bob & darren why 12-out-of-N

Bob Valentine wrote,

>As an example, I was working through some 12-out-of-N tunings in 31tet,
>trying to maximize the existance of certain intervals in the tuning. I
>noticed that

> ( 7 * 7 ) mod 31 = 18

>which in this tuning means that a chain of 7 "7/6"'s is equal to "3/2"
>(after octave reduction).

Interestingly, this seems to be exactly the property that Jon Wild's
11-of-31 scale was most geared toward exploiting, given its chain of 9
"7/6"'s or its possible interpretation as a chain of 11 "7/6"'s when one
note is added. Take note, Darren McDougall (none of the JI interpretations
of Wild's scale can exploit this property as well as the original 11-of-31
tuning)!

>Producing a 12-out-of-N scale demonstrating this property can be done,
>but selects a different subset of the vertices than one might
>choose for having nice major triads, or 4:5:67 chords.

Absolutely. Interestingly, the two approaches are not always unrelated. Your
observation about 31-tET means that in 31-tET, the unison vector 3^8*7^(-7)
(or 839808:823543) vanishes. Meantone temperament has more nice major and
minor triads than JI (in a 7-tone diatonic scale) _because_ in meantone a
chain of four "3/2"'s is equal to "5/4" (after octave reduction). At the end
of this list's tenure on the Mills server, we (Carl Lumma, Dave Keenan, Paul
Hahn, . . .) began a discussion about the best 12-note scales for 7-limit
tetrads, and even given a tight (�3 cent) tolerance on the 7-limit
consonances, it seemed clear that the solution would _not_ be in JI but
would exploit a tuning where stacking two "3/2"'s and two "5/4"'s would
equal "7/4" (after octave reduction), and that the result would allow three
"4:5:6:7" chords and three "1/7:1/6:1/5:1/4" chords. Carl Lumma was
responsible for this insight, but his result seemed to be traceable to
Fokker, and Dave Keenan showed how the tight tolerance could be acheived. In
72-tone equal temperament (my suggestion for an ET version of Keenan's
non-JI non-ET solution), the result (0 5 12 16 23 30 35 42 49 53 58 65) can
be diagrammed as follows:
/,'/ \`.\ /,'/
30-/---\--0--/-
/|\/ \/| /
/ |/\ /\|/
53--------23--------65--------35
/|\ /|\`. /,'/ \`.\ /,'/
/ | \ / | \ 5--/---\--49 /
/ 16--------58 \ | / \ | /
/,'/ \`.\ /,'/ `.\|/ \|/
30-/---\--0--/------42--------12
/|\/ \/| /
/ |/\ /\|/
53--------23--------65--------35
/|\ /|\`. /,'/ \`.\ /,'/
/ | \ / | \ 5--/---\--49 /
/ 16--------58 \ | / \ | /
/,'/ \`.\ /,'/ `.\|/ \|/
30-/---\--0--/------42--------12
/|\/ \ | /
/ |/\ /\|/
----65--------35
/,'/ \`.\ /,'/

Incidentally, the equivalent scale in 31-tone equal temperament appears to
be best 12-tone scale for tetrads in 31-tET as well, still giving six
tetrads, though in 31-tET more of the other intervals are consonant (and so
more connections would appear on this diagram) than in 72-tET, though the
maximum deviation from JI doubles.

Back in the world of 17-cent tolerances, my decatonic solution for
integrating large numbers of 7-limit tetrads into a pseudo-diatonic scale
exploits the fact that in certain tunings, such as 22-tone equal
temperament, a "7/5" equals a "10/7", and two "4/3"'s equal a "7/4" (see
http://www-math.cudenver.edu/~jstarret/22ALL.pdf).

Hi, it's me again.

> Bob Valentine wrote,
>
> >As an example, I was working through some 12-out-of-N tunings in 31tet,
> >trying to maximize the existance of certain intervals in the tuning. I
> >noticed that
>
> > ( 7 * 7 ) mod 31 = 18
>
> >which in this tuning means that a chain of 7 "7/6"'s is equal to "3/2"
> >(after octave reduction).
>
Paul H. Erlich wrote,

> Interestingly, this seems to be exactly the property that Jon Wild's
> 11-of-31 scale was most geared toward exploiting, given its chain of 9
> "7/6"'s or its possible interpretation as a chain of 11 "7/6"'s when one
> note is added. Take note, Darren McDougall (none of the JI interpretations
> of Wild's scale can exploit this property as well as the original 11-of-31
> tuning)!
>
Let's see if I understand now:
Pythagorus gave us a chain of eleven "3/2"'s and a pythagorean comma.
Quarter comma mean tone gave us eight "5/4"'s and four commas of Didymus.

Is 11-of-31 a collection of chains of very close approximations to several JI
intervals? So instead of 11 pure fifths and nothing else (Pythag.); or 8 pure
thirds, 8 acceptable fifths and whatever else (QCMT); does 11-of-31 favor a
*multitude* of intervals in such a way that none are pure but all are
acceptable albeit with some wolves due to commas that are easily avoided?

DARREN McDOUGALL

P.S.
I have been quietly reading this list for over a year; at last I have actually
written to it. Since posting my questions about "why n from m-tET" I have
learnt more in this last week from your replies than in the previous 12 months.
Thankyou all for your help. You will no doubt get to read some more wierd
questions in the future.

Darren -- re your later masseage -- 5/4 does not appear in Pythagorean tuning, and
neither does the comma of didymus (syntonic comma).

I wrote,

>> Interestingly, this seems to be exactly the property that Jon Wild's
>> 11-of-31 scale was most geared toward exploiting, given its chain of 9
>> "7/6"'s or its possible interpretation as a chain of 11 "7/6"'s when one
>> note is added. Take note, Darren McDougall (none of the JI interpretations
>> of Wild's scale can exploit this property as well as the original 11-of-31
>> tuning)!

Darren wrote,

Let's see if I understand now:
>Pythagorus gave us a chain of eleven "3/2"'s and a pythagorean comma.
>Quarter comma mean tone gave us eight "5/4"'s and four commas of Didymus.

Only if you limit yourself to 12 tones per octave to you get those numbers of
intervals and intervals off by those commas (except that in meantone the
diminished fourth is a 5/4 off by a diesis, not a comma of Didymus (syntonic
comma)).

>Is 11-of-31 a collection of chains of very close approximations to several JI
>intervals? So instead of 11 pure fifths and nothing else (Pythag.);

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#.

>or 8 pure
>thirds, 8 acceptable fifths and whatever else (QCMT); does 11-of-31 favor a
>*multitude* of intervals in such a way that none are pure but all are
>acceptable

Yes. The lattice diagrams that I drew for this scale and for others show all the
approximations to 7-limit JI intervals that are contained in these scales. Where a
note appears more than once in the diagram you have a "pun", not possible in JI.
If you don't understand the diagrams, I'd be happy to help.

>albeit with some wolves due to commas that are easily avoided?

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.