Lesson 3
Four voices, many rules
One prohibition took about a fifth of the options. There are half a dozen more, they all apply at once, and the question worth asking is not what each forbids but what is left standing when they run together.
The most constrained moment in the style
The dominant seventh resolving to the tonic is where part-writing is at its most specified, so it is the right place to run the experiment. Take the chord in root position, take every arrangement of it across four voices, take every arrangement of the tonic that follows, and apply the rules.
There are 780 such pairs. That is the whole space in this key and register, not a sample.
One rule from the standard list is missing and its absence is deliberate. Textbooks also forbid the melodic augmented second, and that cannot be detected honestly from MIDI numbers: an augmented second and a minor third are the same three semitones and differ only in spelling. Implementing it would mean inventing spellings and then reporting the consequences of my invention as a result.
All the rules at once
Every resolution of a dominant seventh, filtered until the style is what is left.
Wide bar: arrangements still legal. Narrow bar inside it: those whose tonic chord has all three of its notes. Watch the second bar disappear faster than the first.
- Arrangements
- 780
- Still legal
- 42
- With a complete tonic
- 4
- Removed
- 94.6%
Lesson 1 measured one prohibition and found it took about a fifth of the options. Six of them together take 95%, and what survives is not a style so much as a short list. Notice also that the rules do not divide the space evenly: the overlap rule removes more than four hundred arrangements on its own, and the doubling rule removes nothing at all, because a dominant seventh has four distinct notes for four voices and there is nothing to double.
They do not share the work evenly
From 780 to 42, which removes 94.6% of the options, and the interesting part is how unequally the rules contribute.
The overlap rule alone removes more than four hundred arrangements, far more than the famous prohibition on parallel fifths, which takes 114. The overlap rule is also the one nobody outside a theory class has heard of. Fame and effect are unrelated here.
At the other end, the prohibition on doubling the leading tone removes nothing at all, and it is left in the figure rather than tidied away. A dominant seventh has four distinct notes for four voices, so nothing is doubled and there is nothing for the rule to catch. It is a real rule with no purchase on this particular situation, which is worth seeing: a rule set does not apply evenly to the cases it is taught for.
And then the rules break a rule
Watch the narrow bar inside each bar in that figure. It tracks how many of the surviving arrangements end on a complete tonic triad, with all three of its notes present, and it collapses faster than the wide one.
What survives, and what it is missing
The legal resolutions, and the note most of them had to give up.
- Legal resolutions
- 42
- With a complete tonic
- 4
- This one is missing
- fifth
- Largest leap here
- 5 semitones
Thirty-eight of the forty-two legal resolutions end on a tonic chord with no fifth in it. Switch to the complete ones and there are four, every one of which needs a voice to jump a fifth or more to get there. This is why part-writing textbooks carry an explicit clause allowing an incomplete tonic at exactly this cadence. The clause is usually read as the rules being generous. It is closer to the opposite: the rules are strict enough to force a chord that another rule would call defective, and the exception is there to repair the damage.
Thirty-eight of forty-two have no fifth
Obeying the leading-tone rule and the seventh rule at the same time uses up the voices. The leading tone must go up to the tonic, the seventh must come down to the third, the bass takes the root, and the fourth voice has nowhere useful left to be. What comes out is a tonic chord with its fifth missing: 90% of every legal resolution.
Four resolutions do produce a complete triad, and every one of them needs a voice to leap a fifth or more to manage it. That is the trade, and it is audible in the figure.
Which explains something that looks like generosity in the textbooks. They carry an explicit clause permitting an incomplete tonic at exactly this cadence, and it reads as the rules relaxing. It is closer to the reverse. The rules are strict enough to force a chord that another part of the same tradition calls defective, and the clause is there to repair the contradiction they created.
What a rule set is for
Forty-two options out of 780 is not a description of a style. It is close to a specification of one, and that is what a rule set of this kind does: it narrows a space until what remains is recognisable, and the narrowing is the content.
Lesson 1 argued that one convention can be exact, have a real mechanism, and still be a choice. This lesson adds the part that only shows up in combination. Individually these rules are defensible and mostly mild. Together they are so constraining that they generate their own exception, and no single rule in the list is responsible for that.
It is also why part-writing is teachable at all. A student is not being asked to acquire taste; they are being handed a filter narrow enough that passing it produces something that sounds like the repertoire. Whether that is a good way to learn to write music is a separate question, and this course is not going to settle it.