Virtual pitch — where it appears
Named by 9 essays across 3 fields — each of them below, with the objects they name alongside it.
The root an ear supplies
A major triad's notes fit 4:5:6 with nothing missing and a fundamental two octaves below the bass. A minor triad's fit two different series with two different answers a major sixth apart, and the model cannot choose between them. The ambiguity theorists argued about for two centuries is a computable quantity, and the spectrum invented to remove it is one no object produces.
A stack that is not thirds
Build every chord that repeats a single interval and score them all. Eight of the eleven three-note stacks are smoother than the major triad; the four-note stack of fourths is smoother than a dominant seventh by a third and has no fundamental any matcher can find. Two absences, both computable, and both are exactly what the chord was adopted for.
The pitch that moves the wrong distance
Take three partials two hundred hertz apart and move every one of them up by forty. The spacing has not changed, so anything reading the pitch off how often the waveform repeats must give the same answer as before. The pitch moves to 204 — the shift divided by the harmonic number — which is what a harmonic template predicts and what listeners report. Push the shift to a hundred and a second reading overtakes the first, so there are two pitches and neither is the spacing. This is the measurement that closes the question, and it closes it by ruling out one mechanism rather than by choosing between the two that are left.
The note that sounds twice
A triad has three notes and a four-part texture has four voices, so one note is doubled — and the voicing model used here leaves the choice free because the rules have an opinion about it. Asked properly, the arithmetic agrees with the treatises for the first time in nine essays: root, then fifth, then third. For a minor triad it does not agree, and for a symmetric chord it correctly has nothing to say.
A bell is never minor alone
A tuned bell carries two thirds: the tierce, a minor third above the prime, which is why a bell is called minor, and the deciem, a major third above the nominal. Run the masking count across every register and every stroke and the minor one is never delivered without the major one. Struck softly a bell delivers both; struck harder, only the major; and below C2 the tierce is never delivered at all. The deciem survives because it is masked by one mode barely louder than itself, the tierce by two that outweigh it.
A bell can be made minor by the third nobody tunes
A low bell struck hard delivers its major third, the deciem, and loses its minor one, the tierce, to masking. A founder could flatten the deciem from a major to a minor third above the nominal, which puts it exactly an octave above the tierce. The masking count says that move costs the deciem only one or two decibels of stroke at the top of its window, so the flattened deciem still outlasts the tierce in every register. A bell tuned that way reads minor in every cell of register and stroke that delivers a third at all. In-between tunings read as neutral thirds, so the move only works if it goes all the way.
The major-third bell has no mirror problem
A minor bell struck hard loses its tierce to masking and delivers its major deciem instead. Raise the tierce to a major third and the two thirds agree — but the raised tierce sits closer to the quint, and it was fair to fear that the quint would bury it as the prime buried the minor one. It does not. The prime is what masks a tierce, and moving away from it gains four to twelve decibels of stroke while the quint stays nine decibels too weak to matter. Tuned to agree, the major bell delivers its colour in 141 of 190 cells; the minor bell, made consistent, in 131.
The raised tierce points the bell at its own note
A bell's four lowest modes — hum, prime, tierce and quint — can imply a pitch of their own. With a major tierce they are the second, fourth, fifth and sixth harmonics of the prime two octaves down: the strike note's own name, with one empty slot. It was the minor tierce that pointed elsewhere all along. Its lower modes are the fifth, tenth, twelfth and fifteenth harmonics of the note a major third below the strike, and in 82 of 190 struck bells that is the series the delivered modes fit. Raising the tierce removes a competitor rather than adding one.
A ring of minor bells rings a second scale a major third below
A minor-tierce bell's four lowest modes, read as a residue, imply a note a major third below the bell's strike note. Put eight such bells in a ring tuned to a major scale and the implied notes are that scale moved down a major third — a ring in C carries the scale of A♭ major underneath it. Only three of the eight land on notes the ring rings, from the mediant, the submediant and the leading note; the tonic, the octave, the supertonic, the subdominant and the dominant all imply notes the ring never sounds, each a semitone from two that it does. Above a tenor on C3 the lower modes deliver the implied note at nearly every stroke. A reading that weights the whole spectrum still hears none of it.
Named alongside it
The objects these essays reach for when they reach for this one.
MaskingRegisterSpreading functionResidue pitchDynamicsTriadChord qualityRoughnessVoicingAutocorrelationDoublingFrequency ratio