I am learning level 8 theory. There was an exercise where I had to write each interval below the given note. The interval given was a diminished 1. In general, how would I solve this? Here is a similar picture of the exercises I had:
You would actually solve it exactly as you would any other diminished interval. Let's say you want to write a diminished first (also known as a diminished unison) below C.
- First, you want to determine what the note name will be without any accidentals. Since you're looking for a unison, it will be the same pitch as C: C!
- Next, let's determine what quality unison this is from C. Obviously C to C is a perfect unison. (Recall that there are no major or minor unisons, fourths, fifths, or octaves!)
- This is a perfect unison, but you want a diminished unison. Just as we lower the resulting pitch a half step to move from, say, a perfect fifth to a diminished fifth, we'll lower the resulting tone in our example down to a C♭.
As such, your diminished unison below C is a C♭. Even though this is enharmonic to B, B–C would be a minor second, not a diminished unison.
Now let's see these steps for your first example looking for the major 10:
- Write the note name that is a 10th below F: D.
- Now, measure this interval: D up to F is a minor tenth.
- We have a minor tenth, but we want a major tenth. To make a minor interval major, we have to make it larger. So, we lower the D to D♭ to make the major tenth.
Make sure you do not alter the given pitch. We could have raised the written F to F♯, but this is typically against the rules.
A quick note on whether this diminished unison is above or below the C in question.
When teaching intervals, we assume octave equivalence. We know this for at least two reasons: first, because we equate simple intervals (intervals of an octave or less) with their compound (intervals larger than an octave) siblings; and second, because the patterns that arise when we invert intervals are not possible without octave equivalence.
As such, in my opinion it's necessary that a diminished unison is spelled the same way as the compound version of that interval (i.e., that interval plus an octave). The diminished octave above C is unequivocally C♭, and thus the same is true for the diminished unison.
The main trouble that some people find with Diminished unisons is that the starting note is above the other note. This has been the source of much debate among theorists, with some arguing that diminished unisons are therefore the same as augmented unisons when written harmonically.
Suppose that you want a diminished fifth written above the root G. 5th is D, diminish it BY DECREASING THE DISTANCE BY 1 SEMITONE, changing the ending note. This is going to become a problem later. Suppose I want, now, a diminished 1st above the note G. G-G is a unison, diminish it to G♭. Note that this end note is below its root G. Diminished 5th below G? G-C is fifth, diminish it to C♯. G to C♯. But what about the diminished unison BELOW G? G-G is the unison, and to diminish the note, we've got to raise the G to G♯. Except, now our diminished unison BELOW the note is actually HIGHER than its starting note, making NO SENSE AT ALL.
So do diminished unisons exist? In theory, it's possible to construct one, but it makes little sense, and theorists argue over whether the phrase "descending diminished unison" is even correct.
Your picture shows a diminished 12th; that should be easy. Unisons can be augmented but I've never seen a diminished unison. Kosta and Payne say such an interval does not exist. (Apparently, from a perusal of the NET, some people call C to Cb a diminished unison but that makes little musical sense.)
I believe the question in the example is unfair, and if you brought that up to the teacher you would look like you knew even more theory than what you have studied in class. As explained here on this thread, the idea of a diminished unison is quite debatable and as such is not a suitable question for a student to answer. If I faced your situation, I would leave the answer blank and go talk with the teacher.
In my opinion, a diminished unison is like dividing by zero. An answer cannot be found. The idea of an interval is the distance between two notes. We can adjust those distances and make them smaller or larger. No matter what is done to a unison, it is made larger. C to Cb is an augmented unison. C to C# is an augmented unison.