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Why it's impossible to tune a piano
#2910621 11/11/19 12:44 AM
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AWilley Offline OP
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I wanted to share an image of an interesting graph I've been playing around with. It shows the width of several slow-beating intervals in cents. In an ideal world you'd want these lines to all stay close to zero. As you can see that's quite impossible.
[Linked Image]
A few observations:
  • The 2:1 octave (solid blue) is always tuned wide.
  • The treble is tuned so that the 4:1 double octave (yellow) and 3:1 twelfth (black) are pure.
  • The temperament region is a blend of 4:2 and 6:3 octaves with pure 12ths. Then the 6:3 octave goes pure in the tenor.
  • 8:4 octaves are always narrow; 10:5 octaves are very narrow don't have a chance of ever being pure on this piano.
  • In the tenor the 3:1 twelfths start to go wide, while the 6:2 twelfths pass through pure on their way to wide in the low bass
  • The low bass on this piano is dominated by pure 8:4 double octaves and 8:1 triple octaves. There are a lot of choices on how to tune down here...6:3 octaves and 6:2 twelfths could be made pure with less "stretch" forcing the double and triple octaves narrow. Pure 8:4 octaves would force everything else wider. Trying to do 10:5 octaves would be way too much stretch on this piano.

I realize there's a lot going on in the graph, but I've tried to color code things a bit. I also put in some reference lines showing how many cents wide/narrow equals 1 beat per second. This is calculated from the left note of the interval...see how at A4 (note 49) 1 beat per second equals 4 cents.

Once you understand what is going on it is fun to imagine how the graph would change if you choose to tune different intervals pure. Pure 2:1 octaves, for instance, would push everything below the blue line down...the fifths would be 3 cents narrow (almost 1 beat per second in the middle). Pure fifths in the center would make the octaves, twelfths, and fourths too wide.

Anyway, I hope somebody finds this fun or interesting.


Anthony Willey, RPT
PianoMeter
Willey Piano Tuning
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Re: Why it's impossible to tune a piano
AWilley #2910634 11/11/19 02:18 AM
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Thanks for providing your findings!

Interesting (whether it means anything or not) that the general trend of your curves is the opposite or inverse of the "Railsback curve".


Keith Akins, RPT
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Re: Why it's impossible to tune a piano
AWilley #2910657 11/11/19 05:21 AM
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Are these values according to a Piano Meter curve?


Chris Leslie
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Re: Why it's impossible to tune a piano
AWilley #2910695 11/11/19 08:51 AM
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Nice graph.

However it does not prove "Why it's impossible to tune a piano".

The fact that you made this plot already proves that you tuned a piano :-) That is, you brought the strings to some carefully planned tensions in order to plot this graph.


[Linked Image][Linked Image][Linked Image][Linked Image]
Re: Why it's impossible to tune a piano
Chris Leslie #2910713 11/11/19 10:05 AM
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Originally Posted by Chris Leslie
Are these values according to a Piano Meter curve?

Kind of, but not exactly. There's nothing special about this particular tuning. It's using different "weights" on the different intervals and a slightly different process for calculating the tuning. (The graph from pianometer would look similar though.) Also this is a calculated tuning for a piano, not measured. I was trying to find ways to visualize it.

The similarities to Railsback are not coincidental :-)

I'm a bit curious what people think about tuning triple octaves in the low bass. I realize it's more stretch than the regular 6:3 octaves and it makes sense to me conceptually, thinking about what type of chords people play when they're using those low bass notes, but it's not something I'd ever thought to have attempted aurally.


Anthony Willey, RPT
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Willey Piano Tuning
Re: Why it's impossible to tune a piano
AWilley #2910722 11/11/19 10:55 AM
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It is not easy. But I would say it is possible.

Re: Why it's impossible to tune a piano
Hakki #2910797 11/11/19 02:25 PM
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Originally Posted by Hakki
It is not easy. But I would say it is possible.

I mean impossible in the same sense that it's impossible to shoot a perfect bullseye from 20 feet using a shotgun. You might hit the exact center of the target with 1 pellet. You might hit the bullseye with multiple pellets. You might have a nice general distribution of pellets around the center of the target. But you will never be able to hit the center with every pellet.


Anthony Willey, RPT
PianoMeter
Willey Piano Tuning
Re: Why it's impossible to tune a piano
AWilley #2910833 11/11/19 04:15 PM
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That graph is a good illustration of why I, as an aural tuner, will tune by compromising several intervals as I expand so that none are too noisy. Relying on just a single interval or octave type means that other intervals may be excessively noisy. I guess that principle applies to some ETD software as well, or I hope so.


Chris Leslie
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Re: Why it's impossible to tune a piano
AWilley #2910987 11/12/19 12:01 AM
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Wow. This is so cool. Makes me really want to rely on twelfths more than I currently do in the middle of the piano.

It'd be interesting to see what this would look like for a concert grand. I'm imagining all the curves flattening out some in the bass and high treble.


Bob Runyan, RPT
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Re: Why it's impossible to tune a piano
AWilley #2910993 11/12/19 12:39 AM
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OK, two more images. These show why some pianos are harder to tune than others.
[Linked Image][Linked Image]
Note the spread in the bass, and the tenor break around Note #34 on the spinet.


Anthony Willey, RPT
PianoMeter
Willey Piano Tuning
Re: Why it's impossible to tune a piano
AWilley #2918849 12/02/19 01:53 AM
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[Linked Image]
Here's another image, again Yamaha U1. It's still pretty busy (sorry) but this time I've adjusted the darkness of the lines to rougly correspond to how strong each harmonic is. So strong harmonics (dark lines) should be more in tune, while weak harmonics (very light) don't matter as much and are allowed to be out of tune. Sharing because I find it interesting and thought others might as well.


