Intervals and cantus firmus
I have just started learning music theory and I have learnt intervals and I have gotten up to learning the cantus firmus.
I have two questions: On the piano keys, when I count interval steps, am I to include both the black and white keys or only the white keys. The instructions says for cantus firmus, I am only to use a diatonic scale, but for counting intervals am I still to count the black keys also? So even on a diatonic scale F to E is 11 half steps which is a major seventh.
The cantus firmus below is an example composed by Heinrich Schenker and has the following notes and the intervals from note to next note counting both white and black keys on the piano:
c1 d f e f g a g e d c
2,3,11,1,2,2,10,9,10,10 (intervals of above notes)
Why is Schenker using both major seventh (11 half steps) and minor seventh (10 half steps)? I was told at the beginning of the lesson to avoid dissonant intervals and I believe the seventh is considered dissonant.
Here are the links to the example I was using:
openmusictheory.com/cantusFirmus.html trinket.io/music/da93d4d902
I am still a beginner to music theory, but I hope I made myself clear.
Thanks.
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You are using two different interval systems. Counting in semitones (comparable to counting all successive black and white keys on a piano) is one interval system and it is mostly associated with atonal music. Since it is used with atonal music, you would not typically use it with cantus firmus exercises written by Schenker since his expertise is in tonal music.
Tonal interval systems, on the other hand, do not count every possible note between any two given notes. Therefore the interval between C and G is a fifth, and not a seventh (as it would be if you counted all the semitones)
F to E is a seventh because it includes 7 letter names - C D E F G A B. It is a major seventh because E is the seventh note of F major scale.
I don't understand your numbers. How is F to E 11 semitones?
There are (for now) two types of intervals: harmonic and melodic. When you measure F (up) to E as a major seventh, you're measuring a melodic interval because the two pitches occur successively (like in a melody). When F and E are played together, this can still be a major seventh, but it will be a harmonic major seventh because the two pitches are played simultaneously (like a harmony).
When you're writing counterpoint to a cantus firmus, you're really allowed to use most melodic intervals except for the tritone.
But the dissonances that you're thinking of avoiding---sevenths, etc.---are the harmonic intervals that occur between a pitch in the cantus firmus and the pitch that you write in counterpoint against it.
Thus Schenker moving F down to E (a minor second) is okay because it is a melodic minor second. But when Schenker writes an F, you don't want to write an E at the same time, because this would create a harmonic minor second, which is a dissonance.
A major seventh is indeed eleven semitones. The name of the interval is determined by how it is spelled, so F up to E is a seventh, but F up to F flat would be a diminished octave, even though it is the same distance in semitones. So going from F up to E is a major seventh. Going from E up to F would be a minor secomd (one semitone). Hang in there. You'll get it! It will all be second-nature pretty soon.
Richard's explanation of how this all fits into counterpoint is spot on.
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