Calculation of Modes

Gregg Gibson said:
> > > To give some idea of just how immense this universe is - and remember
> > > these are _real_, usable resources, not purely theoretical, aurally
> > > imperceptible variations - compare the number of modes (I here use the
> > > term loosely merely to indicate a collection of seven notes used in a
> > > melody) consisting of 12 notes taken 7 at a time:
> > >
> > > nCr where n = 12 & r = 7
> > > = 792
> > > with the number of modes consisting of 19 notes taken 7 at a time:
> > > nCr where n = 19 & r = 7
> > > = 50,388

Graham Breed replied:
> > Really, you should define modes relative to a tonic, so the numbers
> > get smaller. With 12, you're choosing 6 from 11, which gives 462
> > modes. 19C6 is 12376. The contrast is still striking. Including
> > all the transpositions, you get 462*12=5544 and 12376*19=235144.

Gregg Gibson replied:
> Thanks very much for your correction regarding the method of finding
> numbers of 12-tone versus 19-tone modes. You are quite right that the
> tonic must be omitted from the calculation. The corrected numbers would
> be nCr (n=11, r=6) = 462 and nCr (n=18, r=6) = 18564 (not 12376, you
> must have used n=17), or a factor of about 40 to 1. Since I am concerned
> here with modes, not keys, the transpositions I neglect.

I would like to thank Graham Breed again for pointing out to me that in
the calculation of the numbers of modes in a given system, the tonic
must be excluded from the algorithm. This is because, I take it, the
tonic is by definition present in each mode, and so does not vary. This
is one of those obvious points that it is easy to overlook. I have
wondered off and on for years why my informal estimates of the numbers
of 12-tone equal modes were in the vicinity of 400 - 500, not 792 as my
formal calculations suggested. Now I know!


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From: Gregg Gibson
Subject: 22-tone equal & chromaticism
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