
Why are pianos traditionally tuned “out of tune” at the extremes? - amelius
https://music.stackexchange.com/questions/14244/why-are-pianos-traditionally-tuned-out-of-tune-at-the-extremes
======
vnorilo
Related: in a grand piano, the hammers strike the string at a node of the 5th
harmonic (thus avoiding excitation), which reduces the beating from thirds
that do not match the harmonic series (due to equal temperament)

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bloak
It seems that no one objected to the (slightly off-topic) claim that "the vast
majority of western music uses equal temperament". As I understand it, equal
temperament is just a way of tuning keyboard instruments so that they sound
the same in every key; a string quartet or a choir doesn't use equal
temperament. (And some people claim that J. S. Bach, who insisted on tuning
his own keyboards, didn't use equal temperament, either.)

~~~
systoll
It's a way of tuning so that all semitone intervals are an equal [2^(1/12)]
frequency ratio. The vast majority of western music, _today_ , uses equal
temperament. This includes most recorded performances of JS Bach's pieces.

It wasn't particularly common in Bach's time, though, and the math to do it
'properly' wasn't even known outside of China.

Today, though, it's everywhere. Modern Woodwinds are equal temperament due to
the placement of the holes. To match that, the entire orchestra is in equal
temperament, regardless of the requirements of the instrument.

Guitars are in equal temperament due to the placement of their frets -- to do
otherwise would require frets that bend between strings.

In isolation, people will [attempt to] sing in equal temperament, because
that's what 'all' accompaniment is like.

Barbershop quartets, acapella groups, and to some extent string quartets have
a tendency to shift _harmony notes_ toward simple fractions of the root note
of the chord, on the fly, even as the overall melody runs in equal
temperament. This isn't a different 'tuning', per se, but it is a thing.

~~~
mrob
Barbershop is notable for using the harmonic seventh, which doesn't exist in
the standard 12 note scale. Its frequency is 7/4 times the frequency of the
root, so a harmonic 7th above C is somewhere between A and B-flat.

[https://en.wikipedia.org/wiki/Harmonic_seventh_chord](https://en.wikipedia.org/wiki/Harmonic_seventh_chord)

~~~
yesenadam
From that page: the harmonic seventh is '"sweeter in quality" than an
"ordinary" minor seventh'. Well, as an ex- trumpet and trombone player,[0]
what it sounds is out-of tune, and is generally avoided, although the out-of-
tuneness can be compensated for by adjusting the trombone slide. It doesn't
sound particularly bluesy to my ear. Sure, sliding around the flat 7 and flat
5 areas sound bluesy, but just that note as a melody note sounds very flat.

[0] Playing brass instruments is a matter of using your lip to find the
various harmonics, and using the valves to flatten them by various amounts.

~~~
mrob
I don't think there's anything inherently wrong with natural harmonics in a
melody. The opening and ending of Britten's Serenade for Tenor, Horn and
Strings, Op. 31 uses them, and it sound good to me. Because it's notated as
equal temperament, the score is ambiguous, and there are two ways of playing
it:

[https://www.youtube.com/watch?v=WxVnSkX4Fco](https://www.youtube.com/watch?v=WxVnSkX4Fco)
(13th harmonic high note version)

[https://www.youtube.com/watch?v=mkLyK-
oSQ7A](https://www.youtube.com/watch?v=mkLyK-oSQ7A) (14th harmonic high note
version)

I suspect that brass players are unusually sensitive to deviations from equal
temperament because they have to work so hard to overcome them if they want to
play with other instruments (including other brass instruments of different
sizes).

~~~
yesenadam
>I don't think there's anything inherently wrong with natural harmonics in a
melody.

Not sure what you mean by 'natural harmonics', i.e. which ones you mean. I was
just talking about how the Ab harmonic (on trombone/trumpet) sounds very flat.
It's avoided because it sounds out of tune. I'm not sure what something being
'inherently wrong' would even mean. In music, if it sounds good, it is good.

>I suspect that brass players are unusually sensitive to deviations from equal
temperament because they have to work so hard to overcome them if they want to
play with other instruments

I don't know what you're referring to there - I've never had or even heard of
that problem. And I hadn't noticed or heard, and don't believe, that 'brass
players are unusually sensitive to deviations from equal temperament'.

~~~
mrob
By "natural harmonics" I mean notes at the resonant frequencies of the
instrument. Several of them are obviously different from equal temperament,
which means they will sound bad when played with other instruments. When brass
players talk about "good intonation", they often mean adjusting these notes
into equal temperament with valves/slides/embouchure/hand stopping. Because
brass players actually have to pay attention to the tuning, I wouldn't be
surprised if they were more sensitive to it than people who played fixed pitch
instruments like piano. The Britten piece is notable for the parts where the
horn player is instructed not to adjust the pitches to equal temperament, and
because it's a solo it still sounds good.

------
xchip
Related: If you have a piano you can check that by using my online tuner that
shows you the power of all the 88 notes a piano has. This means you can look
at the fundamental frequency and its harmonics too.

