Scale Patterns, Sets and the Diatonic Interval Vector — What Is Rare About the Major Scale

Department of Music | Stage: Albedo (Intermediate) | Estimated duration: 60 minutes

Objectives

After this lesson, you will be able to:

Build any scale on any root from its step pattern, and spell a seven-note scale with one letter per degree
Tell a scale’s step pattern, its pitch-class set and its interval-class vector apart, and say what each one forgets
Compute the vectors of the major, pentatonic and whole-tone scales, and use the common-tone theorem to say how many notes two keys share
Recognise the deep scale property, Myhill’s property, maximal evenness and generation by fifths, and say which of them the major scale, the pentatonic and the whole-tone scale have
Trace how GA computes these properties, and where its results depend on the transposition or differ from the definitions

1. A Scale as a Step Pattern

A scale lists notes in order within an octave. Its step pattern lists the semitones from each note to the next, ending with the step back to the octave, so the steps add up to 12.

Scale	Step pattern	On C
Major	2 2 1 2 2 2 1	C D E F G A B
Natural minor	2 1 2 2 1 2 2	C D E♭ F G A♭ B♭
Major pentatonic	2 2 3 2 3	C D E G A
Whole tone	2 2 2 2 2 2	C D E F♯ G♯ A♯

Written in whole steps (W) and half steps (H), the major pattern is W W H W W W H. Wikipedia’s “Major scale” notes that “a major scale may be seen as two identical tetrachords separated by a whole tone”: W W H, then W, then W W H.

To build a scale on another root, apply the same steps from that root. A seven-note scale then takes one letter per degree, and each accidental follows from the letter (MUS-007). E major is E F♯ G♯ A B C♯ D♯. F major is F G A B♭ C D E: the fourth degree is a semitone above A, and it is written B♭, not A♯, because A is already the third degree.

On the guitar, a step pattern on one string is a list of fret distances. On the A string, the major pattern from the open string gives frets 0 2 4 5 7 9 11 12: A B C♯ D E F♯ G♯ A. The same distances from fret 3 give C major: frets 3 5 7 8 10 12 14 15.

Practice Exercise

Build B♭ major and D natural minor from their patterns, with one letter per degree.

Solution: B♭ major: B♭ C D E♭ F G A, steps 2 2 1 2 2 2 1. D natural minor: D E F G A B♭ C, steps 2 1 2 2 1 2 2. In both, each letter appears once.

2. Pattern, Set and Root

A step pattern says nothing about where to start; a root fixes the notes. The pitch-class set keeps the notes and forgets the root, the order and the octaves (MUS-020): C major is {0, 2, 4, 5, 7, 9, 11} with C = 0.

Two scales can share a set and differ in their pattern. A natural minor, A B C D E F G, holds the notes of C major. Its pattern, 2 1 2 2 1 2 2, is the major pattern read from its sixth step: the steps A–B, B–C, then C–D through G–A. Reading one pattern from another step gives a mode; MUS-006 counts modes as rotations.

Transposing a scale adds the same number to every pitch class. It keeps the pattern and changes the set: the twelve major scales are twelve different sets. A scale that maps to itself under a transposition has fewer: there are only two whole-tone scales, {0, 2, 4, 6, 8, 10} and {1, 3, 5, 7, 9, 11} (MUS-006, MAT-022).

The rest of this lesson studies properties of the set, which the root does not change: its interval vector, the sizes its intervals come in, and how evenly it spreads around the octave.

Practice Exercise

Which major scale holds the notes of E natural minor, and from which of its degrees does E minor start?

Solution: E natural minor is E F♯ G A B C D, the notes of G major. E is the sixth degree of G major, and E minor’s pattern, 2 1 2 2 1 2 2, is G major’s read from that degree.

3. The Interval Vector of a Scale

The interval-class vector counts, for every pair of notes in a set, the interval class between them, from 1 (a semitone or a major seventh) to 6 (a tritone) (MUS-020). A set of n notes has n(n − 1)/2 pairs: 21 for seven notes, 15 for six, 10 for five.

