Acoustics & Music Theory
why it sounds like that
Scales and modes

A degree is where it goes next

The first essay on scales beyond twelve said a mode is not a set and listed four things a set cannot record. This one makes the sharpest of them arithmetic. Specify a mode as an ascent and a descent — which is how a living tradition specifies one — and it becomes a directed graph on its degrees; sixty-three such graphs collapse onto a single pentatonic set, sixty-two of them with an ascent that is not the descent reversed, and every standard measure of a scale returns the same value for all of them.
15 min read 5 figures How many of the twelveNotation hides thingsThe instrument is not the model

Assumes: A scale is not a set of pitches

The first rung of this ladder ended with a list of four things a pitch set cannot record about a mode: which degrees are structural, which appear going up and which coming down, the characteristic phrases, and the intonation. It named them and did not measure any of them.

The seventh rung ended by recording a debt in writing: what it means for a degree to be defined by where it goes next. This is that debt.

Making it arithmetic turns out to be easy, and the reason it is worth doing is what comes out — a number, for how much of a mode a set is.

A mode as a graph

Take the second item from that list, because it is the one with a definite form. An aroha is the sequence of degrees a raga uses ascending; an avaroha is the sequence descending. Both are lists, and a list of degrees is a statement about which degree may follow which.

So a mode specified that way is a directed graph on its degrees, and the graph is derivable from data that is already written down.

Two modes, one set, and every measure of a set that cannot tell them apart. Raga Bhupali and Raga Deshkar drawn as the moves each allows: an arrow from one degree to another means the tradition's ascent or descent goes that way. Raga Deshkar's ascent omits Re, so the two graphs differ by an edge. Below, every standard measure of a scale, evaluated on both — and they are identical in every row, which the drawing checks before it is made. An ascent and a descent that between them use all 5 degrees can be chosen in 63 ways, 62 of them asymmetric. That is how many modes collapse onto one pitch set under the simplest order model there is, and a census over subsets counts the set once.
Fig. 1 Two ragas as the moves each allows. An arrow arching above the row is a step the ascent makes; one below is a step the descent makes. The two graphs differ by a single edge — Deshkar’s ascent goes from the second degree straight to the third, skipping Re — and beneath them is every quantity computed here about a scale, evaluated on both. Each row is identical, and the equality is checked by computation rather than claimed.

The lower half of that figure is the point. Pitch classes, step pattern, interval vector, prime form, generator, well-formedness, maximal evenness: seven properties, two modes, and no disagreement anywhere. This is not a shortcoming of a badly chosen property. It is a structural fact about all of them.

Why every property here is blind to it

The reason is worth stating plainly, because it explains why the blindness is universal rather than accidental.

Every scale property in this collection is computed from an unordered set. The interval vector counts how many times each interval class occurs among the pairs — and pairs are unordered, so reversing the scale leaves every count where it was. Myhill’s property asks how many sizes each generic interval takes, over all starting positions, which is the same multiset read from either end. Maximal evenness is a property of positions on a circle. Well-formedness asks whether one generator produces the set.

Not one of them contains an operation that distinguishes before from after. That is not a defect in the design — it is what made a census over all 349 shapes in twelve and a sweep over every universe from four to thirty tractable at all, because there are far fewer sets than orderings and a property defined over orderings could not have been enumerated.

Raga Bhupali. Raga Bhupali: its degrees in cents above the tonic, drawn against the twelve equal steps of a keyboard. vadi Ga, samvadi Dha — the weight of the raga sits in the lower half. Source: Hindustani classical practice; the pitch set is the common pentatonic.
Fig. 2 The first of the two, drawn the way any scale is drawn here: degrees in cents above the tonic against the twelve equal steps. Deshkar’s picture is identical — the same five pitch classes, the same intervals, the same everything a census can read. What separates them is that Bhupali’s weight sits in the lower half of the octave with Ga as vadi and Dha as samvadi, and Deshkar’s sits in the upper, with the emphasis reversed and different notes approached from different directions. Nothing on this axis can carry that, which is the essay’s whole subject stated as an absence.

The site has met this once before from a different direction. The census cannot tell which note is home — one property tuple covers all eighty-four rotations and transpositions of the diatonic set, so a major scale and its own Aeolian mode are one object to it. That is rotation-invariance. This is time-invariance, and they are two faces of one thing: a set has no distinguished element and no distinguished direction.

