Parallel and relative are two different maps

A mode is described in two incompatible ways and both are correct.

Relative: the same seven notes, started on a different degree. C major and D Dorian have identical pitch content and different tonics. This is what a mode is, structurally — the census cannot tell the two apart at all — and it is how the first rung of this ladder introduced it.

Parallel: the same tonic, with one or more notes changed. C major and C Dorian have the same home and differ by two notes — E becomes E♭ and B becomes B♭. This is how modes are actually taught, thought about and used, because a musician who wants a Dorian sound on C does not want to think about B♭ major.

The two descriptions do not merely differ in emphasis. They are two coordinate systems on the same object, and the celebrated fact about modes belongs to only one of them.

The chain is the chain of fifths #

Read the changed notes down the figure: F♯, B, E, A, D, G. Those are descending fifths — the chain of fifths, walked from its sharp end.

That is not a coincidence and it is not a fact about modes. It is a restatement of what a chain of fifths is. The diatonic set is seven adjacent positions on the circle of fifths; moving one step flatwards drops the sharpest note and adds the next flat one, which changes exactly one pitch class. Doing that seven times walks the collection from F♯ major through to C♭ major, and if the tonic is held fixed while the collection moves, the seven positions are the seven modes on that tonic. Walk far enough in either direction and the chain arrives back where it started, twelve steps and one comma later — which is a fact about tuning rather than about modes, and is the reason the parallel chain is a segment of something longer.

So brightness order, the one-note chain, and the chain of fifths are three descriptions of one thing. A musician who learns modes as “Lydian, Ionian, Mixolydian, Dorian, Aeolian, Phrygian, Locrian — flatten one more note each time” has learned the circle of fifths without being told.

And it disappears without the generator #

The test is to run the same procedure on a scale that is not a chain.

Read down the brightness order, the steps are three, four, three, three, three and two — eighteen changed notes over six steps against the diatonic chain’s six. No step moves a single note, and no reordering of the rows would produce a chain of single moves, because a single-note move between two rotations requires the two collections to be adjacent in some generated sequence and there is no generator here to be adjacent in.

The one-note chain is a property of well-formedness, and rotation has nothing to do with it. The previous rung measured four ways in which harmonic minor’s rotations are not a family; this is the fifth and it is the one that most affects how they can be taught, because the mnemonic that makes the diatonic modes easy simply does not exist for them.

The characteristic degree falls out of the chain #

Every mode is taught with a “characteristic note” — the degree that gives it its flavour: Dorian’s raised sixth, Mixolydian’s lowered seventh, Lydian’s raised fourth, Phrygian’s lowered second. Those are four separate pieces of received wisdom, and the parallel chain produces all of them at once.

The characteristic degree of a mode is the note that changes on the step that produces it. Mixolydian is Ionian with B lowered to B♭, so its characteristic note is the seventh. Dorian is Mixolydian with E lowered to E♭, so its characteristic note is the third — which is not what is usually taught, and the reason is that Dorian is more often compared to Aeolian, one step further down, where the changing note is the sixth.

A characteristic degree is a comparison, not a property, and which degree gets the name depends entirely on which mode it is being compared against. The chain makes that visible: every mode has two neighbours and therefore two characteristic notes, one for each direction.

Two coordinates, and what each is good for #

Once the two systems are separated it becomes clear that each answers a different question and neither answers both.

The relative coordinate is right for pitch content. Which notes are available, which key signature to write, which instruments have the notes, whether a passage can be played on the white keys — every one of those is a question about the collection, and the collection is what the relative description names.

The parallel coordinate is right for sound. How bright the mode is, which degree gives it its character, what changes when a piece goes from major to Mixolydian — every one of those is a question about intervals above the tonic, and only the parallel description makes them visible.

Seven neighbouring collections are the relative axis in its plainest form: each shares six notes with the next and differs by one. Hold the tonic fixed instead of letting it move with the collection and the same figure becomes the parallel axis — which is the whole of the difference between the two maps, and is why one of them is drawable as a set of sets and the other is not.

The two are related by an exact statement that is worth writing down. The mode on degree d of the collection k fifths sharp of C is the same object as the mode on C of the collection k − (position of d in the chain) fifths sharp. That is arithmetic, it is one line, and it is the translation between the coordinates.

One coordinate has a property the other has not, and it decides which collections have modes at all.

