One dominant among seven

The seven rotations of the diatonic set contain the same seven triads. Which degree each triad sits on changes with the rotation, and that is the whole difference — so the question of what a mode can end on is a question of bookkeeping, and the bookkeeping can be done exhaustively.

Three, then two, then one #

A dominant needs three things, and each of them removes a different rotation.

A major triad on the fifth degree. Three have it: Ionian, Lydian and Locrian. That count is not obvious in advance and it is not three because three is a natural number of anything — the triad on the fifth degree of a rotation is major exactly when the two degrees above the fifth are four and seven semitones above it, which is a statement about where the collection’s two semitones fall relative to that degree, and the semitones fall where the chain of fifths puts them. The other four have a minor triad there, except Dorian, which also has a minor one, and Mixolydian, whose fifth-degree triad is minor — which is exactly the fact that makes Mixolydian sound like Mixolydian.

A tonic triad to resolve to. Locrian’s first degree carries a diminished triad, because its tonic has no fifth above it. A cadence onto a diminished triad is not a cadence; it is an unresolved chord arriving at the end of the phrase, and a diminished triad is the roughest of the four qualities by the site’s own measure. Two left.

And a fifth degree that is not the parent collection’s own tonic. Lydian fails this one, and it is the only rotation that can, because the fifth degree of rotation r is the collection’s own first note exactly when r is three. In F Lydian the triad on the fifth degree is C major — the tonic triad of the collection F is a rotation of. So a V–I in Lydian is C major to F major, which is not a dominant resolving to a tonic; it is a plagal cadence in C, heard the wrong way round. The chord that ought to establish F is the chord that most strongly establishes C.

One left, and it is Ionian.

What the other six do instead #

Every mode has repertoire and every repertoire ends its phrases, so the six without a dominant do something else. Computing the voice-leading distance from each of a mode’s six non-tonic triads to its tonic says what is available, and the answers are not what a distance-minimising account would predict.

Mixolydian ends on ♭VII–I — in G Mixolydian, F major to G major. Its voice-leading cost is 6 semitones, the largest of the six available moves in that mode and twice the price of Ionian’s V–I.

Phrygian ends on ♭II–I — F major to E minor in E Phrygian, at 4 semitones, and it is the one move in the whole table that approaches the tonic by a semitone from above.

Dorian’s characteristic chord is its major IV — G major to D minor in D Dorian, at 4 semitones — and the raised sixth degree that makes that triad major is the single note distinguishing Dorian from Aeolian.

Aeolian has ♭VII–i at 5 and ♭VI–i at 1, and the repertoire overwhelmingly uses the first.

The bass line is the other half #

Voice-leading distance treats a chord as a set of pitch classes and assigns the voices to minimise total motion, which deliberately throws away which note is at the bottom. For cadences that is throwing away half the evidence, and the modal table makes the omission visible.

In Ionian’s V–I the bass falls a fifth or rises a fourth. In Mixolydian’s ♭VII–I it rises a whole tone. In Phrygian’s ♭II–i it falls a semitone. Those are three quite different gestures and the voice-leading number does not distinguish them at all — it reports 3, 6 and 4 for reasons that have nothing to do with the bass.

There is a general point here about measurement that this site keeps arriving at from different directions. A metric that has been carefully defined and correctly computed can still be the wrong quantity, and the way to tell is that its ordering disagrees with an ordering nobody disputes. Nobody disputes that V–I is a stronger ending than iii–I. The metric says iii–I moves less. Both are right, and the second is not about endings.

The cheap moves are not the endings #

Set the six non-tonic triads of each mode beside their voice-leading costs and a pattern appears that is worth stating on its own.

In Ionian, the cheapest move to the tonic triad is iii–I at one semitone. Nobody cadences on iii–I. In Aeolian the cheapest is ♭VI–i at one semitone, and ♭VI–i is used as a deceptive move — a chord that arrives instead of the ending. In Lydian the cheapest is iii–I at one, again unused.

