This article presents Werckmeister temperament viewed by fifths instead
of in chromatic order.  Refer to the previous article by Hans van Oost,
"Werckmeister Tuning for Wilhelm Bruder Organ", in 010222 MMDigest.

The principle of the Werckmeister tuning is the division of the
Pythagoraic comma (24 cents) into four 5th intervals, placing these
narrowed fifths between C-G-D-A and B-F#.  The Werckmeister fifth has
an interval of 1.49492696 approximately, whereas the frequency ratio
of the "just" (zero beat) fifth is exactly 1.5.

Based upon the modern standard pitch A=440 Hz and tuning the tenor
notes by fifths, from MIDI 48 (C) to 59 (B), these frequencies result:

   48 C   131.7021 Hz
 w 55 G   196.8850
 w 50 D   147.1644
 w 57 A   220.0000
   52 E   165.0000
   59 B   247.5000
 w 54 F#  184.9972
   49 C#  138.7479
   56 G#  208.1219
   51 D#  156.0914
   58 A#  234.1371
   53 F   175.6028

The frequency ratios of some common major chords are

          F-major      C-major       G-major       D-major
 fifth   C 1.5        G 1.4949      D 1.4949     A  1.4949
 third   A 1.2528     E 1.2528      B 1.2571     F# 1.2579
 root    F 1          C 1           G 1          D  1

The frequency ratios of some common minor chords are

          D-minor      A-minor       E-minor
 fifth   A 1.4949     E 1.5000      B 1.5000
 third   F 1.1932     C 1.1973      G 1.1932
 root    D 1          A 1           E 1

In comparison, "just" triads with zero beats would be in ratios

            major    minor
 fifth     1.5000   1.5000
 third     1.2500   1.2000
 root      1        1

If tuned with equal temperament, like a piano, then

            major    minor
 fifth     1.4983   1.4983
 third     1.2599   1.1892
 root      1        1

Summary: In comparison with equal temperament tuning, the Werckmeister
temperament gives a noticeably better approximation to the major and
minor thirds which predominate in most songs, and therefore it is
better suited for instruments which play songs in the key of C or
related keys.  The Werckmeister temperament was used by many firms in
Waldkirch 80 to 100 years ago, and it is most likely the temperament
used for band organs built in America in the same years.

Robbie Rhodes