Week 6 — Chapter · Intervals I

Course: MUS 156 Music Theory I · Kenosis College
Instructor: Professor Hemsworth
Unit: Learning Unit 6 of 16 (Week 6) · two sessions per week
The unit's primary reading · ~30–40 minutes · SLOs 3 and 7 · Read this before the first session if you can, or right after — everything else in the module builds on it.

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You already hear this

Sing the opening of the Twinkle tune — the one every human being on earth can produce on demand. The first move is two repeated notes; the second move is a jump. Now sing the beginning of Amazing Grace: "A-ma…" — also a jump, and your ear knows instantly that it's a smaller jump than Twinkle's. You just measured two musical distances and compared them, accurately, using no equipment except the ear you walked in with.

That distance between two notes is called an interval, and intervals are what this unit and the next are about. Here's why they matter more than almost anything else in this course: everything after the midterm is built out of them. A chord is a stack of intervals. A melody is a chain of them. A harmony vocal is one long interval held against a tune. When you can name and spell intervals fluently, the rest of music theory stops being new material and becomes arithmetic you already know, wearing different hats.

By the end of this chapter you will be able to: find an interval's size by counting letters; find its quality by counting half steps — and keep those two measurements strictly separate; name and spell perfect, major, minor, augmented, and diminished intervals up to the octave; spell a named interval above or below any note; and invert any interval by rule. None of it requires new hearing. Your ear has ranked these gaps for years; we're issuing the names.

Where we are

Unit 1 gave every note an exact address. Units 2 and 3 gave every note a length. Units 4 and 5 organized the addresses into keys — major and minor scales, spelled with every letter used exactly once. This unit measures between the addresses. It leans hard on two old friends: the half step from Week 1 (the smallest keyboard move, your quality ruler) and the major scale from Week 4 (which turns out to be a pre-built bank of intervals). The midterm in unit 8 — two units away, with its review materials opening this very week — certifies all of it.

Two rulers, one name

An interval's full name has two parts, in this order: quality, then size — a major third, a perfect fifth. Typed the course way: M3, P5. Each part comes from its own ruler, and the two rulers never share a job.

Ruler one — letters give the size. Count the letter names from the lower note to the upper note, counting both ends. C up to A: C-D-E-F-G-A, six letters — a sixth. E up to B: E-F-G-A-B, five letters — a fifth. F up to D: F-G-A-B-C-D — a sixth. Two notes on the same letter are a unison; eight letters, like C4 up to C5, make an octave.

Three facts about size, each worth its own sentence:

• Both ends count. C to D is a second, not a first. You're counting fence posts, not the gaps between them.
• Accidentals are invisible to size. C to E is a third. C to E♭ is a third. C to E♯ is a third. The letters decide, and only the letters.
• Size alone is half a name. Two intervals can share a size and sound completely different — which is why the second ruler exists.

Ruler two — half steps give the quality. Once the letters have fixed the size, count the half steps between the two notes (the Week 1 keyboard move: every key to its immediate neighbor). The half-step count tells you which third, which fifth: the quality.

Here are the two thirds side by side:

METER: free    CLEF: treble
| C4/w E4/w | C4/w Eb4/w |

Two thirds on the treble staff: first C4 up to E4 — four half steps, a major third (M3) — then C4 up to E♭4 — three half steps, a minor third (m3). The letters say "third" both times; the half steps tell the two thirds apart. Play them and your ear confirms it: the major third is the brighter one.

The method, as a procedure you will run hundreds of times:

1. Count letters, both ends included → size.
2. Count half steps → quality.
3. Say the name quality-first: minor sixth, perfect fourth, m6, P4.

Do the steps in that order every time. Step 2 is never allowed to overrule step 1 — that rule has its own section below, because breaking it is the most common interval error in the world.

The two families

Sizes divide into two families, and each family runs its own quality ladder.

The perfect family: unisons, fourths, fifths, and octaves. These sizes take the quality perfect at their home spacing — a perfect fourth (P4) is 5 half steps, a perfect fifth (P5) is 7, a perfect octave (P8) is 12, a perfect unison (P1) is 0. Stretch a perfect interval one half step without changing the letters and it becomes augmented (A); squeeze it one and it becomes diminished (d). The ladder: d → P → A. There is no such thing as a major fifth or a minor fourth — those names mix families, and a mixed name is always a spelling error, never a rare interval.

