Tortuous Tuning: Newsletter Challenge (June 2018)

Yes. The chromatic scale is a compromise, and the further from the middle 'A' frequency to which most instruments are tuned, the greater the departure is from mathematical perfection and the greater the compromise. The wide range of frequencies that a piano encompasses makes the tuning an even greater challenge. Luckily, I guess, human hearing allows a fair amount of compromise.

Well, that one was solved quickly. The (long-winded) answer is now posted.

This topic got me thinking, "Why are there 12 notes in an octave?" Is it just a historical accident, or if we ever meet extraterrestrials, will they have music with 12 notes per octave, or in other words, is there a physical reason for the number 12?

Here is an interesting video that explains where 12 comes from. (It starts out kind of slowly, but have patience, it is very interesting once it gets going.)

It WAS interesting, and I like the use of repeated color spectra to show other octaves, but somehow I didn't really understand why 12 instead of some other number. I really like the decimal system, so why not 10?

I looked at at least one other YouTube video explaining why 12 notes, and it clarified nothing, at least not to my satisfaction!

I'm sure that at least part of what sounds good depends on what we are accustomed to hearing. I clearly remember hearing some oriental music that had what sounded to me as strange pitches/intervals. I presume that those pitches sounded normal to the musician playing the instrument.

"so why not 10?" 10 is not very compatible with the 1/2 and the 2/3 ratios. 12 is very compatible with both.

I fully understand the importance of a 1:2 ratio as far as octaves are concerned, due to wave interference and overtones/harmonics.

I have yet to fully grasp the importance of 1/2 and 2/3 ratios with an octave. This is something that has bothered me for many years!

Oh. I have no answer for that. Sorry.

Tones that are small number ratios in frequency, e.g., 1:2, 2:3, 3:4, ..., are pleasing to the ear, a term referred to as consonance.

There is a trig identity that the sum of two sinusoids is equivalent to the product of two other sinusoids. So if you sound two notes together, the sum sounds like a third frequency modulated by a fourth frequency:

http://hyperphysics.phy-astr.gsu.edu/hbase/trid.html

The result is that when two notes are sounded together, you hear a "beat tone", which is the difference term. If the two frequencies are close together, the beat tone is a low frequency and is not very pleasing (e.g. B and C on the piano). If the frequencies are related by a small number ratio, e.g., 2:3, 3:4, ...etc, each frequency will be a harmonic of the beat frequency, the mixture of tones will sound "uniform" and be pleasing to the ear. (consonance)

https://en.wikipedia.org/wiki/Consonance_and_dissonance

This example is a consonant mixture of 200 Hz and 300 Hz producing a beat tone of 50 Hz. The sum of these two tones is equivalent to 250 Hz modulated by 50 Hz. 200 Hz and 300 Hz are multiples (harmonics) of the 50 Hz "beat" frequency.

Thanks! Some day, I hope to get out a couple of signal generators, a pair of earphones, and my 'scope, and experiment with that. 'Till then I'll take your word for it, and check out your links...

If you have Matlab installed on your computer (or like me, Octave, which is free!), you can write some software and play the sound through your computer speakers.

% COPY THIS CODE:

Sec = 1; % number of seconds to play

FS = 8000; % sampling rate

k=1:Sec*FS;

f1=200; % tone frequencies

f2=300;

x1=sin(2*pi*f1*k/FS);

x2=sin(2*pi*f2*k/FS);

y=x1+x2;

soundsc(y,FS)

figure;

plot(y);

grid on;

% END OF CODE

https://octave.en.softonic.com/download

Now that you mention it, I actually did that about 25 years ago! While programming a machine that was controlled via a SCSI port, I needed two more control lines than I had available on the port, so I made a two-tone detector circuit board for those last two functions, and put the appropriate tones at the appropriate places in the controller software. It operated flawlessly for at least 10 years... Done in BASIC before I had a compiler available.

In the challenge, replace the preface “While tuning your piano...” with “While cutting your hair...” or maybe “While removing your wisdom teeth...” and then adjust the remainder of the challenge text to align with the preface.

Yeesh! NEVER tell a client you are charging him for something that cannot be accomplished.

One reason it could be considered impossible to “accurately tune a piano” is due to the anharmonicity of real strings. The harmonics of an ideal string are integral multiples of the fundamental frequency, but with a real string this is not the case (due to stiffness and end effects).

So taking two notes an octave apart, if the fundamental of the higher note was exactly twice the frequency of the lower, the first harmonic of the lower would be "out of tune" with the fundamental of the higher.

To complicate matters, the degree of anharmonicity varies with the type, thickness and to some extent tension of the string, so varies from one note to to the next along the piano keyboard.

