Progress in Tuning?
Reflections on Tuning
2011-08-03 — Original posting (on Blogger)
2014-10-28 — Re-posting as is (WordPress)
2016-06-20 — Brushed up for better readability
- Introduction to the Topic of Tuning
- The Fascination of Tuning
- Inherent Limitations
- Dealing with the Limitations
- Early Tunings
- An Explosion of Tuning Schemes
- What’s the Effect, in the Extreme?
- Post-Meantone Tuning Schemes
- Bach and Tuning
- Switching Between Tunings?
- Basic Steps
- Criticizing Tuning…
Introduction to the Topic of Tuning
My original intent was to write a blog post about how we tune our harpsichord (i.e., both what type of tuning we aim for, as well as how we achieve that technically). However, then I realized that I can’t discuss the technicalities of tuning without first defining what I mean with the various terms. People who know how to tune a harpsichord should stop reading here. This note will sound like a bag of trivialities to them!
I’m writing this text for people who have no idea about the tuning of keyboard instruments. Also, I’m just talking about our own experience with this topic. A comprehensive discussion would be way beyond the scope of this note. This can be found on Wikipedia or in specialized literature (see also below). I’m not trying to cover this subject scientifically. I also don’t try convincing anybody to tune his/her keyboard instrument in any particular way: that’s a musician’s personal decision!
The Fascination of Tuning
This is a fascinating topic, even though listening to a tuning professional working on your piano might sound like a boring, uninteresting task. In the end one nevertheless expects to have a “perfectly tuned instrument”! My interest in tuning stems from various areas / personal experience:
- There is no violin playing (even for beginners like myself) without being able to tune the instrument. There, you learn to listen to intervals (fifths in this case). And you learn to alter the interval until there are no residual interferences. That’s tuning to perfect, “pure” intervals. It means that even after I stopped playing the violin I was still familiar with assessing the purity of a tuning interval (unison, octaves, fifths, fourths).
- A real eye-opener (OK, ear-opener) for me was the encounter with pre-baroque music (e.g., by William Byrd and Orlando Gibbons) on a Virginal with Meantone temperament tuning. After having heard the very same music played on a modern piano, this felt like listening to completely different music. I could hardly believe that the two versions were one and the same composition! To me, that became what all this is about!
- Finally, I met Lea, and she owned a harpsichord. This happened to be at a time when I was playing with the thought of purchasing a kit for building my own instrument. I’m glad that I stayed away from this idea! I’m still grateful that when I asked her whether I could try tuning the instrument, she had the trust and the audacity to let me give it a try! OK, initially she would tune the first octave, and I would do the rest. This did cost us a couple broken strings. But after a while, I tuned the instrument quite frequently.
But before exploring my / our tuning methods, let me discuss some basics. Tuning a keyboard instrument is subject to a number of mis-perceptions, such as
- At its core, tuning is not about creating “pure” intervals (largely, at least).
- There is no such thing as “pure”, “clean” tuning — all tuning schemes have “impurities” one way or another.
- With a piano tuned by standard methods (especially on small instruments) not even an octave may be a “clean” interval.
The gist of the problem is that “pure” intervals work with simple frequency ratios, such as 2:1 (octave), 3:2 (fifth), 4:3 (fourth), 5:4 (major third), and so on. However, as we’ll see below, this system is inherently inconsistent.
If you add pure fifths to a given tone and walk through all chromatic tones, you will never return to a pure octave of the starting note. The conductor of a chorus in which I once participated once reported that years ago he once tuned a keyboard instrument (probably a harpsichord) by adjusting to pure fifths, i.e., A – E – B – F# – C# … up to G – D, and then he naturally (or naïvely) expected to find a pure fifth between D and A. For him, a world collapsed when he realized that there was a (relatively) huge difference between what he heard and what he expected. In frequency numbers, the error may look small, but for the ear, that residual fifth is very much off.
Short numeric excursion: a pure fifth is defined as two tones with a frequency ratio of 3:2. 12 (pure) fifths “stacked on top of each other”, i.e., going through all the tones when tuning as shown above, yields (3/2)^12, or 129.746… — but if you start at the lowest A on your keyboard and then go up by 12 fifths, you reach an A seven octaves above the starting tone, and seven octaves is a frequency ratio of 2^7, or 128.000; this error of 129.746…/128.000 or 1.013643… is called the “Pythagorean comma“. This is almost a quarter of a half-tone interval (24/100, or 24 Cent), i.e., it is a very audible error!
