Well I don't see any point in arguing about it. I gave you the math and you can do with it what you will. Stretch is a proven phenomenon whether you think your computer generated tone is close or not. The formula I gave you will produce the stretch you disbelieve. A few last comments and then I'm done.

I myself don't think 55 cents (quarter tone) sharp at C8 sounds good but I am regularly tuning between 25-35 cents higher than theoretical pitch and my customers like the results. Those customers include professional orchestra musicians (indeed I am/was a pro french horn player) so they are used to listening for intonation. My mentor who once tuned satisfactorily for Stravinsky and tuned aurally for 25+ years also approves of my tunings. I also believe you are adding stretch to your tunings because if you truly tuned dead octaves all the way up and down then the piano would sound like crap and you wouldn't have customers very long.

If you looked at the trial version of Tunelab them plug a mic into your laptop and and check your own aural tunings. If you loaded Tunelab Pocket you are already set up. Don't forget to calibrate your device. You might be enlightened or you might continue to disbelieve the reality of the measurements.

I always enjoy reading your posts BDB. Have a good day.

I've been tuning by ear for over 20 years. I've been fiddling with the free trial version of tunelab pocket and tried Bill's little test. Results turned out very close to his readings. I then measured one of my tunings on a 6' grand, and yes C8 ends up about 35 cents sharp. It did not surprise me at all, my mentor always told me that stretch is not something that you do. When your tuning is good, it is naturally streched. Thanks Bill and John for putting in words what I have been doing all along.

I have to applaud BDB's consistency:

"That would be absurd!"
"There is no way one would come up with a C8 that high.."
"The only thing I can assume is that the measurements are wrong.."
"This does not make sense.."
"The chart means nothing to me"
"I am not convinced that it could read either."
"My math was not wrong.."
"I doubt that.."


.... A long time ago...

I was at school on a bassoon scholarship, even though I started out as a physics major.

I had played in band and orchestra through high school and played in a "reading" summer band for pay. Senior year I decided to join the choir. (I'm sure there were girls involved in my choice!) The director soon found that I could sight-read and sing on pitch, even though my vocal skills were limited. During that time I bought my first Korg tuner, which I loaned to the oboe player to set pitch in the orchestra.

Being a supporting instrument, I worked hard at blending in, both pitch and tone. The bassoon in band gets used both with the brass and the woodwinds, so I had a lot of practice matching pitches to come up with the "best" fit.

When I went to college, I continued to sing in the choir, as well as play in the band. Even though I had chosen the school for science, the music department payed my way. After two years I was looking to switch schools to pursue a degree in bassoon performance.

Which brings me to Jordan. He came into the choir - a young singer with a big voice. It's just, that... he sang flat. Almost all the time. He was sure he was on pitch. We talked, we tried different approaches. He just couldn't hear it. Finally, I brought my Korg to choir. After class, I showed him how when I played the piano, the needle pointed close to zero. (How often did THAT piano get tuned?) I showed him how I could sing into it and get it to hover around zero. (try it sometime... it's hard!) He tried... He tried again.... He tried again.

"There MUST be something wrong with that thing. It can't be right."

It was a long season in the tenor section that year.


---coda---

I came back to visit a few years later. Jordan was still there. He came up to me and told me that I had been right, how could I stand singing next to him all that time? I still don't know what the breakthrough had been for him, but yes, he could now sing on pitch.


BDB, if you really want to understand this, the only thing I can suggest at this point is to further your education about overtones and inharmonicity. There are plenty of links to physics sites that have animated figures that may allow you to reach a breakthrough in understanding of the behavior of strings in the piano.

You know, none of you guys ever answer the interesting questions, like how you can tell the 16th partial from the 17th.
{523.3*8 = 4186.4 -16th partial
523.3*8.5 = 4448.05 -17th partial
(523.3*8)*2^(1/12) = 4435.3 -16th partial + 100 cents}
That question begs another, which is how these machines can distinguish the 16th partial, which should be very, very weak, if it exists at all, from noise.

