Acoustics & Music Theory
why it sounds like that
Intervals and chords

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.
15 min read 8 figures The same shape, movedHow many of the twelve

Assumes: The same seven, started later

A mode is a rotation, and nothing about the set can prefer one rotation to another. That is a statement about the set. It is not a statement about the rotations, and the rotations are not all alike: two questions asked of each of the seven give different answers, and both questions have one-line definitions.

Does the first degree have a perfect fifth above it? And where does the tritone land?

Which rotations have a fifth above their own tonic. Each rotation drawn by scale degree, with the degree a perfect fifth above the tonic marked. 1 of the 7 does not have one: Locrian. A tonic without a fifth above it has no triad of its own, which is a stronger disqualification than any preference.
Fig. 1 The seven rotations by scale degree, with the tonic and the degree seven semitones above it marked. Six have one. Locrian’s fifth degree is six semitones above its tonic, not seven, so the interval is a tritone and the triad on its own first degree is diminished.

One of seven has no fifth

The diatonic set contains six perfect fifths and one tritone. That is the interval vector’s last two entries — ⟨2, 5, 4, 3, 6, 1⟩, six of size five semitones and one of size six — and it is the same in every rotation, because the vector is a property of the set.

Six fifths and seven degrees means one degree is left out, and the degree left out is whichever one the tritone starts on. Rotate the set so that degree is first and the tonic has no fifth above it.

That is Locrian, and the consequence is not a matter of taste. A tonic with no perfect fifth above it has no triad of its own — the chord on the first degree is B–D–F, which is diminished, and a diminished triad is unstable by every measure this site has: it is the roughest of the four triad qualities, it contains the one interval that has to resolve, and there is no voice-leading move that makes it sound final.

The triad on every degree of every rotation. The quality of the triad built on each degree of each rotation, computed from the set rather than tabulated. The triad on the fifth degree is major in 3 of the 7: Lydian, Ionian, Locrian. Of those, Locrian has no triad on its own first degree to resolve to, and Lydian has its fifth degree on the note the parent collection is built from, so the cadence would confirm the parent rather than the mode. What is left is Ionian — one rotation of the 7, and the others have to end some other way.
Fig. 2 The quality of the triad on each degree of each rotation. The first column is the tonic triad: three rotations have a major one, three a minor, and Locrian has neither. That single cell is why six of these are modes with repertoire and the seventh is a curiosity.

Notice how little is needed for the argument. No preference rule, no claim about what listeners like, no appeal to tradition. The set has six fifths, there are seven degrees, and one degree is therefore not the bottom of one — which is a counting argument, and it would hold in any universe and for any well-formed scale.

The missing fifth is where the chain stops

There is a second way to see the same count, and it is the one that generalises.

The diatonic set is a chain of six fifths: F–C–G–D–A–E–B. A chain of k notes contains k − 1 links, so six of the seven degrees are the bottom of a fifth and one — the top of the chain, B — is not. Rotate to B and the tonic is at the end of the chain with nothing above it.

The rotations of the diatonic set, brought to one tonic. The same rotations started on the same note rather than on their own, ordered by how many of their degrees are raised. Every step lowers exactly one note by a semitone, and the notes it lowers, in order, are F♯, B, E, A, D, G — which is the chain of fifths read backwards. A rotation is a rotation whatever the scale; the chain is a property of a set generated by a single interval, and it disappears the moment the set is not.
Fig. 3 The chain read the other way, which is where the missing fifth comes from. Brought to one tonic and ordered by brightness, every step down the list 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. So the seven rotations are seven consecutive cuts of one chain, the darkest is the one that has given up the most links, and Locrian is the end of the line: the rotation whose own fifth is the last note the chain had left to lose.

That makes it a theorem rather than a fact about seven. In any scale that is a chain of k fifths, exactly one of the k + 1 degrees has no fifth above it, and it is always the last link. The pentatonic has five degrees and one such — E, in the C pentatonic — so the pentatonic has a Locrian too, and it is as unusable for the same reason.