There is a second demonstration available for nothing, and it is the harsher one.

A census over every equal division from four to thirty counts sets, and a raga is not a set: it is a set plus a grammar of approach, emphasis and direction. Subsets of twelve number 4,096; directed graphs on seven labelled degrees number two to the forty-ninth. There is no census of the second kind and there will not be one.

That is not a criticism of the sweep. It is a statement of what its domain is, and the point of putting the two figures next to each other is that a result can be complete and still be about something narrower than its subject.

How much is thrown away, counted

If a set discards ordering, how much ordering is there to discard? The question has an answer for the simplest possible order model.

Let a mode be an ascent and a descent. Each is a subset of the degrees, in order; each must contain the tonic; each must have at least three notes to be a mode of anything; and between them they must use every degree of the set, since a degree in neither is not in the mode at all.

Count them for five degrees: eleven admissible ascents, eleven descents, and sixty-three of the hundred and twenty-one pairs use all five degrees. Sixty-two of those sixty-three have an ascent that is not the descent reversed.

Sixty-three modes, one set.

That is a floor rather than an estimate, and the reasons it is a floor are all in the direction of more:

  • It permits no degree to be visited twice. A vakra raga’s ascent turns back on itself — up to the fourth degree by way of the second — and once revisiting is allowed the count is unbounded.
  • It has no vadi and samvadi in it. Bhupali and Deshkar differ in emphasis as well as in ascent, and the emphasis alone distinguishes ragas whose ascents agree.
  • It has no phrases. A pakad is a specific figure, which is an ordering of a length the model has no term for.
  • And it has no intonation, no ornament and no time of day.

The number to take away is not sixty-three. It is that a census counting subsets of an equal division is counting one object where a tradition has dozens, and the ratio is at least that and probably not close to it.

What the site’s machinery would have to become

It is worth being precise about what would be required to see any of this, because the answer is not “a better property”.

Raga Deshkar. Raga Deshkar: its degrees in cents above the tonic, drawn against the twelve equal steps of a keyboard. vadi Dha, samvadi Ga — the same set, weighted in the upper half, and a different raga. Source: Hindustani classical practice; the same five notes as Bhupali.
Fig. 3 The second of the two, drawn identically because it is identical in everything the drawing carries. Vadi Dha, samvadi Ga — the weights are Bhupali’s exchanged — and of Deshkar’s sixty-two seven-note lines, thirty-eight have a retrograde the raga does not admit. Play one of those backwards and the pitches are identical, the interval multiset is identical, every property in the table at the top of this essay is identical, and the result is not Deshkar. That is an asymmetry which is audible, countable, and invisible to every instrument this collection owns.

A property that distinguished Bhupali from Deshkar would have to be a function of the transition graph rather than of the pitch set, and that changes the size of the space it is enumerated over. Subsets of twelve: 4,096. Directed graphs on seven labelled degrees: two to the forty-nine. There is no census of the second kind and there will not be one.

So the alternative is not a bigger enumeration but a different kind of question — not how many objects are there but what does this particular object do, which is what a tradition’s own theory asks and what a set-based theory was built to avoid asking.

That is the honest summary of the relation between the two. A census is a good instrument that answers a question about a space; a raga is not a point in that space and no amount of resolution will make it one.

The nearest thing the site has to an ordered property

There is one quantity in the scale machinery that is not quite blind to this, and it is worth examining because it fails in an interesting way.

The nearest thing the collection has to an ordered property is the evidence-accumulation curve for a tonic, which has a clock in it and distinguishes earlier from later. It cannot see any of this either: what it measures is the accumulation of evidence for a key, so it is sensitive to the order degrees arrive in only through their running frequencies, and two orderings with the same counts are the same to it. Even the machinery with a clock reduces to counting.

So even the machinery with a clock in it reduces to counting. That is the general shape of what goes wrong: any property that summarises a sequence by tallying its members has thrown the sequence away, and nearly every tractable property does exactly that.

And the chain of fifths that generates these sets has one parameter and produces sets: the pentatonic at five links, the diatonic at seven. Nothing in it can produce an ascent that differs from a descent, because it has no term in which such a difference could be expressed.

The one place ordering does appear here

There is an exception in this collection and it is instructive, because of what it cost.

