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A well-temperament with _flat_ major thirds

🔗Paul H. Erlich <PErlich@xxxxxxxxxxxxx.xxxx>

12/7/1999 3:23:25 PM

Although Bach's predilection for sharp major thirds almost certainly came
from the fact that sharp major thirds are required to close a meantone
tuning at 12 pitches, we may imagine creating a well-temperament using some
other number of pitches. A fun example is 26, since 26-tET's major thirds
are about as flat as 12-tET's are sharp. We can arrange this by setting the
C-E major third to be exactly 386 cents, and having the circle of fifths
progress smoothly from largest fifths (696.63 cents) adjacent to D and
smallest fifths (687.99 cents) at the opposite side of the circle, using a
sinusoidal function. The resulting tuning is:

Note Cents
C 0.0
C# 71.4
Cx=Dbb 105.0
Db 128.9
D 193.0
D# 257.1
Dx=Ebb 281.0
Eb 314.6
E 386.0
E# 438.6
Fb 458.0
F 504.1
F# 577.1
Fx=Gbb 616.8
Gb 637.7
G 696.4
G# 764.8
Gx=Abb 793.0
Ab 821.2
A 889.6
A# 948.3
Ax=Bbb 969.2
Bb 1008.9
B 1081.9
B# 1128.0
Cb 1147.4

The usual meantone keys will sound very nice, having major thirds of about
382-386 cents, except for E-G# at 379 cents (similar to the 19-tET value).
On the other side of the circle of fiths, the five notes which can be named
using either double-sharps or double flats form an excellent representation
of the Thai pentatonic scale, with steps of 176 and 336 cents (the Thai
7-tET uses 171 and 343) -- note that here the major third is 2*176=352
cents. Adding B# and Fb to this collection produces a diatonic scale (Cx
major or Dbb major) which is nearly equally-spaced and may approach the
diatonic tunings in some Scandinavian regions (according to Parncutt). Also,
due to the 26-tET behavior "on average", there are many powerful 7-limit
relationships across the width of the circle, e.g. the near-7:4 interval
from C to Ax=Bbb. There are a few comma-like intervals to play around with
too: E#-Fb and B#-Cb are each 19.4 cents