The major/minor family: seconds, thirds, sixths, and sevenths. These sizes come in major at their home spacing and run a four-rung ladder: d → m → M → A. Shrink a major interval one half step (letters unchanged) and it's minor; shrink the minor one more and it's diminished; stretch a major one and it's augmented.

The reference spacings, all measured in half steps:

Interval	Half steps	Interval	Half steps
m2	1	P5	7
M2	2	m6	8
m3	3	M6	9
M3	4	m7	10
P4	5	M7	11
—	—	P8	12

Don't memorize this table tonight. Learn the next section instead — it rebuilds the whole table from something you already own.

The major scale is an interval bank

Here is the payoff for the scale-spelling you did in unit 4. Measure from the tonic of any major scale up to each scale degree, and every interval you get is major or perfect — no exceptions, in all fifteen keys:

KEY: C major    METER: free    CLEF: treble
| C4/w D4/w E4/w F4/w G4/w A4/w B4/w C5/w |

The C major scale climbing the treble staff from C4 to C5. Measured from the bottom note: C4 to D4 is a major second, to E4 a major third, to F4 a perfect fourth, to G4 a perfect fifth, to A4 a major sixth, to B4 a major seventh, to C5 a perfect octave.

In degrees and solfège (sing it — MUS-155 will, in Aural Skills I):

From do up to…	Interval	In C
re (^2)	M2	C4 → D4
mi (^3)	M3	C4 → E4
fa (^4)	P4	C4 → F4
sol (^5)	P5	C4 → G4
la (^6)	M6	C4 → A4
ti (^7)	M7	C4 → B4
do (^8)	P8	C4 → C5

This gives you a spelling machine: to spell a major or perfect interval above any note, treat that note as do and walk its major scale. A major sixth above E♭4? Walk E♭ major: E♭-F-G-A♭-B♭-C — C5. Need a minor interval instead? Spell the major one and lower the top note a half step, keeping the letter: a minor sixth above E♭4 is C♭5 (not B4 — the letters committed to C the moment you counted six of them). Augmented? Raise the major or perfect version a half step. Diminished? Lower the minor or perfect version one more. The whole quality ladder hangs off the scale you can already spell.

The trap: why counting keys fails

Now the section this unit is famous for. Someone hands you this: C4 up to E♯4. Name it.

The tempting method — the one nearly every beginner and nearly every chatbot uses — is to count piano keys: C4 to C♯4 to D4 to D♯4 to E4 is five half steps, and five half steps is the spacing of a perfect fourth. Answer: P4. Confident, quick — and wrong.

Run the real method, in order. Letters first: C-D-E. Three letters. This interval is a third, and nothing about its sound can ever change that, because size comes from the page, not the piano. Now the second ruler: five half steps on a frame that the letters already fixed as a third. A major third is 4; this one is stretched a half step wider. Augmented third: A3.

METER: free    CLEF: treble
| C4/w E#4/w | C4/w F4/w |

Two intervals on the treble staff that the piano cannot tell apart: C4 up to E♯4 — three letter names, an augmented third — then C4 up to F4 — four letter names, a perfect fourth. Same two piano keys, same sound, two different written intervals with two different names.

Why does the piano get fooled? Because E♯4 and F4 are enharmonic twins — Week 1's one-key-two-names rule. The key is the same; the spelling is not, and the interval's name lives in the spelling. So A3 and P4 here are two names for one sound, and which name is correct depends entirely on which spelling is on the page in front of you. Enharmonic means "sounds alike," never "equals" — and that rule now covers intervals, not just notes. (Next unit takes this idea somewhere genuinely fun: a famous sound that owns two names with two different musical jobs. For now, one discipline: the letters name it, every time.)

This is also this week's AI-audit bait. Chatbots count keys. Ask one to name C4-to-E♯4 and there's an excellent chance it announces "a perfect fourth" in a perfectly confident sentence. As of this unit, you can out-spell it — and your tutorial's scored audit step runs on exactly that skill, using your own two-ruler arithmetic as the authority.

