Mathematical ET vs. actual practical piano tuning

Edited: October 3, 2021, 10:20 AM · Someone on this forum recently asserted that pianos are not tuned to pure ET - they are biased in favour of the more common keys, and if a big concert were to be performed in, say Db, then a tuner would come and bias the piano in favour of that key. Indeed my old head of music claimed that for him a piece in Db resonated differently from a piece in A, and I assume that these biased intervals were what he was hearing.

If all this is true, it raises the question, does this system of biasing have a name? It strikes me that it could perhaps be related to Bach's well-tempered system (which may have been Werckmeister III), but a little bit tighter?

Replies (39)

October 3, 2021, 10:25 AM · This is the first I've heard of this and I've had a lot of nerdy conversations with my piano tradesman who has probably 50 years of experience. I suppose it is *possible* to bias a piano, but I don't know why anyone would. "Pieces in A" don't remain in A very long unless it's Elizabethan music. I can't imagine anyone wanting the middle part of a Beethoven sonata to sound wretched just so that the first page of the exposition sounds a tiny bit brighter. Take the Waldstein for example. It starts out in C major but by the middle of the second page you're very solidly in E major.

Having said that, my daughter thinks our piano sounds particularly good in F major.

October 3, 2021, 11:40 AM · Actually the OP is right, its common to bias piano temperament slightly towards the common keys, and piano tuners for big concerts will bias a piano tuning to the piece that is about to be played, I don't know where Paul gets his non information, but its not from experts.
October 3, 2021, 12:10 PM · I played with a synthesizer before and the synthesizer tune to my violin.
October 3, 2021, 12:44 PM · Again, I'm not sure where Lyndon is getting HIS information regarding piano tuning.

There might be a few oddball tuners who do this, but it is definitely NOT mainstream.

It also doesn't really make that much sense. For example suppose a pianist has a solo recital with several pieces in different keys (no one is going to program an all-g minor concert, for example). Let's say Scarlatti in G-minor, Haydn in D, Shostakovich in, well, whatever. What is the technician supposed to do--re-tune the piano before every work? Are you kidding? And while key coloration might have been on the mind of some composers, it wasn't uniform with respect to stylistic period. You can't say that Bach and Rachmaninoff had the same ideas on how the different key relationships within a movement should have been perceived.

Even for a concerto, it doesn't make sense. A typical major concerto will have three movements in at least two different home keys. Tuners aim for slightly wide 4ths, slightly (almost imperceptibly) narrow 5ths, good double and triple octaves, and gradually increasing beat rates for major thirds and major sixths.

The typical audience member does NOT want to hear different "flavors" of keys, in spite of being told that they "should" by some people with strong opinions about past practice. They want to hear consistency, because inconsistency is quickly perceived as being "out of tune."

By far the most important thing in tuning, especially a concert piano, is unison stability.

But what do I know? I'm just a Registered Piano Technician.

Edited: October 3, 2021, 1:44 PM · There is a rather interesting book "My Life With the Great Pianists" by Franz Mohr, who was a Steinway piano technician to the highest level of their performing artists.

I recall it for two things, first he wrote of his adventures traveling with Arthur Rubinstein's piano for his world-wide concert tours. Also, Mohr was a devout Christian, which he did not fail to emphasize in the book. The book was recommended to me by a violinist friend who is also a devout Christian. In fact he may have loaned it to me (however, my Amazon records show that I did purchase it in 2002.)

The great cellist, Janos Starker included an amusing piano-tuning incident in his autobiographical book "The World of Music According to Starker." It is part of Chapter 9 about his African Adventures.

Starker and his pianist, Gunther Ludwig on a concert tour of Africa were scheduled for two concerts in Angola, in the capital, Luanda (on the north coast) and the southern coastal city of Mocamedes. (I had not realized that Angola is a large country, twice the size of Texas - and flight time between these cities was several hours.) When they arrived at the first concert venue it only required Gunther to hit two notes to realize the piano was horribly out of tune. Those in charge said the tuner had just left but they called him back and he worked at the piano. When Starker and arrived for their c0ncert that night, they found the piano even more out of tune than earlier. Nevertheless, the gave their concert and no one in the audience noticed anything wrong. The next day, after their flight they checked the piano in Mocamedes and found it to be even worse than the one in Luanda. They learned that hosts had flown the same tuner down from Luanda. So they just just sort of grinned and bore it.

