Temperaments

As this is almost a current topic, I thought I'd detail some well
temperaments I worked out a while back. They're all based on the black note
scale being Pythagorean, and the white notes tempered to something like
meantone.

The first has C-G-D-A tempered. Quite daring as those fifths are so out.
It means they reach meantone size, so it's a bit like meantone without
wolves.

0 90.2 192.2 294.1 384.4 498.0 588.3 696.1 792.2 888.3 996.1 1086.3 1200
= -10 -8 -6 -6 -2 -12 -4 -8 -2 -4 -14
+2 -8 -6 -4 -4 +0 -10 -2 -6 = -2 -12

Now, F-C, G-D-A-E and B-F# are tempered. This gets close to equal, with one
fewer tempered fifth than Valotti & Young:

0 94.9 199.2 298.8 393.7 502.7 593.0 702.0 796.9 896.5 1000.8 1095.7 1200
= -5 -1 -1 -6 +2 -7 +2 -3 -3 +1 -4
+3 -2 +2 +2 -3 +5 -2 +5 +0 = +4 +0

Finally, F-C, G-D-A and E-B tempered. This is symmetric about D. It's a
bit like meantone, what with the fifths being spread out to favour the
thirds. I think this is my favourite of the three, although it's so long
ago I played with them I can't be sure.

0 96.1 198.0 300.0 396.1 503.9 594.1 702.0 798.0 894.1 1002.0 1092.2 1200
= -4 +2 = -4 +4 -6 +2 -2 -6 +2 -8
+6 +2 +8 +6 +2 +10 = +8 +4 = +8 -2

If any of these have names, I'd like to know them.

Now, while I'm at it, here's a lattice for a meantone-like just intonation:

B
/
/
/
G
/
F#----/-C#
/ \ / /
/ \ Eb/
/ \ /
D-------A
/ \ /
/ G#\ /
/ / \ /
Bb/-----F
/
E
/
/
/
C

For those who don't read lattices, it's tuned so that D-A is a just fifth,
and F-A and A-C# are just thirds. Then, F#-C# and Bb-F are tuned to be just
fifths. Then, G# is tuned so that Bb-G# is a just 7/4, and Bb7 is the
7-limit version. C and E are tuned in thirds from that. And Eb, G and B
are similarly tuned so that Eb-C# is a just 7/4.

This sounds pretty much like meantone, with the bonus of a few just chords
in there. Fifths that aren't just are worse than 1/4 comma meantone. C-G
and E-B are 7.7 cents flat. The others, except the wolf G#-Eb, are 6.1
cents flat.

This technique of just thirds with just 7-limit intervals can be made closer
to meantone. However, I find nothing wrong with this tuning. So, the
question is, do the piano tuners on the list think it could be tuned by ear?
And, do techniques like this have any historical significance?

Graham
http://www.cix.co.uk/~gbreed/tuning.htm

Graham Breed wrote,

> Now, while I'm at it, here's a lattice for a meantone-like just intonation:
>
> B
> /
> /
> /
> G
> /
> F#----/-C#
> / \ / /
> / \ Eb/
> / \ /
> D-------A
> / \ /
> / G#\ /
> / / \ /
> Bb/-----F
> /
> E
> /
> /
> /
> C
>
> For those who don't read lattices, it's tuned so that D-A is a just fifth,
> and F-A and A-C# are just thirds. Then, F#-C# and Bb-F are tuned to be just
> fifths. Then, G# is tuned so that Bb-G# is a just 7/4, and Bb7 is the
> 7-limit version. C and E are tuned in thirds from that. And Eb, G and B
> are similarly tuned so that Eb-C# is a just 7/4.
>
> This sounds pretty much like meantone, with the bonus of a few just chords
> in there. Fifths that aren't just are worse than 1/4 comma meantone. C-G
> and E-B are 7.7 cents flat. The others, except the wolf G#-Eb, are 6.1
> cents flat.
>
> This technique of just thirds with just 7-limit intervals can be made closer
> to meantone.

How?

> However, I find nothing wrong with this tuning. So, the
> question is, do the piano tuners on the list think it could be tuned by ear?

Of course! If you give us one that's closer to meantone, I may have to use it on my
piano.

I wrote:

> > This technique of just thirds with just 7-limit intervals
> can be made closer
> > to meantone.

I've lost the bit of paper I originally worked this out on. I can't find a
closer scale, so it looks like I was wrong. Although I haven't looked at
scales where any of the good thirds are allowed to be tempered.