Dimished seventh type chords in 46-et

Since 46-et tempers out 126/125, it has the 6/5-6/5-6/5-7/6, septimal
semicomma type diminished seventh chord. This is a 12-12-12-10 chord,
and of course has four versions depending on the position of the 10
(the 7/6.) On the other hand, 46 not only has good 7-limit harmony, it
also has a good 17th overtone, and hence it has its version of the
12:14:17:20 diminished seventh chord. This becomes 10-13-11-12, with
all the various thirds distinguished, with four versions, and then
four more versions for the inverted chord. How other chords, such as
10-10-13-13, 11-11-11-13 or 10-11-12-13 might sound is a question for
experimentation.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
>
> Since 46-et tempers out 126/125, it has the 6/5-6/5-6/5-7/6, septimal
> semicomma type diminished seventh chord. This is a 12-12-12-10 chord,
> and of course has four versions depending on the position of the 10
> (the 7/6.) On the other hand, 46 not only has good 7-limit harmony, it
> also has a good 17th overtone, and hence it has its version of the
> 12:14:17:20 diminished seventh chord. This becomes 10-13-11-12, with
> all the various thirds distinguished, with four versions, and then
> four more versions for the inverted chord. How other chords, such as
> 10-10-13-13, 11-11-11-13 or 10-11-12-13 might sound is a question for
> experimentation.

I've experimented with some of these before and can report that some
very effective results can be obtained with them. Consider the chord
E:G:A#/Bb:C#/Db in different contexts:

In C major, the top note of 10:12:14:17 resolves very nicely to
9:12:15:18; since 10:17 is wider than 3:5, it more convincingly
resolves by expanding to an octave.

In F major, the top note of 15:18:21:25 resolves well to 16:16:20:24.
Since 15:25 (alias 3:5) is narrower than 10:17, it more convincingly
resolves by contracting to a fifth.

You also suggested that 46-ET can be used above the 13 limit if one is
willing to forego consistency. You already know that, because the
amount of inconsistency at the 17 limit is very slight, it should
present little problem. The numbers seem to indicate that the chief
problem would be with prime 19; however, I've tried a couple of chords
containing 19 and judged them acceptable to my ear, both melodically
and harmonically.

These are a few of the reasons that I consider 46 my favorite ET
between 31 and 72.

--George