Inspired by Brad Lehman - a good way of tuning 2/7-comma meantone

Hi all.

Yesterday, Brad Lehman's ideas on tuning by ear (especially the trick with
tuning some intervals temporarily pure) were so inspiring to me that I made
a new method of tuning 2/7-comma meantone. I can say I was quite successful
as there IS a meantone very similar to 2/7-comma which has some very
specific properties regarding beat rates in minor triads. For example,
intervals like C-Eb, Eb-G and C-G have all identical beat rates. For
comparison, the size of the minor second in this meantone is ~120.952 cents;
in 2/7-comma meantone, it is ~120.948 cents. Because this is a pretty small
difference indeed, I can happily use the property of identical beat rates in
minor triads if I want to tune 2/7-comma meantone on an acoustic instrument.
Okay, let's see how it works.

- 1: A4 = 439Hz for best results, then A3 an octave lower
- 2: C4 beats -7/3Hz to both As (i.e. 140 bpm negative)
- 3: E4 temporarily pure = 5/4 to C4
- 4: C#4 pure = 5/6 to E4
- 5: E4 changed to make E4-C4 beat the same as C4-A3
- 6: A3-F#4 beats opposite A3-C#4 (i.e. F#4-A4 beats twice A3-C#4)
- 7: G4 temporarily pure = 6/5 to E4
- 8: Eb4 pure = 4/5 to G4
- 9: G4 changed to make G4-Eb4 beat the same as Eb4-C4
- 10: B4 temporarily pure = 5/4 to G4
- 11: G#4 pure = 5/6 to B4
- 12: B4 changed to make B4-G4 beat the same as G4-E4
- 13: D4 temporarily pure = 6/5 to B3
- 14: Bb3 pure = 4/5 to D4
- 15: D4 temporarily pure again = 4/5 to F#4
- 16: F4 pure = 6/5 to D4
- 17: D4 finally changed to make D4-F4 beat the same as F4-A4
- 18: Having already tuned A3-A4, now you can tune all the other octaves to
complete the task.

The resulting scale looks like this:

! m2scra.scl
!
Rational approximation to 2/7-comma meantone (1/1 = 262.9333Hz)
12
!
25/24
19825/17748
11811/9860
19685/15776
1979/1479
49475/35496
5895/3944
49125/31552
6585/3944
35307/19720
58845/31552
2/1

If you compare this to a regular chain of 2/7-comma meantone fifths, you
find that even the largest errors are smaller than 1/40 of a cent! (provided
all the overtones of the instrument are tuned to 100% exact harmonics, which
is impossible, of course). Even if you start with A4 = 440Hz instead of
439Hz, the errors are still pretty small (about 1/20 of a cent).

Petr

Simply fabulous Petr! Can you mayhap suggest proportional beating scenarios
for my 79 tone proposal?

Cordially,
Ozan

----- Original Message -----
From: "Petr Pa��zek" <p.parizek@chello.cz>
To: "Tuning List" <tuning@yahoogroups.com>
Sent: 16 Nisan 2006 Pazar 19:18
Subject: [tuning] Inspired by Brad Lehman - a good way of tuning 2/7-comma
meantone

> Hi all.
>
> Yesterday, Brad Lehman's ideas on tuning by ear (especially the trick with
> tuning some intervals temporarily pure) were so inspiring to me that I
made
> a new method of tuning 2/7-comma meantone. I can say I was quite
successful
> as there IS a meantone very similar to 2/7-comma which has some very
> specific properties regarding beat rates in minor triads. For example,
> intervals like C-Eb, Eb-G and C-G have all identical beat rates. For
> comparison, the size of the minor second in this meantone is ~120.952
cents;
> in 2/7-comma meantone, it is ~120.948 cents. Because this is a pretty
small
> difference indeed, I can happily use the property of identical beat rates
in
> minor triads if I want to tune 2/7-comma meantone on an acoustic
instrument.
> Okay, let's see how it works.
>
> - 1: A4 = 439Hz for best results, then A3 an octave lower
> - 2: C4 beats -7/3Hz to both As (i.e. 140 bpm negative)
> - 3: E4 temporarily pure = 5/4 to C4
> - 4: C#4 pure = 5/6 to E4
> - 5: E4 changed to make E4-C4 beat the same as C4-A3
> - 6: A3-F#4 beats opposite A3-C#4 (i.e. F#4-A4 beats twice A3-C#4)
> - 7: G4 temporarily pure = 6/5 to E4
> - 8: Eb4 pure = 4/5 to G4
> - 9: G4 changed to make G4-Eb4 beat the same as Eb4-C4
> - 10: B4 temporarily pure = 5/4 to G4
> - 11: G#4 pure = 5/6 to B4
> - 12: B4 changed to make B4-G4 beat the same as G4-E4
> - 13: D4 temporarily pure = 6/5 to B3
> - 14: Bb3 pure = 4/5 to D4
> - 15: D4 temporarily pure again = 4/5 to F#4
> - 16: F4 pure = 6/5 to D4
> - 17: D4 finally changed to make D4-F4 beat the same as F4-A4
> - 18: Having already tuned A3-A4, now you can tune all the other octaves
to
> complete the task.
>
> The resulting scale looks like this:
>
> ! m2scra.scl
> !
> Rational approximation to 2/7-comma meantone (1/1 = 262.9333Hz)
> 12
> !
> 25/24
> 19825/17748
> 11811/9860
> 19685/15776
> 1979/1479
> 49475/35496
> 5895/3944
> 49125/31552
> 6585/3944
> 35307/19720
> 58845/31552
> 2/1
>
> If you compare this to a regular chain of 2/7-comma meantone fifths, you
> find that even the largest errors are smaller than 1/40 of a cent!
(provided
> all the overtones of the instrument are tuned to 100% exact harmonics,
which
> is impossible, of course). Even if you start with A4 = 440Hz instead of
> 439Hz, the errors are still pretty small (about 1/20 of a cent).
>
> Petr
>
>

Ozan wrote:

> Simply fabulous Petr! Can you mayhap suggest proportional beating
scenarios
> for my 79 tone proposal?

Actually, I could try. The only thing which complicates the matter a bit is
the large number of tones. FYI, the 12-tone system took me almost two days
to develop into the final form. :-D
I was very lucky, as I've said, as there is another very similar meantone
which DOES have these properties by itself (you can find more about these
synchronous types of meantone in message 62320). Another problem is that I
do these things just by hand and head as I haven't found an universal method
for finding similar beat rates in scales. If I managed to find one, then I
could try to make a small utility which could do these tasks for me much
faster. Even more, neither do I have a 79-tone keyboard nor a 79-tone "way
of keyboard thinking", which makes it impossible for me to prove if my
assumptions here or there are right. But if I managed to find something like
a common kind of procedure for all of these tunings, maybe things could
change. Maybe when I finish my second year at school in June, then I can
think about the question of such a procedure which could be easily
transcribed into a small piece of code. Sadly, as I'm not a great
programmer, I always write my software for the old-fashioned QBasic which
runs under MSDos. So if I wanted to give such a program to you, for example,
someone would have to rewrite it into a more usual form beforehand.

Petr