October 3, 2021, 1:12 PM · Stretched tuning is used in tuning pianos. Octaves are not pure, but are made slightly wide. Thus they are not purely in equal temperament in a mathematical sense.
October 3, 2021, 1:17 PM · I have heard this, but it seems like more work than it's worth. A concert will be played in so many keys as to make it pointless anyway. Maybe if the pianist was going onstage to play solely Bach chorales in one key...

The vast majority of pianos probably approximate ET, with some tweaking on the basses and high end.

October 3, 2021, 1:40 PM · I'm no expert (like Scott is) but it seems to me that octave-stretching has little to do with temperament.
October 3, 2021, 1:57 PM · octave stretching does effect temperament. Imagine that the octave c4 through b4 were tuned in equal temperament. C5 was then stretched a bit higher. D 5 higher as well. We then examine the octave d4 through d 5. We would find that c5 would not be equal tempered with respect to the lower notes.
October 3, 2021, 2:00 PM · I am referring to how tuners used to tune by ear, I guess many now don't tune by ear but rely on electronic tuners tuned to ET, this is a shame, as to the Beethoven reference, obviously Beethoven wasn't writing for ET but rather a tempered tuning that emphasized C major to the detriment of C sharp major etc, Beethoven temperament would have been much more tempered than Piano tuning I was referring to And Bach's temperament even more pronounced.

As to Scott, I wouldn't consider someone an expert on tuning that has to rely on electronic devises to tune.

October 3, 2021, 2:22 PM · "As to Scott, I wouldn't consider someone an expert on tuning that has to rely on electronic devises to tune."

Lyndon, in order to become an RPT, you have to demonstrate your aural tuning ability in a four-hour test.
While it's true that many tuners don't understand the theory and method of aural tuning, you can't accuse me of that. So, as I've said in other threads, and to which you will likely reply with an epithet, go get your RPT, and attend countless classes on tuning at PTG conventions, tune a thousand pianos, and then tell me you're an expert.

So please don't discount my understand and ability to tune a piano. Stick to what you know, which is selling old violins and insulting people on the forums who contradict you.

On the subject of stretch: octaves on the piano tend to be stretched for different reasons, and they have little to do with ET. On the low end, they are stretched due to varying amounts amounts of inharmonicity.
Large pianos are stretched more than small ones because those longer strings are capable of generating higher partials, and those trend sharper and sharper. Small pianos tend to be less stretched because the shorter bass strings are less capable of producing those too-sharp high partials.

The high end is a little different: The human ear tends to want to hear those high octaves to be overly wide.
If you listen to electronically generated tones that are mathematically octaves (double the frequency), you will likely judge them too narrow. For a great real-world example of this, listen to the opening theme of
Schitt's Creek. Right at the very end, there's an octave jump--I think it's a triangle or something? It's obviously an octave leap, but the upper note seems flat. It's not--it's a theoretically perfect octave leap up.

Whatever the stretch, we aim for the same things: wide 4ths, slightlier narrow 5ths, octave agreement among single, double, and triple octaves (which is always a compromise), consistently increasing major interval beat rates, perfect unisons.

Edited: October 3, 2021, 3:14 PM · You're actually way off, I'm a keyboard builder and know full well how to tune a harpsichord by ear as well as tune ET by ear if I have to. i admit what I was saying about Piano tuning comes from discussions with piano tuners in the 80s, I guess you modern guys with your electronic gizmos do things different.
October 3, 2021, 3:16 PM · This is an example of a clavichord I tuned by ear to historical temperament of the type Bach used

https://www.youtube.com/watch?v=4gLClRJP0MI

Edited: October 3, 2021, 3:44 PM · Scott, can a piano have internal resonances, from the soundboard and other non-string parts, that could "favour" particular keys such as C or G, but perhaps not so much keys such as F# or B minor (for example)?

John Lill, the concert pianist, once related in a radio programme how he was on tour in the Far East and was due to play Beethoven's 4th at his next venue, in a concert hall he didn't know, and with an orchestra he had never worked with, but he did know the conductor. Travel problems meant he wasn't able to get to the concert hall in time for the usual run-through rehearsal, and in fact didn't arrive until the concert was about to start, with no time to warm up. More importantly, he didn't have an opportunity to try out the piano in the warm humid climate.