Contour is a property of an ordering — the sign sequence of a melody’s intervals — and it is the only quantity here that a retrograde changes. It is enumerable at short lengths, and the enumeration works precisely because contour throws away everything except one bit per move.

That is the trade. A property that sees order is affordable only if it discards nearly all of the pitch information, and a property that keeps the pitch information is affordable only if it discards the order. Nothing here has both, and the two halves of this collection are the two sides of that.

A raga is not on either side. It keeps the pitch information to a finer resolution than twelve, and it keeps the ordering to a finer resolution than a sign sequence, and it is specified by a tradition rather than enumerated by anybody.

What a graph does buy

Setting out what the model cannot do makes it sound useless, and it is not. Two things follow from the graph that the set does not give.

A count of admissible melodies. Given the graph, the number of distinct n-note lines a raga admits is a walk count on it, and it differs between two ragas that share a set. “A substantial fraction” is a phrase, and the walk count is a number:

notes in the line Bhupali Deshkar Deshkar as a share
3 14 11 79%
5 42 26 62%
7 126 62 49%
10 648 226 35%

The gap widens with length, which is what a missing edge does: each additional note is another chance to need the edge that is not there. At three notes the two ragas admit nearly the same melodies and at ten Deshkar admits about a third of Bhupali’s. So the two are hardest to tell apart in exactly the fragments a pitch set would be tested on and easiest to tell apart over a phrase.

The mechanism is sharper than “a missing edge” as well. In Deshkar the only arrow into Re is from Ga above it, and the only arrow out is down to Sa — Re is a purely descending degree, reachable one way and leavable one way. It has not been removed from the set and it has been removed from every ascent, which is precisely the distinction a set cannot hold and a musician states in one sentence.

An asymmetry that is audible, and it can be counted too. Of Deshkar’s sixty-two seven-note lines, thirty-eight — 61 per cent — have a retrograde the raga does not admit. Play one of those backwards and the pitches are identical, the interval multiset is identical, every property in the table at the top of this essay is identical, and the result is not Deshkar.

Bhupali gives zero. Its ascent is its descent reversed, so its graph is symmetric and every line’s retrograde is another of its lines — which makes Bhupali exactly the one of the sixty-three enumerated below whose ascent is the descent reversed, and Deshkar one of the sixty-two that are not. The two ragas this essay compares are one from each side of that count, which was not why they were chosen.

So the listening test has a size on it: pick a seven-note Deshkar line at random and there is a three-in-five chance its retrograde is inadmissible. That is a demonstration anybody with the graph can construct, and it needs no acoustics at all — the only difference between the two lines is the direction of time.

An asymmetry that is audible. A raga whose ascent omits a degree used in the descent has the property that the same interval is available in one direction and not the other. Playing a line and its retrograde produces one that is idiomatic and one that is not, and the two are pitch-identical. That is a listening test anybody can construct from the graph, and it is the cleanest demonstration that ordering carries content — there is no acoustic difference at all between the two lines beyond the direction of time.

Three rotations of one seven-note set are three different objects to a listener and one object to every property in the table, which is the same failure one size up — and the collection has met it before without naming it as this.

4 ways to fill the same fourth. Each row is a tetrachord: a span of 498 cents — a pure fourth — with two notes inside it, drawn in cents from the lower bound. The bounding notes never move, which is what makes the tetrachord a unit; everything that varies is interior. jins rast puts them at 204 and 355 cents, giving steps of 204, 151, 143; jins bayati puts them at 151 and 294 cents, giving steps of 151, 143, 204; jins hijaz puts them at 90 and 384 cents, giving steps of 90, 294, 114; jins nahawand puts them at 204 and 294 cents, giving steps of 204, 90, 204. Against the twelve equal steps below, 2 of 4 land within twenty-five cents of a semitone at every degree; the worst miss is 49 cents, which is a quarter of a semitone and has no name on a keyboard. Nothing in this construction is a subset of an equal division, so nothing the census here does applies to any of it.
Fig. 4 The one place ordering does appear in this collection, and what it cost. A maqam’s tetrachords are units with names, and joining two of them is a grammatical operation rather than a selection from a set — which is why the maqam essays could say things about approach and emphasis that the scale essays cannot. The price was giving up the census: there is no enumeration of tetrachord pairs comparable to the enumeration of subsets, and the arguments that field makes are about particular objects rather than about a space. That is the trade this essay is about, made once already and made without comment.

