Intervals and chords

A tuned close seventh chord low on the keyboard flutters at one rate

A dominant seventh tuned 4 : 5 : 6 : 7 in close position repeats exactly at a frequency two octaves below its bass, and below B2 that frequency is too slow to be a pitch. The question left was whether the repeat is heard as a flutter. In every auditory filter the chord reaches, its envelope pulses at that one rate, forty times over the threshold for detecting a fluctuation in the filter where it is strongest. The tempered chord fluctuates too, but a third as deeply and at rates that change from filter to filter. Tuning adds something here rather than sharpening it, and it adds it only within two cents of 7 : 4.

15 min read 6 figures The ear is not a microphone Small whole numbers The limits of the listener

Assumes: A tempered seventh chord already has the tuned one's low note · A tuned seventh chord does not beat; it buzzes

A tempered seventh chord already has the tuned one’s low note asked whether a dominant seventh tuned to the harmonic series has a note below it that the tempered chord lacks, and found that in the range of pitch it does not: the tempered chord’s autocorrelation peaks at the same place, a few cents sharp and a little lower. Tuning sharpened a low note the chord already implied. But in close position — C, E, G and B♭ within one octave, tuned 4 : 5 : 6 : 7 — the series fundamental is two octaves below the bass, and when the bass is below about 120 hertz that frequency is under the 30-hertz floor where a periodic sound stops carrying a pitch. There, the essay found, the tuned chord still repeats exactly, but at a rate rather than a note. It ended on whether that slow repeat is heard.

A tuned seventh chord does not beat; it buzzes had given the answer a name before the question was asked: a tuned chord’s partials do not beat against each other, but the whole chord fluctuates together at the one frequency they share. This essay computes the fluctuation, and it is the first result in these chords where tuning adds a quality the tempered chord does not have, rather than refining one it has.

Fig. 1 The dominant seventh in close position over C2, tuned 4 : 5 : 6 : 7 and tempered: the modulation spectrum of the envelope in the auditory filter centred at 93 Hz, as a modulation index at each rate up to 40 Hz, against the published threshold for detecting amplitude modulation. The tuned chord fluctuates at 16.4 Hz, its series fundamental, with an index of 1.33 — 42 times the threshold — and all of its fluctuation is at that rate and its multiples. The tempered chord’s strongest fluctuation is at 15.4 Hz with an index of 0.34, and less than half of its fluctuation is at one rate and its multiples.

Where a flutter lives

A fluctuation is not in the chord’s spectrum. It is in what each auditory filter makes of the spectrum. A filter centred at some frequency passes the chord’s partials near that frequency, weighted by its rounded-exponential shape, and when two or more partials pass together the filter’s output rises and falls at the differences between them. That rise and fall is the envelope, and a steady fluctuation of the envelope at a few to a few dozen times a second is heard as flutter, or at the faster end as roughness. It is the same object the beat a tuner can actually use priced for two partials, extended to a filter full of them.

For the tuned chord every partial is a whole multiple of the series fundamental, so every difference between two partials is too, and every filter’s envelope is exactly periodic at that fundamental. For a tempered chord the partials are not multiples of any audible frequency, and each filter’s envelope wanders at whatever combination of differences its partials produce. So the computation is direct: for filters spaced one equivalent rectangular bandwidth apart from 60 hertz to 1.5 kilohertz, form the envelope of the chord’s partials passed through each, take its modulation spectrum, and read the depth at each rate as a modulation index — twice the spectral line over the mean. The strings are the struck strings of these essays, sixteen partials a note at 80 decibels. The detection threshold is the published temporal modulation transfer function, an index of 0.03 at slow rates rising above 50 hertz, the same curve the beat essays used.

