haven’t started using this. One drawback in my view is that it does not give any detail about any of the 268 built-in temperaments, so it acts like a black box.
It turned out that at least some of the iOS tuning apps work so well that one can indeed use them to tune the entire keyboard. You no longer need to tune octaves. Of course, one should still tune any extra stops (in unison, or in octaves for 4′ or 16′ stops) directly, not via the app.
Bernhard Billeter. (1979). Anweisung zum Stimmen von Tasteninstrumenten in verschiedenen Temperaturen. Kassel: Verlag Merseburger Berlin GmbH.
Lehman, B. (2005a). Bach’s extraordinary temperament: our Rosetta Stone—1. Early Music, 33(1), 3–24. https://doi.org/10.1093/em/cah037
Lehman, B. (2005b). Bach’s extraordinary temperament: our Rosetta Stone—2. Early Music, 33(2), 211–232. https://doi.org/10.1093/em/cah067
Billeter, B. (2007). Wie hat Bach seine Cembali gestimmt? Schweizer Musikzeitung, 10(4), 13–14.
Clerc, P.-A. (2007). J. S. Bach et les tierces justes: la solution proposée par Émile Jobin. Schweizer Musikzeitung / Revue Musicale Suisse, 10(7/8), 26–27.