Ranking each mode’s six moves and asking where its own cadence sits makes the pattern exact rather than anecdotal:

Not one mode cadences on a move from the cheap half of its own six, and three of the six cadence on the single dearest move available. Averaged, the cadences cost 4.67 semitones of voice motion and the moves nobody uses cost 3.22 — the endings are forty-five per cent more expensive than the non-endings, systematically and without exception.

So the relationship is not that distance fails to predict finality. It is that distance anti-predicts it, and a model that chose a mode’s cadence by minimising voice motion would get all six wrong.

The bass does not rescue it either, which is worth reporting since the section above nominates it as the other half. The used cadences move their bass by 2.83 semitones on average and the unused moves by 3.53, so the cadences are the smaller bass motions and the separation runs the wrong way again. What distinguishes them is the kind of bass motion rather than its size — a fifth in Ionian, a tone in Mixolydian, a semitone from above in Phrygian — and none of those is a distance.

Which leaves the finding as a negative with a shape. Two quantities this collection can compute, both carefully defined and correctly evaluated, and neither of them picks out the cadences; one of them picks out their opposite. Whatever makes a move an ending is not a property of how far the notes travel, in either voice or bass, and the search for it should look somewhere other than a metric on chord space.

The moves that are actually used as endings cost 3, 4, 5 and 6. Sorting the six candidate moves of each mode by cost and asking where the used one lands gives the same answer in every mode: never first, usually last or next to last.

This is the same result the standard pass found in the cadence essay and could not explain: measured with this site’s own solver, the authentic cadence is 3 semitones, the plagal is 3, and the deceptive is 5 — so the deceptive cadence is not the near-copy of the authentic one it is usually described as, and the plagal ties the authentic. Distance does not grade a cadence.

The modal table says why not, and the reason is almost embarrassing in its simplicity. A cadence is a change, and a small change is a poor ending. What makes V–I final is not that the voices move little; it is that the two chords share exactly one note, that the leading note moves by a semitone into the tonic, and that the bass moves by a fifth. Cheapness in total motion is orthogonal to all three.

Why the leading note is the whole of it #

Of the three components above, one does most of the work, and the modal table isolates it cleanly.

Exactly two of the seven rotations have a note a semitone below their tonic: Ionian and Lydian. Exactly two have a note a semitone above it: Phrygian and Locrian. The other three — Dorian, Mixolydian, Aeolian — have neither, because the diatonic set has two semitones and each of them serves two rotations.

So the three modes with no semitone next to their tonic have to end with a whole-tone move in the bass and no semitone anywhere — which is exactly what ♭VII–I in Mixolydian is, and exactly why it sounds like an ending of a different kind rather than a weaker version of the same kind.

And Phrygian’s ♭II–i is the mirror image: a semitone approach from above, into a minor tonic, with the bass falling by a semitone. It is as strong an ending as Ionian’s and it is built from the other semitone of the same set.

There is one more consequence of the two-semitone count and it explains a piece of standard advice that is usually given without a reason. A mode with no semitone next to its tonic has to establish that tonic by repetition, by register and by the bass, because it has no note whose ordinary tendency points at home. That is why modal writing leans on drones, pedal points and ostinati — devices that assert a tonic by insisting on it rather than by approaching it — and why the same devices are rare in common-practice tonality, which has a cheaper mechanism available.

Two semitones, four rotations that can use one of them, three that cannot. That is the entire structural account of modal cadence, and every number in it was counted rather than remembered.

The same census on the harmonic minor #

The essay’s arithmetic assumes a rotation of the diatonic set. Applying it to the scale that minor-key repertoire actually uses shows what the raised seventh bought and what it cost.

Harmonic minor is a manufactured object and it is worth being precise about what it manufactures. Its tonic triad is minor and the triad on its fifth degree is major, which is the combination no rotation of the diatonic set offers: Aeolian’s fifth-degree triad is minor, so a minor key using only its own notes has no dominant either.

Both minor scales in ordinary use are repairs to the same missing chord. The natural minor is Aeolian and has no dominant; harmonic minor manufactures one by raising the seventh degree; melodic minor raises the sixth as well, to remove the augmented second the first repair created. Every fact in that sentence is visible in a triad table and none of it needs a preference rule.

