moving on to "scales"

Dan Stearns wrote,

>I'd also be interested in some things like, relative to some of
>the 20-tET work that I've done, what say the most concordant stack of
>near 240¢ fifths of an octave are for instance.

If you'd like to think of it as a 5-tone scale that continues many octaves
up and down, I'd use an octave-equivalent diadic discordance measure. For
that purpose, let me just simplistically use "invert" the interval into
first half-octave of the harmonic entropy curve, and then into the second
half-octave, and average the two values. Then, starting with an
equipentatonic (5-tET) scale, the local minimum of total diadic harmonic
entropy is

A C D E G
0 308 504 700 1008

which is a standard pentatonic scale consisting of the following fifths:

C---700---G---696---D---696---A---700---E

Both minor thirds are 308¢, and the major third is 392¢.

Now, in anticipation of future discussions with Pierre, I'll do the same
thing for an equiheptatonic (7-tET) scale. The local minimum comes out as:

C D E F G A B
0 193 386 501 697 889 1086

which has these fifths:

F---699---C---697---G---696---D---696---A---697---E---700---B

The two minor thirds involving D are 308-309¢, the other two minor thirds
are 311¢, the C-E major third is 386¢, and the other two major thirds are
388-389¢.

So, if you can imagine a 7-dimensional diadic discordance surface, and you
place a ball at the point representing the equiheptatonic scale, it will
roll down into one of the seven basins representing the seven modes of the
diatonic scale, all tuned in this pseudo-meantone fashion.

By the way, here are the results for putting the ball on various ETs (I've
called these result "relaxed" ETs in the past) -- again, all modes of these
scales are equally valid:

2: 0 616
3: 0 389 811
4: 0 313 576 889
5: 0 308 504 700 1008
6: 0 224 400 624 800 1024
7: 0 193 386 501 697 889 1086
8: 0 113 227 424 616 812 926 1119
12: 0 100 200 300 400 500 600 700 800 900 1000 1100

For 9-11 and >13, at least one pair of notes made it over the hump and
settled into a unison.

Wow. This is a very strong justification of the usual pentatonic and
diatonic scales (in particular, pseudo-meantone tunings), of 12-tET, and of
a few scales I've never seen before (let's start playing with them!), purely
on grounds of diadic concordance.

Since I've been looking lately at approximateing other
ETs in 31tet, I was surprised and interesteed to see
how close the 31tet approximations come to these relaxed
versions.

> >> 4: 0 314 574 888

31tet : 0 310 581 890

> >> 8: 0 116 312 428 620 813 929 1124
>

31tet : 0 116 310 426 619 813 929 1123

> >> 6: 0 197 388 620 811 1008

31tet : 0 194 387 619 813 1006

Bob Valentine

Hmmm, looks like JdL and Paul may be finding something
going on in the tuning algorithm, maybe 31tet provides
a clue (maybe not). In the prior post I missed

> 10: 0 78.3 194 387.3 501 581 697.3 777.3 890 1084.3
31tet 0 77 194 387 503 581 697 774 890 1084

I don't know what this all means, but whatever 31 seems
to do 'right' is rated very highly by the relaxation
algorithm.

Bob Valentine