source file: mills2.txt Date: Mon, 22 Jul 1996 09:17:32 -0700 Subject: puzzle solution From: COUL@ezh.nl (Manuel Op de Coul) 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 Received: from eartha.mills.edu [144.91.3.20] by vbv40.ezh.nl with SMTP-OpenVMS via TCP/IP; Mon, 22 Jul 1996 18:24 +0100 Received: from by eartha.mills.edu via SMTP (940816.SGI.8.6.9/930416.SGI) for id JAA08071; Mon, 22 Jul 1996 09:24:44 -0700 Date: Mon, 22 Jul 1996 09:24:44 -0700 Message-Id: <199607221622.AA12816@interlock.wdni.com> Errors-To: madole@ella.mills.edu Reply-To: tuning@eartha.mills.edu Originator: tuning@eartha.mills.edu Sender: tuning@eartha.mills.edu