"...next question: how many (unique six-tone sets) are there?"

Would it be cheating if I ran over and got my Forte book out & looked in the index? I know there's not **that **many; I think Schoenberg and his crew used up all of the good ones. Ha ha; Schoenberg was interested in hexads which had certain constant characteristics under inversion. One of these he called "The Miracle Row."

I'm going to fly by the seat of my pants & discuss fifths, cycles, and the "12 limit." As you may have guessed, when intervals other than the fifth are projected, they tend to "poop out" in a short time. The next interval Hanson discusses is the major third; it yields a cycle of only three notes before re-connecting with itself at the octave. There are four adjacent versions of this, hence the four augmented chords (3x4=12).

But we forget, don't we? that the "12 cycle" we take for granted is itself imperfect, the stacked circle of 3/2 fifths not really perfectly "reconnecting" with itself after 12 notes have been generated. Thus, Pythagorus coined the term "close enough for rock & roll."

What if we did the same thing with the major third, and used a stack of 'just' major thirds (5/4) to generate our cycle? Where will it stop, this Pythagoran Wheel of Fortune?

I decided to do it with cents. We know that an octave is 1200 cents, and that a 'just' major third is 14 cents flat of our equal-tempered third, which is 400 cents, making the 'just' M3rd about 386 cents (386.3 is closer).

So, how many times do we have to "stack" this 5/4 major third before it (apparently) comes back to its starting point? We're shooting for some multiple of 1200 cents, since that's the octave.

Okay, 1200 + 1200 is 2400, + 1200 is 3600, 4800, 6000, 7200, 9600, 10,800, 12,000, 13,200, 14,400, 15600, 16,800, 18,000, 19,200, 20,400, 21,600, 22,800, 24,000, 25200, 26400, 27,600, 28,800, 30,000, 31,200, 32,400, **33,600,** 34,800, 36000,

Now, we start adding up the 5/4s, or 386.3s. That gives us 386.3, 772.6, 1158.9, 1545.2, 1931.5, 2317.8, 2704.1, 3090.4, 3476.7, 3863.0, 4249.3, 4635.6, 5021.9, 5408.2, 5794.5, 6180.8, 6567.1, 6953.4, 7339.7, 7726.0, 8112.3, 8498.6, 8884.9, 9271.2, 9657.5, 10043.8, 10430.1, 10816.4, 11202.7, 11589.0, 11975.3, 12361.6, 12747.9, 13134.2, 13520.5, 13906.8, 14293.1, 14679.4, 15065.7, 15452.0, 15838.3, 16224.6, 16610.9, 16997.2, 17383.5, 17769.98, 18156.1, 18542.4, 18928.7, 19315.0, 19701.3, 20087.6, 20473.9, 20860.2, 21246.5, 21632.8, 22019.1, 22405.4, 22791.7, 23178.0, 23564.3, 23950.6, 24336.9, 24723.2, 25109.5, 25495.8, 25882.1, 26268.4, 26654.7, 27041.0, 27427.3, 27813.6, 28199.9, 28586.2, 28972.5, 29358.8, 29745.1, 30131.4, 30517.7, 30904.0, 31290.3, 31676.6, 32062.9, 32449.2, 32835.5, 33221.8, 33608.1, **Wait! Bingo! 33,608.1 is almost equal to 33, 600: only 8.1 cents off!** I say, let's go for it, before our calculators run out of battery power.

So how many cycles of 5/4s is that? Let's see...33,608.1 divided by 386.3 is 87!

So our Pythagoran "major third" scale is 87 notes per octave!