A strange 9-limit temperament

Paul Erlich's "contest" has led me into some strange regions of tuning space, but none stranger than this one. This is my best and final attempt to find a tuning with 12 notes per octave or less, that approaches the proportion of 7-limit harmonies of his 10 of 22-tET (or twin chains of similar sized fifths). Unfortunately it fails his melodic criteria for being a generalised diatonic, but harmonically I am almost certain that the tuning I am about to present is the best possible.

It has 12 notes, and while it only gives 8 complete 7-limit tetrads, no better than any 12 note tuning with twin chains of meantone-sized fifths, it has 4 additional 7-limit triads (5:6:7 and 1/7:1/6:1/5) and 4 additional 5-limit triads. What's more, 4 of the 8 tetrads and all of the 7-limit triads can be extended to 9-limit pentads and tetrads respectively.

So from a 9-limit perspective what we really have is 4 complete 9-limit pentads, 4 complete 7-limit tetrads, 4 9-limit tetrads (5:6:7:9 and 1/9:1/7:1/6:1/5) and 4 5-limit triads. There are equal numbers of majors and minors in all cases.

The errors in all intervals are 12 cents or less, except for the 5:7 which has a 17.5 cent error since it is approximated by the half octave as usual.

The tuning has three sizes of fifth:
Small 665.186 a -37 cent wolf
Medium 705.214 very good at +3.3 cents
Large 713.957 a +12 cent semi-wolf

Is there general agreement on how small the error in a fifth has to be, to not be considered a wolf? One can reduce those semi-wolves a little at the expense of the 7:4's and 9:5's. Full error list shown below.

As a cycle of 12 fifths it goes:
M M S M M L M M S M M L
\_____________________/

As offsets from 12-tET it is
C +12.19
C# +17.41
D -17.41
D# -12.19
E -6.98
F +6.98
F# +12.19
G +17.41
G# -17.41
A -12.19
A# -6.98
B +6.98

The accurate chords are:
4 5 6 7 9
------------------------------

Topic No. 5
G B D F
C E G Bb D
A C Eb G
C# E# G# B
F# A# C# E G#
D# F# A C#

1/9 1/7 1/6 1/5 1/4
-------------------------------
F# A C E
A C# E G B
E G# B D
C Eb Gb Bb
Eb G Bb Db F
Bb D F Ab

4 5 6
-----------------
F A C
B D# F#

1/6 1/5 1/4
-----------------
F Ab C
B D F#

Errors in the intervals of the 12 pentads and tetrads are:

2:3 4:5 5:6 4:7 5:7 6:7 4:9 5:9 6:9 7:9
3.26 -5.48 8.74 12.00 17.49 8.74 6.52 12.00 3.26 -5.48

Errors in the intervals of the 4 triads are:

2:3 4:5 5:6
12.00 3.26 8.74

The main price paid for all this is the two wolves D-A and G#-D#.

Ever seen anything like this before? Care to check my calculations?

Melodically it's not terribly interesting (as a 12 tone scale) and what I expect would happen is that one would use various diatonic or blues or jazz scales as subsets of the 12.

The smallest ET that this wierd tuning has any relation to is 126-tET. A reasonable approximation to the tuning would be obtained by 11,11,7,11,11,12,11,11,7,11,11,12 steps of 126-tET and the max error (apart from the 7:5's) would only go up to 12.33 cents (unfortunately in the semi-wolf fifths).

It has a vague resemblance to a maximally even 12 of 34-tET but this has only two sizes of fifth and the max error is 19.4 cents (in the 7:4) with 15.5 in the 7:6. So we can see that the major benefit of semi-wolfing those two other fifths is to improve the 7:4's and 7:6's (on chords which don't involve those semi-wolves) to acceptable errors.

This seems to be a tuning that falls in the gaps between n of m-tET's, justs, well-temperaments and meantone. I don't know what to call its "kind" of tuning. Do you? What it is is a 3 dimensional lattice 2 x 2 x 3 where one dimension is a 2-cycle and the other two dimensions are open.

It's interesting that it gets its large number of accurate 7-limit intervals by approximating fifths and major thirds in two different ways.

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