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Recently when playing in a piano trio as a cellist, my teacher told me that a note can have a small range of correct pitches depending on the chord.

Shouldn’t everybody try to play exactly in tune with the piano since it has the definitive pitch for each note?

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    You may find youtu.be/QaYOwIIvgHg useful (Not claiming any knowledge of these instruments myself! ) – Rusi Nov 13 at 6:00
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    The piano doesn't have the definitive pitch for each note. It's really a compromise, where each and every semitone has been shared equally between each octave. So it works in every key equally. Violins, cellos et al, can move certain notes slightly to make them sound better in tune for the key they're in at the time, so a C played in one key won't be exactly the same pitch as a C in another key - necessarily - hence what teacher said. However - the question needs asking! If piano is playing simultaneously, it makes sense. Otherwise, doesn't usually happen. – Tim Nov 13 at 8:18
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    @Tim not to mention that the 2 or 3 strings per piano key are very slightly detuned from each other to enhance the sustain and the perceived sound. – Carl Witthoft Nov 13 at 14:12
  • Even when the piano is playing, if the string part is the "lead," we often adjust accidentals a bit to emphasize their departure from the current key. This makes the deviations more apparent to the audience, without going painfully off-pitch. – Carl Witthoft Nov 13 at 14:14
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    @CarlWitthoft - so very slightly, but that's not to do with anything except producing a richer sound by introducing very slow beats. Certainly not in line with any temperament - unless it's honky-tonk piano, which always brings out a bad temperament in me when I have to listen to it. – Tim Nov 13 at 14:20
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The piano might not even be playing. If it is, it is indeed probably the best reference.

However in other circumstances it is possible that you would play a slightly different pitch for (example) an E natural as the major third on a C chord, compared to the fifth on an A chord.

To understand why this is, you need to understand how intervals (like major third, perfect fifth) are derived from the harmonic series, and the fact that dividing an octave into 12 equally spaced pitches is a bit of a compromise (a necessary one that allows us to play a piano in 12 keys without needing to retune it when we change!) Here is an explanation.

However stringed (and other) instruments are microtonal in nature, and the player should tune by listening to the other players, and also by having some knowledge of the function the note she is playing has harmonically.

When the harmony you are playing is not strict major/minor (for instance a sax section playing jazz voicings with lots of "colour" notes and no fifth) it's even a bit more hard to pin down as the harmonies are intentionally "slightly out of tune" anyway. Playing in a sax/brass section is a lot more sensitive to tuning when the arranger wrote parts that have major triads or even simpler harmony. (This from experience!)

It's a pretty deep subject and one that takes a bit of getting your head around, but in general listening to what sounds best is a good starting point. (People with more experience of chamber music may know of more precise methods than I do.)

  • The more I read and think about this, I don't think it's true that the intervals were derived from the harmonic series, as common as that assertion is. I think rather that the intervals have adjusted to (and away from) the harmonic series over time as the degree and nature of harmonic elements in music have changed. As to chords with lots of color notes, I suspect that attention to using 9:8 or 10:9 whole steps (and even 8:7) would tighten up the sound quite a bit, but I haven't done any detailed work on that hypothesis. – phoog Nov 13 at 20:06
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This is really going to depend on the music. For example, if you are playing a first-inversion major chord (for example, you have a G♯ in an E-major chord) then your pitch should be lower than it would otherwise be (assuming that the E and B played by the other strings are in tune with the piano). However, nobody will notice this much unless the chord is sustained (and even then it will only be very noticeable if it is played with little or no vibrato).

If the strings are playing a sustained passage while the piano is flying around in sixteenth notes, then you don't need to worry nearly as much about matching the E on the piano; instead, you should pay attention to the sonority of the string chord. By contrast, if the piano has a sustained note, you may want to match it. If the cello part doubles the bass part in the piano, you will almost certainly want to match the piano. If the cello part takes on more of a tenor role or especially a solo role, then you may want to inflect your tuning a bit more, if the piano part is written to allow it.

The real thing to keep in mind, of course, is that you should do what makes the music sound best. That will vary from one piece to the next, but you should be alert for passages where it sounds better when you tune to your other colleagues rather than to the piano, and that is, I suspect, what your teacher was trying to get at.

As an example, picked out of thin air (or, more precisely, the first result from a search for mozart piano trio sheet music), consider this trio in E major, K 542:

Piano trio K 542

Much of the cello part here is in unison or octaves with the piano, and will have to be tuned to the piano. The C♯ at the third beat of measure 20, however, is not doubled, so you could tune it acoustically to the E in the left hand of the piano part. An acoustic minor tenth has a frequency ratio of 12:5, which is roughly 16 cents wider than than an equal minor tenth. Since the upper note of this interval is played by the piano, you would lower the cello note to make the interval wider.