A collection with symmetry has fewer modes than it has notes, because a rotation that maps the set to itself is not a new mode. The relative map is therefore not always seven-fold: it is as wide as the collection’s rotations, and for the symmetric sets it collapses — which is a limit on the relative axis that the parallel axis, holding the tonic fixed, does not have.

How far apart two modes are, in each coordinate #

Having two coordinates means having two distances, and they disagree in the way one would expect.

Relatively, all seven modes are at distance zero from each other: identical content, seven different tonics. There is nothing to measure.

In parallel, they are as far apart as their positions in the chain: Ionian to Mixolydian is one note, Ionian to Dorian is two, Ionian to Locrian is six. That is a Hamming distance on pitch-class sets and it is exactly the chain position difference, because each step changes one note.

So the parallel coordinate carries all the distance information and the relative coordinate carries none — which is a good reason to think the parallel one is the real description and the relative one is a fact about notation. Against that: the relative description is what makes a mode cheap, because a Dorian passage in D needs no accidentals, and every notational system in use is built around collections rather than around tonics.

The coordinate that measures distance and the coordinate that saves ink are different ones, and a musician has to hold both. That is the whole of why modes are hard to teach.

Where the confusion costs something #

Three places, all of them ordinary.

The relative minor is not the parallel minor and both are called “the minor”. A major key’s relative minor shares all seven notes and has a different tonic; its parallel minor shares four notes and has the same tonic. By voice-leading distance the parallel is nearer — the tonic triads are one semitone apart — and by shared content the relative is nearer, and calling both of them “the minor” is a genuine ambiguity that costs students a term.

Modal interchange is a parallel operation described in relative terms. Borrowing ♭VI from the parallel minor is described as “borrowing from A♭ major” — which is true and useless, because the reason it works is that ♭VI is a semitone-lowered degree over an unmoved tonic, and A♭ major has nothing to do with it. The same chord in an actual modulation to A♭ would take several bars of evidence to be heard as one; as an interchange it takes a beat, because the tonic never moved.

And “play D Dorian over a D minor chord” hides which fact is doing the work. It is a relative instruction — use the notes of C — and the reason it works is parallel: Dorian’s raised sixth degree is the note that distinguishes it from Aeolian, and it is the note the player is being told to include.

The same two coordinates, one level up #

Nothing about this is peculiar to modes. The two coordinates are the two independent directions in which a collection of seven notes can be moved, and they show up again wherever that object does.

A key change is a relative move: the collection slides along the chain and the tonic goes with it. A modal shift is a parallel move: the collection slides and the tonic stays. The two operations are the same operation on the collection and differ entirely in what happens to the tonic, which is not part of the collection at all.

That is why a piece can be ambiguous between the two readings — a passage that adds a B♭ to C major is C Mixolydian if C stays home and a move toward F major if it does not — and why the ambiguity is not resolvable from the notes. The pitch content is identical in both readings. What decides is where the phrase ends, which chord is held, and which note the bass rests on, and every one of those is outside the set.

This is the same ambiguity a modulation has, seen from the other side, and it is why a key change takes a computable number of bars to be detectable: the evidence that distinguishes a modal inflection from a modulation is not in the notes that changed, it is in what the music does afterwards.

Brightness is a real quantity and a crude one #

The ordering used throughout this essay is by the sum of the pitch classes above the tonic, which is as crude a statistic as it sounds and gets the answer right.

It gets it right because of the chain. Each step flatwards lowers exactly one note by one semitone, so the sum falls by exactly one at each step, and the ordering by sum is the ordering by position in the chain. Any statistic that decreases when a note is lowered would produce the same order.

That is worth being explicit about, because “brightness” is usually presented as a perceptual claim and here it is an arithmetic one. Nothing in this essay says that Lydian sounds brighter than Phrygian. What is shown is that the seven modes admit a total order with a one-note step between neighbours, and that the order coincides with the one musicians describe in the language of brightness. Whether the perceptual quality tracks the arithmetic one is a psychoacoustic question this site has not measured, and the obvious confound is that the modes at the two ends are also the ones with the tonic in a tritone.

A prediction the arithmetic makes #

If the chain is doing the work, then a scale that is almost a chain should have an almost-chain, and the amount of “almost” should be measurable.

