Re: TD 861: Keenan Pepper's Noble Tuning -- corrected diagram

Hello, there, and here is a corrected version of a diagram in my
article remedying two errors regretfully found after posting. These
errors involve weighting factors for intermediate ratios between 5:4
and 9:7.

In this diagram, the y/x ratio for 19:15 should be 1/2, not 2/3; and
for 33:26 it should be 2/3, not 3/4 -- the latter of which would give
(4*5+3*9):(4*4+3*7) or 47:37, ~414.163 cents, a point on the continuum
a bit on the other side of 29-tET.

The emended diagram follows along with the introductory paragraph from
the original post, and with an added point for x/y=3/4 as 47:37: since
I got it wrong the first time, I might as well include it and get it
right this time.

. . . .

Here the main number line shows the sizes of major thirds in cents,
with the notations above this line indicating Keenan's tuning (KP) and
some equal temperaments. Fractions and interval ratios below the
number line show weighted values for y/x and resulting integer ratios,
with the Noble Mediant (NM) occurring at y/x=Phi. At the top of the
diagram, two number lines show approximate distances from the "valley"
ratios of 5:4 and 9:7.

Distance from 5:4 (~386.314)
+21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|--
Distance from 9:7 (~435.084)
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|--
-28 -27 -26 -25 -24 -23 -22 -21 -20 -19 -18 -17 -16 -15 -14 -13 -12 -11

407.55 409.76 413.79 416.38 417.39 423.53
53 41 29 KP 46 17
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
407 08 09 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24
4/9 1/2 2/3 3/4 1/1 4/3 Phi 2/1
81:64 19:15 33:26 47:37 14:11 51:40 NM 23:18
407.82 409.24 412.75 414.16 417.51 420.60 422.49 424.36

Most respectfully,

Margo Schulter
mschulter@value.net