I have a recording of a vocalist accompanied by guitar. The vocalist claims not to have a good voice but swears he is "singing in tune". Is there anyway to determine if this is true and, if in this case it is not, is there anyway of highlighting where the voice is not in tune and why it is not in tune?
I would try loading the recording into a Melodyne trial, and see if that picks out the voice from the recording reasonably cleanly. If it does, you should be able to see if it is in tune - and maybe even fix the tuning, if you wish!
Being in tune on one level means if the pitch to sing is
A4 - 440 Hz - then the singer is supposed to sing
A4 - 440 Hz. Most can't measure Hz with their ears, you would use some electronic device to measure it. But this kind of measure will not account for vibrato, slides, partially voiced notes, etc. which are all part of good, expressive singing. In other words lots of good singing goes in and out of tune ...tastefully.
If you can't hear it, you might re-assess your actual concern. If you can't hear it, do you really care? Does your audience? If ear training doesn't interest you, why not use autotune?
So you want to verify the recording? Use melody.ml to separate the vocals from the guitar. Use an app like VocalPitchMonitor to get a graphical representation of the vocal pitch. Off-pitch spots should be obvious.
With one instrument, unless someone has (really precise) perfect pitch, the absolute pitch of one note the person is singing is difficult to observe, nor does it usually make a difference.
However, since music is made out of more than just single notes, the relationship between different pitches is objectively measurable and meaningful.
Musical intervals are consonant when their frequencies of the notes are a simple rational number. This ranges from the "perfect" octave (2:1), perfect fifth (3:2), major third (5:4), down to such dissonances as the chromatic semitone (25:24) or tritone (not even close to anything reasonable).
Of course, perfection is impossible in real life, but luckily from psychoacoustics we know that the brain will attempt to interpret two pitches as if they were a nearby consonance. The error forms the phenomenon of beating (wiki), and the further off it is, the stronger the effect - until it is so severe it's just not a musical interval any more.
There are two ways of forming a relationship: two notes played after one another (melody), and two notes at the same time (harmony). If these relationships conform reasonably to what was intended by the composer, then they are in tune.
Suppose the composer is writing something tonal and wants a perfect fifth (sample). No amount of so-called "artistic licence" will get me to accept a wolf fifth (sample) as anything but an error - even though this is "only" about a quarter of a semitone off. This is what is meant by out of tune.
In short, when there is an accompaniment, the singer can be objectively judged on whether they are in tune by measuring the frequencies of both instruments and comparing the ratio to the expected simple rational numbers that the written musical intervals imply.
A small amount of difference is expected (and required for tempering), but in the immortal words of Stewart J., lightly paraphrased, "I know it when I hear it".