The theorem has a converse at seven notes and it is the stronger half. Not only does every chain of six fifths give a count of one; among all 462 seven-note subsets of the twelve, nothing else does. The seven sets that score one are the seven rotations of the diatonic set and there are no others, so the count is a complete characterisation at this cardinality rather than a symptom. That is a fourth way of picking the diatonic set out of the universe, alongside the two-step-sizes property, maximal evenness and the chain itself, and it is the cheapest of the four to test: one lookup per degree.

Run the same count on scales that are not chains of fifths and the number stops being one, which is a useful check that the argument is really about the fifth. Every figure below is computed rather than quoted, and the four agree with the counting argument to the unit:

  • Harmonic minor: three of its seven degrees have no fifth above them, because it is not a chain of anything.
  • Melodic minor: three as well.
  • The whole-tone scale: six of six. It contains no perfect fifth at all, which is most of why it cannot establish a tonic by any ordinary means.
  • The octatonic: four of eight.

That reading is tempting and it is wrong, and the census says so in one line. Running the count over every seven-note subset of the twelve that contains the tonic — all 462 of them — the distribution is 7 sets with a count of one, 84 with two, 210 with three, 140 with four and 21 with five.

Exactly seven sets score one, and they are the seven rotations of the diatonic set. So among seven-note scales the count is not an approximate measure of anything: a count of one is the diatonic set, and nothing else in the universe achieves it. That is a sharper result than the essay was reaching for and it is worth having on its own — the property “every degree but one is the bottom of a perfect fifth” picks the diatonic set out of 462 candidates without any other criterion.

But the count does not measure distance from generation, and the counterexample is exact. There are fourteen well-formed seven-note sets in twelve. Seven of them are the diatonic rotations and score one; the other seven are the chromatic cluster and its rotations — 0,1,2,3,4,5,6 and the rest — which are generated by the semitone, are as well formed as anything can be, and score five.

generator degrees with no fifth above
the diatonic set 7 semitones 1
the chromatic cluster 1 semitone 5

Both are perfectly generated and they sit at opposite ends of the range. So what the count measures is not generation but how much of the set lies on a chain of fifths specifically — which is the same thing only for scales that happen to be built out of fifths. The harmonic minor, the whole-tone scale and the octatonic score badly because they are not chains of fifths; the chromatic cluster scores badly for the same reason and is a chain of something else.

Where the tritone is

The second question is more productive, because unlike the first it has seven different answers.

Where the tritone sits in each rotation. The two degrees a tritone apart, in each rotation. They are the same two notes every time — the set does not change — and which degrees they land on changes everything: Lydian at 1 and 4; Ionian at 4 and 7; Mixolydian at 3 and 7; Dorian at 3 and 6; Aeolian at 2 and 6; Phrygian at 2 and 5; Locrian at 1 and 5. Only where they are the fourth and the seventh do they resolve inwards onto the tonic and its third, and in 2 of the 7 the tonic is itself one end of a tritone.
Fig. 4 The two degrees a tritone apart, in each rotation. It is the same pair of pitch classes every time — F and B in the C collection, unmoved — and the degrees they occupy run from 4 and 7 in Ionian down to 1 and 5 in Locrian.

Read that column and most of the behaviour of the seven modes falls out.

Ionian: degrees 4 and 7. Neither is the tonic, and both are a semitone from a note of the tonic triad — the fourth degree lies a semitone above the third, the seventh a semitone below the eighth. So the tritone can contract inwards onto the third and the tonic, which is what a dominant seventh does and is the single most characteristic gesture in three centuries of European music.

Lydian: degrees 1 and 4. The tonic is in the tritone. Whatever the tritone does, it does to the tonic, and the note it wants to move to is the seventh degree of the parent major — which is to say, the tritone in Lydian pulls away from home rather than towards it.

Locrian: degrees 1 and 5. The tonic is in the tritone again, and this time the other end of it is the degree that should have been the fifth.

