Coleman7 Well Temp. (long)

[Jim]

Hi Everyone:

Since I did have some requests for details of the Cole7 Well, Here it
comes.  For simplicity use the following cents deviations from your
best ET if you have an electronic tuning device.

  F    #     G     #     A     #     B     C     #     D     #     E

  3   -1     2     0     0     0    -2     3     0     1     0    -1


For a practical aural procedure after having established a good Equal
temperament, Tune C to F 4th so that it is narrow at half the beat
      which it had in ET as a wide interval or compared to its ET
      neighbors.
Next, tune the F4 to C 4th so that it is again in ET relationship.
      Tune the F3 to C 5th so that it is again in ET relationship and
      the F-F octave is similar to other octaves.
          This leaves the C to G 5th narrow by 5 cents
Next, tune G to D 4th so that it is almost as wide as the C-G 5th when it
          becomes less narrow in the process.
          This leaves the D to G 4th wide by 4 cents and the C to G 5th
          narrow by 4 cents.
Next, tune G3 to G4 so that G3-C4 = C4-G4 in beat speed, both at twice
          normal ET speed.
Next, Bring up the D to G 4th so that the A-D 4th and the D-G 4ths are
          equal.
Next, tune E4 to B3 4th so that it beats only half as much but on the
          narrow side.
Next, tune F#3 to C#4 5th so that it beats only half as much as ET 5th
          and the F# to C# 5th will beat half as much as F-C 5th and the
          F#3 to B 4th will be 1 1/2 times as fast as ET.
Next, tune B3 to F#3 4th to beat half as much as an ET 4th while the B3
          to E 4th will be 1 1/2 the speed of an ET 4th.

The A, A#, C#, D#, and G# remain the same as in equal temperament.

Here are the cent widths results concerning  4 intervals.

           F    #    G    #    A    #    B    C    #    D    #    E
Thirds     11   15   10   17   14   15   16   10   17   12   16   15
Fourths   -1    1    3    2    3    2    3    2    1    3    2    3
Fifths    -2   -1   -3   -2   -3    1   -1   -3   -2   -3   -2   -3
Sixths    14    17   13   19   15   18  18   13    16   13   19   17


What I like about this Well temperament over the Coleman4 is that there
are no 4 cent wide 4ths and no 4 cent narrow 5ths. However, there is
one wide 5th (A#) and one narrow 4th (F).  That doesn't bother me as
much as a 4 cent narrow 5ths or a 4 cent wide 4th. The C M3rd is the same,
but the F and G M3rds are slower which makes the key of C sound bigger and
more harmonious. The cost of this is found in the wider M3rds (16 and 17
cent 3rds, not nearly as harsh as Pthagorean 3rds) on G#, B, C# and D#.

I point out that this Well Temp. was not derived in the normal historical
sense, but was devised to accomplish the above results by making the
deviations from equal temperament.  If I figure out a sensible way to
tune this directly rather than going from ET to Cole7, I may post that at
a later date.

An interesting thing which I noticed is that when playing a M3rd in ghost
tone fashion (that is: holding down F3-A3 silently and striking A5
sharply), the ghost tone rings louder than when doing the same test for
the F#3-A#3 M3rd and striking F#5 sharply. This is not as significant
in the slower beating intervals like 4ths and 5ths, but it is there to
some degree, and may indicate why Pthagorean 5ths were popular for so
long. (I just threw this last in for the purists out there)  You will
notice also that the M6ths on C, F and G are not as wide as ET, and that
also contributes to the bigger (more harmonious) sound of the primary
keys C, F and G.

Jim Coleman, Sr.