As you know, Beethoven's 4th has what must be one of the most difficult openings for any piano concerto - unaccompanied quiet chords each of which must be perfectly balanced both within itself and in relation to the other chords. And you pray that the humidity and heat aren't going to mess with the action of a piano you've never played on before - the tuning in these circumstances would be a relatively minor consideration in comparison. As it happened, the action was acceptable, as was the tuning, but it was a potentially nightmarish situation to be in.

October 3, 2021, 4:38 PM · I agree Scott that stretch tuning has little to do with equal temperament. I do want to make clear though that if octaves are not of equal size, then you cannot tune in equal temperament. It can be close to equal temperament, but it will not truly be equal temperament from a mathematical perspective.
October 3, 2021, 5:36 PM · I should admit that my accounts of slightly tempered, not ET tuning are from older piano tuners in the 80s, I guess things may have changed for the worse, given how horrid ET actually is.
Edited: October 3, 2021, 6:14 PM · My piano tuner is an old-time tradesman, he's a gentleman in his 70s who has been doing this all his adult life. He tunes pianos at both of the nearby universities. About two years ago, I asked him to tune my piano and I saw him using an app on his phone, and I had never before seen him using a device other than something to establish A=440.

I immediately recalled a conversation that I had with my dad's piano tuner, ca. 1980, a man named Frank Caliguri, who told me that I should never ever hire a piano tuner who uses an electronic device. Well, that was 1980, and the "electronic device" of the day was the Conn Strobotuner. So even though tuner technology has advanced in the intervening 40 years, I admit that I was somewhat apprehensive seeing my technician using his phone, partly because I remember the Conn Strobotuner as having a pretty good reputation for accuracy. So I worked up the nerve to ask my current technician about it, and his reply was that he never thought he would ever use an electronic device either, but that the newest phone apps are actually more accurate than trying to estimate, for example, the beat frequency in a major third in the middle of the piano, which I believe is a necessary skill in setting equal temperament aurally.

So, my question for Scott is whether my technician is right or whether I should be concerned. I must admit that after he's finished, the piano sounds just fine to me, just like before. I can't tell a difference. For all I know the app might be able to handle octave stretching, anharmonicities --- the whole enchilada. All that remains is knowing how to pull the hammer, which I'm not about to try to teach myself, and I certainly have no inclination toward DIY any of my other maintenance tasks. I do fill the humidistat tank. :)

October 3, 2021, 6:53 PM · I once read a book whose title, I believe, is simply "Piano". It was a woman's journey to discover why the grand piano that was delivered to her home in Montana during the winter did not sound anything like the piano she had fallen in love with at the piano store in NYC. Eventually she tracked down the person who had actually tuned the piano and he said, "Oh, that was my Shubert tuning." Based on the comments above, that sounds more like a sales gimmick than something a real performer would want their piano tuned as, but it does illustrate a tuning that compliments certain Western music.
October 3, 2021, 7:44 PM · Paul,
For one thing, the software has come a long way since the 80's. Just because it's an app doesn't mean it's not extremely accurate. The top two, Cybertuner and Veritune, cost, respectively, $1000 and $800.

One of the reasons so many use them is that tuning a piano by ear is one thing, but doing a large raise in pitch takes much more time and guesswork. Almost every piano I come to needs a pitch raise, and the software makes this much more easy. I don't want to be in the customer's home for hour after hour. Even most of the big-shot instructors in the PTG use electronic tuning devices now.

Yes, these apps do calculate inharmonicity and stretch, and are analyzing and correcting as you go.

Tuners vary in how diligent they are. Just having an app, even if it theoretically gives perfect tunings, doesn't make one a good tuner. You can still be sloppy, and it takes quite a while to figure out how to set a stable tuning. Most tuners don't rely on apps, but verify the tuning by ear. Doing it your whole life isn't necessarily a qualification either--many of the old-timers got their certification by doing an "good old boy" type of test, instead of the relatively stringent, objective tests given these days. Someone who is 80 might have been doing marginal tunings their entire life. And many tuners start cutting corners to fit more tuning in in a day. One tuner in my area is infamous for not tuning the high or low octave if he thinks he can get away with it.