Flip it: interval inversion

Take an interval and move its bottom note up an octave so the pair trades places — that's inversion. C4 up to A4 is a major sixth; move the C above (now A4 up to C5) and the gap that's left is a minor third. Every inversion obeys two rules:

1. The sizes sum to 9. Sixths become thirds (6+3=9), seconds become sevenths, fourths become fifths, unisons become octaves. (Why 9 and not 8? Because when you split an octave at a note, that note gets counted by both halves — the fence-post rule again.)
2. Qualities trade partners: M ↔ m, A ↔ d, and P ↔ P. Major flips to minor and back; augmented flips to diminished and back; perfect flips to perfect. The perfect family is the one that inverts into itself — part of why it earned the name.

METER: free    CLEF: treble
| C4/w A4/w | A4/w C5/w |

An interval and its inversion on the treble staff: first C4 below A4 — a major sixth — then the same two letter names flipped, A4 below C5 — a minor third. Six plus three is nine; major traded for minor.

Worked flips: M3 → m6. m7 → M2. P4 → P5. A2 → d7. If you ever invert a perfect interval and get something that isn't perfect, stop and re-add your sizes — something slipped.

Why bother? Two reasons you'll use this week. First, inversion is a checking tool: any interval answer can be verified by flipping it and seeing whether the flip obeys both rules. Second, it's a shortcut for spelling downward — which is the next section, and the skill the bandstand actually asks for.

Spelling downward

"Give me a third below the melody." "The bass sings a fifth under that." Real requests point down at least as often as up, so you need to spell intervals below a given note as fluently as above. Two honest methods — learn both, keep whichever your brain likes:

Method 1: two rulers, pointed down. A major third below C5. Letters first, counting down: C-B-A — the answer is an A of some kind, that's now committed. Half steps second: four half steps down from C5 lands on the key shared by G♯ and A♭ — and since the letters already said A, the answer is A♭4. Not G♯4. Same piano key, wrong letter, wrong answer.

Method 2: invert and drop. To go down by an interval, go up by its inversion, then drop an octave. A major third below C5? The inversion of M3 is m6: a minor sixth above C5 is A♭5; drop an octave: A♭4. Same answer, always — the two methods check each other.

METER: free    CLEF: treble
| C5/w Ab4/w |

On the treble staff: C5, then the note a major third below it — A♭4. Three letter names counting down (C, B, A), four half steps, and the A-spelling rather than the G♯-spelling because the letters commit first.

More worked drops, each done letters-first: a perfect fifth below D5 is G4 (D-C-B-A-G, seven half steps). A major second below C5 is B♭4 — and notice the octave number: the moment you step down from any C to any B, Week 1's boundary rule fires and the number drops. A minor second below G4 is F♯4 (G-F: the answer is an F-something; one half step down from G4 is the key F♯/G♭ shares — the letters chose F♯4).

The two traps, named so you can watch for them: the letter trap (right key, wrong spelling — G♯4 where A♭4 was committed) and the octave trap (crossing the B/C boundary going down and keeping the old number). Both are exactly the errors your AI coach will make this week; both are exactly the errors you can now catch.

Why this makes you a better musician

Here is what interval names buy you, concretely, this month.

You're learning a melody by ear — the way you've always done it — but now, instead of fishing for every next note, you catch that the hook opens with a leap of a fifth, and your hand or voice goes there on the first try, in any key you start from. That's the shape of the line, captured once, portable everywhere. When the worship leader says "sing a third below the melody," you now know that's a real spelling question — which third changes note by note — and you can write the harmony line before you sing it, instead of hunting for it out loud during soundcheck. Tuning works the same way: a harmony stops being "somewhere up there" and becomes a sixth, so lock it wide and bright. And when the riff needs to reach the band before rehearsal, "the chorus jumps a P5 off the D" is nine characters that hand four people the exact move. The studio — Sound to name — is this skill made physical: the class builds a bank of interval anchors out of songs you already carry, so that every gap you'll ever need has a hook attached to it in your own memory.

Key terms

• Interval — the distance between two notes, named by quality plus size.
• Size — the letter-count of an interval, both ends included; ignores accidentals entirely.
• Quality — the half-step refinement of a size: perfect, major, minor, augmented, or diminished.
• Perfect family — unisons, fourths, fifths, octaves; quality ladder d → P → A.
• Major/minor family — seconds, thirds, sixths, sevenths; quality ladder d → m → M → A.
• Major (M) — the home quality of the 2-3-6-7 family, matching the major scale above do.
• Minor (m) — one half step narrower than major, letters unchanged.
• Augmented (A) — one half step wider than perfect or major, letters unchanged.
• Diminished (d) — one half step narrower than perfect or minor, letters unchanged.
• Inversion — flipping an interval by moving its bottom note up an octave; sizes sum to 9, qualities trade M↔m, A↔d, P↔P.
• Melodic interval — two notes sounded one after the other (a leap in a tune).
• Harmonic interval — two notes sounded together (a harmony against a melody).

