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2 types of intervals

I have done very little piano of late. But I’d started to work on the grade 5 theory materials, using a book called “master your theory”. Much of the material I already know, but I haven’t utilised it for quite some years.
Today I was reminded there are 2 different ways of naming intervals. Aurally, and thereticalalusing theoryy might have more technical names, but I can’t remember those, so if anyone knows, feel free to comment.
Aurally naming intervals is fairly easy and logical for me. I just use the chromatic scale to help me. There is a pattern to them, and if you can remember the pattern, and becomes fairly easy to reconize intervals. As an example of the chromatic system I use, starting on C, a semitone or half-step above C is a minor second, a tone or whole step above C is a major second; one tone and one semi-tone above C is a minor third, 2 tones above C is a major third; then 2 tone and one semi-tone above C is a perfect fourth. The fourth, fifth and octave intervals are “perfect” intervals. The pattern is always minor, followed by major. Looking at a whole, the pattern is as follows: Minor second, major second, minor third, major third, perfect fourth, tritone, perfect fifth, minor sixth, major sixth, minor seventh, major seventh, perfect octave. Is you can reconize a perfect fifth by ear, you can usually work out the interval by singing above or below the notes to work it out. I have perfect pitch, so I don’t usually have too. But this is how I taught myself to reconize a minor sixth which gave me a lot of problems while studying for my grade 8 piano.
Now, if you aren’t bored with reading yet, to the theoretical naming of intervals. This all depends on the key it’s written in. For the sake of keeping it simple I will use C as the key. The perfect intervals remain the same; that is, perfect fourth, fifth and octave. DEPENDING on the notes used, will depend on how the interval is named. A C to E for example, is a major third, because in C, an E is a third above C. A C to D sharp though, is you heard it, it would be a minor third, because using the chromatic scale, the pattern is minor major, and counting up the chromatic scale, it would be a minor third. However, in this case, it is named as a augmented second. Because the distance between a C and a D is a second. However, the note used in this example is a D sharp. That’s one semi-tone or half step above a D. This makes for a larger, or augmented interval, in this case augmented second.

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