Seven rotations that are not seven modes

The seven rotations of the diatonic set are seven modes, and everything the last three rungs said about them relied on properties of the set they rotate: it is a chain of fifths, it has two sizes of every step, its interval counts are all different. Harmonic minor is what minor-key repertoire actually uses and it has none of those properties.

Its seven rotations have names — Phrygian dominant, Lydian ♯2, Ukrainian Dorian, and so on. Whether they are modes in the same sense is a question with an answer, and the answer is no in four measurable ways.

The point is not that the rotations are illegitimate. Two of them are in daily use. It is that “mode” has quietly come to cover two different kinds of object, and every property the diatonic modes have as a family is a property they inherit from the set they rotate rather than from the act of rotating.

That order is the circle of fifths run backwards, which is what makes the parallel arrangement a scale of brightness rather than a list. Read this way the seven rotations are seven points on one axis — and the reason they are not seven modes is that the axis has no tonic on it anywhere.

A fourth that might be four, five or six #

The third rung of the scale ladder is about what two sizes of every step buys: it is what makes an interval name useful. A third is three semitones or four; either way it is a third, and a musician learns to call both of them thirds without being told why that is allowed.

In harmonic minor it is not allowed. Its generic seconds are 1, 2 or 3 semitones. Its fourths are 4, 5 or 6. Its fifths are 6, 7 or 8. Its sevenths are 9, 10 or 11. Only the thirds and the sixths behave.

That is not a subtlety. “A fourth” in harmonic minor picks out a set of three different intervals, one of which is a tritone, and the qualifiers musicians actually use — perfect, augmented, diminished — are doing work that in the diatonic set is done by the scale’s own structure. In a well-formed scale the two sizes alternate predictably and a musician can tell which one they have from where in the scale they are. In harmonic minor they cannot, because there are three and no simple rule for which.

All fifteen have two members and every one of them is, or contains, the tritone above the tonic. A single degree never decides, which is the sharpest statement of this essay’s title: seven rotations of one set cannot be told apart by any note taken on its own, and what tells them apart is an interval rather than a pitch.

One step is responsible for all of it. Harmonic minor’s step pattern is 2-1-2-2-1-3-1, and the 3 is the augmented second the raised seventh creates. A single oversized step is enough to give four of the six generic intervals a third size, which is a good measure of how tightly Myhill’s property constrains a scale.

Where the third size comes from, exactly #

It is worth tracing the propagation, because it shows that the four failures are one failure counted four times rather than four separate defects.

A generic interval of size d, measured from degree i, is the sum of d consecutive steps starting at i. Harmonic minor’s steps are 2-1-2-2-1-3-1. The three appears once. So a generic interval has a third size exactly when some of its seven windows contain the 3 and some do not and the ones that do not already differ from each other.

• The second is a single step, so it takes the values 1, 2 and 3 directly.
• The third is two consecutive steps: 3, 3, 4, 3, 4, 4, 3 — two sizes, because the window containing the 3 pairs it with the 1 beside it and lands at 4, where two other windows already are.
• The fourth is three steps: 5, 5, 5, 6, 5, 6, 4 — three sizes, and one of them is a tritone.
• The fifth: 7, 6, 8, 7, 7, 7, 6 — three.
• The sixth: 8, 9, 9, 9, 8, 9, 8 — two, because the 3 is always accompanied by both of the 1s beside it in a six-step window.
• The seventh: 11, 10, 11, 10, 10, 11, 9 — three.

The two survivors are the thirds and the sixths, which are inverses of each other and therefore one fact rather than two. Harmonic minor has exactly one generic interval class that still names a distance, and it is the one European harmony leans on hardest — which is a coincidence worth noticing and not worth building on, since the sixth survives for a reason about window widths and nothing to do with harmony.

Three rotations with no fifth #

Exactly one rotation of the diatonic set has no perfect fifth above its own tonic, and the count is one because the set is a chain of six fifths over seven degrees.

Harmonic minor is not a chain of anything, and the count is three.

Melodic minor gives three as well. The whole-tone scale gives six of six, because it contains no perfect fifth at all. The octatonic gives four of eight.

The number of rotations with no fifth is a cheap measure of how far a scale is from being generated, and running it over every seven-note shape rather than over the scales in use makes the separation absolute rather than wide.

