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I know that while, for many instruments, enharmonic notes are a single pitch and their differences are more of an interpretation, helping music to remain consistent (it would make more sense for a scale to have C and D♭ than it would to have C and C♯), there are certain instruments that can cover a more-continuous, less-discrete range of pitches. The violin is one such example and while a majority of music written for the violin still treats enharmonic notes as the same pitch, there is music where a distinction between E♯ and F exists (I believe that E♯ is slightly higher than F by an interval called Pythagorean comma).

I want to develop the kind of intonation for this on my violin but I'm not entirely sure how. When I practice intonation, I reference an electric tuner, as sadly, I didn't start playing music until adulthood and missed the window of opportunity for absolute pitch. This works fine for most notes but it doesn't work for enharmonic ones; I've looked all over the place and I haven't found a single tuner that distinguishes between enharmonic notes (I'm not even sure such a thing exists). How do violinists get this kind of intonation?

Addendum: I recently spoke with a violin teacher about this. According to her, because the Pythagorean comma is so small, it has no real practical use in music that is physically played by a person. It's used in 'conceptual' and 'abstract' stuff like the math behind different tuning systems.

Apparently I had misinterpreted information like 'violinists can play B♯ and C as individual notes;' I had thought 'can' in this context referred to a skilled violinist having the capacity to intentionally play them individually, while the word 'can' was actually referring to it technically being possible, as violins have a non-discrete gradient of pitches. While a well ear-trained violinist could carefully make adjustments to their finger positions (more or less a slight shift in the muscles) to get the particular pitch, it's virtually impossible for a person to develop their intonation the point of reliably playing a Pythagorean comma simply because of human error.

She also went on to say that the smallest interval I'd ever really need to learn is a quarter tone (half a semitone) and that's more just if I was interested in certain types of Middle Eastern music.

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    By not using a tuner! Tuners generally speaking are using 12tet, so will not help you to develop the important part used by musos - their ears! Playing up and down scales in all keys will help (if you listen hard) to put your fingers in the subtly different places for certain notes in certain keys. After a while it will become second nature, depending what key you play in, as to exactly where you finger F# or Gb - depending whether that's a M3 or P5, or whatever in whatever key. Eschew reliance on the tuner!! – Tim Jun 16 at 8:23
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    I don't believe it's well defined. If E# "means" the 12th step in a circle of 5ths starting F then yes, it will be sharper than F by a Pythagorean comma. However if it "means" A's augmented 5th (F (natural) being the minor 6th) it's flatter probably by more than a comma – Rusi Jun 16 at 11:35
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    This calculator gives me 2 commas flat. IOW E# seems to vary 3 commas (at least) around F. – Rusi Jun 16 at 11:42
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    @Rusi is correct in that there are different versions of E♯. Thus you can not make a tuner that makes a difference between E♯ and F, because which version of E♯ should be indicated on the tuner? There is no right version. Tuners are adjusted to the system of 12 equal half steps in one octave and therefore enharmonic notes on the tuner are regarded as the same note. Tim is also correct. – Lars Peter Schultz Jun 16 at 11:44
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    Some evidencr that Mozart's father taught Mozart to play the violin with sharps one comma flatter than flats – Rusi Jun 16 at 12:15
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Intonation in classical music is one of the things where there isn't a single, catch-all rule that you can just apply to get it “right”. Really, it's part of an interpretation. Perhaps the only thing we can say universally is that it should have a purpose.

12-edo is a good starting point. Although I often rant against the dominance of 12-edo, it can't be denied that it works remarkably well for most Western music. Non-professional string players often deviate from 12-edo by too much, in a way that does more harm than use musically. So, that's not really the reason against using a tuner or piano as pitch reference: likely enough, getting closer to 12-edo would be in most cases be an improvement. I will admit that this is also still the case for my playing.

The real problem with a tuner is that it doesn't teach you the purpose of notes and their intonation. A central device of classical music is the contrast between tension and resolution, dissonance and consonance. And this is something in which both the composer and the performer are involved. As a rule of thumb, I'd say as a performer you should try to understand the composer's intention as best as possible, and then make it really clear to the audience. Many great performers do this by exaggering everything – not specific to intonation, perhaps the best way to see it is if you look at tempo fluctuations. Here's an extreme example (possibly exaggerated for the sake of both satire and education):

Notice how all the fast passages are played even faster then they would be according to metronome, and the slow ones stretched even further with fermatas.

