What a raised seventh is for

The harmonic minor scale is introduced, almost universally, as a list. Take the natural minor and raise the seventh degree. The result has a gap of three semitones between its sixth and seventh notes, which is larger than any step in any other common scale, and it produces the distinctive sound that turns up in everything from Bach to surf rock.

Presented that way it is arbitrary, and students reasonably ask why anybody would want a scale with a hole in it. The answer is that nobody wanted a scale. Somebody wanted one chord to be a different quality, and the scale is what was left afterwards.

The chord that has to be major #

Stack thirds on the fifth degree of a natural minor scale and the result is a minor triad. In A minor: E, G, B. Stack them on the fifth degree of a major scale and the result is a major triad, and it contains a note a semitone below the tonic.

That semitone is what a dominant chord is for. It provides the leading note, and — once a seventh is added — the tritone whose two notes each move by a semitone into the tonic triad. A minor v chord has neither. Its third is a whole tone below the tonic and pulls at nothing.

So a composer wanting a cadence in a minor key with the same force as a cadence in a major key has to raise the seventh degree of the scale, from G to G sharp in A minor, and thereby convert the v chord into a V.

Everything else follows. Raising G to G sharp leaves F and G sharp adjacent in the scale, three semitones apart. That interval is the augmented second, and it is not a decision anyone made — it is the residue.

Which computation produced the number #

The figure does not label the chords from a table. It builds each one and reads off its quality.

For each degree $i$ of a scale $S$ it takes the notes $S_i$, $S_{i+2}$, $S_{i+4}$, wrapping round the octave, and reduces them to intervals above the root. A triad whose intervals are $\{0, 4, 7\}$ is major, $\{0, 3, 7\}$ minor, $\{0, 3, 6\}$ diminished and $\{0, 4, 8\}$ augmented.

Run on the natural minor $\{0,2,3,5,7,8,10\}$ the fifth degree gives $7, 10, 2$, which is $\{0, 3, 7\}$ above its root: minor.

Run on the harmonic minor $\{0,2,3,5,7,8,11\}$ the same degree gives $7, 11, 2$, which is $\{0, 4, 7\}$: major.

The rest of the table falls out of the same procedure, and one of its entries is worth noticing on its own. The harmonic minor’s third degree produces $\{0, 4, 8\}$ — an augmented triad, which no other common scale contains as a diatonic chord. It is a genuine consequence of the raised seventh, it is rarely used, and any figure that decided chord quality from the third alone would have called it major and been wrong.

The repair is forced, and so is the bill #

Two of the claims above can be made stronger than “this is what happened”, because the space of alternatives is small enough to enumerate.

Raising the seventh is the only repair there is. The triad on the fifth degree is built from the fifth, seventh and second degrees, so its third is the distance from the fifth degree to the seventh. With the tonic at 0, the supertonic at 2 and the dominant at 7 — the three notes a minor key cannot move without becoming a different key — that distance is four semitones if and only if the seventh degree is at 11. There is nothing to search: no other alteration of any other degree can make that chord major.

The augmented second is the generic outcome and the melodic minor is the only escape. Hold those four notes fixed, let the third, fourth and sixth degrees be anything, and there are eighteen seven-note scales. Nine of them have a minor third and are therefore candidates for a minor key. Of those nine, exactly one has no step of three semitones in it, and it is 0, 2, 3, 5, 7, 9, 11 — the ascending melodic minor.

So the melodic minor is not one remedy among several that a tradition happened to settle on. It is the unique way of having a major dominant in a minor key without a three-semitone step anywhere, and its price — a major fourth degree and a diminished sixth — is the price of the only option.

And the augmented triad on the third degree is not a quirk of the harmonic minor at all. Every one of the nine has it, without exception, and the reason is one line: a minor tonic puts the third degree at 3, a raised seventh puts the seventh at 11, and 3 to 7 is four semitones and 7 to 11 is four semitones. Any minor scale with a major dominant has an augmented mediant, necessarily. The section above reports it as a curiosity of one scale and it is a theorem about the repair.

