Post by Deaf BootsThe pitch-distance 3rd is that between "I" and "iii" ultimately
expressed as "a" frequency ratio (i think!). But "the" 3rd as two
combined notes is "I" AND "iii" sounded together. In C Major E is "a"
3rd from C while C+E is "the" 3rd in that scale. Please correct me if
I'm wrong.
Would it be correct to say that G-B is also a 3rd as pitch-distance? How
about G+B, that interval surely can not be "the" 3d as a combination of
2 notes in the same scale, or can it?
Is this question inspired by the idea that there's ambiguity around the use of the word "third" (or "3rd")? If not, then please disregard this answer. :) But, if so, then here's my take on that ambiguity.
To me, there's only one concept to which we apply the term "third", and that concept is, "an ascending or descending interval (or pitch distance if you prefer that term) that involves three note letters, in alphabetical order, inclusive". So, a "third" (if that's all you know about it) is a general interval. You know its quantity (a third; three note letters), but not its quality (whether it's major, minor, diminished, etc.). So, A natural up to C natural is a third. But, by the way, we also know that it's a minor third. D# down to Bb is a third. But, by the way, we also know that it's an augmented third.
Intervals also crop up in scale degree names. A major heptatonic scale can be formulated with the string of numbers "1 2 3 4 5 6 7". The implication is that that's a cycle, rather than a ladder. In other words, the 1 loops (descending) around to the 7; and the 7 loops (ascending) around to the 1. Each number is a code telling us what interval that scale degree is from the root. So, "1" is a unison from the root. It *is* the root. 2 is a major second above. 3 is a major third above. If we wanted the third scale degree to be a minor third above the root then we'd use the scale formula "1 2 b3 [...]". In scale degrees, "3" does not mean "the third degree of the scale". It means "the degree of the scale that's a major third above the root. And, as I say, "b3" means "the degree of the scale that's a minor third above the root". Similarly, the convention is that 4, 5, 6, and 7 encode, respectively, perfect fourth, perfect fifth, major sixth, and major seventh. You can alter those numerals (with sharps and flats) to encode something different. So, you don't pronounce "1 2 3 [...]" as "first, second, third, [...]", but rather as "one, two, three, [...]". And you pronounce "1 2 b3 [...]" as "one, two, flat-three, [...]". Again, "three" means "a major third above the root", "flat-three" means "a minor third above the root", and so on. That way you're using numerals to encode an interval just like you can use terms such as "major third" to do the same thing in a different context. That's my understanding, anyhoo.
My understanding of the upper- and lower-case Roman numerals is that they're used to encode triads. So, "I" is a major triad whose root is scale degree 1. "i" is a minor triad whose root is scale degree 1. And "iii" is a minor triad whose root is scale degree is 3. "bIII" is a major triad whose root is scale degree b3. For triads, we say, "the root, third, and fifth". But there's not enough info in that, out of context, to actually for the triad. That's because, here, the word "third" is a general interval. We know its quantity but not its quality. If, however, we know from context that we're talking about a C major triad, for example, or "bIII", then from that context we know everything we need to know about the "root, third, and fifth" to play the chord. You could also think of the word "third", when used in the sense of a general interval, as just a regular ordinal number (ordinal numbers are "first, second, third"; existing to place things in order. Cardinal numbers are "one, two, three"). In other words, you're talking about whatever note is the third of some heptatonic scale formed on the root note. But, again, the scale degree "3" (a cardinal number), is NOT ambiguous like the word "third" is. "3" is unambiguous simply because of the convention that we take it to mean "the scale degree a major third above the root". We don't (or, IMHO, shouldn't) assume any such thing about the word "third" when used like that, unqualified. If someone says "a third" (which can only mean an interval), then don't assume anything more about it than its quantity; if you know its quality from context, then you're not assuming anything.