The reason I ask these questions is because it seems unlikely that one could not, when listening carefully in a harmonic context, distinguish between a pure note and a pure note plus a quarter tone. I will admit that it is possible, if you have a tin ear, but not likely.

I will believe one of you understands inharmonicity when one of you does the actually calculations that are associated with inharmonicity. I will believe that your machines measure that when the results that they give are reasonably close to what the calculations. Otherwise, it is more likely that the machines are measuring noise.

Have any of you done any of the things I have done? Have you gotten hold of a tone generator? Have you compared that to your tuning? I have put my theories and yours to the test. Have you?

Tuning is not something that comes out of a box. Tuning existed lang before there were ever such boxes. The ability to distinguish intervals is something that seems to be built into our hearing.

I do have some conclusions:

1. Spending a lot of time worrying about the top octave is a waste of time. Most people do not play up there, and the tone is so short that fine points are lost. Still, one should develop one's ear for such things., in case you end up doing a lot of tuning for jazz artists, like I do.

2. Inharmonicity is vastly overemphasized as a factor in piano tuning, especially if you only are worried about it in the top octave.

3. Inharmonicity makes more of a difference around the break. It makes some pianos difficult to tune there. There are very good makes of piano that have problems there.

4. That sort of problem can be improved immensely with some easy rescaling, as in this comparison:

5. Having done this sort of rescaling, I can say that the results are that the piano is easier to tune, probably due to an improvement in the inharmonicity, but that most people would not hear much of a difference when they play it.

Incidentally, the story about the mentor tuning for Stravinsky brought up memories. Back when I was a young whippersnapper, I tuned for someone Stravinsky wrote a concerto for. He is gone now, but I still tune that same piano at the same venue, six times just this month. Actually, my experience with tuning for big-name musicians is that some of them care how well the piano is tuned, but most of them do not, and that those who do care have tolerances a lot looser than my own.

When you ask an ETD to listen for C8, that's ALL it's listening for. Does not matter which note you play. So if you play C4, the 16th partial IS C8. Therefore, it registers on the ETD. Any note that has C8 for a partial will register. What is so difficult to understand?

I am just taking a brief lunch break and really wish I had more time to answer everything right now. I also knew a guy who consistently sang loudly and 20 cents flat. He would drag down everybody else's pitch. He used to sing with the symphony chorus when it was a "come on, come all" type of group. Fortunately, things finally progressed to the point where you had to pass a real audition and he was excluded. I had tried to help him several times and he protested that he was a "leader" in the group and that no one else had ever told him he sang flat (although plenty of people did say it behind his back).

I only ever dabbled in scale design but smoothing out inharmonicity was never the target, smoothing out tension was. I did know of a PhD who was hell bent on reducing inharmonicity in the wound strings. He did not turn out to be a scale designer whose work anyone ever bought and used.

As far as I know, the pitch at C8 would correspond to the 12th partial of C4, not the 16th. Am I wrong? In any case, the calucluated programs do not go past the 8th partial in consideration as far as I know, so yes, whatever lies beyond that is trivia or is it?

I also demonstrated at the last PTG Convention how I tune the low bass, the first octave. In any but the largest pianos, you will hear a clear conflict between the 9th, 10th, 11th or 12 partials from the note in the first octave and the 5th, 6th, 7th or 8th partials of the note in the second octave when tuning octaves in the low bass. Take tuning F1 fro a previously tuned F2, for example when all higher notes have already been tuned (at least all higher notes in the mid range). On most any piano, you will hear a faint but pleasant "jingle" when the octave is at or about its optimum width.

My suggestion is to make that "jingle" which is in fact a rapid beat "match something that you have going in the midrange". That is how I put it because the exact partials which are clearly audible will vary from piano to piano but there will always be that faint, rapid beat which actually sounds pleasant and musical to the ear. Taking the example, F1-F2, listen for that faint, rapid beat and match it as best you can with the F3-A3 3rd, the C3-A3 6th, the F2-A3 10th, anything you can find related to the Key of F Major that you can make equal beating with that faint jingle and still leave a good sounding single octave. This will give a broad, F major chord a beautiful resonance because the rapidly beating intervals are all supporting each other.