The first figure is the filter centred at 93 hertz, over C2, where the tuned chord’s fluctuation stands furthest above threshold. That filter passes chiefly the fifth and sixth members of the series — the fundamentals of E2 and G2 — with C2 and B♭2 on its skirts, and they are unresolved, several harmonics in one filter, exactly the situation which harmonics carry the pitch described at the bottom of the bass guitar’s range. Their envelope pulses once per period of the series fundamental, 61.2 milliseconds, with an index of 1.33. The tempered chord’s partials in the same filter produce an envelope fluctuating at 15.4 hertz, a third as deep, with the rest of its fluctuation spread over other rates.

Every filter, one rate

Fig. 2 The close dominant seventh over C2, tuned and tempered: in each auditory filter from 60 Hz to 1.5 kHz, how far the strongest fluctuation of the envelope stands above its detection threshold, labelled with its rate. The tuned chord fluctuates 19 to 28 dB above threshold in every filter, always at 16.4 Hz or 32.7 Hz. The tempered chord fluctuates 13 to 22 dB above, at rates that change from filter to filter — 16.8, 31.2, 35.7, 33.2, 11.9 Hz.

Across the whole range the tuned chord fluctuates well above threshold, by 19 to 28 decibels, and always at 16.4 hertz or at twice it. The higher filters pass higher partials of several notes, and the partials that fall together there are two series members apart as often as one, so the doubled rate appears; but no filter fluctuates at any rate that is not a multiple of the fundamental. The tuned chord’s flutter is the same flutter everywhere it is heard.

The tempered chord fluctuates above threshold too, by 13 to 22 decibels, which is worth saying plainly: a tempered close seventh chord low on a piano is not a steady sound. But its rates are 16.8 hertz in the lowest filter, 31.2 in one near 170 hertz, 35.7 in one near 320, 11.9 near 1.2 kilohertz. Each filter’s partials make their own set of differences, and no two filters agree on a rate.

That is the difference a listener could hear. A fluctuation at one rate across the whole sound is one flutter, a property of the chord. Fluctuations at different rates in different frequency regions are a texture of several slightly different flutters, which is closer to a roughness than to a pulse. Roughness can be computed as the sum of pairwise fluctuations, and in that sum the two chords differ little, as the earlier essays found; the regularity is what differs.

Fig. 3 The close dominant seventh over eight basses from C2 to C3, tuned and tempered: the depth of the strongest fluctuation in the filter where it stands furthest above threshold (solid) and the share of the envelope’s fluctuation at one rate and its multiples (dashed). The tuned chord flutters with an index of 1.06 to 1.33 at every bass up to B2, entirely at its series fundamental; the tempered chord at 0.44 to 0.56, with 57 to 86 per cent at one rate. At C3 the series fundamental is 32.7 Hz, inside the range of pitch, and the tuned chord’s depth falls to 0.74.

Across the bass the tuned chord’s flutter is steady in depth and exact in rate. From C2 to B2 its strongest fluctuation has an index between 1.06 and 1.33, always at the series fundamental, which rises from 16.4 to 30.9 hertz, and all of its fluctuation is at that rate and its multiples. The tempered chord’s strongest fluctuation stays between 0.44 and 0.56, and between 57 and 86 per cent of it is at one rate.

At C3 the series fundamental reaches 32.7 hertz, and two things happen together. The flutter’s depth falls to 0.74, because the partials two series members apart are now more often in separate filters, and the rate enters the range where a periodicity is heard as a pitch, which is where the earlier essay found the tuned chord’s low note — and the tempered chord’s blurred version of it. The flutter and the pitch are the same periodicity on either side of the 30-hertz floor. Below it the tuned chord has something the tempered chord lacks; above it, both have the same thing to different degrees.

Why close position, and how deep is deep

The flutter belongs to close position for the reason the earlier essay’s last figure gave. A close 4 : 5 : 6 : 7 chord’s series fundamental is two octaves below its bass; an open 2 : 5 : 6 : 7 chord’s is one octave below. Over C2 the open chord’s periodicity is at 32.7 hertz, and its envelope does pulse there — with an index of 0.94, all at that rate — but 32.7 hertz is already a rate the ear hears as a low pitch rather than as a pulsing. Over C3 the open chord’s periodicity is at 65 hertz, far above anything heard as flutter. Only close position, low on the keyboard, puts a tuned chord’s single periodicity into the range of rates between a slow beat and a pitch, and only there is the periodicity heard as a movement of the sound rather than as a note.

