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In a message dated 10/4/02 9:58:22 AM Central Daylight Time,
jkanter@rollingball.com writes:
> You're saying basically "I'm not careful with numbers, therefore others make
> mistakes too." I won't issue any new charts with your data until you
> publish scans of Owen's letter.
>
> Jason
>
Well, I just tried to scan one of Owen's pages but my scanner won't work, so
you can either believe what I said or not. Owen listed the offsets from E to
F, don't ask me why. He also listed some notes as flats and some as sharps.
I was eating lunch and in a hurry. So, yes, I mixed up a few numbers.
I'll say it again. The EBVT has never substantially changed since I first
began tuning it 10 years ago. What I started with was ET and narrowed some
3rds and widened others to get the Well Tempered effect. As I began to
notice that some of the 5ths turned out pure, I eventually specified them
that way. As some of the 3rds and 6ths routinely turned out Equal Beating, I
eventually specified them that way too.
These are the figures Owen came up with. I haven't tried them with an FAC
program yet and probably won't have time to do that today either. I have
very carefully checked to see that they are what Owen wrote.
C +3.8
C# -0.8
D +0.9
D# +3.1
E -2.0
F +1.8
F# -2.7
G +3.1
G# +1.2
A 0.0
A# +2.4
B 0.0
They look good to me, though. I don't know why you'd be saying this is far
different from what I've always said. The main thing is the Equal Beating
FA, GB, CE 3rds and GE 6th all beating at 6 beats per second. The rest falls
into line as a consequence.
I admit that I am not a "numbers" guy. Even if I claim to be a "letters" guy
instead, the task of actually spelling out on paper what I do as an aural
tuner and getting it all down just the way it's supposed to be is not easy at
all. You've been a great help to me so far and I appreciate it. Why not try
these correction figures first before you go making another graph and tell me
if what you get matches the description on my website.
Here is that bearing plan in plain text:
Aural Instructions for the EBVT:
1. Tune A4 to A-440 pitch source.
2. Tune A3 to A4, stretching the octave to a a 6:3 type.
3. Temper F3 from A3, a wide 3rd, estimating it to beat at 6 per second,
just slightly slower than it would be in Equal Temperament (ET).
4. Tune C4, a pure 5th from F3.
5. Temper E4 a wide 3rd from C4, making it also beat at 6 per second. (About
half the speed of the same interval in ET.)
6. Temper G3 from E4, a wide Major 6th so that it beats exactly the same as
the F3-A3 and C4-E4 3rds. (6 beats per second). This will make the G3-C4 4th
be tempered a little more than in ET.
7. Temper D4 from both G3 and A3 so that both the G3-D4 5th and the A3-D4
4th beat at exactly the same mild rate. (Each one just slightly more tempered
than in ET).
8. Temper B3 from G3, a wide Major 3rd so that it beats exactly the same as
the F3-A3, C4-E4 and G3-B3 3rds and the G3-E4 6th. (Approximately 6 beats per
second). Now, the F3-D4 6th will also beat rapidly but a little faster than
the previous series of Rapidly Beating Intervals (RBI). It will probably beat
at 8 beats per second.
9. Tune F4 a pure 4th from C4. This will make a pure sounding 4:2 octave
from F3-F4.
10. Temper by 1 beat per second, Bb3(A#3) a 5th from F4.
11. Listen to the resultant 3rd, Bb3(A#3)-D4. Temper C#4 from A3, a wide
3rd, so that the A3-C#4 3rd beats exactly the same as the Bb3(A#3)-D43rd.
This will probably be about 9 beats per second, about the same as in ET.
12. Tune F#3 a pure 5th from C#4.
13. Tune G#3 a pure 4th from C#4.
14. Tune D#4 from both G#3, a pure 5th. The resultant A#3-D#4 4th will
probably also be pure, resulting in a total of 5 pure 5ths, F-C, F#-C#,
C#-G#, G#-D#, & D#-A#.
15. When expanding the octaves do so in a manner which will cause the 2nd
inversion major triads of C, G, D, A, E, B to have the 3rd & 6th beat exactly
the same. Example: to tune E3, form a beatless octave with E4, then sharpen
or flatten until the 6th, E3-C#4 beats exactly the same as the 3rd, A3-C#4.
The octave should be stretched enough so that there is a very slight beat in
the single octave (approximately 1/2 beat per second) and that the tempered
5th will beat slower than the 4th.
When both the 4th and the 5th were pure in the temperament, the octave will
naturally be stretched a little less than an octave whose 4th and 5th are
tempered.
16. When expanding the outer octaves, try to reconcile the double octave and
the octave and 5th (12th) so that the double octave and the 12th beat exactly
the same. (A very slight beat, almost inaudible). This will naturally result
in octaves which vary slightly in size up and down the keyboard. These will
quite effectively be "Tempered Octaves". (See my complete article on <A HREF="http://www.billbremmer.com/TemperedOctaves.htm">How to
Tune Tempered Octaves</A>).
***************************************************
Incidentally, Kent Swafford sent me a private note telling me he got the same
results. He said however, that the final 4th, Bb3-Eb4 would not be pure but
have a slight beat. That's fine, still 4 pure 5ths as always. Sometimes
that 4th ends up quite pure sounding, sometimes it doesn't but either way,
follows the rules for Well Tempered Tuning.
Just keep in mind that all of the people who designed temperaments in the
past had to work through these kinds of problems too and many of them
produced several variations and changes along the way. What I have now,
hopefully, is a correct theoretical model. Once I am sure of it, then when I
have to make small modifications to accommodate scaling and/or Inharmonicity
irregularities, I have a foundation from which to do that.
I appreciate your help and patience, believe me. I'll have to reinstall my
scanner program to get it to work, I've had this trouble with it before.
When I get all of Owen's documents scanned, I'll send them to you and will
post them on my website. But I only have so much time to spend on this.
Bill Bremmer RPT
Madison, Wisconsin
www.Billbremmer.com
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