Re: the "7 +/- 2" rule

[Paul Hahn:]
>Not for my sake. I have several problems with it:

(a) It's still quite uneven, even without the overlap turning 14 pitches
into 13.

(b) Too many pitches anyway (cf. the "7 +/- 2" rule).

Hmm... Could someone flesh-out "the "7 +/- 2" rule" for me?

Thanks,
Dan

See Miller, George A.
"The Magical Number Seven, Plus or Minus Two: Some Limits on Our
Capacity for Processing Information", _The Psychological Review_
vol. 63, 1956, pp. 81-97.

Read it at http://www.well.com/user/smalin/miller.html

Manuel Op de Coul coul@ezh.nl

[manuel.op.de.coul:]
>See Miller, George A. "The Magical Number Seven, Plus or Minus Two:

Thanks.

Dan

PS - Has anyone put forth any "generalized diatonic scales" consisting of 6, 8, or 9 notes?

On Wed, 16 Jun 1999, D. Stearns wrote:
> Has anyone put forth any "generalized diatonic scales" consisting of 6,
> 8, or 9 notes?

Search the list archives from last August. Look for the word
"nonatonic".

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Dan Stearns wrote,

>Has anyone put forth any "generalized diatonic scales" consisting of 6, 8,
or 9 notes?

9 notes is the most common among these proposals. Balzano's stunningly
beautiful (but ill-founded, IMO) proposal was a 9-out-of-20tET scale; see
"The Group-theoretic Description of 12-fold and Microtonal Pitch Systems" in
_Computer Music Journal_, Vol. 4, No. 4, Winter 1980. Both Bohlen and Pierce
proposed 9-tone ones as the best "generalized diatonic scales" for their
13th-root-of-3 tuning (these scales repeat at the 3:1 instead of the 2:1).
See "Harmony and New Scales" by Mathews, Pierce, and Roberts in _Harmony and
Tonality_ edited by Sundberg, and Bohlen's web site. Goldsmith's generalized
diatonic is 9-out-of-16-tET with inharmonic timbres; see "An Electronically
Generated Complex Microtonal System of Horizontal and Vertical Harmony" by
Goldsmith in _Journal of the Audio Engineering Society_ Vol. 19 (1971) no.10
pp. 851-858. I mention all of these in the part of my paper that you quoted.

In 12-tET we have symmetrical 6- and 8-tone scales (aka the augmented and
diminished scales) which have as many (or more) consonant triads as the
diatonic scale (Blackwood discusses these and gives a few examples from
famous composers) but, due to their symmetry, are unable to project the
sense of a "home" or tonic. Passages in the diminished scale are extremely
common in 20th century movie scores, non-serial classical, jazz, and even
progressive rock; the augmented scale is much less common (Blackwood
suggests that it's because it's very hard to write a good melody without
major seconds -- hmmm . . .)

[Paul Erlich:]
>Balzano's stunningly beautiful (but ill-founded, IMO) proposal was a
9-out-of-20tET scale;

Thanks for the references Paul. I do remember this one from your paper...
If you don't mind could you perhaps tell me a bit more about it? I don't
quite understand how the 'minor third times the size of the major third
equals the size of the entire set' of a 7-out-of-12 (in 12-tET) leads to a
9-out-of-20 in 20-tET.

>In 12-tET we have symmetrical 6- and 8-tone scales

Yes, but how about any non-12, 6 or 8 note "generalized diatonic scale"
proposals along the lines of the others you cited?

Dan

Paul H. Erlich [TD222.16]

>Dan Stearns wrote,
>>Has anyone put forth any "generalized diatonic scales" consisting of 6, 8,
>or 9 notes?

>9 notes is the most common among these proposals. Balzano's stunningly
>beautiful (but ill-founded, IMO) proposal was a 9-out-of-20tET scale;

In the harmonics theory after the 7 note major scale the two notes that
are the most significant are Eb and Bb at 7/6 and 7/4. That makes 9
notes but I don't know whether you would think of it as a scale or not.

