Week 11 — Chapter · Roman Numerals and Function

Course: MUS 156 Music Theory I · Kenosis College
Instructor: Professor Hemsworth
Unit: Learning Unit 11 of 16 (Week 11) · two sessions per week
The unit's primary reading · ~30–40 minutes · SLOs 5, 4, and one component of SLO 8 · Read this before the first session if you can, or right after — everything else in the module builds on it.

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The fifty-song trick

Here is a party trick you have already heard performed. A musician plays four chords in a loop, and the room starts calling out songs — a pop anthem, a worship opener, a country ballad. Ten songs in, the room realizes they're all the same four chords, and everybody laughs.

Now the part nobody explains: play those four chords in C, and the room hears the songs. Play them in G — every letter different — and the room hears the same songs. The letters changed; the songs survived. The letters were never what made it work.

What survived is what each chord does: one is home. One leans so hard toward home it practically falls there. One packs the bags for the trip. Your ear has tracked those jobs in every song you've ever loved. This unit names them — and hands you the naming system working musicians use: Roman numerals, which name a chord by scale degree and quality instead of letters, so the name survives every key change on earth.

By the end of this chapter you will be able to: analyze diatonic triads and sevenths in major and minor with Roman numerals, letting case and symbols carry the quality; sort any diatonic chord into its function family and follow the functional cycle; attach inversion figures to a numeral and name the bass note they promise; explain why V is major in minor keys and why vii°7 lives in harmonic minor; and describe harmonic rhythm without confusing it with tempo. Then you'll aim all of it at your own Chart song — milestone 2 is this week.

Where we are

Units 4 and 5 gave you every key; units 9 and 10 taught you to build every triad and seventh, in every inversion, with figures and chord symbols. So you can already spell everything this unit touches. What you can't yet say is what any of it is for — and the answer is context: which degree of which key the chord is built on, and what job that degree does. Roman numerals are how analysis writes the context down. This is the unit where chords stop being objects and start being sentences.

Roman numerals: the job title system

A Roman numeral tells you two things at once:

1. The numeral itself = the scale degree of the chord's root. A chord built on ^1 is some kind of I; on ^5, some kind of V; on ^6, some kind of VI or vi.
2. The case and symbols = the chord's quality. UPPERCASE means major (I, IV, V). lowercase means minor (ii, iii, vi). A lowercase numeral with ° means diminished (vii°). An uppercase numeral with + means augmented (III+ — rare, real, minor-key resident).

That's the entire code. It reads at a glance and types almost as fast: I ii iii IV V vi viio is how you'll type the major-key set in this course (the ° becomes a small letter o; more typing conventions below).

Apply the code to C major — the harmonization you built by hand in unit 9, now wearing name tags:

Degree	Chord	Spelling	Quality	Numeral
^1	C	C–E–G	major	I
^2	Dm	D–F–A	minor	ii
^3	Em	E–G–B	minor	iii
^4	F	F–A–C	major	IV
^5	G	G–B–D	major	V
^6	Am	A–C–E	minor	vi
^7	B°	B–D–F	diminished	vii°

Say the quality parade out loud — major, minor, minor, major, major, minor, diminished — because it is the same parade in every major key. That's the point of the whole system. In E♭ major, ^2 carries F–A♭–C: different letters, same job, same name — ii.

Worked example. In G major, name the chord on ^6. Count up: G–A–B–C–D–E. Build from the key's own notes: E–G–B. The third, E to G, is minor — so lowercase: vi. Notice you never needed to say "E minor" at all. A musician who knows the key hears "six" and builds it themselves.

Typing convention (for every AI session and quiz): numerals type as plain letters with figures attached — I, ii6, V7, V65, i64, viio, viio7, III+. Half-diminished types viiø7, or vii half-dim 7 if your keyboard has no ø. Never type viio7 for the half-diminished chord — ° and ø are different chords, and this unit shows where each one lives.

Misconception check. "The case is just style — everyone knows what you mean." In C major, vi is A–C–E, a minor chord. VI names an A major chord — one C major cannot build without an accidental. The case isn't calligraphy; it's the quality, half the name. vi and VI are "there" and "their": same out loud, different on the page — the distinction this course has policed since unit 1.