Which computation produced the numbers

The graphs are derived from the tradition data this site already stores: the consecutive pairs of the ascent and the consecutive pairs of the descent, read as edges. Nothing was invented for this essay.

The invariants are the same functions the scale essays use, called on the pitch-class set each mode collapses to, and their equality is tested by comparing serialised values rather than by inspection. If a future change to any of those functions made them distinguish the two, the figure’s caption would say so — it reads the comparison rather than stating it.

The count of sixty-three is a full enumeration over subsets: for each of the eleven admissible ascents and eleven descents, the pair is kept if the union of the two is the whole set. The asymmetric count is the same enumeration with the pairs whose ascent equals its descent removed.

The walk counts are the number of paths of the stated length in the directed graph, computed by iterating a vector of per-degree counts through the adjacency once per note, with every degree admissible as a starting point. Two things about that are conventions rather than results. A line here may sit on one degree for as long as it likes only if the graph has a self-edge, and neither of these graphs does, so a repeated note is not counted as a move — which undercounts real melodic material in both ragas by the same construction. And the count is over lines that stay inside one octave, because the graphs are on five degrees rather than on a register; a real ascent crosses into the octave above and the edge that carries it is the one from the top degree back to Sa, which neither tradition record here contains.

Maqam Rast, Arabic theory. Maqam Rast, Arabic theory: its degrees in cents above the tonic, drawn against the twelve equal steps of a keyboard. the third and the seventh sit halfway between major and minor — by convention, exactly halfway. Source: the quarter-tone convention fixed at the Cairo congress of 1932.
Fig. 5 One maqam in two codifications, which is a different kind of disagreement from the one this essay is about: here the two differ in the cents of their degrees while agreeing about the path. Set beside the two ragas, which agree about the cents and differ about the path, the pair marks out the two independent axes a mode has — and a pitch set records neither of them completely.

Whose music, and when

Bhupali and Deshkar are Hindustani, the pair is a standard teaching example, and the pitch set is the common pentatonic. They are used here because they are the clearest documented case of two named modes on one set — not because the phenomenon is Indian.

It is not. Arabic maqam practice specifies sayr, the characteristic path through a maqam, and two maqamat can share a jins inventory and differ in it. Persian dastgah has the same structure. Byzantine and Ottoman modal theory both specify movement as well as material. And European practice had it too, before it stopped: a medieval mode was defined by its final, its ambitus and its repercussa — the reciting note — and by melodic formulae, and the reduction of “mode” to “which seven of the twelve” — the reduction that makes the seven rotations of one set into seven modes — is a specifically post-Renaissance simplification that this site has otherwise inherited without comment.

That last point deserves to be made explicitly. The set-based account is not the general case with the others as variants. It is the special case that arose in one tradition alongside a keyboard, and its convenience is what made two centuries of the combinatorics in this collection possible.

What the picture cannot show

It cannot show a performance. An aroha is a summary of what a raga does, written down by theorists; what a performer plays is a great deal richer and is not a graph traversal. Treating the ascent as a rule rather than as a description is the same error as treating a scale as a set, one level up.

The graph is first order. It records which degree may follow which and nothing about longer dependencies, and a pakad is a figure several notes long. A model that captured phrases would be a grammar rather than a graph, and the count above would be very much larger.

It has no drone. Every raga is performed over a fixed tonic sounding continuously, which makes each degree a simultaneity with the tonic as well as a step in a path — and the vertical arguments this site is full of therefore apply to it too, at every moment. A model that has only the path has thrown away half of what a listener is given.

And it has no time in it. Duration, emphasis by length, and the pauses that shape a phrase are all absent, and in a tradition where a note may be held for a minute they are not details.

The ladder from here

beyond-twelve has now paid both of the debts it recorded. The tetrachord as a unit of construction is written; a degree defined by where it goes next is this.

What is owed now is different in kind, and it is a measurement rather than an argument: two ragas on one set are distinguishable by ear, quickly and reliably, by listeners who know the tradition. Nothing in this collection has asked what that discrimination is using — whether it is the ascent, the emphasis, the phrases or the intonation, and in what proportion. That is an experiment rather than a computation, and it is the only way to find out which of the four items on the first rung’s list is doing the work.

