Re: Chain-of-4:7s tunings

Now we're getting "way out there".

Here's a periodic table of EDOs (ETs) that have reasonable 4:7s. Errors are
between -18.8c and +16.9c.

----- x (by 6's)
|
| 0
Y 5 11
(by 10 16 22 28
5's) 15 21 27 33 39
20 26 32 38 44 50 56
[19]25 31 37 43 49 55 61 67
24 30 36 42 48 54 60 66 72 78 84
29 35 41 47 53 59 65 71 77 83 ... (+16.9c)
34 40 46 52 58 64 70 76 82 ...
39 45 51 57 63 69 75 81 ...
44 50 56 62 68 74 80 ...
49 55 61 67 73 79 ...
48 54 60 66 72 78 ...
53 59 65 71 77 ...
58 64 70 76 ... (+0.4c)
63 69 75 ...
68 74 ...
73 ...
(-18.8c)

Note that all EDOs of 24 or more tones appear at least once. 19-EDO isn't
intended to be in the table but is shown in [] to show where it would be
(-21.5c error).

The formula is N = 5Y + 6x, x and Y are integers where Y > 0 and
-Y/6 <= x <= 3Y/2. Ideally Y divisions would make up 2^17/7^6 (187.0c) and
X divisions would make up 7^5/2^14 (44.1c). The ratio is around 4.24 and
EDOs which are multiples of 26 (Y=4, x=1) have very good 4:7s (+0.4c). The
number of divisions making up the 4:7 will be 4Y+5x.

Of those with reasonable consistent 5-limit intervals, we only have 0, 22,
24, 26, 27, 29, 31, 34, and 36 and above. Looking at all 7-limit intervals
and enforcing consistency we lose 24, and 29 and 34 look pretty marginal.
This leaves Paul Erlich's list: 22, 26, 27, 31, plus 36 and above.

The formula suggests an 11 tone scale with two step sizes. This has 5 big
and 6 little steps as opposed to the 4 big and 7 little we came up with for
chain-of-minor-thirds. In case you're wondering, if we do force this scale
into 19-EDO (where Y=5 and x=-1) we don't get the same 11-tone scale we got
for the chain-of-minor-thirds. In fact it ends up having 3 step sizes.

Here's the chain-of-4:7s scale in 26-EDO (26-tET).

C C# Cx Db D D# EB Eb E E# Fb F F#
* * * * *

Fx Gb G G# Gx Ab A A# BB Bb B B# Cb(C)
* * * * * * (*)

Here's a 7-limit lattice of it when in 26-EDO.

Fb--------Cb--------Gb
,' `. ,' `. ,'
C---------G---------D---------A
,' `. ,' `. ,' `. ,'
D#--------A#--------E#--------B#
,' `. ,' `. ,'
Fb--------Cb--------Gb

Note the wraparound where Fb = Ex, Cb = Bx, Gb = F#x.

There are no thirds (unless you can tolerate 29c and -39c errors in major
and minor thirds respectively!). There are only 4:6:7 and 1/7:1/6:1/4
"triads". 14 of them, using only 11 tones. While the 4:7s are excellent
(+0.4c error) the fifths have a -9.7c error and the 6:7s therefore have a
-10.1c error.

I haven't looked at embedding this scale in any of the other suitable EDO's
(22,27,31) yet, to see what other useful linkages it might form.

Regards,
-- Dave Keenan
http://dkeenan.com

Dave Keenan wrote,

>Of those with reasonable consistent 5-limit intervals, we only have 0, 22,
>24, 26, 27, 29, 31, 34, and 36 and above. Looking at all 7-limit intervals
>and enforcing consistency we lose 24, and 29 and 34 look pretty marginal.

34 is not consistent in the 7-limit, but 29 is consistent all the way
through the 15-limit.