Dealing with the Limitations
How does one deal with this inherent error in tuning? At least since the late classic period (the late 18th century), the method of choice is to tune all fifths too small by 1/12th of the Pythagorean comma. This is called “equal temperament tuning”. It can be viewed as “cleaning” an apartment by spreading the dirt (errors, imperfections) evenly on all surfaces. There will be no accumulation of dust standing out anywhere, but on the other hand nothing will be really clean. This method has the big advantage that one can play all keys equally well, i.e., they will all sound equal in character and equally (im)pure. From a purity / sound quality point-of-view it doesn’t matter whether a piece is played in C major or in C# major.
Note that ironically, with equal temperament tuning, while all fifths are a little too small, all thirds are too large and actually substantially more impure than the fifths!
Prior to the advent of electronic aids, equal temperament tuning was anything but trivial (see an upcoming note). Today, one would simply tune the first octave by adjusting each half-tone in succession to “0” on the tuning meter — this completely eliminates the accumulation of errors as one progresses from one tone to the next in tuning by ear—see the note on practical aspects of tuning.
In pre-baroque times there was no such thing as a notion of being able to play in all keys of the chromatic circle. Equal temperament tuning (and the methods to achieve it) were not been invented yet. Instead, people used special recipes for tuning, e.g., for tuning in Meantone temperament (mitteltönige Stimmung) one would start with setting the C, tune a pure third to E, then make that third a little too large, such that the error is just not audible. Then, starting again with C, you tune the fifths to G, D, and A such that the fifths C-G, G-D, D-A, and A-E all have the same error — and so on for the other notes.
Such recipes have been written down, and that’s why we have an idea about tonal qualities in medieval / pre-baroque times. Another source for such information is from some medieval organs (e.g., an organ in the monastery in Muri / Freiamt, CH, in Switzerland) where the original length of the pipes for some stops (flue pipes) has not been altered (or can still be reconstructed / measured), and so the original tuning is still available.
An Explosion of Tuning Schemes
Many tuning schemes / recipes existed in pre-baroque times. As the music evolved, people invented and used new schemes, the tuning of keyboard instruments evolved along with the music. One early scheme for organs was written down by Arnolt Schlick in 1511, and Andreas Werckmeister (1645 – 1706) described a whole series of different tuning schemes. The organ tunings of the famous Silbermann family of organ makers have been reconstructed at least in approximation.
In baroque times, composers started exploring all keys of the chromatic circle, and from the classic period onward one should assume that keyboard instruments use(d) equal temperament tuning (as in the modern piano).
What’s the Effect, in the Extreme?
Does one hear the difference between various types of tuning? Oh yes — see also the example above!
- When you hear music played on keyboard instruments with Meantone tuning you should immediately note the difference in tonal quality. Music of the 16th century is based upon / exhibits those beautiful, (almost) pure major thirds in keys close to C major. However, it is impossible to tune all major thirds as pure intervals — some keys are clearly impossible to play. Fifths are of lesser (relative) importance in that period — most of them are actually even smaller than with equal temperament tuning. The exception: one particular fifth (G# – D#), called “wolf fifth” (Wolfsquinte), is way too big and really offends the ear. Another prominent feature in Meantone tuning is that the major seventh is smaller than with equal temperament tuning, hence does not sound like a lead tone.
Post-Meantone Tuning Schemes
This is the extreme example, but it’s also interesting to follow the evolution:
- Arnolt Schlick‘s organ tuning is similar to Meantone tuning, though it moderates the “wolf fifth” and shifts it to C# – G#
- The organ tuning used by the Silbermann family is conservative and retains an almost full “wolf fifth” G# – D#. When Bach played such organs, he jokingly selected keys where the wolf fifth was most audible, just to pull the organ maker’s leg!
- Andreas Werckmeister described several schemes. Here (essentially for the first time), most keys are playable. At the same time one will easily note that the all sound different! F major and C major are especially pure (the former features a pure third and a pure fifth), the other keys vary, with H major being the worst.
- After Werckmeister, there have been numerous proposals for “moderate” schemes (Kirnberger, Vallotti, etc.) that make tuning more compatible with more recent music (i.e., the use of “exotic” keys), especially for organs. They avoid “bad spots” (keys), at the expense of some of the individuality and variation that the Werckmeister tuning schemes offers.
- Equal temperament tuning, finally: here you should not hear any difference in (im)purity between tonalities. However, you may miss the colors of baroque tuning, and the pure, clean chords of pre-baroque tuning schemes.
Bach and Tuning
An interesting question is how Bach was tuning his instruments — I’ll return to this in a separate blog entry. The key point here is that there is no single, ideal method:
- If one plays 16th century music with equal temperament tuning, one loses essential aspects of the tonal quality of that music.
- Meantone tuning is preferable for 16th and early 17th century music, but hardly usable for baroque and later music.
