A scale without an octave, and the spectrum that asks for it

The octave is the one interval nothing on this site questions. Every scale repeats at the 2:1, pitch classes exist because the 2:1 collapses, and the circle of fifths is a circle rather than a line for the same reason. It reads like an axiom.

It is not one. The octave’s special status has a mechanism, the mechanism has conditions, and the conditions can be removed.

That is the control this essay needs. A harmonic spectrum has no reason to like 3:1 better than 2:1, so a scale built on the tritave is not a rearrangement of the usual preferences — it is a scale for a spectrum that does not exist on a string.

Every partial of the upper note is at 2n times the fundamental, and 2n is always an integer, so the upper tone’s spectrum is a subset of the lower’s. Nothing new is added. That is the mechanism, and it needs the lower note to have partials at even multiples for the upper note’s partials to land on.

The spectrum that removes it #

A cylindrical pipe closed at one end supports only odd harmonics. The clarinet is the standard instrument of this kind, and its spectrum is nearly a pure odd series: the second, fourth and sixth partials are almost absent.

The wells move. A spectrum with only odd partials prefers only odd ratios, which is exactly the condition Bohlen–Pierce was designed around — so the scale without an octave is not an arbitrary construction but the scale a stopped pipe would have chosen, had anybody let it.

For a spectrum containing only odd multiples, the octave stops being special. The upper note’s partials sit at 2n times the lower fundamental — all even — and the lower note has no even partials for them to coincide with. Nothing lands on anything.

The interval where coincidence does happen is the 3:1. The upper note’s partials are at 3n times the fundamental, and where n is odd, 3n is odd, so every one of them lands on a partial the lower note has. The 3:1 does for an odd spectrum exactly what the 2:1 does for a full harmonic one.

The 3:1 is an octave and a fifth, and it goes by the name tritave in the literature this essay is about.

What the roughness model says, unprompted #

That argument is a counting argument and it is worth checking against the sensory model, which knows nothing about it.

The wells this curve produces sit on 7/5, 11/7, 5/3, 9/5, 11/5, 7/3 and 13/5. Every one of those is a ratio of two odd numbers, which is what the counting argument predicts and what nothing in the computation was told. Not one of them is 2:1, 3:2 or 5:4 — the intervals that are consonant for everything else on this site do not appear at all, because the coincidences that build them require even partials.

Two of the seven are worth a note, because they are what an honest scan produces rather than what a tidy account would want. 11/7 and 11/5 involve an eleventh partial and 13/5 a thirteenth; the list handed to this curve runs to thirteen, so those coincidences are available to it. Shortening the list to seven partials removes them, and what is left is 7/5, 5/3 and 7/3 — which are exactly the intervals the Bohlen–Pierce literature names as its consonances. Which wells exist is therefore a property of how many partials a timbre actually has, not of the ratios in the abstract.

Shortening the list also produces one well the counting argument did not predict, and it is the most interesting thing on the curve: a minimum six cents below 2/1. The octave is back, and by the opposite mechanism. For an odd spectrum the upper note’s partials at a 2:1 fall at 2, 6, 10 and 14 times the lower fundamental, and the lower note’s are at 1, 3, 5 and 7 — so every upper partial sits almost exactly halfway between two lower ones. Nothing coincides, and nothing is close enough to grate either. The 2:1 is smooth here because the two spectra maximally avoid each other rather than because they agree.

That is a different kind of consonance and it is worth keeping separate. Coincidence gives fusion — two notes heard as one object. Avoidance gives only the absence of roughness, with the two notes staying audibly two. Both show up as wells in a curve that measures roughness and nothing else, and reading them as the same thing is the standing risk of a purely sensory account.

The chord the coincidences build is 3:5:7, the odd-spectrum counterpart of 4:5:6.

Thirteen steps, and why thirteen #

Having a tritave to divide, the question becomes how many equal steps to divide it into, and it is the same question as nineteen, thirty-one or fifty-three with a different equave.

The tritave is 1901.96 cents, so thirteen steps are 146.30 cents each — larger than a semitone by half again. Those steps land close to the odd ratios the curve identified, which is the same criterion by which twelve is a good division of the octave, applied to a different equave and a different set of target ratios.

Applying a criterion means running it on the alternatives, and that has not been done above: thirteen has been shown to be good, not to be chosen. So here is every division of the tritave from five to thirty, scored by the mean error in cents with which its steps approximate the seven odd ratios in the figure.

