This web page goes into great detail about this important subject but the principles involved should be grasped by all instrumentalists.
‘He that touches pitch shall be defiled therewith.’
(Ecclesiasticus Chapter XIII Verse i)
Frank Morelli (http://www.morellibassoon.com) used to sell a CD on his website which would play any particular fixed note that you could check your intonation with. When I first came across this I sighed deeply and muttered something unprintable about Americans. How earnest they can be! Just imagine hordes of over-keen bassoon students religiously sitting in their living rooms playing long notes in unison with the CD. But this is not the point of it. Please read on…
I believe this CD is no longer available but an extremely useful resource is this tone generator: https://www.szynalski.com/tone-generator/. This or something similar can be used in conjunction with a tuning machine. You will need to use a contact microphone to connect your instrument to the tuner.
Now let’s get the the tone generator playing, for example, the note middle C. (the volume should be turned up quite a bit). Shut your eyes; play any C; listen till it sounds in tune; open your eyes and, surprise, surprise, the needle on your tuner is in the middle.
Play tenor G or F in the bass clef (so as to make a perfect fifth) and the reading will also be where you’d expect it (well, almost).
Now let’s move on to thirds and sixths: eyes shut again; play a tenor E and get the chord to sound resonant and warm. You might be able to hear the ‘resultant’ (the low note that is generated when two higher notes are played together also known as a Tartini tone) which should be a low C. (There is a detailed website on the theory behind resultants or interference beats and how they work at http://www.phys.unsw.edu.au/jw/beats.html). Now open your eyes and see how far the needle is from the centre and in which direction it is deflected. For many who are not sure what to expect, this can be a true epiphany – a revelation that that will revolutionise your playing for evermore.
Now let’s try the same with a high A (the resultant should be a low F) a middle A flat (resultant: A flat below the bass clef) and a tenor E flat.
Once you realise just how far you need to ‘bend’ a note to make it fit into a chord, you’ll appreciate how important it is to play on reeds that allow this amount of flexibility.
Because of the complicated mix of overtones on a bassoon, it can be difficult to get some of these two-note chords to ‘gel’ as well as some others – but don’t worry about this. Then try this out on various different notes on the tone generator. One word of warning: wait till there is no-one else in your house or flat before you do all this or you’ll drive them mad.
I remember when I had my epiphany. I was at a lecture given long ago by the highly respected English flute player and teacher Trevor Wye ( http://www.trevorwye.com/ ) who rigged up a big sine wave generator and a flautist wired up to a big pitch meter. The flautist ‘bent’ her note till we all agreed that it sounded in tune and then the reading from the pitch meter was shown to us.
There is some detailed information on different temperaments at http://www.vibrationdata.com/piano.htm.
One advantage I had in my position as second bassoon in an opera orchestra over my colleagues in symphony orchestras is that more often than not the orchestral parts are ‘double’ parts (i.e. first and second parts printed together) so it’s comparatively easy to work out what the harmonies are and therefore which way, if any, I should be ‘bending’ my note.
A word of warning here about leading notes (the seventh note in a scale): you might hear people talk about sharpened leading notes; this refers to the raising of the note by a semitone. For example, in the key of G minor the leading note is often F sharp not F natural but that F sharp should still be played to sound in tune within the chord (as likely as not to be a major third above a D natural).
People can get very worked up about intonation. My advice as a second bassoonist is to talk about it as little as possible. Generalised comments like ‘so and so always plays sharp’ are counter-productive and best ignored, hence the warning at the beginning of this entry taken from the book of Ecclesiasticus in the Apocrypha.
In much the same vein, Long John Silver in Robert Louis Stevenson’s ‘Treasure Island’ says: ‘You can’t touch pitch and not be mucked’.