# Interval direction

Could someone tell me the usual terminology for moving an interval up vs moving an interval down vs moving an interval distance? When we write things like 1 b3 5 we mean moving up modulus octaves, right? But when we say an interval is a third, do we mean it is a +3 (like [1] to [3]), or a -3 (like [1] to [b6]) or do we mean "a third" as in a distance measure or magnitude |3| rather than a vector quantity?

• Another ambiguity that this question overlooks is that "a third" can reasonably be major or minor. It may or may not be obvious from the context. If you move from C by a third you could end up at E, E flat, A, or A flat. May 28 at 9:26

Saying that an interval is "a third" only specifies the magnitude, not the direction. However, without context, intervals are presumed to be measured from the lower note. To specify direction, one would say "an ascending third" or "a descending third".

Scale degrees are presumed to be within an octave unless otherwise specified. But direction can be ambiguous. 1 b3 5 is presumed to be a series of monotonically ascending pitches/intervals. 1 b7 b6 would be presumed to be a series of monotonically descending pitches. But 1 6 5, for example, would be ambiguous.

• Furthermore, "a third" is ambiguous even as to magnitude; it could refer to a major or minor third. May 28 at 9:28

It is important to distinguish between harmonic intervals (simultaneous), and melodic intervals (consecutive).

Harmonic intervals are counted from low to high, so are implicitly upward (unless otherwise specified) — like a magnitude.

``````The interval between C4 and E4 is a major third.
''    ''       ''   C4 and A4 is a major sixth.
''    ''       ''   A3 and C4 is a minor third.
``````

Melodic intervals are given a direction from the first note to the second, either ascending or descending (up or down) — like a vector.

``````The interval from C4 to E4 is an ascending major third.
''    ''     ''  E4 to C4 is a descending major third.
''    ''     ''  C4 to A4 is an ascending major sixth.
''    ''     ''  A4 to C4 is a descending major sixth.
''    ''     ''  A3 to C4 is an ascending minor third.
''    ''     ''  C4 to A3 is a descending minor third.
``````

Compound intervals reduce the name of the interval to be within one octave. This is usually more useful; it makes them easier to work with / think about.

``````The interval between C4 and E5 is a major tenth,
or a compound major third.
''    ''       ''   C4 and E6 is a major seventeenth,
or a compound major third.
etc.
``````

NB intervals are ordinal numbers, not cardinal numbers.

• "Harmonic intervals are counted from low to high, so are implicitly upward": conventionally, perhaps, not necessarily. There is nothing wrong with saying "the second violin is a major third below the first." The interval between E4 and C4 is also a major third regardless of whether it's melodic or harmonic. May 29 at 6:25
• "Melodic intervals are counted from first note to second" - does that really matter? I'd still calculate the interval using the lower note as datum point. Adding 'down' or 'up' qualifies. Seems easier that way, and I guess many would agree.
– Tim
May 29 at 8:41
• I usually count from the finger furthest away or closest on the violin fingerboard ;)
– Emil
May 29 at 8:55
• @Tim — It matters because it is useful. You can describe a melody of several notes, using a sequence use melodic intervals. May 29 at 9:37
• True, but once one has that sequence, it doesn't really matter how one arrived at it. And counting backwards isn't as easy as forwards.
– Tim
May 29 at 12:54

Intervals are always calculated from bottom to top. But, as phoog points out, a 'third' is too vague as it stands. C>E is a third, but so is C>E♭. That's without considering C>Ex, or C>E♭♭. (Major 3, minor 3, augmented 3 and diminished 3 in that order). So 'third' needs to be specified.

It's easy to say 'up a M3', or 'down a M3' to anyone who understands - it's unequivocal: C>E, or C>A♭.

Just be aware that the inverted interval is different - C>E is M3, while E>C is m6. The 'rule' is simple - it's the 'rule of 9'. (3+6=9), M becomes m, aug becomes dim (and vice versa). But any given named note to any other named note will always have the same interval name, totally regardless of key.

• "Intervals are always calculated from bottom to top": how would you describe the first two intervals in "Mary Had a Little Lamb"? May 28 at 10:22
• @phoog - aha! To calculate the interval between 1st and 2nd notes, I'd use the lower one as base point, and work up. So M2, if it's the same tune as you use. I haven't sung it since I was a kid ! There's always going to be a lower note of two - unless it's a unison...
– Tim
May 28 at 10:31
• Sure, but if you start on E, the second note is D, a major second below E. It's difficult for me (1) to believe that you would take exception to that statement, and (2) to accept that it constitutes "calculating from bottom to top." May 28 at 10:53
• @phoog - I've been calculating intervals for a long, long time. So long I don't need to actually work them out any more. I'm describing it like that because it's easier for folk who haven't reached that stage yet. Easier counting up than down, same as alphabet forwards instead of backwards?
– Tim
May 28 at 14:57