C D Eb E F G A Bb B
24 27 28 30 32 36 40 42 45

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[Paul Erlich:]
>>Balzano's stunningly beautiful (but ill-founded, IMO) proposal was a
>>9-out-of-20tET scale;

Dan Stearns wrote,

>Thanks for the references Paul. I do remember this one from your paper...
>If you don't mind could you perhaps tell me a bit more about it? I don't
>quite understand how the 'minor third times the size of the major third
>equals the size of the entire set' of a 7-out-of-12 (in 12-tET) leads to a
>9-out-of-20 in 20-tET.

Dan, it's funny, but when I first heard of you, it was about your album
_Opus Contra Naturem_ (sp?) in 20-tET. I was sure that you were using
Balzano's theories, because Balzano rejects the frequency ratios inherent in
the _natural_ harmonic series. Apparently, I was wrong!

Balzano depicts the diatonic scale using a triagular lattice, as I do in
figure 3 of my paper. The difference is that he views the intervals not as
approximations to frequency ratios, but as operations on the group C-12 (a
circular group of 12 elements). He thinks that the fact that the operations
of moving by zero to three minor thirds and moving by zero to two major
thirds together generate an unambiguous representation of C-12 (which
obviously relies on the fact that the size of the minor third times the size
of the major third equals 12) is somehow significant. His paper is so
beautifully presented that you can easily forget to question that point, but
it's really just a numerological game like Yasser's.

Anyway, 7-out-of-12 is related to the the fact that 3 (size of minor third)
plus 4 (size of major third) equals 7 (another numerological game, despite
all the fancy group-theoretic footwork which I won't go into here but is
quite beautiful for its own sake). So 20, which is generated by the 4- and
5- step intervals (since 4*5=20), has a 9-tone diatonic, since 4+5=9.
Further Balzano microtonal systems would be 11-out-of-30, 13-out-of-42,
15-out-of-56, and 17-out-of-72. The basic "consonant intervals" in
17-out-of-72 would be 133 cents and 150 cents. Clearly Balzano's theory,
like Yasser's, runs aground when these implications are fleshed out.

Balzano later proposed that tetrads be used instead of triads in
9-out-of-20, to fill in that big empty 13-step hole left by his original
triads. But that ruins the beauty of his original theory. Zweifel has taken
Balzano very seriously and has published papers about 11-out-of-20, etc. in
Perspectives of New Music.

Although Balzano's paper explicitly mentions frequency ratios as the basis
of previous studies and then presumes to ignore them, there's a big lie
involved. He assumes total octave equivalence, but the octave of course is
2:1, the simplest frequency ratio after 1:1. This he never mentions.

To me the clincher is that, although his theory claims to cover the whole
system of tonality and key-relations of common-practice music, is that that
system arose in conjunction with meantone temperament (as well as fuzzier
tunings for non-fixed-pitch instruments) and that 19- or 31-tone equal
temperaments were deemed superior than 12 but rejected for considerations of
convenience. His whole theory hinges on the number 12 in such a strong way
that it completely crumbles if you use 19 or 31 instead.

Sure, Balzano's scales have important properties in common with the diatonic
scale, such as the fact that modulation by the generating interval produces
one chromatic shift. But any MOS scale has these properties, so there's no
reason to single out 9-out-of-20.

>Yes, but how about any non-12, 6 or 8 note "generalized diatonic scale"
>proposals along the lines of the others you cited?

I can't think of any.

[Paul Erlich:]
>but it's really just a numerological game like Yasser's.

Thanks Paul for the quick rundown on Balzano's 9-out-of-20. I think that
Blazano's "numerological game" (1/2+1/3rd=5/6ths, etc., etc.) could at
least be said to cull much more manageable/palatable scales (or sub-sets)
than Yasser's does... and even if this "runs aground when these
implications are fleshed out" (at least inasmuch as where all the logic
doesn't really much hold together as the out-of-n's keep increasing in size
is concerned...), I don't think that this should really much derail that
particular usefulness.

Dan