What the gap costs #

An augmented second is awkward to sing, and that is not a matter of taste either.

Melodic writing in the European tradition moves predominantly by step, and a step in that tradition is a tone or a semitone. Three semitones is a leap, but it is spelled as a step — F to G sharp is written on adjacent lines and spaces — so it reads as a step and does not behave as one. Singers approaching it from either side have to make a leap they were not led into.

The traditional remedy is to raise the sixth degree as well when the melody is going up, giving $\{0,2,3,5,7,9,11\}$: the melodic minor ascending. That removes the gap by moving the wall rather than the note.

And then the melodic minor is traditionally described as reverting to the natural minor on the way down, which is the part of the story that students find hardest to believe and which is the least well supported.

The chord it borrows #

There is a sharper way to state what the raised seventh does, and it makes the whole business look less like an alteration and more like a loan.

The dominant seventh chord in A minor is E, G sharp, B, D. The dominant seventh chord in A major is E, G sharp, B, D. They are the same four notes. A minor key with a raised seventh does not have its own dominant; it has the major key’s dominant, taken over unchanged.

That is why a minor-key cadence sounds as final as a major-key one: the approach is identical, and only the arrival differs. It is also why the Picardy third works — ending a minor-key piece on a major tonic chord. If the dominant was borrowed from the major key anyway, arriving in the major is not much of a further step, and seventeenth-century composers ended minor pieces that way as a matter of routine.

The general observation is that a key in tonal practice is not a fixed collection at all. It is a tonic plus whatever apparatus is needed to point at it, and the apparatus is shared between the major and minor forms.

And a chord that cannot decide #

The raised seventh produces a second chord worth having, on the seventh degree with a seventh added: G sharp, B, D, F in A minor. Its intervals above the root are $\{0, 3, 6, 9\}$ — four notes in equal steps of three semitones.

That equal spacing has a consequence. Transpose the chord up by three semitones and the pitch classes are unchanged: B, D, F, A flat is the same set. So there are only three distinct diminished sevenths in the whole system, not twelve, and each of them can be spelled four different ways.

Each of the four spellings points at a different tonic, and the chord itself gives no clue which. That makes the diminished seventh the standard pivot for a sudden change of key across the whole nineteenth century: it is reached in one key and left in another, and nothing in the chord objects.

None of this was designed. It is what happens when a note is raised to make one triad major, and it is the second-order consequence of a first-order repair.

Three scales or one practice #

The tidy account — three minor scales, each with its own name and its own use — is a nineteenth-century pedagogical construction. The practice it describes is older and messier.

What composers of the seventeenth and eighteenth centuries actually did was raise the sixth and seventh degrees when the line was going up towards the tonic and leave them alone otherwise. That is a rule about voice leading, not about scales: a raised note is raised because it is about to resolve upward, and there is no reason to raise it when it is not. That is a claim about where the voices go, and it is testable against any score.

The three-scale account converts that into three fixed collections and then has to explain why one of them behaves differently in each direction. The voice-leading account does not need the explanation, because direction was the criterion in the first place.

The distinction matters because it changes what counts as an exception. Under the three-scale account, a descending raised seventh is a deviation requiring justification, and there are a great many of them. Under the voice-leading account there is no rule being broken — the note was raised because the line went up, and when the line goes up on the way down, it is raised then too.

The other repair #

There is a second solution to the same problem, and European practice used it too.

Leave the seventh alone and the chord on the seventh degree is a major triad a whole tone below the tonic. Moving from that chord to the tonic is a perfectly good cadence; it is simply a different one, without a leading note, and it is what a great deal of modal and folk repertoire uses. It is the cadence of Renaissance polyphony before musica ficta practice sharpened everything in sight, and it is the cadence of most rock music written in a minor key — where the chord a whole tone below is so common it has its own name.