Thirteen does not merely win. Its nearest genuine rival is three times worse. Nothing between five and thirty gets under sixteen cents except thirteen, its own double at twenty-six, and thirty — and the divisions immediately on either side of it, twelve and fourteen, are at 48.9 and 38.4, which is to say they are not close to anything. Thirteen sits in a hole of its own.

That deserves one comparison, because a number is not impressive until it is beside something. Score twelve equal semitones the same way, against the eight ordinary five-limit ratios, and it comes out at 9.8 cents mean and 15.6 worst. Thirteen equal divisions of a tritave fit the odd ratios better than twelve equal semitones fit the ordinary ones — 5.6 against 9.8 — and it is isolated among its neighbours in the same way twelve is among its own (9.8 against 27.7 for ten, 29.4 for eleven, 27.9 for thirteen).

One target is handled badly and the table should say so: 27/25, the first step of the scale, misses by 13.1 cents, which is five times the error on 7/5 and is the whole of the worst-case column. Every other one of the seven is inside seven cents.

The result is the Bohlen–Pierce scale, arrived at independently three times: by Heinz Bohlen in 1972, by Kees van Prooijen in 1978, and by John R. Pierce at Bell Labs in 1984. Bohlen’s route was combination tones, Pierce’s was the odd-harmonic argument above, and the fact that three people looking for the same thing found the same thing is the strongest evidence available that the criterion is doing real work.

The scale itself #

Thirteen equal steps of a tritave are the raw material rather than the scale, in the same way that twelve equal semitones are not a scale. Something has to be selected from them.

The standard selection is the nine-note Lambda scale, at steps 0, 1, 3, 4, 6, 7, 9, 10 and 13 — which gives step sizes, in units of the 146.30-cent step, of 1, 2, 1, 2, 1, 2, 1, 3. Two sizes for the seconds, which is the same criterion the diatonic scale meets, with one larger step to close the equave.

That selection is not arbitrary either. It is chosen so that the chord 3:5:7 can be built on several of its degrees, exactly as the diatonic scale is selected so that triads can be built on all seven. The construction is the same construction; only the equave and the target chord have changed.

The generator is worth naming too. The diatonic scale is seven consecutive fifths; Lambda is generated by three steps of the thirteen, which is 438.9 cents — very close to 9/7, one of the odd consonances. So the system has a chain of generating intervals, a set of scales cut from it, and a chord type they are built to support, which is the entire apparatus of ordinary tonal theory reproduced from different premises.

What a scale like this keeps and what it loses #

Almost every piece of structure this site has built survives the substitution, which is the surprising part.

The ordinary diatonic scale drawn as positions on a circle of twelve is the same kind of object as a Bohlen–Pierce scale drawn on a circle of thirteen: a selection from an equal division of a repeating interval. What differs is only which interval repeats — and everything else about how a scale is described survives the substitution unchanged, which is the useful part of taking the octave away.

There are chords — 3:5:7 is the “major” triad of the system and 5:7:9 its “minor” — and they have the same relationship to the equave that 4:5:6 and 10:12:15 have to the octave. There are scales chosen from the thirteen, of which the nine-note Lambda scale is the standard one, and it has two step sizes. There is a chain of generating intervals. There is a notion of key and of transposition.

What is lost is note identity across the equave. Two notes a tritave apart are treated as the same note in this system, and two notes an octave apart are not — they are unrelated pitches with no special relationship. That is deeply strange to hear and it is exactly what the spectrum implies: an octave adds nothing to a harmonic tone and adds a whole new set of frequencies to an odd one.

It also loses every instrument. A Bohlen–Pierce scale on a piano is inaudible as anything but out-of-tune noise, because a piano string has even partials and the whole argument fails. The scale requires clarinets, or synthesis, or something else with a closed-pipe spectrum, and the two most-recorded pieces in the idiom are for clarinet ensemble for precisely that reason.

The other route to the same scale #

Pierce’s argument is the one above: odd spectrum, tritave, odd ratios. Bohlen reached the same thirteen steps twelve years earlier from something else entirely, and the coincidence is worth recording because two independent derivations of one object are the strongest evidence a construction of this kind can have.

Bohlen was working from combination tones — the extra frequencies the ear generates when two loud tones are present, the most audible of which sits at the difference between them. His criterion was that a chord should be one whose combination tones reinforce its own notes rather than introducing new ones, and he asked which chords satisfy it.