Chords have jobs: the three families

Play C major's I chord and stop. Nothing itches. Play V and stop — the room leans forward, waiting. Play IV: neither home nor desperate, just going somewhere, bags packed. Your ear has always sorted chords into those three feelings. Here are the names:

• Tonic function (T) — home. Rest, arrival, the place phrases end. I is home itself; vi and iii are in the family — home-flavored, restful-ish — the couch cushions rather than the couch.
• Predominant function (PD) — packs the bags. These chords point away from home and toward the lean: ii and IV. They rarely end anything; their whole job is setup.
• Dominant function (D) — the lean that has to land. V7 and vii°, the two chords that own the key's tritone (more below; the plain V triad hasn't got it — the seventh is what adds fa). When you feel a song about to arrive, this family is why.

And the road they usually travel, which this course calls the functional cycle:

T → PD → D → T. Home, away, lean, land.

Play the plainest lap — I, IV, V, I — and you'll recognize the chassis under thousands of songs:

KEY: C major    METER: 4/4    CLEF: bass
| <C3,E3,G3>/w | <F3,A3,C4>/w | <G2,B2,D3>/w | <C3,E3,G3>/w |

Four whole-note chords on the bass staff: C–E–G (I, home), F–A–C (IV, packing), G–B–D (V, leaning), and C–E–G again (I, landed). One complete lap of the functional cycle in C major.

Family members substitute for each other: swap the IV for a ii and play I–ii–V–I — same trip, different scenery. That's what makes function such a powerful listening tool: you can predict a progression's shape before you've identified a single chord.

Now the famous loop from this chapter's opening: I–V–vi–IV. It doesn't run the textbook lap — the dominant arrives early, hands off to the tonic family's cousin (vi) instead of home, and the loop restarts before anything fully lands. That's not the cycle failing; that's a songwriter using its gravity to keep a chorus airborne. Function isn't a rulebook music obeys; it's the physics music plays with. (What we call the landings is next unit's whole story.)

Misconception check. "Function = position in the bar." A chord's function is its job, not its seat. V is dominant whether it falls on beat one, beat three, or the last eighth note before the chorus; I is tonic at any tempo, in any bar. When chords change is a real and separate fact — it has its own name, harmonic rhythm, waiting at the end of this chapter.

Minor keys: one sharp changes the story

Build triads from A natural minor — nothing but the key signature's notes — and label them with the code:

Degree	Spelling	Quality	Numeral
^1	A–C–E	minor	i
^2	B–D–F	diminished	ii°
^3	C–E–G	major	III
^4	D–F–A	minor	iv
^5	E–G–B	minor	v
^6	F–A–C	major	VI
^7	G–B–D	major	VII

Read the case doing honest work: home is lowercase now (i — minor tonic), the majors moved to III, VI, VII, and ^2 turned diminished. The guitar loop Am–F–C–G that lives in half the minor-key songs you know? i–VI–III–VII — four chords straight off this table, which is why it flows so smoothly: natural minor gliding in a circle, nothing forcing the issue.

But listen to what this table is missing. Play v → i (E–G–B to A–C–E) and be honest: does that land, or just… arrive? Now raise one note — G to G♯ — and play V → i. That lands. Unit 5 taught you this move melodically: when minor music drives home, ^7 rises — te becomes ti. Harmonically, that single raised note does three things to the table:

• v becomes V — E–G♯–B, major, the real dominant, with the leading tone inside it.
• vii° appears on the raised note — G♯–B–D, diminished, the dominant family's second member, now available in minor exactly as in major.
• III becomes III+ — C–E–G♯, augmented, the + symbol's one diatonic residence in this course. You'll meet it rarely; recognize it, name it, don't fear it.

So minor-key analysis speaks two honest dialects: the natural-minor circle and the harmonic-minor drive. The numeral reports what actually sounds. G natural in the chord on E? Write v. G♯? Write V. Most charts live in the second dialect the moment the music heads home — "the five chord is major, even in minor keys" is one of the most-used facts in a working musician's pocket.