The price is the subject of the next rung, and it is large: harmonic minor is not generated, is not well formed, is not deep, and its rotations are not modes in the sense this ladder has been using.

What the picture cannot show #

Seventh chords are not counted here. Every triad above is three notes. Adding the fourth note changes several of the answers: Dorian’s fifth-degree seventh chord is a minor seventh and Mixolydian’s is a minor seventh too, but Mixolydian’s tonic seventh chord is a dominant seventh — a tonic that is spelled like a dominant, which is one of the most characteristic sounds in blues-derived music and is invisible in a triad census. That is a real omission and the reason for it is that seven modes times seven degrees times two chord sizes is a table nobody reads.

A triad table is not a repertoire. Nothing here says how often ♭VII–I is used in Mixolydian, only that it is available and expensive. The usage claims in this essay are the ordinary ones from the literature on modal practice, and this site has no corpus to check them against.

Chords are pitch-class sets here. Voicing changes everything about a cadence — which note is in the bass, which is doubled, how the voices are spaced — and the whole of that is discarded before any number above is computed. A V–I with the leading note in an inner part is a different ending from one with it in the soprano, and both are “3 semitones”.

And the third criterion is a judgement. Ruling Lydian out because its fifth degree is the parent tonic is defensible and it is not a theorem: a piece can establish F as home by other means and then use C major as its dominant, and some do. What is computed is that Lydian’s dominant is the parent key’s tonic; whether that disqualifies it is an argument, and it is made here rather than proved.

Nor is a mode obliged to cadence at all. A cyclic form does not cadence and a great deal of modal music is cyclic. Asking which modes can make an authentic cadence presupposes that an authentic cadence is what a piece wants, which is a fact about one repertoire of about three hundred years.

A count that had to be run twice #

One thing in this essay was got wrong on the first attempt and it is worth recording, because the correction is the sort a table produces and prose does not.

The first version of the criterion said that a mode can cadence if the triad on its fifth degree is major and its tonic triad is a triad, which gives two — Ionian and Lydian — and the essay was drafted around two. Only when the fifth-degree triads were written out with their actual note names did it become obvious that Lydian’s dominant is the parent key’s tonic, which is not a technicality but the entire reason Lydian collapses into its parent when played without care.

The number moved from two to one because the table was drawn. That is the argument for computing a census rather than reasoning about it: the reasoning had already reached “two of the seven”, and it was wrong in a way that reads perfectly well in a sentence.

Whose music this is a claim about #

The three centuries in which V–I is the default ending are exactly the three in which Ionian and Aeolian crowd out the other five. That is the correlation this essay’s arithmetic predicts, and the direction of the causation is not settled by anything here: a repertoire that wants a dominant will end up in the one mode that has one, and a repertoire that has settled on that mode will find the dominant waiting.

The interesting evidence is the modes that did not disappear. Dorian and Mixolydian survive in folk repertoire across Europe for the whole period, in music that does not use functional harmony and therefore never needed a dominant. And when modal harmony returns to art music — through Debussy, through Vaughan Williams, through jazz after 1959 — it returns with exactly the endings this table predicts: the ♭VII, the ♭II, the modal IV, and a marked avoidance of the leading note. Lydian returns too, and it returns in exactly the guarded way the third criterion above would predict: as a colour over a pedal, or with the fifth degree deliberately withheld, because sounding a plain triad on it hands the piece back to the parent key.

The last of those is the tell. Modal writing in the twentieth century avoids the leading note deliberately, which means it is a resource being declined rather than one that is absent. In Dorian and Mixolydian there is nothing to decline.

The ladder from here #

Everything so far has been about the seven rotations of one well-formed set. The next rung asks what happens when the same rotation trick is applied to a scale that is not well formed — harmonic minor, whose seven rotations have three tonics with no fifth, four generic intervals in three sizes apiece, and no useful sense in which they are seven modes of one thing.