And the four in between put it on neither the tonic nor its fifth: Dorian at 3 and 6, Phrygian at 2 and 5, Mixolydian at 3 and 7, Aeolian at 2 and 6. In all four the tritone is a fact about the middle of the scale rather than about home, which is one reasonable description of what modal, as opposed to tonal, harmony sounds like.

Every mode of this collection contains the same one tritone — F and B — and what changes from mode to mode is which scale degrees they are, and therefore what they are a tritone away from. On a keyboard the two notes never move.

The same argument, one level up

The two questions above are the two cases of a single one: which intervals does the tonic have above it?

A rotation’s tonic sees the six other notes at six distances, and the multiset of those distances is different for every rotation — it is the row of the interval table starting at that degree. That is the object a mode actually is, and it is worth writing out for once.

The mode is its tritone. Every set of degrees that lies inside at least one of the seven modes — 510 of them — scored by how many modes it leaves standing, with the 15 minimal sets that decide printed in full. All 15 have two members and every one of them either is a tritone or contains the tritone above the tonic. The only degree every mode has is the tonic itself; there is no degree belonging to just one mode. Averaged over every order the degrees could arrive in, 5.67 of the seven have to have been heard before the mode is settled — most of the scale, because the deciding pair arrives when it arrives.
Fig. 5 The same argument run exhaustively, which is the strongest form it takes. Of the 510 sets of degrees that lie inside at least one of the seven modes, the minimal ones that pin a mode down are fifteen — and all fifteen have two members, and every one of them either is the tritone or contains it. So the tritone is not one diagnostic among several. It is the diagnostic: a listener who has heard two notes and knows which mode is in force has heard the tritone, and one who has heard six notes without it has not narrowed the field at all.

In Ionian the tonic sees 2, 4, 5, 7, 9, 11 — a major second, major third, perfect fourth, perfect fifth, major sixth, major seventh. In Locrian it sees 1, 3, 5, 6, 8, 10. Those are two different intervallic environments and neither is the set’s environment, because a set has no environment; it has only the pairs.

It is worth noticing that the tritone question and the fifth question are the same question. A degree with no fifth above it is a degree whose seventh semitone is missing, and in a set with one tritone that degree is the one the tritone starts on — so “where is the tritone” and “which rotation has no fifth” are one fact read at two levels. The first names a rotation and the second names a pair of degrees in each rotation, and there is no additional information in the second beyond where the first one’s answer sits.

A mode is a set plus a choice of vantage point, and the interval content seen from that vantage point is the whole of what changes. Everything this ladder does from here is a consequence of that sentence.

Reading the whole table from the tonic

Both questions so far are single entries in a larger table: for each rotation, which of the eleven intervals appear above its tonic. Written out, the table explains several things that are usually given as separate facts.

The seven modes, brightest first. The same seven pitch classes started on each of its degrees in turn, ordered by how many of their notes are raised. Each row differs from the one below it by exactly one note, and that note moves down one semitone each time.
Fig. 6 The seven rotations as note rows, brightest first, which is what the tonic sees. Each row differs from the one below it by exactly one note and that note moves down a semitone each time — so a rotation’s tonic sees the six other notes at six distances, and the multiset of those distances is different for every rotation. That multiset is what a mode actually is: not a starting point on a fixed set, but a complete description of what home has above it, which is why two questions — does the tonic have a fifth, and where is the tritone relative to it — turn out to be two readings of one row.

The rotation whose tonic sits at the sharp end of the arc sees the most intervals below the notes around it and the fewest above; the rotation at the flat end sees the reverse. That is the same ordering as brightness — Lydian at one end, Locrian at the other — arrived at by counting rather than by listening, and it is why the brightness ordering and the chain of fifths are the same ordering.

The practical consequence is a rule of thumb that is exactly true: a mode’s character is the position of its tonic in the parent collection’s chain of fifths, and every named modal alteration — a raised fourth, a lowered seventh, a lowered second — is a step along that chain. There is nothing else it could be, because the chain is the only order the set has.

