Symmetric Scales: Build Whole-Tone and Half-Whole Collections from Any Pitch

--- title: "Symmetric Scales: Build Whole-Tone and Half-Whole Collections from Any Pitch" slug: symmetric-scales-whole-tone-half-whole description: "Learn to construct the whole-tone and octatonic (half-whole) collections by repeating a single step pattern. Discover why only two whole-tone sets and three octatonic sets exist under transposition, practice checking note membership, and explore how these pitch-class sets generate augmented triads and diminished-seventh chords. Includes free worksheets for step-loop building and membership checks." category: Music Theory ---

What makes a scale "symmetric"?

You already know scales built from a specific pattern of half and whole steps: the major scale, the natural minor, the modes. Each of those has a characteristic uneven sequence — major goes W-W-H-W-W-W-H — and that unevenness is exactly what gives them a sense of direction, of home and away, of tension and resolution.

A symmetric scale is different. It uses the same interval over and over, or repeats a short pattern without interruption, until the steps bring you back to where you started. Two symmetric collections appear constantly in music after about 1880, and both are worth understanding from the ground up: the whole-tone collection (six notes, all whole steps) and the octatonic collection (eight notes, alternating half and whole steps).

Neither collection is exotic just for novelty's sake. They have clear internal logic, they are easy to build from any starting pitch, and they connect directly to the pitch-class thinking that underlies a lot of twentieth-century harmony. This article builds both collections from scratch, explains why there are only two distinct whole-tone sets and only three octatonic sets, and shows how the collections generate specific chord types. Practice materials are in Repeat the Steps, Find the Loop and Does the New Note Belong?.

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The whole-tone collection

The whole-tone collection has one rule: every step is a whole step, meaning 2 semitones.

Start on C and add 2 semitones six times:

C → D → E → F♯ → G♯ → B♭ → (C)

The sixth step brings you back to C, one octave higher. The complete set of pitch classes is {0, 2, 4, 6, 8, 10} in integer notation (where C = 0, C♯/D♭ = 1, and so on up to B = 11).

The whole-tone collection starting on C. Every step is exactly 2 semitones — six whole steps return to C an octave higher.

Count the badges: six steps of 2 semitones, summing to exactly 12 — one octave. That sum of 12 is not a coincidence. For any repeating-step pattern, the steps must total 12 for the collection to "close" — to arrive back at the starting pitch class. If they summed to less, the pattern would stop mid-octave; if more, it would overshoot.

The step formula lets you build the collection from any note. Starting from F♯ gives F♯-G♯-B♭-C-D-E-F♯, which uses the same six pitch classes {0, 2, 4, 6, 8, 10}. Starting from G♯ gives G♯-B♭-C-D-E-F♯-G♯ — still the same six. Every even-numbered starting pitch lands back in this one set.

Why the formula is useful

A pitch-class set is a collection of pitch classes considered without regard to which octave they occur in. Once you know that a piece or a passage is drawn from the whole-tone collection starting on C, you can immediately name every note that belongs: any of C, D, E, F♯, G♯, or B♭ (and their enharmonic equivalents such as A♭ for G♯). Any other note — D♭, E♭, F, G, A, B — does not belong.

An original melodic line built entirely from whole-tone Set 1 (pitch classes 0, 2, 4, 6, 8, 10). Every note verified: C=0 ✓ E=4 ✓ F♯=6 ✓ B♭=10 ✓ G♯=8 ✓ D=2 ✓.

This two-bar line uses only whole-tone notes. The melodic surface may feel floating or ambiguous — that is a direct consequence of the uniform step pattern, which leaves no asymmetry for the ear to use as a landmark.

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The two whole-tone sets

Here is a fact worth pausing on: there are only two distinct whole-tone collections in the twelve-note chromatic system.

That is surprisingly few. A major scale can start on any of 12 pitches; a natural minor likewise. But a whole-tone collection started on C and a whole-tone collection started on D are the same six pitch classes, just listed in a different order. Transposing the pattern by 2 semitones — by one whole step — gives back the identical set.

