[r-t] Summary falseness of royal methods

[Richard Smith]

Don Morrison wrote:

> When applying the well known false course head "group" letters to
> royal methods some changes are needed, since the partitioning into
> sets of related false course heads was designed for major. Well known
> is that the in and out of course chunks that are always present
> together in major need not be together in royal; and that some of the
> in course chunks divide into two chunks which no longer have to appear
> together.

Of the 25 tenors-together falseness groups (A-U minus J and
Q, together with a-f), L, P and U (in-course), and K, N and a
(out-of-course) each split into two chunks.

L splits:  126543; 136245, 142563
P splits:  154326, 164352; 156342, 164523
U splits:  134256, 135426, 142356, 143652, 152436, 163254; 135264, 142635

K splits:  134526, 146352, 152346, 164253; 154362, 164325
N splits:  153264, 162435, 135462, 143625; 152634, 165234, 135624, 145632
a splits:  123465; 123546, 126453


> Is it also true that some of the chunks that in major can appear
> separately, in royal will always appear together? For example, is it
> true that in royal the in course B (24365) and D (32546, 46253) chunks
> will always appear together? If so, what other bits, potentially
> separate in major, coalesce in royal?

This is true, and it is the only (tenors-together) example
of two groups coalescing in this manner.

> I guess another way to ask this, is in royal there really is a
> different partitioning into false course head groups than in major.
> Closely related, but different. What is it?

There are 40 tenors-together falseness groups, and another
646 solely splits tenors (i.e. equivalent to X, Y, Z on
eight).

Of the 25 major groups, 15 are already single-parity
(if you ignore split tenors fchs): A, G, I, L, M, P, R, S,
U, a, b, c, d, e, f.  This leaves only ten which split
through parity: B, C, D, E, F, H, K, N, O, T.  The six
further splits noted above occur, and the in-course
coalescence of B and D reduces the number by one.

   25 + 10 + 6 - 1 = 40

> And what happens at higher stages? It appears the in course B and D
> are no longer coalesced there.

This is the only change.  There are 41 tenors-together
falseness groups for maximus and all higher stages.

RAS