grayness!

OVERTONES UNDERTONES AND EQUALTONES:

Walking an U-series into an O-series, or vice versa, involves passing
through a gray E-series zone.

U/O-"equal series" occur by letting x=2+(sqrt(2)) where "n" and "x"
are any given numbers, and an under to over series is defined as n*x
with a sequential numerator rule of +(x-2) and a sequential
denominator rule of -1. So letting x=2 would gives the corresponding
n-under series, and incrementally increasing the value of x by
rationals works n towards ever more accurate approximations of its
over series. If x=2+(sqrt(2)), the maximum underlying errors for any
given fraction of "n" is only ~2�. This would be virtually
indistinguishable from their corresponding n-tETs. Relaxing
x=2+(sqrt(2)) so that x=3.5 allows for the simplest rational
interpretation of an E-series.

Though I use three rational U/O-equal variations -- x=3, x=3.5, and
x=4 which reset the 1:2^(1/2) as 5:7, 12:17, and 7:10 respectively --
only the x=3.5 variation is essentially a rational "well temperament"
of its corresponding n-tET. The x=3 and x=4 variations are more like
some very colorful rational recasting of the corresponding n-tET, and
could be said to represent a sort of O/U interpretational extreme.

Here's the U/O-equal series where x=3.5 and n=1,...,12 (note that the
O/U-equal series are simply the inversions of these).

1/1 2/1

1/1 17/12 2/1

1/1 24/19 27/17 2/1

1/1 31/26 17/12 37/22 2/1

1/1 38/33 41/31 44/29 47/27 2/1

1/1 9/8 24/19 17/12 27/17 57/32 2/1

1/1 52/47 11/9 58/43 61/41 64/39 67/37 2/1

1/1 59/54 31/26 13/10 17/12 71/46 37/22 11/6 2/1

1/1 66/61 69/59 24/19 15/11 78/53 27/17 12/7 87/47 2/1

1/1 73/68 38/33 79/64 41/31 17/12 44/29 13/8 47/27 97/52 2/1

1/1 16/15 83/73 86/71 89/69 92/67 19/13 14/9 101/61 104/59 107/57 2/1

1/1 87/82 9/8 31/26 24/19 99/74 17/12 3/2 27/17 37/22 57/32 117/62 2/1

While I think most everyone would agree that these are not equal
temperaments, they are all but sonically indistinguishable from
them... and while I doubt that many would consider these to be JI
scales or sets, they are both rational and derived by walking one
series into the other...

November in New England is my favorite time of the year. When I'm out
hiking and I see the great gray skies churning through the now
leafless trees, and I feel that crisp snap of early winter winds...
everything just seems so super intense... the density of New England
in November, the visceral physical and emotional impact... that's what
I want to see and hear in music...

on a clear day you can see for miles!,

--Dan Stearns