For the major scale, count each class on C D E F G A B:

Class 1, semitones: E–F and B–C. 2.
Class 2, whole steps: C–D, D–E, F–G, G–A and A–B. 5.
Class 3, minor thirds: D–F, E–G, A–C and B–D. 4.
Class 4, major thirds: C–E, F–A and G–B. 3.
Class 5, perfect fifths: F–C, C–G, G–D, D–A, A–E and E–B. 6.
Class 6, the tritone: B–F. 1.

The total is 21, and the vector is <2 5 4 3 6 1>.

Scale	Notes	Pairs	Vector
Major	C D E F G A B	21	<2 5 4 3 6 1>
Harmonic minor	A B C D E F G♯	21	<3 3 5 4 4 2>
Major pentatonic	C D E G A	10	<0 3 2 1 4 0>
Whole tone	C D E F♯ G♯ A♯	15	<0 6 0 6 0 3>

The pentatonic has no semitone and no tritone; MUS-006 computes the same vector for the minor pentatonic. In the whole-tone scale every note has a whole step on each side and a major third on each side, which makes six pairs of class 2 and six of class 4. Its three tritones are C–F♯, D–G♯ and E–A♯, and it has no odd interval class at all.

Practice Exercise

Compute the vector of the six-note scale C D E F G A.

Solution: Fifteen pairs. Class 1: E–F. Class 2: C–D, D–E, F–G and G–A. Class 3: D–F, E–G and A–C. Class 4: C–E and F–A. Class 5: C–F, C–G, D–G, D–A and E–A. No tritone, since B is missing. The vector is <1 4 3 2 5 0>, and its entries add up to 15.

4. Deep Scales and Common Tones

The six entries of <2 5 4 3 6 1> are six different numbers. Wikipedia’s “Common tone (scale)” calls this the deep scale property: “containing each interval class a unique number of times”. The pentatonic’s vector repeats 0, the harmonic minor’s repeats 3 and 4, and the whole-tone scale’s repeats 0 and 6: none of them is deep.

The common-tone theorem. Transpose a set by n semitones. A note b of the transposed set is a + n for some note a of the set, and b is also in the original set exactly when the set holds the interval going up n semitones from a. So the number of common tones is the number of such pairs:

for n from 1 to 5, each pair is one interval of class n, so the count is the vector’s entry for class n;
for n from 7 to 11, each pair is one interval of class 12 − n;
for the tritone, n = 6, each tritone counts from both of its ends, since a + 6 + 6 = a: B up to F and F up to B. The count is twice the vector’s entry.

Wikipedia states the theorem for the diatonic scale: “However many times an interval class occurs in a diatonic scale is the number of tones common both to the original scale and a scale transposed by that particular interval class.” For C major:

Transposition (semitones)	Keys	Common tones
0	C	7
1 or 11	D♭, B	2
2 or 10	D, B♭	5
3 or 9	E♭, A	4
4 or 8	E, A♭	3
5 or 7	F, G	6
6	F♯	2: B, and F, written E♯ in F♯ major

The two neighbours of C on the circle of fifths, F and G, keep six notes: going from C major to G major changes only F to F♯, one finger moved by one fret. Wikipedia: “Six of seven possible common tones are shared by closely related keys”. Because the scale is deep, the transpositions by 1 to 5 semitones keep five different numbers of notes, the same up or down.

The tritone row needs care. Wikipedia gives the tritone 1 common tone, “as there is only one tritone in a diatonic scale”, and its table lists B for F♯ major and F for G♭ major on separate rows. Counted as pitch classes, C major and F♯ major share both B and F; F♯ major spells F as E♯. Each tritone counts from both of its ends. That makes two notes, the same count as a transposition by a semitone; Wikipedia’s 1 holds only for notes spelled the same. The doubled tritone count can repeat another one: the deep property guarantees different counts only for interval classes 1 to 5.

The whole-tone scale shows the opposite case: “every even transposition of the whole tone scale is identical with the original and every odd transposition has no common tones whatsoever” (Wikipedia, “Common tone (scale)”).