Melodic minor is the case to try. Its step pattern is 2-1-2-2-2-2-1, which is the diatonic pattern with one step moved — it fails Myhill’s property on two generic intervals rather than four, and it is the nearest thing to a well-formed seven-note scale that is not one.

That is the expected result and it is worth having as a check rather than as a surprise. How close a scale’s rotations come to a one-note chain is a measure of how close the scale is to being generated, and the three scales measured here order the same way on both questions.

Both halves of that are now countable, so it is worth doing rather than asserting. Myhill’s property asks, for each of the six generic intervals, whether it comes in exactly two specific sizes. The diatonic set answers yes six times; melodic minor answers yes four times and no twice; harmonic minor answers yes twice and no four times. That is the ordering on the left-hand question, and it is where the essay’s “two rather than four” comes from.

The right-hand question is a shortest path. Take the seven rotations as seven points, the distance between two of them as the number of notes that differ, and ask for the cheapest ordering — a Hamiltonian path, and with seven points all 5,040 of them can simply be tried. The diatonic’s cheapest chain costs six, which is the floor, since six steps cannot each cost less than one. Melodic minor’s costs thirteen and harmonic minor’s fourteen. Same order on both questions, as promised, and now with a gap in it: melodic minor is barely nearer a chain than harmonic minor, while both are more than twice as far from one as the diatonic set.

The search turns up something the brightness ordering hides, though. Brightness is not the cheapest ordering once the generator is gone. For the diatonic set the two coincide — the brightness order costs six, and nothing costs less. For melodic minor the brightness order costs sixteen and the cheapest costs thirteen; for harmonic minor, eighteen against fourteen. So a fifth of the changed notes in those figures are an artefact of insisting on the brightness axis, and the more usual reading — that these rotations are simply further apart — is only three-quarters of the story. The other quarter is that the axis they are being laid out along has stopped being the right one, which is exactly what it means for the chain of fifths to have been doing the ordering all along.

What the picture cannot show #

Neither coordinate contains the tonic. Both descriptions assume a tonic has already been established, by whatever means. The set cannot supply one and supplying one costs between a sixth and a third of the sounding time on the crudest model. Everything here is downstream of that.

The chain has ends. Seven steps of the parallel chain exhaust the modes of the diatonic set, and continuing flatwards leaves the collection entirely: the next step after Locrian lowers the tonic itself. The chain is a segment rather than a cycle, and the segment has exactly the length of the scale.

And nothing here is about voicing or register. Two pieces both in C Mixolydian can sound nothing alike, and the difference will be in the bass, the harmonic rhythm and which degrees are emphasised — none of which is in a pitch-class set. The parallel coordinate is a better description of a mode’s sound than the relative one; it is still not a description of a mode’s sound.

Whose music this is a claim about #

The parallel description is modern. Modal theory before the seventeenth century is entirely relative — a mode is identified by its final and its ambitus within one collection, and the idea of transposing all seven modes onto one note is a nineteenth-century pedagogical device that became the twentieth century’s default.

That history explains a durable confusion. The names Dorian, Phrygian and the rest are inherited from a system in which they meant “the collection, read from this degree”, and they are now used in a system where they mean “this pattern of intervals above whatever note is home”. The names survived a change of coordinate system, which is why a modern musician can use them fluently and still find the medieval sources baffling.

The jazz usage is a third thing again: there, a mode name usually specifies a set of notes to play over a stated chord, which is the chord-scale reading and is neither of the two coordinates above. It is worth noticing that the chord-scale reading is parallel in form — the chord supplies the tonic and the mode name supplies the intervals above it — which is why the modal jazz of 1959 onwards could treat the seven as seven colours on one note in a way that no earlier repertoire did.

The one place all three usages agree is the ordering. Whatever a tradition means by Lydian, it puts it at the sharp end, and whatever it means by Phrygian, it puts it at the flat end. That agreement is not a coincidence between traditions; it is the chain of fifths showing through three different vocabularies.

The ladder from here #

This ladder has seven rungs and the last four have all been about what changes when the vantage point moves. What has not been examined is the other axis of the same map: not which degree of a collection is home, but which collection, and how far apart two collections are.

That is the key-relations ladder, and its next rung finds that the three standard ways of measuring the distance between two keys rank the twelve in three different orders — with the disagreement concentrated in exactly one family of keys.