And that also is the main factor I see to intonation in classical music: it should put extra tension on leading notes in dissonance chords, but make resolutions extra harmonious. We can summarize this to three points:

  • “Foundation” notes, in particular the tonic and its fifth or the subdominant, should be played in straight Pythagorean or 12-edo (which is in that case almost exactly the same). This is based on very simple frequency ratios 3:2 and 3:4, which gives the music a solid framework. It's easily to hear those intervals: a slightly detuned fifth has a notably beat in it, because the 3rd overtone of the fundamental will go in and out of phase with the 2nd overtone of the fifth.
  • “General mood” notes, most prominently the thirds of major and minor chords, should be played in such a way to bring out that mood best. Here we're already getting into ambiguous-interpretation territory. Like the Pythagorean fifths, major and minor chords can be derived as just-intonation ratios, namely 5:4 for the major third and 6:5 for the minor third. I highly recommend you practice hearing those as well, and detecting the beat when they're a little bit off. And in many cases, this ratio is also what you should go for in an actual piece. In particular the 4:5:6 major chord is a very stable, resonant sound. But this is actually more of a concern for middle voices, which have the third of a concluding chord more typically than the main-melody or bass voice does.
    The 5:4 third is a little narrower than the 12-edo major third, while the 6:5 is a bit wider than the 12-edo minor third. IOW, 12-edo exaggerates the difference between major and minor thirds – not much (15 ct), but it is audible. It could be argued that this is a good thing, but IMO it depends. In fact one could thing that always using an over-wide major third has a bit of a silly, restless caricature effect, similar to how Yo Yo Ma exaggerated the tempo expressions in the Sesame Street example I gave above. That appears to be not really the case in this case, but perhaps only because we're so used to 12-edo instruments like piano that we harly notice it anymore.
  • Melodic leading notes, typically appearing on dissonances, should highlight the need for resolution. In particular, the ⅶth-degree note leading up to the tonic is very often played significantly higher than its 12-edo frequency, let alone the one you get from stacking a JI major third on top of the fifth. The idea is that this should suggest the tone “gravitates” towards its resolution. An extreme proponent of this thinking was Pablo Casals, who would sometimes intonate leading tones as much as 50ct sharp from 12-edo. Again, you could consider this a bit caricature-ish, but I think it can definitely work if only it's done judiciously, i.e. not according to a fixed rule but always in the way that conveys what you're trying to express musically.
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    Hearing beats is all very well when there are others playing the other notes which may, or may not, set up those beats. However, when one plays alone, no beats will ever be heard, on the assumption single notes are played. The OP sounds a little inexperienced - most seasoned players wouldn't be concerned using tuners for this purpose - therefore will probably only be playing single notes. And, again, tuners will not be a gret help if one wants to play a note sharper or flatter than it is normally in 12tet, although I suppose one could watch the needle! That apart, good answer! – Tim Jun 16 at 13:51
  • While I will say this does seem like good advice, I feels like it kind of misses the point of my question. How can I improve my intonation so that I can play D♭ as a distinct note from C♯? – Kevin Jun 18 at 2:43
  • @Kevin well that's the point of my answer: it's no good playing D♭ different from C♯ just for the sake of... annoying pianists, or whatever. Instead, what you should do is train your ear to figure out for each note why it should be intonated at a particular pitch. For that, I explained the three categories of notes and how each of them should be approached, in terms of intonation. Until you feel comfortable in that system, I suggest you err on the side of 12-edo, i.e. to not play D♭ as a distinct note from C♯. – leftaroundabout Jun 18 at 8:12
  • @leftaroundabout Ah, that clears it up. I was a bit confused about why the accentuation of things for their function was necessary. I see now. – Kevin Jun 18 at 8:51
  • That said: I personally like meantone temperaments, in which the major third is better approximated than in 12-edo. In those, C♯ is generally a little bit lower than D♭. But in practice, this mostly amounts to watching out for C♯ notes which act as the third of an A-major chord, and play those a little bit flatter than you'd play another C♯ or D♭. But the same thing applies to E notes which appear as the third of a C-major chord: you may also want to play those a bit lower than the E which is the fifth of an A-major chord, despite the name being the same in both cases. – leftaroundabout Jun 18 at 9:21
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What you can do is be aware of the leading tones (up or down) and support their function by exaggerating a little bit the distance where you place the finger on the fingerboard: for ♯ (sharps) a little bit higher, for ♭ (flats) a little bit lower.

  • Unless you're looking for justly tuned thirds, in which case the sharps should be loweer and the flats higher. – phoog Jun 17 at 3:12
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    Who decided they could delete my comment? Only made in jest! – Tim Jun 17 at 5:56
  • @Tim I think it was me. Sorry. – luser droog Jun 18 at 4:52

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