How does counting E twice explain the discrepancy between the 5th between C and G vs. the 3rd between C and E, 3rd between E and G?

Question

I still don't grasp the "source of the discrepancy". "the E got counted twice when we went C,D,E and then E,F,G, but only got counted once when we went C,D,E,F,G." — So what? How does this expound the discrepancy? All emboldings are mine.

Impaled on a Fencepost |

The music theorists of the Middle Ages committed a fencepost error that’s too entrenched to dig up now. Consider the chord made of the notes C, E, and G (a C major triad). If music theory nomenclature for intervals made sense, the distance from C up to E plus the distance from E up to G would equal the distance from C up to G. And that’s true if you measure the intervals by counting upward steps. The problem comes when you describe intervals with what I suppose might be termed “ordinal nomenclature”: going from C to E is called going up by a third (because you count 1,2,3 when you play C,D,E) and going from E to G is called going up by a third for the same reason, but going from C to G is called going up by a fifth (because you count 1,2,3,4,5 as you play C,D,E,F,G).

The source of the discrepancy should be clear: the E got counted twice when we went C,D,E and then E,F,G, but only got counted once when we went C,D,E,F,G. So in music theory, when you stack a third on top of a third, you get a fifth. We musicians are stuck with nomenclature that essentially makes us say “3+3=5” so many times that we eventually stop noticing we’re saying it. (Of course, saying “a third plus a third is a fifth” is confusing on a different level, since it sounds like “1/3 + 1/3 = 1/5”. But I digress.)

Answer 1

"The source of the discrepancy should be clear"

Well, first things first, the discrepancy in question is that in musical terms, a third plus a third is a fifth. As the article mentions, 3 plus 3 equals 5 in interval speak, which is a discrepancy from the usual arithmetic idea of addition.

Now, the source of that discrepancy is that musical intervals are indexed ordinally - that is, two of the same note are a unison, or a first apart. There is no interval of a zeroth, and this is where the problem begins. Since the unison acts as the additive identity, it is possible to intervallically "add 1" to a note over and over again and still end up on the exact same frequency.

Had the music theorists of yore thought of zero-indexing their intervals, intervals would in fact be additive! Suppose instead that two notes of the same frequency were considered to be a perfect zeroth, and the doubling of frequency a logically-named heptave. In that case, we would be going up a major second from C to E, another (minor) second from E to G, and end up with C-G being a perfect fourth. The shoe fits: 2 + 2 = 4 in this case!

^{Side note: while intervals cannot be put together arithmetically, there is a way to count intervals that is zero-indexed: by converting frequency ratios into a number of semitones (or, for extra precision, cents), addition is preserved! Observe that C to E is 4 semitones, E to G is 3 semitones, and C to G is 7 semitones. 4 + 3 = 7, and thus the math works out right with half-step counting!}

Answer 2

What the quotation is complaining about is that if you count the notes of the third from C to E ...

C = first note
D = second note
E = third note

Thus, three notes in total.

... and then you count the third from E to G ...

E = first note
F = second note
G = third note

Thus, three more notes.

Then the distance from C to G really ought to be a "sixth", because 3 + 3 = 6. However, C to G constitutes only five notes.

C = first note
D = second note
E = third note
F = fourth note
G = fifth note

The "problem" is that the E was counted twice when we added thirds.

C = first note
D = second note
E = third note and fourth note
F = fifth note
G = sixth note

What the quotation is missing is that intervals aren't additive. They only count notes from a given starting point.

Answer 3

Musical intervals count ordinal start and end points not the actual "space" between tones:

ORDINAL: 1       2       3       4       5

TONES  : C       D       E       F       G

RULER  : 0       1       2       3       4
           --1--   --2--   --3--   --4--

If you treat musical intervals as spatial distances, then indeed it doesn't make sense. But, that is not what they are.

It might make more sense if you think of it in terms of ordinal numbers for the scale degrees above the tonic, and then extend that to ordinal numbers above any given tone in lieu of a tonic.

So, in C major the E is the third ordinal position. If you transpose that to start on E going up to G, think of the E as ordinal 1 and the G is the third ordinal above it.

It's kind of like saying in plain English "take a given tone and the third scale step above it" and then condensing it to just "a third" for convenience.

The tip off should be the "-nd", "-rd", "-th" ending of the interval names. Those are ordinal names. We have, for example, fifths, not fives. And of course it isn't a faction like "one fifth."

There are some oddities in the terminology. Musical intervals are very often referred to as distances, but if you really dig into it, we can see they aren't distances. Also, there isn't a musical interval of a first, it's called a unison. Those kinds of terminology inconsistencies aren't unique to music.

The whole 3+3 thing doesn't make sense. If you counted the "spaces", it would be 2+2=4, but because the ordinal numbers start with 1 you add 1 to get the musical interval, 2+2+1=5, a musical fifth. Taking two ordinal threes and then adding them together just doesn't make sense.

Answer 4

The problem is that this isn't a proper counting system starting with 0. So every additional interval only adds "n-1" where "n" is the name of the interval.

An octave means "8," but it's only 7 notes up. It's the 8th note. The next octave is. . . 15ma, the 15th note.

Answer 5

I still don't grasp the "source of the discrepancy"

The discrepancy is between the fact that two stacked thirds form a fifth, and the OP's unfulfilled expectation that it should be expressed by addition of corresponding numbers, whereas in fact 3 + 3 ≠ 5. Possibly people who first named the intervals didn't consider performing such mathematical operations.

If music theory nomenclature for intervals made sense...

That's the key sentence. Music notation was first developed, to notate the music. Theoretical analysis came later. It could be compared to languages. For instance, in this site we use language in which the words "red" and "read" can be pronounced the same, while "read" and "read" can be pronounced differently. One can find more examples where conventions used for historical reasons don't follow simple expectations, both in languages, and in music notation.

There are some theoretical developments that fix the alleged "mistake". In particular, music set theory names intervals according to the number of semitones, with the root corresponding to 0, not 1; this allows for addition.

Answer 6

Put simply: C>E is a third (of some sort) as E is the third note from C, counting C as the first. E>G is also a third (of some sort) as G is the third note from E, counting E as the first. And, as we know, C>G is a fifth (of some sort), counting C as the first.

We could go more silly, and say that C>D is a 2nd, D>E is a 2nd, E>F is a 2nd, F>G is a second, G>A is a second, A>B is a second, and B>C is also a 2nd. That makes C>C a 14th, not what we know to be an 8th.

Even sillier, mathematically, we could say 1+2=3, 2+3=5, therefore 1+3= you work it out, I can't.

It's not quite as black and white (or grey!) as that.

We appear to use two distinct counting systems. One starts at O (think measuring with a rule, or timing with a stopwatch), the other at 1 (think counting money, counting sheep, but stay awake...), and I believe the two got interbred.

It all happened so long ago, that it really doesn't make sense to do anything about it. Apparently scientists found that positive and negative (electicity) really are the wrong way round. Should we all change them? Certainly not! It's taken this long for some of us to understand intervals as they are, please don't consider changing, for any reasons, let alone logic or pedancy...

Can't do any more, just been called back to the funny farm, where all is normal.