Which sets are deep? Six different whole numbers add up to at least 0 + 1 + 2 + 3 + 4 + 5 = 15, so a deep set has at least 15 pairs, and therefore at least six notes. A count over the 4096 pitch-class sets finds 48 deep sets, twelve transpositions in each of four set classes:

Forte class	Example	Vector	A chain of
6-1	C C♯ D E♭ E F	<5 4 3 2 1 0>	semitones
6-32	C D E F G A	<1 4 3 2 5 0>	fifths, F to E
7-1	C C♯ D E♭ E F F♯	<6 5 4 3 2 1>	semitones
7-35	C D E F G A B	<2 5 4 3 6 1>	fifths, F to B

So among seven-note sets, only the diatonic scale and the chromatic segment of seven notes are deep. Wikipedia’s “Common tone (scale)” states the pattern this way: “In twelve-tone equal temperament, all scales with the deep scale property can be generated with any interval coprime with twelve”. Read with the table, each deep set is a chain of one such interval, not of every one: the semitone or the fifth, the only two up to inversion. Multiplying every pitch class by 5, which MUS-020 calls M5, swaps the counts of classes 1 and 5: it sends each chromatic row of the table to the row of fifths of the same size.

Practice Exercise

How many notes do C major and E♭ major share, and which?

Solution: E♭ is 3 semitones above C, and the major scale has four intervals of class 3, so they share four notes: C, D, F and G. E♭ major replaces E, A and B with E♭, A♭ and B♭.

5. Two Sizes per Step, and Even Spacing

A generic interval “is the number of scale steps between notes of a collection or scale”; a specific interval is the number of semitones (Wikipedia, “Generic and specific intervals”). In the major scale, a third, two steps, is a minor third or a major third: 3 or 4 semitones.

Generic interval (steps)	1	2	3	4	5	6
Major	1, 2	3, 4	5, 6	6, 7	8, 9	10, 11
Major pentatonic	2, 3	4, 5	7, 8	9, 10
Whole tone	2	4	6	8	10

Myhill’s property is having “exactly two specific intervals for every generic interval” (Wikipedia, “Generic and specific intervals”). The major and pentatonic scales have it; the whole-tone scale, with one size per step, does not.
Maximal evenness (Clough and Douthett, 1991) is a different condition: every generic interval comes in one size or two consecutive sizes, so that the notes are “spread out as much as possible” (Wikipedia, “Maximal evenness”). The major, pentatonic and whole-tone scales are maximally even; the harmonic minor is not, since its seconds are 1, 2 and 3 semitones. Neither property implies the other. The whole-tone scale is maximally even without Myhill’s property, and the seven-note chromatic segment C to F♯ of §4 has Myhill’s property without being maximally even: its seconds are 1 or 6 semitones, two sizes but not consecutive ones.
A generated collection is “formed by repeatedly adding a constant interval in integer notation, the generator” (Wikipedia, “Generated collection”). The major scale is a chain of seven notes linked by six fifths: “F-C-G-D-A-E-B”. A chain is well formed when the generator always spans the same number of steps: in the major scale, each fifth of the chain spans 4 steps. “The major and minor pentatonic scales are also well formed.” The six notes F C G D A E give C D E F G A, whose steps are 2 2 1 2 2 3: a fifth spans 4 steps from C to G but 3 from F to C, so this chain is generated and not well formed. Wikipedia’s “Maximal evenness” ties well-formedness to Myhill’s property, “a scale with Myhill’s property is said to be a well-formed scale”, and says that the whole-tone scale “is not well-formed since each generic interval comes in only one size”. Its “Generated collection” calls the same scale a “degenerate well-formed collection”, in which every step is the generator. For sets of two or more notes, once the degenerate chains are set aside, well-formedness and Myhill’s property mean the same thing.

Scale	Deep	Myhill’s property	Maximally even	Generated
Major	yes	yes	yes	yes, seven notes in fifths
Major pentatonic	no	yes	yes	yes, five notes in fifths
Whole tone	no	no	yes	degenerate, whole steps
Harmonic minor	no	no	no	no
C D E F G A	yes	no	no	yes, six notes in fifths

Of these five scales, only the major scale has all four properties. Rothenberg propriety adds one more test: an interval spanning more steps should never be smaller than one spanning fewer. The major scale passes, but the tritone is both a fourth, F–B, and a fifth, B–F, so Wikipedia’s “Rothenberg propriety” calls it proper but “not strictly proper because the three step intervals and the four step intervals share an interval size (the tritone)”; “the major pentatonic scale is strictly proper”.

These are facts about structure. They do not prove that the major scale sounds balanced; they show what is rare about it among the 4096 sets.

