How wrong must a note be, to be considered out of tune?

[marbak]

Hello, I'm currently on an assignment where I'm needed to analyse the frequency spectrum of a piano note to determine if it is in or out of tune. I'm not a person with a whole lot of musical knowledge, so I was hoping I could have a quick question answered by  someone who knows it better than I.

I've been able to find the correct frequencies for the various notes, (C4-B5), but the question is as follows: How wrong (assuming in percent is most accurate) can a note be, before being considered out of tune?

for instance would a C4 that's 0.2Hz from ideal be considered out of tune?

[perfect_pitch]

That's technically an ambiguous question, considering that many countries have a different idea on what A is. Many countries, A can be 446 hertz, here in Australia the average is around 442 Hz.

But it ALSO depends on the persons ability to hear. For me, I can hear the difference between a note if it's a single Hz out of tune, and can tune intervals to within 1/40th of a semitone at around the Middle C register.

Others would probably have much greater trouble getting them that accurate however... so others might not be able to identify them as being out of tune.

[marbak]

Thank you for your reply, the chart I've been using has 440Hertz set as A4, but the exact number isn't too important for me. Since the whole assignment is more about the mathematical aspect of signals. Sound and in this case piano notes are just used as examples of such. But good to know they can be different in case I find conflicting sources, though while the exact frequency is different, is the difference between notes constant? so that it's just a constant shift in the notes from start to end?

[iansinclair]

Ir isn't so much a matter of how far off a particular tone is from some arbitrary standard (A has varied, historically, from at least as low a 430 hz -- still found on some European organs -- to say 446) as it is how far off it is relative to other notes -- particularly, in the case of a piano, to the other strings of the same note.

And it does depend some on the perception of the tuner.  Among the three strings of a piano note in the middle registers, a difference of as little as 0.5 hz may create a noticeable "beat" in the note.  Less difference may just create a sensation of uneasiness (hard to describe).  It is harder to discern out of tune intervals.  Octaves aren't that hard -- although it is common to tune octaves slightly "broad".  Lesser intervals are harder -- and indeed the definition of "out of tune" begins to get rather badly confused with what is the correct frequency ratio for an interval -- the whole discussion regarding even tempered vs. well tempered scales, for instance.

I don't think that helped much...

[quantum]

IMO there is an element of listener perception in the detection of tuned vs out-of-tuned pitches.  A scientific measurement may tell you the quantitative values of how much a pitch is out of tune, however a realistic scenario in a musical applications involves human ears.  This is regardless if the music is acoustic or electronic, the destination still lies with the human ear - and thus the evaluative process takes place within the human form, and not through scientific measurement instruments. 

That said, there can be great variability from person to person on the consideration of what is "in tune."  I know professional musicians that throw the term "in tune" around a lot, yet cannot quantify its meaning or distinguish between tuned and out-of-tuned unisons on the piano.  If a person learns what to listen for, it is very possible to detect minute differences in tuning.

For real world musical applications perfect tuning is not exactly a prerequisite for the end result to be perceived as "in tune" by the listener.  There is somewhat an amount of forgiveness in this aspect.  Some musicians may be highly sensitive to tuning and make frequent adjustments to their instruments throughout a performance.  However, other musicians may take it in stride and be willing to accept some degree of out-of-tuneness. 

There is discussion within the digital-sampling community whether creation of digital instruments with tuning that is too perfect detracts from the musical application of such instruments.  Some suggest that what makes the experience of listening to an acoustic instrument vivid and engaging, as opposed to its digital counterpart, is a certain degree of tuning imperfection and the manner in which pitches interact within that state of imperfection.   

[oxy60]

Keyboards are tempered instruments. Remember that good violinists will play the same written note and pitch it differently depending on harmony. We want the 5ths, octaves and 12ths to be clean. Otherwise we'll sound like an elementary school orchestra.

First one must define what "in tune" means.

[j_menz]

A piano "note" is made up of one, two or three strings - a note can be "out of tune" if those strings are out of tune in relation to each other. Your premise of "a piano note" is flawed and you need to take that into account.

Beyond that, assuming it is in tune by itself, notes are in tune not by reference to some standard but by reference to one another. You could drop them all by 40Hz or more and the thing would still sound "in tune". The in-tuneness with respect to one another depends on the tuning system and the ear of the listener. In some tuning systems, it also depends on the key to which it was tuned vs the key you are playing in.

In short, what seemed like a simple question in fact isn't.

[silverwoodpianos]

There is one important thing missed in this thread. It is the fact that when measuring frequencies we are referring to pitch levels.

A440 means that A beats @ 440 cycles per second. This is what is called International Standard Pitch which was agreed to in 1939 at an international conference. This technical standard was taken up by the International Organization for Standardization in 1955 and reaffirmed by them in 1975 as ISO 16.

Interesting to note that the original standard was 439, but there were complaints about this. The “diapason normal” varied from 440 and the culprit was thought to be the temperature the French took the measurement at. The other complaint about the French using 439 was that 439 is a prime number.

So back to the OP question.  I believe the assignment has been misunderstood because of a lack of knowledge about music, musicology, and what defines pitch.

The answer would be once A440 varies from 440cps it is out of tune according to the international standard frequency.

Now it may not be out if tune with the rest of the instrument but that is not what is being asked.

The first sentence of the OP reads:

Hello, I'm currently on an assignment where I'm needed to analyse the frequency spectrum of a piano note...(snip)

The OP goes on to state that he/she has found the frequencies for other notes. This has nothing to do with being in tune or out of tune.

 It has to do with the measurement of pitch, which is a frequency cycle.

[j_menz]

So back to the OP question.  I believe the assignment has been misunderstood because of a lack of knowledge about music, musicology, and what defines pitch.

