When working out intervals in a musical score, you need to check if the notes are in different clefs, or need to be transposed, before you can begin to work out the interval.
Intervals can be harmonic or melodic.
Melodic means “in the melody” and refers to notes which one instrument plays, one after the other. A melodic interval transposed into concert pitch (or another key) does not change its number or quality, so melodic intervals do not need any special working-out methods.
Harmonic means “in the harmony” and refers to notes played by two separate instruments (or e.g. piano), at the same time. Harmonic intervals must be transposed into concert pitch before the interval can be worked out.
Here are two notes for clarinet in A, and cor Anglais.
A#-D# may look like a perfect 4th, but it’s not!
Both of these are transposing instruments, so the notes that you would hear are not the same as what is written. We need to transpose each note into concert pitch, using the same clef, and then work out the interval.
The clarinet in A needs to be transposed down a minor 3rd, and the cor Anglais needs to be transposed down a perfect 5th. This gives us D# and B#, which is a major 6th (same as D-B is a major 6th).
This example is from Brahms’ Variations on a Theme of Haydn.
What’s the interval between the bassoon and double bass notes marked with the circles?
Bassoons (and trombones and cellos) often have clef changes between bass, tenor and treble clefs. There is a tenor clef in bar 98, so this note is Ab below middle C. (Remember that the C clefs show you the specific position of middle C).
The double bass note is F, but the double bass sounds an octave lower than written.