Re: ordering 'diatonically delineated' scale steps

[Graham Breed:]
>Split it again, and you get 3+7. Or again, to get 3+10: stop whenever you
have enough notes.

Has anyone come up with a (relatively speaking) reliable, or consistent way
to '_diatonically_ delineate' the ordering of the W's and h's, or L's and
s's, or whatnot (thereby - at least in theory anyway - consistently
rendering a tonic/major scale analogue) in all these different
something-out-of-something scales?

It would seem to me (and I want to be wrong here...) that the consistency,
and reliability of maximal evenness, where say n/s*d is the total number of
(non-octave duplicating) pitches in a set (the 2/1 equally divided by "n")
divided by the total number of (non-octave duplicating) pitches in the
subset (the "s"cale) times the scales (_0,_1,etc.) "d"egrees (and the
scale, is an '"n"e-out-of-an-"n"' so to speak...), is going to be pretty
hard to beat for general usefulness...

By replacing "s" -- which I'll 'assume' is derived [(n/s)*(n/12*7/12*7)]:

n
/ \
f F
/ \ / \
ff Ff=fF FF

with n/s*4 (where 4 is (n/12*7)/(12*7) rounded to the nearest whole
number), the 7e-out-of-12e 0, 2, 3, 5, 7, 9, 10, 12, WhWWWhW, is
'transformed' into the 0, 2, 4, 5, 7, 9, 11, 12, WWhWWWh of diatonic
7-out-of-12e...

Where the 12e diatonic major scale is thought of as 5W+2h, or say 5 by 7
where 7=0 [Carrying out a 5 by 7 sequence where 7=7 very clearly delineates
the fifth parameters of the periodic table.], the W's and h's would agree
with the number sequence:

05-03-01-06-04-02-00-05
10-08-13-11-09-07-12
15-20-18-16-14-19
25-23-21-26
30-28-33
35-40

I use a bunch of other methods (both similar and dissimilar to the above
methods) for a variety of similar purposes, but they all seem to lack a
certain (diatonic) 'universalness' which is so easily found in the
maximally even '"n"e-out-of-"n's..."'

Dan

Dan Stearns wrote,

>It would seem to me (and I want to be wrong here...) that the consistency,
>and reliability of maximal evenness, where say n/s*d is the total number of

>(non-octave duplicating) pitches in a set (the 2/1 equally divided by "n")
>divided by the total number of (non-octave duplicating) pitches in the
>subset (the "s"cale) times the scales (_0,_1,etc.) "d"egrees (and the
>scale, is an '"n"e-out-of-an-"n"' so to speak...), is going to be pretty
>hard to beat for general usefulness...

I've said it before, I think maximal evenness is not a useful characteristic
of the diatonic scale, does not necessarily lead to generalized-diatonic
scales, and misses out on good generalized-diatonic scales. Aside from my
paper, the most recently discussed example was 7-out-of-24, where the
maximally even scale, which has three modes with two identical tetrachords,
is far less common than the usual Arabic scale, which has five modes with
two identical tetrachords. Identical tetrachords are a way of providing the
scale with the capacity for near-equivalence at ratios of 3, the same way
you automatically assumed they have the capacity for equivalence at ratios
of 2. This near-equivalence is a cognitively powerful feature; evidence for
its importance is the fact that untrained singers are often off by a ratio
of 3. The break in the near-equivalence (in the usual diatonic scale, the
tritone) provides a unique structural characteristic that allows a sense of
tonaily to be established, while many maximally even scales lack this unique
structural feature.

[Paul H. Erlich:]
>I've said it before, I think maximal evenness is not a useful
characteristic of the diatonic scale, does not necessarily lead to
generalized-diatonic scales, and misses out on good generalized-diatonic
scales.

I think that you are probably quite right, but I still think it (maximal
evenness) is useful (and it certainly is general...) for quickly
generating subsets, or scales with a pretty high hit-to-miss ratio... for
example: 0, 2, 4, 6, 7, 9, 11, 13 was a 13e scale that I kept coming to
long before I knew that it was a 7e-out of-13e, but once I saw this
relation, I was then able to quickly generate a bunch of other e-out-of-e's
that I ended up using pretty quickly.

Dan