Why the leading note is only in two of them

One entry in that list deserves its own paragraph, because it is the only interval whose presence or absence has a standard name.

A leading note is a degree a semitone below the tonic. The diatonic set has two semitones, so exactly two rotations have their tonic a semitone above one of the set’s members: Ionian and Lydian.

The triads each scale builds on its own degrees. Every chord that can be stacked in thirds on each degree of each scale, with its quality worked out from the intervals the scale actually supplies. The quality of the chord on each degree is a consequence of each scale's own step pattern rather than of a convention.
Fig. 7 The triads of three scales on the same tonic. The natural minor has no leading note — its seventh degree is a whole tone below the tonic — and the harmonic minor is the repair, raising that degree and changing the triad on the fifth from minor to major in the process.

That is the structural reason the harmonic minor exists. Aeolian has no leading note, because its seventh degree is the parent key’s fifth, a whole tone below its tonic. Raising it manufactures one — at the cost of a three-semitone step between the sixth and seventh degrees, which breaks Myhill’s property and takes the scale off every list the census ever ran.

The most-used scale in minor-key European music is one that fails all four structural properties, and it fails them in order to obtain one interval that a rotation could not supply. That is a better argument for the importance of the leading note than any number of assertions about tendency tones.

What a fifth above the tonic is actually for

It is worth asking why the fifth, of all the intervals, should be the one whose absence disqualifies a rotation. Three answers, and they are independent of one another, which is the interesting part.

It is the most consonant interval after the octave. Every roughness curve this site has drawn puts its deepest well below the octave at 702 cents, for every harmonic spectrum tried. A tonic triad without a fifth is a chord with no consonance holding it together, and a diminished triad is measurably rougher than the other three qualities at every register.

It is the interval a fundamental’s own partials supply. The third partial of any harmonic tone is a twelfth above it, which is a fifth plus an octave, so a note sounded alone already contains its own fifth. The missing-fundamental ladder is built on how strongly the ear uses that fact. A tonic whose fifth is absent from the scale is being contradicted by its own spectrum.

And it is the generator. The whole set is a chain of fifths, so the fifth is the interval that relates the scale’s members to each other structurally. The degree with no fifth above it is the degree with no structural neighbour in one direction.

Those three are separate arguments — one psychoacoustic, one physical, one combinatorial — and they select the same interval. That coincidence is not explained here and is arguably the deepest thing in the subject; it is the reason a purely combinatorial ladder like this one can say anything about music at all.

The first eight partials of a string. A string vibrating in one, two, three and more equal parts, with the frequency ratio and the nearest named note beside each. The seventh partial is a third of a semitone flat of anything on a keyboard, which is a fact about strings rather than about tuning.
Fig. 8 The first eight partials of a single note. The third is a fifth above the second, which is the interval this essay has been counting, arriving from physics rather than from the chain. A rotation with no fifth above its tonic is at odds with what its own tonic is already producing.

What the picture cannot show

Every count here assumes equal temperament and twelve positions. A tritone is “six semitones” only in a universe of twelve equal steps. In meantone the two tritones of the diatonic set — F to B as an augmented fourth and B to F as a diminished fifth — are different sizes, 590 and 610 cents in quarter-comma, and the argument about symmetry that makes this essay tidy is a fact about temperament rather than about the scale.

A structural disqualification is not a prohibition. Locrian has been written in, deliberately, by composers who wanted exactly what it lacks, and the diminished tonic can be made to work by never sounding it as a triad. The claim here is that the rotation gives its user nothing to end on, not that nothing can be done about it.

And the tritone’s “resolution” is a convention of one repertoire. Where the two notes go, and whether they go anywhere, is a claim about eighteenth-century voice leading. In modal counterpoint the same interval is simply avoided between the outer voices, and in a great deal of later music it resolves nowhere at all. What is computed here is where the interval is; what it does is repertoire.

