Correction on "scales"

There was a slight error in my program (I was shifting the second
half-octave of the curve by 2 cents). Here are the corrected results:

starting with an equipentatonic (5-tET) scale, the local minimum of total
diadic harmonic entropy is

A C D E G
0 309 504 699 1008

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

C---699---G---696---D---696---A---699---E

Both minor thirds are 309¢, and the major third is 390¢.

for an equiheptatonic (7-tET) scale, the local minimum comes out as:

G A B C D E F
0 191 387 503 696 888 1004

which has these fifths:

F---699---C---697---G---696---D---695---A---697---E---699---B

The two minor thirds involving D are 308-309¢, the other two minor thirds
are 312¢, the C-E major third is 385¢, and the other two major thirds are
387¢.

All ETs:

2: 0 616
3: 0 400 800
4: 0 314 574 888
5: 0 309 504 699 1008
6: 0 197 388 620 811 1008
7: 0 191 387 503 696 888 1004
8: 0 116 312 428 620 813 929 1124
10: 0 78.3 194 387.3 501 581 697.3 777.3 890 1084.3
12: 0 100 200 300 400 500 600 700 800 900 1000 1100

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

This is good stuff, Paul E.!

On Thu, 24 Aug 2000, Paul H. Erlich wrote:
> 3: 0 400 800
> 4: 0 314 574 888

Remember when John DeLaubenfels was talking about the augmented triad
and the diminished seventh being best left in their 12TET forms, and I
said I agreed about the Aug but not the d7?

Heh.

> 6: 0 197 388 620 811 1008

Wholetone, with one tone almost exactly a 8/7 and the rest (roughly)
meantonish.

> 8: 0 116 312 428 620 813 929 1124

This looks (after you mess around a bit to find the right mode) like a
pseudomeantone diatonic (major) scale, with a minor sixth added
in.

> 10: 0 78.3 194 387.3 501 581 697.3 777.3 890 1084.3

This has one nearly-just 9/8; if you make put it between your 3/2 and
4/3 it has pretty good tetrachordality (one 4/3 contains alternating
diatonic- and chromatic-seeming semitones, and the other would too
except for one missing pitch).

--pH <manynote@library.wustl.edu> http://library.wustl.edu/~manynote
O
/\ "Well, so far, every time I break he runs out.
-\-\-- o But he's gotta slip up sometime . . . "