Anthony Willey, RPT
PianoMeter
Willey Piano Tuning
Re: Why it's impossible to tune a piano
AWilley #2918857 12/02/19 02:24 AM
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Fortunately, I do not read these charts, so like the bees that cannot fly according to the laws of aerodynamics, I tune pianos.


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Re: Why it's impossible to tune a piano
BDB #2918926 12/02/19 07:45 AM
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Originally Posted by BDB
Fortunately, I do not read these charts, so like the bees that cannot fly according to the laws of aerodynamics, I tune pianos.


It's an old myth that's incorrect - see https://rationalwiki.org/wiki/Bumblebee_argument

Re: Why it's impossible to tune a piano
AWilley #2918975 12/02/19 10:24 AM
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Anthony,

The title on the graph says "width from pure." What is "pure?"


Bob Runyan, RPT
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Re: Why it's impossible to tune a piano
bobrunyan #2919008 12/02/19 12:00 PM
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Originally Posted by bobrunyan
Anthony,

The title on the graph says "width from pure." What is "pure?"


Beatless.


Anthony Willey, RPT
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Willey Piano Tuning
Re: Why it's impossible to tune a piano
AWilley #2919026 12/02/19 12:46 PM
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... but any of these intervals can be tuned beatless for a given note (at least in theory). I guess what I am getting at is what does zero represent in the graph.


Bob Runyan, RPT
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Re: Why it's impossible to tune a piano
AWilley #2919039 12/02/19 01:27 PM
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Originally Posted by AWilley
Originally Posted by bobrunyan
Anthony,

The title on the graph says "width from pure." What is "pure?"


Beatless.


You do not tune octaves beatless?


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Re: Why it's impossible to tune a piano
bobrunyan #2919074 12/02/19 02:33 PM
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Originally Posted by bobrunyan
... but any of these intervals can be tuned beatless for a given note (at least in theory). I guess what I am getting at is what does zero represent in the graph.


Only one interval can be tuned precisely beatless at a time, unless you're at a lucky point where two of the lines cross each other, then in theory those two intervals could be tuned beatless.

Zero represents beatless. So if the 2:1 octaves were all tuned beatless, that top blue line would be placed on the horizontal zero line, and every line below that would be pushed down in the "narrow" zone. (4:2 octaves would be narrow, twelfths would be narrow, etc.

Originally Posted by BDB
You do not tune octaves beatless?

What kind of octaves? 4:2 octaves are slightly wide of beatless in the midsection and 6:3 octaves are nearly beatless in the tenor. If I didn't favor the 12ths so much the 4:2 octaves would be closer. The 2:1 octaves are always wide.

As a concrete example, in the image above, the 4:2 line is at about +0.7 cents for the A3-A4 octave. The coincident harmonic is at A5 where 1 Hz equals about 2 Cents, so the A3-A4 octave is about 0.35 beats per second wide (or 1 beat per 3 seconds). If I remember correctly that's pretty close to the target of the Baldassin Sanderson aural temperament where you use the F3-A3 F3-A4 test to tune the octave about 0.5 beats wide.

Anyway the point of this is that it's impossible to tune all the slow-beating intervals beatless, and we have to settle for a "close enough" compromise. Thus the slightly facetious title of this thread.


Anthony Willey, RPT
PianoMeter
Willey Piano Tuning
Re: Why it's impossible to tune a piano
AWilley #2919083 12/02/19 03:03 PM
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What kind of octaves? Why, beatless ones, of course!


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Re: Why it's impossible to tune a piano
AWilley #2919946 12/05/19 01:26 AM
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OK, here's the same Yamaha U1 tuning, but with a lot of the data removed from the image. Now it's only showing the octaves. Hopefully without all the extra lines it will be easier to see what's going on.
[Linked Image]
  • In the treble the 2:1 octaves are strongest, but not as important as other intervals (12ths, double octaves) so they're tuned wide. (Rememeber the darkness of the lines is related to the strength of the interaction of the harmonics.)
  • In the temperament region the 2:1 octaves are still strongest, but we're still not tuning them pure. 4:2 octaves are tuned slightly wide, and 6:3 octaves are tuned slightly narrow
  • On this graph you can see the effect of the transition from plainwire to copper wound strings. At note 33 (F3) the 6:3 octave is tuned pure, while 4:2 and 2:1 spike wide, and 8:4 and 10:5 spike narrow. This is because the lowest plainwire has higher inharmonicity relative to its neighbors.
  • In the tenor, the 6:3 octaves are being tuned basically pure. The 4:2 octaves continue to get wider as you go down. 4:2 and 6:3 octaves are pretty strong here.
  • In the low bass the 10:5 octave is strongest, but we don't tune that pure because that would be too much stretch for this piano. This tuning makes a compromise between the 6:3 and 8:4 octaves, but neither is pure. Note that at A0 there is a 12 cent difference between the two...probably one of the reasons the low bass is where different people tune differently. It matters a lot what you're listening to. (FYI the 10:5 line continues downward and ends at -22 cents at A0.)
  • Looking again at the darkness of the lines, the 8:4 octave seems to be weaker than all the other octave types. I suspect this is due to the 8th harmonic being weak because the hammer strike point is near 1/8 of the length of the strings.


Anthony Willey, RPT
PianoMeter
Willey Piano Tuning
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