Which is to say the raised seventh is not a correction to a defect. It is one of two available cadential mechanisms, and it dominated European art music for roughly three centuries, and it does not dominate anything else.

Where the model stops #

Scales are a description, not a cause. No composer of the period selected a scale and then wrote in it. Notes were raised because of what the voices were doing, and the collections in these figures are a summary of the outcome. Treating a scale as the input reverses the direction of the explanation, which is the fundamental problem with all scale-first pedagogy.

Triad quality is not the whole of function. The figures ask what quality each chord has and stop there. Whether a chord functions as a dominant depends additionally on where it sits in a phrase, what precedes it, whether a seventh is present, and what the bass is doing. A major triad on the fifth degree is a necessary condition for a tonal dominant, and nothing like a sufficient one.

The augmented triad on the third degree is a real prediction and a rare event. The computation says the harmonic minor contains one. Repertoire contains very few, because composers writing in minor keys mostly did not use the raised seventh in that chord. The scale is a fiction whose consequences are not all exercised.

Nothing here concerns intonation. Whether the raised seventh is played high — leaning towards the tonic, as string players and singers routinely do — is a separate question, and the answer is not the same in every tradition or century. Everything above is in twelve equal semitones, and that is an assumption that most of the repertoire in question predates.

“Melodic minor descends as natural minor” is a rule about exercises. It is reliably true of scale practice and unreliably true of repertoire. It survives in teaching because it makes the three-scale system consistent, which is a property of the system rather than of the music.

Whose music, and when #

The raised seventh in minor keys is a feature of European art music from roughly 1600 to roughly 1900, and its rise and fall both have visible causes.

Before that, the practice known as musica ficta had performers raising notes that were not written, according to conventions that the sources describe incompletely and inconsistently. The scholarly literature on exactly which notes were raised in fifteenth- and sixteenth-century repertoire is enormous and unresolved, which is itself evidence that the practice was a performance convention rather than a scale.

Through the eighteenth and nineteenth centuries the raised seventh in minor is close to universal at cadences and optional elsewhere. Bach’s minor-key fugue subjects routinely include the augmented second and treat it as an expressive interval rather than an obstacle.

After about 1900 it thins out rapidly, along with functional harmony generally. Twentieth-century writing in minor keys — including nearly all popular music — largely uses the natural minor and the ♭VII chord, which is the modal cadence returning.

The augmented second also has a life outside all of this. It appears as a structural interval in several maqamat — Hijaz has one between its first and second degrees, which is a much more prominent position than the harmonic minor gives it — and in Jewish liturgical modes, in Romani repertoire across central and eastern Europe, and in Andalusian and Flamenco practice. In those traditions it is not a residue of a raised note; it is a characteristic interval that the mode is partly identified by, and treating it as a defect to be patched says more about the European framing than about the interval.

One further piece of evidence sits in the notation itself. In a minor key signature the seventh degree is not sharpened — A minor’s signature has no accidentals at all, and every raised G is written in as an accidental where it occurs. That is not an oversight. It is notation recording the practice accurately: the note is raised at particular moments and for particular reasons, and a key signature would assert that it is raised always, which is false.

Compare that with the major keys, where every note of the collection is in the signature because every note really is fixed. The asymmetry between the two conventions is the strongest available evidence that a minor key was never understood as a fixed set of seven pitches, whatever the three-scale account implies, and that what the voices are doing has been the operative criterion the whole time.

The ladder from here #

Later rungs on this anchor: the modes of the melodic minor and what they are used for, which is a twentieth-century jazz question with a seventeenth-century answer. Musica ficta, and what can and cannot be recovered about performance practice from notation that deliberately omits it. The ♭VII cadence in modal and popular repertoire, and why it feels final without a leading note. The diminished seventh chord, which the harmonic minor produces on its seventh degree and which is symmetrical in a way that makes it ambiguous. And the maqamat built round the augmented second, where the interval is the point rather than the price.

Nobody wanted a scale with a hole in it. Somebody wanted a chord to be major, and the hole is what that cost.