Modes — the series
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The same seven, started later
A mode is not a new scale. It is the same seven notes with a different one treated as home, and the entire change in character comes from reassigning which degree the semitones fall next to.
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What a raised seventh is for
The harmonic minor is usually taught as a scale with an odd gap in it. It is better understood as a repair to a single chord — raising the seventh degree turns the dominant triad from minor to major, and everything else about the scale is the bill for that.
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What a tonic costs in seconds
The standard key-finding algorithm cannot be run on a pitch-class set at all — a flat histogram has no variance and the correlation is undefined. Give it durations and it answers with the parent key for all seven modes identically, and it takes between 15.8 and 30.0 per cent of the total time spent on one note before it names that note instead.
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The mode with no fifth
Six of the seven rotations have a perfect fifth above their own first degree and one does not, which is a structural disqualification rather than a preference. And the set's single tritone sits on a different pair of degrees in every rotation — on the fourth and seventh in Ionian, on the tonic itself in two others — which decides most of what each mode can do.
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One dominant among seven
The triad on the fifth degree is major in three of the seven rotations. One of those three has no triad on its own first degree, and one has its fifth degree on the note the parent collection is built from — so exactly one rotation of the seven has a dominant, and the other six end by moves that cost twice as much in voice leading.
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Seven rotations that are not seven modes
Rotate harmonic minor and the result is not a family the way the diatonic modes are a family. Four of its six generic intervals come in three specific sizes rather than two, so a fourth might be four, five or six semitones; three of its seven rotations have no fifth above their own tonic; and four have a tonic inside a tritone. The word mode has been doing two different jobs.
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Parallel and relative are two different maps
Bring the seven modes to one tonic and order them by brightness, and each step lowers exactly one note by a semitone — and the notes it lowers, in order, are F♯, B, E, A, D, G, 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 belongs to the generator, not to rotation.
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The one note that decides the mode
A key signature names the seven and not the rotation, so the mode has to be heard. Enumerate every set of degrees that lies inside at least one of the seven modes — 510 of them — and count how many modes each leaves standing. Fifteen minimal sets decide, every one has two members, and every one is the mode's own tritone. The only degree all seven share is the tonic; no degree belongs to a single mode; and averaged over the orders the degrees could arrive in, 5.67 of the seven have to have been heard first.
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The set with fewer modes than notes
Eight earlier essays rotate one collection, and the seven modes were never a result — they are arithmetic. A set has as many modes as it has notes divided by the order of its own transposition symmetry, so the whole-tone scale has six notes and one mode, the octatonic has eight and two, and the question asked just before — how much has to be heard before the mode is settled — is not merely hard for those collections but undefined. Seventy-five of the twelve-note universe's 4,095 subsets are built this way and they reduce to sixteen step patterns. Five, seven and eleven notes cannot be among them, which is why every seven-note scale has seven modes before anybody plays one.