KEY: a minor    METER: 4/4    CLEF: bass
| <E3,G3,B3>/h <E3,G#3,B3>/h |

Two half-note chords on the bass staff: E–G–B, then E–G♯–B — A minor's dominant triad before and after the seventh degree rises. One accidental turns v into V: the difference between drifting home and driving there.

One spelling discipline, because a chatbot will get this wrong for you eventually: that raised note is G♯, never A♭. Same piano key, different written notes — and only one of them is ^7. Enharmonic means "sounds alike," never "equals," and in harmonic analysis the spelling is the meaning.

Misconception check. "Minor key, so minor five chord." Only while the music floats in natural minor. The moment it drives home, ^7 rises and V is major — E–G♯–B in A minor, D–F♯–A in G minor, B–D♯–F♯ in E minor. If you write v where the music sounds V, your analysis says "drifting" where the song says "arriving."

The dominant pair: V7 and vii° share an engine

Unit 10 gave V its seventh: in C major, V7 = G–B–D–F, the chord that begs hardest for home. Now spell its neighbor on ^7: vii° = B–D–F.

Side by side, vii° is the top three notes of V7 — they share B, D, and F, and buried in both is the interval doing the actual work: B to F, the tritone. Unit 7 taught you that six-half-step itch; every major and harmonic-minor key hides exactly one, between ^4 and ^7, and both dominant-family chords own it. The shared tritone is the shared job — it's why V7 and vii° are teammates and why either can hand the music home.

KEY: C major    METER: 4/4    CLEF: bass
| <G2,B2,D3,F3>/w | <B2,D3,F3>/w |

Two whole-note chords on the bass staff: G–B–D–F, then B–D–F. The vii° triad is V7's upper three notes — including the tritone B–F that gives both chords their lean toward C.

So is vii° "just V7 missing a note"? Closer than most rumors get: the notes agree, the tritone agrees, the job agrees. But vii° is its own chord — its own root, its own bass options, its own figured life (vii°6 spends its career walking between I and I6, and the studio will make you hear it do exactly that). Cousins who carpool. Not the same person.

Sevenths in minor — and vii°7 finds its home. Stack sevenths on A natural minor and read the parade: i7 (A–C–E–G) · iiø7 (B–D–F–A) · IIIM7 (C–E–G–B) · iv7 (D–F–A–C) · v7 (E–G–B–D) · VIM7 (F–A–C–E) · VII7 (G–B–D–F). Two deserve a second look. iiø7 — half-diminished on ^2 — is the minor-key predominant with the velvet ache in it. And VII7 is a dominant-seventh-quality chord natural minor builds with no accidentals at all, sitting on the subtonic.

Then raise ^7, and the dominant machinery proper clocks in: V7 = E–G♯–B–D. And on the raised note itself, the chord unit 10 told you was homeless in major keys finally gets its address: vii°7 = G♯–B–D–F — diminished triad, diminished seventh, every note diatonic to A harmonic minor.

Here is the full circle, worth memorizing as a pair of sentences. In any major key, the seventh chord on ^7 is viiø7 — half-diminished — because the key supplies only a minor seventh above that root (B–D–F–A in C major). The fully diminished vii°7 is diatonic to harmonic minor — raised ^7 as its root, diminished seventh above (G♯–B–D–F in A minor). One interval of difference, two different homes. If a plain major-key analysis ever shows you vii°7, somebody — possibly a chatbot — misspelled the ø.

Misconception check. "vii° is just an incomplete V7, so I can write either." They overlap in notes and agree in function, but analysis names what's actually on the page: no G in the chord means it isn't V7. Write vii° (or vii°6, per the bass), and save "they're cousins" for the rehearsal conversation.

Figures ride along

Units 9 and 10 taught the figures as a bass-naming system: triads take 6 (third in the bass) and 6/4 (fifth in the bass); sevenths take 7, 6/5, 4/3, 4/2 as the bass walks down through the chord's members. In analysis, the figures simply attach to the numeral, so one compact symbol names the chord, its quality, its job, and its bass note:

• ii6 in C major: the ii chord (D–F–A) with its third, F, in the bass. Predominants love first inversion — the bass slides more smoothly into V.
• i6/4 in A minor: the tonic triad (A–C–E) with its fifth, E, in the bass.
• V6/5 in C major: V7 (G–B–D–F) with its third, B — the leading tone itself — in the bass, aimed straight up at C.
• V4/2 in C major: V7 with its seventh, F, in the bass, which unit 13 will teach you tends to step down. For now: 4/2 means the seventh is downstairs.

