source file: mills2.txt
Date: Sun, 3 Mar 1996 10:16:44 -0800

Subject: Linus, Balzano

From: non12@delta1.deltanet.com (John Chalmers)

Linus: Congratulations on your daughter's success!
 
Re Balzano: B seems to have independently discovered the principle 
of propriety, though he called it "coherence. Basically it means that 
the scale has no overlapping interval classes, i.e. that the largest 
2nd is less than or equal to the smallest third for all melodic seconds 
(intervals between adjacent tones) and melodic thirds (intervals between 
every other tones), irrespective of their acoustical size  and similarly 
for all scale interval classes.
 
His other contribution to scale theory is his application of Group Theory 
to scale generation. In analogy with the major mode in 12, Balzano created 
a class of scales partitionable into chains of triads of the form 0 k 2k+1 
and 0 k+1 2k+1, where k and k+1 are the number of scale degrees in the 
mediant and conjugate-mediant ("Thirds") and 2k+1 is the dominant of 
the triads, following Riemann's and Lewin's terminology. The "chromatic" 
sets C thus have k(k+1) or k^2 +k tones, the generators (G) of the 
diatonic sets have 2k+1 tones  and the diatonic sets (D) have 2k+1 tones.
The K and k+1 are also the generators of groups of cardinality C. 
 
Hence for 12-tet, k = 3, k+1 = 4, 2k+1 = 7, k^2+k = 12. Balzano studied 
the case where k = 4, 5, and 6 and mentioned k=8 briefly. These correspond 
to equal temperaments of 20, 30, 42, and 72 tones (56 is presumably 
included). The diatonic sets and their generators have 9, 11, 13, and 17 
tones respectively (15 for k = 7). 
 
There are several problems with this theory. The set of predicted scales 
misses several harmonically much better tunings by 1 degree with the  exceptions of 12 and 72-tet (19, 31, 41, also 29 and 43 are better by most
criteria than 20, 30 and 42). Secondly, the triads are continually 
shrinking as k gets larger, and the number of tones in the scale compared 
to the number in the chromatic set also decreases. For example, in 72-tet, 
only 17 of the 72 tones are in the scale, leaving the remaining 55 tones 
as auxiliaries, alternates, or ornaments. The triads 0 8 17 and 0 9 17 are 
essentially tone clusters of 0 133.3 283.3 and  0 150 283.3 cents. The 
scale itself, consisting of 4 repeated blocks of  5 4 4 4 degrees and a 
final interval of 4 degrees, is coherent (actually, strictly proper), 
however. It is an MOS (as are all of Balzan's scales), though not a 
"deep scale" (Winograd, Gamer, L= C/2 or C/2+1). 
 
One might modify the theory and increase the number of "thirds" in the 
basic chords and harmonize with 7th chords, whole-tone scales, etc. 
(as did Yasser in 19-tet). Another possibility is to lengthen the chain 
of generators to produce larger MOS's. Eleven tones out of 20-tet is a
 better analog of the major scale than are 9, which really belong in 
16-tet, assuming a fifth-like generator. By 11 out of 30, one is already 
into an essentially non-diatonic realm of scales.
 
All in all, Balzano's is an extremely interesting theory, which should be 
tried compositionally (I believe a visiting composer at UCSD did write a 
piece using larger chords in 20-tet, but I've not heard it).
 
--John

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