A (factitious) "two and only two sizes" for every 1 - ? EDO�

If "d" = division (any so defined equidistant {d}ivision of the octave), "F"

= 7 (a {F}ifth), "f" = 5 (a {f}ourth), and "O" = 12 (an {O}ctave), and you

take the arithmetic means of f[d] and F[d] and use them as a �median� F/f @

7.1/35 and 4.34/35: (f � O x F) + (F � O x F)/(f � O x f) + (F � O x f)�

And if d = (d � O) X F/(d � O) X f, (d � O) X F = {[(d � O) X F] � (O X

F)}/ {[(d � O) X F] � (O X f ), and (d � O) X f = {[(d � O) X f ] � (O X

F)}/{[(d � O) X f ] � (O X f )}, you could use ({[(d � O) X F] � (O X F)} +

{[(d � O) X F] � (O X f )) and ([(d � O) X f ] + {[(d � O) X f ] � (O X f )}

as {H}orizontal and {v}ertical {c}oordinates on a I�IV�V triad square where

the first (root), third and fifth of the I, IV and V, delineate the

1st�3rd�5th, 4th�6th�8th, and 5th�7th�(9th - 7) of a + w + w + h + w + w + w

+ h diatonic heptad�

And if Hc 1, 2 and vc 1, 2 are �literally� represented as decimal

fractions/mixed decimals where:

0 = the one of the tonic (the d 1st)

0 + {[(d � O) X F] � (O X F)} = the third of the tonic (the d 3rd)

0 + {[(d � O) X F] � (O X F)} + {[(d � O) X f] � (O X F)} = the fifth of the

tonic (the d H5th)

0 + [(d � O) X f ] = the one of the subdominant (the d 4th)

0 + [(d � O) X f ] + {[(d � O) X F] � (O X F)} and {[(d � O) X F] � (O X F)}

+ [(d � O) X f ] = the third of the subdominant (the d 6th)

0 + [(d � O) X f ] + {[(d � O) X F] � (O X F)} + {[(d � O) X f] � (O X F)}

and {[(d � O) X F] � (O X F)} + {[(d � O) X f] � (O X F)} + [(d � O) X f ] =

the fifth of the subdominant (the d 8th)

0 + [(d � O) X f ] + {[(d � O) X f] � (O X f )} = the one of the dominant

(the d v5th)

0 + [(d � O) X f ] + {[(d � O) X f] � (O X f )} + {[(d � O) X F] � (O X F)}

and {[(d � O) X F] � (O X F)} + [(d � O) X f ] + {[(d � O) X f] � (O X f )}

= the third of the dominant (the d 7th)

0 + [(d � O) X f ] + {[(d � O) X f] � (O X f )} + {[(d � O) X F] � (O X F)}

+ {[(d � O) X f] � (O X F)} - d and {[(d � O) X F] � (O X F)} + [(d � O) X

f] � {(O X F) + [(d � O) X f ]} + {[(d � O) X f] � (O X f )} - d = the fifth

of the dominant (the d 2nd)

Then an internally consistent representation (representation being the key

word here, least this all be some manner of numeric sophistry) of diatonic

seconds should be achieved from rounding off the decimal fractions/mixed

decimals of �w�w�h�w�w�w�h�* to the nearest integer.

Respectfully,

Dan Stearns

*Actually this should be: + Lw + Sw + Lh + Sw + Lw + Lw + Sh. As there would

be both a {L}arge and {S}mall diatonic �half-step�, and a {L}arge and

{S}mall diatonic �whole-step� derived from a sort of �ordinal� interval

where the arithmetic mean of F + f squared = [(+ Lw + Sw + Lh + Sw + Lw + Lw

+ Sh) � (v5 - H5)], and d is always comprised of mF + mf squared (4,900

�ordinal� intervals) arranged + 841 + 840 + 349 + 840 + 841 + 841 + 348�

Eventually the integer representations of the w and h diatonic seconds will

round L and S up and down by one digit; when�? I don�t know� But I suspect

it would be well beyond any 'utilitarian' representation of d. (3 is the

first equidistant division of the octave to demarcate the integers of w and

h.)