11-limit, 31 tones, 9 hexads within 2.7c of just (was: Strict JI considered undesirable)

Why would anyone bother with strict JI beyond the 5-limit? For the cost of only a 2.7 c error in any 11-limit interval you can have a LOT more consonances.

Thanks to Paul Erlich for pointing out the unevenness of my earlier 22 tone 11-limit tuning and asking why I stopped at 22 tones. For no good reason as it turns out. You can trash that 22 tone tuning in favour of the strictly proper 31 tone tunings below.

The lattice below is really two lattices. One must choose between Gbb and Ex (they are only 13 cents apart). I think it is obvious to choose Gbb since this gives 5 otonal hexads and 4 utonal. When this choice is made we are left with 31 tones in 4 step sizes, 30.4 c, 36.2 c, 49.2 c and one step of 55.0 c (between Cb and B#). It may be considered as a detempered 31-tET.

Sorry about the width (85 characters). I hope I managed to avoid word-wrapping it outgoing. If it looks like a mess, it might come good if you widen your window. Otherwise you're going to have to paste it somewhere else and delete some returns. When it is right it looks like a diagonal stack of 6 hexagons (with flies buzzing around them).

(Ex)-----Bx
/ \ / \
G#/ \ D#/ \ A#
Bb F C / G \ / D \
Fx------Cx------Gx------Dx
/ \ / \ / \ /
(Ex) A / Bx\ E / \ B / \ F#/ C#
Cb Gb Db / Ab \ / Eb \ / \ /
5 G#------D#------A#------E#------B#
/ \ / \ / \ / \ /
/ 7 \ Fx Bb/ Cx\ F / Gx\ C / Dx\ G / D
/ 11 \ Abb Ebb / Bbb \ / Fb \ / \ /
4-------6-------9 A-------E-------B-------F#------C#
otonal hexad / \ / \ (Ex)/ \ Bx /
legend G# Cb/ D#\ Gb/ A#\ Db/ E#\ Ab/ B# Eb utonal hexad
/ Cbb \ /(Gbb)\ / \ / legend
Bb------F-------C-------G-------D 1/9-----1/6-----1/4
/ \ / \ Fx / \ Cx / Gx Dx \ 1/11/
A / E \Abb/ B \Ebb/ F#\Bbb/ C# Fb \1/7/
/ \ / \ / \ / \ /
Cb------Gb------Db------Ab------Eb 1/5
/ \ / \ G# / \ D# / A# E# B#
Bb / F \ / C \ / G \Cbb/ D (Gbb)
/ \ / \ / \ /
Abb-----Ebb-----Bbb------Fb
\ A / \ E / B F# C#
Db\ / Ab\ / Eb
\ / \ /
Cbb----(Gbb)

!
! keenan5.scl
!
11-limit, 31 tones, 9 hexads within 2.7c of just, Dave Keenan 27-Dec-99
31
!
36.19153216 ! Bx
85.39311378 ! C#
115.8026469 ! Db
151.994179 ! Cx
201.1957607 ! D
231.6052938 ! Ebb
267.7968259 ! D#
316.9984075 ! Eb
353.1899397 ! Dx
383.5994728 ! E
432.8010544 ! Fb
468.9925866 ! E#
499.4021197 ! F
548.6037013 ! Gbb
584.7952335 ! F#
615.2047665 ! Gb
651.3962987 ! Fx
700.5978803 ! G
731.0074134 ! Abb
767.1989456 ! G#
816.4005272 ! Ab
18/11 ! Gx
883.0015925 ! A
932.2031741 ! Bbb
968.3947062 ! A#
998.8042393 ! Bb
1048.005821 ! Cbb
1084.197353 ! B
1114.606886 ! Cb
1169.590467 ! B#
2/1 ! C

The lattice below is a near miss. 8 complete 11-limit hexads. At least it only has 3 step sizes, 30.4 c, 36.2 c, 49.2 c. It may be useful when fewer than 31 tones are to be made available.

We can't put B# here because it clashes with Cb ( ) Fx------Cx------Gx
/ \ / \ / \
A / \ E / \ B / \ F#
Gb Db / Ab \ / Eb \ / \
5 C#------G#------D#------A#------E#------B#
/ \ / \ / \ / \ /
/ 7 \ / Fx\ Bb/ Cx\ F / Gx\ C / G D
/ 11 \ Dbb Abb / Ebb \ / Bbb \ / Fb \ / Cb
4-------6-------9 A-------E-------B-------F#
otonal hexad / \ / \ / \ /
legend C# G# / D#\ Gb/ A#\ Db/ E#\ Ab/ B# Eb utonal hexad
Fb / Cbb \ / Gbb \ / \ / legend
Bb------F-------C-------G-------D 1/9-----1/6-----1/4
/ \ / \ Fx / \ Cx / Gx \ 1/11/
A Dbb/ E \Abb/ B \Ebb/ F#\Bbb/ Fb Cb \1/7/
/ \ / \ / \ / \ /
Gb------Db------Ab------Eb 1/5
C# / \ G# / \ D# / \ A# / E# B#
Bb F / C \ Fb/ G \Cbb/ D \Gbb/
/ \ / \ / \ /
Dbb-----Abb-----Ebb-----Bbb------Fb------Cb
\ / \ A / \ E / B F#
Gb\ / Db\ / Ab\ / Eb
\ / \ / \ /
Fbb-----Cbb-----Gbb ( ) Can't put Dbb here because it clashes with C#

Regards,

-- Dave Keenan
http://dkeenan.com

On Tue, 28 Dec 1999 21:35:49 -0800, David C Keenan <d.keenan@uq.net.au>
wrote:

>Why would anyone bother with strict JI beyond the 5-limit? For the cost of
>only a 2.7 c error in any 11-limit interval you can have a LOT more
>consonances.
>
>Thanks to Paul Erlich for pointing out the unevenness of my earlier 22 tone
>11-limit tuning and asking why I stopped at 22 tones. For no good reason as
>it turns out. You can trash that 22 tone tuning in favour of the strictly
>proper 31 tone tunings below.

Wow! This looks really cool, I'll have to try it out! It's interesting how
Gbb works out to be the 11/8 in this scale as well as in 50-tet and 31-tet.
(It's Ex in 43-tet.) Could this be considered a 31-note well-tempered
scale? (What's the worst error of the "misspelled" intervals?)

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