Degrees of temperaments

[Jim Coleman, Sr.]

Here are some thought which may encourage some to try variations from 
Equal Temperament.

Historical temperaments by degrees

Since for some time I have been using the Coleman XI Temperament for 
special customers with a great measure of success, it occurred to me
recently that if one wished for an even milder departure from equal
temperament, one could use half the deviation values of the Coleman 11.

After this a stupendous thought occurred, it followed that if one wished 
for an even bolder temperament, one could multiply the deviations. 
Now with the advent and popularity of the SAT III, it is quite simple 
to add greater variations in even steps until one achieves the desired 
amount of deviation from equal temperament. Here is how that can be 
done with the SAT III:

Normal Cole 11:
C     C#     D     D#     E     F     F#     G     G#     A     A#     B
3.0   0.0    0.0    0.0  -3.0   4.0   -2.0   2.0   0.0   -1.0   2.0   -4.0

Half of Cole 11 (new name Coleman 10)
1.5   0.0    0.0    0.0  -1.5   2.0   -1.0   1.0   0.0    -.5   1.0   -2.0

Dr. Sanderson has made it possible to convert any tuning in memory to an 
historical tuning by one of two methods: Call up the historical tuning 
before calculating the FAC tuning and the tuning with the historical 
variations will be included, OR, add the historical deviations after the 
FAC tuning has been calculated (this will not permanently change the 
page of memory).

On the page "1-7t" (equivalent to page 203) input the values of Cole 10 
in the position of octave C1 through B1. 
On the same page of memory, input the values of the Cole 11 in the 
position of the octave C2-B2.

When using any normal ET tuning, you can call up either of these two 
variations by holding down the blue SHIFT button and touching the PAGE/UP 
button once to get the Cole10 variations or by touching the PAGE/UP 
button twice to get the Cole 11 temperament variations which would be 
added to that ET tuning.

Now you get 4 different degrees of historical temperaments from one set 
of FAC measurements. For example, on page 1 you can compute the regular 
FAC tuning and add in the temperament "1t" to make the tuning milder 
than the regular Coleman 11(highly recommended for beginners). OR, on 
page 2 you could instead add in TMPT "2t" to make it a regular Cole 11. 
OR, on page 3 you could select TMPT "2t" before calculating the FAC 
tuning and then add in the  TMPT "1t" afterwards. OR, on page 4 you 
could invoke the TMPT "2t" before calculating the FAC tuning and then 
add in TMPT "2t" again to make a very bold historical temperament.

With the above tuning on page 1 the M3rds graduate from 11 cent widths to
16 cent widths in the sharp keys and from 11.5 cents to 16 cents in the 
flat keys. (Note: a nominal value for equal temperament M3rds is 14 
cents wide, but a more accurate value is 13.7 which of course because of 
inharmonicity is seldom that exact in pianos)

With the above tuning on page 2 the M3rds graduate from 8 cent widths to
18 cent widths in the sharp keys and from 9 cent widths to 18 cent widths 
in the flat keys.

With the above tuning on page 3 the M3rds graduate from 5 cent widths to
19 cent widths in the sharp keys and from 6.5 cent widths to 20 cent 
widths in the flat keys.

With the above tuning on page 4 the M3rds graduate from 2 cent widths to
22 cent widths in the sharp keys and from 4 cent widths to 22 cent widths 
in the flat keys. The 4ths and 5ths on the flat side of the circle of 4ths 
and 5ths are all at 2 cents except for the F-C 5th which is at -4 cents 
width. When this type of 4th or 5th is on the sharps side of the circle, 
this is what Bill Bremmer refers to as "Reverse Well Temperament". On 
this page the 4ths and 5ths are all either 4 cent or 6 cent widths except 
for the 5th B-F# which is 2 cents wide. The other 5ths are all narrow and 
all the 4ths are wide.

If you tune like page 4, here is what 3 of the types of intervals will be 
like:

Sharp side 5ths 4ths 3rds   Flat side 5ths 4ths 3rds
C           -4    4    2    F          -4   -2    4
G           -6    4    2    Bb          2   -2   10
D           -4    6   10    Eb          2    2   18
A           -6    4   16    Ab         -2    2   20
E           -4    6   20    Db         -2   -2   22
B            2    4   22    Gb          2   -2   22

For those who are new to Historical tunings, it might be well to start 
with the Coleman 10. It will give a definite flavor to each of the 
tonalities and it does not depart from equal temperament much more than 
a novice attempt at equal temperament. The keys of C, F and G will 
definitely sound better than ET. The other keys will add more character 
than ET presents. If one wants to try the avante guard, then the results 
will be more like the paragraph immediately above. In this case there 
will be 3 to 5 M3rds which are closely akin to the Pythagorean 3rds.

With the additive capability of the SAT III one can produce 4 varieties of
temperament with just two twelve note memories. With this principle in 
mind, one could take any of the more robust temperaments and make
a table of values which are just half of the original values and produce
a milder form and still maintain the relational characteristics of the 
famous original temperaments. It might be fun to do this with the Paul 
Bailey modified meantone, or the Vallotti Well which is still a little 
too strong for my tastes in some of the key tonalities. Please note, these 
last two temperaments are very good, it's just that I'm not personally
into this much contrast in temperaments. Many just love them. One 
person remarked about the Paul Bailey temp. that "it feeds her soul."

One might also use reverse values in this same manner which is 
what I used to have to do in RCT to get back to an original ET after 
having converted a tuning to a well temperament.

There are many other possibilities such as subdividing the variations 
of a very strong historical temperament into thirds instead of a half and
then gradually working up to where you can relate well to the full bodied 
original. To do this with just two stored temperaments, the first one 
would be one third of the values of the original deviations. The second 
one would be two thirds of the values of the original dev. The full values
would be achieved by combining these two sets as in the example of
the page 3 of memory above. I would not recommend going beyond
this for the more bold temperaments.

Paul Bailey suggested that one could lose some of the advantges of
equal beating relationships found in some the very good well 
temperaments when this is done. This however could be a door opening up 
other possibilities of relationships which none of us have thought of yet.

Jim Coleman, Sr.