Do this for as many strings in the first octave that you can hear that faint beating on and you will have a beautifully resonant piano in general and optimally tuned low bass. This conflict between the octaves will generally measure out to between 12 and 28 cents. As, Accordeur pointed out, the ETD can only identify the partial it is set on and nothing else. This phenomenon is certainly not random noise. From studying these low, wound strings, it appears to me that they have, in fact a less steep inharmonicity curve than the plain wire from the midrange does.

It's only a theory of course but I have long thought that even though virtually no one can single out the 12th partial of C4 just by listening, it does, in fact influence at least some tuners in how high they tune their 7th octave. I do not discount the 7th octave at all, it is as important as any other part of the piano when someone does play in that range and it resonates with whatever is tuned below it. Certainly, to tune a 7th octave as high as I have been talking about, you have to "work up to it". You can't tune 2:1 octaves from the midrange all the way through the 6th octave and then suddenly take off, that will not work. Curiously, however, the PTG "Master Tuning" does exactly the opposite of that: it is tuned with optimum stretch until the 7th octave and then the 7th octave is tuned at 2:1 to the 6th. Virtually no one would tune this way in the real world.

I always enjoy reading your posts Mr Bremmer. A lot of what you write, even though I did not know it, is what I have been doing. I agree that even tough we can't always single out the high partials, they do influence the end result. I am personally a jazz player and to dismiss the last octave to me is like blasphemy. I think most classical players would agree.

I don't believe BDB is actually a piano tech.

I believe that he is actually a pre-teen messing with us.

I know this because I have heard teenagers before. Spending a lot of time listening to teenagers is a waste of time, their attention span is so short, than any real point in the discussion is lost.

paraphrased from:
"Spending a lot of time worrying about the top octave is a waste of time. Most people do not play up there, and the tone is so short that fine points are lost"

I am so sure of this, that I'm convinced that there's nothing he can write to change my mind...


... just a little satire...

Here's what I hope will come across as the harmonic series in two different pictures.

Partials 1-12


Partials 8-20

I'll look some more, but here is a piano note, looks to be somewhere around 670hz. As you can see from the first 7 partials, even though the amplitude of each partial drops, the noise is mostly clustered in the lower range of this note. Even though they are weaker, they are easy to pick out in this picture.


Ah, here we go... G1 out to the 18th partial

Ah ha, so I suspected, I never knew for sure about what happened after the 8th partial but obviously, just about every note of the scale gets thrown in there at some point. This makes it easy for an ETD to latch on to the wrong one when trying to substantially change the pitch of a wound string. It really may not know which partial it is listening to in this instance, just like BDB hinted at but in a different context. I have known people to break wound strings because of this.

During the tuning I just did on a Walter Console, I heard that "jingle" when I tuned the C2 from the C3 and traced it to the note which corresponds to G5 (sorry, I have to learn the rest of the partial series before I can even tell you what partial that is) but it was quite audible. I measured tha conflict at 17.8 cents which gave it quite a "purr". A much more pleasant sounding conflict was heard within the same octave that corresponded to the note E4 and measured about 10 cents, just the way I tune that C4-E4 3rd. All in all, it was quite a nice sound.

Off to my last tuning and dinner. Maybe will write more tonight, maybe will have company, don't know yet.

Are you sure they didn't break strings because the machine was set to the wrong note?

Look at the second picture above. Can you find the partials? Can you count from left to right to find any specific partial? I assume that any of the modern software can find them as well. Once locked on, it can tell how flat or sharp from the calculation and track your progress as you manipulate the pin.

I would bet that anything from this century for tuning pianos would be able to be a bit more sophisticated than just listening at a certain Hz. level to find a partial...