Many notes of a piano use two and three strings per note. I understand that the two of a pair or the three of a triplet are intentionally tuned to slightly different frequencies, so the beat(s) between the strings in a set give a fuller sound.

After spending 40 years as an engineer automating various processes I have always wondered why piano tuning has never been automated. There are now simple apps for a smart phone that listen to the frequency of a guitar string and indicate which way to turn the tension knob but the technology exists to go much further.

If I were writing a brief for the design of an automatic piano tuner my first draft would include:-

A couple of sophisticated microphones that could be positioned local to the player and at sweet spots in the audience.

A set of tension adjustment actuators (one for each string) so that human error is eliminated from string tension adjustments.

A second set of actuators to depress the keys with varying degrees of speed and strength to add consistency and repeatability to the striking of keys These could also be programmable to replicate the style of an individual player by including feedback to record their depression off the keys.

A computer with a library of different piano nuances so that the customer would choose the style of tuning i.e Classical, Modern, Jazz. This would be used after basic scale tuning to tune the 3rd and 5th harmonics of chords. It may even be possible by analysing the chord repetition frequency of different composers to tune for Schubert, Mozart, Gershwin or Rachmaninoff.

What would certainly be possible is that when the player comes upon a piano that perfectly fits his or her requirements, that piano could be recorded and reproduced on their own instrument. The fact that individual string wires vary by make and age would be cancelled because the program would be tuning to reproduce a specific predefined sound and any variations would thus be nullified. Concert venues that re-tune their pianos to the requirements of individual performers could do so quickly and easily from a data set provided by the artiste.

Bare in mind that I do not play the piano so this first draft could be greatly improved by others.

I remember seeing an autotune device for a guitar a while back. Keeping a guitar in tune is a problem for professional performers. If the strings are new, they tend to stretch for a while, and if the performance is outside, there may be a temperature difference that changes the tuning.

The two parameters that determine the pitch of a string are length and tension. When a string stretches or the temperature changes, the tension changes changing the pitch.

There's a bridge that automatically controls the tension on the strings.

https://www.guitarplayer.com/miscellaneous/wilkinson-atd-ht-440-self-tuning-bridge

This would probably be impracticable for a piano, which may have up to 236 strings.

http://www.piano.christophersmit.com/strings.html

This actually was the device I was thinking of. It's completely passive, a simple mechanical design, and one of those "Why didn't I think of that" kind of inventions.

The pitch of a string is determined by its length and tension. Normally, to tune a guitar string, it is stretched with the tuning peg until the tension, along with its length, results in the desired pitch. But this tension and the pitch changes if the temperature varies or the string stretches or slips.

The Evertune bridge maintains a constant tension regardless with changes in temperature and string aging. Once it is set with the adjustment screw, the spring tension is transferred to the string and the pitch stays constant because the string tension is determined by the spring mechanism and not by the stretching of the string.

There's probably no reason why this couldn't be incorporated into a piano.

The EverTune mechanism keeps a string's tension constant by using a spring and lever system.

https://en.wikipedia.org/wiki/Evertune

Interesting. I hadn't heard of that... Also interesting that the Wiki article apparently hasn't been updated since 2011...

It might be difficult to get them to fit the strings of a piano, especially for those notes that have two and three strings for a single note.

I have no idea whether it is significant, but the pitch of a string not only depends on the length, mass per unit length, and tension, but also on the stiffness. Stiffness is temperature dependent, so a more nearly perfect device might include some form of temperature compensation...

Perfect doesn't exist in reality, so the answer to the Challenge is NO.

I try not read too much into these puzzles. The clues are purposefully vague, which is why they're puzzles, and it's more trying to figure out what the puzzler has in mind.

The post by jhhassociates reminded me of seeing this Evertune gadget, which I thought was very clever at the time, and I thought it might be interesting to share. It removes the by far biggest factor that causes a guitar to get out of tune, change in string tension.

By the way, you can buy them now. They're way too pricey for me, but I can see how it might be worth it to a professional entertainer.

I'm going to give the "safe" answer: Yes, and, no. If one has the precise definition of what being in tune is, and also has a precise measuring device, Then it would be possible to sync the two. But, if the tolerance is not specified, who can say if the instrument itself can be tuned that precisely? You might keep over/undershooting the sweet-spot. If that kind of precision is not detectable by the the human ear, who's to say whether it's in tune or not, or if it's acceptable or not? The key is about how close you can/need-to get to the defined standard (whatever that is).

"Perfect" tuning would be if the ratios of the pitches of the scale notes were rational numbers, likely impossible to achieve mechanically, but easily achieved electronically. Rational ratios are pleasing to the ear, for some reason that probably nobody understands.