Practice Exercise

Does A harmonic minor have Myhill’s property? Is it maximally even?

Solution: Neither. Its seconds are A–B, 2 semitones, B–C, 1, and F–G♯, 3: one generic interval in three sizes. Wikipedia’s “Maximal evenness” gives this example.

6. Where GA Stands

GA is the music-theory library and chatbot of the GuitarAlchemist ecosystem. The facts below are read from its code at commit 5c3a52a. This lesson documents that code and does not change it. It has not run GA or its tests: the counts below come from a line-by-line Python transcription of the methods cited. The files it cites are unchanged on GA’s main at 0843879. Learn’s music-theory-ga course compiles an earlier version of GA, a826864, which has IsDeepScale but not ScaleStructuralProperties. Its lesson 4 already quotes IsDeepScale and Wikipedia’s definition of a deep scale; this section goes on to the common tones and to the other properties.

A scale is built from notes. Scale takes a list of notes and computes their pitch-class set and vector. Its named scales are spelled out: Major is “C D E F G A B”, MajorPentatonic “C D E G A”, WholeTone “C D E F# G# A#”, and Minor.Natural “A B C D E F G”. A test checks that the major scale’s vector is <2 5 4 3 6 1>, and another that A minor and C major share a set, as in §2. The scale also exposes six structural properties, and GA’s MCP server prints them for a 12-bit scale id in its GaScaleProperties tool.

IsDeepScale is right, and no code refers to it. IsDeepScale checks that the vector’s six values are distinct, and the vector always holds six values, zeros included, so the pentatonic’s two zeros count as a repeat. Transcribed over the 4096 sets, it is true for exactly the 48 sets of §4. Its documentation points to the common-tone theorem. No other code file in GA names it, and no test checks it.

Myhill’s property and Rothenberg propriety follow the definitions. HasMyhillProperty collects the sizes of each generic interval and requires exactly two; transcribed, it holds for the major and pentatonic scales and not for the whole-tone scale. GetRothenbergPropriety compares the largest size of each generic interval with the smallest size of every larger one; transcribed, it returns Proper for the major scale and StrictlyProper for the pentatonic, as Wikipedia does. The Myhill test is named Dorian_HasMyhillProperty_ReturnsTrue and builds Scale.Major; its comment reads “Major / Dorian (1709)”, but 1709 is the id of C Dorian, a different set from C major’s 2741. All five tests of the file build Scale.Major, so none of them checks a scale for which HasMyhillProperty or IsWellFormed is false.

IsWellFormed tests generation, not well-formedness. The method’s summary defines a well-formed scale as one “generated by repeatedly stacking a single generator interval mod 12”, and the method accepts any chain of one interval. That is §5’s generated collection. Transcribed over the 4096 sets, 89 pass IsWellFormed without Myhill’s property:

13 sets of at most one note: the empty set and the 12 single notes;
16 equal divisions of the octave: the tritone, the augmented triad, the diminished seventh, the whole-tone scale and the full chromatic, which Wikipedia calls degenerate;
60 chains of fifths of 4, 6, 8, 9 or 10 notes, such as C D G A and §5’s C D E F G A, each of which has a generic interval in three sizes.

No set has Myhill’s property and fails IsWellFormed. For the major scale the method reports generator 5, the first that works in the order 1 to 11: the chain of fourths B E A D G C F, which is §5’s chain of fifths read backwards.

The evenness score depends on the transposition. GetMaximalEvennessDiscrepancy takes the pitch classes of a set of n notes in ascending order from C, compares them with the points 0, 12 / n, 2 × 12 / n and so on, and returns the root mean square of the differences. Its summary cites Clough and Douthett and promises “0.0 for perfectly even sets like Whole Tone or Augmented”. Transcribed, rounded to four places:

the whole-tone scale scores 0.0000 on C and 1.0000 on C♯, and the augmented triad 0.0000 on C and 1.0000 on C♯;
GA’s own Scale.Augmented, C D♯ E G G♯ B, is the six-note augmented scale, not the triad, and scores 0.7071;
the twelve major scales score from 0.2857, B♭ major, to 1.1780, F♯ major; C major scores 0.4041.