Part 8 of 14

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

3 ways to fill the same fourth. Each row is a tetrachord: a span of 498 cents — a pure fourth — with two notes inside it, drawn in cents from the lower bound. The bounding notes never move, which is what makes the tetrachord a unit; everything that varies is interior. diatonic (ditonic) puts them at 90 and 294 cents, giving steps of 90, 204, 204; intense chromatic puts them at 81 and 231 cents, giving steps of 81, 151, 267; enharmonic puts them at 63 and 112 cents, giving steps of 63, 49, 386. Against the twelve equal steps below, 1 of 3 land within twenty-five cents of a semitone at every degree; the worst miss is 37 cents, which is a quarter of a semitone and has no name on a keyboard. Nothing in this construction is a subset of an equal division, so nothing the census here does applies to any of it. A scale built downward from a fourth Part 7 — A tetrachord is a bounded span with two free notes inside it, which is a different generator from a chain of fifths and a different object from a subset of an equal division. At the maqam tradition's own quarter-tone resolution the construction admits thirty-six fillings of the fourth and 1,296 octaves built from them; the census here can see six of the first and 2.8 per cent of the second, and one of the octaves it cannot see is an ordinary mode of a living tradition. one measured slendro is the scale least committed to a spectrum. Each scale drawn, placed by how smooth it is against 2000 random scales of the same size, under 4 spectra. Low is smooth. Every one of them lands in the smoothest third under every spectrum, so the account that a spectrum chooses a scale survives as a statement about levels. What separates them is the length of each row's line, which is how much the answer moves when the instrument changes: one measured slendro spans 2.9 percentile points and Rast, Turkish theory spans 17.6, a factor of 6.0. The scale usually offered as the case for spectral matching is the one whose standing barely depends on the spectrum, and Rast, Turkish theory is the one that depends on it most. The scale least committed to its own instrument Part 9 — The essays on scales beyond twelve draw scales without spectra and spectra without scales, and none of them had put a tradition's own degrees through the roughness model. Doing it for six scales and five spectra says the account survives — every tradition lands between the eighth and the twenty-sixth percentile of random scales of its size, and under a pure tone not one of them does. But the scale whose standing moves least when the instrument changes is the gamelan's, at 2.9 percentile points, and the one that moves most is the five-limit diatonic at 17.4. The scale the argument is usually made about is the one it explains least.

Where this goes next

Maqam Rast, Arabic theory. Maqam Rast, Arabic theory: its degrees in cents above the tonic, drawn against the twelve equal steps of a keyboard. the third and the seventh sit halfway between major and minor — by convention, exactly halfway. Source: the quarter-tone convention fixed at the Cairo congress of 1932. A scale is not a set of pitches Two ragas can have identical pitch sets and be different ragas. Two national theories of one maqam put its third degree thirty-five cents apart. Both facts are fatal to the idea that a mode is a collection of notes, and both are ordinary in the traditions concerned. 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. 3 ways to fill the same fourth. Each row is a tetrachord: a span of 498 cents — a pure fourth — with two notes inside it, drawn in cents from the lower bound. The bounding notes never move, which is what makes the tetrachord a unit; everything that varies is interior. diatonic (ditonic) puts them at 90 and 294 cents, giving steps of 90, 204, 204; intense chromatic puts them at 81 and 231 cents, giving steps of 81, 151, 267; enharmonic puts them at 63 and 112 cents, giving steps of 63, 49, 386. Against the twelve equal steps below, 1 of 3 land within twenty-five cents of a semitone at every degree; the worst miss is 37 cents, which is a quarter of a semitone and has no name on a keyboard. Nothing in this construction is a subset of an equal division, so nothing the census here does applies to any of it. A scale built downward from a fourth A tetrachord is a bounded span with two free notes inside it, which is a different generator from a chain of fifths and a different object from a subset of an equal division. At the maqam tradition's own quarter-tone resolution the construction admits thirty-six fillings of the fourth and 1,296 octaves built from them; the census here can see six of the first and 2.8 per cent of the second, and one of the octaves it cannot see is an ordinary mode of a living tradition.

What links here

Essays that reach for this one mid-argument — the half of a link its own author cannot write down, the 8 sharing most with it of 13.

What this makes readable

Essays that declare this one a prerequisite.

The objects named here

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

ArohaEnumerationIntervalMaximal evennessRagaScale degreeSet classWell-formedness