Lesson 2
Cadences
Endings are the conventions doing the most work in this music, and they come pre-sorted by strength in every textbook. That ranking is worth taking apart, because almost none of it survives being measured.
Five formulas
A cadence is a stereotyped way of stopping. There are five that get taught, they are defined by which chords and which notes are on top and in the bass, and like the rule in lesson 1 they are exact enough to implement.
They also arrive ranked. The perfect authentic cadence is the full stop, the half cadence does not end anything, and the rest sit somewhere between. That ordering is presented as a property of the cadences. So: which measurable thing is it a property of?
The voicings in both figures are searched rather than chosen. Every arrangement of both chords is generated, the pairs that break the parallel rule or fail to resolve a leading tone are discarded, and the smoothest survivor is what gets measured. Hand-picking examples would have let me pick flattering ones.
Five formulas, and the rank they are given
Everything but the rank is measured. Click one to hear it.
V to I, both in root position, soprano landing on the tonic. Rank is what theory teaching gives it, 1 being most final; a question mark marks the three whose order changes between authors.
Roughness of the final chord across every legal voicing of each formula. They overlap almost completely, which is what rules the measure out.
- Total voice motion
- 8 semitones
- Same for all five
- yes
- Legal voicings
- 10
- Quoted rank
- 1
The smoothest legal version of every one of these costs exactly eight semitones of voice movement. Not approximately: the same number, five times. So voice-leading economy is not what separates a full stop from a comma. Roughness does not do it either, since the ranges overlap. The one measurement that does split the list is whether a leading tone is present to resolve, and that divides them into two groups rather than ranking five.
Three measurements, and what they do
Voice-leading economy does nothing. The smoothest legal version of all five costs exactly eight semitones of total movement. Not close to each other: the same integer, five times. Whatever separates a full stop from a comma, it is not that one takes less effort to arrive at.
Roughness does nothing either. The second panel shows the dissonance of the final chord across every legal voicing of each formula, and the ranges overlap almost completely. A measure whose spread within one cadence type dwarfs the difference between types is not measuring cadence type. What it is mostly measuring is spacing and register.
The leading tone does something. It is present in the authentic cadences and the deceptive one, absent from the plagal and the half. That is a real acoustic difference and it traces back to Harmony lesson 4, where the tritone in the dominant seventh turned out to be the only interval in the key with a forced resolution.
So one measurement earns its place. Note what it buys, though: it splits five cadences into two groups. It does not rank them, and it says nothing whatever about the distinction the textbooks care about most.
Which brings us to one note
The perfect and imperfect authentic cadences are the same progression. V to I, both chords in root position, leading tone resolving, in both. The difference is which chord tone the soprano lands on: the tonic makes it perfect, the third makes it imperfect.
The difference is one note
Same progression, same voice leading cost. One is taught as an ending.
- V to I
- both root position
- motion 8 = 8
- leading tone resolves
- same three chord tones
- soprano C5 vs E4
- Chords
- identical
- Total motion
- 8 and 8
- Leading tone
- resolves in both
- Difference
- one note
Both are V to I in root position, both resolve the leading tone, both cost eight semitones of movement, and both end on the same three notes. The soprano lands on the tonic in one and the third in the other, and that is the whole of it. One is taught as a full stop and the other as a comma. If you can hear the difference, and most listeners raised on this music can, then what you are hearing is a convention you learned rather than an acoustic fact about the chord.
Nothing measurable separates them
Same two chords. Same three pitch classes at the end. Same eight semitones of movement. Leading tone resolving in both. Roughness ranges overlapping. One is taught as an ending and the other as a pause, and every quantity available says they are the same event.
And yet the distinction is not imaginary. Play both. If you were raised on this music you will almost certainly hear the second as unfinished, and composers have relied on that for four hundred years. Something is doing the work; it is just not anything in the sound that can be counted.
What is doing the work is that you know the tune is supposed to end on the tonic. That is a fact about the scale-degree hierarchy of a particular tradition, learned by exposure, and it is exactly the kind of thing this course exists to name. A listener without that exposure has no reason to hear one as more closed than the other.
One thing the search turned up that the textbook description gets slightly wrong. These are usually described as differing only in the top note, and I went looking for a pair that literally did: same bass, tenor and alto, soprano on the tonic in one and the third in the other, both properly voice-led from a root-position dominant. There are none. Holding the lower three still forces the leading tone somewhere it cannot resolve. The chords are identical and the voice leading has to differ, which is a smaller point than the lesson but a real one.
The ranking is not even agreed
There is a tell in the ranking itself. Every source puts the perfect authentic cadence first and the half cadence last, and the three in the middle move around depending on who is writing. The figure marks those three.
A measured quantity does not behave like that. Nobody disagrees about which of two intervals has more shared partials. Disagreement that persists across careful authors for centuries is what a convention looks like from the inside: everyone can hear the effect, and the ordering is a description of a practice rather than a reading off a dial.