The A in measure 21 will need to be tuned to the A in the right hand, but the G♯ in measure 22 could be low, because this is the first-inversion chord mentioned in the first paragraph. An acoustically pure major third is roughly 14 cents narrower than an equal-tempered major third. If the violin's B and E are a perfect fifth apart, the violin's B will be two cents higher than the piano's B (this accounts for the difference between the minor third's 16-cent adjustment and the major third's 14-cent adjustment). The frequency ratios for the B and E are 48:5 (6:5 plus three octaves) and 32:5 (8:5 plus two octaves). (Although the notes do not sound simultaneously, there will probably be some G♯ still reverberating when the violinist plays the E.)

In practice, you might find these lowered notes sound flat and dull because, well, they are indeed flat compared to other C♯s and G♯s that you'll hear from the piano. The solution in that case may be to reduce the adjustment rather than to eliminate it altogether. That is, you may find it helpful to compromise between melodic considerations and acoustically "pure" harmonic intervals. Lowering the notes very slightly from the equal-tempered pitch might be an improvement even if going all the way to the acoustically pure pitch isn't.

  • Roughly how many cents does a vibrato on violin or cello vary from the absolute spot on pitch - on any note? I guess the answer would bring the pitch to match to either/both 'notes' in any temperament, but need to ask. – Tim Nov 13 at 22:40
  • @Tim it's not that simple, of course. Even if the theoretical pitch is within the range of the vibrato, the perceived pitch is going to be near the center of that pitch oscillation. But the main reason I brought up vibrato is that it prevents chords from having the acoustical buzz that arises when they are acoustically pure. – phoog Nov 13 at 23:02
  • @phoog thank you for your expert response. I’m wondering, are there timbral resources at the string player’s disposal—in the sense of adjustments to the upper partials only—which may evoke a psychoacoustic (illusory) effect of flatness without actually impinging on the fundamental? – Tekhnee Nov 19 at 18:08
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    @Tekhnee I'm no expert on stringed instruments, but the timbre is affected by pretty much anything you do with the bow, including changing the speed, the pressure, and the distance from the bridge. The distance has a particularly strong effect on the timbre. Whether any of these create a psychoacoustic effect of flatness I cannot say. – phoog Nov 19 at 19:15
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This question is related to many others about leading tones and equal temperament with good answers and also wikipedia information like this here https://en.wikipedia.org/wiki/Equal_temperament that render excellent explanations with background information so I can give you a short answer without repeating the entire history of music and temperament.

It is a fact that wind instruments and also the human voice like string instruments don’t play the 12 tones always exactly in the same pitch depending their function as leading tones.

E.g. melodic:

F# as a 7th in G-major scale will lead to G (root of G) while Gb as a 4th in Db will lead to F (3rd of Db).

Harmonic:

F# as 3rd in a D7 (dominant chord of G) has not the same function like Gb as 7th in Ab7 (dominant chord of Db).

The mentioned instruments above can adjust (adapt) the leading tones and also other intervals like 3rd or 5th relating to their melodic and harmonic function.

A brass band leader has always to correct his bandsmen: This note with a sharp has a tension upward, try to play it higher (or opposite: play a flattened accidental note lower).

As the piano has fixed but well tempered tuning the pianist can’t do this adjusting - but the other musicians are able to play the notes according to the piano.

Shouldn’t everybody try to play exactly in tune with the piano since it has the definitive pitch for each note?

They should and they will do it, they even don’t have to mind it, they do it by the ear and they find the tuning. (In opposite to some amateur bands ... what makes them sound sometimes so horrible.)

  • The abbreviation for equal temperament is ET. The first T in TET stands for tone, which is only necessary when identifying a particular equal temperament. For example, 12-TET, stands for twelve-tone equal temperament, where the 12-T stands for twelve-tone. – phoog Nov 13 at 20:09
  • "This note with a sharp has a tension upward, try to play it higher (or opposite: play a flattened accidental note lower)": this melodic approach is diametrically opposed to the modifications dictated by harmonic considerations: Sharpened notes are usually the upper pitch of a major third or sixth or the lower pitch of a minor third or sixth, so they should be lower, while the opposite is true of flattened notes. Perhaps these two considerations cancel each other out, leaving us just with equal temperament. – phoog Nov 13 at 20:14
  • @phoog - this answer is only concerned with 12tet, there are no other references. Most of the time 12tet is the basic premise that tuning is discussed from - equal temperament using numbers other than 12 cannot really be compared with the other generally known tuning systems. – Tim Nov 13 at 22:36
  • @Tim what you say is true. Is my comment inconsistent with that? I'm only saying that TET isn't the abbreviation for "equal temperament." It's actually the abbreviation for "tone equal temperament," which is not a particularly useful phrase. Useful abbreviations, therefore, are 12-TET and ET. By itself, TET, isn't particularly useful. – phoog Nov 13 at 22:47
  • So I’ll gonna edit this TET! – Albrecht Hügli Nov 16 at 7:53

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