This is called transpositional symmetry. In pitch-class set theory (see Straus, Introduction to Post-Tonal Theory, 4th ed.), two sets are transpositionally equivalent if one is a transposition of the other by some interval n. The whole-tone set is transpositionally symmetric because T₂ (transposition by 2 semitones) maps the set onto itself: every element shifts by 2, but the collection as a whole stays the same.

Transposing whole-tone Set 1 up 2 semitones (T₂) returns exactly the same six pitch classes, reordered. The set is invariant under T₂.

Shift whole-tone Set 1 up 2 semitones and you get the same six pitch classes back. Shift again by 2 semitones — same set. Only when you shift by an odd number of semitones do the pitch classes change:

Transposing by 1 semitone (an odd number) moves every pitch class to the other whole-tone set.

T₁ (shifting by 1 semitone) lands on D♭, E♭, F, G, A, B — pitch classes {1, 3, 5, 7, 9, 11}. That is the other whole-tone set. And T₁ of that set lands back on Set 1. The two sets partition the chromatic scale into even-numbered and odd-numbered pitch classes.

There are only two whole-tone collections. Set 1 uses the six even pitch classes {0,2,4,6,8,10}; Set 2 uses the six odd ones {1,3,5,7,9,11}.

The consequence: every transposition of the whole-tone pattern lands in one of exactly these two collections. You cannot construct a third. Tymoczko (A Geometry of Music, ch. 4) discusses this property; the minimal transposition that returns a symmetric scale to itself can be called its period: the whole-tone collection has period 2.

All 12 pitch classes arranged in a ring. Cyan = whole-tone Set 1 (even); coral = whole-tone Set 2 (odd).

The pitch-class circle makes this tangible: the twelve positions alternate between Set 1 (cyan) and Set 2 (coral). Every even-numbered clock position is in Set 1; every odd-numbered position is in Set 2.

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The octatonic collection

The octatonic collection alternates two step sizes: one half step (1 semitone), then one whole step (2 semitones), repeating until the pattern closes. This is called the "half-whole" ordering; the reverse (whole-half) is also possible and produces the same pitch-class set from a different starting point.

Starting on C with the half-whole pattern:

C → D♭ → E♭ → E → F♯ → G → A → B♭ → (C)

Eight distinct pitch classes: {0, 1, 3, 4, 6, 7, 9, 10}. The step sizes: 1-2-1-2-1-2-1-2. Four pairs of (1+2) = four pairs of 3 semitones = 12 total. ✓

The octatonic (half–whole) collection starting on C. Steps alternate: 1 semitone, 2 semitones, repeating four times. Eight notes, step sum 4 × 3 = 12.

The coral badges mark the half steps (+1); the cyan badges mark the whole steps (+2). Eight notes instead of six — the collection is denser, covering two-thirds of the chromatic scale rather than one-half.

An original melodic line using octatonic Set A. Every note belongs to {0,1,3,4,6,7,9,10}.

This line moves through octatonic pitch classes in a non-scalar order. Notice how a melody can skip freely within the collection while all eight candidates are available: the set gives the composer a wider pool of pitches per phrase than the whole-tone collection.

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Three distinct octatonic sets

The octatonic collection's period is 3, not 2: transposing by 3 semitones (T₃) returns the identical pitch-class set. So from twelve possible starting pitches, we get not twelve distinct collections but twelve ÷ 4 = three (since each set appears four times — at T₀, T₃, T₆, and T₉).

The three distinct octatonic (half–whole) collections. Every starting pitch belongs to exactly one of these three sets.

• Set A (from C): {0, 1, 3, 4, 6, 7, 9, 10}
• Set B (from D♭): {1, 2, 4, 5, 7, 8, 10, 11}
• Set C (from D): {0, 2, 3, 5, 6, 8, 9, 11}

Together these three sets cover all twelve pitch classes (with each pitch class appearing in exactly two sets — because each set has eight members and three sets × eight members = 24 = 2 × 12).

Open Music Theory v2 (Gotham et al.) states that there are exactly three octatonic pitch-class sets. The result follows from the period-3 symmetry and the arithmetic of 12 ÷ gcd(12, 3) = 12 ÷ 3 = 4 repetitions per collection.

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Comparing the two collections

The step formula determines everything about a collection's structure. Two properties follow immediately: how many notes the collection has, and how many distinct sets exist.

