What is 78 ÷ 3?
You can solve it directly: 78 ÷ 3 = 26.
You can also break it apart: 78 ÷ 3 = (30 + 30 + 18) ÷ 3 = 10 + 10 + 6 = 26
Today this is often sold as a “modern” way to teach arithmetic. It’s not modern at all, this is just mental math. It’s the way people have always calculated in the street, in the shop, in the kitchen, at the table. Break the problem into friendly pieces and get the answer.
I disagree with this method being taught to kids early on. This is the kind of method the brain should discover after it has learned arithmetic. It should not replace arithmetic. Teach the child long division. Teach the child multiplication. Teach the child the structure. Then the shortcuts will come naturally. Do not give children somebody else’s clever solution, give them the machine that produces cleverness:
3 goes into 7 two times. That gives 6. Subtract it and you have 1 left. Bring down the 8 and you have 18. Then 3 goes into 18 six times. That gives 18. Subtract it and you have 0.
The answer is 26. No need to remember special cases or to try and find magic numbers that make it easy. Sometimes those don’t exist. And sometimes the result has decimal digits. The same machine works for any other division problem. The shortcut only works when the numbers are friendly. The algorithm works when they are not.
That is the difference between cleverness and structure.
In Byzantine music much of the traditional system is mental math dressed as sacred science:
- “That is how I heard the interval.”
- “This is the true soft chromatic.”
- “This is the correct vocal style.”
- “Take honey for the voice.”
Harmless when spoken by a grandpa, but extremely harmful when spoken by a teacher in class.
Traditional Byzantine music teachers focus on memory, habit, folklore, taste, and ego. It only gets worse, since this mindset is now rebranded as the contemporary teaching method of the same traditional practice. It’s the same: It still produces singers who imitate, teachers who intimidate, and theorists who cannot build.
A man saying “this is how I heard it” has not explained an interval. Reporting a memory is fine as an anecdote but never acceptable as a method. A teacher saying “this is the correct style” has not explained technique. By playing “ownership” games he neutralizes the student’s curiosity and motivation and just forces something arbitrary, mystical and custom. A school saying “these are the true intervals” has not solved the problem but hidden it. And how do you know those were the exact intervals? Chrysanthos himself didn’t!
This has consequences:
- It produces musicians who can repeat but cannot reason.
- It produces chanters who can police style but cannot explain structure.
- It produces theorists who decorate confusion with terminology.
- It produces shopkeepers, rather than scientists. Or, to use their own term, “lower clergy”.
Musical shopkeepers keep receipts, repeat prices, guard the counter, and -most of the time- give correct change to customers. Sometimes they don’t. They’ve mastered a habit, they remember the prices, they mechanically produce intervals that are a hit or a miss, hoping they will sound decent.
But ask them “what is diatonic” and they will freeze. Ask them to harmonize, and they panic. Ask them to explain a cadence, and they reach for authority: “I’ll have to check that and I will get back to you”.
Mastering Byzantine Polyphony respects human intelligence enough to not go down that dead end. In monophonic chant, vagueness can survive for centuries. A singer can slide, bend, ornament, delay, and hide behind style. In polyphony, vagueness collapses immediately.
Mastering Byzantine Polyphony teaches a system which makes people less dependent on memory and more dependent on method. Just like the long division. It gives the student the mechanism so the student can analyze, harmonize, compose, correct, compare, and disagree intelligently.
Instead of preserving confusion, we rescue the music from the confusion. Instead of creating shopkeepers, we create scientists.