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Hi everyone,

You may have come across this How to Tune a Piano with an Electronic tuner article by Juan Olalla. The author explains how he calculates a piano's inharmonicity by comparing the theoretical frequency of F4 to that of F3's second partial. That can be accomplished by manually setting a tuner (for guitars, etc.) to F4 and playing F3. Then we divide the difference between the theoretical frequency and the measured frequency by 12. Finally, we add that value to the theoretical frequency of each note in the temperament octave.

Does that make sense to you, either as an idea or how I presented it?

The author mentions recalibrating every note to reflect the inharmonicity factor. Can we simply add the inharmonicity to the initial callibration of A440, or must we add it to the theoretical frequency of every note individually.

Thank you very much for any feedback you can offer,

Steven

Am I wrong, or is this exactly what the SAT range of tuners have built in...?!

<http://www.accu-tuner.com/AT4.html>

I tried something similar some years ago. I took a $100 tuner, and wanted something that would set a decent temperament - nothing more. I tuned C4 to the machine, tuned F3 and F4 by ear, measured the deviation of both F's with the machine, applied the deviation to each note, graduating either side of C4, while tuning the balance of the F-F temperament with the machine. It resulted in a decent temperament some of the time, and a lousy temperament other times. It was not reliable and I dropped it.

Learning to recognize when an octave is in tune is hundred times more useful and very probably learned faster.

Even the recognized ETD dont hear if an octave sound good or no.

Remember that IH is logarithmic across the compass. Therefore the octave spread must be apportioned across the scale that way.

Inharmonicity depends on the stringing scale of the piano. It may be logarithmic for the harmonic series of an individual note (I have not analyzed the function), but it is not logarithmic across the compass.

As said above, inharmonicity depends on the stringing scale of the instrument. An individual note's iH is logarithmic. A piano's iH can be reasonably accurately modelled by a 4th order polynomial - if you have determined the iH of sample notes on the piano. Guesses don't work well. Aural tuning is better, but an good ETD can get close.

When dealing with purely theoretical half steps, you would multiply by the 12th root of 2.

So, A4 = 440*[(1/12 root 0f 2)^0]
A#4 = 440*[(1/12 root of 2)^1]
B4 = 440*[(1/12 root of 2)^2]

and so on until

A5 = 440*[(1/12 root of 2)^12]


But, because of inharmonicity, the actual frequency of A5 will be higher that 880.

Let's use round numbers just to make things easy. Let's say that A5 is 882.

Then, 882 divided by 440 = 2.00454545455 or there abouts.

So, A4 = 440*[(1/12 root 0f 2.00454545455)^0]
A#4 = 440*[(1/12 root of 2.00454545455)^1]
B4 = 440*[(1/12 root of 2.00454545455)^2]

and so on until

A5 = 440*[(1/12 root of 2.00454545455)^12]

Of course in reality, this is still not accurate enough. There will be variations in inharmonicity. But this should give better results that simply dividing up into 12.

Thank you for the helpful replies. My quest is in finding a simple way to set the temperament. I can do beatless at this point. What I can't do is hear 7 beats per second, or accurately gauge increasing beat rates of contiguous 3rds, etc. I'll practice, but what is the easiest approach/temperament to setting the temperament octave for those with difficulty gauging beat rates. I don't care if its equal or well (even some meantones might be ok).

Thanks again!

This will get you started.

http://www-personal.umich.edu/~bpl/larips/practical.html

If any of this was simple everyone would be a piano tuner. The only real difference between setting a temperament and rocket science is that pianos don't blow up if you do it wrong. OK. 'Slight exaggeration, but it isn't simple and there are no legitimate shortcuts.

You could also try setting a D3 to D4 temperament. Tune A4 to a reference. Tune A3 to A4. Then, tune D3 to A3 and so on.

The thirds and sixths are slower because you are starting lower.

Then as your ear improves with experience, you will begin to hear the faster bps more easily.

Good luck!

