The fifth and the fourth of the fifth and the fourth file attachment/etc.

I've put together a very simple Excel 'interpretation' of this post, but as
I do not have a personal web site, etc., I don't believe that I can post it
in any sort of an easily accessible format... However, I would be more than
happy to send it to anyone who might be interested - in the form of a file
attachment, via private email.

ANY n-tET =/< 794:
The following is a curious 'mathematical condition' that I believe 'allows'
diatonic heptads with five whole steps and two half steps (formed from each
scale degree of a major scale that is derived from the three triads of the
I-IV-V) where W=W, h=h, and W>h (i.e., the five whole steps and two half
steps are always the _same_ 'n-specific' _two_ whole numbers) in every
n-tET =/< 101 [Save 1 and 2-tET, where both the whole and the half steps
round to zero, and 8-tET, where both the whole and the half steps round to
one.]... and in a modified, and perhaps somewhat less esthetically
satisfying variation,--every n-tET =/< 794.

EXTRAPOLATING NUMERICAL RELATIONSHIPS OF THE I-IV-V:
If the familiar diatonic major scale of 12-tET is thought of in terms of
I-IV-V triads (where 1, 3, 5 + 4, 6, 8 + 5, 7, 9 (2) = 1st, 2nd, 3rd, 4th,
5th, 6th, 7th, 8th), it is seen that horizontal major thirds plus
horizontal minor thirds, and vertical fourths plus vertical seconds,
separate the intervallic relationships of this 'triad square':

1st 3rd 5th

4th 6th 8th

5th 7th 9th

where:

1st (+3rd=) 3rd (+b3rd=) 5th
(+4th=) (+4th=) (+4th=)
4th (+3rd=) 6th (+b3rd=) 8th
(+2nd=) (+2nd=) (+2nd=)
5th (+3rd=) 7th (+b3rd=) 2nd
[9th]

THE FIFTH AND THE FOURTH OF THE FIFTH AND THE FOURTH:
If you take the fifth and the fourth of an equal division of the octave
(where "n" is the equal division of the octave, n / 12 * 7 = "F"; the fifth
of n, and n / 12 * 5 = "f"; the fourth of n), and then take the 'fifth and
the fourth' of that fifth and the fourth... you could then call the fifth
of the fifth (FF) and the fifth of the fourth (Ff) of "n," the horizontal
major thirds plus horizontal minor thirds, and likewise the fourth (f) and
the fourth of the fourth (ff) of "n" could be taken as the vertical fourths
plus vertical seconds where:

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

and:

n/12*7=F
n/12*5=f

F/12*7=FF
F/12*5=fF

f/12*7=Ff
f/12*5=ff

Using the example of 12-tET (where n=12) this would be:

12 / 12 * 7 = 7
12 / 12 * 5 = 5

7 / 12 * 7 = 4 & 1/12th
7 / 12 * 5 = 2 & 11/12ths

5 / 12 * 7 = 2 & 11/12ths
5 / 12 * 5 = 2 & 1/12th

and:

12
/ \
7 5
/ \ / \
'4' ('3'='3') '2'

where FF+Ff and f+ff are '4'+'3' and 5+'2':

C (+'4'=) E (+'3'=) G
(+5=) (+5=) (+5=)
F (+'4'=) A (+'3'=) C
(+'2'=) (+'2'=) (+'2'=)
G (+'4'=) B (+'3'=) D

THE MEAN FIFTH OF 5 AND 7-tET:
Using the arithmetic mean fifth (@ 41/70ths):

3/5ths + 4/7ths
---------------
2

and the arithmetic mean fourth (@ 29/70ths):

2/5ths + 3/7ths
---------------
2

of the whole numbers 5 and 7 (i.e., 5-tET & 7-tET) for the F and f
multiples of n/12, where:

1 & 5/7ths * 4 & 1/10th = 7 & 1/35th (12/7*4.1=~7.02857)
1 & 5/7ths * 2 & 9/10ths = 4 & 34/35ths (12/7*2.9=~4.97143)

and:

7 & 1/35th = F
4 & 34/35ths = f

would furnish diatonic heptads with five whole steps and two half steps
where W=W and h=h (when the n-Whole steps and the n-half steps are rounded
to, _and taken as_, the nearest integer) on each scale degree of the I-IV-V
('triad square') major scale in any n-tET =/< 101.

xF*xf*12:
To render any n-tET =/< 794 (795-tET would render a W>W where the whole
number whole step 57 exceeds 56 by one and twenty-seven fifty-thirds of a
cent) capable of the above... You could recast the "x" (arithmetic mean)
"F" (fifth) and "f" (fourth) of 5 and 7-tET:

(7/12 + 1/420) * 12 = ~7.02857
(5/12 - 1/420) * 12 = ~4.97143

as:

(41/70 + 1/14268) * 12 = ~7.02941
(29/70 - 1/14268) * 12 = ~4.97059

where 420 is derived F*f*12, and 14,268 is derived xF*xf*12.

-Dan

p.s. Although I've yet to find any real super-useful, practical application
for 'this'--a persistent 'gut feeling' keeps telling me: "Stick with it...
it's worth it... (etc., etc.)" So I do my best to do as I'm told!