The top tuning apps can easily give you historical tunings--the fact that they are software is neither here nor there. My Veritune, for example, comes pre-loaded with no less than 46 well temperaments, and numerous other types including Just, Pythagorean, different meantone types, quasi-equal etc.

Here's the issue: the typical tuner NEVER gets asked for any of these. If they did, I'd load it up and tune, no problem. The type of musicians and audience that care about these esoteric tunings is a very tiny niche.

The bottom line: what do most pianists and audience members refer to when they think the piano is "out of tune"?

It's the unisons.

Edited: October 3, 2021, 8:21 PM · Scott, thanks for that informative reply, it's consistent with what my technician said. All the pros in the area use him for their home pianos. And I agree with you about the unisons, that you'd hear those first, that seems kind of obvious to me. When Emanuel Ax was here for a solo recital my piano tech appeared on stage during the intermission with a few of his tools. He played a couple of intervals in the middle of the piano, but he didn't change anything. Afterward I asked him what was up and he said that Ax had asked him to check a couple of notes but he thought they were fine.

I would love to get a (less expensive) tuning app to do my dad's harpsichord over the holidays. Are there any really cheap ones you can recommend? I don't need a sparkling tune, and it's a simple instrument, there are no unisons to contend with and not very much range at that. But I bet my dad would be tickled to be able to try a few of his favorite tunes from the "Fitzwilliam Virginals" book. And there is no way in hell that I'm ever going to set any kind of temperament by ear alone. Never.

October 3, 2021, 9:12 PM · The husband of the last pianist I played with (weekly for about 5 years) used a computer app to tune her piano and he did an excellent job of it. This app accounted for variables such as the length of the strings (size of the piano) and other physical characteristics I now forget.

At home we have always used a professional tuner, but the husband's result was just as good.

Edited: October 4, 2021, 5:42 AM · Normally I don't like it when people ask a question and then never come back in again, but it has its advantages - I find that whenever I respond to someone's comments it results in thread drift that doesn't help me.

This one is interesting to read. I gather that the basic answer is that we are as close to ET as we can get.

I ask because I have books on Baroque, Classical and Romantic performance, and mention of temperament stops dead in the baroque book, and I wanted to know how Bach's temperament (and also the concert pitch question) resolved itself during the Classical and Romantic periods. Nada. Mozart and Beethoven were amateur/private organists. So they used organs with baroque temperaments? I wondered if such organs still exist, but then I decided if the pipes have pistons for precise tuning, we can't know much about them.

There's a theory that the WTK (das wohltemperierte Klavier) was not written to equate all the keys but, the opposite, to exploit the different resonances caused by his temperament. That might aid us in deciding which temperament he used. Equal temperament had been known about since the 16th century, but was not liked and not used.

October 4, 2021, 8:42 AM · Bach's student Kirnberger comes the closest to a Bach WTK tuning with his Kirnberger 3 temperament, a lot of tuning devices have it programmed in. by Mozart and Beethoven, a Valotti type temperament would have been more likely
October 4, 2021, 9:11 AM · Horowitz was reported to have had his favorite piano tech to tune the instrument to a slightly sharper pitch to make the piano stand out above the orchestra. I think Paganini may have did something like that on the violin for the same effect.
October 4, 2021, 9:20 AM · "Horowitz was reported to have had his favorite piano tech to tune the instrument to a slightly sharper pitch to make the piano stand out above the orchestra."

Yuk. Pavarotti's perfect pitch made him sing at the same pitch no matter how flat the orchestra had drifted in the heat, and it sounded dreadful at times!

Edited: October 4, 2021, 9:38 AM · Are we singing now? Sorry for the contribution.
Edited: October 4, 2021, 9:46 AM · To my recollection, Ruggiero Ricci, in an interview for "The Way They Play", said that in the greenroom he would tune his violin a shade sharp of the orchestra's pitch out there on stage so as to get more projection.

It looks like it's a trick that's been around for a while.

Here's another but quite different trick. It is not unknown in string quartets for the cello and viola to tune their C-strings a shade sharp on perfect fifths tuning. This is so that the resonance of the E partials from those strings will match the pitch of the open E's of the violins, helping the overall resonance of the ensemble.

