skills/musical/references/intervals.md

# Intervals

An interval has two parts, and both must be right: a **number** (how many letter
names it spans) and a **quality** (how many semitones, given that number).

## Number

Count letter names inclusively, starting at 1 on the lower note.

C to E is a third: C(1) D(2) E(3). C to Eb is *also* a third - flattening the
top note changes the quality, never the number. C to D# is a second, not a
third, because D is the second letter.

This is why spelling matters. Eb and D# are the same pitch but produce
differently-named intervals from C.

## Quality

| Number | Diminished | Minor | Major | Perfect | Augmented |
|---|---|---|---|---|---|
| Unison | - | - | - | 0 | 1 |
| 2nd | 0 | 1 | 2 | - | 3 |
| 3rd | 2 | 3 | 4 | - | 5 |
| 4th | 4 | - | - | 5 | 6 |
| 5th | 6 | - | - | 7 | 8 |
| 6th | 7 | 8 | 9 | - | 10 |
| 7th | 9 | 10 | 11 | - | 12 |
| Octave | 11 | - | - | 12 | 13 |

Numbers are semitones. Unisons, fourths, fifths and octaves are perfect and
never major or minor. Seconds, thirds, sixths and sevenths are major or minor
and never perfect.

One step narrower than minor is diminished; one step wider than major is
augmented. One step narrower than perfect is diminished; one wider is augmented.

## Compound intervals

Beyond the octave, add 7 to the number: a ninth is a second plus an octave, an
eleventh a fourth, a thirteenth a sixth. The quality carries over unchanged - a
major ninth is a major second an octave up.

## Inversion

Invert by moving the lower note up an octave. The numbers sum to 9, and the
quality flips: major to minor, augmented to diminished, perfect stays perfect.

A major third inverts to a minor sixth. A perfect fourth inverts to a perfect
fifth. A tritone inverts to a tritone - the only interval that inverts to its
own size, though an augmented fourth properly inverts to a diminished fifth.

## Grading procedure

Given two notes and the learner's answer:

1. Count the letter names yourself and fix the number.
2. Count the semitones yourself and fix the quality.
3. Compare both parts against their answer, separately.

An answer with the right quality word and the wrong number is wrong. So is the
reverse. Say which half failed - they are different mistakes with different
fixes.

## Worked cases

**Right pitch distance, wrong number.** C to D#, learner says minor third.
Verdict: wrong number. C to D is a second, so C to D# is an augmented second.
It sounds identical to a minor third, and the pitch reasoning was sound - the
letter counting was not. If the exercise gave them C to Eb, minor third is
correct.

**Perfect where perfect cannot go.** C to E, learner says perfect third.
Verdict: wrong quality - there is no such thing. Thirds are major, minor,
diminished or augmented. C to E is a major third. This is a vocabulary gap, not
a counting error; fix the vocabulary.

**Descending misread as ascending.** Asked for G down to C, learner says perfect
fifth. Verdict: correct. Direction does not change the interval's name. If they
answered perfect fourth, they measured C up to G instead - name that, because
the confusion recurs.

**The diminished fifth.** C to Gb, learner says augmented fourth. Verdict: wrong
spelling. Both are six semitones, but C to G is a fifth, so C to Gb is a
diminished fifth. C to F# is the augmented fourth. Same sound, two names, and
which one is correct is decided entirely by the letters in front of them.

**Compound.** C4 to E5, learner says major third. Verdict: incomplete rather
than wrong. It is a major tenth. Accept the reasoning, supply the octave, and
check whether they knew it was compound or missed the register.