Step patterns compared. Whole-tone: six 2-semitone steps (sum = 12). Octatonic: four pairs of 1+2 semitones (sum = 12). Both close in one octave.

The whole-tone collection has six notes because 12 ÷ 2 = 6. Its period is 2, so twelve starting pitches divide into 12 ÷ 6 = 2 distinct sets.

The octatonic collection has eight notes because four pairs of (1+2) = 12. Its period is 3, so twelve starting pitches divide into 12 ÷ 4 = 3 distinct sets.

Quick reference: whole-tone versus octatonic at a glance.

Notice the "Repeats at" row: T₂ for whole-tone, T₃ for octatonic. Transposing the whole-tone collection by any even number returns the same set; by any odd number, you get the other set. Transposing the octatonic collection by any multiple of 3 returns the same set; by 1 or 2 (anything not divisible by 3), you move to one of the other two sets.

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How to build any collection: the recipe

Both collections follow the same construction algorithm.

Three steps to build any symmetric collection from scratch. The sum check: 6 × 2 = 12 for whole-tone; 4 × (1+2) = 12 for octatonic.

Step 1 — pick a starting pitch. Any pitch class from 0 to 11. For the octatonic collection, the starting pitch determines which set you are in: starting on C, E♭, F♯, or A all give Set A.

Step 2 — add the steps in order. For whole-tone: add 2 repeatedly. For octatonic (half-whole): add 1, then 2, alternating. Keep track of the running pitch class, wrapping back to 0 when you pass 11.

Step 3 — stop when you return. When the next step would bring you back to the starting pitch class, the collection is complete.

The Repeat the Steps, Find the Loop handout shows the step-loop algorithm worked out for both collection types. The single student task is to pick a starting pitch, apply the pattern, work out the complete member list, and name which of the two or three sets you landed in.

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Chords from symmetric collections

One of the most useful properties of these collections is the chord types they contain.

Augmented triads from the whole-tone collection

Every augmented triad draws three notes equally spaced by 4 semitones. Since the whole-tone collection contains only even pitch classes, and 4 is even, the three notes of an augmented triad always land inside the same whole-tone set.

An augmented triad built from whole-tone Set 1. Three of the six pitch classes — C=0, E=4, G♯=8 — equally spaced at 4 semitones.

C — E — G♯: pitch classes 0, 4, 8. All in Set 1 {0, 2, 4, 6, 8, 10}. The whole-tone collection contains exactly two distinct augmented triads: C–E–G♯ and D–F♯–B♭, which together account for all six pitch classes of Set 1.

Diminished-seventh chords from the octatonic collection

Every diminished-seventh chord draws four notes equally spaced by 3 semitones. Since the octatonic collection's period is 3, the T₃ symmetry means the four notes of a diminished-seventh chord are invariant under T₃ — the chord maps onto itself when transposed by 3 semitones.

A diminished-seventh chord from octatonic Set A. Four pitch classes (0, 3, 6, 9) equally spaced at 3-semitone intervals. T₃ maps this chord to itself.

C — E♭ — F♯ — A: pitch classes 0, 3, 6, 9. All in Set A {0, 1, 3, 4, 6, 7, 9, 10}. Three distinct diminished-seventh chords exist (one per octatonic set), and together they cover all 12 pitch classes with no overlap — which is why the diminished-seventh chord was such a versatile pivot in late nineteenth-century harmony.

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Membership check: does a note belong?

Once you know which set a passage is drawn from, you can check whether any candidate note belongs. The algorithm is simple.

1. Identify the set. If you know the starting pitch and collection type, look up the pitch-class set from the tables above, or derive it using the step formula.
2. Find the pitch class of the candidate note. Pitch class = MIDI note number modulo 12 (C4 = 60, so pc 60 mod 12 = 0 = C). Or just count semitones from C.
3. Check set membership. Is that pitch class in the set? If yes, the note belongs; if no, it does not.

A complete inventory: two whole-tone sets (cyan) and three octatonic sets (coral). Every pitch class appears once in each type's inventory.