But if you tune the piano perfectly in C, if a song were played in the key of D, the ratios would be wrong. For example, the note G, which is the "perfect 5th" in the key of C, has a frequency of 3/2 times the frequency of C. The note A, which is the corresponding note in the key of D, has a frequency of (5/3) / (9/8) or 40/27 the frequency of D.

If you tune it perfectly in one key, it will be out of tune in another key. The compromise is to make the ratios between each half-tone the same, which would be the twelfth root of 2. This makes it equally "out of tune" in all keys, but close enough.

Exactly, you can transpose a musical piece to another key, only if you use the tempered tuning, i.e. constant ratio between the semitones (a power of the twelveth-root of 2). This has already been mentioned. One could further read https://en.wikipedia.org/wiki/Pythagorean_tuning

Nevertheless, the question is: why should we REALLY care about transposing to another key? I understand possibly the need to adapt to a singer's vocal range or something, but in classical music I guess this is mostly irrelevant. I don't play piano myself, but I play some guitar (which is tempered by design) but still the usage of open strings make the selection of key relevant to some extend and thus part of the experience. (Not so in jazz maybe.) Some other musical instruments are not tempered by design, like recorder, flute etc, so in an orchestra, even if you have the piano tempered, some of the rest of the instruments (bowed excluded of course) could sound out of tune.

What I want to say, is that the music works - especially the classical ones - were written for a specific key, and I have a hunch that the composers actually wanted the piece be heard in a certain chromatic way. I once asked a composer and told me that choosing the key was part of the composition, and he would choose according to what sounded "nicer". He didn't want or couldn't explain me any further (or didn't ask enough). Maybe this particular composer was non "typical" (neither his piano's tuning), but anyway, I suggest you to consider this aspect too.

I wonder if a professional piano tuner tunes according to who uses the piano and which music is supposed to be played on it. Whether this is true or false, I guess that he already knows what he is supposed to do (e.g. tune equal intervals or else start with the 3rds and 5ths etc), and therefore, there is not a "tortuous procedure" at all, except from the fact he has to manually turn and turn about some hundred of screws!

You've touched on some interesting points here. String and wind instruments may not be tempered in the same way as a piano, but the difference is they can adjust pitch on the fly, or at least tune to a fixed instrument like a piano or organ -- strings can adjust their tuning pegs, wind instruments can change their wind pressure or adjust tuning slides, etc. In a piece with winds/strings and a piano, all the players tune to the piano with the knowledge that the latter can't adjust pitch.

I've done a fair bit of composing -- there are lots of reasons to use different keys. A big part of it has to do with instrument ranges, or using open strings, as you mentioned. Using a relatively lower key often results in chords that sound muddy or unclear, which is a good argument for using a relatively higher key for that music.

If you listen to and study music long enough, keys sort of take on their own "personality" based on where the pitches fall, and how different instruments sound in those keys. D major is close to C major and Eb major but it's almost always a bit brighter-sounding than those two, for example. It's hard to explain beyond that.

Many songs have a change in key and therefore cannot be played on an instrument built to play in only one key. Key changes (or modulation) are a part of music.

http://www.pianotv.net/2016/02/12-amazing-key-changes-pop-music/

If it's a 'she' I'm not going to argue the point.

A piano is esentially a box with a lot of strings under tension. Get one string to desired tension and it must have some small effect on the overall shape of the frame. No problem so far. Move on to second string, and surely it must also alter the shape of the overall frame and thus the tension in that first string ? I guess I am wrong since the question could have been phrased for any stringed instrument.

"The treble pitches have three unison strings, the tenor range uses two unison strings, and the bass strings use only one string. The end result is that for 88 notes, there can be as many as 236 strings. Each string has a tension of 160-200 pounds, resulting in a total string tension of 35,000 pounds!"

http://www.piano.christophersmit.com/strings.html

Pianos are really heavy due to the cast iron frames that keep them from imploding!

This is the closest answer to the original query. The next response addends the approximate tension. Yes, the cast iron frame bends as the tension is altered. However, what everyone else has missed is that the wrest pins are screwed into a wooden wrest plank. This wooden plank gives more than the cast iron frame and does so in a quasi-plastic manner, ie, the fibres of the timber deform more and more over time.

So, a good piano tuner will do a rough tuning and then a fine tuning - it you've paid for that - and over time, the wrest plank deforms and keeps on deforming; so, the piano is only correctly tuned for that one instant just after the piano tuner has finished the fine tuning.

This reply, while interesting historically, does not deal with the original query. It deals with some quaint notion of what is acoustically pleasing by not having unusual beats or harmonics.