Toward a Better Music Theory (2013)
The main problem I see with (occidental) &quot;music theory&quot; is its historical baggage. Most of it is useful, beautiful by its own merits but, specially if you have a scientific&#x2F;logic way of looking things, much of its structure and nomenclature really gets in the way. After many years of doing false starts I sat one day, started from scratch, looked at just the facts (= math), and then everything fell together:<p>- The Pythagoreans dealt for centuries with the known conundrum of figuring out irrational numbers. I had a computer so I just run a script for measuring the total error with different bisecting rational quantizations (up to 256 IIRC) of note frequencies in the equal-tempered scale within one octave (2^(1&#x2F;N) splits). Oh, peaks at 12, 19 at 24 on the low side of the counter. That is why they settled on 12 notes, middle eastern has 24 (just double the resolution) and why 19 also sounds good [1]. 12 seems like the minimum acceptable then. Ok, 10 minutes, move on.<p>- Since we know we can only approximate to irrationals and our brain tries to makes things even, moving through scales with increasing intervals (2^(n&#x2F;12)) will normally accumulate &quot;tension&quot;. Half notes sound spooky. Whole notes are a bit better. Too bad sound perception is logarithmic (= tempered scale) and has non-linear compression on both amplitude and frequency (Fletcher-Munson, etc). Double the frequency is an octave. Perfect fit, brain happy (2^(12&#x2F;12)). Try to split an octave in half? Again, fitting a peg in the only wrong hole available when using half-note resolution.  Welcome to accidentals and the Circle of Fifths. &quot;Try&quot;, because of course you can&#x27;t: 2^(7&#x2F;12) = 1.498307.... Close to 150%, but no cigar. Close enough for a crippled monkey brain that can only hear up to 20kHz or so. They even named it &quot;Perfect&quot;. Moving up: F to C, G to D, A to E...<p>- Music would be very dull and inadequate if we cannot shift notes up or down. However our brain still recognizes the patterns. We can even play those patterns at the same time, even on different instruments. So that pattern with the same scale on a G sounds good with the other guy playing C (harmonies). What if the guy playing the C wants to play like the guy on the G? Enter modes... Oh yeah, cool Myxolydian tunes. And minor scale too - don&#x27;t forget that looks THE SAME as an Aeolian.<p>- Your new singer cannot hit that low notes. What about now moving everything up then? Transpose. We have &quot;physical instruments&quot; like a guitar. Easy, just play the same pattern some frets up. Done. Poor piano player who has a &quot;logical instrument&quot; with notes separated nicely in two different kinds based on half notes and that ugly 1.498307 fraction. Good that he is also good at math. So that C scale becomes a D. He just replaces on the fly some white keys with black keys... F# and C#. But wait a sec. Didn&#x27;t the Circle of Fifths also moved F to C too? Does it continue like that for all notes? Yes it does! Transposing has the same structure as modes the other way around! So what if instead of moving up we move down, or play the same pattern in the original C scale? For every raised (sharp) accidental we get it&#x27;s complement lowered (flat) one. Of course, since 7 + 5 = 12! The Circle of Fifth is complete. B to E, A to D, G to C... That G pattern moved down to C? G major scale (aka Ionian) has one accidental: F#. The first flat is Bb. C scale with B lowered to Bb... Yep,  Myxolydian indeed!<p>- We can also use those interval numbers to name chords when we play notes together. We can also name the scales when adding and removing notes, maybe borrowing from other scales. Sounds pentatonic or exotic. The sky is the limit.<p>- Piano player now teases the guitar player. Now it&#x27;s her turn to do the math. Looks like &quot;physical&quot; and &quot;logical&quot; instruments are also complementary. What is easy on one needs thinking on the other, and vice-versa. Win-win.<p>Honestly, that covers like 95% of practical music note theory. Of course there is much more to it (note, rhythm, history, etc). If they have just started explaining it like that without all the mumbo-jumbo it would have made sense since day #1 and I would have saved many years of my life hitting my head in the wall and trying to learn things I forgot within a week. Perhaps the main issue is that not many music teachers are good at math or they would think their students would get scared if they told them it&#x27;s all math underneath?<p>[1] - <a href="https:&#x2F;&#x2F;en.wikipedia.org&#x2F;wiki&#x2F;19_equal_temperament" rel="nofollow">https:&#x2F;&#x2F;en.wikipedia.org&#x2F;wiki&#x2F;19_equal_temperament</a>