12 from 31, wolves and commas

Paul Erlich wrote:

> Note that in Pythagorean tuning the diminished fourth is only 2 cents off a
> 5/4
> and the diminished seventh is only 2 cents off a 5/3. This was often
> exploited
> around 1450; a Pythagorean chain from G-flat to B would give "just" D-major,
> A-major,
> E-major, F#-minor, C#-minor, and G#-minor triads, but the wolf would be right
> there
> between B and F#.
>
I checked out the maths (the 5/4 vs dim4th): you are absolutely correct. Quite
a surprising and lucky phenomenon this.

> I think of a wolf as a dissonant interval where you want a consonant one. So
> I
> would need to know the musical context to know if any wolves would come up or
> not.
>
When I tune my synth to QC mean-tone, there are four intervals that look like a
third but sound awful (not surprising). When tuned to Pythag, there are four
intervals (those dim4ths) that look and sound like thirds (nice surprise) : I
guess I wouldn't call *them* wolves.

Concerning 12 from 31, there must be intervals that on a keyboard sound
different in different places -- like the way QCMT has 8 good thirds and 4 bad.
What I am now wondering (and I hope this is true) is that in the places on the
keyboard where the intervals sound unlike what is expected, do we get another
approximation of a just interval? -- like those lucky dim4ths in Pythag?

DARREN McDOUGALL

Darren McDougall wrote,

>When I tune my synth to QC mean-tone, there are four intervals that look
like a
>third but sound awful (not surprising). When tuned to Pythag, there are
four
>intervals (those dim4ths) that look and sound like thirds (nice surprise) :
I
>guess I wouldn't call *them* wolves.

>Concerning 12 from 31, there must be intervals that on a keyboard sound
>different in different places -- like the way QCMT has 8 good thirds and 4
bad.
>What I am now wondering (and I hope this is true) is that in the places on
the
>keyboard where the intervals sound unlike what is expected, do we get
another
>approximation of a just interval? -- like those lucky dim4ths in Pythag?

Sometimes. Let me assume you mean a 12-out-of-31 chain of fifths, which is
nearly identical to QCMT. Some nice surprises in either: an augmented second
is nearly a just 7/6, an augmented sixth is nearly a just 7/4, and those
awful diminished fourths are actually very close to a just 9/7 -- pretty
dissonant on its own but nice in certain chords like 4:5:6:7:9. Did you mean
a different 12-out-of-31?