BDB asked the question, "You know, none of you guys ever answer the interesting questions, like how you can tell the 16th partial from the 17th?" I have to admit, I don't count them and I don't pay attention to the chromatic ones, just octaves 5ths and 3rds. It may be true that anything beyond the 8th partial which would be 3 octaves above the fundamental is not very important for the midrange and above but as I mentioned here recently, the partials which lie in the 3rd octave above (8th partial and above) the fundamental for notes in the 1st octave are clearly audible and easily detectable by an ETD. They are enough different from the 4-8th partials of the notes in the 2nd octave to produce rapid beats. Taking control of those rapid beats can enhance the resonance of the entire piano and thus produce a better quality tuning overall.

I suggest that anyone who has not been aware of this upper partial "clash" experiment by listening carefully to the octaves around F1-F2, a few above and a few below and try to detect a faint, rapid beat. It has been my experience that even in the same piano, the partials which are strongest are not always the same from note to note and some low bass notes don't always exhibit a very strong high partial. If the string is "dead" in those upper partials, you don't have anything to work with, you just tune what sounds like a good octave. But when you do hear a clear resonance, you can match it to the rapidly beating intervals which occur in the upper 3rd, 4th and lower 5th octaves. As I mentioned yesterday, I heard one at G5 when I played C2-C3. The difference between the 12th partial of C2 and the 6th partial of C3 (thanks to Ron for the chart) was 17.8 cents, a very rapid beat when the octave was optimally tuned.

What I still do not understand is what it means have gotten a note 108 cents above pure C8. If the 16th partial of C4 is that pitch because of inharmonicity, then the 15th partial (523.3*7.5 = 3924.75) plus about 300 hz inharmonicity would have been closer to C8 than the pitch that was measured. The machine should have registered that partial instead.

That would probably be best answered by the programmers that made the machines. I could see how that could get confusing for someone using the machine as a direct-partial-reference in the higher numbered partials. Thankfully, during tuning, most techs will be referencing lower partials.

But back to this:


As you can see, a piano string's spectrum has an expected pattern. Even if a partial is weak or missing, it is possible to find the location NO MATTER WHAT THE STARTING PITCH. Looking at the picture, (Or "listening" in a similar manner) it is simply a matter of counting up to the partial number you want THEN measuring the pitch. (As opposed to looking for a partial at a specific Hz. measurement.)


But even if you'd like to discount the measurement for a single string's inharmonicity, I gave you the information for tuning strict 2:1 (matching the fundamental of the upper note with the second partial of the bottom note)octaves from the middle that end up way above what you thought was possible.

(From page 2)
Even if you take a very conservative route, here's where you will end up:

A4 set to 440 (measured second partial = 1.08)
A5 set to 1.08 (measured second partial is 7.1)
add 1.08 and 7.1 to get
A6 set to 8.18 (measured second partial 20.54)
add 8.18 and 20.54 to get
A7 set to 28.72

So even setting 2:1 octaves from the middle up brings us to an A7 setting of about 29 cents.
**************************************

At this point in the conversation, there was a confusion between hz and cents... we got sidetracked for awhile. Let's see if I can figure this out a different way.

Baldwin studio again.
Math without inharmonicity:
C4 261.6 Hz.
C5 523.3 Hz.
C6 1046.5 Hz.
C7 2093 Hz.
C8 4186 Hz.

Start at C4 261.6 Hz, measured the second partial of that string 523.45, just a smidge higher than without inharmonicity.

Tune C5 to that smidge higher amount to make a pure octave, measure second partial to be 1047.7 Hz.

Tune C6 to 1047.7 - Because the upper partial of the lower string matches the lower partial of the upper string, it "sounds" pure.
Measure second partial of C6 at 2106 Hz.

Tune C7 to 2106 Hz. Measure the second partial of C7 to be 4274 Hz.

Tune C8 to 4274, matching the fundamental pitch to the second partial of C7, a beatless octave. And there we are at a pitch a full 88 Hz. above what would occur without the inharmonicity of this piano. (That corresponds to about 36 cents at that pitch.)

Ah, but the machine knows about inharmonicity, so it is listening near where the 16th partial of C4 ought to be, not just the pitch of C4 x 16. At least that's how my ETD works. But you are sort of right too because if the ETD was an old Conn Strobotuner, then it would register the 15th partial as the 16th.

Robert Scott
Ypsilanti, Michigan