The answer would be once A440 varies from 440cps it is out of tune according to the international standard frequency

Lovely for A, assuming you (I think wrongly) assume all three strings should be tuned to exactly 440; useless for any other key.

[hfmadopter]

Lovely for A, assuming you (I think wrongly) assume all three strings should be tuned to exactly 440; useless for any other key.

I think in the purest sense of the meaning in terms of pitch, then  "in tune" means all three strings would be for instance A 440. In terms of my ear it comes across as mighty pure. I tune one string to A 440 exactly by digital instrument and don't measure the other two generally unless they are giving me trouble and I want to see a reference number. I tune those to what my ear likes to hear but in terms of tone I seem to favor a slight difference. So I guess my piano is not tuned to A 440 in reality. I've asked the question here before if others tune all three strings exactly the same or not and got no reply. They probably think I'm crazy.

Tuning too purely to me gives a less resonating result according to my ear. If we could tune purely then tuning just octaves should work and we know it doesn't.

 On another note, my G1 is an absolute bear to work with. That register from G1 to E 2, to get a result friendly to my ear in several circumstances of playing is basically my nemesis in tuning my piano. I can get that G in sync with the rest of the keys and then a C chord sounds hideous down there for instance. I can get the G and D to harmonize but add the B and it's mud but the B works with the rest of the piano. Something in those strings right there at that register is haunted by a demon I think ( I think the demon is rust ). Fortunately not a lot of pieces use a tonic chord position down that deep in the register and so I let it stand as long as I get most of the notes to work/ harmonize with the rest of the piano.. Back in the summer I released most of those strings and re twisted them, maybe I've over or under done a couple of them.  At least they sing now where they used to deliver a dull thud and were even less tunable.

[silverwoodpianos]

One more time using different words;

At international standard pitch of A440 each key in the tempered scale on a piano has a particular frequency cycle. For example the lowest key beats at 27 cps while the highest key is more than 4000cps.

If a key deviates from the demand of a particular frequency cycle in the pitch of A440 it is then considered out of tune to that particular scale frequency.

One can have whatever pitch frequency level one chooses to have (within reason) 440, 441,442,443, 452,415,439, etc.etc.

The particular frequency of each key inside those pitch levels is determined by the mathematics of the scale chosen.

A deviation of a half cent cannot be heard by the human ear. But that does not mean it isn’t there.

 Today we use tempered intonation instead of just intonation. This means that the musical scale is tempered slightly out of tune so playing in each key sounds relatively fine to most lay people.
Theoretically the entire piano is slightly out of tune. But once again that is not the question being asked.

The question is how to determine if the frequency of a particular key of the piano is out of tune. The first thing to determine is what pitch level has been agreed to. Most likely the instructor is referring to international standard pitch.

Something else;

This has nothing to do with three strings of a unison. It has nothing to do with the particular key A440 on the piano; that is simply a chosen standard of pitch frequency cps for a particular key out of 88 total...

If three strings do not match the same frequency they are considered out of tune with each other.

Whether those same three strings are at the correct pitch for the scale set on the instrument is to be determined by a tuning fork or tone generator.

[iansinclair]

Perhaps the problem word is "considered".  From a musician's point of view, it's out of tune if you can hear that it's out of tune.  For a thoroughly trained classical musician, that will be less than half a cent (I differ there with silverwoodpianos) when compared with a precision tone generator (not working with intervals here at all).  For a lay person of ordinary background, it will be considerably more.  For a person whose ears have been assaulted by iWhatnots and rockbands for a decade, the question is probably moot.

Then the next question is... how stable is that?  Your average crystal controlled oscillator is good to perhaps one part in 10,000.  A top notch one may be good to 1 in 100,000 or better.  An atomic resonator in a fully controlled environment will be good to one in a billion or better.  A piano string?  If you change the temperature a few degrees the frequency will change significantly -- but so will the whole piano.  A violin string?  That's why there are no frets on violins.  And so on.

So a one sentence answer would be: a note is out of tune when the person listening to the note can hear that it's out of tune.

An aisde to j_menz: bass strings are a bear, aren't they?!

[j_menz]

An aisde to j_menz: bass strings are a bear, aren't they?!

I don't actually do my own tuning, but they are the ones I am always least satisfied with.

On the three strings being tuned to the same frequency, my understanding is that a slight variation is desirable and used to shape the  decay pattern/attack of the note.  It would be rare that a well tuned note would have them exactly the same.  On silverwood's analysis, that would mean that all well tuned pianos are out of tune, which seems a somewhat unweildy result.

[hfmadopter]

An aisde to j_menz: bass strings are a bear, aren't they?!

j_menz is least satisfied with the result of  bass string tuning . However, I think they are a bear , yes ! At least so on my particular Henry F. Miller anyway..

Edit:
I should go on to add that there are few things less desirable on a camping trip than a bear entering your camp site except perhaps one entering your tent and if you have to face that bear you have a serious struggle on your hands. It would be a tad worse than having a sticky wicket ?!
David G.

[thesixthsensemusic]

I don't actually do my own tuning, but they are the ones I am always least satisfied with.

On the three strings being tuned to the same frequency, my understanding is that a slight variation is desirable and used to shape the  decay pattern/attack of the note.  It would be rare that a well tuned note would have them exactly the same.  On silverwood's analysis, that would mean that all well tuned pianos are out of tune, which seems a somewhat unweildy result.

Exactly, the same principle applies to sound synthesis with digital synthesizers, often using 3 oscillators simultaneously, and detuning them with one slightly higher, one centred, and one slightly lower, is indeed essential in generating sounds with a rich tone colour. I am quite sure that it's the same in a piano, which basically has 3 'mechanical' oscillators per note for most of the keyboard range...