puzzle solution

The first series is the number of different rotational and inversional
equivalent triads in 3-tET, 4-tET, etc. The second one is the number
of different triads without the inversional equivalence.
12-tET is the only one which has the same number of notes as number of
triads (with inversional equivalence). Peter Schat's compositional system
is based on this fact, see his book "De Toonklok", Meulenhoff/Landshoff,
Amsterdam, 1984. There is an English translation now but I don't know
the publisher; its title is probably "The Tone Clock".

I have found formulas for the functions, given here. The division is
integer division, for example: 5 / 3 gives 1 as result. N is the number
of notes per octave. It would be possible to do without the
if-statements, using integer roundoff of some fractional multiplication,
but that would go at the expense of clarity.

function Triad_Count (N : Positive) return Natural is
Result : Natural;
begin
Result := (((N - 3) mod 6) + 3 * ((N - 3) / 6)) * ((N + 3) / 6);
if (N + 3) mod 6 = 0 then
Result := Result + 1;
end if;
return Result;
end Triad_Count;

function Second_Triad_Count (N : Positive) return Natural is
Result : Natural;
begin
Result := (N mod 3) * (N / 3) + 3 * (N / 3 - 1) * (N / 3) / 2;
if N mod 3 = 0 then
Result := Result + 1;
end if;
return Result;
end Second_Triad_Count;

Here are some results:
1: 0 0
2: 0 0
3: 1 1
4: 1 1
5: 2 2
6: 3 4
7: 4 5
8: 5 7
9: 7 10
10: 8 12
11: 10 15
12: 12 19
13: 14 22
14: 16 26
15: 19 31
16: 21 35
17: 24 40
18: 27 46
19: 30 51
20: 33 57
21: 37 64
22: 40 70
23: 44 77
24: 48 85
25: 52 92
26: 56 100
27: 61 109
28: 65 117
29: 70 126
30: 75 136
31: 80 145
32: 85 155
33: 91 166
34: 96 176

Manuel Op de Coul coul@ezh.nl

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By the way, be careful of one thing: I found out after I grabbed a free copy
of the MIDI-file spec, that the words of the spec itself is copyrighted,
although although its author has placed the ideas it presents in the public
domain. So you might want to be cautious of the possibility that the same might
hold for MIDI tuning dump, sample dump, and such.


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