For the ordinary major triad the answer is well known: 4:5:6 generates a difference tone at 1, two octaves and a fifth below the root, which reinforces the whole chord. Bohlen’s question was whether there is a chord that does the same thing without using the octave, and 3:5:7 is the answer — its difference tones fall at 2 and 4, which are outside the chord, and at ratios that reinforce rather than muddy it.

From that chord he worked backwards to an equave and a division, and arrived at thirteen steps of the 3:1. Van Prooijen, in 1978, arrived at the same division a third way, by looking for equal divisions whose steps approximate ratios of odd numbers — which is the same criterion by which 19, 31 and 53 are good divisions of the octave, with the primes restricted.

Three criteria — spectral coincidence, combination tones, and rational approximation — converging on one division is not proof that the scale is musically viable. It is good evidence that the division is the right answer to the question that was asked.

The claim this makes about the ordinary case #

The point of a system like this is not that anyone should use it. It is what it settles about the system everybody does use.

If the octave were an axiom — a fact about pitch perception independent of what is being listened to — then no change of spectrum could dislodge it, and the tritave scale would be a curiosity with no bearing on anything. If it is a consequence of harmonic spectra, then changing the spectrum should move it, and the prediction is testable.

The clarinet’s own curve over an ordinary octave already shows the effect in miniature: an interval that is consonant for a string is not consonant for a clarinet, for a reason that can be read off the partial list. Removing the even partials entirely takes that from a modification to a replacement.

So consonance is a fact about spectra rather than about small integers as such, and octave equivalence is a fact about spectra too. The small integers keep turning up because strings and open pipes have harmonic spectra, and strings and open pipes are what most instruments are.

Somebody built the instruments #

The system would be a thought experiment if nobody had played it, and several people have, which supplies the only evidence available about how it actually sounds.

The instruments are the constraint. A Bohlen–Pierce clarinet — built to play the thirteen steps and to have the closed-pipe spectrum the argument requires — was made by Stephen Fox in Toronto from 2006 onward, and it is the reason there is a repertoire at all rather than only a literature. Earlier work, including Pierce’s own, was synthesised.

What players and listeners report is consistent and mixed. The 3:5:7 chord is described as stable and restful by people who have spent time with it, and as thin or hollow by people who have not — which is what would be expected if the consonance is real and the familiarity is not. Elaine Walker, Charles Carpenter and Todd Harrop have written the bulk of the music, and the honest summary of the corpus is that it is small enough that no general claim about the system’s expressive range is supportable.

The negative result is worth as much as the positive one. A Bohlen–Pierce piece played on ordinary instruments, or synthesised with a full harmonic spectrum, sounds simply wrong — not exotic, wrong — and that is the prediction the whole argument makes. If the scale sounded equally good on a piano the spectral account would be in trouble.

Where the argument stops #

Three things this account does not establish.

It does not establish that the octave is only a spectral effect. Octave equivalence appears in behavioural tests with pure tones, where there are no partials to coincide, and it appears in the vocal ranges of men and women, which differ by roughly an octave and make octave-doubled singing the default. Those are independent sources of the same convention, and they would survive a change of spectrum.

It does not establish that the Bohlen–Pierce scale is learnable in the sense that matters. The categories a listener sorts intervals into are acquired, and a listener with a lifetime of twelve-tone exposure hears this system through those categories, badly. The experiments that would settle whether its consonances are as available as the ordinary ones require listeners who do not exist.

And it does not establish that a real clarinet is odd enough. The even partials of a clarinet are small rather than zero — the bore is not perfectly cylindrical, the bell radiates, and the reed’s motion is not symmetric — so a real clarinet retains a weak preference for the octave. The pure case is a synthesised one, which is the standard position for an argument of this kind: the model is exactly right about an object that does not exist, and approximately right about the ones that do.

What the picture cannot show #

A dissonance curve is about two notes, and it is a curve of roughness rather than of preference — a distinction the fourth rung of this site’s consonance ladder makes at length, and one that bites here: a spectrum’s wells say where a scale could be built without beating, not that anybody would want to build one there. The other assumption worth naming is that the octave is exactly 2:1 for a listener, which it turns out not to be. The tritave scale’s claim is about a whole system — chords, progressions, a sense of home — and none of that is visible in a curve with two tones in it. Whether 3:5:7 functions as a resting place in a piece, rather than merely scoring well in a computation, is a question about music that has barely been written.

The curve also cannot show the thing every listener notices first, which is that these intervals sound unfamiliar rather than consonant. That reaction is real data about listeners and no data at all about the intervals, and separating the two is the hardest methodological problem in the whole subject.