Summary

• An interval's name is quality + size: M3, m6, P4, A3, d5. Say both parts or you haven't named it.
• Two rulers, in order: letters give the size (count both ends; accidentals invisible), half steps give the quality. The second ruler never overrules the first.
• Two families: perfect (1, 4, 5, 8; ladder d→P→A) and major/minor (2, 3, 6, 7; ladder d→m→M→A). Names that mix families are spelling errors.
• The major scale is an interval bank: do up to each degree gives M2, M3, P4, P5, M6, M7, P8 — in every key. Minor, augmented, and diminished are half-step nudges off that bank, letters unchanged.
• C4 up to E♯4 is an augmented third, never a perfect fourth — five half steps, but only three letters, and letters commit first. Counting keys is how chatbots get it wrong.
• Inversion: bottom note up an octave; sizes sum to 9; M↔m, A↔d, P↔P.
• Spelling below: count letters down first (they commit the letter), then half steps — or go up by the inversion and drop an octave. Watch the letter trap (G♯4 for A♭4) and the octave trap (the B/C boundary going down).

Check your understanding

Work these before peeking — they're ungraded, and worked answers follow at the bottom of the page.

1. Name the interval from G3 up to E4 — size first, then quality, then the full name in course typing.
2. A friend counts six half steps from D4 up to G♯4 and six from D4 up to A♭4 and announces "same interval either way." What's right and what's wrong in that claim? Name both intervals.
3. Using the interval bank: spell a major sixth above E♭4. Then turn your answer into a minor sixth above E♭4, and explain why the letter of the top note doesn't change.
4. Invert these three and name each result: m3 · P5 · A2. One of the three answers proves that "inversion flips every quality" is false — which one, and why?
5. Spell a perfect fourth below F♯4, showing the letter count and the half-step count separately. Then check yourself with the invert-and-drop method.

Answers (worked reasoning — read after attempting)

1. Letters, both ends: G-A-B-C-D-E — six, a sixth. Half steps: G3 up to E4 is 9. A sixth at 9 half steps is the major spacing: major sixth, M6.
2. Right: the two intervals sound identical — G♯4 and A♭4 are enharmonic twins on one piano key. Wrong: they are not the same interval, because the letters differ. D4 up to G♯4 is D-E-F-G, four letters at six half steps: an augmented fourth (A4). D4 up to A♭4 is D-E-F-G-A, five letters at six half steps: a diminished fifth (d5). Same sound, two different written intervals — the E♯ rule in new clothes. (Much more on this famous pair next unit.)
3. Walk E♭ major six letters: E♭-F-G-A♭-B♭-C5 — a major sixth above E♭4 is C5. Lower the top note a half step keeping the letter: C♭5, the minor sixth. The letter can't change because size counted six letters, E to C; make it a B and you've silently rebuilt the interval as a fifth.
4. m3 inverts to M6 (3+6=9, m↔M). P5 inverts to P4 (5+4=9, P↔P). A2 inverts to d7 (2+7=9, A↔d). The middle one is the proof: the perfect fifth inverted and stayed perfect — the perfect family flips into itself, so "inversion flips every quality" fails exactly there.
5. Letters down from F♯4, counting both ends: F-E-D-C — four letters, so the answer is a C of some kind. Half steps: a perfect fourth is 5; five half steps down from F♯4 lands on C♯4. Check by inversion: the inversion of P4 is P5; a perfect fifth above F♯4 is C♯5; drop an octave: C♯4. Both methods agree — and note the letter trap you dodged: D♭4 names the same key and is still the wrong answer.

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Next in this module: the Readings & Resources page if you want other angles on intervals, then Tutorial 6 — your AI tutor session, due before Week 6's second session so the studio can build on it. And note the week's headline in the overview: the midterm's study guide, prep tutorial, and practice exam all open with the second session.

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