[https://htmlpreview.github.io/?https://github.com/aguaviva/G...](https://htmlpreview.github.io/?https://github.com/aguaviva/GuitarTuner/blob/master/GuitarTuner.html)

------
gtrubetskoy
The piano keyboard (usually) has 7 octaves. Frequency doubles every octave,
therefore if the lowest key is C, then the highest C all the way on the right
is

    
    
       C*2^7
    

You can also follow fifths and arrive at the same note after 12 fifths. A
fifth of a note is 3/2 of its frequency. Thus, starting from the same C, you
arrive at

    
    
       C*(3/2)^12
    

But...

    
    
       (3/2)^12 = 129.7, while 2^7 = 128.
    

And that's (roughly) the problem that Bach's temperament addresses by ever so
slightly adjusting the frequency so that in any key it sounds "right".

~~~
dontreact
I don't follow this. The answer contained in the link seems more relevant as
it has to do with the fact that the harmonics of real metal strings are not
ideal and so the detuning at both ends compensates for this.

It seems like you are talking about what would happen if you tuned a piano
using something other than equal temperament. Why is the iterated fifths
relevant at all in this case?

~~~
jedimastert
Generally, Pythagorean tuning uses perfect fifths, as it's the easiest to hear
and to match using harmonics. That example is just showing that if you using
consecutive intervals (such as fifths) then other intervals (such as octaves
and thirds) won't line up.

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fxj
Even electric (sampled) pianos are tuned that way. The difference from the
well-tempered tuning is given by the Railsback Curve
([https://en.wikipedia.org/wiki/Piano_acoustics#The_Railsback_...](https://en.wikipedia.org/wiki/Piano_acoustics#The_Railsback_curve)).
For the highest and lowest notes it is as much as 30 cent. It comes from the
inharmonicity of the strings.

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taylodl
Are they stretch tuning electric keyboards so if they accompany pianos they
both sound good together?

~~~
teilo
Aside from sampled instruments, there's the entire world of synthesis,
including the endless number of virtual instruments that have become staples
in modern popular and electronic music. As to your question: they are not
generally stretched tuned. Many instruments support alternate temperaments
through the use of .tun files, but temperament (with a few outlying
exceptions) only speaks to the interval deltas of the notes within an octave,
not of the entire range of the instrument.

However, it doesn't matter. The reason for the stretch-tuning of a piano is
due to the physical limitations of a vibrating string being unable to produce
the same harmonic intervals at the extremities of the instrument's range, and
does not apply to electronic instruments which, particularly in additive
synthesis, can generate whatever harmonics are desired, including harmonic
configurations that no physical instrument would be able to produce.

~~~
8bitsrule
I ran across this Red Bull video the other day; right at the beginning Mark
Verbos shows how his new synth design has (if I understand correctly) hardware
sliders to allow the user to adjust individual harmonics ... and circuitry
(Hadn't seen the like of that before.)

[https://www.youtube.com/watch?v=AFxxvCXf-k0](https://www.youtube.com/watch?v=AFxxvCXf-k0)

More on Verbos:[http://daily.redbullmusicacademy.com/2018/11/studio-
science-...](http://daily.redbullmusicacademy.com/2018/11/studio-science-mark-
verbos-modular-synth)

~~~
yesenadam
"hardware sliders to allow the user to adjust individual harmonics" \- Ah,
just like traditional church organs.

------
teilo
The best guitar players also do a form of stretch tuning, depending upon the
range of the piece that they are playing. Just as on a piano, the high end of
the instrument will sound flat if they didn't. This is a limitation of all
stringed instruments.

~~~
mrob
Inharmonicity increases with string thickness and decreases with string length
(or effective string length when you're fretting it). It's most noticeable if
you play high pitched notes on the E2 string. However, slight inharmonicity is
just part of the sound and not a flaw:

[http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.515...](http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.515.8900&rep=rep1&type=pdf)

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asplake
Wonderful! One of weirdnesses I’d noticed but never got to the bottom of

~~~
derriz
Indeed - I've been playing piano and guitar (badly) on and off since
childhood. I'm surprised I never wondered about this. It also explains why
tuning guitars purely by harmonics doesn't "work" \- you end up with the
bottom and top E strings sounding out of tune.

~~~
beat
Guitars are even worse, due to fretting. The act of fretting a note stretches
the string, which raises its pitch, and the amount varies by string and by
fret. So guitars are _never_ truly in tune. Not only are harmonics not
sufficient, but neither are electronic tuners! A guitar that is "in tune"
according to an electronic tuner is always sharp in practice.

And it gets worse from there! The tuning of a guitar string varies as it
decays. This is true of piano strings too, but not to the same degree. A
freshly struck string is vibrating more widely than a gentle or decaying note,
so it's stretching itself sharp. You can see this on a fast digital tuner -
the note will go sharp at first, and then settle to a slightly lower pitch. So
when you tune to an electronic tuner, are you tuning the initial pitch, or the
decayed pitch? This variation might be 20 cents or more.