Edited: October 4, 2021, 10:33 AM · miscl. comments;
Playing slightly sharp for projection is a common trick, sometimes done unconsciously. Orchestra section players should do the opposite. If you can't hear yourself during the loud tutti parts you are probably in tune.
E.T. has equal half-steps on the pitch scale, Not the frequency scale. The perceived pitch difference is proportional to the 12th root of 2. That is an irrational number that does not exactly match any Pythagorean fraction.
The Helmholtz "cents" system divides the octave into 1200 little pieces. The E.T. half-step is 100 cents. The Pythagorean half-step is smaller than that. The Just half-step is wider:
Depending on the key, Cellos and Violas can tune the C string a little sharp, because the C tuned by perfect fifths will be 6 cents flatter than the perfect fifth C.
Edited: October 4, 2021, 10:49 AM · Kurt Sassmannshaus's general advice to his students is "Tune to the highest A that you hear in the orchestra."
October 4, 2021, 11:17 AM · Joel, I think your last paragraph is great advice. On my D'Addario micro tuners those 6 cents seem to be about "2 ticks" to the right if I want Pythagorean tuning - at least that is what my ears tell me.

I frequently played with pianists for over 50 years and when playing viola or cello I would always tune my C string to a piano C. Which one? Well, given the chance I would tune the C to a piano C I would likely have to match with my open string in whatever music we were going to play.

My ears could not easily hear the discrepancies of the D and E strings, however, there were times when I would check my G against the piano.

For other than the open lowest string we string players can make our own intonation on the fly - and even beyond that, vibrato and the way ears interpret it cover many sins.

October 4, 2021, 4:54 PM · and,-- I have read that when working with a pianist, Primrose would tune to the piano D, then perfect fifths for the other strings.
October 5, 2021, 1:28 PM · When I play the viola in orchestra I often find that my open G and C strings sound flat. I don't know why that is. Maybe I am too accustomed to equal temperament or something.
October 5, 2021, 8:26 PM · Reminds me of a viola joke:
Q: How many violists do you need to change a light bulb?
A: One to change it, all the rest to argue how Primrose would have done it.
Edited: October 6, 2021, 12:23 AM · @Paul - me too. From the violin section you can't help noticing how the violas and cellos often make a grab for the C peg when they hear their open strings. Violins too with their open G although to a lesser degree, and it's because we all tune in perfect fifths down from the A string. It really beats me why most conductors don't seem to notice. The "close fifths" tuning that Trevor mentions isn't just for string quartets.
October 6, 2021, 1:05 AM · For me this video from Augustin Hadelich about intonation was very clarifying. This is video 1:
https://youtu.be/mkHSzmeE18Y

There is also a part 2 and 3 for who is interested

October 6, 2021, 9:19 AM · Yeah the Part 2 is especially relevant but the whole series is good.

Part 2: Intonation with piano, or other strings such as cellos
https://www.youtube.com/watch?v=WGH3PZgbapE

Part 3: Thirds and sixths
https://www.youtube.com/watch?v=Rp3OuEJAD0s

October 8, 2021, 4:56 AM · Now I've started playing in orchestras again, my open D and G both sound out of tune. They aren't, but they definitely sound flat... don't know whether this is due to a complex point of temperaments, harmony, or just the rest of the orchestra being a bit out of tune...
October 8, 2021, 5:45 AM · In a less than top orchestra it's the winds that have the most problems..and they are loud! Each wind instrument has a different structure, and requires continual adjustments in the embouchure.
And they all go sharp as the air column warms up..

And for us, pure fifths are a fraction larger than ET fifths.
I adopted a Wittner tailpiece with integrated tuners to do fine adjustments nearly "on the fly".

October 11, 2021, 1:06 PM · @Paul Deck, I noticed this too, many years ago. I read somewhere - I think it was in the book of interviews of the Guarneri Quartet, that Michael Tree talked about tuning the G "slightly tight" to the D, and then the C likewise to the G. Also Joel Quivley's comment about Primrose - I read that in the book "Playing the Viola". From there, I've taken to tuning the D perfectly to a tuner, then tuning the G ever so slightly sharp - maybe a 1 to 1.5 cents, and the C about 2 cents sharp. For the A to sound a perfect fifth above the perfectly tuned D, it ends up about 1.5 cents sharp. All this is with a tuning app set to ET. This method has served me well in quartets and orchestras. I don't play much with pianists - I would imagine having to adjust more in that case.


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