- Baroque tuning schemes (such as the Werckmeister schemes) add essential (and intended) sound characteristics (“colors”) to baroque music. However, they are hardly adequate for pre-baroque music, let alone music of the Vienna classic or later periods.
- One can view some of the “moderate” (late) baroque tunings as viable compromise (especially for organs)—but they are hardly satisfactory for either 16th century or classic and later music.
- Equal temperament tuning may be prerequisite / mandatory for late classic, romantic and later music, but loses / suppresses essential characteristics of baroque and earlier music.
Switching Between Tunings?
Unfortunately, re-tuning to switch between temperaments is time-consuming by itself. On top of that, the changes are large enough to make it necessary to tune the instrument twice—e.g., after switching from Meantone to Werckmeister tuning. One possibly even needs to insert a resting period of a couple of hours. In practice, one tries avoiding a switch between temperaments.
Here’s an interesting thought. A key element of baroque music is the basso continuo, where the right hand plays chords over a single bass line in the left hand. With this, the chords fall into a pitch range in which the ear is most sensitive to tonal purity—hence this range is where one starts tuning, see my upcoming blog entry on technical aspects of tuning. The classic period saw a shift towards playing chords (either plain or broken, e.g., as “Alberti bass”) in the left hand. This served as accompaniment to melodies in the right hand. It means that in the classic and later music chords typically (or at least partially) fall into lower pitch. There, interferences from impure chords are much slower and less audible. Maybe this facilitated the spreading of equal temperament tuning?
As this note is getting long, I also decided to create a separate note explaining (my view & experience on) practical aspects of (harpsichord) tuning. Most of the information above I either collected from Wikipedia, or extracted from Bernhard Billeter, “Anweisung zum Stimmen von Tasteninstrumenten in verschiedenen Temperaturen“, Merseburger Verlag, Berlin/Kassel, 1979 & 1989 (ISBN 978-3875371604).
All of the above is about how to tune / set up the keys in the first octave in a keyboard instrument, typically a .. a’. That’s the part of the tuning where one deals with the inherent impurities of the tonal system. Once the tuning in that first octave is complete, one tunes the remaining keys as pure octaves (or octaves from octaves, etc.), based on the initial / starting set. On instruments with multiple stops (organs, most harpsichords) one would then progress to tuning the other stops in unison with the base stop. This last step is definitely the easiest one.
On pianos, there is only one stop, but most keys actually use two or three strings. Here again, the last (and easiest) step is to tune those extra strings in unison with the initial one. For adjusting the first / initial string, professionals mute the additional strings with a special “rubber comb”.
A last point: to me, an octave is an octave, and as such should be pure (a frequency ratio of 2:1), i.e., when playing an octave there should be no audible interference beats. Typically, we have our piano tuned once a year. We went through a number of tuning professionals. Over years, the results never really satisfied, especially in the bottom (left-most) part of the keyboard. None of the octaves was really clean / pure. At least once I called back the tuning professional, without real improvement. “No, this is correct, this is the way it is”, or “This is the way one must do it” were about the best explanations I received. It took us years until we found a piano tuner who could at least give us an explanation for this:
- Compared to a grand piano, small, vertical pianos have short bass strings. It is relatively difficult to produce a pure, full and clean sound from such short strings. This also means that tuning these strings is inherently tricky.
- Overall, these short strings often produce a dull, unclear sound. Piano tuners apparently learn to tune octaves a tad short. On purpose they tune “short octaves” in order to produce a somewhat brighter (clearer?) sound.
- Initially I didn’t pay attention to how they tune the bass notes. But at some point I realized that (at least for small instruments) piano tuners do not use octaves to adjust the bass notes but rather tenths (octave + third). This apparently makes it easier to judge the relative pitch. But maybe it causes the tuner to ignore the resulting impurity in octaves?
On the bright(er) side, I was able to convince our current piano tuner to try going for pure(r) rather than short octaves. In general I found the result pretty satisfactory. The rest is probably limitations of our instrument. You might say “how meticulous, overly demanding and unpleasant for the tuner!” — however, I’m sure I’m not alone!
Many people probably don’t pay attention, and so don’t notice such tuning deficiencies. However, I have at least one other reference, in my late father-in-law, a professional musician. He was flute teacher. He also played recorder, other wind instruments, violin. And he used to be chorus master. After the piano tuner had gone, he would regularly inspect all keys. He took a soft pencil and marked with an “X” all bass keys that he judged to be incorrect. Then — to the tuner’s dismay — he called the piano shop to have the tuning corrected. This was with a grand piano!
Bernhard Billeter. (2010). Anweisung zum Stimmen von Tasteninstrumenten in verschiedenen Temperaturen. Kassel.
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