any n-tET =/< 794

ANY n-tET =/< 794:
The following is a curious (to me anyway...) 'mathematical condition' that
I believe 'allows' diatonic heptads with five whole steps and two half
steps (formed from each scale degree of the major scale) 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. (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':

1 3 5
4 6 8
5 7 9

where:

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

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 / 12 * 7, and n / 12 * 5 equal the fifth and the fourth of an
n-tET), 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/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 FF+Ff and f+ff would be '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 1 and 27/53rds 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. As my methods (and no doubt means) of checking "every n-tET =/<
101..." and "every n-tET =/< 794..." are probably more than a bit suspect;
someone with the computer savvy (and the math know-how) may want (if indeed
they were both interested and so inclined...) to give this some 'proofing.'