A mostly irrational quasi-meantone tuning

Hi there.
I've made a new 24-tone scale which is meant for two keyboards. It's
essentially a chain of fifths from Bbb to C## (i.e. from 8 flats to 9
sharps). The interesting thing is that all the fifths are smaller than the
just 3/2 - some by 1/9 of the Pyth. comma, others by its 2/9 or 3/9 (= 1/3).
The tempering is so strong that it behaves much more like a meantone rather
than a positive or well-temperament. So this is my first quasi-meantone
tuning which is not made purely of rational intervals but is mostly
irrational because the basic tempering unit here is 1/9 of the Pyth. comma.
Here is the "key-to-degree" mapping for the two manuals. The degree 24 (or
0) should be C. The keys are numbered from 1 (meaning the C# key) to 12
(meaning the C key). The symbol "U" stands for "upper manual" and "L" stands
for "lower manual".

Degree Key Manual
1 1 U
2 1 L
3 2 L
4 2 U
5 3 L
6 3 U
7 4 U
8 4 L
9 5 L
10 5 U
11 6 U
12 6 L
13 7 L
14 7 U
15 8 U
16 8 L
17 9 U
18 9 L
19 10 L
20 10 U
21 11 U
22 11 L
23 12 L
24 12 U

! qmtp.scl
!
Quasi-meantone irrational tuning with 1/9 Pyth. comma as basic tempering
unit
!August 2004 - Petr Parizek
24
!
74.58499
121.50501
151.77665
196.09000
268.06832
314.98834
8192/6561
43046721/33554432
461.55165
505.86500
580.44999
627.37001
652.42831
699.34833
67108864/43046721
6561/4096
890.22500
934.53835
967.41665
1011.73001
1083.70833
1130.62835
1155.68665
2/1

Petr

--- In tuning@yahoogroups.com, Petr Paøízek <p.parizek@t...> wrote:

> The tempering is so strong that it behaves much more like a meantone
rather
> than a positive or well-temperament.

It seems to me it *is* a meantone; it is an irregular meantone on 24
notes. As such, it doesn't circulate--it has 23 meantone fifths of
varied sizes, and a wolf 36.5 cents sharp.

An alternative meantone suitable for 24 notes would put the fifth
at exactly (144688/11)^(1/23), which is 696.413 cents. This is in the
5, 9, and 11-limit poptimal range, and gives a wolf "fifth" of exactly
11/7. If you can accept an even flatter fifth, another possible "wolf"
is 8/5, which corresponds to a meantone fifth of 10240^(1/23), or
695.057 cents, and fans of sharper meantones might prefer
(73728/7)^(1/23), which is extremely close to 2/9-comma meantone at
697.178 cents, and which gives a "wolf" of exactly 14/9.

From: "Gene Ward Smith" <gwsmith@s>

> An alternative meantone suitable for 24 notes would put the fifth
> at exactly (144688/11)^(1/23), which is 696.413 cents. This is in the
> 5, 9, and 11-limit poptimal range, and gives a wolf "fifth" of exactly
> 11/7.

By the way, I think it must have been a typing mistake that you wrote
"144688" instead of "114688".
I can tell you that you got it again. I knew about this possibility and
therefore I decided to make such a temperament where the interval of C##-Bbb
is close to 11/7. And what came out was just the temperament I sent here.
Also, this tuning has many Pythagorean diminished fourths which are used as
major thirds. This is an interesting similarity to a meantone with 1/4 Pyth.
comma tempering (I think someone else has also mentioned this meantone but I
don't know who it was, again).
Petr