ramblin' (tuning by pure 5ths)

[Jim]

Hi Frank and all:

As I had promised others earlier, I am now sending a tuning system for
aurally tuning a temperament which I have proven out myself aurally
independently and in comparison with the stretched Sat tuning used
previously.

                          Pure5ths Temperament

Basically the principles of setting a pure 5ths temperament are very much
akin to setting regular aural temperaments. However, more use of the 6th
10th test intervals are common. In general the 3rds and 6ths come out
just like they do in normal tuning. The octave spread is the single 
biggest
difference. The 4ths are all a little faster than in normal equal temp.

Since two contiguous 5ths add up to a little more than an octave, the
octave needs to be stretched by almost the amount of beat that is normally
in two 5ths. Since a 4th plus a 5th normally makes up one octave in
width, and the beat is removed from the 5th, the 4ths will have almost
twice as much beat as they would normally. Due to the effects which
inharmonicity has on the beat speeds of normal 5ths, they usually have
an average beat of less than .5 bps. If that is subtracted from the 5ths
and added to the 4ths which usually average around 1 bps, you can see that
the 4ths will now beat around 1.5-2. bps. Now, when you think about a 4th
plus a 5th equalling an octave (like F3-A#3 plus A#3-F4 being a test for
the F3-F4 octave), it is easy to see that if the A#3-F4 5th is to be pure,
then the F3-A#3 4ths will have to take on the beat which was subtracted 
from the 5th A#-F4. So it follows that the 4ths will be faster than 
normal. Now when you consider that an octave is short 2 half steps of 
equalling 2 stacked 5ths, the octave will need to be stretched a little
more. How much you ask? It turns out to be about 1.5 to 2.0 bps. When the
octave is stretched, the 4ths also are stretched about the same amount. If 
you tried to figure all this out on paper, you would end up with a larger
stretch, but due to the influence of inharmonicity, it turns out to be a
little less. After tuning a number of pianos this way, it was discovered
that the 4ths and the octaves have about the same beat rate and most 
of the 4ths are less than 2.0 bps.

                           Tuning procedure

1. Tune A4 to 440
2. Tune D4 to A4 pure (make F3-D4 equal to F3-A4.  (The 6th 10th test)
3. Tune A3 to A4 wide almost 2 bps. Compare F3-A3 and A3-A4 (3rd Tenth)
4. Compare A3-D4 4th the same speed. Use 3rd 6th test but with > diff.
5. Tune F3 wide by 7 bps.
6. Tune A#3 to F3 wide almost 2 bps
7. Tune F4 to A#3 pure. Make C#3-A#3 equal C#3-F4. (The 6th 10th test)
     Compare width of F-F octace with width of A-A octave. Check both with
     the 3rd-10th tests and with the minor 3rd-M6th tests for 6-3 octave.
     The A-C should be half bps slower than the C-A test. You can tempor-
     arily raise the C4 to slow down the beats to better hear them.
8. Tune C#4 so that the A-C# M3rd fits evenly between the F3 3rd and the
     C#4 M3rd. This will accurately locate the C#4.
     If the C# M3rd does not fit between the A3 M3rd and the F4 M3rd, then
     Retune the F4 to balance the 3 3rds of the A-A octave, and then
     Retune the F3 to make the same sized F-F octave as the A-A octave.
     Retune the A#3 as pure 5th to F4 and note speed of F-A# 4th
9. Tune F#3 as pure 5th to C#4      (Use 6th 10th test)
     Compare F#3 M3rd with F3 M3rd
10 Tune E4 as pure 5th to A3        (Use 6th 10th test)
11 Tune G3 to D4 as pure 5th        (Use 6th 10th test) 
12 Tune B3 to balance between F#3 and E4. Each should be around 2.0- bps.
     Compare G-B M3rd with F-D 6th
13 Tune C4 as pure 5th to F3        (Use 6th 10th test)
     Compare C4-E4 M3rd with C#4 M3rd.  If you goofed, it will talk to you.
14 Tune D#4 as M3rd to B3 to fit with neighbors
15 Tune G#3 as pure 5th from D#4    (Use 6th 10th test)  and comp. M3rds

I predict that at first you will not like the wider octaves and the wider
4ths, but as you tune up the scale by pure 5ths and stretched octaves, you
will like what it does for you in the 5th octave and the 6th octaves. And
you will be hard put to tell the difference in the 3rds and 6ths. They
should all progress evenly just as in your normal equal temperament. 
The 17th do get a little faster in the upper octaves.

On larger pianos, another variation would be to first tune pure 5ths down
contiguously from A4 to D4 to G3 to C3 to F2. Then by tuning stretched
octaves up from each of these, you can get a pretty good idea how much
stretch they all need when you see how the F3-A3 comes out. It should be
right at 7 bps. If not, you can recheck your octave widths. I still
think it is better to first set up the contiguous M3rds as in the
Baldassin/Sanderson Temperament. I haven't found anything to beat that 
yet.

The minor 3rd-M6th test for octaves is very useful in this temperament,
because it already gives more stretch to the octave than usual. Just make
the m3rd a little slower than the 6th by about a half bps.

Jim Coleman, Sr.

PS I would sure like to hear back from the brave sould who actually try
this. But be sure to play some music on it before you tell me how much
you do not like the single octaves. JWC