For Clough and Douthett, maximal evenness is a yes-or-no property, and all twelve major scales have it. By GA’s score, 472 seven-note sets that are not maximally even come out more even than F♯ major. The tool’s own description suggests “1709 for Dorian or 2741 for Major”: the two ids hold the notes of B♭ major and of C major, one scale in two transpositions, and score 0.2857 and 0.4041. No test checks the score.

Fixing any of this is for GA’s owners; this lesson only describes it.

Practice Exercise

What do GA’s IsWellFormed and HasMyhillProperty return for C D G A, and why do they disagree?

Solution: IsWellFormed returns true with generator 5: from A, adding 5 semitones three times gives A D G C. HasMyhillProperty returns false: the seconds C–D, D–G, G–A and A–C are 2, 5, 2 and 3 semitones, three sizes. The set is a chain of one interval, which is all that IsWellFormed checks, but the fourth spans two steps from A to D and one from D to G, so the chain is not well formed.

7. Proposed Experiment (Not Yet Run)

Status: not run. This section proposes an experiment for Learn’s music-theory-ga lab, which compiles GA. The lab’s GA pin would first move to 5c3a52a, since ScaleStructuralProperties does not exist at a826864. Nothing in this section is a measurement:

the predictions come from §3 to §6 and are written down before any run;
the counts and scores come from the Python transcription of §6;
a later version of this lesson will report the results.

Every step calls GA’s types in the lab’s own process, never a running MCP server or a language model.

Three vectors. Print IntervalClassVector and IsDeepScale for Scale.Major, Scale.MajorPentatonic and Scale.WholeTone. Prediction: <2 5 4 3 6 1> and true, <0 3 2 1 4 0> and false, <0 6 0 6 0 3> and false.
The deep census. For every pitch-class set, read IsDeepScale, and for each deep set, IsWellFormed and its generator. Prediction: 48 deep sets, 24 of six notes in classes 6-1 and 6-32 and 24 of seven notes in classes 7-1 and 7-35; all of them pass IsWellFormed, with generator 1 for the chromatic classes and 5 for the others.
The common-tone theorem. For every non-empty set and every n from 1 to 11, count the pitch classes common to the set and its transposition by n, and compare with the vector, doubled for n = 6. Prediction: no exception among the 4095 sets; for C major, the counts for n = 1 to 11 are 2 5 4 3 6 2 6 3 4 5 2.
The evenness score. Call GetMaximalEvennessDiscrepancy on the twelve major scales and the two whole-tone scales. Prediction, to four places:

Major scale Score Major scale Score

C 0.4041 F♯ 1.1780

C♯ 0.5151 G 0.5151

D 0.6389 A♭ 0.4041

E♭ 0.3194 A 0.7693

E 0.9035 B♭ 0.2857

F 0.3194 B 1.0400

The whole-tone scales score 0.0000 on C and 1.0000 on C♯, and 472 seven-note sets that are not maximally even score below 1.1780.
Generated versus well formed. Over all 4096 sets, the empty set included, count the sets for which IsWellFormed is true and HasMyhillProperty false, then the reverse. Prediction: 89 and 0.

Major scale	Score	Major scale	Score
C	0.4041	F♯	1.1780
C♯	0.5151	G	0.5151
D	0.6389	A♭	0.4041
E♭	0.3194	A	0.7693
E	0.9035	B♭	0.2857
F	0.3194	B	1.0400

Practice Exercise

Step 4 predicts exactly 1.0000 for the whole-tone scale on C♯. Why?

Solution: Its notes in ascending order are 1, 3, 5, 7, 9 and 11, and the points they are compared with are 0, 2, 4, 6, 8 and 10. Each note is one semitone from its point, so each squared difference is 1, and the root mean square of six 1s is 1.

8. Common Pitfalls

Taking the pattern for the set. A natural minor and C major share a set, not a pattern.
Spelling with the wrong letter. F major has B♭, not A♯: one letter per degree.
Taking deep for even. The whole-tone scale is perfectly even and not deep; the deep set C D E F G A is not maximally even.
Counting the tritone once. In pitch classes, a transposition by a tritone keeps twice as many notes as the set has tritones: C major and F♯ major share B and F.
Reading GA’s IsWellFormed as Carey and Clampitt’s well-formedness. It checks only that a set is a chain of one interval.
Comparing GA’s evenness scores across transpositions. The score is measured from C: B♭ major and F♯ major are equally even and score 0.2857 and 1.1780.