Reading them is a two-step reflex: numeral → which chord and job; figure → which member is in the bass. One standing convention, restated because it matters more every unit: inversion figures use slashes, never hyphens — a hyphen (4-3) means a suspension, a different event that belongs to unit 14.

Misconception check. "V6/5 is a different chord from V7." Same four notes, same quality, same dominant job — different floor plan. The figures change the bass note, not the chord's identity. One dominant seventh, four addresses: V7, V6/5, V4/3, V4/2.

Harmonic rhythm: how fast the chords walk

One more tool — the one that makes analyses sound like music instead of spreadsheets. Two songs can share a tempo and feel completely different because their chords change at different rates. That rate is harmonic rhythm.

Play the cycle with one chord per bar:

KEY: C major    METER: 4/4    CLEF: bass
| <C3,E3,G3>/w | <F3,A3,C4>/w | <G2,B2,D3>/w | <C3,E3,G3>/w |

Four whole-note chords on the bass staff — I, IV, V, I in C major, one chord per bar: a harmonic rhythm of one change per measure.

Now the same lap at the same tempo, chords changing twice as fast:

KEY: C major    METER: 4/4    CLEF: bass
| <C3,E3,G3>/h <F3,A3,C4>/h | <G2,B2,D3>/h <C3,E3,G3>/h |

The same four chords on the bass staff, now as half notes — two chords per bar. The beat never changed; the harmony walks twice as fast. That difference is harmonic rhythm.

The beat didn't move. The harmony did. A hymn changes chords nearly every beat; a pop chorus holds each chord a bar or two; a ballad's bridge might sit on one chord for eight bars while the vocal does all the walking. When you draft your Chart song's progression this week, the second question after "which chords?" is "how often do they change?" — two songs with identical numerals and different harmonic rhythms are two different songs.

Misconception check. "Harmonic rhythm is just another word for tempo." Tempo is how fast the beat runs; harmonic rhythm is how fast the chords walk. Clap the beat with one hand and wave each chord change with the other — two speeds at once, which is exactly the point.

Why this makes you a better musician

Every skill here converts to bandstand currency at full exchange rate. You can call a progression after two hearings — "it's a one–five–six–four" — because you're listening for jobs, not letters, and there are only three jobs. You can transpose instantly: when the singer says "can we try it in E?", a progression stored as numbers re-spells itself, and the studio makes you prove it at the whiteboard. You can talk harmony the way bands talk — "to the four," "hang on the five," "half-diminished two going to five" all mean something exact to you now. And when a chart shows that G♯ in an A-minor song, you know it's the dominant's engine, not a typo. Unit 15 will hand you the Nashville Number System — the numbers-first chart language of working rooms everywhere — and you'll discover you've been fluent since this morning. This week, milestone 2 aims all of it at your own song: your Chart's progression, drafted in numbers, by your own ear.

Key terms

• Roman numeral (analysis) — a chord label giving the root's scale degree (the numeral) and the quality (case + symbols), valid in every key.
• Case (upper/lower) — uppercase = major (I, IV, V); lowercase = minor (ii, iii, vi, i).
• ° and + — the diminished and augmented symbols: vii° and ii° (minor); III+ (harmonic minor).
• Tonic function (T) — the home family: I, with vi and iii as relatives.
• Predominant function (PD) — the setup family, pointing toward the dominant: ii and IV.
• Dominant function (D) — the leaning family that resolves home: V (and V7) and vii°.
• Functional cycle — the standard trip T → PD → D → T: home, away, lean, land.
• The dominant pair — V7 and vii°: three shared notes, the key's tritone (^4–^7), one shared job.
• viiø7 vs. vii°7 — half-diminished on ^7 belongs to every major key; fully diminished vii°7 to harmonic minor.
• Figures (in analysis) — bass-naming numbers on the numeral: 6, 6/4; 7, 6/5, 4/3, 4/2. Slashes always; a hyphen means a suspension (unit 14).
• Harmonic rhythm — the rate at which chords change, independent of tempo.