Because of this, how to tune a guitar _well_ is very much a matter of taste,
context, and experience. I use harmonics and electronic tuners to get myself
in the ballpark, then start fine-tuning based on the guitar itself (each one
has its own quirks), and the material I'm planning to play. On acoustic
guitar, I tend to focus on getting the B string in tune with the D and A
strings first, by the quality of octaves for open C and D chords (which also
gets the A and D in tune with each other). Then I focus on getting the low E
in tune with an octave E on the D string. Then get the high E in tune in
unison with E on the B string. Finally, get the G string in tune with G on the
low E, an octave down. This means my G string is usually a bit flat relative
to the D and B strings, but that's okay - it's in tune for G chords, and being
a little flat is good for E major and D chords. I might adjust a little if I'm
playing in C/Am.

James Taylor has an _excellent_ YouTube video about tuning guitars
consistently with electronic tuners. It's very much to his taste and the
specific guitars he uses, but his principles are sound. And if you try it on
an acoustic with good intonation, you'll immediately hear that "James Taylor"
sound.

~~~
SyneRyder
To save others a click, I think this is the relevant James Taylor video that
you were talking about:

[https://www.youtube.com/watch?v=V2xnXArjPts](https://www.youtube.com/watch?v=V2xnXArjPts)

And thanks for the video reference & your comment, this helps me feel better
about tuning my guitar, I thought I was just really bad at it.

~~~
beat
You're welcome! The most important takeaway, I think, is that to a certain
degree, "in tune" is a matter of opinion (for fretted instruments, anyway).
You can significantly change the tonal quality of the instrument while
remaining "in tune". That's the neat thing about the James Taylor video...
follow his method, and your guitar suddenly gets that James Taylor sound, very
rich and resonant.

Try experimenting with modal tunings like DADGAD, too. It's much easier to get
them "in tune", and you hear this beautiful resonance that guitars can make.

I brought this up on Facebook, and a friend who is an excellent player
responded with his own tuning method. He tunes the A string to a reference
(tuner, piano), and then tunes every other string to a fretted A note that is
in tune with the open A string. This is probably more "in tune" than the
highly resonant approach that I use.

------
kop316
I hate to say....the explanation is a bit wrong.

Music is based off of ratios in between each other. An Octave is 1:2 (For
example, A1 = 440 Hz, A2 - 880Hz, etc.). But the fundamental ratio for modern
music is the Fifth. Thr first eight fifths of F is C, G, D, A, E, B. Re
arranged, that is also C, D, E, F, G, A, B (a Major Scale). If you go to 12
Fifths from F, you get the chromatic scale. This will work for any starting
note, and is why we have 12 Major Scales.

However, we typically say a fifth has a 1:1.5 Ratio. This is an approximation
due to the overtone series. When a string vibrates (or a horn vibrates, take
your instrument), it vibrates at a fundamental frequency and several harmonics
above it as well. I cannot find it exactly, but I believe it is closer to
1:1.48 to make overtones work. This means the math does not work out between a
Fifth and an octave. To solve this issue, we use the approximation on the
piano so that the fifth and Octave line up exactly for the ratios (we tune
fifths slightly sharp and octaves slightly flat).

Now remember how I said music is all based off of ratios? Due lot this, we
need a common fundamental frequency that everybody can agree to tune to. The
musical world as decided on the A=440 Hz (which is close to the middle of the
piano). Due to the fact that the reference frequency is in the middle of the
piano, as you get out further, the approximation rears its ugly head.

This is why pianos are tuned "out of tune" at extremes.

~~~
kop316
As a fun fact too, the overtone series is also why you have different size
pianos. The longer string gives you a different overtone, and if i recall
right, the overtone is "more in tune" overtone series (if there are any piano
folks in here and I got that wrong, please forgive me, I am a mere trombone
player).

~~~
TheOtherHobbes
Inharmonicity is a trade off between the ratio of string length to fundamental
and stiffness (which is related to thickness.)

You could make a piano with really, really long bass strings, and the
overtones would be more linear. There would also be a lot more fundamental,
which is almost non-existent on the lowest octave or so of a grand piano.

Native Instruments actually sells a sample pack called The Giant which is
taken from an experimental long-string piano.

I don't think it sounds all that good, because there's something just right
about the colour of the "imperfect" bass strings on a fine grand. (That could
just be acculturation, but I'm not completely convinced that's the case.)

Contrariwise, uprights tend to sound boxy and constrained in the bass because
the strings are shorter than on a full-sized grand, and the overtones are even
louder and even less linear.

You can't do much at the top end, because nicely linear strings - like the
ones on a steel guitar - would have to be very thin and they'd be too fragile
to survive piano hammers.

The closest approximation would be a hammer dulcimer, which has a much sweeter
and more open top end than the percussive plink of a piano, but doesn't go
quite as high.

