Eh. Calling it a “fifth” is making it clear that the label is an ordinal number: first second third fourth fifth ...
You need to think of it as “if the bottom note was the first note of a scale, where in the scale would the top note be?”
As with many questions of indexing, off by one errors are tricky.
It’s a system that confuses names for notes in a scale with names for intervals between notes. You’d rather they called them by cardinal numbers representing some kind of “distance”, instead of a count starting at one.
But that would be applying a later mathematical understanding on the earlier system. If that’s what you want, you should just use a log scale and count twelfth roots of two.
Ideally we’d switch all our indexing to start at zero, and use half-open intervals everywhere. Start at 0 AD, call the ground floor of a building “0”, start spreadsheets with row 0, switch Matlab to index from 0, et cetera. This is pretty unlikely to happen though.
The intervals in a 7-note scale inherently don’t add up like that, because they’re not based on even divisions. So regardless we need to have 12 different named intervals for various numbers of semitones:
Reducing all those ordinal numbers by one really doesn’t help all that much. You still have to remember how the “minor” and “major” labels interact for every interval in the scale, and remember that sometimes the interval between the same two notes is given multiple names depending on the key, etc., which is all horribly confusing mess.
> Reducing all those ordinal numbers by one really doesn’t help all that much. You still have to remember how the “minor” and “major” labels interact for every interval in the scale, which is a horribly confusing mess.
Those interactions are pretty intuitive. Where defined, major + minor = perfect (considering an octave as perfect), perfect + major/minor = major/minor. As long as you remember which notes exist, you can't get it wrong, so you'll never get confused by a piece of arithmetic in an actual piece.
The “perfect” intervals are just ±5 (ratios very close to 3:2 and 4:3). The “major” intervals are –3, –1, 2, 4 (approx. ratios of 5:3, 15:8, 9:8, 5:4). The “minor” intervals are –4, –2, 1, 3 (approx. ratios of 8:5, 16:9, 16:15, 6:5).
Then it’s easy to see that your “major + minor = perfect” formula only works for some intervals, Etc. Overall the simple heuristics are more obfuscatory than helpful IMO.
> Then it’s easy to see that your “major + minor = perfect” formula only works for some intervals, Etc.
Where does it go wrong? Do those cases come up in practice?
Counting up and down the scale is a core use case for a notation for intervals. It absolutely needs to be well-supported. A 12-semitone approach is never going to match the usability of even the existing system.
You need to think of it as “if the bottom note was the first note of a scale, where in the scale would the top note be?”
As with many questions of indexing, off by one errors are tricky.
It’s a system that confuses names for notes in a scale with names for intervals between notes. You’d rather they called them by cardinal numbers representing some kind of “distance”, instead of a count starting at one.
But that would be applying a later mathematical understanding on the earlier system. If that’s what you want, you should just use a log scale and count twelfth roots of two.
Ideally we’d switch all our indexing to start at zero, and use half-open intervals everywhere. Start at 0 AD, call the ground floor of a building “0”, start spreadsheets with row 0, switch Matlab to index from 0, et cetera. This is pretty unlikely to happen though.