The diatonic set is the only seven-note shape in the twelve with a single fifthless rotation. Not the best of the ones anybody uses — the only one there is. Twelve shapes manage two, thirty manage three, and harmonic minor is in that middle crowd rather than at the bottom of it.

The other two counts are worth putting beside it, because they say the measure is not a coincidence and they are kinder to harmonic minor than the sections above have been. Sixteen of the 66 shapes contain a single tritone, so that count alone singles out nothing. And on Myhill’s property — how many of the six generic interval classes come in more than two sizes — the distribution is savage: sixty of the 66 shapes fail on all six classes, one fails on two, three fail on four, and two fail on none.

So harmonic minor’s four failing classes are not a poor score. Three shapes of 66 do that well or better, and harmonic minor is one of them — it is the third-closest seven-note shape to the diatonic set on the property this rung’s first argument is about, and the sixty that fail on everything are what a seven-note scale normally looks like.

The two shapes with no failing classes at all are the diatonic set and the chromatic run of seven adjacent semitones, and that pair is the whole of Myhill’s property at this size. Both are generated — one by the fifth, one by the semitone — which is the same theorem arriving from the other end. Every criticism this rung makes of harmonic minor is a criticism of not being one of two shapes, and that is worth saying before the criticisms rather than after.

There is a reading of the whole rung in that last table, and it is the one the census supports. The question “are harmonic minor’s rotations modes in the same sense” has been answered no, four times, on four measures — and the population says the honest form of the answer is that nothing is a mode in that sense except the diatonic set. The properties the diatonic modes have as a family are not properties a seven-note scale can have with a bit of luck; they are properties one shape of 66 has, plus a chromatic run nobody builds music on. So the vocabulary problem the names give away is not harmonic minor’s failing. It is that a word coined for the one shape that supports it was then applied to every other scale anybody rotated, and there was never a second candidate for it to fit.

The consequence for the rotations is direct. A rotation with no fifth above its tonic has a diminished or augmented triad on its first degree, so it cannot be a key of anything; it can only ever be a set of notes played over a chord that is somewhere else. Three of harmonic minor’s seven are in that position before any question of taste arises, against one of the diatonic set’s seven.

And four with the tonic in a tritone #

The diatonic set contains one tritone, so two of its seven rotations have the tonic at one end of it. Harmonic minor contains two tritones, so four do.

Both counts come from the interval vector, which is ⟨3, 3, 5, 4, 4, 2⟩ for harmonic minor against ⟨2, 5, 4, 3, 6, 1⟩ for the diatonic set. Two tritones rather than one; and two pairs of repeated counts, which is why it is not deep and therefore why a listener counting intervals could not tell how far a harmonic-minor passage had been transposed.

The names give it away #

There is a piece of evidence about all this that requires no computation, and it is the names.

The rotations of the diatonic set are called Ionian, Dorian, Phrygian, Lydian, Mixolydian, Aeolian and Locrian — seven independent names, none of them defined in terms of another. The rotations of harmonic minor are called Phrygian dominant, Lydian ♯2, Ukrainian Dorian, Ionian ♯5, Locrian ♮6 and altered diminished: five of the six are named as alterations of a diatonic mode, by the alteration.

That is what a vocabulary does when the objects it is naming are not a family. The diatonic modes are named as seven things because they behave as seven things; harmonic minor’s rotations are named by their distance from the seven, because that is the only structure available to describe them by.

What a scale-degree name promises #

Behind the arithmetic is a claim about notation, and it is the reason this rung is filed where it is.

Writing a note on a stave assigns it two things: a letter — which generic interval it is from the tonic — and an accidental. In a well-formed scale the letter alone almost determines the distance, because there are only two possibilities and the scale’s structure says which. That is why a melody can be read, transposed and sung from a stave with five lines and seven letters: the notation is carrying the well-formedness.

In harmonic minor the letter determines much less. Four of the six generic classes need the accidental to be read before the distance is known, and one of the resulting intervals is a tritone. Sight-reading a harmonic-minor line is measurably harder than sight-reading a major one and the usual explanation is unfamiliarity; this suggests a structural component as well.

What the rotations are for #

None of this says the rotations are useless. Two of them are in constant use and the reason is worth stating precisely, because it is not the reason the diatonic modes are used.