Summary

• A Roman numeral names a chord's root degree (the numeral) and quality (case; ° and +). The major-key parade — I, ii, iii, IV, V, vi, vii° — is identical in all fifteen major keys, which is why numbers survive key changes and letters don't.
• Three function families: tonic (I, vi, iii — home), predominant (ii, IV — setup), dominant (V, vii° — the lean). The functional cycle T→PD→D→T is the chassis under an enormous share of real music, including loops that ride it without fully landing.
• Natural minor's table — i, ii°, III, iv, v, VI, VII — explains the smooth minor loops (i–VI–III–VII); raising ^7 (unit 5's ti) creates the driving dialect: V major, vii° on the raised note, III+ the rare augmented resident. The numeral reports whichever seventh degree actually sounds.
• V7 and vii° are the dominant pair: vii° is V7's upper three notes, and both own the key's tritone (^4–^7). Major keys build viiø7 on ^7; vii°7 is diatonic to harmonic minor, its true home.
• Figures attach to numerals and name the bass: ii6 (third), i6/4 (fifth), V6/5 / V4/3 / V4/2 (V7's third, fifth, seventh downstairs). Slashes, never hyphens.
• Harmonic rhythm is the speed of chord change, not of the beat. Same tempo, different harmonic rhythm = a different-feeling song. (One more color while we're here: build a seventh on harmonic minor's own tonic — do, me, sol, ti — and you get the minor-major seventh, i(M7): a minor triad wearing a major seventh. Film composers love it. You'll hear it, you can now name it, and it will never appear on a quiz or exam in this course.)

Check your understanding

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

1. In E♭ major, spell the chord that the numeral ii names, give its quality, and name its function family.
2. In G minor, spell the chord on ^5 twice — once as natural minor builds it, once as the music builds it when driving home — and write the numeral for each version. Which single note changed, and what does unit 5 call that change in solfège?
3. Spell vii°7 in C minor, and explain in one sentence why this exact chord cannot be built diatonically in any major key.
4. In D major, a chart's analysis shows V6/5. Spell the full chord and name the bass note. What one word describes that bass note's relationship to the key (unit 5 vocabulary)?
5. Two praise songs both run at 72 beats per minute. Song A changes chords every two beats; song B holds each chord for a full bar. Which has the faster harmonic rhythm, and why is "Song A is faster" the wrong way to say it?

Answers (worked reasoning — read after attempting)

1. ^2 in E♭ major is F; the key's own notes give F–A♭–C. F to A♭ is a minor third: minor, so lowercase — ii. Function family: predominant, the bags-packed setup crew.
2. Natural minor: D–F–A, minor — written v. Driving home, ^7 rises: D–F♯–A, major — written V. The changed note is F→F♯, and unit 5's solfège calls the move te → ti. One accidental; drifting versus arriving.
3. C harmonic minor raises ^7 to B natural; thirds inside the scale give B–D–F–A♭ — diminished triad, and B up to A♭ a diminished seventh. No major key can build this on its ^7: a major key supplies only a minor seventh above that root (B–D–F–A in C major), which makes viiø7. The fully diminished version needs harmonic minor — its diatonic home.
4. V7 in D major is A–C♯–E–G; the figure 6/5 puts the third in the bass: C♯. That bass is D major's ^7, a half step under the tonic — the leading tone (unit 5's word) — which is why V6/5 sounds aimed straight at D.
5. Song A has the faster harmonic rhythm — its chords change more often. But the tempo is identical, so neither is "faster" as music. Harmonic rhythm measures the walk of the harmony, tempo the run of the beat — two independent speeds, both audible at once.

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Next in this module: the Readings & Resources page, then Tutorial 11 — due before Week 11's second session so the studio can call progressions at full speed. And keep your Chart song within reach: milestone 2 lives inside this week's assignment.

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