I don't mean ear training - naming intervals, or playing back melodic phrases by ear. I mean - becoming better at noticing when you are perhaps 20 or 30 cents out on a note. For instance if I play certain octave intervals on saxophone they do tend to be slightly wide. It is all to easy to get used to that. I am interested in how to become more aware of it.
You mention that you are not talking about traditional interval, transcription, or melodic dictation ear training exercises. You also mention that you want to improve the ability to notice differences in expected intervals, like noticing if an interval is out of tune, in which direction, and by how much.
I've found that tuning a string instrument (in a particular way, which is what this answer is about) is a great way to develop and improve that ability. Just doing it once at the start of each practice session will cause a noticeable improvement, so it can be a small part of your ear training or overall practice routine and still be more than worth it.
You use both your ear and a tuner. Your ear first to tune as accurately as you think you can, and then the tuner to see how much you were off and in which direction, if at all. You can do this on a string by string basis, or only one time once you finished with all strings.
Just tuning it in any random way isn't very useful though, at least for the purposes of the question and exercise. Even if you only use your ears, tuning using unisons and octaves exclusively has very limited usefulness and results. It's better than relying on a tuner, but not by much (in my opinion, anyone disagrees on this one?).
What I do is grab any guitar-like instrument I see close (or the one that I'll be practicing with at that time, or whatever), then:
Detune each string randomly. Ideally you want some strings to be sharp and others flat, so you can practice all directions, so don't be so random about it. Don't overdo it, be careful. You just want to be at most two semitones flat or sharp. You can go flatter if you want to, but going sharper can put excess stress on the guitar, so avoid going beyond two semitones sharper than the usual tuning the instrument is designed for.
Use a reference to tune one of the strings. I use a tuning fork (normally an A fork to tune the 5th string of a classical guitar) but you can use audio, another instrument, or whatever you like.
Tune all the strings using the string of step 2 (or other strings you already tuned in later iterations) as reference.
- First select which string you want to tune next. It's easier with strings that are adjacent to the reference string of previous steps, but you can practice with any string once you feel comfortable tuning the adjacent ones.
- Don't use the 5th fret method (for strings tuned in fourths) aka the octave / unison method, or the harmonics method yet.
- Play the two strings. Using the reference string, well, as reference, is the target string in tune? Is it sharp? Flat? By how much? On a typical guitar, using the 6th and 5th strings as example (E and A respectively) you should be hearing a perfect 4th interval if both strings are in tune. If the instrument is tuned in fourths and you consciously tuned some strings a little sharp and some strings a little flat semi-randomly, you'll be hearing some tritoneoid and some major thirdish intervals. If the string is not in tune (most likely scenario), make corrections. Repeat this step until you feel like both strings are in tune (the first times there will be a big range at which you will not be able to tell, that's ok).
- Now use the unison or octave test (5th fret test on typical guitars) to do smaller corrections. Now that you are comparing a note with itself in two different strings, it'll be much easier to hear if the target string is out of tune. Make further corrections if needed.
- Now use the harmonic test. Pay attention to the pitches, but also the speed of the beatings (which are easier to hear using harmonics, and are the actual beauty of the harmonic method imo, but a lot of teachers skip the beating part, I don't think beatings are even mentioned in the article I linked for the harmonic method, but I'm mentioning them here anyway). The beatings become slower as the strings become more in tune, until they become so slow they are infinitely slow at perfect tune (no beatings at all). The beatings become faster as the strings become more out of tune, until the beating enters the pitched sound range around 20Hz. Make sure you notice both! Make further corrections as needed.
- You can now check how close you are with the tuner. Or you can wait until all strings are done to do the tuner check. Make last corrections with the aid of the tuner if needed.
- Very important: Pay attention to the continuous interval change, the complete gradient. Notice how a major second turns into a minor third, then slowly into a major third, to then arrive in the exact mid (hopefully!) of the target perfect fourth. To achieve this play the string continuously and rhythmically at the same time as you make changes in tuning. Notice how slightly sharp or flat stuff sounds like. Notice how all degrees of sharpness and flatness sound like. Notice how it feels to slowly and eventually arrive to a pitch that is in tune (this sense will develop and improve with time). This is the heart of the exercise.
Repeat step 3 until all strings are in tune.
This can be a tiring process for some people (I find it very relaxing and therapeutic, because of the amount of concentration you can achieve, meditation-like if you get really into it), so once a day should be more than enough. As always, if you do it 2 or more times a day you'll improve much faster, but at the same time it increases the chances of a burn out. Consistency is better than brute force in the long run.
Eventually you'll become pretty fast, that's the moment you really want to start thinking about doing it multiple times a day, and with different tunings: drop-n, thirds (open), fifths, whatever, the point is to target different intervals. You can also practice different intervals by fretting a reference string, and using that new fretted pitch as reference.
Now I am aware that tuning is far from an absolute anyway - there are various systems, and a minor third in a blues band may be a very different thing from a minor third in a barbershop quartet (and even depending on what its function is in the chord also).
If you use a 12 tone equal temperament instrument and tuner, you can use those as base or reference from which you "measure" other temperaments. Flatter in some cases, sharper in others, but you have an actual base that can be used to identify and compare differences.
But even so - everything depends on having a well developed sense of "relative pitch" - far more than just being able to name intervals, but a really fine tuned sensitivity to fine variations of pitch, relative to the rest of the musical setting.
Do you want to avoid interval training? Or are you doing it already? Because it is the base in which all classical ear training is based upon. Intervals are part of our most basic measurement system, that we'll use to communicate, classify, abstract, and more importantly: reason about aural relationships between pitches.
In other words, you first need to know how a perfect fifth sounds like, before you can objectively judge how sharp or flat a potential perfect fifth might be. In the case of the proposed exercise above, and assuming a typical guitar with standard tuning and adjacent strings order, if you want to tune the strings without the unison, octave, or harmonics methods you need to at least roughly know how the perfect fourth and major third intervals sound like. Otherwise you go directly to the other methods that rely on unison, octave, and beatings; which is ok, but music is made of many other intervals, so relying only on octaves makes no sense.
It would be extremely interesting to investigate this ability scientifically by means of a note produced electronically and adjusted with a slider or similar. Maybe it has been done.
That's exactly what we are trying to achieve with this tuning exercise, but acoustically.
One way to take these type of ear training exercises to a completely other level fairly easily (as in, make them much more efficient, improve much faster and accurately, without much more effort) is to sing them too. At least the larger intervals, since at the start singing anything smaller than a semitone can be very challenging, but trying to sing smaller intervals that are out of tune (like slightly out of tune unisons) is very fun!