Key Terms

Term	Definition
Step pattern	The semitones from each note of a scale to the next, back to the octave, such as 2 2 1 2 2 2 1
Pitch-class set	A scale’s notes without root, order or octave, such as {0, 2, 4, 5, 7, 9, 11}
Interval-class vector	The number of pairs of notes in each interval class from 1 to 6, such as <2 5 4 3 6 1>
Deep scale	A set whose vector holds six different numbers: in twelve notes, the classes 6-1, 6-32, 7-1 and 7-35
Common-tone theorem	A transposition by n semitones keeps as many notes as the vector counts for the interval class of n, twice that for the tritone
Generic interval	The number of scale steps between two notes
Myhill’s property	Every generic interval comes in exactly two sizes
Maximally even	Every generic interval comes in one size or two consecutive sizes
Generated collection	A set formed by stacking one interval, such as the chain of fifths F C G D A E B
Well formed	Generated, with the generator always spanning the same number of steps; for chains of two or more notes that are not degenerate, equivalent to Myhill’s property

Self-Check Assessment

1. Build E♭ major from its pattern and give its pitch-class set.

E♭ F G A♭ B♭ C D, with steps 2 2 1 2 2 2 1. With C = 0, its set is {0, 2, 3, 5, 7, 8, 10}.

2. Why does a modulation from C major to G major change one note, while C major and F♯ major still share two?

G is a transposition by 7 semitones, interval class 5, and the major scale holds six intervals of class 5, so six notes stay and one changes, F to F♯. F♯ is a transposition by a tritone; the scale has one tritone, B–F, which counts from both ends, so B and F stay. Spelled in F♯ major, F is E♯.

3. Is the major pentatonic deep? Does it have Myhill’s property? Is it maximally even?

It is not deep: its vector <0 3 2 1 4 0> holds 0 twice. It has Myhill’s property, since each generic interval comes in two sizes: 2 or 3, 4 or 5, 7 or 8, 9 or 10. It is maximally even, since each pair of sizes is consecutive.

4. In GA at 5c3a52a, what does GetMaximalEvennessDiscrepancy return for C major and for B♭ major, and what does the difference show?

0.4041 and 0.2857, to four places. The two scales have the same pattern and are equally even; the score differs because it measures each set from C.

Pass criteria: Build and spell a scale from its pattern on any root. Compute a scale’s vector and use it to count common tones, the tritone included. Say which of the deep, Myhill, maximally even and generated properties a scale has. Say what GA computes for each, and where its result depends on the transposition or differs from the definition.

Research Basis

Wikipedia, “Major scale”: the two tetrachords, and maximal evenness.
Wikipedia, “Interval vector”: the definition, and the deep scale property of the major scale and its modes.
Wikipedia, “Common tone (scale)”: the common-tone theorem and its table, closely related keys, the deep scale property, generation by an interval coprime with twelve, and the whole-tone scale’s transpositions; after Johnson, Foundations of Diatonic Theory, 2003, and Gamer, 1967.
Wikipedia, “Generic and specific intervals”: generic and specific intervals, and Myhill’s property.
Wikipedia, “Maximal evenness”: Clough and Douthett’s definition, the harmonic minor, the whole-tone scale, and Myhill’s property as well-formedness.
Wikipedia, “Generated collection”: the chain of fifths, well-formed and degenerate collections, after Carey and Clampitt, 1989.
Wikipedia, “Rothenberg propriety”: the major scale proper but not strictly, the pentatonic strictly proper.
GA source at commit 5c3a52ab40d0d433f8f68dbbd0590f70c8d8cf26: every code fact in §6 links to its line.
Learn, music-theory-ga lesson 4 at commit 0c918c8206dc9dda0241d097e6b0c65e897c92ec: its account of IsDeepScale.
Experiment: proposed in §7, not run; this lesson contains no measurement of its own.
Provenance: hand-authored by a Claude Code session (Opus 5.5) from the Streeling curriculum plan, not produced by the Seldon course pipeline; under review.
Belief state: T(0.85) F(0.02) U(0.10) C(0.03).