Close position also crowds the partials. Four notes within an octave put their low partials only a few hertz apart in absolute terms, and the auditory filters at the bottom of the keyboard are wide in semitones, so a single filter passes several of them at once. That is what makes the envelope a deep pulse rather than a gentle ripple: the more partials a filter passes at once, the more sharply their sum peaks once a period and falls between.

The depths are large on the scale that matters. At 16 hertz the detection threshold is an index of about 0.03, so the tuned chord’s 1.33 is 42 times the threshold and the tempered chord’s 0.34 is 11 times it. Both are plainly audible as fluctuation; the tuned one is about 12 decibels deeper, which is the difference between a modulation a listener notices and one a listener cannot ignore. The beats between two partials that the earlier essays counted have an index of at most one, when the two partials are equal, and a few tenths when one is much weaker than the other. The tuned close chord’s flutter is deeper than any beat between two partials can be, because it is the pulse of many partials in step.

Only within two cents

The earlier essay moved the seventh from tempered to 7 : 4 with the third and fifth tuned and found the chord’s periodicity improving smoothly at every step, within a hundredth of exact over ten cents around the target. The flutter behaves quite differently.

Fig. 4 The close dominant seventh over C2, third and fifth tuned, with the seventh moved from tempered to 40 cents flat: the depth of the strongest fluctuation and its regularity, measured over two seconds. The depth sits between 0.67 and 0.74 from tempered to 28 cents flat, rises to 1.12 at 30, 1.33 at 7 : 4 and 1.20 at 32, and is back to 0.74 by 35. The dial moves the seventh.

With the seventh tempered, or anywhere from tempered to 28 cents flat, the flutter’s depth sits between 0.67 and 0.74 — a little deeper than the fully tempered chord, because the tuned third and fifth already share the series. It rises to 1.12 at 30 cents flat, 1.33 at the seventh partial and 1.20 a cent past it, and has fallen back to 0.74 by 35. The deep, single flutter exists only within about two cents of 7 : 4. Six cents short of the target the chord is as periodic, by the earlier essay’s measure, as it will ever usefully be, and its flutter is no deeper than a tempered seventh’s.

The reason is time. The periodicity is read over a few tens of milliseconds, the delay at which the sound resembles itself, and a few cents of mistuning is a fraction of a cycle over that delay. The flutter is a regularity sustained over seconds: a seventh a few cents off makes its partials drift against the others’ by a few tenths of a hertz, and over two seconds that is enough to smear the envelope’s single line into several. The measure here uses a two-second window, and a listener integrating over a longer stretch would find the peak narrower still. This is the timing the pitch that does not wobble found for a vibrato’s smear of periodicity, applied the other way round: what the ear integrates over a long window rewards exactness far more than what it reads in a short one.

So the flutter is a tuning cue, and a sharper one than the beating. Beats are arithmetic: the slow beats of an open seventh chord tell a tuner how far off the seventh is and fall to nothing only at the target. The flutter of a close chord low on the keyboard does the converse. It is absent until the seventh is nearly exact, and then it appears, as a single pulse at a rate set by the bass.

What the tempered chord does instead

Fig. 5 The envelope in the filter centred at 93 Hz over the first half-second of the close dominant seventh over C2: tuned above the line, tempered below it. The tuned chord’s envelope is one pulse every 61.2 ms, repeating exactly; the tempered chord’s rises and falls at a similar rate but no two cycles are alike.

Drawn as a waveform, the difference is unmistakable. In the filter at 93 hertz the tuned chord’s envelope is a train of identical pulses, one per series period, each almost silent between peaks — which is why its modulation index exceeds one, since a pulse train’s fundamental line is larger than its mean. The tempered chord’s envelope rises and falls at about the same rate, but its peaks differ in height and spacing, and it never falls near silence. It is the envelope of three partials slightly out of step, and it is what a listener hears as a tempered chord’s shimmer in the bass.