While I wouldn't recommend it for a piano, Kirnberger III is easy to tune, because you don't have to count beats. Basically, you tune C-E as a pure, beatless M3, then it is just pure, beatless ascending P5s from E to B to F#, descending P5s from C to F to Bb to Eb to Ab to Db. Then, tune G as narrow ( flat ) as you can stand, then tune D and A so that the fourths D-G and E-A sound equally bad. You are done. I've been tuning this temperament since 1970 on harpsichords and clavichords.

Once you've got this down, you could modify Kirnberger III to Vallotti, and from there to Young 1799. Don't do Vallotti's Young variant, it's a cop-out!

What you say there mean "counting" beats. Tuners do not count beats, really, they compare them, so even to tune a pure M3 you evaluate beating.

I guess you are correct Isaac. I just meant that the number of beats you are counting is zero. Zero is a number of beats. As westerners, we have the Arabs to thank for the number " zero ". Bloody Romans couldn't figure it out, though the Mayans and the Sumarians also thought of it. Sorry to get off topic.

Steven, you will need a reasonably accurate tuner to use the method that the author explains. The tuner displayed in the article (Korg OT120) does not have the accuracy, versatility or the consistency that is required. You might as well just tune to the theoretical tuner's notes and the results will be just as inaccurate and inconsistent because the resolution of the tuner is so poor.

The author is trying badly to explain how to set a 2:1 F4 to F4 temperament octave with equal spacing semitones. The inharmonicity offset the author suggests should be added proportionally across the octave, i.e. all of it to the lower F and none to the higher F. You can't add it to the initial A440 calibration because that will just offset the calibration equally across the whole tuning range rather than changing proportionally for each note.

You have two choices. Practice hearing, counting and comparing the beats. Or buy a decent ETD. I would do the former, then eventually get the latter.

You need to be able to control the pin well enough to get the beatrates you want. This control takes practice. You need to be able to be listening to the right partial matches to find the beatrates, which is something else that takes practice. Basically, practice.

Did I mention practice?

Try this: (if the piano is already roughly in tune, go right ahead. Otherwise, you need the two octaves A3A4 and A4A5 to be in tune, to a reasonable degree. You know how to do that? If not, you shouldn't be tuning intervals anyway)

Slowly press and hold down F3A3, so that the dampers lift but the hammer doesn't strike. Now, play a loud staccato blow on A5. This will sympathetically excite the 5th partial of F3 and the 4th partial of A3, so that they ring. The two partials will have a beat. Zone your brain into what this beat sounds like, isolate it. Count it. You may as well get some kind of second source (watch/metronome) here, although counting beats isn't strictly necessary once you can hear and understand and manipulate them in comparison to each other. How many beats are there per second? It's difficult to count seven, but not impossible. Try tapping out seven to the second with your hands.

Now, play the interval, with your mind still zoned into that beating partial. It'll be there again, the same rate, but with the rest of the harmonic series of both notes mixed in. Ignore the rest of the sound. You just want that A5 beat.

Now you can manipulate the F3 string. Go to beatless. Move a bit flat. Hear the beat increasing. Try and get it roughly to 7.

Then you can tune the octave up, and compare F3A3 to F4A4. A lot faster up there. Can you fit the C#4 in between? For every M3 interval, you can ghost the notes as I described above, and hit the note for the relevant partial match to find where to zone in your hearing. Or just listen to two octaves above the higher note.

Give it a go. Don't get frustrated. It took me months to really be able to do this at all quickly and reliably. And months beforehand to be able to reliably tune an octave. And don't talk to me about unisons, they're still not really quite good enough all the time!

Practice. Don't look for short cuts. If you want to learn to tune aurally, it's the only way.

If you want an easy to set temperament, investigate the Charles E. Moscow Equal-Beating Temperament of 1895. You can find info at the www.rollingball.com site under Pythagorean Temperaments. All 5ths in the temperament octave beat at either 0.0 bps, or 1.4 bps. The 5ths that beat, beat equally. There could not be a simpler temperament to tune.

It is normal to have trouble, there are 2 beat sets going together at slightly different rates that creates an accélération - slowing impression.

One way to hear more clearly is that play the second note a little later. Less noise and it is easier to hear the beating that is quite by then

Good point. I am leaning to do that.