'tonic of a major scale'

I haven't read that particular book, but from what you've posted I can see why it's confusing.

We measure the size of intervals from the lower note up. We CAN go from the top down (and often do for the purposes of music theory exams), but I've always found it easiest to go up, and I've never met anyone who's found it easier to go down.

The lower note is the tonic of the scale you'll use to measure the interval. If that note happens to be G#, you're correct - you'd use a G# major scale to calculate the interval size. But since G# contains a double sharped F, it can be confusing to do that... which is why I'm guessing the author talked about enharmonic notes.

So I'll guess what the author proposes, and then I'll give you the method I teach. To wrap up, I'll explain why you can't just count tones/semitones.

Let's say the interval in question is G# to C.

1. If you consider the enharmonic equivalent to G#, that's Ab.
2. Since you've raised the letter name of the lower note (from G to A), you also need to raise the upper note's letter name (from C to D). Finding the enharmonic equivalent means the top note is now Dbb (D double flat)
3. Count the letter names. A to D is a fourth. The interval Ab to Dbb is therefore some kind of fourth.
4. Adjust the interval for the accidental on either note - I'll start from the bottom: lowering the A note to Ab means you've increased the interval size. Ab to D is an augmented fourth.
5. Adjust the interval for the accidentals on the other note. We've changed D to be Dbb; it's been lowered by two half steps. Lowering an augmented fourth by one half step gives you a perfect fourth; lowering it one more gives you a diminished fourth. That's the answer.

The method I teach:

1. Ignore the accidentals. Our original interval is G# to C; G to C is a fourth.
2. C is in the key of G, so our starting point is a perfect fourth.
3. Only G has an accidental, so only one adjustment is needed: G has been raised, and since it's the lower note, the interval has gotten smaller. Lowering a perfect fourth by a half step gives you a diminished fourth. That's the answer.

Why we can't count half steps:

Interval size is one of those areas of theory that depends on WRITTEN notation, rather than sound. If you count half steps from G# to C, you'll get four half steps. Converting that back to an interval, you'll find that four half steps is a major third. But that's the wrong answer - Ab to C is a major third (as is G# to B#), but neither one is the same as G# to C - interval size is determined by the way notes are written, and if one's a space and the other's a line, they can't be a third.

Make sense?

Interval size is one of those areas of theory that depends on WRITTEN notation, rather than sound. If you count half steps from G# to C, you'll get four half steps. Converting that back to an interval, you'll find that four half steps is a major third. But that's the wrong answer - Ab to C is a major third (as is G# to B#), but neither one is the same as G# to C.
I can deal with this, it's weird, but it's a rule; it's consistent, I'm happy.
1. Ignore the accidentals. Our original interval is G# to C; G to C is a fourth.
2. C is in the key of G, so our starting point is a perfect fourth.
3. Only G has an accidental, so only one adjustment is needed: G has been raised, and since it's the lower note, the interval has gotten smaller. Lowering a perfect fourth by a half step gives you a diminished fourth. That's the answer.

Thanks, this makes sense once you've learnt the major scales. I assume at step two if you had F to B you would adjust for B not being in the key of F and raise the interval to an augmented fourth.

Actually, their example for 'if the lower note of an interval is not the tonic of a major scale' is for finding the note at an interval: a 6th above G#, they go to Ab, the sixth of which is F and the 6th of G# has to be an E, therefore E#. They do also suggest a method similar to yours for finding an interval, though they don't give steps, just an example (Cx to B,C to B is a seventh, then drop two semitones to get a diminished 7th).

I suppose what gets me confused is that I understand how the intervals work and how to measure them, but I don't get why there are these odd ways of approaching working them out. I think seeing step #2 in your explanation helps, in that they're ways to manipulate the scales and keys you already know, i.e. the scales work as 'times tables' for intervals. Once you know them you don't need to work out intervals directly and can use these as shortcuts.

You've got it - if the interval is F to B, it's some kind of fourth. B isn't in the F scale, but Bb is - so the upper note is raised a half step. That makes the interval bigger, and F-B is an augmented fourth.

The reason we have these odd ways of working things out has to do with the development of music (and music theory) over time. A thousand years ago we didn't know about sharps and flats. When we started using them (for flats, that's about 800 years ago; for sharps, about 550) they were added one by one. Although the written record of what happened is spotty, we've still got ways of figuring out the development - you can find very old church organs with just one black key per octave (Bb).

Our modern tuning method, called 12TET (twelve tone equal temperament) is a fairly recent development. It isn't what Mozart or Bach used. And not too long before Bach's day, F# and Gb were actually different pitches - you can find old instruments (especially virginals) with 14, 17, or even 19 keys to the octave.

Music theory grew up around these more complicated tuning systems. When we simplified things to what we use today, the way we labeled intervals was already established.

Music theory grew up around these more complicated tuning systems. When we simplified things to what we use today, the way we labeled intervals was already established.
Thanks, it is helpful to be reminded it isn't intentionally perverse.
Although the written record of what happened is spotty, we've still got ways of figuring out the development - you can find very old church organs with just one black key per octave (Bb).

I remember a chorister friend (who unfortunately lives too far away to quiz about music theory) getting very excited about seeing 14th C antiphonaries in Italy.

I love old music manuscript. I own a handful of 18th century scores, and it's really fascinating to see how our notation has changed, even in pretty modern times. In the 17th century we hadn't really evolved a common standard for our key signatures yet, so you might see the key of A noted with a single sharp (G# - I guess they figured you were smart enough to know that if G was sharp, so were F and C). A century earlier, the sharp itself hadn't evolved yet - you might see a natural with an 'x' through it, indicating it wasn't natural, but it wasn't flat either. Back another hundred years and we had only four staff lines, and some clefs now obsolete.

Every generation makes new music, then wrestles with how to write it, and both in turn affect how theorists think about it. Today we're struggling with efficient and effective notation for highly chromatic pieces, microtonal works, complicated polyrhythms, bitonality, etc. A few hundred years from now historians will find some of our efforts and view them as ridiculous :)