That shimmer is worth distinguishing from the pulse, because it is the sound a keyboard player hears every time and the pulse is one most keyboard players never hear. A close-position seventh chord low on a piano, or a low close chord on an organ, has a restlessness that players often attribute to the lower register itself — “muddy” is the usual word. The arithmetic separates two things that word covers: the tempered chord’s irregular fluctuation, which is real and well above threshold, and the tuned chord’s regular one, which is deeper and would be heard as a single pulsing rather than as mud.

For anyone tuning such a chord by ear, the two sounds make a usable signal. With the third and fifth set, lowering the seventh changes the texture of the tempered shimmer very little until the last few cents, and then the chord’s bottom locks into one pulse at a rate that is a fixed fraction of the bass: a quarter of it, for a close chord. A choir holding a low close seventh chord, or an organist setting a just-tuned rank against itself, has in that pulse a target that is absent everywhere except at the target — the opposite of a beat, which is present everywhere except there — and the two cues together bracket the right tuning from both sides.

A struck string keeps it

Fig. 6 The close dominant seventh over C2 through its first three seconds: the depth of the strongest fluctuation in the filter where it is strongest, tuned and tempered. The tuned chord’s index falls only from 1.33 to 1.24; the tempered chord’s stays between 0.37 and 0.50, its rate wandering.

The flutter does not fade with the strike. The trace of the beating found an open seventh chord’s slow beats concentrated in its first half-second, because they came from high partials that die first. The flutter lives in the filters near the chord’s own low notes, whose partials decay slowly, so the tuned chord’s index falls only from 1.33 to 1.24 over three seconds. The tempered chord’s fluctuation also persists, between 0.37 and 0.50, with its strongest rate moving as partials drop out.

So on a struck instrument a tuned close chord low on the keyboard would pulse for as long as it sounds, and a sustained instrument would pulse indefinitely.

Whose chords

The chord is the close-position dominant seventh, and the claim is about it below B2 — the register of a piano’s left hand, an organ’s pedal and manual bass, a low brass or string choir, and a barbershop quartet’s lowest voicings, with the bass near the bottom of the voice. The claim is not that such chords are tuned to the series: on a piano they cannot be, and the tempered chord’s irregular fluctuation is what a piano plays. It is that where they are tuned — by voices, by an organ tuned in just intonation, by an electronic instrument — a close seventh chord low enough acquires a single regular flutter, and that the flutter is a cue for tuning sharper than beating, because it appears only at the target.

What the picture cannot show

Whether a flutter at one rate is heard as one thing. The model shows one rate in every filter for the tuned chord and different rates for the tempered one. That a listener groups fluctuations at one rate across filters into a single percept is established for simple modulated tones; whether it holds for a chord’s filters, each passing different partials, is an inference.

The window. The flutter’s regularity is measured over two seconds. How long a listener integrates a fluctuation to hear it as regular is not established here, and the narrowness of the peak around 7 : 4 depends on it.

The envelope’s phase. The partials are given the phases a struck string gives them, all starting together. A voice or an organ pipe starts each partial at its own phase, which changes the shape of the pulse and so its index, though not its rate or its regularity.

Still open: whether the flutter binds the chord

A fluctuation shared by every filter at one rate is the cue the root an ear supplies and the essays on auditory scene analysis would treat as evidence that the filters hear one source, since independent sources rarely fluctuate in step. If so, a tuned close chord low on the keyboard should be heard as more fused — harder to hear out its individual notes — than the tempered chord, whose filters fluctuate independently. The computation is a comodulation measure: the correlation between filters’ envelopes, which the tuned chord makes perfect, set against how many of the chord’s notes a listener can hear out, which four notes cannot get below the seventh partial and its neighbours have never asked of a chord.