Is it correct to refer to a third/fifth/etc even if they're on black keys?
As I understand it, the terms "third", "fifth", "seventh", etc. all refer to the n-th white key relative to the root note.
To demonstrate this, let's observe a C major chord (interval 0-4-7):
Root: C
third: E (this is the THIRD white key relative to the root)
fifth: G (this is the FIFTH white key relative to the root)
Now let's take a C# major (interval 0-4-7):
Root: C#
third: F
fifth: G# (This is not a white key)
Because a major chord always has the interval 0-4-7, the 'fifth' (note with interval 7 relative to root) is on a black key.
Is G# still called the 'fifth'?
Does a 'third' always refer to interval 4 and a 'fifth' always refer to interval 7? Even in other scales/keys/modes?
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"As I understand it, the terms "third", "fifth", "seventh", etc. all refer to the n-th white key relative to the root note."
Not quite They refer to the n-th LETTER NAME relative to the lower note.
C - G is a 5th. So is C? to G?, C? to G?, C double-sharp to G double-flat and any other combination of C and G notes you like to pick (though that last example, C double-sharp to G double-flat, might be difficult to name as what SORT of 5th!)
it's all about spelling. And spelling is all about what the interval DOES. In isolation, is the interval from that white note near the piano keyhole to the next-but-two black one C to F? or C to G?? Or even B? to E double-sharp (OK, unlikely, that one!) Until the interval has context, until it DOES something, we have no way of knowing.
As I understand it, the terms "third", "fifth", "seventh", etc. all refer to the n-th white key relative to the root note.
Interval naming and sizes aren't specific to an instrument. "...white key relative..." how would that work for non-keyboard instruments? Basically, the intervals are named by the letters and the specific quality comes from the number of half steps between the tones.
You're on the right track with 0-4-7. Those are the interval sizes in half-steps. But that isn't enough to get the interval names. You need to spell it with letters. That part - the spelling with letters - is important and it leads to a mistake in your example.
C# F G# is sized in half steps as 0-4-7, but the intervals are not a third and a fifth, not with the spelling you give. C to F is a fourth of some quality. A perfect fourth would be 5 half steps, but in this case the sharp on C makes the interval one half step smaller, it becomes diminished, so it's a diminished fourth.
If you want to take C E G - interval above root of major third and perfect fifth - and raise it a half step and keep the same interval naming, you spell it C# E# G#. Starting with root C# any third above it must be letter E, accidentals are next applied to size it to the specific kind of third: E# for a major third, E for a minor third, Eb for a diminished third, etc.
Your use of the white key on keyboard effectively serves the purpose of providing letters, but I think it leads to confusion with enharmonic spellings like E# versus F natural...
...that image in truth is wrong. The key labeled F can be F or E# or Gbb, etc. etc. Drop the method of keyboard reference and get comfortable with just reciting the gamut of letters. For practical purposes you want to be able to recite by thirds and fifths in ascending and descending order.
Starting with gamut A B C D E F G...
all fifths are perfect except B,F which is a diminished fifth
all fourths are perfect except F,B which is an augmented fourth
all thirds are major except those encompassing either half-step B,C or E,F
all sixth are major except those encompassing both half-steps B,C and E,F
...then just keep track of any accidentals applied and the corresponding changes in interval quality.
Yes. The intervals are always relative to the root. There are some interesting cases where there are two possible names for the same interval depending on how the note is named (even though it's the same note). So, for example, with the root C you would call Gb the diminished fifth. The same note spelled F# would be an augmented fourth. Changing the root to C# you get a diminished fifth is G but the augmented fourth will be called F## (and now you know why there are double-sharp and double-flat spellings for notes).
As I understand it, the terms "third", "fifth", "seventh", etc. all
refer to the n-th white key relative to the root note.
This is not correct. The interval distance is determined by the letter name distance which is the number of letters you encounter between the two notes which is very easy to see on a staff. This loops if you go to an octave or above. An example on C would be:
C to C (same octave) 1st (unison or prime)
C to D 2nd
C to E 3rd
C to F 4th
C to G 5th
C to A 6th
C to B 7th
C to C (next octave) 8th (octave)
On the staff it looks like this for C:
X: 1
K: C
L: 1/4
%%score (T1 T2)
V:T1 clef=treble
V:T2 clef=treble
[V:T1] C D E F | G A B c
[V:T2] "1st"C "2nd"C "3rd"C "4th"C | "5th"C "6th"C "7th"C "8th"C
So any C to any D has to be some kind of 2nd. C to F can't be a 3rd because it's a letter distance of a 4th (C to F). To make it a 3rd, you have to convert it to its enharmonic equivalent of E? which is a 3rd (a major 3rd to be specific).
Don't let the keyboard layout fool you into false premises!
Use the guitar and it should become clear that what you start from in your question is dubious. On guitar there are no 'black' or 'white' notes (keys on piano). And you are probably aware that when you play , say, C E G somewhere, by moving everything up one fret, you get C? E? G?. Up to wherever, use the same shape, and you always have a major triad - on whichever strings you choose. And - that F isn't in the triad - it gets called E?. Honest!
To simplify even more - take your 0-4-7 formula, and use only one string. Let's say open G. 0=G, 4=B, 7=D. now add whatever number you wish to those start numbers, and play. You make that triad again every time, so whatever the note names, 0=root, 4=third (M3) and 7= fifth (P5).
Using the white keys on piano to ascertain intervals isn't always the best approach, as you may now be aware! However, if you insist, then simply count the number of smallest steps (semitones, or each guitar fret) for half the answer. The other half will be revealed once you've established what the lower and upper notes are called. And simply playing them cold, and listening to them aint gonna work! They need to be written down, in a specific key, on a stave. Otherwise they could have any of two or more different names, depending on situation and their own names.
Just realised I haven't answered the question!
Yes, the black keys work in their own right, just as the white keys do, although they usually have ? or ? in their names. And when you consider that F is enhamonic with E?, why shouldn't they? Any black key can be part of any interval, including 3rds and 5ths (and everything else) Yes!
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