There is a test for the difference and it is a question about duration. Ask how long a piece stays in the object. A Dorian tune is Dorian from beginning to end and its tonic never moves. Phrygian dominant lasts for as long as the dominant chord underneath it lasts, which in a bebop line is two beats, and then the notes change because the chord changed. A mode is a property of a piece; a chord-scale is a property of a bar.

Phrygian dominant — the rotation on the fifth degree — is the scale over a dominant seventh chord resolving to a minor tonic, in flamenco, in Middle Eastern and Balkan repertoire, and in every bebop line over a V7 in a minor key. It is used because it contains a particular chord, not because it is a mode in which pieces are written.

Ukrainian Dorian — the rotation on the fourth degree — is likewise a scale over a chord: a minor triad with a raised fourth and a major sixth.

That is the difference. A diatonic mode is a home: pieces are in Dorian for their whole length. A rotation of harmonic minor is a chord-scale: a set of notes appropriate to one harmony for the duration of that harmony, after which the notes change. The word “mode” covers both and they are not the same object, and every structural measurement in this essay is a measurement of that difference.

What the picture cannot show #

Melodic minor’s two forms are one scale here. The ascending and descending forms differ by two notes, and every count above uses the ascending one. A scale that changes depending on which way the line is going is not a pitch-class set at all, and none of this machinery can represent it — which is a limitation of the machinery and also a reasonably strong hint that the object is not a set of pitches in the first place.

A property is not a verdict. Harmonic minor fails all four of the census’s tests and is one of the most-used scales in European music. That is not a contradiction; it is a demonstration that the four properties measure nameability and transferability rather than usefulness. The scale was built to solve a specific problem and it solves it.

Nothing here counts how often each rotation is used. The claim that Phrygian dominant and Ukrainian Dorian are the two in circulation is the ordinary one from the literature and this site has no corpus to check it against — as with every usage claim it makes.

And the count of tritones is a count in equal temperament. In meantone, harmonic minor’s two tritones are different sizes and the augmented second is a wolf-adjacent interval whose size depends entirely on the temperament — between 269 and 300 cents depending on the fraction of a comma. The scale was in use throughout the period when that was true, and every essay that draws it in twelve equal is drawing a later object.

The four properties, one row at a time #

For completeness, harmonic minor against the census that found the diatonic set:

• A chain of one interval? No. There is no interval whose repeated stacking produces this set.
• Two sizes of every step? No, as above — four of six fail.
• As evenly spread as possible? No. The maximally even seven-note set is the diatonic set, and harmonic minor’s 3 is the largest step any seven-note scale in use has.
• Deep? No. Its interval vector ⟨3, 3, 5, 4, 4, 2⟩ repeats twice.

Four properties, four failures, and a scale that half of European art music is written in. That is the strongest single check available on what the census actually measures — and the answer is that it measures a family of structural conveniences, every one of which harmonic minor forfeits in exchange for one chord.

A scale can be worth using and structurally poor, and the two are decided by different questions. It is the same lesson the roughness ladder reached about the triad from the opposite direction: a criterion that names exactly one object is a report about the criterion at least as much as about the object.

Whose music this is a claim about #

Harmonic minor as described here — a fixed seven-note scale with a raised seventh — is a textbook object of the nineteenth century onwards. What composers before then wrote is a minor key with a raised seventh at cadences, which is a rule about voice leading rather than a scale, and the two are not the same claim.

The distinction matters for the argument. If harmonic minor is a scale, then its properties are properties of something people composed in, and its failures are surprising. If it is a bookkeeping device for “the minor key, plus the accidental the dominant requires”, then its failures are exactly what one would expect of a scale assembled from a chord requirement — and the fact that four of its six generic intervals stop naming distances is not a defect but evidence that it was never meant to be read as a scale at all.

The second reading is better supported by everything above, and it explains the naming: nobody names the rotations of an object they think of as a scale by their distance from another scale’s modes.

The ladder from here #

One more question about rotation is left, and it is the one that makes the diatonic modes cohere. Brought to a common tonic, the seven modes form a chain in which each step lowers exactly one note by a semitone — and the notes it lowers are the chain of fifths read backwards. Do the same to harmonic minor’s rotations and each step moves two, three or four notes at once.

The famous chain is not a property of rotation. It belongs to the generator, and the next rung separates the two coordinates it has been confusing.