I don't remember where I learned this, but what you said is more accurate than what I said -- the concept was definitely known well before Beethoven, but my understanding is it wasn't the standard tuning on keyboard instruments until much later, and came to its fruition with all the atonal music of the early 20th century. I think composers even went so far as to assign emotions/moods to various keys based on each key's sound. E.g. "E-flat major is austere, D-minor is sad," etc.
More generally, I'd be curious to know how they'd practically tune a keyboard to 12TET before the electronic chromatic tuner got around. Start with Pythagorean fifths then compress ever-so slightly? How'd you keep them… equal?
Oo I can take this one! I'm not a registered piano technician, but I've tuned my piano (and helped a few friends) for some 15 years. Aurally (as opposed to electronically) tuning a piano is actually pretty straightforward. I'll stop short of saying it's easy, but once you learn the method, it's very sensible and just a matter of practice.
The general idea is to achieve consistent beat rates for a given interval -- major thirds being the most useful for its relatively high beat rate compared to other equally tempered intervals -- as you play it chromatically. By that I mean play A and C#, then Bb and D, then B and D#, etc. and if the beat rate hardly changes (but does consistently climb) then you've achieved equal temperament.
Say you've got A tuned to a reference (tuning fork). Then set the A above that so there's no beating, since octaves are always perfectly 2:1 regardless of temperament (until the extremes, when you need to stretch a bit, but I digress). Then tune the C# and then tune the F. Basically it's an augmented triad, or a stack of three major thirds (including from F back up to A). Get all of them to beat by about the same amount, but the higher ones just slightly faster than the lower ones. The fact that this feat utilizes two A's is the key to pulling it off. You frame out the octave and then fill in the augmented triad.
But now you need to do the next set: Bb, D, F#, Bb. How to get here without another external reference? Well, well, well. We have our ways. The perfect fourth between A and D is an option, but be careful not to make it actually a perfect integer (no beating), as that wouldn't be equal tempered; it should beat maybe about half as fast as the nearby major thirds, IIRC.
> In equal temperament, all perfect fifths are “contracted”, while all perfect fourths as “expanded”.
Minor thirds are contracted, while major sixths are expanded. Major thirds are expanded, while minor
sixths are contracted.
The piano tech must have knowledge of the approximate beat rates the intervals of equal temperament in
the temperament octave:
The beat rate of perfect fourths within the temperament octave may be about 1 beat per second.
The beat rate of perfect fifths within the temperament octave may be about 1/2 beat per second.
The beat rate of F3-A3 major third is about 7 beats per second and that of higher thirds are faster.
In my previous comment, my memory was a bit off when I said "about half the beat rate"... that's the difference in rate between fourths and fifths, apparently.
> Example: to check the tuning of D4 within the temperament octave,
play A3-D4 and G3-D4. The fourth should beat faster than the fifth. If the fifth is too fast and the fourth too pure
perhaps the D4 is flat; if the fourth and fifth beat at the same rate, perhaps the D4 is flat; if the fourth beats too
fast and the fifth is too pure, perhaps the D4 is sharp.
> you
can use more and more checks as one progresses through the sequence and tune each new note as a “best
compromise” with all the previous notes, that is, each new note will not depend only on the last note tuned, so
there will be more of a chance that errors will not accumulate in the later notes tuned.
I still feel that way about certain keys, even playing on an exactly equal tempered keyboard. I don't think the degree to which certain intervals might vary between keys is necessarily the important factor.
Those descriptors ("austere," etc.) have always struck me as subjective -- I'm not one to tell people what mood they're getting from certain keys. But a root major chord will have a much different feel in, e.g., C#-major on an 18th-century tuning than in equal temperament.
I have this CD [1] in a box somewhere but can't find it on Youtube. It's a few Beethoven sonatas in the temperament he would've used. Just sounded out-of-tune to me in certain parts (especially during the Waldstein), but I don't have perfect pitch. The booklet that came with that CD is really helpful in understanding all this, and I think that might be where I learned about that Owen Jorgensen tome.
There's no shortage of similar experiments on Youtube. This one [2] has a wild one in just intonation, but I doubt that temperament was still used when Mozart was composing.
Thanks for that link, I don't know if it demonstrates "just intonation" though? But the 1/4 Comma Meantone tuning just sounds horrible the moment a diminished chord comes into the picture.
You're right, I mis-typed -- it's meantone, not just intonation. Someone elsewhere on this HN thread argues that's the tuning Mozart himself would've been using for that piece, which is bizarre to think about once you hear it on Youtube like that!
This is truly a fascinating world. People who argue that we should be using the original temperaments that composers used do have an argument. Imagine, for example, if Shakespeare were "tuned" to be in modern UK English rather than the English of its time.