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I have never heard the inharmonicity change while tuning. The tension changes we make while tuning are a thousand fold or more below what would appreciably change the frequency relations amongst the partials. Unison coupling CAN change the energy distribution (Olek I think you use the term "spectra' for this), amongst the partials.

Also in my opinion, as Isaac reports and suggests, it won't be only tuning... meaning that correct frequencies (which would make a piano sound "in tune") should go along with tone quality (read color) and energy circulation (resonance).

Think of a singer: he/she might be "perfectly" in tune and yet sound terrible... Pro-singers normally depart from an amateur approach and start a process that goes far beyond their (perhaps natural) talent, they will learn how to manage their spectral content and their own flows of energy, so I do not think we can separate these three issues, again pitch, color and energy: taken individually, every single issue might reduce performances, but these issues all together can achieve the best performance.

Then I too would suggest to evaluate iH fluctuations (approximations) and influence also in consideration of other factors that apparently contribute to shaping the tone and making energy flow, all factors being related: active-pin//active pin-block//string-3-lenths-tensions//loads//and all the other details that produce the sound, hammers and dynamics of the piano action.

From the field: I recently tuned a baby-G for a colleague, so that in real time he could follow his ETD and record the job. I can confirm that the ETD was not sensible to some variations, both on single string's pitch and unisons, that for my ear would be determinant. To be clear, in order to manage and control partials, I - surely like others - will have to take into account infinitesimal variations that were far beyond the performance of that ETD, perhaps differences that we believe to be "...too small of a difference to be significant".

In this sense, I would not take the worse of what theory, maths and "science" can offer, for instance I would not confuse ETDs figures, how exact they seem to be, with the exactitude needed and that we are able to achieve aurally.

Regards, a.c.
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I do appreciate the smallness. However, I am a literalist, and if a claim is made that iH doesn't change with a small pitch change, and then is found in fact to do so, I am interested in that, regardless of the order of magnitude. Thanks again for your test.

Is this for real?

One incomprehensible post...

followed by a post that draws an incorrect conclusion from the evidence it cites...

followed by a post containing suggested reading, the contents of which indicate how to calculate cents with nomograms, log tables, or a slide rule.

Is this for real?

One incomprehensible post...

followed by a post that draws an incorrect conclusion from the evidence it cites...

followed by a post containing suggested reading, the contents of which indicate how to calculate cents with nomograms, log tables, or a slide rule.

Is this for real?

One incomprehensible post...

followed by a post that draws an incorrect conclusion from the evidence it cites...

followed by a post containing suggested reading, the contents of which indicate how to calculate cents with nomograms, log tables, or a slide rule.

Here is a reference on inharmonicity from the Verituner Manual, © 2012 JWS

"Inharmonicity varies with pitch on the same string

The inharmonicity can vary slightly for a single string as its pitch is changed. This can be a problem when tuning a piano which is far out of tune at the start. To see this, I show below the inharmonicity for a single A4 string as a function of its frequency.

Offset (in cents) Fundamental B value

0 439.955 0.000749032
-10 437.445 0.000758101
-20 434.805 0.000770484
-30 432.353 0.000775297
-40 429.717 0.000789170
-50 427.322 0.000801439

With a 50 cent starting offset: The B value changes from 0.000749032 to 0.000801439, which represents a change in the frequency of partial f4 from 1769.2 to 1769.9 Hz (or 0.7 cents), and for f6 from 2673.6 to 2675.9 (1.5 cents).

For a 20 cent deviation: The B value changes from 0.000749032 to 0.000770484, which changes f4 from 1769.2 to 1769.5 Hz (or 0.3 cents), and f6 from 2673.6 to 2674.6 (0.65 cents). "

My point is that, even for small changes in overall pitch of the piano, the iH changes enough to cause audible differences in the upper partials, especially when one tunes beatless unisons.

The portion of the piano scale with the highest IH is the treble. From note 55 or so on up there is so little energy at even the 2P so as to render the IH differences amongst unison strings inaudible compared to the false beating that is almost always present in that portion of the compass in some amount.

The data you present prove that when doing a fine tuning, IH does not change in an audible amount or even a measurable amount. The thread is about a tuning that sounds too "in tune".

In the wound strings there are often IH differences amongst unison strings and they cannot be reconciled by unison technique.

I maintain that the effect the OP notices is a function of the instruments design and not the result of tuning style.

MWM,

I don't understand where you get your Hz values for the 4th and 6th partials.

You posted the Verituner data, so let's use that. A4 is 50 cents flat, and according to the Verituner, partial 1 is reading a value of 427.322 Hz. The inharmonicity of the string at that tension, was given as 0.000801439.

Wouldn't one expect partial 4 to be somewhere around 1712 Hz and partial 6 to be somewhere around 2570 Hz. Where do you get 1769 Hz and 2673 Hz?

MWM,

I don't understand where you get your Hz values for the 4th and 6th partials.

You posted the Verituner data, so let's use that. A4 is 50 cents flat, and according to the Verituner, partial 1 is reading a value of 427.322 Hz. The inharmonicity of the string at that tension, was given as 0.000801439.

Wouldn't one expect partial 4 to be somewhere around 1712 Hz and partial 6 to be somewhere around 2570 Hz. Where do you get 1769 Hz and 2673 Hz?

I haven't been following this discussion too much, but today I skimmed through and thought I'd add a bit about ETD inharmonicity. There has long been reported different FAC numbers from college techs returning to the same instrument different times of the year...

Now we have a tunelab test that seems to indicate another platform measuring differently depending on pitch.

I did some tests with Dave Carpenter (verituner) a few years after the Verituner was in production to see if we could force that platform to measure differently based on pitch. We de-tuned a piano string in steps down to around -80cents and couldn't see any difference on that machine (or any more difference than between multiple tests at pitch). He had a unit that was able to show the raw data that the machine collected.

This was only one string, so it didn't test the whole piano being at a different pitch, nor did it test the possibility that a change in humidity might be influencing the entire system...

MWM,

I don't understand where you get your Hz values for the 4th and 6th partials.

You posted the Verituner data, so let's use that. A4 is 50 cents flat, and according to the Verituner, partial 1 is reading a value of 427.322 Hz. The inharmonicity of the string at that tension, was given as 0.000801439.

Wouldn't one expect partial 4 to be somewhere around 1712 Hz and partial 6 to be somewhere around 2570 Hz. Where do you get 1769 Hz and 2673 Hz?

Not wanting to argue specifically, but in the end iH is a result from string stretch under tension, so certainly a clearer tone is heard, but the mechanical behaviour of the string is changed too, its energy is better transmitted. There is a (small) part of iH that is due to the amount of resiliency of the bridge/backscale, so justness wise I suppose there is a limited effect but tonally it is large.

Anyway for instance tuning a piano at 415hz 435 or a tension it was not intended for is not helping the instrument .
Old strings get hard and stiff , tension may help them to gain a minimal resiliency and to tone better.

I had experiences with soft strings that gave not enough iH to me, unless they are used with low tensile stress. We are used to iH, sometime too much probably.

Nevermind,

I see what's happening. The Verituner data is pointing out the error that would be introduced if one simply used the IH values read from the piano strings when they were 50 cents or 20 cents flat. This is an argument for remeasuring after the first big pitch raise. I have no problem with that.

But what we're talking about here is the error that would be introduced by the IH if you measured the string at initial state where it were already very very close to pitch, say just 1 cent flat. The effect is numerically calculable, sure, but it's infinitesmal and inaudible.