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According to my reference
"By an interval in music is meant the difference in pitch between any two notes."

In my book this difference is a cardinal number ... C to G is 7 semitones.

It is therefore total rot to suggest that "Intervals are ordinal."

Shame on you prof for not majoring in mathematics like some of us.

Forget about labels for a moment. How do you think invervals?

For example, for me augemented is:

x-4-4-x, meaning that the octave just takes you back to where you started and it is divided evenly.

diminished:

x-3-3-3-x

major scale:

x-2-2-1-2-2-2-x(1)

etc.

Hi GaryD

Here’s a schedule of the absurd interval names with which we burden our students ... the nonsensical fine line drawn between the "enharmonically equal intervals" will never be
understood by our Computer generation ... only old fuddy-duddies like us (count me out!) can possibly defend calling the same single semitone interval in the one case AUGMENTED PRIME (#) and on the other MINOR 2ND (b).

intervalnames

Diminished and augmented is just another way of saying add or subtract a semitone.

JS Bach’s monumental WTC (Well-tempered Clavichord) is founded on the acceptance of a keyboard octave of 12 equally spaced notes (thus well-tempered) to gain free modulation into 12 different major and minor keys ... at the cost of a slight tweaking of certain notes to achieve the equal semitone spread ... but we have got so used to this "off-key" tweak that we no longer notice ... violinists on the other hand, with their absolute pitch range, might well wince in adjusting to some of the sounds coming from the well-tempered piano.

If everyone would just add semitones in their heads, (and label the names modulo 12), everything would be EASY ! That's what computer MIDI does, and what I thought as a kid decades ago. But doing that in real time is for computers, not humans...

Now I also think 'intervallic reading' is a great help to us fallible humans... only it gets even more complicated in non-C key signatures - to keep track whether the # is in the key signature or not etc. Or, easier back to addition modulo 12 ? What's easier, what pro musicians (not us amateur geeks) do ?

e.g.
A 5th above G# is D# (and they are NOT in the key signature of G Major)
A 5th above F# is C# (and the first is, and second is NOT in G Major)
OR
G# 8+7=15 -12=3 D#
F# 6+7=13 -12=1 C#

And... I'd still want to understand in what way pre-well-tempered intervals differed - in what way the Circle of Fifths didn't close. I understand the basic definition there was 1 tone = 2 fifths - 1 octave ? and the major third was 2 such tones, or the harmonic 5/4 (quite far from such approximations) ?

btb, I can't tell whether you are just trying to provoke an argument here, but if so I'll take the bait.

As Kreisler said, C to G is a 5th because G is the 5th note in the letter sequence beginning with C. In the preceding statement, "C", "G" and "5th" are meant in the general sense, not specifically C natural or perfect 5th. It holds for Cb to G#, Cx to G natural, etc. The 5 is an ordinal number.

As you said, C natural to G natural is 7 semitones. This 7 is a cardinal number.

These are two different, equally valid ways of thinking about intervals. Each has its appropriate use, and the two are not in conflict.

Mustn’t twist your tail Ferdinand ... at least you concede that an interval of 7 semitones (C-G) is a cardinal number ... sorry if I insist that an interval by definition can’t be anything but a cardinal number.

Use of the number series (minor 2nd etc. ) is a ill-judged dodge by musicologists, who had run out of suitable number formats for their musical lingo ... only for the dusty blighters to add the heavy-going I, II, III, IV, V, VI, VII, VIII format to differentiate the triads ...
(not forgetting so-called "minors" with the lower case i, ii, iii, iv, v, vi, vii, viii.)

Under the moggy blundering legacy of our musicologist forebears we must forgive the learned Professor for saying "Intervals are ordinal".

On a side note, we can prove mathematically that this approximation is the absolute best we can do.

We've come a long way!

Real cool ... here’s a scientific analysis of the major scale notes . It will be noted that the 4th and 5th are so close to being true that we regard them as "consonant" notes ... it’s interesting to be reminded that the 6th and 3rd are most out of whack.

well-tempered