I had never heard of this "movement" -- good name for it.

The author does a very nice job deconstructing it. The important thing is relative relationships between frequencies, not absolute frequencies. And there is certainly nothing important about "integer" frequencies, which are only so relative to the arbitrary unit of time we measure in.

He also nicely comments that the potentially interesting thing to look at is alternative scale definitions. When we hear pitches played together, it is pleasing if they are "consonant". This means that they have a ratio of frequencies that is composed of small integers, so that they come back "in sync" with each other. The simplest example is doubling frequency 2:1, so the lower note completes one cycle for every two of the higher -- of course, this is what we call an "octave". Other small ratios (staying within one octave) that might sound musical are 3:2, 4:3, 5:3, 5:4, 6:5, etc. The examples given, respectively, are what we call a perfect fifth (e.g C to G), a perfect fourth (C to F), a sixth (C to A, much less common), a major third (C to E), and a minor third (C to Eb). A perfect fifth and perfect fourth are conjugate, such that used sequentially they make an octave (3/2 * 4/3 = 2). The root note with a perfect fifth plus a major/minor third construct the prototypical major/minor chords bearing the name of the root.

The trick is to try to divide the octave evenly into some number of steps (piano keys) that hits all of these nice integral ratios. It turns out not to be possible. 12 "semitones" turns out to be the smallest number of divisions that works reasonably well, as an approximation. Dividing the octave into N "equally spaced" (note: equally spaced by multiplication, not addition -- our hearing is logarithmic in frequency) means that relative to the (arbitrary) base frequency f_0, each successively higher note f_i in the "octave" above will have the relationship f_i = f_0 * 2^(i/N), that's the base frequency times two to the power of (i/N). For i = 0, you get 2^0 = 1, i.e. f_0 is just the base frequency. For i = N, you get (i/N) = 1, so 2^(i/N) = 2, so f_N is twice f_0 (the octave pitch). If we choose the value N=12, then f_7 is (1.498 ~ 3:2 "fifth"), f_5 is (1.335 ~ 4:3 "fourth"), f_9 is (1.682 ~ 5:3 "sixth"), f_4 is (1.260 ~ 5:4 "major 3rd"), and f_3 is (1.189 ~ 6:5 "minor third"). Another option that works well is N=19, and it has been used historically (and is still I believe) in certain cultures/instances. In any case, this is an approximation, although a few "cents" (hundredths of a semi-tone) deviation can generally not be resolved by humans. If one goes the other way, and tries to maintain only a sequence of idealized fifths, etc., then the octaves get out of sync, or you can't change keys without a new piano, or etc.

Clearly all 12 (or 19) notes don't sound good together simultaneously, so one selects a subset of the 12 that do sound good as a "scale". This is generally chosen to be 8 notes. Counting in semi-tones for a 12-tone system from the root (f_0), the 2nd note is a "Whole step" (two semi-tones) up (f_2), the major third (third note in scale) is another whole step up (f_4), the perfect fourth is a "Half step" (single semi-tone) up (f_5), the perfect fifth goes whole step again (f_7), as do the sixth and seventh notes (f_9 and f_11), and the cycle closes with a final half-step up (f_12) to the octave (eighth scale tone). Starting from a "C" on the piano, one gets this "major: Half/Whole scale pattern (WWHWWWH) by playing only the white keys. It's a Whole step when you skip a black key, of a half step when you don't. Note that these pitches have internal relationships above and beyond their relation to the root. For example, the seventh (f_11) is a major third (two whole steps) above the perfect fifth (f_7), or a perfect fifth (7 semitones) above the major third (f_4). By flatting the major third to a minor third (f_3 instead of f_4) and moving down its relative perfect fourth (5 semitones above) AND relative perfect fifth (7 semitones above) in parallel (had to check that

one), i.e. flatting the root's sixth (f_8 instead of f_9) and the root's seventh (f_10 instead of f_11) we get the minor scale with sequence (WHWWHWW). Alternatively, the original major scale step pattern already contains the minor scale sequence (in a different "key") if we start on the sixth rather than the root. In other words, starting on an "A", the white keys give a minor scale when played through the octave. Also interesting to note that in the basic major chord, e.g. C-E-G (f_0,f_4,f_7), the f_7 (G) is relatively a minor third (3 semitones) above f_4 (E). Likewise, in the basic minor chord, e.g. C-Eb-G (f_0,f_3,f_7), the f_7 (G) is now relatively a major third (4 semitones) above the f_3. So, both "major" and "minor" chords contain one "major" and one "minor" interval.

CV -- thanks for the original post. Hope that was interesting, although perhaps you knew most/all of it very well already!