Whose music this is a claim about

The seven rotations are not seven equally used objects, and the usage is roughly what the two questions above predict. Ionian and Aeolian dominate European art music after 1600, and both have a tonic triad and a tritone away from the tonic. Dorian and Mixolydian are the workhorses of folk repertoire across Europe and of a great deal of jazz — both have a tonic triad, and both differ from a common-practice key by exactly one note. Phrygian is regionally concentrated, strongly, in Iberian and Balkan practice. Lydian is rare as a whole-piece mode and common as a colour, which is what one would expect of a mode whose tonic is in the tritone.

And Locrian is not a mode anybody’s tradition uses, which is the only place where a structural argument and a repertoire count agree completely. It is worth noticing that the agreement is suspiciously clean: a structural criterion that ruled out three of the seven, or that failed to rule out any, would be far more informative than one that rules out exactly the one everybody had already agreed to leave alone. What makes the argument worth making is that it is derived rather than fitted — nothing in the counting knew which mode was unused.

The caveat is that all of this is a claim about traditions built on the diatonic set. A tradition whose scale is not a rotation of a well-formed set has no such table, and the questions asked here would have to be asked in cents rather than in degrees.

The ladder from here

Two of the seven have a leading note; one of the seven has a tritone on degrees 4 and 7. Both counts point at the same object, which is the cadence — and the next rung asks how many of the seven can make one, with the triad on the fifth degree computed rather than remembered. The answer is smaller than three.

Part 4 of 9

One essay in the series on modes. The essays either side of this one:

The price of a tonic. Every note of Dorian is given the same duration except its tonic, which is lengthened; the horizontal axis is the share of the total that goes to it. The key-finder answers with the parent key until 25.0 per cent of the time is spent on the modal tonic, and with D minor above it. At the left-hand edge every note has equal weight, which is the pitch-class set itself — and with every weight identical the correlation is not merely low but undefined, because a flat histogram has no variance to correlate with anything. What a tonic costs in seconds Part 3 — 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. The triad on every degree of every rotation. The quality of the triad built on each degree of each rotation, computed from the set rather than tabulated. The triad on the fifth degree is major in 3 of the 7: Lydian, Ionian, Locrian. Of those, Locrian has no triad on its own first degree to resolve to, and Lydian has its fifth degree on the note the parent collection is built from, so the cadence would confirm the parent rather than the mode. What is left is Ionian — one rotation of the 7, and the others have to end some other way. One dominant among seven Part 5 — 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.

Where this goes next

The seven modes, brightest first. The same seven pitch classes started on each of its degrees in turn, ordered by how many of their notes are raised. Each row differs from the one below it by exactly one note, and that note moves down one semitone each time. 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. Every interval, and what it becomes when it is turned over. Each interval within the octave paired with its inversion — the interval left when the lower note is raised by an octave. The two always add to twelve semitones, so exactly one interval can be its own inversion, and it is the one at six. The interval that inverts to itself Six is the only number that divides twelve into two equal halves, so the tritone is the only interval unchanged by being turned over. That symmetry is not a curiosity — it is why one tritone belongs to two dominant chords, and why they resolve to keys a tritone apart. The seven modes, brightest first. The same seven pitch classes started on each of its degrees in turn, ordered by how many of their notes are raised. Each row differs from the one below it by exactly one note, and that note moves down one semitone each time. Nothing in the census knows which note is home All four properties six earlier essays are about are invariant under rotation and transposition — one property tuple over all eighty-four rotations and transpositions of the diatonic set. So the census that separates 349 shapes cannot separate a major scale from its own Aeolian mode, and everything that makes one note a tonic is outside it.

What links here

Essays that reach for this one mid-argument — the half of a link its own author cannot write down.

The objects named here

The third way in, after the field and the series: the things themselves, and every essay that touches each one.

Interval patternLeading noteModePerfect fifthRotationTriadTritone