Example: is A♭ in the whole-tone collection starting on C? A♭ = pc 8. The whole-tone Set 1 is {0, 2, 4, 6, 8, 10}. Is 8 ∈ {0, 2, 4, 6, 8, 10}? Yes — A♭ belongs (it is enharmonically G♯). Is D♭ in that set? D♭ = pc 1. Is 1 ∈ {0, 2, 4, 6, 8, 10}? No — D♭ does not belong to Set 1.

The Does the New Note Belong? exercise builds this check into four question types: filling in a missing member, yes/no candidate tests, transposition comparisons, and a short composition in the octatonic bank.

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Pitch-class set notation: a brief primer

Integer notation is the standard tool for describing these collections precisely. The conventions come from post-tonal music theory as formalized in Allen Forte's The Structure of Atonal Music (Yale UP, 1973) and Joseph Straus's Introduction to Post-Tonal Theory (4th ed., W. W. Norton, 2016):

• Pitch class 0 = C (and any octave of C: C2, C4, C7, ...)
• Pitch class 1 = C♯/D♭
• Pitch class 2 = D
• ... and so on up to pitch class 11 = B

Transposition T_n shifts every pitch class up by n semitones, wrapping at 12. T₂ applied to {0, 2, 4, 6, 8, 10} gives {2, 4, 6, 8, 10, 0} — the same set, reordered. T₁ gives {1, 3, 5, 7, 9, 11} — the other set.

The set class labels used in pitch-class set theory are 6-35 (the whole-tone hexachord, in Forte's numbering) and 8-28 (the octatonic octachord). These labels are reference identifiers, not performance instructions — but knowing them lets you look up analytical literature that uses them.

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Putting it into practice

The two worksheets for this topic take you from concept to pencil-and-paper fluency.

Repeat the Steps, Find the Loop (Handout 482) shows the step-loop algorithm worked out for both collection types and asks you to work out the complete member list from any starting pitch. Audio examples let you hear the characteristic sound of each collection: the whole-tone melody drifts without gravitational pull; the octatonic line has sharper interval contrasts from its alternating half steps.

An original melodic idea using whole-tone Set 1. Every note's pitch class is verified to belong to {0,2,4,6,8,10}.

Does the New Note Belong? (Exercise 186) has four sections:

• Section A — Fill in a missing member of a given collection row.
• Section B — Yes/No: does this candidate note belong to the named collection?
• Section C — Transposition checks: does T₂ or T₁ keep or change the set?
• Section D — Compose a short line using only the octatonic bank C/D♭/E♭/E/F♯/G/A/B♭.

Both worksheets include a listening component — audio examples of the complete collections — so you can connect the theoretical pitch-class set to its sound before attempting the written tasks.

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Summary

The whole-tone and octatonic collections are symmetric in the strict mathematical sense: each is invariant under a regular transposition. That single property — the period — determines how many distinct sets exist, what chords the collection contains, and how the music built from it sounds.

Whole-tone collection:

• Step pattern: 2-2-2-2-2-2 (six whole steps)
• Members from C: C D E F♯ G♯ B♭
• Pitch classes: {0, 2, 4, 6, 8, 10}
• Only 2 distinct sets (period T₂)
• Characteristic chord: augmented triad

Octatonic (half-whole) collection:

• Step pattern: 1-2-1-2-1-2-1-2 (alternating)
• Members from C: C D♭ E♭ E F♯ G A B♭
• Pitch classes: {0, 1, 3, 4, 6, 7, 9, 10}
• Only 3 distinct sets (period T₃)
• Characteristic chord: diminished seventh

Learning to build either collection from scratch — picking a starting pitch, applying the step loop, stopping when you return — is the foundational skill. Once you can do that, membership checks, transposition comparisons, and chord identification all follow from the same arithmetic. The worksheets give you the repetitions needed to make that arithmetic automatic.

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Sources

• Forte, Allen. The Structure of Atonal Music. Yale University Press, 1973.
• Gotham, Mark, et al. Open Music Theory, version 2. VIVA/Open Textbook Library, 2023.
• Straus, Joseph. Introduction to Post-Tonal Theory, 4th ed. W. W. Norton, 2016.
• Tymoczko, Dmitri. A Geometry of Music. Oxford University Press, 2011.